Culinary fluid mechanics and other currents in food science

Abstract

Innovations in fluid mechanics have refined food since ancient history, while creativity in cooking inspires science in return. Here, we review how recent advances in hydrodynamics are changing food science, and we highlight how the surprising phenomena that arise in the kitchen lead to discoveries and technologies across the disciplines, including rheology, soft matter, biophysics and molecular gastronomy. This review is structured like a menu, where each course highlights different aspects of culinary fluid mechanics. Our main themes include multiphase flows, complex fluids, thermal convection, hydrodynamic instabilities, viscous flows, granular matter, porous media, percolation, chaotic advection, interfacial phenomena, and turbulence. For every topic, we first provide an introduction accessible to food professionals and scientists in neighbouring fields. We then assess the state-of-the-art knowledge, the open problems, and likely directions for future research. New gastronomic ideas grow rapidly as the scientific recipes keep improving too.

Full text

Culinary fluid mechanics and other currents in food science
Arnold J. T. M. Mathijssen ,1, ∗Maciej Lisicki ,2, †Vivek N. Prakash ,3, ‡and Endre J. L. Mossige 4, §
1Department of Physics & Astronomy, University of Pennsylvania,
209 South 33rd Street, Philadelphia, PA 19104, USA
2Institute of Theoretical Physics, Faculty of Physics,
University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
3Departments of Physics, Biology, Marine Biology and Ecology,
University of Miami, 1320 Campo Sano Ave, Coral Gables, FL 33146, USA
4Department of Physics, University of Oslo (UiO), Sem Saelands vei 24, 0371 Oslo, Norway
(Dated: January 31, 2022)
Innovations in fluid mechanics have refined food since ancient history, while creativity in cooking
inspires science in return. Here, we review how recent advances in hydrodynamics are changing food
science, and we highlight how the surprising phenomena that arise in the kitchen lead to discoveries
and technologies across the disciplines, including rheology, soft matter, biophysics and molecular
gastronomy. This review is structured like a menu, where each course highlights different aspects
of culinary fluid mechanics. Our main themes include multiphase flows, complex fluids, thermal
convection, hydrodynamic instabilities, viscous flows, granular matter, porous media, percolation,
chaotic advection, interfacial phenomena, and turbulence. For every topic, we first provide an
introduction accessible to food professionals and scientists in neighbouring fields. We then assess
the state-of-the-art knowledge, the open problems, and likely directions for future research. New
gastronomic ideas grow rapidly as the scientific recipes keep improving too.
CONTENTS
I. Introduction 2
II. Kitchen Sink Fundamentals 3
A. Eureka! 4
B. Navier-Stokes equations 4
C. Drinking from a straw: Hagen–Poiseuille
flow 5
D. Onset of turbulence: Reynolds number 5
E. Bernoulli principle 5
F. Pendant drop: Surface tension 6
G. Wetting and capillary action 7
H. From jets to drops: Plateau-Rayleigh
instability 7
I. Hydraulic jumps in the kitchen sink 8
J. How to cook a satellite dish 9
K. Washing and drying hands, skincare 9
III. Drinks & Cocktails: Multiphase Flows 10
A. Layered cocktails 10
1. Inverted Fountains 10
2. Internal Waves 11
3. Kelvin–Helmholtz Instability 11
4. Rayleigh-Taylor instability 12
B. Tears of wine 12
C. Whisky tasting 13
D. Marangoni cocktails 14
E. Bubbly drinks 15
∗amaths@upenn.edu
†mklis@fuw.edu.pl
‡vprakash@miami.edu
§endre.mossige@gmail.com
F. Foams 16
G. Ouzo effect 18
IV. Soup Starter: Complex Fluids 19
A. Food rheology 19
1. Linear viscoelasticity 20
2. Non-linear viscoelasticity 21
B. Mixing up a sauce 23
C. Suspensions 23
D. Emulsions 24
E. Cheerios effect: capillary floating 25
V. Hot Main Course: Thermal Effects 25
A. Feel the heat: energy transfer 25
B. Levitating drops: Leidenfrost effect 26
C. Heating and Boiling: Rayleigh-B´enard
convection 27
D. Layered latte: double-diffusive convection 27
E. Tenderloin: moisture migration 28
F. Flames, vapors, fire and smoke 29
G. Melting and freezing 29
H. Non-stick coatings 30
VI. Honey Dessert: Viscous Flows 30
A. Flows at low Reynolds number 30
B. Fundamental solution of Stokes flow 31
C. Coffee grains in free fall 32
D. Slender body theory 33
E. Lubrication theory 33
F. Pot stuck to stove top: Stefan adhesion 34
G. Viscous gravity currents 34
Making the perfect crˆepe 34
H. Viscous fingering 35
I. Microbial fluid mechanics 35
J. Microfluidics for improved food safety 35
K. Ice creams 36
arXiv:2201.12128v1 [physics.pop-ph] 27 Jan 2022

2
VII. Coffee: Granular Matter & Porous Media 36
A. Granular flows and avalanches 36
B. Hoppers: grains flowing through an orifice 37
C. Brazil nut effect 38
D. Brewing coffee: porous media flows 38
E. Coffee ring effect 40
VIII. Tempest in a Teacup: Non-linear Flows,
Turbulence and Mixing 41
A. Tea leaf paradox 41
B. Secondary flows 42
C. Turbulent jets emanating from tea kettles 42
D. Sound generation by kitchen flows 42
E. Making macarons: chaotic advection 44
F. Sweetening tea with honey: mixing at low
Re and high Pe 45
IX. Washing the dishes: Interfacial flows 45
A. Greasy galleys smooth the waves 45
B. Splashing and sloshing 46
C. Dishwashing and soap film dynamics 46
D. Ripples and waves 47
E. Rinsing flows: thin film instabilities 48
F. Dynamics of falling and rising drops 48
1. Immiscible drops 48
2. Miscible drops 49
X. Discussion 50
A. Summary 50
B. Learning from kitchen experiments 50
C. Curiosity-driven research 52
D. Conclusion 52
Acknowledgements 52
References 53
I. INTRODUCTION
The origins of fluid mechanics trace back to ancient
water technologies [Fig. 1a], which supplied our earliest
civilizations with reliable food sources [1]. Subsequently,
as soon as the water flows, surprising phenomena emerge
beyond number. This naturally sparked the interest of
the first inventors, since the kitchen can serve as a lab-
oratory [2] that is accessible to people of different back-
grounds, ages, and interests. As such, the scullery is a
source of curiosity that has driven innovations through-
out history [3]. The problems that emerge while cook-
ing have led to creative solutions, which have not only
improved food science but also led to breakthroughs in
modern engineering, medicine, and the natural sciences.
In turn, fundamental research has improved gastronomy,
and thus the cycle continues. Hence, science and cooking
are intrinsically connected across people and time.
Today, numerous chefs have written extensive cook-
books from a scientific perspective. Well acclaimed is the
work on molecular gastronomy [4], which turned into a
scientific discipline, as reviewed by Barham et al. [5]. An-
other recent movement, known for using advanced equip-
ment including centrifuges and blow torches, is called
modernist cuisine [6]. With its striking photography,
sometimes tricked, it also connects science with art in
the field of fine dining [7]. The book by McGee [8] is
particularly influential too: Celebrity chef Heston Blu-
menthal stated it is “the book that has had the greatest
single impact on my cooking”, and then he wrote eight
books himself. Another excellent cookbook containing
various experiments and scientific diagrams was written
by L´opez-Alt [9]. One of the first people to approach
cooking systematically, a century earlier, was the ‘king
of chefs and chef of kings’ Escoffier [10], whose 943-page
culinary guide still remains a golden standard in haute
cuisine [11].
In the scientific community, a wave of excitement hit
when Kurti and Kurti [12] solicited recipes or essays on
cooking from the members of the Royal Society. Sci-
ence can improve cooking, but they also showed that
food can lead to better science, a notion that was not
taken very seriously at the time. Later, in her essay
‘food for thought’, Dame Athene Donald FRS pleads
that the scientific challenges are as exciting in food as
in any more conventional area. They should not be over-
looked or, worse, sneered at [13]. Her early vision sparked
scientists to regard food as an interdisciplinary research
topic. Food science now spans across many fields, includ-
ing materials science [e.g. 14], food chemistry and phys-
ical chemistry [e.g. 15], nutrition genetics [e.g. 16], food
engineering [e.g. 17], food microbiology [e.g. 18], food
rheology [e.g. 19], soft condensed matter [e.g. 20–22] and
biophysics [e.g. 23,24].
However, to the best of our knowledge, and despite the
overwhelming number of surprising hydrodynamic effects
that emerge in the kitchen, there is no comprehensive re-
view of fluid mechanics in gastronomy and food science.
Therefore, we aim to address this topic here in a manner
that first provides a broad overview and then highlights
the frontier of modern research. As such, we aim to con-
nect the following communities:
First, for chefs and gastronomy professionals, fluid
mechanics can make or break their culinary creations.
Hydrodynamic instabilities can ruin a layered cocktail
[§III A], while the Leidenfrost effect helps with searing
your steak [§V B], and baristas learn about percolation
to perfect their coffee [§VII D]. We will discuss these ex-
amples here, and quite a few more. Indeed, throughout
this Review we aim to connect the science with food ap-
plications. We also point out some common mistakes in
cooking and think of new ideas for recipes.
Second, for food scientists, it is important to unravel
hydrodynamic effects in order to develop better food
processing technologies [25]. For example, microfluidic
techniques are now extensively used for edible foam gen-
eration and emulsification [26,27], but also bioactive
compound extraction and the design of novel food mi-
crostructures [28]. More generally, fluid mechanics de-

3
scribes the transport of mass, momentum and energy,
which is to be optimised in food processing [29] and food
preservation [30,31]. We will highlight a number of un-
expected flow phenomena, and their relation with food
science technologies.
Third, from the perspective of medicine and nutri-
tion professionals, flow physics has led to novel health
care solutions and provided insights on the physiology
of digestion [32]. For example, flow devices can detect
food-borne pathogens or toxins [33], which is essential
for food safety [34] and food quality control [35]. Similar
technologies can equally be used for in vitro fertilization
for agricultural animal breeding, or other applications in
animal health monitoring, vaccination and therapeutics
[36]. Moreover, using next-generation DNA and protein
sequencing with nanopore technology [37], the field of
foodomics could help with improving human nutrition
[16]. In this article we will reflect on more of these food
health innovations.
Fourth, for engineers and natural scientists, kitchen
flows have led to breakthrough discoveries, and continue
doing so. To name a few here, Agnes Pockels established
the modern discipline of surface science after her obser-
vations of soap films while washing the dishes [§IX C],
and Pyotr Kapitza discovered the roll wave instability
while under house arrest [§IX E]. Most universities and
labs were closed during the COVID-19 pandemic, which
again lead to an unasked-for wave of kitchen science [38].
Moreover, culinary flows have given rise to engineering
applications in completely different fields. The piston-
and-cylinder steam engine was inspired by Papin’s pres-
sure cooker [§II A], and inkjet printers rely on capillary
breakups observed in the sink [§II H]. Indeed, because of
the low activation barrier, the kitchen is a hotspot for
curiosity-driven research where new ideas arise.
Not least, for science education, the kitchen can serve
as an exceptional classroom [39–41] or indeed a lab [2].
Being a natural gateway to learning about fluid me-
chanics, food science demonstrations equally connect to
numerous other disciplines. Examples include teaching
oceanography [42], chemistry education [43,44], geology
[45], soft matter physics [46], and the science of cooking
for non-science majors [47]. Recently, based on their suc-
cessful edX and Harvard University course, Brenner et al.
[48] connected haute cuisine with soft matter science. In-
deed, a lot of science awaits to be discovered during our
daily meals.
This Review is structured like a menu: We begin with
washing our hands in §II about kitchen sink fundamen-
tals, where we provide a brief introduction to fluid me-
chanics that is accessible to scientists across the disci-
plines. Then we are ready to pour ourselves a cocktail,
which we discuss in §III concerning multiphase flows.
The first course might be a consomm´e, so in §IV we focus
on complex fluids and food rheology. The main course
is often hot, so we review thermal effects in cooking in
§V. Tempted by dessert with honey and ice cream, we
consider Stokesian flows in §VI. We then brew a coffee
FIG. 1. Hidden channels. (a) The Shushtar Historical Hy-
draulic System in Iran, listed by UNESCO as ‘a masterpiece
of creative genius’. This irrigation system, dating back to the
5th century B.C., features canals, tunnels, dams, and water
mills, which all work in unison. Image by Iman Yari, licenced
under CC BY-SA 4.0. (b) The first discovered specimen of
a Tantalus bowl (Pythagorean cup) from the late Roman pe-
riod, catalogue no. 9 of the Vinkovci treasure. This silver-gilt
bowl empties itself when filled above a critical level by a hid-
den siphon, soaking a greedy drinker in wine. Image credit:
Damir Doraˇci´c, Archaeological Museum in Zagreb, from Vuli´c
et al. [49]. (c) The moka pot, a traditional stove-top coffee
maker, where the boiling water percolates through the coffee
by following the red arrows. From Navarini et al. [50].
after the lavish meal, the thought of which sparks inter-
est in granular flows and porous media, as discussed in
§VII. Pouring another cup of tea, we discuss different as-
pects of non-linear flows and turbulence in §VIII. Once
the meal comes to an end, it is time to wash the dishes,
which brings our attention to interfacial flows in §IX. We
conclude the Review with an extensive discussion in §X.
II. KITCHEN SINK FUNDAMENTALS
We begin this Review by introducing the basics of fluid
mechanics in the context of food science. Flow experts
may choose to skip this section, or to refresh their mem-
ory while enjoying the various anecdotes. Starting with
surprising aspects of hydrostatics, we quickly transition
to the hydrodynamics of wine aeration, hydraulic jumps
and satellite dishes, to name a few. Some of these con-
cepts are not as simple as appearance makes believe. In
the words of Drazin [3],

4
A child can ask in an hour more questions
about fluid dynamics than a Nobel prize win-
ner can answer in a lifetime.
As things get harder in the later sections, we will often
refer back to these kitchen sink fundamentals.
A. Eureka!
In his work “On Floating Bodies”, Archimedes of Syra-
cuse (c.287–c.212 BC) described the principles of hy-
drostatics. The buoyancy force on an immersed object
equals the weight of the fluid it displaces. For a cube
of volume L3, aligned with gravity, this force is just the
pressure difference between the top and bottom surfaces,
ρgL ˆ
z, times the area of this surface, L2, where ρis the
fluid density and g=−gˆ
zis gravity. More generally,
we can integrate the pressure p(z) = patm +ρgz over
the surface Sof an object of any shape, which gives the
buoyancy force
Fb=ZS−p dS=−ZV
ρgdV =−ρgVdisp,(1)
where we used the divergence theorem. Since Greece is
a sea-faring nation, Archimedes’ principle is important
to describe the stability of ships: If the center of gravity
is above the metacentre (which is related to the centre
of buoyancy of the displaced water), the boat will topple
[51,52]. Try floating a cup upside-down in the kitchen
sink. Another classic experiment is throwing a stone out
of a boat (upright cup). Does the water level rise? Buoy-
ancy is also essential in heat convection [53], as described
in §V C.
The concept of pressure became more established be-
cause the works of Evangelista Torricelli (1608-1647), of-
ten credited as the inventor of the barometer, and Blaise
Pascal (1623-1662). Pascal’s law states that a pres-
sure change at any point in an enclosed fluid at rest
is transmitted undiminished throughout the fluid [54].
This principle lies at the heart of many applications with
siphons and ‘communicating vessels’ including water tow-
ers, modern plumbing and water gauges.
While hydrostatics may seem simple compared to hy-
drodynamics, it can still be rather counter-intuitive. For
example, gravity can readily be defied when turning a
glass of water up-side down, aided with a special trick
[55]. One of the ‘oldest pranks in history’ is the Pythago-
ras cup, also known as the Tantalus bowl, which is a
drinking vessel with a hidden siphon [Fig. 1b]. While
described in ancient literature, the first specimen of a
Tantalus bowl was discovered only very recently, in the
Vinkovci treasure [49]. It functions normally when filled
moderately, but if the liquid level rises above a critical
height the bowl will drain its entire contents. Therefore,
the bowl concretises a “Tantalean punishment”, taking
away pleasure from to those who get too greedy! The
same siphon method is used in modern washing machines,
certain toilet flushing systems, and anti-colic baby bot-
tles [56]. Another example of non-intuitive hydrostat-
ics is Heron’s fountain, attributed to Heron of Alexan-
dria (c.10-c.70 AD). It can spout water higher than its
reservoirs without pump, so it appears to be a perpet-
ual motion machine, but in the end it is just very clever
hydraulics. Furthermore, in his treatise on pneumatics,
Heron describes a total of 78 different inventions and dis-
coveries made by himself and earlier ancient philosophers
[57].
The use of pressure in the kitchen expanded with the
invention of the steam digester by Denis Papin (1647-
1713). His improved designs included a stream-release
valve to prevent the machine from exploding, an essen-
tial feature in all modern pressure cookers, and coffee
makers like the moka pot [see §VII D, Fig. 1c]. Besides
kitchen appliances, the steam digester was the forerunner
of the autoclave to disinfect medical instruments, and the
piston-and-cylinder steam engine. Pascal’s principle also
underpins the hydraulic press, a force amplifier capable
of uprooting trees, invented by Joseph Bramah (1748-
1814). Bramah invented and improved many other culi-
nary technologies during the Industrial Revolution, in-
cluding high-pressure public water mains and the beer
engine [58].
B. Navier-Stokes equations
Moving on from hydrostatics, we shift our attention
to fluids in motion, which are described by the famous
equations named after Claude-Louis Navier (1785-1836)
and George Gabriel Stokes (1819-1903). For most simple
liquids that are approximately incompressible, like water,
the Navier-Stokes equations can be written as
ρDu
Dt =−∇p+µ∇2u+f,(2a)
0 = ∇·u.(2b)
Here u(x, t) is the flow velocity at position xand time
t, and p(x, t) is the pressure field, ρis the fluid density,
µis the dynamic viscosity, ν=µ/ρ is the kinematic
viscosity, fis a body force (usually gravity) acting on
the fluid, and the operator D/Dt =∂/∂ t + (u·∇) is the
material derivative. Physically, Eq. (2a) stems from the
conservation of momentum, essentially Newton’s second
law applied to an infinitesimal fluid parcel, and Eq. (2b)
describes the conservation of mass.
Today, over 200 years after their formulation, the
Navier-Stokes equations have still not been solved in gen-
eral. This is mainly due to the non-linearity of the con-
vective term, (u·∇)u. Even basic properties of their
solutions have never been proved, particularly the ‘exis-
tence and smoothness problem’. One of the seven Mil-
lennium Prizes of $1 million can be earned with a correct
solution [59]. Having said that, in specific cases, solu-
tions to the Navier-Stokes equations can in fact be found
analytically.

5
FIG. 2. Original drawings by Reynolds [66], showing laminar
pipe flow, turbulent flow, and turbulent flow observed by the
light of an electric spark.
C. Drinking from a straw: Hagen–Poiseuille flow
One example where an exact solution is known is the
flow of drinking through a straw. This current is driven
by a difference in pressure, ∆p, where we consider a cylin-
drical tube of radius Rand length Lwith negligible grav-
ity. The volumetric flow rate (the flux) going through the
tube is described by the Hagen–Poiseuille equation,
Q=πR4∆p/(8µL).(3)
The expression was first deduced experimentally, in-
dependently by Gotthilf Hagen (1797-1884) and Jean
Poiseuille (1797-1869), and soon after it was confirmed
theoretically [60]. Perhaps the best-known theory [61]
was derived by Sir George Stokes (1819-1903) in 1845,
but he did not publish it until 1880, supposedly because
he was not certain about the validity of the “no-slip”
boundary condition of vanishing velocity at the walls [60].
Stokes also derived the exact flow velocity ueverywhere
in the pipe. Starting from the Navier-Stokes equations
(2) in cylindrical coordinates (ρ, θ, z ), assuming that the
flow is steady, axisymmetric, and that the radial and az-
imuthal components of the velocity are zero, one finds
the parabolic flow profile,
uz=−∆p(R2−ρ2)/(4µL),(4)
which, as expected, is strongest at the center line. The
consequences of the Hagen–Poiseuille equation (3) can
be substantial: It is 16 times harder to drink through a
straw that is 2 times thinner, to achieve the same flux.
This fourth-power scaling is even more problematic for
microscopic flow channels, in the field of microfluidics
[see [62–65] and also §VI J]. For large pipes or fast flows,
conversely, the flow is no longer laminar so the hydraulic
resistance also increases. This is what we discuss next,
in §II D.
D. Onset of turbulence: Reynolds number
Following the work by Reynolds [66], the transition
from laminar to turbulent flow can be observed directly
in the kitchen sink. When the tap is opened a little, the
flow is laminar and follows parallel streamlines. The wa-
ter column is clear and can be used as an optical lens.
Opening the tap further, the image begins to fluctuate.
Then bubbles appear and begin to jump around vigor-
ously, blurring the water. When the tap is opened all
the way, the column turns completely opaque and white
(due to entrainment of air bubbles causing Mie scatter-
ing, which is roughly independent of the wavelength of
light, as opposed to Rayleigh scattering that turns the
sky blue). This transition from laminar (ordered) to tur-
bulent (disordered) flow [Fig. 2] depends on the relative
magnitude of fluid inertia to viscous dissipation in the
fluid as described by the Reynolds number,
Re ≡Inertial forces
Viscous forces =ρU0L0
µ.(5)
The critical Reynolds, Rec, number can be measured di-
rectly from this faucet experiment: The characteristic
length scale L0is often chosen to be the diameter of the
faucet nozzle, d∼1 cm, and the velocity scale U0can
be determined easily by holding a cup under the faucet
at the onset of turbulence [67]. One should find a volu-
metric flow rate of Q∼1.8 cm3/s, which corresponds to
Rec≈2300 in pipe flow [68,69].
Interestingly, one can also determine the diameter of
the valve inside the faucet. Without seeing it, the on-
set of turbulence can still be heard, as a hissing sound.
Using the same relation as before, Re = 4Q/(νπd), with
Re known, we can find the critical Qat which the sound
emerges, and thus compute the valve diameter. Because
the valve is usually smaller than the nozzle, this happens
at a lower flow rate. In medicine, this listening technique
called auscultation [70] can be used to detect narrowing
of blood vessels, sounds referred to as bruit or vascular
murmurs [71–73] and, similarly, obstructions of the air-
ways in respiratory conditions [74–76]. In §VIII D we will
talk more about hydrodynamic sound generation.
E. Bernoulli principle
In his book “Hydrodynamica”, Bernoulli (1700-1782)
found that pressure decreases when the flow speed in-
creases. More generally, Bernoulli’s principle is a state-
ment about the conservation of energy along a stream-
line. The Swiss mathematician Leonhard Euler (1707-
1783) used this principle to derive the modern form of
the Bernoulli equation,
1
2u2+Ψ+w= constant along a streamline,(6)
where Ψ = gz is the potential force due to gravity, wis
the enthalpy of the fluid per unit mass, and where it is

6
FIG. 3. Wine aeration. (a) Oxygen injection using the Ven-
turi effect: The wine moves down into a narrow funnel by
gravity. In the funnel the liquid accelerates, which lowers
the pressure compared to the surrounding atmosphere, as de-
scribed by the Bernoulli principle [Eq. (6)]. Hence, air bubbles
are drawn in, which aerate the wine. (b) Wine decanter: By
pouring and swirling the liquid around, ripples form that mix
in oxygen efficiently. (a,b) From Vintorio Wine Accessories,
with permission.
assumed that the flow is steady and that friction due to
viscosity is negligible. For an incompressible fluid this
is w=p/ρ, which is often appropriate for water. For
a compressible fluid additional information is required
about its thermodynamic energy [54].
Despite its apparent simplicity, the Bernoulli princi-
ple is a powerful tool in many applications. On the one
hand, it can be used to compute flow rates by measuring
a pressure difference, for example to determine the speed
of an aircraft with a Pitot tube [77]. On the other hand,
it can be used to compute pressure differences by mea-
suring flow rates, for example to determine the pressure
distribution around a plane wing (airfoil) and thus the lift
force [77]. Together with the principle of mass conserva-
tion, the Bernoulli equation also explains the “Venturi
effect”, the decrease in pressure in a pipe constriction.
This effect is utilized in Venturi gauges to measure the
flow rate through pipes; in the kitchen, it is exploited in
wine aerators [Fig. 3a], where the low pressure is used to
draw in air bubbles. The mixing with air can improve
wine flavour [78,79], which was already known to Louis
Pasteur, who famously wrote “C’est l’oxyg`ene qui fait
le vin” [80], or “It’s the oxygen that makes the wine”.
The Venturi effect is also used in gas stoves and grills,
where inspirators mix air and flammable gas. Read more
about the use of fire in the kitchen in §V F. Finally, the
Bernoulli effect can be used to make food grippers, for
instance to handle sliced fruits and vegetables [81,82].
F. Pendant drop: Surface tension
The Bernoulli and Navier-Stokes equations describe
the motion of fluids, but they don’t say much about the
interesting phenomena that occur at interfaces: How can
droplets hang upside-down from a tap? Water molecules
are attracted to each other by cohesion forces, particu-
larly by hydrogen bonds [83,84]. This cohesion leads to
a surface tension, γ, a force per unit length, which acts
as if there was a taut elastic sheet covering the liquid
interface [85]. Indeed, surface tension acts to minimise
the surface area, so free droplets tend to be spherical.
This inward force is balanced by a higher pressure inside
the droplet. The pressure difference across the water-
air interface, called the Laplace pressure, is given by the
Young-Laplace equation,
∆p=−γH, (7)
where His the mean curvature of the interface. For
spherical droplets of radius R, we have H= 2/R. The ex-
pression is named after Thomas Young (1773-1829) and
Pierre-Simon Laplace (1749-1827).
In the kitchen, the surface tension of water-air in-
terfaces can be measured with a dripping tap experi-
ment: A drop of radius Rhanging from a faucet balances
surface tension (Fγ∼2πRγ) with the force of gravity
(Fg=4
3πR3∆ρg). Therefore, the pendant drop will fall
if it grows larger than R∗∼λc, the capillary length
λc=pγ/(∆ρg).(8)
Then, by measuring this critical droplet radius, the sur-
face tension can be estimated using the known values for
gravity Gand the density difference ∆ρbetween water
and air. For a more accurate result, one can measure
the weight of an ensemble of (say) a hundred drops and
take the exact shape of the nozzle and the pendant drop
into account [86]. For water-air interfaces, we then find
the surface tension γ≈0.07 N/m. This value (double-O-
seven) is easy to remember because the non-dimensional
parameter that characterises the importance of gravity
compared to surface tension is called the Bond number,
Bo ≡Gravitational forces
Capillary forces =∆ρgL2
0
γ=L0
λc2
,(9)
named, however, after an English physicist Wilfrid Noel
Bond (1897-1937). Here, L0is a characteristic length
scale such as the drop radius R. Note that the Bond
number is also called the E¨otv¨os number (Eo). Surface
tension can lead to a vast number of counter-intuitive
phenomena, and new effects are still discovered every day.
This is particularly important for nanotechnology and
miniaturization in microfluidics, where the Bond num-
bers are inevitably small. Also in microgravity experi-
ments the effects of surface tension are amplified [87], as
highlighted by wringing out a wet cloth in the Interna-
tional Space Station [88]. An alternative simple method,
accessible for students, is to use a smartphone camera to
measure the shape of a hanging droplet [89].

7
G. Wetting and capillary action
Besides cohesion forces that lead to surface tension,
liquid molecules are also subject to adhesion forces, when
they are attracted to other molecules [83]. This can be
observed directly when looking at a droplet sitting on
a surface [84]. The line where the liquid, gas and the
solid meet is called the contact line or the triple line.
The contact angle, θc, is defined as the angle between
the liquid-gas interface and the surface. If the surface
is hydrophilic (attracts the water), the droplet spreads
out and wets the surface [90], leading to a small contact
angle of 0 ≤θc<90◦. But if the adhesion forces are
weak such as on waxed surfaces (hydrophobic surface) the
cohesion forces can keep the drop together, leading to a
large contact angle of 90◦< θc≤180◦. The equilibrium
contact angle is found by minimising the total free energy,
for example using calculus of variations. This leads to the
Young equation,
cos θc=γSG −γSL
γLG
,(10)
where the interfacial energies γij encode the relative
strengths of the cohesion and adhesion forces between
the three phases i, j ∈(Liquid, Gas, Solid). Note, with-
out subscript we imply γ=γLG, and the gas phase is
sometimes also called the vapour phase. This equation
may be improved by accounting for advanced effects like
interface curvature and contact line dynamics [91,92].
The degree to which a drop will wet a substrate can be
estimated with the spreading parameter [84], given by
S=γSG −(γLG +γSL).(11)
When S > 0, the drop spreads indefinitely towards a
zero equilibrium contact angle, as is the case of silicone oil
spreading on water. When S < 0, the drop instead forms
a finite puddle, as when a drop of cooking oil is placed
on a bath of water in a simple kitchen experiment. Due
to its weight, the drop deforms the water surface. The
amount of distortion depends on the relative magnitude
of buoyancy to surface tension forces, as characterized by
the Bond number [Eq. (9)].
Capillary action. Leonardo da Vinci (1452-1519) first
recorded that liquids tend to flow spontaneously into con-
fined spaces, an effect now called capillary action [84].
When dipping a narrow glass tube or a straw into wa-
ter, the liquid will rise up until it reaches a constant
height h. The narrower the capillary, the more the liq-
uid ascends. This is somewhat unexpected because the
Hagen–Poiseuille equation (3) suggests that the flow rate
should decrease for smaller tube radii, if the pressure
difference were constant. Moreover, the flow even moves
against the hydrostatic pressure imposed by gravity. This
effect is also explained by intermolecular adhesion and
cohesion forces.
The height to which the water rises in a capillary
can be calculated by balancing the hydrostatic pressure,
∆pg=ρgh, with the Laplace pressure as in Eq. (7). If
the capillary is cylindrical with inner radius Ri, and the
meniscus has a spherical shape, its radius of curvature is
R=Ri/cos θcin terms of the contact angle. Combining
these ingredients yields Jurin’s law [93],
h= 2γLG cos θc/(ρgRi).(12)
For a typical glass microchannel of Ri= 50µm, the water
can rise up to ∼30cm and much more for thinner tubes.
Hence, capillary action has many applications [84]. For
example, if you like growing your own basil, plants use
capillarity for transporting water from the soil to their
leaves, together with other mechanisms including osmosis
and evaporation [94,95].
Conversely, if the contact angle θc>90◦for hydropho-
bic surfaces, hturns negative in Jurin’s law [Eq. (12)]
so liquid is expelled. Then one can observe dewetting,
the process of a liquid spontaneously retracting from
a surface [96–98]. Consequently, thin liquid films are
metastable or unstable on these surfaces, as it breaks
up into droplets. Therefore, dewetting is often not de-
sirable in industrial applications because it can peel off
protective coatings or paint [99], and inhibit lubrication
in machinery [see §VI E]. Dewetting is also important
in solid-state physics because it can damage thin solid
films. This effect can also be turned into an advantage for
making photonic devices and for catalyzing the growth
of nanotubes and nanowires [100]. An exemplary fine-
dining accessory exploiting the effects of surface tension
is the ‘floral pipette’ [101], which presents a novel means
of serving small volumes of fluid in an elegant fashion.
H. From jets to drops: Plateau-Rayleigh instability
Dripping kitchen taps offer a direct example of an
important hydrodynamic instability: When a vertical
stream of water leaves a worn tap, it narrows down be-
cause it is stretched by gravity. Once the liquid cylinder
is sufficiently thin, we observe its breakdown into droplets
before hitting the sink [Fig. 4a]. Plateau [102] was the
first one to describe it systematically, which then enticed
Lord Rayleigh [103] to provide a theoretical description
and a stability analysis of an inviscid jet. He showed
that a cylindrical fluid column is unstable to disturbances
whose wavelengths λexceed the circumference of the
cylinder. The most unstable mode for a jet of radius R
has the wave number k= 2π/λ ≈0.697Rand from the
growth rate of this mode, a typical jet breakup time can
be estimated as τb∼3pρR3/γ, which in a kitchen sink
is typically a fraction of a second. Kitchen jets exhibit
also cross-sectional shape oscillations attributed to cap-
illary waves which can easily be seen in home-made ex-
periments [104]. By measuring these variations of jet ec-
centricity, occurring at a frequency proportional to τ−1
b,
one can measure the surface tension [105].
The physics and stability of liquid jets are fundamen-
tally important for a number of applications, as reviewed

8
FIG. 4. Jets in the kitchen sink. (a) Example of the Rayleigh-
Plateau instability, where a thin jet from a faucet breaks into
droplets. Image by Niklas Morberg, licenced under CC BY-
NC 2.0. (b) A circular hydraulic jump forms when a thicker
liquid jet impinges on a planar surface. (c) A triangular hy-
draulic jump, seen from below through a glass plate. The
impinging jet is the black center region, the jump line is the
black line surrounding it, and an additional roller vortex is
marked with a red dye. (b,c) From Martens et al. [112].
by Eggers and Villermaux [106]. The break-up of jets
has some universal features, which result in a number of
self-similar solutions. The scale invariance is manifested
both in the conical shape of the tip of a French baguette,
and in a bimodal distribution of droplets produced, inde-
pendently of the initial conditions, in the pinch-off pro-
cess. The latter is an important technological problem
in ink-jet printing [107,108]. In microfluidic systems,
the surface-tension assisted breakup is also used to cre-
ate mono-disperse droplets [109], and drinking (lapping)
mammals including cats and dogs may adjust their jaw-
closing times to the pinch-off dynamics to maximise wa-
ter intake [110]. Citrus fruits can eject high-speed mi-
crojects from bursting oil gland reservoirs, up to 5000
g-forces, comparable to the acceleration of a bullet leav-
ing a rifle [111].
I. Hydraulic jumps in the kitchen sink
When a thicker water jet (that does not break up) im-
pinges on the kitchen sink, it will first spread out in a thin
disk at high velocity. Surprisingly, at some distance from
its origin, the thickness of the film suddenly increases;
a hydraulic jump is formed [Fig. 4b]. Again, Leonardo
da Vinci was the first known to study hydraulic jumps
[113], and Lord Rayleigh [114] first described it mathe-
matically. He postulated that a balance between inertia
and gravity forces lead to the jump. In other words, a
jump is expected when the Froude number,
Fr ≡Inertial forces
Gravitational forces =U0
√gL0
,(13)
transitions through unity, since the flow in the thin film
continually loses momentum as is spreads out radially.
Here, U0is the velocity at the free surface of the film, g
is gravity, and L0is the film thickness. Rayleigh did not
include effects of surface tension [§II F] in his analysis.
However, he wrote that “On the smallest scale surface-
tension doubtless plays a considerable part, but this may
be minimised by increasing the stream, and correspond-
ingly the depth of the water over the plate, so far as may
be convenient”. Watson [115] included effects of viscos-
ity in his description of circular jumps, and two years
later, in 1966, Olsson and Turkdogan [116] performed
complementary experiments to validate Watson’s theory
and hypothesized that surface tension contributed to the
loss of kinetic energy at the jump. Bush and Aristoff
[117] expanded Watson’s theory by exploring the role
of surface tension on the formation of circular hydraulic
jumps, and Mathur et al. [118] found surface tension to
dominate over gravity for films on the micrometer scale.
However, all of the above-mentioned authors regarded
capillary pressures (due to surface tension, see Eq. (7)) as
being negligible compared to hydrostatic pressures (due
to gravity) on the scale of kitchen sink hydraulic jumps.
Bhagat et al. [119] observed that when a strong jet im-
pinges on a planar surface, the radius of the disk (inner
region before the jump) is independent on the orienta-
tion of the surface. They concluded that gravity is not
causing the hydraulic jump when the film is sufficiently
thin, as it is when produced by a strong jet. Instead,
a capillary pressure competes with the transport of mo-
mentum, and this balance is characterised by the Weber
number,
We ≡Inertial forces
Cohesion forces =ρU 2
0L0
γ,(14)
where γdenotes surface tension and ρdenotes the density
of the impinging liquid. For weaker jets, however, gravity
can no longer be neglected. By balancing the energy at
the jump and adopting the approach by Watson [115] by
assuming a boundary layer flow inside the film, Bhagat
et al. [119] found the following more general criterion for
a circular jump to occur:
We−1+ Fr−2= 1,(15)
where the first term is associated with capillary waves
and the second is associated with gravity waves.
As we have seen, the rich physics characterizing cir-
cular hydraulic jumps has attracted researchers for cen-
turies, and the degree to which surface tension controls

9
these jumps remains an active research topic. Duchesne
et al. [120] and Bohr and Scheichl [121] consider a static
control volume and argue that surface tension has a neg-
ligible influence as it is fully contained in the Laplace
pressure, while Bhagat and Linden [122] come to a dif-
ferent conclusion by an energy-based analysis. Another
aspect of hydraulic jumps concerns the influence of dif-
ferent surface coatings on the jump radius and shape,
and Walker et al. [123] showed that when a water jet
impinges on a shear thinning liquid [see §IV A], the ra-
dius becomes time dependent. Later, the same group
used viscoelastic liquids to enhance the degree of parti-
cle removal through inducing normal stresses that ‘lift’
the particles away from the substrate [124,125]. Finally,
it is also possible to create polygonal jumps, either by
leveraging hydrodynamic instabilities in viscous liquids
[112,126–129], as displayed in Fig. 4c, or by utilizing
micro-patterned surfaces [130].
J. How to cook a satellite dish
The importance of parabolas to focus light rays was al-
ready known since classical antiquity: Diocles described
it in his book On Burning Mirrors, and legend has it that
Archimedes of Syracuse (c.287–c.212 BC) used these to
burn down the Roman fleet [131]. The latter is probably
fictional, but Archimedes did write that the surface of a
rotating liquid forms a paraboloid [Ibid]. At hydrostatic
equilibrium, the gravitational force on a fluid element is
balanced by the centripetal force and buoyancy [Fig. 5a],
such that the liquid height profile is given by
h(ρ)−h(0) = ω2ρ2/(2g),(16)
where ρis the radial distance from the rotation axis,
ωis the angular velocity, and the corresponding focal
distance is f=g/2ω2. Liquid-mirror telescopes use ex-
actly this concept: the Large Zenith Telescope [Fig. 5b] is
made of a 6-meter pool of rotating liquid mercury [132].
Note, the earliest known functional reflecting telescope
by Isaac Newton (1642–1727) used a spherical mirror be-
cause paraboloids are hard to fabricate [133]. For modern
large telescopes, the parabolic mirror is sometimes made
by spinning molten glass in a rotating furnace. You can
try to do this yourself in the kitchen, by melting some
wax (or gelatin) and letting it cool on a record turntable.
Once it has solidified you could even coat it with reflec-
tive paint. Parabolic reflectors are also widely used in
solar cookers and various renewable energy technologies
[134].
Equation (16) holds under the assumptions that the
whole fluids rotates as a rigid body and that there is
no local rotation of neighbouring fluid elements. Then
the flow is called irrotational [135]. In closed containers,
one must also account for surface tension and potential
dewetting when the bottom becomes dry in the middle of
the vessel [136]. For higher rotation speeds, however, this
static description is no longer valid as the flow becomes
FIG. 5. Liquid mirrors. (a) Diagram of a rotating liquid that
forms a parabolic reflector by the principle of hydrostatic equi-
librium. The black arrows denote the force balance between
gravity, rotation and buoyancy. (b) The Large Zenith Tele-
scope uses this principle. It was one of the largest optical tele-
scopes in the world. Diameter: 6.0 m, rotation period: 8.5 s,
mercury thickness: 1.2 mm, accessible area of sky: 64.2 deg2.
Person for scale. From Hickson et al. [132].
rotational. In fact, this transition is also highlighted by
a symmetry breaking, leading to the formation of polyg-
onal rotating structures [137,138] before all symmetry is
lost in turbulence at even higher rotation speeds.
This local rotation of fluid elements is quantified by the
fluid vorticity [135], defined as ω=∇ × u. The Navier-
Stokes equation (2) for an incompressible fluid (∇·u= 0)
can be recast upon taking a curl of both sides, giving
Dω
Dt = (ω· ∇)u+ν∇2ω.(17)
Here the first term of the right-hand side accounts for the
stretching or tilting of vorticity due to the flow gradients,
while the last term describes the diffusion of vorticity
in the fluid. Vortices formed across all scales, from at-
mospheric to molecular processes, are described by their
velocity or vorticity distribution. The simplest models
assume an axisymmetric velocity field in which the fluid
circles around the vortex axis. However, the flow field in
most practical cases is more complex, as secondary flows
arise due to this circling motion. We describe them in
more detail in §VIII B where we discuss how stirring the
tea may impact the dynamics of submerged tea leaves.
Secondary flows in the vertical direction arise also in
bathtub or sink vortices [139].
K. Washing and drying hands, skincare
In Shakespeare’s Scottish play, Lady Macbeth repeat-
edly washes her hands ‘for a quarter of an hour’ to cleanse
away her murderous guilty conscience. At the start of the
COVID-19 pandemic, her troubled soliloquy was used
to satirise WHO posters that offer personal hygiene in-
structions in public restrooms [140]. Jokes aside, washing
hands with soap is “a modest measure with big effects”
to combat pathogen dispersal [141], which is of particu-
lar importance for the food industry [142]. Also coron-
aviruses can be cleaned off the skin with soap, or with

10
hand sanitisers that contain sufficiently high concentra-
tions of agents such as ethanol or isopropanol [143–145].
Dancer [146] also reminds us that besides our hands, we
should not forget to clean the surfaces that we touch,
following the legacy of Florence Nightingale (1820-1910),
often called the founder of modern nursing. Despite the
importance of proper sanitation, its hydrodynamics is
not so well explored. Mittal et al. [147] recently wrote
“Amazingly, despite the 170+ year history of hand wash-
ing in medical hygiene [148], we were unable to find a
single published research article on the flow physics of
hand washing.” There is of course a large body of litera-
ture about micelle formation and multi-phase flows [§III],
foaming [see §III F] and the physics of micro-organisms
[§VI I], but connecting this network of knowledge in the
context of personal hygiene is only just starting.
Motivated by this gap in the literature and without
access to a lab due to current stay-at-home orders, Ham-
mond [149] conducted a theoretical assessment of hand
washing. Using a lubrication approximation, he came a
long way in describing how and when a virus particle
is released from our hands when we rub them together.
Hammond found that the rubbing speed needs to exceed
a certain value set by the depth of the surface undula-
tions of the skin, which in his model are represented by
sinusoidal waves. Surprisingly, he found that multiple
rubbing cycles are needed to remove a particle. More
generally, the study of washing biological surfaces could
widen our understanding of hydrodynamic interactions
between particles, rough surfaces and fluid flows, with
important implications in the food industry, for example.
A natural extension of this work is to include viscoelastic
effects, which might more realistically represent the ma-
terial properties of the soap film. Moreover, two recent
review papers that discuss the biological physics and soft
matter aspects of COVID-19 were written by Poon et al.
[144] and Bar-On et al. [145].
After washing our hands, it is essential to dry them
properly [142,150]. When we use a towel, the water gets
pulled into the fabric by capillary action [see §II G]. This
only works well if the towel is more hydrophilic (water-
loving) than the surface of our hands. Paper and cotton
cloth are especially hydrophilic, aided further by the large
surface area of the fibres. Another method is to dry hands
by evaporation. Whereas evaporation has been studied
extensively on idealised surfaces [see §VII E], not so much
is known about wetting and evaporation on soft materials
like the skin [151,152]. An ongoing debate is whether the
dispersal of viruses and bacteria can be stopped more
efficiently by warm air dryers, or jet dryers, which on
the one hand may avoid having to touch surfaces but
on the other hand could cause pathogen aerosolization
[147,153–155], which is especially problematic in food
processing plants [156].
A common medical condition that comes with wash-
ing and drying hands frequently is xeroderma, or dry
skin [157]. This can lead to symptoms including itch-
ing, scaling, fissure, or wrinkling [158,159]. These prob-
lems can often be alleviated with moisturisers or emol-
lients, but in more severe cases an effective treatment
requires understanding the underlying biophysical mech-
anisms [160]. Liquid transport has been studied in the
networked microchannels of the skin surface [161], as well
as the physics of stratum corneum lipid membranes [162],
and more generally soft interfacial materials [163]. Con-
necting the disciplines of physics and medicine will be-
come increasingly important in future research.
III. DRINKS & COCKTAILS: MULTIPHASE
FLOWS
After washing our hands, it is time to start dinner with
a beverage of choice. In this section we review a wealth of
hydrodynamic phenomena that emerge in these drinks,
such as shock waves in the tears of wine, effervescence
in Champagne, or ‘awakening the serpent’ during whisky
tasking. These multi-phase flows [164,165] have seen
rapid scientific advances recently, and they are applied
extensively in industrial processes. Perfect to contem-
plate while waiting for the main course to arrive, or to
impress at a cocktail party.
A. Layered cocktails
A classic example of a culinary multi-phase fluid is
a layered cocktail [Fig. 6a]. For instance, an Irish flag
cocktail is made by first pouring cr`eme de menthe, then a
layer of Irish cream, topped off with orange liqueur. This
beverage is called stably stratified, because each layer is
less dense than the one below it. Multiple coloured lay-
ers can be formed using a density chart for the differ-
ent liquid ingredients, also called a specific gravity chart
[166]. Stratification is essential for life on Earth, both
in the atmosphere [167], where sharp cloud layers can
be observed, and in the ocean [168], where water layers
can be characterised by large gradients in density (pycno-
cline), but also gradients in temperature (thermocline) or
salinity (halocline), immediately impacting environmen-
tal stratified flows [169] and more generally geophysical
fluid mechanics [170].
1. Inverted Fountains
The cocktail layers will separate readily if the ingre-
dients are immiscible, such as, say, lemon water and
rose oil. However, if the liquids are miscible, it is rec-
ommended to pour the layers slowly (ideally along the
side of the glass with the help of a spoon) because oth-
erwise the layers will mix. We can understand this tur-
bulent and miscible mixing process as an ‘inverted foun-
tain’ [171,172], where the lighter fluid is forced down
into the heaver fluid, opposed by buoyancy [173]. The

11
(inverted) height of the fountain, zf, and thus the mix-
ing volume, depends strongly on the Reynolds number
[Eq. (5)], where U0is the pouring velocity and L0is the
radius of the injected jet, but also the densimetric Froude
number, where gin Eq. (13) is replaced by |g0|, the re-
duced gravity due to buoyancy, given by
g0=g(ρi−ρa)/ρa,(18)
in terms of ρiand ρa, the densities of the injected and
the ambient fluid. Conventionally, g0is negative for in-
verted fountains. For large Froude numbers, Turner [171]
showed that the fountain height is given by
zf≈2.46L0Fr.(19)
This classical result predicts a linear relation, which
agrees well with modern experiments. Note that differ-
ent scaling laws have been derived for weaker fountains
with a smaller Froude number, as reviewed by Hunt and
Burridge [172].
Note, we already mentioned the Froude number in the
context of hydraulic jumps [§II I]. Interestingly, it turns
out that hydraulic jumps also occur in layered miscible
fluids, as described by Wood and Simpson [174]. Re-
cently, new models and experiments were developed to
describe such internal hydraulic jumps, which do not
rely on the Boussinesq approximation and account for
entrainment effect as shear increases [175–179].
2. Internal Waves
Once the cocktail layers are established, more interest-
ing flow phenomena can be observed. When the glass
is slightly disturbed, internal waves can be seen, which
are gravity waves (not to be confused with the gravita-
tional waves in general relativity) that propagate inside
the fluid instead of on its surface [181,182]. Specifically,
they are called interfacial (internal) waves when they
propagate horizontally along an interface characterised
by a density gradient, dρ/dz < 0. Consider a fluid parcel
in a continuously stratified fluid, with a smooth density
profile ρ(z), that is in hydrostatic equilibrium at z0. If
the parcel of density ρ0=ρ(z0) is displaced a vertical
distance ∆z=z−z0where the surrounding fluid has
a different density ρ(z0+ ∆z), it will feel a gravitational
restoring force [Eq. (1)]. To first order, that leads to sim-
ple harmonic motion [183] with an oscillation frequency
called the Brunt–V¨ais¨al¨a frequency,
f=1
2πs−g
ρ0
dρ
dz .(20)
These internal oscillations are typically slow compared
to surface waves at the liquid-air interface, because the
density gradient between the liquid layers is much smaller
[see §IX D]. Internal waves are common in oceanography
[184] and atmospheric science [170], where they lead for
example to rippled clouds or lenticular clouds [185].
FIG. 6. Multi-phase cocktails. (a) B-52 shot made by lay-
ering Kahlua, Bailey’s Irish Cream and Grand Marnier, with
a splash on top. Image from J. D. Baskin on Flickr, licenced
under CC BY 2.0. (b) Evaporation-induced Rayleigh-Taylor
instability. Colours indicate the ethanol concentration, from
blue (low) to red (high), measured with Mach–Zehnder in-
terferometery. Image courtesy of Sam Dehaeck. (c) Kelvin-
Helmholtz instability waves formed at an oscillated water-oil
interface. From Yoshikawa and Wesfreid [180].
3. Kelvin–Helmholtz Instability
Fluctus clouds look like breaking ocean waves in the
sky. They are caused by the Kelvin–Helmholtz (KH) in-
stability [186,187], when two fluid layers move alongside
each other. Indeed, the same can be observed when a
layered cocktail is sheared. By rotating the glass, a ve-
locity gradient ∂u/∂z is created between the stratified
layers. This shear is especially pronounced if the liquid
spins up by friction with the bottom wall instead of the
side walls [see §VIII A]. The velocity gradient drives the
KH instability, while it is opposed by buoyancy, quanti-
fied by the density gradient ∂ρ/∂z. The ratio of these
two forces is encoded by a dimensionless quantity named
the (densimetric) Richardson number,
Ri ≡Buoyancy forces
Shear forces =g0
ρ
∂ρ/∂z
(∂u/∂z)2.(21)
The fluid layers are unstable when the shear is large
enough, when Ri .1, depending on the system con-
figuration. In a setup resembling our cocktail, the in-
stability was characterised recently in a spin-up rotating
cylindrical vessel by Yan et al. [188], and in an oscilla-
tory cylindrical setup by Yoshikawa and Wesfreid [180],
as shown in Fig. 6c. Naturally, the KH instability will
cause the stratified layers to mix with one another, as re-
viewed by Peltier and Caulfield [189], or cause emulsion
formation if the layers are immiscible [§IV D]. If the lay-
ers are immiscible, surface tension will stabilise the short

12
wavelength instability on top of buoyancy, which strongly
affects emulsion formation [190,191]. Therefore, the KH
instability is important for many processes in industry
and food science. Think about making mayonnaise with
a blender, for example. From a fundamental point of
view, understanding these flows is intrinsically connected
with the heart of theoretical physics: symmetries. Only
recently, Qin et al. [192] described that the KH instability
results from parity-time symmetry breaking. Moreover,
the Kelvin-Helmholtz instability also features in the mag-
netohydrodynamics of the sun [193], ocean mixing [170],
relativistic fluids [194] and superfluids [195].
4. Rayleigh-Taylor instability
Until now we have discussed stably stratified cocktails.
Yet, when a heavier fluid sits on top of a lighter fluid, the
latter pushes into the former by gravity, so the mechani-
cal equilibrium is unstable. Any small perturbation will
lead to a familiar pattern of finger-like structures with a
mushroom cap, as seen in Fig. 6b. This phenomenon is
explained by the Rayleigh-Taylor (RT) instability, which
was first discovered by Lord Rayleigh [196], and later
described mathematically by Taylor [197] together with
systematic experiments by Lewis [198]. Many develop-
ments followed, and Chandrasekhar [199] extended the
theoretical description in his famous book. Like the KH
instability, the RT instability is relevant across the disci-
plines, from the astrophysics of supernovae [200,201] to
numerous technological applications [187].
The RT instability arises because the system seeks to
minimise its overall potential energy. Its onset is primar-
ily governed by the Atwood number,
At = ρh−ρl
ρh+ρl
,(22)
the non-dimensional difference between the densities of
the heavier and the lighter fluid, ρhand ρl. We think
this number is named after George Atwood FRS (1745-
1807), who also invented the Atwood machine, but we
struggled with finding the original source. To describe
the RT instability more generally, one must account for
the fluid viscosities and surface tension [202], and poten-
tial effects due to fluid compressibility [203]. Moreover,
the dynamics depend strongly on the initial conditions:
They begin with linear growth from perturbations, which
transitions to a non-linear growth phase involving char-
acteristic structures of rising ‘plumes’ and falling ‘spikes’.
Subsequently, these plumes and spikes interact with each
other through merging and competition, and roll up into
vortices. The final stages are characterized by turbulent
mixing [see §VIII].
In the words of Benjamin [40], the RT instability is
a fascinating gateway to the study of fluid dynamics.
It can readily be observed in kitchen experiments, but
it also occurs spontaneously without us even noticing
FIG. 7. Shock waves in tears of wine. (a) Schematic of a
conical-shaped glass of inclination angle α, showing a one-
dimensional thin wine film traveling up an inclined flat glass
surface. The film height h∗is exaggerated for clarity. (b)
Experiment using 18% ABV port wine and α= 65°. Swirling
the wine around the glass creates a front that forms out of
the meniscus. The draining film advances up the glass and
destabilises into wine tears. (c) The formation of a reverse-
undercompressive (RUC) shock. (a-c) From Dukler et al.
[207].
[Fig. 6b]: In a well-mixed (non-layered) cocktail, the alco-
hol evaporates faster than water. Hence, at the air inter-
face, a water-rich layer develops that is denser than the
bulk mixture, which gives rise to the RT instability [204].
The plumes of such ‘evaporating cocktails’ are observed
using a Mach–Zehnder interferometer. By demodulat-
ing the fringe patterns using a Fourier transform method
[205], it is possible to compute the refractive index field,
and hence the local ethanol concentration. Evaporation-
induced Rayleigh-Taylor instabilities also occur in poly-
mer solutions [206], so your cocktail need not necessarily
be alcoholic.
Interestingly, the RT instability is often inseparable
from the Kelvin-Helmholtz instability. RT flows cre-
ate velocity gradients that trigger KH billows, while KH
flows create density inversions that trigger RT fingers.
Moreover, the RT instability is closely related to the
Richtmyer–Meshkov (RM) instability, when two fluids of
different density are impulsively accelerated [201], and to
the Rayleigh-B´enard convection, where an instability oc-
curs due to heating a single liquid from below or cooling
it from above, which we describe in more detail in §V C.
B. Tears of wine
One of the most surprising phenomena in multi-phase
flows is the Marangoni effect, named after Marangoni
[208]. You may have seen this effect already in the
kitchen, when a droplet of dish soap falls into a bowl
of water sprinkled with pepper: Within the blink of an

13
eye, the pepper moves to the edges by an outward flow
along the liquid-air interface. Another striking example
is adding food colouring drops to a bowl of milk, where
poking it with a soap-covered cotton bud generates beau-
tiful flow patterns (try it!). These Marangoni flows arise
because the surfactant molecules in the soap lower the
surface surface tension [209], leading to a difference in
surface tension along the interface, of ∆γ=fγ, where
the factor f∼10−1for most soaps. Consequently, the
water without soap pulls more strongly on the water with
soap, generating a current from regions of lower to higher
surface tensions. The flow strength can be estimated
roughly as u≈∆γ/µ, so even small fractions fgive rise
to fairly strong flows, using γ∼0.07 N/m and viscosity
µ∼0.9 mPas for water. Of course, more detailed calcu-
lations must take other influences into account, includ-
ing solubility, surface contamination and system geome-
try [209–213]. The ratio between advective and diffusive
transport over a characteristic length scale L0is given by
the Marangoni number,
Ma ≡Advective transport rate
Diffusive transport rate =∆γL0
µD ,(23)
where Dis the diffusivity of the surfactants or any addi-
tive that changes the surface tension.
Fortunately, the Marangoni effect does not only oc-
cur with soap, but also with edible ingredients. In fact,
the phenomenon was first identified by James Thomson
(1822-1892) in the characteristic “tears” or “legs” of wine
[214], and indeed other alcoholic drinks including liquors
and whisky [see §III C]. These tears are formed because
the alcohol is more volatile than water, and it has a lower
surface tension [215]. To see this, pour yourself a glass of
wine. In the thin meniscus that the wine forms with the
glass surface, the alcohol evaporates faster than the wa-
ter, so the surface tension here is higher than in the bulk.
The wine is then pulled up the meniscus, forming a thin
film that starts climbing up along the side of the glass.
After a few seconds, the film forms a ridge approximately
1 cm above the meniscus. This ridge becomes unstable
under its own weight as more wine climbs up, so it col-
lapses into “tears” that fall down towards the meniscus.
Large tears can fall back into the bulk, but small tears
can also be pulled up again by the continuously climb-
ing film that replenishes the ridge. This can cause the
tears to bounce up and down, especially at the meniscus.
The effects is beautifully imaged using the Schlieren or
shadow projection techniques [216].
So far, so good. But the mechanism by which the
droplets form and collapse is more complex. As the wine
film climbs to its terminal height, when the Marangoni
stresses are balanced by gravity, this transient station-
ary state is subject to various hydrodynamic instabilities
[217–219]. Besides alcohol concentrations differences, the
evaporation also induces temperature gradients that lead
to additional Marangoni stresses [220]. The ridge insta-
bility that triggers the formation of wine tears was also
studied and analysed with a Plateau-Rayleigh-Taylor
FIG. 8. Whiskey webs. Different patterns emerge after letting
whiskey droplets of different alcohol percentages evaporate on
a glass surface. At 35 % ABV (left), the deposits are evenly
distributed, while at 10 % ABV (right), the deposits are
distributed preferentially near the rim of the drop. At inter-
mediate (20 % ABV) concentration (middle), the deposits
form complex ‘whiskey web’ patterns. From Williams et al.
[224].
theory [221]. Yet, the dynamic formation of the ridge
itself is still not well understood.
Until now we have discussed a wine film that spon-
taneously climbs up a dry wine glass, but it is common
practice among connoisseurs to swirl the wine around.
This often creates a wet coating film much higher than
the terminal climbing height, which can give rise to rather
different behaviours. Dukler et al. [207] showed that
such swirled films can feature a ‘shock wave’ that climbs
out of the meniscus [Fig. 7], again driven by Marangoni
stresses due to evaporation. This wave can be observed
as a ridge that propagates upwards, where the wine film
above the shock front is thicker than below it. Specif-
ically, the dynamics can be described as a reverse un-
dercompressive (RUC) shock. This type of shock wave
is unstable: Small inhomogeneities in the wine film are
amplified into thick drops, which then fall down as tears.
As described previously for rising films driven by thermal
gradients [222,223], different shock morphologies can oc-
cur in other circumstances. This is of great scientific and
technical interest, including dip-coating and painting ap-
plications. Moreover, it would be interesting if people’s
dining experience could be improved by developing a new
dish that uses the shock wave as a surprise effect.
C. Whisky tasting
When we taste whisky or whiskey, we use all of our
senses [225]. Note: the spelling whiskey is common in
Ireland and the United States, while the term whisky is
common for produce from the UK and most other coun-
tries. While complete volumes have been written about
the art of whisky tasting, see e.g. [226], we would like
to highlight some of its hydrodynamic aspects, which are
clearly visible. That is, a good deal of information may be
gained by assessing the appearance and dynamics of spir-
its [227,228]. The following tests can give clues about
the whisky quality and vintage before any smelling or
tasting.
As a first examination, it is customary to inspect the

14
tears that we described in §III B. This gives an indica-
tion of the whisky’s alcohol content, its viscosity and its
surface tension, which in turn depend on the exact chem-
ical composition. When the tears run down slowly, with
thick legs, it indicates that it will give more texture in
the mouth [229]. Conversely, if the legs are thin and
run quickly, the whisky is likely to be younger and of a
lighter body. This is because the texture changes during
the aging process, as viscous natural oils and other com-
pounds are released from the wooden casks [230], which
inevitably also influence the whisky colour. The rheo-
logical and thermophysical properties are also affected
by storage and temperature [231]. However, it would
be wrong to claim that a whisky with more pronounced
tears is automatically sweeter or better in quality, since
the tear formation is a purely physical phenomenon. In-
deed, the tears vanish when the glass is covered, since
the evaporation-induced Marangoni stresses disappear.
A second experiment is the ‘beading’ test [227,232,
233]. When a whisky bottle is shaken vigorously, a foam
can appear on the liquid interface if the alcohol concen-
tration is higher than approximately 50% alcohol by vol-
ume (ABV). The beading is not necessarily more pro-
nounced at higher concentrations, but it is not observed
below a certain percentage. Beading can also say some-
thing about the age of the whisky: The bubbles tend
to last longer in older vintages because the compounds
released from the wooden casks can stabilise the foam.
Read more about foam stability in §III F.
A third inspection method is called whisky viscimetry
[229,234]. When adding a little water to the whisky,
small vortices called ‘viscimetric whirls’ appear when the
liquids of different viscosities mix with one another. Con-
noisseurs sometimes refer to this phenomenon as ‘awak-
ening the serpent’ [234]. These vortices only last for a few
seconds, but again they tell us something about the tex-
ture of the whisky. The more persistent the whirls, the
thicker the mouthfeel and the higher the alcohol concen-
tration. To the best of our knowledge, this effect has not
been quantified systematically in the scientific literature.
Depending on the distillation method, spirits reach an
initial strength of ∼70% ABV (pot still) or even higher
(column still). Most whiskies are then diluted down to
∼60% ABV prior to storage in casks. After maturation,
they are often mixed with more water to ∼40% ABV,
the minimum in most countries. There are several rea-
sons for this dilution: First, it can enhance the flavour
because many of the taste-carrying molecules, such as
guaiacol, are thermodynamically driven up to the liquid-
air interface at low ethanol concentrations [235]. Second,
the ethanol concentration influences the sensory percep-
tion [225,236], where lower strengths are more palat-
able by most consumers. Third, besides enhancing the
flavour of spirits, dilution can lead to a better mouth-
feel. At 20 °C, the viscosity of pure ethanol and water is
µ1≈1.2 mPa s and µ2≈1 mPa s, respectively, but the
viscosity of an ethanol-water mixture features a maxi-
mum of µ12 ≈3 mPa s at 40-50% ABV [237]. This sur-
FIG. 9. Marangoni-stress powered cocktail boats. (a) A fleet
of cocktail boats with different designs. The fuel (any liquor)
is stored in the central cavity. The thin slit at the rear slowly
releases the fuel into the glass, which establishes a surface
tension gradient that drives the boat forward. (b) Trajectory
in a cocktail glass. This boat is ∼1.5 cm long and fueled by
Bacardi 151 (75% ABV). (a,b) From Burton et al. [101].
prising non-linear effect of binary mixture viscosities is
described in more detail in §IV B.
Remarkably, diluting your drink can also help with dis-
tinguishing whether it is whisky or whiskey: An evapo-
rating bourbon droplet of 40% ABV tends to leave a uni-
form surface deposition [Fig. 8], while a diluted droplet at
20% ABV leaves distinctive patterns called whiskey webs
[224,238,239]. Apparently, Scotch whisky and other dis-
tillates do not feature these web patterns and they are
unique across different samples of American whiskey, so
they could act like fingerprints. Indeed, the flavour pro-
file results from the intricate chemical composition, which
also affect the web patterns through the interplay of bulk
chemistry with surfactants and polymers. Similarly, the
webs do not form in droplets below 10% ABV, where
instead the coffee-ring effect is observed [see §VII E]. In
general, this rich variety of surface depositions results
from a combination of intrinsic (chemical composition)
and extrinsic factors (temperature, humidity) that lead
to an interplay of Marangoni flows and macromolecular
surface adsorbtion. The non-uniform residues are often
undesired in many industrial applications including 3D
printing, so whisky experiments could help us with un-
derstanding and controlling uniform coatings [240]. Dry-
ing drop technologies may also be used for wine and hard
drinks quality control [241].
D. Marangoni cocktails
Another well-known demonstration of the Marangoni
effect is the ‘soap boat’ [242,243]. These boats pro-
pel themselves in the kitchen sink by releasing surfactant
molecules from the back: The surface tension of water is
then higher at the front, so the boat is effectively pulled
forwards. The same effect is also used by water-walking
insects as a quick escape mechanism [244]. However, this
propulsion is short lived when using soap as fuel in a
closed geometry, because the interface becomes saturated

15
with surfactants. A prolonged motion can be achieved by
using other commonly available fuels [245]. Moreover,
continuously moving boats can be made with camphor,
a volatile surfactant that evaporates before the interface
can saturate, allowing for persistent propulsion [246].
Recently, this technology was extended to create
alcohol-powered ‘cocktail boats’ that move around in
your glass [101,247]. We depict them in Fig. 9. A typical
commercial spirit can provide a surface tension difference
up to ∆γ∼50mN/m compared to pure water, but sugar
and other cocktail ingredients tend to reduce this value
somewhat. By collaborating with chefs, various materi-
als were tested to make the boats edible. Gelatin boats
were found to be capable of sustained motion and suit-
able for a wide range of flavourings, but they are suscep-
tible to dissolving and sticking to the glass walls. Wax
boats performed the best, with speeds up to 11cm/s and
travel times up to 2 minutes, but unfortunately they are
not well digestible [248]. It would be interesting if future
research could improve or discover new edible materials.
Marangoni propulsion does not only lend itself to appe-
tizing divertissements. The same mechanism can be used
to create microscopic swimming droplets [249], which can
be used as cargo carriers that move deep inside complex
flow networks [250]. Recently, Dietrich et al. [251] de-
veloped very fast Marangoni surfers that can swim over
ten thousand body lengths per second, and Timm et al.
[252] developed Marangoni surfers that can be remotely
controlled. More generally, similar phoretic effects [253],
where interfacial flows are driven by gradients in con-
centration, electric fields, temperature etc., can be ex-
ploited to make a broad range of self-propelled colloids
that are of extraordinary interest to understand collec-
tive dynamics and emergent phenomena out of equilib-
rium [254–260]. The same mechanisms are also at play
in active emulsions [261], the transport of molecules in
biological systems [262,263], and the fragmentation of
binary mixtures into many tiny droplets, a process called
Marangoni bursting [264].
E. Bubbly drinks
Go ahead and pour yourself some nice sparkling wine
into a glass, and observe the beautiful sight of rising bub-
bles and their effervescence [Fig. 10(a)]. Champagne and
sparkling wines are supersaturated with dissolved CO2
gas, which, along with ethanol, is a product of the wine
fermentation process [265]. When the bottle is uncorked,
there is a continuous release of this dissolved CO2gas in
the form of bubbles. Hence, this physicochemical system
provides a great opportunity to study several fundamen-
tal fluid mechanics phenomena involving bubbles: their
nucleation, rise, and bursting dynamics, which in turn
affect the taste of carbonated drinks [265–271].
Before savoring the wine or bubbles, you need to first
open the cork of the bottle. We are all familiar with the
curious “pop” sound, and of course the dangers of uncon-
FIG. 10. Laser tomography of champagne glasses. (a)
Natural, random effervescence in an untreated glass. Inset:
Growth of bubbles as they rise. (b) Stabilized eddies in a
surface-treated glass. (a,b) Courtesy of G´erard Liger-Belair.
(c) Counter-rotating convection cells self-organise at the air-
champagne interface. From Beaumont et al. [275].
trolled corks flying out! This uncorking process is also
accompanied by the formation of a small fog cloud just
above the bottle opening. It has been shown recently
that uncorking champagne creates supersonic CO2freez-
ing jets [272]. What is the underlying physical principle
behind these interesting phenomena? It turns out that
there is a sudden gas expansion when the bottle is un-
corked (pressure drop from about 5 atm to 1 atm). This
leads to a sudden drop in the temperature (about 90 °C),
resulting in condensation of water vapor in the form of a
fog cloud.
The sudden temperature drop also leads to a drop
in the CO2partial pressure above the champagne sur-
face. Hence, the dissolved CO2in the champagne is no
longer in equilibrium with its partial pressure in the va-
por phase. In fact, just after the uncorking, it turns out
that the champagne is supersaturated with CO2. As de-
scribed by Lubetkin and Blackwell [273], this is quantified
by the supersaturation ratio,
S= (cL/c0)−1,(24)
where cLis the CO2concentration in bulk liquid and c0
is the equilibrium CO2concentration corresponding to
partial pressure of CO2of 1 atm. Just after uncorking,
cL/c0≈5, so S≈4, and the champagne must degas
in order to achieve stable thermodynamic equilibrium.
The gas loss occurs through two mechanisms, by diffu-
sion through the liquid surface (invisible to us), and by
the vigorous bubbling (effervescence) that we can readily
observe and also hear [274] [see §VIII D].
How do these bubbles form in the first place? The bub-
bles do not just pop out of nothing, the CO2-dissolved gas
molecules need to cluster and push their way to overcome
attraction forces that hold together the liquid molecules.
Hence, the bubble formation process is controlled by a
nucleation energy barrier [276]. As described by Jones

16
et al. [277], the critical radius of curvature r∗that is nec-
essary for gas pockets to overcome this barrier is
r∗≈2γ/(patm S),(25)
where the surface tension of champagne is γ≈50 mN/m,
the atmospheric pressure is patm ≈105Pa, and S≈4 at
the time of uncorking. Using these values, we find that
this critical radius r∗for bubble formation is very small
(about 0.25 µm).
The pleasing effervescence (bubbling) that we observe
in champagne can arise from either natural or artificial
sources [265,278], as shown in Fig. 10(a,b). Natural effer-
vescence refers to bubbling from a glass which has not re-
ceived any specific surface treatment. When champagne
is poured into a glass, a majority of bubble nucleation
sites are found on small (100 µm long) hollow, cylindrical
fibre structures which contain trapped gas cavities (lu-
men). Another source of natural effervescence are gas
pockets trapped in tartrate crystals precipitated on the
glass wall. Hence, natural effervescence can vary signifi-
cantly depending on how the glasses are cleaned, dried,
and stored. On the other hand, artificial effervescence
refers to bubbling from a glass surface where precise im-
perfections have been engraved by the glass manufac-
turer. The typical imperfections introduced on the glass
are micro-scale scratches to produce a specific pattern,
which give rise to bubbling phenomena that are markedly
different from natural effervescence [265].
The bubble release mechanism from a fibre’s lumen
has been well studied [265]. After a champagne bottle is
uncorked, the supersaturation of CO2implies that these
CO2molecules will escape to the vapor phase using ev-
ery available gas/liquid interface. The trapped tiny air
pockets on fibre lumens offer gas/liquid interfaces to the
dissolved CO2molecules enabling them to cross the in-
terface to gas pockets. The CO2gas pockets grow in size,
and when it reaches the fibre tip, it is ejected as a bub-
ble. However, a portion of the gas packet is left trapped
behind in the lumen, and the bubble ejection cycle con-
tinues until the dissolved CO2supply is depleted.
After the bubbles form in the trapped gas pockets on
the glass, they rise towards the liquid surface due to their
buoyancy and also grow in size since they absorb the
dissolved CO2molecules. The repetitive production of
bubbles from the nucleation sites has been captured in a
model by Liger-Belair et al. [279], and it has been found
that the bubble radius Rincreases linearly with time t
as:
R(t) = R0+kt, (26)
where R0is the initial bubble radius, k=dR/dt is the
growth rate. Bubble rise experiments conducted with
champagne and sparkling wines revealed kvalues around
400 µm/s and experiments in beer revealed growth rates
of around 150 µm/s, indicating that the physicochemical
properties of the liquids influenced the bubble growth
rate [279].
According to wine tasters, the smaller the bubbles, the
better the sparkling wine. Hence, plenty of attention has
been focused on modeling the average size of the rising
bubbles [Fig. 10(a), inset], which is a resultant of their
growth rate and velocity of ascent. As discussed in detail
by Liger-Belair et al. [280], the average bubble radius is
R≈2.7×10−3T5/91
ρg 2/9cL−kHpatm
patm 1/3
h1/3,
(27)
where Tis the liquid temperature, ρis the liquid density,
gis gravity, kHis Henry’s law constant and his the
distance travelled by the bubble from the nucleation site.
It is interesting to note that so many factors can influence
the average bubble size, which is typically in the sub-
millimeter length-scale in bubbly drinks. The bubble size
in beer is significantly smaller than in champagne, and
the reason for this is that the amount of dissolved CO2
in champagne is about two times higher.
In addition to the visual beauty and fascination, the
bubbles actually play an important role in the drink – the
bubbles have been shown to generate large-scale time-
varying convection currents and eddies inside the glass
[265,275,281–283], often with surprising self-organised
flow patterns [Fig. 10(b,c)]. Since they cause a continu-
ous mixing of the liquid, bubbles are thought to play a
key role in the flavor and aromatic gas release from the
wine-air interface. These release rates are dependent on
the fluid velocity field close to the surface, which is in turn
significantly influenced by the ascending bubbles. As the
bubbles collapse at the air interface, they radiate a multi-
tude of tiny droplets into aerosols [284], which evaporate
and release a distinct olfactory fingerprint [285].
In future, we can look forward to several innovations
in bubbly drinks, where numerous factors must be taken
into consideration – different types of glass shapes, nat-
ural versus artificial effervescence, engraving conditions,
kinetics of flavor and CO2release under various condi-
tions, and sensory analysis.
F. Foams
Bubbles (mentioned in the preceding section §III E)
can burst when they reach the surface, but there is a finite
lifetime associated with this process [265]. Thus, when
the bubble production rate is very fast, greatly exceed-
ing the surface bursting time-scales – then the bubbles
start accumulating on the surface to create layers of bub-
bles called “foams” [Fig. 11] [286–289]. In many beers,
these foams last long since they tend to be stabilized by
proteins – they add to the visual appeal and provide a
creamy texture enhancing the mouthfeel [Fig. 11a]. How-
ever, in champagne, the foam is more fragile and less
stable due to the lack of proteins. Foams are formed
in many other fizzy drinks and also in specially prepared
coffees such as cappuccino. In cappuccino, the foam layer
lasts for a long time since it is stabilized by milk pro-

17
teins. These observations naturally bring up questions
on the mechanisms behind the formation, stability, age
and drainage of foams – we will discuss these aspects
below.
A foam is essentially a dispersion of gas in liquid, and
gas bubbles tightly occupy most of the volume. The liq-
uid phase in the form of films and junctions is continu-
ous unlike the gas phase. Foams are also characterized
by the presence of surface-active molecules called surfac-
tants, which stabilise the bubbles at the interfaces of gas
and liquid [290] (the same type of molecule can also sta-
bilise an oil/water interface in an emulsion [see §IV D]).
Since foams consist of significant quantities of gas, be-
ing hence less dense than the liquid it contains, which
is why a foam floats on the surface of the liquid. An-
other interesting property of foam is the large surface
area per unit volume, since the foam contains a large
number of interfaces. Hence, foams enhance the pos-
sibilities for molecular transfer and find applications in
foods for flavor enhancement (e.g. chocolate or spices)
and also reduce the need for high sugar or salt content
[286]. Foams also have special mechanical properties –
they exhibit both solid-like and liquid-like behavior [291].
If the deformation is not too high, foams can show weak
visco-elastic solid properties and can return to its original
shape. However, if the foam is subjected to a high defor-
mation, it can behave like a visco-plastic solid that can
be sculpted. Foams can flow like liquids and seep through
pores and cavities, so they can be poured into containers
and tubes of various shapes. The foam’s viscous resis-
tance increases less quickly with flow rate compared to a
normal fluid enabling it to reduce frictional losses, hence
foams behave as a ‘yield stress fluid’ [see §IV A] with in-
termittent flow via avalanche-like topological bubble re-
arrangements [292]. Hence, given their unique properties,
in addition to the food industry, foams find applications
in many other areas of science and technology – e.g. cos-
metics [293], cleaning, reducing pollution, surface treat-
ment, fire-fighting, army, and building materials [286].
We will now examine the physical properties that al-
low a foam to exist in equilibrium. There are four rele-
vant length-scales to consider: (i) the meter scale, where
the foam appears to be a soft and opaque solid, (ii) the
millimeter scale, where individual bubbles can be dis-
tinguished in the foam, (iii) the micron scale, which re-
veals liquid distribution between bubbles, and (iv) the
nanometer scale, where molecules (e.g. soap molecules)
at the interfaces (air/water) are relevant. The physics
of foams is hence a very broad subject covering so many
length-scales [286], and here we will only touch upon a
few aspects.
At the scale of the gas/liquid interface, the surface ten-
sion [§II F] and the Young-Laplace law, Eq. (7), deter-
mine the shape of the interface. An interface is flat if
geometric constraints allow it, while the surface of an
interface that is completely surrounded by some fluid be-
comes spherical. The pressure being higher on the con-
cave side, tries to curve the shape while surface tension
FIG. 11. Examples of foams in the kitchen: (a) Beer, (b)Dish
washing, (c) Egg beating, and (d) Chocolate mousse.
tries to flatten the surface.
Foams are prepared using additives that chemically
consist of a polar head and a tail with a long carbon
chain. The head is hydrophilic and the tail is hydropho-
bic [see §II G]. The combination of these properties re-
sults in an amphiphilic molecule (water-loving and fat-
loving). Such a molecule, when dissolved in water, tends
to absorb at the air/water interface. This forms a mono-
layer, which greatly affects the interfacial surface tension
properties. Hence, these molecules are called surface-
active molecules or surfactants. In our everyday expe-
rience, there are several examples where we find many
small bubbles that burst quickly (few seconds), e.g. in
sparkling wine or champagne. Here, the small volume of
gas in the bubble encloses a thin film which is unstable
due to van der Waals forces, and hence breaks. However,
the presence of surfactants such as a dishwashing liquid
carry a small charge giving rise to an electrostatic re-
pulsion, which cancels out the van der Waals forces and
stabilises the thin film – helping the foam last for a longer
period [Fig. 11].
A bubble is essentially a small volume of gas enclosed
by a film of water. The bubble assumes the smallest
possible surface area to contain the gas, which typically
results in a spherical shape for an isolated bubble. The
pressure inside most foam bubbles is only slightly greater
than the atmospheric pressure and is not sufficient to
compress the gas appreciably, so the volume of gas can be
considered to be fixed. If the bubble has a large size, then
the interface becomes deformed and is no longer spher-
ical under external forces such as gravity [294]. When
two bubbles come in contact, they share an interface
and hence change shape to reduce the total interfacial

18
area, and can no longer remain spherical. In general,
when a group of bubbles come together to form a foam,
the conservation of volumes and minimization of area
leads to some simple laws for bubble shape, also known
as Plateau’s laws, first formulated by Plateau [102], and
rigorously proved by Almgren and Taylor [295].
Plateau’s laws are based on an ‘ideal foam’ model,
which makes the following assumptions: (i) The foam
is very dry: i.e. the liquid volume is assumed to be
negligible compared to the total foam volume, (ii) The
foam is at mechanical equilibrium: it is at rest since all
forces within foam are balanced, in a local energy mini-
mum, (iii) The foam energy is proportional to the surface
area of its bubbles, and (iv) The foam is incompressible.
The following are the three Plateau’s laws: (I) Equilib-
rium of faces: The films are smooth, and mean curvature
is determined by the Young-Laplace law [Eq. (7)]. (II)
Equilibrium of edges: Along the edges, the films always
meet in threes, forming angles of 120°= arccos(−1/2).
(III) Equilibrium of vertices: At the vertices, the edges
meet four-fold, forming angles of 109.5°= arccos(−1/3).
These laws are the necessary and sufficient conditions
to maintain the mechanical equilibrium of an ideal foam,
and there are also two-dimensional versions of these laws.
So far, our culinary experience with bubbly drinks have
led us to discuss foams mainly in the context of air bub-
bles in liquid. However, given their lightness and tex-
ture, foams are actually a pleasure to eat! The Choco-
late mousse for desert is one of the most popular exam-
ples of edible foams [Fig. 11d]. These edible foams such
as mousses and breads are typically solids [286,296–298]
with complex mechanical properties [299]. They are often
prepared by solidification of a liquid foam by refrigera-
tion or cooking. Since these foams are solidified before
their collapse, they do not need a stabilizing agent. An-
other interesting point is that air is an important raw
material in these edible foams: air contributes greatly
to increasing the volume of the product, but it practi-
cally doesn’t cost anything. On a final note, many other
popular deserts are edible foams, these include favorites
such as ice cream, meringue, marshmallows, many types
of cakes, baked Alaska, etc.
We have mainly discussed foams which are stable for
a finite duration of time. There are several interesting
examples of dynamic and unstable foams, we show two
popular examples in Fig. 12. We know in general that
bubbles rise due their buoyancy in a liquid, but in Guin-
ness beer there is a collective downward movement of
bubbles, creating a ‘cascade of bubble textures’. It has
been demonstrated that this bubble texture cascade mo-
tion [Fig. 12(a)] arises due to a roll-wave instability of
gravity currents [300], a phenomenon that is analogous
to the roll-wave instability in liquid films that case water
films to slide downhill on rainy days [§IX E]. Further-
more, it has been theoretically shown that these bubble
cascades can occur in systems other than the Guinness
beer [301].
Another interesting phenomenon is ‘beer tapping’ – a
FIG. 12. Examples of dynamic and unstable foaming phe-
nomena: (a) Unique foamy textures in stout with nitrogen
bubbles. From Watamura et al. [301]. (b) Foam overflow
‘volcano’ due to tapping on a beer bottle. Public domain
image.
beer bottle foams up resulting in an overflow when it is
tapped from the top [Fig. 12(b)]. The fascinating fluid
physics underlying this phenomenon was explained re-
cently [302]. It turns out that when the beer bottle is
first hit at the top, a compression wave travels through
the bottle. This wave gets rebounded through the liquid
as an expansion wave. At the base of the bottle, the com-
pression and expansion waves interact to cause ‘mother’
bubbles to break up. This is a rapid process resulting
in the formation of smaller ‘daughter’ bubbles, which ex-
pand rapidly to create foam that starts to overflow [302].
G. Ouzo effect
Ouzo, raki, arak, pastis, and sambuca are popular
aperitifs in Southern Europe. They are known for their
anise aroma and the remarkable change in turbidity:
Clear when pure, they turn milky-white when clear water
or ice is added, which has been termed the ‘ouzo effect’
[303]. The key to this puzzle lies in the chemical com-
position of the drink, being mostly a mixture of water,
alcohol, and essential oils, of which anethole is a promi-
nent part. Anethole (also known as anise camphor) is
highly soluble in ethanol but not in water [304], thus an
undiluted spirit has a completely clear appearance. Upon
the addition of small amounts of water, however, the oils
start separating and create an emulsion of fine droplets
which act as light scattering centres, resulting in the final
cloudiness.
The process is also called louching or the louche ef-
fect, and can be regarded as spontaneous emulsification.
Such emulsions are highly stable and require little mixing
[305]. In these multi-component mixtures, the thermody-
namic stability of the emulsion comes from the trapping

19
between the binodal and spinodal curves in the phase
diagram. The ouzo effect has been widely studied to elu-
cidate its mechanisms [303]. However, the microscopic
dynamics are still under active investigation. Small-angle
neutron scattering studies in Pastis [306] and Limon-
cello [307] measured the size of the demixing oil droplets
to be of the order of a micron, a bit larger than the
wavelengths of visible light, giving rise to Mie scattering
[§II D]. Sitnikova et al. [305] established the mechanism
for oil droplets growth to be Ostwald ripening without
coalescence and observed the ripening rate to be lower at
higher ethanol concentration, with stable droplets reach-
ing an average diameter of 3 microns. Lu et al. [308]
tried to disentangle the effects of concentration gradi-
ents from the extrinsic mixing dynamics by following
the nanodroplet formation in a confined planar geome-
try and observed universal branch structures of the nu-
cleating droplets under the external diffusive field, analo-
gous to the ramification of stream networks in large scale
[309,310], and the enhanced local mobility of colloids
driven by the emerging concentration gradient. The ouzo
effect can be triggered not only by the addition of water
but also by the evaporation of ethanol, e.g. in sessile
ouzo droplets [311–313], leading to an astoundingly rich
drying dynamics involving multiple phase transitions.
The remarkable stability of the spontaneously formed
emulsion gives hope for potential generation of
surfactant-free microemulsions without resorting to me-
chanical stabilisation, for example high-shear stabilisa-
tion that is often used in fat-filled milk formulations
[314]. Thus, the ouzo effect has been used for the cre-
ation of a variety of pseudolatexes, silicone emulsions,
and biodegradable polymeric capsules of nanometric size
[315]. Nanoprecipitation can also be used for drug deliv-
ery and the design of nanocarriers [316]. Particles cre-
ated using the ouzo effect are kinetically stabilised, and
provide an alternative to thermodynamically stabilised
micelles formed using surfactants [317].
IV. SOUP STARTER: COMPLEX FLUIDS
Most foods are neither purely liquid nor solid, but
rather something in between: they are often viscoelas-
tic or complex materials. This strongly affects how we
perceive taste since their flow behaviour is directly linked
with mouthfeel, the oral processing and texture of foods
[318,319]. The rheology of complex fluids are also of ex-
treme importance in the food industry, in terms of trans-
port phenomena, production processes, storage, and pro-
cessing techniques that need to be adopted to the prop-
erties of materials at hand [19,320–322]. Complex fluids
are a bit of a soup sandwich, so we begin this section
with an introduction to food rheology. Then we get into
the thick of it, reviewing the science of food suspensions,
emulsions, and the mixing of sauces. Shall we board the
gravy train?
A. Food rheology
Viscosity quantifies internal friction in a fluid, and it is
often the interplay between the viscosity, elasticity and
inertia of a moving fluid that gives rise to complex phe-
nomena seen everywhere. In essence, viscosity relates
shear stresses to gradients in flow velocity. That is, for
a simple incompressible fluid moving steadily along the
x-axis with a velocity field ux(y) varying along y, the
shear stress component σxy is given by Newton’s law of
viscosity,
σxy =µ(∂ux/∂y),(28)
where the constant of proportionality µis the dynamic
viscosity and the second term is the flow gradient. In a
general coordinate system, the rate-of-strain tensor can
be written as E=∇u+ (∇u)T/2 and the scalar shear
rate as ˙γ=√2E:E.
The stres field σarising in the fluid gives rise to mo-
tion, and the momentum balance is expressed by the
Cauchy equation
ρDu
Dt =∇·σ+f,(29)
which is valid for all fluids. However, the stress field in
Eq. (29) above can on the rate-of-strain in various ways,
depending on the fluid microstructure.
For an isotropic linear viscous fluid, the total stress
combines isotropic pressure with shear stresses as in Eq.
(28), via
σ=−pI+ 2µE,(30)
where Iis the identity matrix. Inserting this expression
into the Cauchy momentum equation (29), we recover the
incompressible Navier-Stokes equations [Eq. (2a)]. Spec-
ifying the relationship of between the stress σand the
flow u(or the tensorial quantities based on this field)
defines the particular fluid model. A Newtonian fluid
is defined by the linear relation (30) between the stress
and the shear, when viscosity is independent of the shear
rate. Most simple liquids are indeed Newtonian, includ-
ing water, alcohol, and most thin oils. However, many
fluids deviate from this linear relation because of their
complex internal structure. Ubiquitous examples are
emulsions, suspensions or polymer solutions [323]. Such
non-Newtonian fluids can feature different types of be-
haviours, including:
•Shear-thinning liquids, whose viscosity decreases
with increasing rate of strain (e.g. yoghurt, mus-
tard, ketchup, clotted cream, but also paint);
•Shear-thickening liquids, whose viscosity increases
with strain rate (e.g. cornstarch in water and sol-
ubilised starches in general [324]); see Fig. 13(a).
•Bingham plastics or yield-stress materials, which
behave as solids at low stresses but start flowing

20
at high stresses (e.g. toothpaste or mayonnaise);
in the latter, rigdes and peaks on the surface show
the existence of a critical yield stress above which
it flows [325–327]; see Fig. 13c.
•Rheopectic fluids, that gradually become more vis-
cous with duration of stress (e.g. whipped cream)
or heat (e.g. pancake batter in a frying pan);
•Thixotropic fluids [328], that become less viscous
over time when agitated (e.g. yoghurt, peanut but-
ter, ketchup, or margarine); this quality is desirable
in spreads, which should stay solid but be easy to
spread on toast; see Fig. 13d.
These fluids obey the same Cauchy momentum equation,
with a different stress tensor σspecified by a constitu-
tive equation that captures the relevant material proper-
ties and relates the stress and stress rate to strain and
strain rate. The topic is vast and discussed widely in
classic textbooks, also in the context of applicability and
structure of different models [329,330]. Typical rheologi-
cal experiments, also widely used in food science, involve
periodic variations of stress or strain, with a harmonic ex-
citation at a given frequency ω, in which the response of
the material is recorded. The latter is subdivided into an
in-phase and out-of-phase components. The amplitude
of the in-phase component is the shear storage modu-
lus G0(ω), which provides information about the energy
stored in a deformation cycle in the material, which corre-
sponds to its elastic energy. The out-of-phase amplitude
G00(ω) is termed the shear loss modulus and is propor-
tional to the energy dissipated as heat over a period of
oscillations, quantifying the viscous nature of the mate-
rial [331].
Below we briefly discuss the two main classes of mod-
els, linear and non-linear constitutive relation, together
with the most popular examples of particular materials
or applications.
1. Linear viscoelasticity
When deformations are small, slow, or slowly varying,
it suffices to use the models assuming a linear constitu-
tive law. Within this approximation, stress and strain
are related by a Volterra integral equation. It is purely
phenomenological and aims only to describe the response
of a material. To probe this response, apart from os-
cillatory rheometry measurements, two typical tests can
be run. The creep test consists of measuring a time-
dependent strain upon the application of a steady stress.
The material starts flowing with a certain delay, which
is grasped by this experiment. In linear approximation,
doubling the stress doubles the strain. This is observed
widely e.g. in processed fruit tissues [333], or in dynamic
rheology measurements of honey [334]. A complemen-
tary experiment, the stress relaxation test, measures the
time-dependent stress resulting from a steady strain. In
FIG. 13. Examples of complex rheological behaviour of fluids:
(a) People walking over a swimming pool full of oobleck, a
mixture of cornstarch and water. Image courtesy of Ion Fur-
janic, director of We are KIX [332]. (b) The Weissenberg rod
climbing effect seen on a 2% solution of high molecular weight
polyacrylamide. From Wikimedia Commons, licensed under
CC BY 4.0. (c) Whipped cream is an example of a Bingham
plastic which remains solid in the absence of stresses but can
be squeezed out onto a slice of pie like a fluid. From Wikime-
dia Commons, licensed under CC BY 4.0. (d) Thixotropic
fluids become thinner with time when they are sheared, and
solidify again at rest. Classic examples are paint or sandwich
spread.
the range of small deformations, many food products can
be aptly described by linear models. Rheology measure-
ments of frankfurters of various composition show a good
linear response for strains up to about 3.8% [335]. Stress
relaxation and creep-recovery tests on oat grains also
show linear behaviour for a range of temperatures and
moisture content [336].
Many non-Newtonian fluids are called viscoelastic be-
cause they exhibit both viscous (creep) and elastic (re-
laxation) effects. Linear viscoelastic response can thus be
described by ordinary differential equations, which can
be rationalised by thinking about their mechanical ana-
logues, called ‘spring-dashpot’ models. In such models,
the fluid response is portrayed by a collection of con-
nected Hookean (elastic) springs and (viscous) dashpots,
producing a combined effect together. The springs relate
stress σwith strain linearly, via
σ=E, (31)
with the spring constant Ebeing an analogue of Young’s
modulus. The springs models an instantaneous deforma-
tion of the material and strain energy stored therein. The
creep response is modelled by a Newtonian dashpot, in
which stress induces strain rate,
σ=ηd
dt,(32)

21
where ηis the viscosity. The ratio of viscosity to stiffness,
τ=η/E, will thus measure the response (or relaxation)
time of a viscoelastic material. This time can be com-
pared to a typical observation (or process) time, τ0, to
yield the dimensionless Deborah number,
De ≡time scale of relaxation
time scale of process =τ
τ0
,(33)
which indicates whether at a given observation times the
material would behave in a fluid-like manner (at low De)
or whether it would exhibit non-Newtonian properties,
with an increasingly manifested elasticity at high De. For
example, viscoelastic ice cream [337] [see §VI K] should
preferably be consumed at high De for practical reasons.
Different combinations of springs and dashpots can be
proposed to describe the linear response of a viscoelastic
material. The simplest models are the Maxwell model,
represented by a spring and a dashpot connected in series,
and the Voigt or Kelvin model, in which a spring and a
dashpot are connected in parallel. By considering the
composite dynamic behaviour, we find the constitutive
relation for a Maxwell material as
σ+η
E˙σ=η˙, (34)
while for a Kelvin-Voigt material we obtain
σ=E+η˙. (35)
These simple models are rather limited. The Maxwell
model aptly describes stress relaxation, being the re-
sponse of a material to a constant strain, for example
in moth bean flour dough [338]. However, for constant-
stress conditions it predicts a linear increase of strain,
which is not seen e.g. in polymers where the strain rate
decreases in time; thus it is not a good model for creep
processes [330]. The Kelvin-Voigt model, on the other
hand, is highly successful at describing creep, e.g. in
potato tissues [339], but it is much less accurate in terms
of stress relaxation.
The simplest model that accurately grasps both creep
and relaxation is the standard linear solid (SLS) model,
or Zener model, which involves two springs and a dash-
pot, connected both in series and in parallel. Again, the
constitutive equation takes the general form
σ+ηf1(E1,E2) ˙σ=f2(E1,E2)+ηf3(E1,E2)˙, (36)
with the coefficients dependent on the exact arrangement
of the elements. A fluid version of the SLS model is the
Jeffreys model. Such models predict the general shape of
the strain curve, both at long times and at instantaneous
loads but still it is a simplified representation with no
direct strategy to model actual materials.
Modifications of the above mentioned models are also
available to better account for various phenomena seen
in actual fluids. The Burgers model incorporates vis-
cous flow into the SLS model. The Generalised Maxwell
Model (GMM) (also called the Wiechert model) involves
combinations of Maxwell elements which account for a
distribution of relaxation times, present e.g. in materi-
als composed of molecular segments of different lengths.
GMM has been successfully used to describe the stress
relaxation behaviour for a variety of semi-solid food prod-
ucts, such as agar gel, meat, mozzarella cheese, ripened
cheese, and white pan bread [340]. Systems with finer mi-
crostructure, such as protein-stabilised oil-in-water emul-
sions, can also exhibit linear viscoelastic behaviour [341].
2. Non-linear viscoelasticity
Some biological fluids can be inherently non-
Newtonian. Rheology measurements of yoghurts show
that overlapping polymer molecules cause viscoelastic be-
haviour even at dilute concentrations (φ < 10−2) and a
sharp increase in viscosity with concentration [342]. Yo-
ghurts are shear-thinning in addition to being viscoelas-
tic. Shear thinning or thickening cannot be explained
using linear constitutive equations, thus more complex
models are needed to quantify their behaviour. Non-
Newtonian properties manifest themselves particularly
in the material properties, which become dynamic quan-
tites, and in particular depend on the shear rate ˙γ. The
non-Newtonian (or shear-dependent) viscosity is defined
as in eq. (28) but now with µ=µ( ˙γ). In the same geome-
try as before, we also define the normal stress coefficients
by
σxx −σyy =−Ψ1( ˙γ) ˙γ2,(37)
σyy −σzz =−Ψ2( ˙γ) ˙γ2.(38)
Together, µ, Ψ1, and Ψ2are referred to as viscometric
functions.
The Weissenberg number is another relevant dimen-
sionless function which quantifies the ratio of elastic to
viscous forces. In a simple shear flow, the latter are pro-
portional to the shear stress σxy , while the former are
represented the first normal stress Ψ1. For a fluid with
characteristic relaxation time λunder shear, we write
this ratio as
Wi ≡elastic forces
viscous forces =σxx −σy y
σxy
=λ˙γ. (39)
Although seemingly akin to De, the Weissenberg number
has a different interpretation, because it captures the de-
gree of anisotropy introduced by the deformation, rather
than the effect of time-dependent forcing [343]. The two
numbers span a phase space interpolating between purely
viscous and purely elastic deformations, with both linear
(typically moderate De and low Wi) and nonlinear vis-
coelasticity (at higher Wi) in between.
A first step into the nonlinear territory is the gener-
alised Newtonian fluid model, in which the stress depends
only on the instantaneous flow, but the viscosity in Eq.
(30) is replaced by a shear-dependent function µ( ˙γ). Its
form is usually derived empirically from the available

22
data. Some common approximations include a power law
fluid with µ( ˙γ) = k( ˙γ)n−1, where kand nare fitting pa-
rameters. If n > 1 the fluid is shear-thickening (dila-
tant), and if n < 1 it is shear thinning. Most fruit and
vegetable purees belong to the latter category [344]. An-
other set of examples are Carreau-Yasuda-Cross models
[345–347], which interpolate between the different zero
and infinite shear rate viscosities (µ0and µ∞, respec-
tively) by µ( ˙γ) = µ∞+ (µ0−µ∞)[1 + (λ˙γ)a](n−1)/a, with
fitting parameters λ, a, n. Such models successfully de-
scribe e.g. the flow of skim milk concentrate [348] or
semisolid natillas (Spanish dairy desserts) [349]. An im-
portant category are yield fluids, which flow only above
some critical stress σ > σc. Within those, Bingham mod-
els satisfy µ( ˙γ) = µ0+σc/˙γ[350]. This type of behaviour
is seen commonly e.g. in tomato pastes [351]. The Her-
schel Bulkley models use µ( ˙γ) = k˙σn−1+σc/˙γ[352],
and have proved useful to aptly describe the rheology of
stirred yoghurts [353].
Before we present other popular nonlinear constitu-
tive relations, we introduce a useful mathematical con-
struct appearing therein: the upper-convected (or Ol-
droyd) time derivative. It describes the rate of change
of a tensorial property of a fluid parcel, written in the
coordinate system rotating and stretching with the fluid.
Typically denoted by a triangle above a tensorial quan-
tity A, it is defined by
O
A=DA
Dt −(∇v)T·A−A·(∇v),(40)
where D/Dt is the material derivative.
The simplest nonlinear model that involves the flow
history is the second-order fluid. In a Newtonian fluid,
the stress tensor (30) is linear in the rate-of-strain tensor
E, which in turn is linear in the (typically small) flow
velocity gradients. Here, we look at the stress tensor as
at an expansion in Eand include also second-order terms,
which results in the following stress-strain relationship
σ=−pI+ 2µE−Ψ1
O
E+ 4Ψ2E·E,(41)
where two new coefficients appear, in addition to viscos-
ity. The first and second normal stress coefficients [331],
are here assumed to be constant material properties of
the fluid. They describe the anisotropic normal stress re-
sponse in a viscoelastic fluid. The nonlinearity enters via
the last term in the stress tensor. The second-order fluid
model accurately describes the flow at high strains and at
moderate Deborah numbers, in the region between New-
tonian fluids and more complicated nonlinear viscoelastic
models.
Another idea to account for nonlinearities consists of
modifying the Maxwell model to include nonlinear stress
response. The upper-convected Maxwell (UCM) model
modifies the stress response while keeping the linear de-
pendence on rate-of-strain via
σ+λO
σ= 2µE,(42)
thus introducing the relaxation time λ. The UCM model
predicts some properties of non-Newtonian fluids (such
as non-zero first normal stress coefficient) while missing
others (no second normal stress coefficient or constant
shear viscosity), which is typically a limiting factor in its
applicability.
A widely used extension of the UCM model is the
Oldroyd-B model [354]. The relevant equation of state
involves nonlinearities both in the stress and strain parts,
and takes the form
σ+λ1
O
σ= 2µE+λ2
O
E,(43)
where now λ1is interpreted as the relaxation time, while
λ2governs the retardation time scale. Microscopically,
the Oldroyd-B model can be thought of as a fluid filled
with elastic dumbbells strained by the flow. This model
is popular for its good approximation of viscoelastic be-
haviour in shear flow. Other constitutive equations have
also been proposed to fit the experimental data but the
choice of a particular model depends on the character of
the problem at hand.
The shear-dependent properties of kitchen matter can
be seen e.g. in the context of mixing and whisking. The
Weissenberg effect [355] is an illustrative proxy for vis-
coelasticity. It is seen when a spinning rod is inserted into
an elastic fluid, as in Fig. 13(b). Instead of the meniscus
curving inwards, the solution is attracted towards the rod
and rises up its surface. This is due to normal stresses
in the fluid acting as hoop stresses and pushing the fluid
towards the rod [356]. When whisking egg whites with a
mixer [357], we see the solution rise up close to the mixer
shaft, rather than move outwards in a parabolic shape
characteristic for Newtonian liquids, described in §II J.
The same effect is observed in sweetened condensed milk
[358], gelatin [359] or dough [298].
Importantly, the viscosity may also change due to a
changing chemical composition under external stimuli
such as heat. This complex landscape is particularly im-
portant in the kitchen environment, where we often work
with thickening agents such as roux or Xanthan gum,
gently heat up egg yolks to make Hollandaise sauce, or
milk for the b´echamel. Composite food products typi-
cally respond in a non-Newtonian manner to deforma-
tion. Dynamic quantities characterising the rheology of
food products are of paramount importance for the pro-
cess of food processing, in which appropriate length and
time scales ought to be chosen for the expected result.
Thus there are numerous papers quantifying the rheo-
logical response; extending yield-stress of Nutella [360],
rheology of chewing gum [361], and rheometry of mayon-
naise [362] or salad dressing [363] are but a few examples
of a wide current aiming to describe transient effects in
the flow of food products [364]. The rheological prop-
erties of food are also important for medical conditions
including dysphagia [365], where fluid dynamics can help
with predicting the ease of swallowing [366].
For a more detailed description of food rheology, we

23
FIG. 14. Blending sauces. Left: pure honey. Middle: 50:50
mixture. Right: pure mayonnaise. Top row: Rheometry ex-
periments of the sauces mixtures flowing down an inclined
plane. While honey and mayo move slowly, the mixture runs
down fast. Bottom row: The same dynamics simulated with
a shear-thinning mixing model and displayed using computer-
generated imagery (CGI). Image courtesy of Yonghao Yue.
recommend reading the following books and reviews [19,
320–322], as well as these works on soft matter [20–22,
367–369], and references therein.
B. Mixing up a sauce
When making a sauce, rather counter-intuitive effects
can emerge: the combination of two thin liquids can sud-
denly lead to a thick mixture, or vice versa. Indeed, as
discussed in §III C, we saw that an ethanol-water blend
has a higher viscosity than both pure liquids. In general,
the viscosity µ12 of most binary mixtures is not a linear
function of their relative composition [370]. Instead, a
first approximation is given by the Arrhenius equation,
ln µ12 =x1ln µ1+x2ln µ2(44)
where xiand µiare the mole fraction and viscosity of the
ith component, respectively. This expression holds for
an ideal binary mixture, where the volume of the com-
ponents is conserved, i.e. the excess volume of mixing
is zero. Building on this work, a more accurate descrip-
tion was given Grunberg and Nissan [371] and Oswal and
Desai [372], which reads
ln µ12 =x1ln µ1+x2ln µ2+x1x2
+K1x1x2(x1−x2) + K2x1x2(x1−x2)2,(45)
where , K1, K2are empirical parameters that account for
molecular interactions. While there is no universal theory
that accurately predicts the viscosity of a liquid blend,
more extended models have been derived that are im-
portant for many industrial applications including food
science [373,374].
In terms of mixing sauces, most ingredients each have
different elasto-viscoplastic properties. To describe the
material properties of food mixtures, Nagasawa et al.
[375] considered a wide range of theoretical models and
derived a viscosity blending model for shear thinning flu-
ids. Using rheometry experiments, they also tested these
models for various sauces including honey, mustard, may-
onnaise, ketchup, hot chilli sauce, condensed milk, choco-
late syrup, sweet bean sauce, oyster sauce, Japanese pork
cutlet sauce, and BBQ sauce. When mixed together, un-
expected behaviours can arise: The top row of Fig. 14
shows experimentally that pure honey flows down an in-
clined slope slowly because of its high viscosity (left) and
pure mayonnaise remains stagnant because of its yield
stress (right) [327]. However, a 50:50 mixture runs down
the slope quickly, with a much lower viscosity µ12 than
its constituents (middle). In the bottom row, the authors
reproduced these surprising dynamics using numerical
simulations combined with high-end computer-generated
imagery (CGI) techniques.
C. Suspensions
Drinks and foods often take the form of a particle sus-
pension [376], with examples ranging from unfiltered cof-
fee and wine to Turkish pepper paste and Kimchi. The
texture and mouth-feel of suspensions depend on their
rheology [see §IV A], which is sensitive to the particle
size, microstructure and concentration. Dilute suspen-
sions such as coffee and juices, are typically Newtonian
fluids, while concentrated suspensions such as pastes and
purees typically display non-Newtonian behavior due to
both long-ranged hydrodynamic interactions between the
particles [see §VI B] and various short-ranged interactions
including friction [377,378]. Indeed, these rheological
properties have been studied to optimise food paste 3D
printing [379].
The influence of internal structure on macroscopic
properties of suspensions has been actively investigated
since the birth of statistical physics. Einstein [380] estab-
lished that the viscosity of a dilute suspension increases
by adding solute according to the Einstein viscosity,
µ=µ01 + 5
2φ,(46)
where µ0is the (dynamic) viscosity of the solvent, and φ
the volumetric concentration of particles. This relation-
ship which was later developed in the context of other
transport coefficients, including the diffusion and sedi-
mentation coefficients of suspended particles, and to ac-
count for higher volume fractions and different interac-
tions between the particles [381]. This is reviewed in the
context of food suspensions by Genovese et al. [382] and
Moelants et al. [376].
When we grind coffee beans or otherwise create a sus-
pension, the particles are not all of the same size, but
instead they follow a size distribution. The width of this

24
size distribution is called the dispersity, and it can be
tuned to control the rheology of a suspension. Notably,
the Farris effect [383] explains how the viscosity of a sus-
pension decreases when the dispersity increases; that is, a
broader distribution of particle sizes yields a lower viscos-
ity as compared to a narrow distribution of particle sizes.
In food science, the Farris effect has been exploited to
adjust the rheological properties of edible microgel sus-
pensions such as cheese [384], and it has been used to
minimise the apparent viscosity of cooked cassava pastes
[385]. Conversely, by narrowing the particle dispersity in
coffee and unfiltered wine, it should be possible to en-
hance the mouthfeel by the opposite mechanism, but we
are not aware of any reports on this topic. The Kaye
effect is a phenomenon that occurs when a complex fluid
is poured onto a flat surface, where a jet suddenly spouts
upwards [386]. Many non-Newtonian liquids feature this
effect, including shampoo, and recent experiments have
explained it by using high-speed microscopy to show that
the jet slips on a thin air layer [387–389].
D. Emulsions
An emulsion consists of two (or more) immiscible liq-
uids [390], where droplets of one phase (called the dis-
continuous or internal phase) are dispersed inside the
other fluid (referred to as the continuous or external
phase). The most typical pair is water and oil, where the
two main classes are water-in-oil (W/O) systems such as
margarine, and oil-in-water (O/W) systems such as milk
[321].
Indeed, emulsions are ubiquitous in gastronomy and
food science [391,392], with everyday products such
as cream, yoghurt, mayonnaise, salad dressing, sauces,
spreads, ice cream, dips and desserts [390]. In these ex-
amples, precise control over composition, functionality
and stability are key requirements for a successful dish,
where manipulation techniques, as well as timing, have
to be adjusted to achieve the desired effect, thus forming
a vast part of modern food science. The physicochemical
stability of many emulsions is also affected by their nat-
ural components, which often are low-processed or raw,
thus heterogeneous and varying in composition.
The process of converting two separate fluids into an
emulsion is called homogenization [393] and is indus-
trially realized with high-energy mechanical methods,
such as blenders or ultrasonics [394], where strong shear
forces break up the dispersed phase into droplets. In
the kitchen, we make use of the same capillary break-
up mechanism when we vividly shake or stir the oil and
water phase, but, as anyone who have tried to make a
Hollandaise sauce would painfully know, such mixtures
are by design thermodynamically unstable and prone to
phase separation, sometimes called a ‘broken sauce’.
In the food industry, phase separation is a major hurdle
as it can severely degrade the food product and shorten
the shelf life, but fortunately stabilizers such as emulsi-
fiers, texture modifiers, ripening inhibitors and weighting
agents can be added to keep the system in a metastable
state (by creating a free energy barrier), efficiently ex-
tending the lifetime to hours, days, months, or even years
[395]. Pickering emulsions [396,397] are stabilised with
solid nanoparticles that sit at the drop interface, with
promising applications in drug delivery and structured
nanomaterials [398,399], and adsorbed protein molecules
stabilise emulsions by lowering the interfacial tension
and by forming viscoelastic networks that act as bar-
riers against coalescence [400,401]. The stabilizing effect
of proteins can be readily observed by adding a small
amount of mustard to vinaigrette dressing. In addition
to be stabilizing, these surface proteins control the com-
plex rheology of emulsions, which is responsible for the
appearance and our sensory perception of food products
[322]. Salad dressing is a widely studied kitchen emul-
sion, which has been examined in the context of its com-
plex rheological response [402] and processing [403], sta-
bility, and linear viscoelasticity [404].
Due to the enormous surface area of emulsion drops,
the overall rheology and stability is controlled by interfa-
cial properties, and in particular, by the surface coverage
and structure of adsorbed protein layers [322]. Animal
proteins such as whey and casein readily form viscoelas-
tic networks with high surface coverage, leading to excel-
lent emulsion stability, while plant-based proteins such as
those from cereals and pulses are less efficient stabilizers,
and this is mainly due to their poor solubility in the aque-
ous phase [405]. While heating can be used to increase
the stabilizing abilities by denaturing the proteins [406],
such treatment can degrade the taste as well as the tex-
ture and the nutritional value of plant based foods. As
such, the ability to control the interfacial properties of
plant-based emulsions without denaturing the protein is
an important goal, and an exciting new direction in food
science for vegetarians and vegans [407].
Bulk-based methods such as Dynamic light scattering
(DLS) have traditionally been used to characterise food
emulsions [395], but they lack optical resolution needed
to resolve details of adsorbed particles, proteins and sur-
factants near individual emulsion drops [408]. Droplet-
based microfluidics is a viable alternative to such meth-
ods, and is characterized by excellent optical resolution,
monodisperse droplet distributions [109], and control-
lable surface chemistry [408]. In recent years, microflu-
idics has contributed to new fundamental understand-
ing of surfactant and protein transport in microfluidic
food emulsions, with a handful of important publications
[27]. Notably, Brosseau et al. [409] extracted the ad-
soption kinetics via so-called microfluidic dynamic inter-
facial tensiometry (µDIT) by measuring the interfacial
deformation as a droplet is passed through a series of ex-
pansions. Another approach was adopted by Muijlwijk
et al. [410], who measured the dynamic interfacial ten-
sion (due to surfactant adsoption) during droplet produc-
tion in a T-junction. Using the same method, they also
showed that convection controls the transport of surfac-

25
tants [411] and proteins [412] during droplet production,
resulting in adsorption-limited kinetics, and for proteins,
they found that the first interfacial layers form within sec-
onds [413]. Finally, they mixed plant-based protein with
animal protein and studied their competitive adsorption
to suspended emulsion drops, with potential implications
in future hybrid protein products [414].
E. Cheerios effect: capillary floating
Before the mixture about to be cooked is stirred well,
we frequently add ingredients by sprinkling or tossing
them on a liquid surface. Sometimes they are deliber-
ately added to form the top layer, as some of us do with
corn flakes. Interestingly, small objects that are more
dense than water may still float at the air-water inter-
face because of surface tension [415]. Moreover, floating
objects tend to aggregate at the surface, brought together
by capillary forces induced by the presence of a curved
meniscus around floating objects. Aptly named the ‘chee-
rios effect’ [416], this is seen not only with corn flakes,
but also e.g. bread crumbles [417], foams, and gener-
ally object that are large enough to create the menisci
of considerable size. The mechanism of lateral capillary
interaction due to interfacial deformation admits a uni-
versal theoretical description for particle sizes ranging
from 10−9to 10−2cm [418]. Initially, the interaction
of widely spaced particles may be regarded as a two-
body problem [419] but eventually multiparticle rafts are
formed [420]. The dynamics of these aggregates is more
complex, since they may undergo internal redistribution
and destabilisation [421]. An interesting example is an
active assembly of dozens of fire ants on a water surface
[422]. The presence of surface tension allows to sustain
deformed surfaces which can support a load of an insect
walking on water [244,423,424], or a biomimetic water-
walking device [425]. Similar behaviour, termed the in-
verted cheerios effect, is seen when water droplets sit on a
soft, deformable substrate, and the induced deformation
drives their assembly [426,427].
By a combination of capillary forces and externally
controlled fields, e.g. electromagnetic field, both static
and dynamic assembly can be achieved in capillary disks
[428,429]. Capillary forces between spherical particles
floating at a liquid-liquid interface have also been quanti-
fied to show a qualitatively similar behaviour [430]. Same
guiding principles are used in micro-scale for colloidal
self-assembly, driven not by gravity but by an anisotrop-
ically curved interface [431]. Finally, in active microrhe-
ology [432–436], an external force field (usually magnetic
or optical) is used to distort surface active or bulk probes
in order to extract viscoelastic responses of complex ma-
terials, with direct applications in food science [437].
A separate class of interfacial interaction involves the
dynamic problem of stone skipping, known in Britain as
‘ducks and drakes’, where the interfacial properties deter-
mine the optimal angle of attack for the most successful
rebound and therefore maximal range [438–440]. More-
over, elastic ‘stones’ have been shown to demonstrate
superior skipping ability by assuming hydrodynamically
optimal shapes during the collision [441].
V. HOT MAIN COURSE: THERMAL EFFECTS
Heat not a furnace for your foe so hot, that
it do singe yourself. – William Shakespeare
The work ‘cooking’ refers to the preparation of food in
general, but specifically the operations involving temper-
ature and heat such as boiling, frying, baking and poach-
ing, to transform food products into a final dish. Thus,
many kitchen flows involve thermal effects which drive
them or alter the behaviour of substances involved. Be-
low we highlight a number of such phenomena.
A. Feel the heat: energy transfer
Heat is transported in fluids in a way similar to mo-
mentum [§II B]. Fluid parcels are advected with the flow,
and additionally exchange heat by conduction. Local
variations in temperature can additionally induce density
gradients, which can drive macroscale convective motion.
Moreover, dissipation of momentum by viscosity acts as
a local source of heat, but in most practical situations its
contribution is negligible.
The relevant quantity characterising the thermal prop-
erties of the fluid is the scalar temperature field, T(r).
The spread of temperature is described by the heat equa-
tion, which for a fluid of density ρand heat capacity at
constant pressure cpcan be written as
ρcp
DT
Dt =k∇2T+h+Q, (47)
where kis the thermal conductivity, governing the dif-
fusive spread of temperature by thermal conduction, h
is a source term accounting for local heating (e.g. by
chemical or nuclear reactions), and Qis the viscous dis-
sipation term, which for an incompressible fluid takes
the form Q= 2µE:E, with Ebeing the symmetric
velocity gradient tensor [see Eq. (30)]. The dissipation,
which scales as µU0/(L0ρcp), is typically small compared
to other terms, and thus is often neglected. In the ab-
sence of local heat sources, the heat equation becomes
simply a Fourier’s diffusion equation, with the thermal
diffusivity DT=k/(ρcp). The dominant (heat) trans-
port mechanism is determined by the (thermal) P´eclet
number,
Pe(T)≡Diffusion time
Convection time =L0U0
D(T)
,(48)
which besides thermal diffusion can equally be used to
characterise molecular diffusive transport.

26
FIG. 15. Leidenfrost effect. (a) Diagram of a droplet levitat-
ing on a cushion of evaporated vapour above a heated surface.
(b) The Leidenfrost effect prevents meat from sticking to a
hot plate. Artwork entitled ‘Bacon Prelude’ by Pedro Moura
Pinheiro, licensed under CC BY-NC-SA 2.0. (c) Star-shaped
oscillations of Leidenfrost drops [444] that make characteristic
sounds. From Singla and Rivera [445]. (d) Time-lapse im-
age of a self-propelled Leidenfrost drop on a reflective wafer
heated at 300 °C. From Bouillant et al. [446].
The concept of diffusion of heat suffices to explain sev-
eral kitchen processes. Baking a cake requires the heat
to reach the inner parts of the dough but changing either
the dimensions of the cake or the amount of batter used
alters the baking time in a way that can indeed be pre-
dicted from the diffusion equation [442]. Similarly, the
problem of perfectly boiling an egg can be quantified in
terms of the energy equation [443] to aid many breakfast
table discussions.
B. Levitating drops: Leidenfrost effect
When grilling steaks, one way to assess whether the
frying pan is hot enough is to sprinkle a handful of water
droplets onto it. When the surface temperature slightly
exceeds the water boiling point, the droplets become vig-
orously evaporating, producing a sizzling sound. How-
ever, if the pan is left on full heat for a while and be-
comes considerably hotter, small droplets change their
behaviour completely and start levitating above the hot
surface without boiling [Fig. 15a,b]. This levitation can
help with preventing the meat from sticking [447].
This phenomenon was first known to be observed by
a Dutch scientist H. Boerhaave in 1732, and later de-
scribed in detail by a German doctor Johan Gottlob Lei-
denfrost in 1756. He provided a record of water poured
onto a heated spoon that ‘does not adhere to the spoon,
as water is accustomed to do, when touching colder iron’
[448]. The Leidenfrost effect, as it was later termed, has
been studied extensively in the scientific context [449–
451], and even became a plot device in Jules Verne’s novel
Michel Strogoff in 1876.
The explanation of this effect boils down to the analy-
sis of heat transfer rate between a hot plate and a droplet.
For intermediate excess temperatures above the boiling
temperature (between 1◦C and ca. 100◦C) the droplets
undergo either nucleate boiling, with vapour bubbles
forming inside, or transition boiling, when they sizzle
explosively upon impact on the plate. However, above
the Leidenfrost temperature, which for water on a metal
plate is approximately 150-180◦C, the heat transfer dy-
namics change when a thin vapour layer is created be-
tween the droplet and the plate. This thin cushion both
insulates the droplet and prevents it from touching the
substrate which would cause nucleation boiling inside the
droplet [Fig. 15a]. Due to the competition between evap-
oration and film draining, the typical thickness of the in-
sulating layer is about 100 µm. Because of this effect, the
lifetime of droplets on a substrate can increase by an or-
der of magnitude [452]. Further increase in the substrate
temperature naturally decreases the lifetime but the de-
crease is slow. The minimum temperature required for
the Leidenfrost effect to occur on smooth surfaces was
characterised recently by Harvey et al. [453].
The presence of a thin lubricating vapour layer, which
is characterised by a low Reynolds number, makes the
droplets highly mobile due to the diminished friction
[Fig. 15d]. As soon as a spontaneous instability causes
a slight difference in the thickness of the vapour layer, a
flow emerges which triggers self-propulsion by rolling mo-
tion [454,455]. The interaction with a structured sub-
strate can also be used to induce directed motion, e.g.
across ratcheted grooves [456–458], and the motion may
further be controlled with thermal gradients [459]. A
video featured on BBC Earth shows how the Leidenfrost
effect can be used to make water run uphill [460]. Be-
sides self-propulsion, the energy injected by droplet heat-
ing can cause droplet vibration with star-shaped droplet
modes [444] that lead to distinct Leidenfrost sounds [445]
[Fig. 15c].
The Leidenfrost phenomenon is seen all across the tem-
perature scale and is controlled mainly by the tempera-
ture difference between the substrate and the droplet,
and surface roughness. Interestingly, the substrate need
not be solid: a similar effect is observed with acetone
droplets (nail polish remover) on a bath of hot water
[461]. More generally, the Leidenfrost state, in which an
object hovers on a solid or on a liquid due to the presence
of a vapour layer, can be seen in a variety of contexts. For
example, it occurs when a block of sublimating solid car-
bon dioxide (dry ice) is placed on a plate at room temper-
ature [462], which is also termed the inverse Leidenfrost
effect [463], or when room temperature ethanol droplet
falls on a bath of liquid nitrogen [464] and start moving.
Furthermore, when two water droplets are placed on a

27
hot plate, the vapour layer between them prevents their
coalescence, which is called the triple Leidenfrost effect
[465]. Frequently demonstrated in popular lectures and
science fairs, Leidenfrost effect allows a person to quickly
dip a wet finger in molten lead or blow out a mouthful
of liquid nitrogen without injury [466].
C. Heating and Boiling: Rayleigh-B´enard
convection
Let’s cook some pasta [467–470]. We place a pot with
water on the stove and start heating. Heat from the
stove is transferred to the water, first through conduc-
tion, and then through natural convection, which we see
as characteristic structures called ‘plumes’ near the bot-
tom wall [Fig. 16]. The fluid layer adjacent to the heated
surface becomes unstable [see §III A, RT instability] and
starts rising, since it is lighter than the bulk fluid. This
fundamental process has been widely studied in many
different configurations. One of the most well-studied is
the Rayleigh-B´enard Convection (RBC) system, consist-
ing of a fluid layer bound between two horizontal plates,
heated from below and cooled from above, as reviewed by
Ahlers et al. [471], Lohse and Xia [472], Kadanoff [473].
It occurs ubiquitously in natural contexts, including as-
trophysics [474,475], geophysics [476,477]) and in en-
gineering applications such as metallurgy, chemical and
nuclear engineering [478,479].
The key non-dimensional parameters governing natu-
ral convection is the Rayleigh number,
Ra ≡heat diffusion time
heat convection time =gβT∆T H 3
νDT
,(49)
and the Prandtl number,
Pr ≡momentum diffusivity
thermal diffusivity =ν
DT
,(50)
where gis gravity, βTis the coefficient of thermal expan-
sion, ∆Tis the temperature difference between the walls,
His the height of the fluid layer, νis the kinematic vis-
cosity of the fluid, and DTis the thermal diffusivity of the
fluid. Note that the Prandtl number only depends on the
inherent properties of the liquid. Most oils have Pr 1,
which means that heat diffuses very slowly in oils. Then,
depending on the magnitude of Ra, we can identify dif-
ferent regimes of natural convection. The heat transfer
through the fluid is solely through conduction till a crit-
ical Ra ∼1708 [482,483]. Beyond this value of Ra, the
convection consists of steady ‘laminar’ rolls [Fig. 16e].
At Ra ∼104, these convection rolls become unsteady,
and beyond Ra ∼105the convection is characterized as
turbulent natural convection [Fig. 16a-d,f].
Now coming back to our water pot, as we continue
supplying heat, the temperature will eventually reach the
boiling point of the water. At this point, the boiling pro-
cess begins with vapor bubbles forming in a superheated
FIG. 16. Rayleigh-B´enard convection. (a) Rising plumes in
a pot of water heated from below, visible because the refrac-
tive index shifts with temperature differences. (b) Contrast-
enhanced magnification. (c) Side-view of mushroom-like
plumes in a high-viscosity fluid. The green line at the bottom
is the boundary layer. (d) Top-view of dendritic line plumes.
(c,d) From Prakash et al. [477]. (e) Temperature field in a
simulation of Rayleigh-B´enard convection at Ra = 5000 and
Pr = 0.7. From Emran and Schumacher [480]. (f) Vortex
structures in a coffee cup with milk at the bottom, which gets
displaced by cold plumes that sink down from the evaporating
interface. Inset: IR thermograph showing convection cells of
colder (downwelling) and warmer (upwelling) regions. From
Wettlaufer [481].
layer adjacent to the heated surface [484]. In this two-
phase system, the vapor bubbles enhance the convective
heat transfer in the standard RBC system [485]. This
boiling process is so complicated that we currently only
have an empirical understanding of the process [484], and
theoretical progress has been lacking. However, cutting-
edge numerical simulations seem to be a promising direc-
tion to model these processes accurately [486]. We put
the pasta into the boiling water and allow it to cook for
approximately 10 minutes. Once the pasta is cooked, we
drain the excess water and serve with a favorite sauce.
D. Layered latte: double-diffusive convection
In §V C we discussed Rayleigh-B´enard convection,
which occurs because the fluid density depends on tem-

28
perature. Often the density also depends on a second
scalar, like salt or sugar concentration. Importantly,
their molecular diffusivity DSis smaller than the ther-
mal diffusivity DT. This explains why your tea cools
down before the sugar diffuses up, but it can also lead
to double-diffusive convection (DDC) that can cause a
range of unexpected phenomena, as reviewed initially by
Huppert and Turner [487], and more recently by Radko
[488] and Garaud [489]. Besides the Rayleigh and the
Prandtl numbers [Eqs. (49), (50)], DDC also depends on
the Schmidt number,
Sc ≡momentum diffusivity
molecular diffusivity =ν
DS
,(51)
where the ratio Le = Sc/Pr is called the Lewis number.
One surprising DDC phenomenon that is readily ob-
served in the kitchen [490] is called salt fingering, which
happens when warmer saltier water rests on colder
fresher water of a higher density [491]. Then, in the
words of Stern [492], “[the] ‘gravitationally stable’ strat-
ification . . . is actually unstable”. If a parcel of warm
salty water is perturbed to move down a bit, it loses
its heat quicker than its salinity, so it will keep sink-
ing further. Hence, salt fingers (vertical convection cells)
spontaneously start growing downwards, accelerated by
thermal diffusion, which gives rise to strong mixing. Kerr
[493] jokes that one might have a Martini cocktail “fin-
gered, not stirred”, which could in fact be quite a specta-
cle with coloured layers [Fig. 17a]. This mixing effect is
likely important for nutrient transport in the ocean and
climate change [494,495]. In astronomy, thermohaline
mixing can occur in evolved low-mass stars [496]. DDC
can equally feature in porous media [497], which might
be relevant for heat and mass transfer in porous materials
under microwave heating [498].
Another striking example of DDC is the formation of
distinct layers in a caff`e latte [499] [Fig. 17b]. To make
one, warm a tall glass with 150 ml of milk up to 50 °C,
and pour 30 ml of espresso at 50 °C into it. The milk
is denser than espresso, so the dynamics will follow an
inverted fountain effect [§III A] leading to stratification
with a vertical density gradient. Subsequently, a hor-
izontal temperature gradient is established because the
glass slowly cools down from the sides. This double gra-
dient leads to stacked convection rolls separated by sharp
interfaces, as seen in creaming emulsions [500]. The cof-
fee pouring (injection) velocity sets the initial density
gradient, and thus the Rayleigh number, which much ex-
ceed a critical value for layers to form [499]. This was
investigated further with direct numerical simulations by
Chong et al. [501], who also discussed the mechanism
how the layers merge over time. Besides caf´e au lait,
similar stacked layers are observed in the ocean, meters
thick and kilometers wide, called thermohaline staircases
[502,503].
FIG. 17. Double-diffusive convection phenomena. (a) A cock-
tail with blue salt fingers, produced by warmer salty water
resting on colder fresh water of a higher density. Image cour-
tesy of Matteo Cantiello (Flatiron Institute). (b) Layered
caff`e latte. Adapted from Xue et al. [499]. Black arrows are
added to highlight the layer boundaries.
E. Tenderloin: moisture migration
So far, we have described various fluid mechanical phe-
nomena related to drinks or liquids. When cooking solid
foods, it is also important to understand moisture mi-
gration to achieve a tender result [470]. We will consider
the example of meat cooking (e.g. tenderloin), where two
important physical aspects include time-dependent pro-
tein denaturation and cooking loss (water loss). To reach
the desired meat textures, the meat must be cooked at
well-defined temperatures to ensure selective protein de-
naturation [504]. Since our focus here is on fluid dynam-
ics, we will discuss water loss during the heat treatments,
which also depends on the temperature [504]. This cook-
ing loss has been described using the Flory-Rehner the-
ory of rubber elasticity [505,506]. This theory mod-
els the transport of liquid moisture due to denaturation
and shrinkage when the protein is heated. The moisture
transport is due to a shrinking protein matrix, similar to
a ‘self-squeezing sponge’. It is assumed that the poroe-
lastic theory applies here. Then, this theory describes
moisture transport by Darcy’s law [see §VII D], where
the fluid flow rate is linear with pressure gradient. The
pressure is due to the elasticity of the solid matrix of the
porous material, an here it is referred to as the ‘swelling’
pressure, pswell. According to Flory-Rehner theory, the
swelling pressure can be decomposed into two compo-
nents,
pswell =pmix +pel,(52)
where pmix is the mixing or osmotic pressure, and pel
is the pressure due to elastic deformation of the cross-
linked polymer gel [323]. In equilibrium, the network
pressure opposes the osmotic pressure, and the swelling

29
pressure is zero. The temperature rise during cooking
causes and imbalance between the osmotic pressure and
network pressure, leading to the expelling of excess fluid
from the meat. Hence, the swelling pressure in meat is
proportional to the difference between moisture content
and water holding capacity. The gradient of this swelling
pressure will drive the liquid moisture flow, and is used
in Darcy’s Law [Eq. (73)]. This model has been found
to agree well with experimental data, proving that the
Flory-Rehner theory provides a sound physical basis for
the moisture migration in cooking meat [506].
F. Flames, vapors, fire and smoke
Next, we consider the flows generated around the hot
cookware, utensils, or hot beverages. There are various
heat transfer processes in the kitchen that can give rise
to vapors, fumes, fires, and smoke. We of course en-
joy the smell of a good dish that is cooking, and unex-
pected smoke is oftentimes an indicator that something
is burning. Hence, in addition to increasing the overall
room/kitchen temperature, the vapors/smoke also play
an important role in giving us positive or negative feed-
back on how the cooking is going. Hot utensils transfer
heat into the surrounding air setting up buoyancy-driven
convection or natural convective flows around them. Fig-
ure 18 shows the beautiful flow of vapors and convection
around a hot espresso cup [507], and around a hot tea
kettle, visualised with Schlieren imaging as reviewed by
Settles [216], Settles and Hargather [508]. Such convec-
tive flows are present around all heated objects, and one
can imagine how different geometries of vessels/cookware
can give rise to complicated flows around them.
The kitchen is our safe place to prepare food, but we
must remember that it is also a place where several safety
hazards exist. At some point in our lives, most of us have
forgotten to turn off the kitchen stove and suffered the
consequences. When food is overheated, it starts to burn
and eventually the temperature gets so high that the car-
bon content gets converted to soot that give rise to smoke
and fumes. Smoke decreases the overall air quality and
inhaling it can adversely affect our health [509], especially
if the burning becomes intense and the heating is contin-
ued. While the dispersion of smoke as a general pollutant
in the atmosphere has been studied extensively, smoke in
the kitchen has also been the subject of several studies
[509,510]. A critical issue related to this is ventilation,
i.e. how well a kitchen is designed to get rid of harmful
smoke and fumes. Most modern kitchens feature a venti-
lation or exhaust ‘hood’ right above the stove. Both Ex-
perimental Fluid Dynamics (EFD) and Computational
Fluid Dynamics (CFD) techniques such have been uti-
lized to study the flow through kitchen hoods in order to
maximise their performance [511]. Fires in the kitchen
are the most dangerous safety hazard [512], and can re-
sult in destruction of property and loss of life. Given the
importance of minimizing safety hazards in the kitchen,
FIG. 18. Buoyancy-driven plumes. (a) Flows developing over
an espresso cup visualised with schlieren imaging. Colors in-
dicate fluid displacement, where red is the highest, and blue
the lowest. From Cai et al. [507]. (b) Plume around a hot
tea kettle captured with schlieren imaging. From Settles and
Hargather [508].
fire engineers rely on fluid dynamics modeling to develop
designs for optimum and safe kitchen ventilation [513–
515].
G. Melting and freezing
We tend to naturally associate cooking with heat. Boil-
ing, melting, freezing, solidifying, dissolving and crystal-
lizing food products need the external heat stimulus to
transform. Thermally driven reconfiguration can happen
on a molecular level but more often it is enough to con-
sider phase transitions due to the heating or cooling of a
substance.
Melting is an exemplary fundamental process of phase
transition. From thawing to cooking, solid substances
are transformed into soft matter or liquid products. The
sole process of melting and flows induced therein are
still of fundamental interest for physicists. The mech-
anism of heat transfer relies, in the simplest case de-
scribed in §V A, on the Fourier law, where the heat flux
qacross a surface is locally proportional to the temper-
ature gradient ∇T. The temperature field thus satis-
fies the advection-diffusion equation (47), which can then
be solved numerically or analytically in specific geomet-
ric configurations. Geometry itself inspires questions on
how the process of thawing can be exploited to create
desired forms, i.e. seen in constantly evolving ice sculp-
tures or in ordinary ice cubes melting in a cocktail. Al-
though the initial shapes of frozen structures may be ar-
bitrary, macroscopic objects such as melting ice cubes
and growing stalactites can approach non-intuitive geo-
metric ideals. For the dissolution of non-crystalline ob-
jects, a paraboloidal shape was shown to be the geometric
attractor [516]. Even the simple melting of icicles is gov-
erned by a combination heat transfer from the air, the
latent heat of condensation of water vapour, and the net

30
radiative heat transfer from the environment to the ice
[517], which emphasizes the complexity of phase change
processes. In kitchen flows, an additional factor is the
microstructure of food products, which renders their flow
non-Newtonian, and their response to temperature vari-
ations non-linear. Even though the flow of molten choco-
late in a fountain is aptly described by a power-law fluid
[518], the process of melting involves different crystalline
phases and is thus highly complex. Subsequent freezing
typically leads to a change in structure and appearance,
with different physical properties and even taste. The
quality of chocolate products, in particular their gloss,
their texture and their melting behaviour, depends pri-
marily on two processing steps: the precrystallisation of
the chocolate mass and the eventual cooling process [519].
These factors must be considered in confectionery manu-
facturing and, fundamentally, in the modelling of crystal-
lization and melting kinetics of cocoa butter in chocolate
[520].
Butter (plant or animal-based) itself, being a vital
product for cooking, responds strongly to external tem-
perature, transiting from completely solid and brittle
when taken out from the refrigerator, to pleasantly
spreadable at intermediate temperatures, to liquid. This
empirical feeling can be related to its viscoelastic char-
acteristics [521,522], which additionally depend on the
substance and on the method of production [523]. Rheol-
ogy and texture are often the basic characteristics when
heating or baking cheese [524], such as in the melting and
browning of mozzarella in the oven [525]. The temper-
ature can be also coupled to the nonlinear viscoelastic
properties, for instance when considering starch gelatin-
isation while making gravies and thick sauces [526] or
when freezing and thawing a gelatin-filtered consomm´e
[527].
H. Non-stick coatings
Many non-stick pans or cookware are designed to be
hydrophobic (water repellent) and oleophobic (oil repel-
lent), which together is called amphiphobic [528,529].
Waterproof fabrics often use chemical coatings such as
polyurethane (PU), polyvinyl chloride (PVC), or fluo-
ropolymers. However, there are many concerns for these
materials regarding toxicity and other environmentally
damaging effects [530], and to limit their destructive im-
pact, the European Union has announced a ban on such
chemicals by 2030 [531]. As such, interest has risen for
purely physical coatings that make use of the “lotus ef-
fect” [532,533]. Like leaves of the lotus plant (Nelumbo
genus), micron-sized structures can be designed that give
rise to superhydrophobicity [534–536]. A droplet that
impacts these surfaces can bounce off without wetting
them [537,538]. Moreover, decorating sub-millimetric
posts with nanotextures can lead to “pancake bouncing”,
where the contact time of droplets with the surface is sig-
nificantly reduced [539,540]. This is important for anti-
icing [541] and self-cleaning surfaces [542], with possible
applications for airplanes and cars. Remarkably, super-
amphiphobic coatings can also be made with candle soot
[543].
VI. HONEY DESSERT: VISCOUS FLOWS
Viscosity controls both slow and small-scale flows,
shaping our notion of ‘heavy’ or ‘thick’ substances such
as honey [544], oil, gravy, or cream. Moreover, viscous
flow theory underlies the behaviour of most composite
flowing food products, such as emulsions, suspensions,
and particle-laden fluid substances. When you are try-
ing to mix together viscous fluids, you might bear in mind
the kinematic reversibility experiment of Taylor [545],
showing that the time-reversible driving force will lead
to time-reversible particle trajectories in Stokes flow [see
Fig. 19a]. When pouring honey or golden syrup on pan-
cakes or toast [Fig. 19b], the apparent coiling of the im-
pinging stream is but one of the many phenomena at the
interplay of gravity, viscosity, and surface tension. Fi-
nally, the complex shapes of sedimenting clouds of food
colouring added to a cocktail can be aptly described us-
ing many-body hydrodynamic interactions of Stokesian
microparticles [378,546], as seen in Fig. 19c. Below, we
will discuss how the concept from Stokes flow manifest
themselves in a myriad of culinary aspects, some of which
are highlighted in this section.
A. Flows at low Reynolds number
Viscous liquids are everywhere in the kitchen, and their
fluid mechanics can be rather non-intuitive. To under-
stand these flows better, we must determine which of the
terms in the Navier-Stokes equations (2) are relevant in
different situations. It is helpful to scale the time by a
characteristic value T0, the velocities by the characteris-
tic speed U0, and distances by the characteristic length
L0. Hence, one can introduce the dimensionless variables
˜
x=x/L0,˜
∇=L0∇,˜
t=t/T0,
˜
u=u/U0,˜p=p−patm
µU0/L0
,˜
f=f
µU0/L2
0
,(53)
where patm is the atmospheric pressure. Then, rewrit-
ing Eq. (2a) in terms of these variables yields the non-
dimensionalised momentum equation,
Re 1
St
∂˜
u
∂˜
t+ (˜
u·˜
∇)˜
u=−˜
∇˜p+˜
∇2˜
u+˜
f,(54)
where the Reynolds number is Re = ρU0L0/µ, as we
discussed before in §II D, and the transient Reynolds
number is ReT=ρL2
0
µT0=Re
St . Here the Stokes number
(St) compares the relative importance of transient iner-
tial forces compared to viscous forces.

31
FIG. 19. Examples of Stokes flows relevant to kitchen setting.
(a) Experimental demonstration of kinematic reversibility in
an annular cylindrical space filled with silicone oil. Upon ro-
tating the inner cylinder slowly a number of times and then
reversing the forcing, the dyed fluid blobs remain unchanged,
thus illustrating the difficulties in mixing viscous fluids. From
Wikimedia Commons, licensed under CC BY 2.0. (b) Coiling
of a liquid rope made of a viscous fluid (corn syrup) demon-
strates the complexities of free-surface gravity flows and also
of pouring honey on the pancakes. From Ribe et al. [547]. (c)
Snapshots of the falling cloud: (left) in point-particle Stoke-
sian dynamics simulation with 3000 particles and (right) in
sedimentation experiments using 70 µm glass beads in silicon
oil. From Metzger et al. [546].
The Reynolds number is typically very small in vis-
cous fluids, or at small length scales. For example,
when humans swim in water with a dynamic viscos-
ity µ≈0.9 mPa s and density ρ≈103kg/m3, we find
Re ∼106. This means that the fluid inertia is much
more important than viscosity. For microbes swimming
in the same medium, however, we have Re ∼10−5, so
the viscosity is overwhelmingly dominant [548]. Then,
Eqs. (2) reduce to the Stokes equations,
0 = −∇p+µ∇2u+f,∇·u= 0.(55)
These equations have several important properties.
Firstly, they are linear, which makes them much easier to
solve than the Navier-Stokes equations. This also means
that, for given set of boundary conditions, there is only
one unique solution [549]. Secondly, the Stokes equations
do not depend explicitly on time. Viscosity-dominated
flows will therefore respond essentially instantaneously
to changes in the applied force, pressure, or the bound-
ary conditions. In other words, disturbances to the flow
field spread much faster than the flow itself. Thirdly,
Eqs. (55) are kinematically reversible, i.e. they are in-
variant under the simultaneous reversion of the direction
of forces and the direction of time. This means that if
the forces driving the flow are reversed, the fluid parti-
cles retrace their trajectories in time. As a consequence,
it is notoriously difficult to mix fluids at low Reynolds
number [see §VIII E]. Apart from beautiful demonstra-
tions of mixing and demixing under reversed forcing in
famous video experiments by G.I. Taylor [545], this is-
sue bears great significance for microfluidic flows, which
are becoming increasingly relevant for food science [see
§IV D and §VI J].
B. Fundamental solution of Stokes flow
The fundamental solution to the Stokes equations, the
Green’s function, is called the Stokeslet. It is the flow
uS(x, t) at position xand time tdue to a point force,
f(x,y, t) = δ3(x−y)F(t),(56)
which has time-dependent strength F(t) exerted on the
liquid, and the force is located at position y(t). For an
unbounded fluid, the boundary condition is u= 0 as
|x| → ∞. Physically, one could think of this point force
as a small particle being dragged through the liquid, such
as a sedimenting coffee grain [see §VI C]. The flow gener-
ated by this point force is given by
uS(x, t) = J(x−y(t)) ·F(t),(57)
where the Oseen tensor Jij (r) has Cartesian components
Jij (r) = 1
8πµ δij
r+rirj
r3,(58)
with indices i, j ∈ {1,2,3}, the relative distance is r=
x−y, and r=|r|. There are different ways to derive
this tensor, as summarised by Lisicki [550].
This Stokeslet solution is powerful, analogous to
Coulomb’s law for electric point charges. Like the Stokes
equations, it also reveals important properties of viscous
flows. First, Eq. (57) shows that the applied force is di-
rectly proportional to the flow velocity. As opposed to
Newtonian mechanics, where the forces are proportional
to acceleration, this reflects Aristotelian mechanics [551],
where there is no motion in the absence of forces. Inertia
vanishes at low Reynolds number, that is, so the dynam-
ics are overdamped. Second, hydrodynamic interactions
are very long ranged. The Stokeslet flow decays as 1/r
with distance, as opposed to gravitation or electrostatics
that both follow inverse-square laws. This has important
consequences. For example, in a particle suspension, the
motion of one particle will produce a flow that moves
other particles, which in turn generate flows that affect
the first particle again [see §IV C].
The fundamental solution can also be used to derive
the widely celebrated Stokes law,
F=−6πµaU,(59)
which describes the viscous drag force exerted on a small
sphere of radius amoving with velocity Urelative to the

32
viscous fluid [54,552]. The flow generated by this finite-
sized sphere can be written [see e.g. 553] in terms of the
Stokeslet flow as
u(x, t) = 1 + a2
6∇2uS.(60)
=U3a
4r+a3
4r3+rU·r
r23a
4r−3a3
4r3,(61)
where we used Eqs. (57-59) for the second step. One can
verify that this expression (61) will satisfy the Stokes
equations and the required no-slip condition on the sur-
face of the sphere, u(r=a) = U. The significance of
Stokes’ law can hardly be overemphasized. Dusenbery
[554] writes that it is directly connected to at least three
Nobel Prizes. Can you name them?
C. Coffee grains in free fall
To optimise our coffee, we may want to assess the size
distribution of the coffee grains after grinding. This can
be achieved in a simple sedimentation experiment, using
the Stokes law [Eq. (59)]. The gravitational force, Fg=
mg, pulling on a spherical grain of size ais proportional
to the mass differential with the surrounding fluid, m=
(ρp−ρ)4
3πa3. The same grain is slowed by a viscous drag
force, given by Eq. (59). When the drag force balances
the gravitational force, the terminal velocity is
U∞=2
9
a2(ρp−ρ)g
µ.(62)
To measure U∞in the kitchen, we can use a mobile phone
to videotape individual grains sedimenting. Eq. (62) can
be rearranged to solve for the particle size a. Conversely,
it is possible to solve for µif ais known, which is the
principle of falling sphere viscometry [555]. Not least,
Millikan [556] used a form of Eq. (62) to find the elemen-
tary electrical charge. Note that the above equation was
developed under several assumptions. In the paragraphs
to follow we assess the validity of these.
The first assumption is that the grain sediments at
low Reynolds number, given by Re = aU∞(ρp−ρ)/µ.
Using Eq. (62), we find that a typical grain of radius
a= 1 mm sediments at 0.1 mm/s, yielding a Reynolds
number just below order unity. We should therefore
be cautious when using Stokes’ law in this case [557].
In order to obtain the range of Reynolds numbers over
which Eq. (62) gives a good approximation, one can turn
to experiments. Numerous reports have been published
on this topic [558,559], and it is generally agreed that
Eq. (62) gives good estimates up to Re ≈1 [560].
The sedimenting coffee grain generates a flow field that
decays slowly with distance from its center [see §VI B].
Close to the particle, viscous forces dominate, while in
the far field the inertial terms dominate. It is the absence
of walls that facilitates this shift from viscous to inertial
dominance, as walls provide friction to the flow. By ac-
counting for inertial effects, considering an unbounded
fluid, Oseen [561] developed the following improved for-
mula for the drag coefficient,
CD=24
Re 1 + 3
16Re.(63)
The first term is the Stokes drag close to the particle,
while the second term stems from inertial effects far away
from the particle. Oseen’s formula agrees fairly well with
experiments up to Re ≈10 [562].
Beyond the Oseen regime, inertial effects in the flow
surrounding the particle can no longer be neglected. This
causes the nearby fluid streamlines to divert from the
particle, causing the flow to separate. Then the pres-
sure drop across the particle is reduced, which leads to a
drag reduction. A simple experiment using coffee grains
released in air (instead of water) can be performed to
observe this behavior. By accounting for inertial effects
around a sedimenting sphere, Stewartson [563] found the
drag coefficient to be approximately 1.06, which may be
compared to the value of 7.2 using Eq. (63) for Re = 10.
For more details, we refer the interested reader to the
excellent review paper on Stokes’ law, and its legacy, by
Dey et al. [562].
Wall effects: We now return to the assessment of the
validity of Stokes’ law for sedimenting coffee grains in
bounded water. In developing Eq. (62), we assumed that
the grain falls without influences of walls. O’Neill [564]
showed that hydrodynamic contributions due to walls can
slow down a sedimenting grain (or a rising bubble) by as
much as 5% when the distance to the walls is ten times
its size. The degree of retardation increases linearly as
the grain gets closer to the vessel walls. Using matched
asymptotic expansions, Goldman et al. [565] obtained a
solution for the drag force acting on a sedimenting sphere
moving parallel to a wall,
FkW
FS
= 1−9
16
a
h+1
8a
h3−45
256 a
h4−1
16 a
h5,(64)
which is normalized by the Stokes drag force [Eq. (59)].
Thorough experiments have been performed to validate
the above result, showing good agreement [566].
For spheres very close to a wall, lubrication effects be-
come important [§VI E], yielding a logarithmic relation-
ship between the force and the distance to the wall [567].
This strong logarithmic dependence can be readily ob-
served when making French press coffee: grains close to
the vessel container sediment much slower than grains
out in the bulk. Recently, Rad and Moradi [568] revis-
ited this problem in non-Newtonian liquids, which has
important biophysical implications.
Collective effects: Our final assessment of Eq. (62)
concerns the influence of multiple particles correlated
by long-ranged hydrodynamic interactions [see §VI B].
When two spheres sediment side-by-side, they fall slower
than in the absence of the other particle, while if they

33
are separated by a vertical line going through their cen-
ters, the opposite is true [377,378]. In a suspension con-
taining several grains, the influence of other particles al-
ways leads to a decrease in sedimentation velocity. This
observation is known as hindered settling [569] and is
mainly due to an upward flow generated by each parti-
cle as it sediments. In a dilute suspension, the hindered
settling velocity depends on the particle concentration
as U≈U∞(1 −6.55φ), as shown by Batchelor [570].
Other effects such as Brownian motion [571] and shape
[558] can also affect the sedimentation velocity. Finally,
if particles sediment towards a surface covered with mov-
ing actuators, a self-cleaning effect can occur where hy-
drodynamic fluctuations repel the particles, leading to a
non-Boltzmannian sedimentation profile [572].
D. Slender body theory
A generalisation of the Stokes law [Eq. (59)] can be
derived for a cylinder of length `and radius a. When
it moves through the fluid along its major axis, it will
experience a viscous drag force
Fk≈2πµ`U
ln(`/a),(65)
as predicted by slender-body theory [573,574]. Because
of the structure of the Oseen tensor, Jij [Eq. (58)], the
cylinder will exert a larger force on the fluid when it
moves sideways,
F⊥≈2Fk.(66)
Despite its simplicity, this result lies at the heart of
numerous transport processes for viscous fluids. The
drag asymmetry for slender filaments enables propulsion
of microorganisms: bacteria rotate their helical flagella
and exploit the rotation-translation coupling due to their
chirality to generate propulsion force [575–577]; ciliates
perform coordinated motion of tiny hair-like appendages
covering their bodies, creating metachronal waves which
transport the fluid along their surfaces [578–580]. More-
over, various experimental designs for artificial swim-
mers, which tend to mimic their biological archetypes,
are based on this hydrodynamic anisotropy [257]. Inter-
estingly, it could be possible to tailor food texture and
rheology by microstructured fibers [581].
E. Lubrication theory
In systems where one dimension is significantly smaller
than the others, its hydrodynamics can often be de-
scribed using lubrication theory. This is relevant in many
contexts where the fluid is confined between surfaces or
forms a thin film on a substrate. The former has indus-
trial applications in fluid bearings, but also in kitchen
flows involving squeezing and spreading of viscous fluids
in thin layers. The key goal there is to relate the geom-
etry to the resulting pressure distribution. The latter,
involving a free surface, is related to the shape evolution
of the fluid surface under the action of external forces or
in response to substrate patterning. This bears impor-
tance for coating and painting, also in the food industry.
In shaping these flows, surface tension and substrate wet-
ting properties may become significant.
From a mathematical point of view, lubrication theory
builds on the separation of length scales, with the fluid
layer thickness h(say, in the direction zlocally perpen-
dicular to the substrate) being much smaller than the
substrate length scale L(in the direction xalong the
surface). Then, the leading-order expansion in the small
parameter ε=h/L yields the equations
∂p
∂z = 0,∂p
∂x =µ∂2ux
∂z2.(67)
This reflects the fact that in the thin gap the pressure
changes very little with height above the substrate, and
its change along the substrate is related to the flow vari-
ation across the gap. In the general three-dimensional
case, as described by Reynolds [582] and reviewed by
Batchelor [54], Oron et al. [583], Szeri [584], this equa-
tion is referred to as the Reynolds equation,
h∂ρ
∂t +ρwab =X
i=x,y
ρua
i∂ih+∂iρh3
12µ∂ip−ρhuab
i
2,
(68)
where a, b denote the top and bottom surfaces, respec-
tively, wis the vertical flow velocity, uiwith i=x, y
denotes the horizontal velocities, wab =wa−wband
uab
i=ua
i+ub
i. This equations may be solved analytically
or numerically for a range of different applications.
Indeed, lubrication theory is used ubiquitously. In tri-
bology, it is crucial for reducing wear and friction between
bearings [585]. In biology, tree frogs can climb vertical
walls using liquid film flows, which has inspired new tire
technology [586], and biomimetic materials that adhere
under wet conditions [587,588]. Thin films significantly
alter the motion of trapped microorganisms [589], gov-
erning their surface accumulation. Note that many bio-
logical fluids are viscoelastic, which changes the Reynolds
equation (68), but the basic concepts still hold. Using
ferrofluids, adhesion can also be made switchable for use
smart adaptable materials [590]. Recent developments on
the nanoscale physics can improve lubricant design [591].
Additionally, lubrication theory has been used to model
air hockey [592] and, in the spirit of kitchen experiments,
Reynolds also used it to determine the viscosity of olive
oil [582]. Pan lubrication is also an important step in
industrial baking [593]. Finally, the lubrication theory
can also be used to model hand washing, as explained in
§II K.

34
F. Pot stuck to stove top: Stefan adhesion
If a pot of pasta overboils, a common subsequent prob-
lem is that the pan is “stuck” to the surface. This effect
does not require any glue or the formation of molecu-
lar bonds – it stems from the viscous liquid film that is
sandwiched between the objects. Josef Stefan [594] first
described this “apparent adhesion”, and later Reynolds
[582] quantified it with a detailed treatise on lubrication
theory [see §VI E].
Making use of the disparity between the radius of the
pan, R, and the width of the liquid film, hR, one can
derive the pressure distribution inside the film described
by the Reynolds equations (68). Integrating this expres-
sion shows that the force required to lift the pan from
the surface, the Stefan adhesion force, is
F=3πµR4
2h3
dh
dt .(69)
This force can be very large for thin films due to the
strong dependence on h, making it hard to squeeze a
viscous liquid through a narrow gap. For pan radius
R∼12cm, film thickness h∼10µm and separation speed
dh
dt = 1mm/s, the force is F∼106N, orders of magnitude
larger than the weight of the filled pan. Conversely, it is
very difficult to bring two surfaces into close contact, as
described by squeeze flow theory, with many generalisa-
tions for viscoelastic fluids [595].
G. Viscous gravity currents
“Open the front door of a centrally-heated house and a
gravity current of cold air immediately flows in.” These
opening words by Huppert [596] describe many processes
in the kitchen: Opening the fridge or the oven door, but
also the spreading of oil in a frying pan. To understand
how long this spreading takes, we need to determine the
evolution of the height profile, h(r, t), which strongly de-
pends on the dynamic viscosity and density of the oil, µ
and ρ, gravity, g, and the pouring rate, Q. The presence
of inertia makes predictions of the radial spreading ve-
locity difficult. If we instead pour the oil really slowly, at
low Reynolds number, assuming that both the oil layer
thickness hand the ‘jet’ radius Rjare small compared
to the current R, then we can use a lubrication approx-
imation [see §VI E] to obtain a simplified version of the
radial force balance:
∂h
∂t −1
3
ρg
µ
1
r
∂
∂r rh3∂h
∂r = 0.(70)
By adding a mass conservation equation to Eq. (70), and
by introducing a similarity variable, Huppert showed that
the radial extent of the evolving puddle, R, is given ex-
actly by:
R= 0.715 ρgQ3
3µ1/8
t1/2.(71)
The dominant balance of forces in Huppert’s analysis is
between a hydrostatic pressure head, that drive the fluid,
and viscous stresses, that slow the fluid. Huppert dis-
carded effects of surface tension, γ, and Eq. (71) requires
that the Bond number be large, Bo = ρR2/γ 1. Many
geophysical flows are characterized by large Bond num-
bers, and the scalings by Huppert have seen widespread
use for predicting spreading rates of saltwater currents
into freshwater, lava flows, and many other geophysical
gravity currents [597–599]. The success of these scal-
ings in miscible fluids might seem surprising. However,
geophysical flows are often characterized by high P´eclet
numbers [Eq. (48)], such that the transport of momen-
tum outpaces the transport of mass [see §V A]. On the
fast time scale of the flow, diffusion does not have suf-
ficient time to blur the interface separating miscible liq-
uids, resulting in the liquids displaying immiscible be-
havior. Recently, Eq. (71) was found to accurately de-
scribe the spreading rate of miscible sessile drops of corn
syrup and glycerol in water [600]. Other important con-
tributions to the study of gravity currents include the
spreading of a saltwater current under a bath of fresh-
water [601], spreading hot plumes in cold environments
[602], and oil spreading on the sea [603]. These and other
works are described in detail in a number of review pa-
pers [597,604] and in the book by Ungarish [605].
Making the perfect crˆepe
In the analysis by Huppert [596], the viscosity is
treated as a constant. If the pan is heated, however,
the spreading problem becomes more complicated as the
viscosity must be treated as a non-linear function of time.
This situation is typical in many cooking processes. For
example, when pouring pancake batter into a hot pan to
make a crˆepe, the temperature of the spreading batter
increases, and this temperature rise is associated with a
reduced viscosity. Later in the spreading process, the
batter starts to solidify, and the liquid-solid transfor-
mation leads to a viscosity increase. Since the current
continually loses momentum as it spreads, the solidifica-
tion can arrest the flow long before it extends the entire
pan. Conveniently, flow arrest can prevented by swirling
the pan, which enhances the spreading rate dramatically.
Motivated by the aim of making a perfectly flat and uni-
form crˆepe, Boujo and Sellier [606] recently approached
the spreading problem with both time-dependent viscos-
ity and gravity. Armed with numerical tools, they identi-
fied different swirling modes and measured the resulting
pancake shape. Interestingly, a swirling mode that is
naturally adopted in pancake making, namely draining
all the batter in one place and then rotating the batter
around the perimeter of the pan in one big swirling mo-
tion, appeared to optimise the pancake shape. To learn
more about the rheology of pancake making and other
cooking processes, see §IV A.

35
H. Viscous fingering
When a less viscous fluid displaces a more viscous one,
such as when we inject water into glycerol, then the
frontal lip of the current is susceptible to Saffman-Taylor
instabilities [607,608] , causing “viscous fingering” pat-
terns to evolve [609]. Viscous fingers can also be created
in the kitchen when mixing water (less viscous) with con-
densed milk (more viscous) when making a banana pud-
ding, or when a sessile drop of water spread on a layer
of honey. The local destabilization of a finger occurs by
a splitting of its tip and results in the formation of two
branches separated by a fjord, leading to an elaborate
pattern formation due to successive tip-splitting [610].
The vast literature on viscous fingering spans from oc-
currences in biological systems [611,612] and porous me-
dia drainage [613], to methods for mitigation in oil ex-
traction processes [614]. Another example with geophys-
ical relevance includes the report by Snyder and Tait
[615], who introduced a miscible current of intermedi-
ate density in between two stably stratified liquid layers.
They found that the interface is unstable when bounded
by more viscous liquids on either side, and that the insta-
bility is suppressed when at least one of the two stratified
liquids are of higher viscosity. A new and refreshing di-
rection in the field concerns so-called particle induced
fingering. Tang et al. [616] discovered that particles can
cause an otherwise stable front between a viscous liquid
such as glycerol displacing a less viscous liquid such as
water to develop viscous fingering patterns. The instabil-
ity arises because of an uneven particle distribution along
the perimeter, which leads to local viscosity differences.
The evolving front propagates faster in low viscosity re-
gions than in regions of high local viscosity, and it is
these differences that allow irregularities at the interface
to develop [617].
Viscous fingers can also appear in the context of
boundary driven lubrication flows. A drop of treacle or
honey squeezed between two glass plates spreads upon
the gravitational settling of the top plate into a circu-
lar structure with its radius Rgrowing with time as
R(t)∼t1/8, in analogy to the viscous gravity flows in
§VI G. Upon levering one of the plates, a draining flow
arises. The droplet front is unstable and thus produces
a network of fingering branches which thin over time as
the gap width increases [618].
I. Microbial fluid mechanics
The remarkable properties of the Stokes equations
[see §VI A] have pronounced consequences for the life
of microscopic swimming organisms, which have to obey
these physical limitations [259,548,575]. One of these
constraints is the scallop theorem, stating that a time-
reversible swimming gait cannot lead to a net displace-
ment. In order to achieve propulsion, microorganisms
they rely on a variety of mechanisms circumventing kine-
matic reversibility of flows. Some organisms use helically-
shaped flagella [576] or cilia [619], others exploit elas-
ticity to break time-reversibility [620], or use the non-
Newtonian properties of the surrounding fluid [621]. Mi-
crobes play an role in cuisine in a spectrum of aspects;
their presence can improve texture and taste of food, but
their growth and development can also cause illness. It is
thus crucially important to understand the mechanisms
of their growth and locomotion to control their influ-
ence on food products and living systems they inhabit.
Even in the absence of a bulk fluid phase, aerosols can
also transport microscopic passengers, such as bacteria or
viruses. Cell motility is particularly important in infec-
tious disease transmission [147], bacterial contamination
by upstream swimming [622,623], and the microbiology
of bacterial coexistence [624]. Microbial dynamics are
critical in food science, and intricate detector designs lead
to sensors capable of detection of harmful microbes. For
example, a porous silk microneedle array can be used to
sense the presence of E. coli in fish fillets [625]. In gen-
eral, a wide spectrum of potential pathogens stimulates
research on diverse biosensors to provide the information
necessary for food safety [626].
J. Microfluidics for improved food safety
Food borne pathogens such as E. coli and Salmonella
bacteria cause approximately 420,000 deaths and 600
million illnesses yearly [627]. Such pathogens often origi-
nate from bacterial biofilms [628,629] in food processing
plants, and on-site rapid detection is therefore necessary
to prevent these harmful bacteria from entering our gro-
cery stores and ending up in foods. The traditional way
of detecting pathogens is by cultivating them in petri
dishes, but slow bacterial growth limits the usefulness
of such ‘babysitting’ in food plants. This has led to al-
ternatives such as nucleic-acid based methods (including
polymerase chain reactions (PCR) [33], which is the pri-
mary method to test patients for COVID-19), but ex-
tensive training requirements, expensive equipment and
labor intensive steps prevent a robust and efficient im-
plementation in food production facilities.
Fortunately, microfluidic-based biosensors can detect
or “sense” pathogens on much faster timescales thanks to
small volumes and flow-mediated transport, with high-
speed imaging enabling real-time monitoring [26,28,
630]. Biosensors work by measuring an electrical or
optical signal induced by a chemical reaction as a tar-
get molecule (pathogen) binds to a bioreceptor molecule
[631,632]. The bioreceptors are designed to exactly
match the surface elements of the pathogen (these ele-
ments are called antigens) like a lock and key fit. Due to
their excellent specificity, monoclonal antibodies (mAbs)
are the most widely used bioreceptor molecules. Figure
20 shows the working principle a biosensor functionalized
with Y-shaped mAbs molecules for detecting E.coli [633].
The speed and accuracy of surface-based biosensors de-

36
FIG. 20. Working principle of an optofluidic pathogen de-
tector. When E.coli bind to Y-shaped mAbs molecules, it
induces a shift in refractive index which can be detected by
an optical sensor. From Tokel et al. [633].
pend on a number of factors where the most important
are the pathogen concentration, the number of available
binding sites on the sensor, the transport of pathogens
from the bulk fluid to the sensor via flow and diffusion,
and finally, the kinetics of the binding reaction. With-
out flow, the sensing time is usually limited by the time
it takes for a pathogen or a virus to diffuse to the sen-
sor, which can take several hours in a microchannel due
to the low diffusivity of such large particles. However,
by leveraging microfluidic flow, the solute transport can
be sped up by several orders of magnitude, leading to
reaction-limited kinetics. We recommend the pedagogi-
cal reviews on transport and reaction kinetics in surface
based biosensors by Gervais and Jensen [634], Squires
et al. [635] and Sathish and Shen [636].
K. Ice creams
Towards the end of the meal or on a hot summer day,
ice cream comes to mind naturally. The origins of the
creamy taste and crunchy texture are less obvious. The
feeling results from the composition and an elaborate pro-
cess of production. From a chemical point of view, ice
cream is an emulsion [see §IV D] made with water, ice,
milk fat and protein, sugar and air. The ingredients are
mixed together and turned into foam upon the addition
of air bubbles. The colloidal emulsion is then frozen to
preserve the metastable mixture. The details of the pro-
cess have been extensively studied within the food science
community [637,638], but also from the physics and gen-
eral science viewpoint [639,640]. Special attention has
been paid to the colloidal character of the emulsion [641].
An appealing texture and rheology are crucial aspects
of ice cream quality. Due to its popularity, ice cream was
apparently the first food product to have its extensional
viscosity measured, as early as in 1934 [642]. Since then,
rheological properties of ice cream have been examined
in detail, and various dynamical models have been pro-
posed to account for their behaviour [643]. The breadth
of related topics has even inspired an interdisciplinary un-
dergraduate course taught within the physics programme
[644].
The production of ice cream involves a flow undergo-
ing a structural and phase transition by a combination of
mechanical processing and freezing. The dynamics of ice
crystallisation therein are not fully understood. Because
an ice cream mix is opaque, in situ crystallisation has
not been observed and its mechanism is debated [645].
In a typical ice cream freezer, ice is formed on exter-
nally cooled walls by surface nucleation and growth, and
is then scraped off to the bulk fluid, where secondary
nucleation and ripening take place [646]. The number
and size of ice crystals formed is also heavily dependent
on the mixture composition, and also affects the melt-
ing rate and hardness of the final food product [647].
Studying the characteristics of ice cream production leads
to the development of novel methods for rapid freezing
and thawing of foods [648]. Although the structure of
ice cream can be practically controlled for a satisfactory
product, a detailed description of the underlying complex
growth process is still under development.
In contrast, one sometimes sees ‘hot ice cream’ that
seems to melt upon cooling. This effect can be achieved
using methyl cellulose, a thickener that acts in high tem-
peratures to produce a product with surprising, ‘oppo-
site’ melting properties [649].
VII. COFFEE: GRANULAR MATTER &
POROUS MEDIA
The Hungarian probability theorist Alfr´ed R´enyi
(1921-1970) once said [650],
A mathematician is a device for turning coffee
into theorems.
In this section we will investigate the fluid dynamics of
coffee. We start with the flow of coffee beans and other
granular materials, including avalanches, hoppers, and
the Brazil nut effect. We then consider brewing coffee
using different methods in the context of porous media
flows and percolation theory, and we finish with the illus-
trious coffee ring effect. In the words of Wettlaufer [481],
there is a “universe in a cup of coffee”.
A. Granular flows and avalanches
Granular materials [651] are found everywhere in the
kitchen, including flour, rice, nuts, coffee beans, sugar
and salt. Indeed, the food industry processes billions of
kilograms of granular material every year [652]. They are
composed of discrete, solid particles (grains) that feature
across a wide range of sizes. Therefore, the grain di-
ameter Dis often denoted on the Krumbein phi scale,
φ=−log2(D/D0), with D0= 1 mm [653]. For exam-
ple, φ=−6 and 6 correspond to oranges and powdered
sugar, respectively.
Granular matter can have surprising properties [654,
655]. One example is the Janssen effect: The hydrostatic
pressure at the bottom of a cylindrical container does

37
not grow linearly with the filling height of grains, unlike
a liquid [see Eq. (1)]. Instead, the pressure saturates
exponentially to a value much less than the weight of
the grains, because they are partially supported by the
vertical silo walls due to friction forces [656]. Another
example is that granular matter can behave like solids,
liquids, or even gases, depending on the amount of kinetic
energy per grain [657–659]. For this reason, granular
flows are hard to predict.
A notorious example thereof is avalanche dynamics
[660–663]. A pile of grains [Fig. 21a] is held together
by ‘chains’ of frictional and compressive forces [664], but
the pile will suddenly collapse if the slope exceeds a
maximum angle, θm. This sliding will only stop after
the slope has reduced below the critical angle of repose,
θr, of which typical values range between 45◦for wheat
flour to 25◦for whole grains [665]. The angle of repose
is important across food industry, from silo roof design
to conveyor belt transport [666] and the geology of hill-
slopes [667] that limits farming. It also sets a fundamen-
tal rule for food plating, which in turn affects the per-
ception of taste [668], and the design of food sculptures
[Fig. 21b]. In this artwork the grains are sticky, but the
weakest links are still susceptible to avalanche dynamics.
From a fundamental point of view, avalanche dynamics
exhibits self-organized critical (SOC) behaviour for rice
grains with a large aspect ratio, as shown by Frette et al.
[662]. Moreover, Einav and Guillard [669] used a column
of puffed rice to investigate the crumbling of a brittle
porous medium by fluid flow. These ‘ricequake’ exper-
iments may give insight how to prevent the collapse of
rockfill dams, sinkholes, and ice shelves. Needless to say,
avalanches can be extremely dangerous, also in food sci-
ence such as entrapment in grain storage facilities [670].
Thankfully there are helpful rescue strategies in case of
grain entrapment [671].
Sometimes it is desirable to make granular matter flow
faster. This can often be achieved with granulation [672],
where a powder or small grains are made to clump to-
gether. Perhaps counter-intuitively, these aggregates can
flow more easily. This has many important examples
in the kitchen, such as granulated sugar: Compared to
powdered sugar, granulated sugar has a smaller angle of
repose and considerably better flow characteristics [673]
because the larger grains are less cohesive and more easily
fluidised [674]. Moreover, it is much easier to compactify
granulated materials than powders, which is of vital im-
portance for making tablets in the pharmaceutical and
food industry [675]. Food for thought when you next
sweeten your coffee.
B. Hoppers: grains flowing through an orifice
An important quantity across food science is how
quickly a granular material can pass through an orifice
[679]. Chefs experience this daily when dispensing spices
or grains from a hopper. The flow rate Qis given empir-
FIG. 21. Granular matter. (a) In a pile of grains, the motion
is quantified by the rate of strain ˙εusing spatially-resolved
diffusing wave spectroscopy (DWS). From Deshpande et al.
[667]. (b) Rice sculpture by doctor and food artist Nawa-
porn Pax Piewpun [676]. (c) The velocity field of grains flow-
ing in a 2D hopper is measured by direct particle tracking.
The seven particles that are marked in colour form a jam-
ming arch, shown in the inset. (d) Force chains in a jammed
state, visualised by photoelastic particles that show intensity
fringes under stress, highlighted in the inset. From Tang and
Behringer [677]. (e) The Brazil nut effect, with 8 mm black
glass beads and 15 mm green polypropylene beads. Evolution
in time from left to right. From Breu et al. [678].
ically by the Beverloo law,
Q=C(µ, θ)ρb√g(D−kd)3/2,(72)
where the constant Cdepends on the friction coefficient
µand the hopper angle theta, where ρbis the grain bulk
density, Dis the neck width, dis the grain diameter, and
the parameter k≈ O(1). This scaling relation can be ex-
plained partially by dimensional analysis and hourglass
theory [680], but the foundations of the Beverloo law are
still under active investigation, as discussed recently by
Alonso-Marroquin and Mora [681]. This problem is hard
to solve because the grains behave both like a liquid and
a solid near the opening [682]. This is the result of jam-
ming [683,684], where the effective viscosity increases
dramatically above a critical particle density, so the flow-
ing grains suddenly form a rigid arch or vault [677], as
shown in Fig. 21c,d. This clogging becomes exponentially
unlikely as the opening size is increased, so all hoppers
have a non-zero probability to clog [685]. Clogging is par-
ticularly common in grain hoppers [686] and microfluidic
devices [687], but also sheep herds and pedestrian crowds
moving through a bottleneck [688]. Jamming is equally

38
essential for food processing [689], for biological tissue
development [690] and for food structuring [691].
The non-linear nature of jamming can lead to sur-
prising consequences. For example, the observed flow
rate does not depend on the filling height of the hopper
[Eq. (72)]. This was assumed to be due to the Janssen
effect [§VII A], but Aguirre et al. [656] showed that the
grain flow rate remains constant even if the pressure at
the orifice decreases during discharge, and that different
flow rates can be achieved with the same pressure. More
counterintuitively, inserting an obstacle just above the
outlet of a silo can in fact help with clogging reduction
[692]. Not least, ‘the sands of time run faster near the
end’, which is caused by a self-generated pumping of fluid
through the packing [693]. You may want to take this
into account for your kitchen timer. By varying the soft-
ness of the grains, Tao et al. [694] showed that clogging
occurs more often for stiffer particles, and that clogging
arches are larger for particles with larger frictional inter-
actions. Understanding jamming dynamics can help with
the design of clog-free particle separation devices [695].
C. Brazil nut effect
When you repeatedly shake a box of cereal mix, the
larger (and heavier) grains often rise to the top. This phe-
nomenon was called the ‘Brazil nut effect’ after the sem-
inal work by Rosato et al. [696], also known as granular
segregation [652,697,698]. One explanation is buoyancy,
but the effect can happen even when the larger grains
are denser than the smaller ones. A second contribut-
ing factor is called percolation (not to be confused with
§VII D), where smaller grains fall into the gaps below the
larger grains during shaking. However, a container shape
can be designed where the larger grains move downwards
[699]. This was explained by a third effect called ‘granu-
lar convection’, where the shaking leads to a flow pattern
of grains moving upward along the walls and downward
in the middle of the container [699]. Soon after, granu-
lar convection was imaged directly using Magnetic Reso-
nance Imaging (MRI) [700,701] and extensive computer
simulations [702]. However, the scientific trail was mixed
up again by the discovery of the ‘reverse Brazil nut effect’
by Shinbrot and Muzzio [703], verified experimentally by
Breu et al. [678]. Here larger but lighter particles can sink
in a shaken bed of smaller grains, which cannot be ex-
plained by granular convection alone, nor by percolation
nor buoyancy. Furthermore, M¨obius et al. [704] showed
that particle segregation depends on the interstitial air
between the grains, and depends non-monotonically on
the density. Huerta and Ruiz-Su´arez [705] then found
that there are two distinct regimes: At low vibration fre-
quencies, inertia and convection drive segregation, where
inertia (cf. convection) dominates when the relative den-
sity is greater (cf. less) than one. At high frequencies,
segregation occurs due to buoyancy (or sinkage) because
the granular bed is fluidised and convection is suppressed.
Interestingly, granular convection can occur even in
very densely packed shaken containers, on the brink
of jamming, where unexpected dynamic structures can
arise under geometrical restrictions [706]. Murdoch et al.
[707] studied granular convection in microgravity dur-
ing parabolic flights, revealing that gravity tunes the
frictional particle-particle and particle-wall interactions,
which have been proposed to drive secondary flow struc-
tures. Recently, D’Ortona and Thomas [708] discov-
ered that a self-induced Rayleigh-Taylor instability [see
§III A 4] can occur in segregating granular flows, where
particles continuously mix and separate when flowing
down inclines.
Granular separation is of vital importance in the food
industry. It might be convenient if grains need to be
sorted, but the effect is often undesirable when we re-
quire an even grain mixture. This is especially problem-
atic when the product must be delivered within a narrow
particle-size distribution or with specific compositions of
active ingredients [652]. However, most food storage fa-
cilities (heaps or silos) and processing units (chutes or
rotating tumblers) are prone to grain segregation [709].
Researchers are learning how to prevent separation, but
this is hard because it strongly depends on details of the
flow kinematics. One technique is to add small amounts
of liquid to make the grains more cohesive [710], but at
the cost of reduced food longevity due to rot. Another
strategy is to use modulation of the feeding flow rate
onto heaps [711]. The Brazil nut effect might also be
suppressed using a system with cyclical shearing, where
grains remain mixed or segregate slowly [712]. There
are also new developments in machine vision systems for
food grain quality evaluation [713]. Granular flows are
often hard to image with opaque particles, but power-
ful techniques to measure the 3D dynamics is using MRI
[700], Positron Emission Particle Tracking (PEPT) [714]
or X-ray Computed Tomography (CT) [715].
In the kitchen, the Brazil nut effect may occur too
when stir-frying. Chefs often toss the ingredients into the
air repeatedly, fluidising the grains, to mix them evenly
and to avoid burning them at very high temperatures.
Ko and Hu [716] recently described the physics of tossing
fried rice. There is also an interesting connection with the
physics of popcorn [717]
D. Brewing coffee: porous media flows
Henry Darcy (1803-1858) was a French hydrologist
who studied the drinking water supply system of Dijon,
a city also known for its mustard. In an appendix of
his famous publication [720], he describes experiments on
water flowing through a bed of sand. From the results
he obtained Darcy’s law, an empirical expression for the
average velocity qof the liquid moving through the bed.
It was refined by Muskat [721], and can be written as
q=−K(∇p−ρg)/µ, (73)

39
FIG. 22. Coffee brewing. (a) Schematic of percolation in an
espresso machine basket. A pressure differential pushes water
down through the pore spaces. Image courtesy of Christo-
pher H. Hendon. Inset: Espresso drops by photographer the-
ferdi, licensed under CC BY-NC-SA 2.0. (b) Diagram of a
French press. The water moves up around the coffee grounds
(blue arrows) by applying a constant gravitational force on
the plunger (red arrows). From Wadsworth et al. [718]. (c)
A cappuccino with latte art in the shape of a phoenix. From
Hsu and Chen [719].
where Kis a tensor that describes the permeability of
the material in different directions, sometimes replaced
by a constant kfor isotropic materials. Note that the
local velocity in the pores is u=q/φ, where φ∈[0,1]
is called the porosity. Darcy’s law can indeed be derived
theoretically [722], and it is integral to many industrial
processes in food science, such as sand filtration and an-
timicrobial water treatment [723,724]. Moreover, it can
be applied to a much broader class of porous media flows
[725–727], which are found everywhere in gastronomy:
Examples include percolation in a salad spinner, drizzle
cake, squeezing a sponge, and not least, making coffee
[728,729].
For espresso brewing [see e.g. 730], one can use the first
term in Darcy’s law for pressure-driven flow [Fig. 22a],
while drip coffee [see e.g. 731] can be described using the
second term for gravity-driven flow in Eq. (73). In both
cases, the permeability Kcan be changed by tamping
the grounds or adjusting the grain size, in order to tune
the flow rate and thus the extraction time [732]. Another
particularly well-studied method of making coffee is using
the Italian ‘moka’ pot [see Fig. 1c]. Its sophisticated de-
sign was patented by Alfonso Bialetti (1888-1970) [733].
Gianino [734] used a moka pot to measure the flow rate
through a bed of grains and applied Darcy’s law to mea-
sure its permeability. Later, Navarini et al. [50] improved
this method by accounting for the decreasing permeabil-
ity as aromatic substances dissolve into water. In a third
study, King [735] investigated the effect of packing and
coffee grain temperature on the permeability. Percola-
tion in a moka pot was visualised beautifully by looking
through the metal using neutron imaging [736].
Most studies on coffee extraction showcase some di-
rect applications of Darcy’s law under idealized condi-
tions, but they do not attempt a description beyond the
Darcy regime [737]. For instance, they ignore the initial
stage of percolation, when the water invades the (initially
dry) coffee grain matrix. This process can be described
with percolation theory [738], where the pores between
the coffee particles can be considered as a random net-
work of microscopic flow channels [727]. If the coffee is
coarsely ground or tamped lightly, the many open mi-
crochannels (bonds) allow the water to find a connecting
path through the coffee. Then this process is first char-
acterised by capillary wetting [739] [§II G], after which
Darcy’s law becomes applicable. For intermediate tamp-
ing, as the permeability Kdecreases, we approach the
percolation threshold. Then the extraction time will sig-
nificantly increase, which can result in over-extraction
[740]. Finally, if the coffee is tamped too strongly or too
finely ground, the water cannot find a path between the
grounds. Then, either there is nothing to drink, or the
pressure builds up until we see hydraulic fracturing [741].
In that case, large flow channels suddenly crack open that
bypass the microchannels [742]. This ‘fracking’ can give
the coffee a bad taste because it leads to an uneven ex-
traction. Symptoms that your espresso is fractured are
liquid spraying through the bottom of the basket, irregu-
lar flow, and a cracked coffee cake. Percolation theory has
many other applications, including predictions for forest
fires, disease spreading, and communication in biology
[743,744].
Coffee can be made in many ways, each involving dif-
ferent fluid mechanics. Here we only mention a few
preparation methods, with some recent results: To make
an espresso, it is important to note that coffee beans are
often prone to variations in quality. Even if the theory
is perfect, the ingredients are not. To overcome this,
Cameron et al. [745] offer advice to systematically im-
prove espresso brewing by proposing a set of guidelines
towards a uniform extraction yield. Interestingly, they
also found that the smallest grains do not give the high-
est yield as they tend to clump together and form aggre-
gates [see granulation in §VII A]. For drip brew coffee it
is common to use a precise temperature-controlled ket-
tle, but Batali et al. [746] surprisingly found that the
brew temperature may not have quite so much impact
on the sensory profile at fixed brew strength and extrac-
tion. Considering the French press, Wadsworth et al.
[718] recently determined the force required to operate
the plunger [Fig. 22b]. They recommend using a max-
imum force of 32 N to complete the pressing action in
50 s, using 54 g of coffee grounds for 1 l of boiling water.
Looking at cold brew coffee, Cordoba et al. [747] recently
evaluated the extraction time and flavour characteristics,
and Rao and Fuller [748] investigated its acidity and an-

40
tioxidant activity. Finally, Greek and Turkish coffee rely
on the sedimentation of fine particles, which we discuss
in §VI C.
After having made the perfect cup, it can be deco-
rated with latte art [Fig. 22c]. Indeed, Hsu and Chen
[719] showed that coffee tastes sweeter with latte art,
which they related to brainwave activity using electroen-
cephalography (EEG). The fluid mechanics of pouring
steamed milk foam into the denser coffee can be described
as an inverted fountain [see §III A], which depends on
the jet diameter and the pouring height via the Froude
number [Eq. (19)]. Large fountains lead to more mixing
and brown foam, while gentle pouring gives white foam.
After much practice, these colours can be combined in
rapid succession to make exquisite patterns, including a
heart and the phoenix [749]. Some people prefer their
coffee without milk, but with a thin layer of espresso
crema [750]. Undesirably, this coffee foam can agglom-
erate along the perimeter. This effect can often be sup-
pressed by heating the cup beforehand since it is caused
by B´enard-Maragoni convection, which we discuss in the
next section.
E. Coffee ring effect
When the last sip of coffee or wine is left overnight,
it dries out creating a stain with a brighter interior and
much darker borders, where most residues are deposited,
see Fig. 23. The coffee ring effect, as it has been termed,
can be observed in almost any kitchen mixture contain-
ing small particles. As explained by Deegan et al. [754],
the coffee ring effect results from the drying dynamics
of a droplet, combined with the pinning of its contact
line with the substrate [755,756]. As the solvent evap-
orates, the outflow of matter decreases the thickness of
the droplet at every point. If the contact line was not
pinned, the droplet would shrink. This additional con-
straint, together with the surface tension requirement to
keep the contact angle fixed [see §II G], induces an out-
ward flow from the interior to replenish the evaporated
liquid at the borders. This flow transports the sediment,
which is then deposited at the outer ring, leaving the
lower-concentration interior.
Instead of a solid particle suspension, the coffee ring
effect also emerges if the dissolved component is another
liquid. In that case, the edge of the puddle remaining af-
ter a volatile solvent evaporates forms either “fingers” or
spherical “pearls”, or some combination of the two [757].
Understanding the underpinning dynamics of coffee ring
formations remain an active research topic. An example
of the many excellent publications in recent years is the
study by Moore et al. [758] on the effects of diffusion of
solute from the pinned contact line to the bulk of the
drop on the pattern formation.
Interestingly, by adding some alcohol to the drop to
make it more volatile, the coffee ring effect can be sup-
pressed by consequential Marangoni flows from the con-
FIG. 23. Coffee ring effect. (a) Schematic of an evaporating
droplet containing a suspension of microparticles. Stronger
evaporation near the contact line drives an internal flow to
the outer edge. From Jafari Kang et al. [751]. (b) A dark
rim is formed by particles that accumulate at the pinned con-
tact line. From Li et al. [752]. (c) Example of a heart-shaped
coffee ring. Image by Robert Couse-Baker, licenced under
CC BY 2.0. (d) Suppressing the coffee ring effect by adding
cellulose nanofibers (CNF) to a drop of 0.1 wt% colloidal par-
ticles. From left to right: CNF concentration of 0, 0.01, 0.1
wt%. From Ooi et al. [753].
tact line to the drop’s interior [759]. These flows are
induced by a surface tension gradient along the surface
of the drop [see §III B], which is in turn caused by nonuni-
form cooling as the droplet evaporates, because the sur-
face tension increases as the temperature decreases. This
is referred to as B´enard-Marangoni convection. The de-
position pattern then depends on the strength of the rel-
ative magnitude of thermocapillary stresses to viscous
ones, as expressed by the dimensionless Marangoni num-
ber, Ma, as defined in Eq. (23) with ∆γ= (∂ γ/∂T )∆T.
For large Ma, the particles end up in the center, for in-
termediate Ma, they are deposited evenly along the sub-
strate, while for small Ma, the coffee ring ring effect
is fully recovered. Therefore, by tuning Ma, for exam-
ple through tuning the alcohol percentage in the coffee,
it is in principle possible to control the deposition pat-
tern. Moreover, the coffee-ring effect can be suppressed
by shape-dependent capillary interactions [760,761].
The applications of the coffee-ring effect and the dry-
ing of thin colloidal films in general go far beyond kitchen
experiments [756,762]. For example, it is the basis of

41
Controlled Evaporative Self-Assembly (CESA), which is
used to create functional surfaces with controllable fea-
tures [763]. The coffee-ring effect can also be used for
controlled inkjetting of a conductive pattern of silver
nanoparticles [764], and particle deposition on surfaces
could be controlled further by light-directed patterning
by evaporative optical Marangoni assembly [765]. More-
over, the effect can be used for nanochromatography to
separate particles such as proteins, micro-organisms, and
mammalian cells with a separation resolution on the or-
der of 100 nm [766]. Interestingly, the growth dynamics
of coffee rings are altered when active particles such as
motile bacteria move around in the evaporating droplet
[767–770], and bacterial suspensions can feature acive de-
pinning dynamics. Hence, in the spirit of frugal science,
it might be possible to exploit the coffee ring effect to
detect antimicrobial resistance [771].
VIII. TEMPEST IN A TEACUP: NON-LINEAR
FLOWS, TURBULENCE AND MIXING
What is turbulence? This intriguing question has fas-
cinated fluid mechanicians throughout history. Leonardo
da Vinci already eluded to two important properties of
turbulence [772]: The generation at large scales, and
the destruction due to viscosity at the smallest scales.
The many scales of motion in turbulence is arguably its
main signature; for example, a volcanic plume spans over
several kilometers, with eddies all the way down to the
Kolmogorov microscales [773]. As the turbulent struc-
tures break up, energy is transferred from large whirls to
smaller ones. The poem by Richardson [774] beautifully
describes this energy cascade:
Big whirls have little whirls
That heed on their velocity,
And little whirls have littler whirls
And so on to viscosity.
The consequences of turbulence are numerous in our lives
of gluttony. It gives us the characteristic sound of a kettle
whistle, it helps with mixing milk into our tea, and it
gives us some frictional losses when biking home from
the restaurant. In this section, we will catch a whiff of
turbulence in the kitchen.
A. Tea leaf paradox
Before diving into chaotic realms, we consider the sur-
prising effects that the non-linearity of the Navier-Stokes
equations [Eq. (2)] can bear in laminar flow. One such
surprises is the “tea leaf paradox”. Biological tissues tend
to be denser than water [775], thus soaked tea leaves will
sink to the bottom of a cup. When the water is stirred
around in circles, the leaves are expected to move towards
the edge of the cup because of centrifugal action. The op-
posite happens, however: The leaves always migrate to
FIG. 24. Tea leaf effect due to secondary flows. (a) Illus-
tration. After stirring the tea, the rotating liquid (primary
flow) slowly comes to a halt because of friction with the walls
(spin-down). This friction also induces a toroidal recirculation
(secondary flow) directed outwards at the top and inwards at
the bottom, which causes the leaves to collect in the center.
(b) Spiral dye streaks due to secondary flows in a cake pan.
Instead of slowing down, the liquid is rotated increasingly
faster (spin-up), so the secondary flow is reversed, such that
the dye spirals outwards at the bottom of the cake pan. From
Heavers and Dapp [779].
the center of the cup, as seen in Fig. 24a. Thomson [776]
first recognised that the solution of this paradox stems
from ‘friction on the bottom’. Later, Einstein [777] gave
a detailed description of the tea leaf experiment itself, in
order to explain the erosion of riverbanks. A d etailed
theoretical treatment was provided later by Greenspan
and Howard [778].
The paradox is resolved with fluid mechanics, as fol-
lows: As the liquid rotates in the cup, the first approx-
imation of the fluid flow is just a solid-body rotation.
Specifically, we have u=Ω×rwith a uniform an-
gular velocity Ω. On the fluid acts a centrifugal force,
Fc∝Ω×(Ω×r). If Ω is constant in space, then this
force does not modify the flow. However, frictional drag
with the cup walls slows the fluid down in the boundary
layer. In particular, near the bottom surface the angu-
lar velocity Ω(r) and thus the centrifugal force will be
less than near the top water-air interface. Consequently,
an in-up-out-down recirculation emerges [Fig. 24a] as the
liquid slowly stops spinning. Interestingly, this recircu-
lation can also be reversed [Fig. 24b], when the liquid is
rotated increasingly faster (spin-up) from rest [778,780].
The tea leaf effect has many applications. In the
kitchen, it can be applied conveniently when poaching
eggs [779,781]: Before cracking the egg into the pot, the
hot water can be gyred to keep the egg whites together
at the center of the pot. But be quick, because the flow
ceases after a time
τE∼rH2
νΩ,(74)
called the Ekman time [778,782], in terms of the kine-
matic viscosity νand His the height of the liquid layer,
the depth. The same technique is also used to separate
out trub during beer brewing [783], and to separate blood

42
cells from plasma in microfluidics [784]. Also in geophys-
ical flows, the same in-up-out-down circulation is seen
in tornadoes [785]. In modern additive manufacturing
(AM) technologies, the Ekman time sets a limit to how
quickly objects can be 3D printed [786].
B. Secondary flows
When discussing the tea leaf paradox, we saw that fric-
tional drag with the surfaces induces a flow structure on
top of the initial rigid body motion. More generally, it is a
very powerful concept to understand fluid flows in terms
of a ‘primary flow’, guessed from simplified or basic phys-
ical principles, and a ‘secondary flow’, a correction due
to high-order effects such as obstacles in the main flow.
Care must be taken with such superpositions, since the
Navier-Stokes equations are non-linear, but often pertur-
bation methods can be followed [787].
Another classic example of a secondary flow is the
emergence of Dean vortices in a curved pipe [788]. Start-
ing from a straight Poiseuille flow [see §II C], the vortices
can be explained using a perturbation method account-
ing for centrifugal forces [789,790]. The relative strength
of the secondary flow is determined by the balance be-
tween inertial and centrifugal forces with respect to vis-
cous forces, which is given by the Dean number,
De = ReqRp/Rc,(75)
where Re is the Reynolds number [Eq. (5)], Rpis the
radius of the pipe, and Rcits radius of curvature. For
small Dean numbers the flow is unidirectional, for inter-
mediate values Dean vortices emerge, and for large De
the flow turns turbulent [791]. This is directly appli-
cable to separating particles by size using inertia [792].
These vortices also emerge naturally in straight channels
when two stratified fluid layers such as air and water flow
through them [793], which is associated with huge pres-
sure losses in (food) industrial pipelines. On the flip side,
Dean vortices may be used along with UV-C to inactivate
microoganisms in fruit juices [794].
Similar calculations can be performed to study sec-
ondary flows in turbomachinery compressors and tur-
bines [795], and oceanic and atmospheric currents with
Ekman layers [782,796,797]. Ekman layers are associ-
ated with transport of biomaterials in the ocean through
so-called Ekman transport processes. The secondary flow
pattern of Ekman transport can lead to upwelling and
downwelling of algae. A thorough understanding of the
underpinning mechanisms is crucial to mitigate the dev-
astating implications of harmful algal blooms. Secondary
flows can also contribute to bridge scour [798], by the re-
moval of sediment such as sand and gravel from around
bridge abutments or piers, leading to one of the major
causes of bridge failure around the world.
C. Turbulent jets emanating from tea kettles
When the water in the tea kettle boils, a turbulent jet
of steam emerges from the spout with a conical profile
[Fig. 25a]. To describe the dynamics of a turbulent jet, it
is useful to decompose the velocities into an average and
a fluctuating component. This averaging procedure is
named after its inventor, Osborne Reynolds (1842-1912),
and is written as ui= ¯ui+u0
iwith i={x, y}for the ve-
locity components in two dimensions [799]. By Reynolds
averaging we arrive at the famous equation for the con-
servation of momentum in turbulent flow,
∂¯ui
∂t + ¯uj
∂¯ui
∂xj
+u0
j
∂u0
i
∂xj
=−1
ρ
∂¯p
∂xi
+ν∂2¯ui
∂x2
j
,(76)
using Einstein notation. The third term is an appar-
ent stress due to turbulent fluctuations, and the remain-
ing ones are the averaged transport terms in the Navier-
Stokes equation [see §II B], where the last (viscous stress)
term can be neglected in inertia-dominated flows. Inter-
estingly, the flux of momentum remains constant beyond
a certain distance from the spout [560]. To maintain its
momentum, the jet must continually entrain ambient air.
This is why blowing on a finger burnt by a hot kettle has
a cooling effect.
Moreover, fluid jets tend to follow a convex surface,
rather than being scattered off, which is called the
Coanda effect [800]. This is sometimes demonstrated by
extinguishing a candle by blowing around a tin can. Simi-
larly, when a steam kettle jet curves around another pot,
it can pose an unexpected safety hazard. An interest-
ing application is robotic food processing using Coanda
grippers [801]. The Coanda effect should not be confused
with the teapot effect [802–804], where a liquid follows
a curved surface like a teapot spout [Fig. 25b], because
this flow is dominated by surface tension and wetting
[§II G]. While the teapot effect can cause a mess when
pouring too slowly, it can be advantageous for coating or
making complex shapes [805], perhaps in novel culinary
decorations.
Other examples of turbulent jets include the stripes
produced by air crafts. According to one study [806],
this warms the planet even more than the carbon emitted
by the jet engines, but fortunately it seems these effects
can be mitigated by avoiding certain altitudes [807]. In
any event, when flying it is best to steer clear off plumes,
such as those emanating from smoke stacks or volcanoes.
As part of a safety assessment, we can use the predic-
tions by Taylor [808] for the shape and final height of
such plumes. Taylor’s theory is valid across many length
scales: It could equally be used to estimate the shape of
a plume rising from a cup of coffee [Fig. 18].
D. Sound generation by kitchen flows
The tea kettle we just discussed can make a pleasant
whistle sound [809,810]. As described by Lord Rayleigh

43
FIG. 25. Tea time. (a) A turbulent jet emanating from a tea
kettle. Image courtesy of Gary S. Settles. (b) The teapot
effect, showing a liquid stream following the curved surface
of the spout (blue arrow). Pouring any slower will make the
liquid stick to the pot entirely. From Scheichl et al. [804]. (c)
Diagram of a tea kettle whistle. The steam passes through
two orifice plates from left to right. From Henrywood and
Agarwal [809].
[811], sound is carried by pressure waves that propagate
according to the wave equation,
∂2p
∂x2−1
c2
∂2p
∂t = 0,(77)
where the speed sound c∼343 m/s in air. Sir Isaac New-
ton (1642-1726) was the first known to measure the speed
of sound, as reported in his book on classical mechanics
Principia [812]. Since then many creative attempts to
measure this quantity have been reported, including the
accurate experiments by the Reverend William Derham
(1657-1735) involving a telescope and gunshots [813]. Ac-
cording to Lord Rayleigh [814], the parameters determin-
ing the sound generation of a whistle are a characteristic
length scale, L0, the frequency, f0, the fluid (steam) vis-
cosity, ν, and the steam jet velocity, U0. They form two
dimensionless groups, namely the Strouhal number,
St = Time scale of background flow
Time scale of oscillating flow =f0L0
U0
,(78)
and the Reynolds number [Eq. (5)]. Henrywood and
Agarwal [809] found that for a typical tea kettle with two
orifice plates [Fig. 25c], the frequency giving rise to the
sound generation is sensitive to the jet diameter, δ, and
not the plate separation distance, so L0is replaced with
δ. Furthermore, the same authors found that the whis-
tle’s behaviour is divided into two regions: For Reδ.
2000, the whistle operates like a Helmholtz resonator,
with an approximately constant frequency (pitch). How-
ever, above a critical Reynolds number, Reδ&2000, the
whistle’s tone is determined by vortex shedding [815],
with a frequency that increases with U0at an approxi-
mately constant Stδ≈0.2. Vortex shedding also occurs
for cables or tall buildings in strong wind, which can
be destructive if the aerodynamic driving frequency res-
onates with the structural eigenmodes [816]. To prevent
damage from happening, newer buildings are designed to
have several eigenfrequencies to effectively dissipate the
energy, or to have roughness elements, as perfected by the
glass sponge Euplectella aspergillum [817]. The sound of
the tea kettle whistle might inspire you to whistle for
yourself while stirring your tea [§VIII A]. The physiol-
ogy of mouth whistling was discussed by Wilson et al.
[818], Azola et al. [819], Shadle [820].
Plink. plink. plink. Another kitchen sound is the mad-
dening noise of a leaky tap [821,822]. The paradox how a
single drop impacting on a liquid surface can be so loud,
compared to a more energetic continuous stream, is still
not fully understood. Franz [823] already discussed that
the droplet can entrain air bubbles, which oscillate to
make sound at a frequency f= (1/2πa)p3γp0/ρ given
by Minnaert [824], where ais the bubble radius, γis
the ratio of specific heats of air, and p0is the pressure
outside the bubble. However, not every drop makes a
sound. Longuet-Higgins [825] and Oguz and Prosperetti
[826] developed the first detailed analytical models to ex-
plain this in terms of the Froude number [Eq. (13)] and
the Weber number [Eq. (14)] given the droplet radius and
impact velocity. The sound volume is set by the wave am-
plitude, but how does sound generated underwater cross
the water-air interface? Prosperetti and Oguz [827] re-
viewed the underwater noise of rain, and Leighton [828]
discusses whether goldfish can hear their owners talking.
Looking at the dripping tap, Phillips et al. [829] tested
previous theories by comparing sound recordings with
direct high-speed camera imaging. They write that the
airborne sound field is not simply the underwater field
propagating through the water-air interface, but that the
oscillating bubble induces oscillations of the water sur-
face itself, which could explain the surprisingly strong
airborne sound. Plink.
It is impossible to cook without making noise, often to
the extend of breaking the sound barrier. Indeed, we al-
ready mentioned the supersonic ‘pop’ made by cracking
open a Champagne bottle [§III E]. Similarly, supersonic
shock waves can be generated by snapping a tea towel
[830,831]. Dropping an object in a filled kitchen sink
can also create a supersonic air jet [832]. By investigat-
ing the popping sound of a bursting soap bubble, Bus-
sonni`ere et al. [833] found a way to acoustically measure
the forces that drive fast capillary flows. Not least, nu-
merous situations in food science involve hydrodynamic

44
cavitation [834,835], which can produce flashes of light
called sonoluminescence [836,837] with internal temper-
atures reaching thousands of degrees Kelvin [838].
The hot chocolate effect [839] occurs when heating a
cup of cold milk in the microwave, mixing in cocoa pow-
der, and tapping the bottom with a spoon: The sound
pitch initially descends by nearly three octaves, compara-
ble to the vocal range of an operatic soprano, after which
the pitch gradually rises again. This happens because air
is less soluble in hot liquids, so it becomes supersaturated
with heating, and adding a fine powder provides nucle-
ation sources for fine bubbles. Air is more compressible
than water, which lowers the speed of sound, and thus
the pitch. The same musical scales are heard when open-
ing a fresh beer [840], which is supersaturated with CO2
[§III E]. The hot chocolate effect was visualised directly
by Tr´avn´ıˇcek et al. [841].
E. Making macarons: chaotic advection
A milk droplet with diffusivity D∼10−9m2/s takes a
long time to mix in a cup of coffee in the absence of fluid
motion, typically τ=L2
0/D, so days, and slower than
the diffusion of heat [§V D]. However, stirring reduces the
mixing time dramatically, down to seconds, as turbulent
eddies stretch the drop into thin filaments so diffusion
can act efficiently [842]. Moreover, hydrodynamic insta-
bilities [§III A] can lead to turbulence, which enhances
the mixing rate by maximizing the exposed surface area
and the concentration gradient between adjacent fluids
[199]. However, turbulence does not occur in viscous flu-
ids at low Reynolds numbers [§II D]. Here, mixing can be
achieved by stretching and folding lamelles of one fluid
into another through ‘chaotic advection’ [843–845].
A culinary example of chaotic advection is making
macarons, where a viscous batter must be mixed gen-
tly to maintain its foam structure [846]. The choice of
stirring protocol has a dramatic impact on the mixing
rate: If we move a rod back and forth, we see no mix-
ing at all because of the scallop theorem [548]. Thus,
we must break time-reversal symmetry, so we change our
strategy and stir in circular patterns instead. Now the
fluid does mix, but slowly, because it is stretched only
linearly. Next, we stir in figure-of-eight patterns [847]
[see Fig. 26a], which speeds up the mixing rate dramati-
cally as this strategy yields exponential stretching [848].
This can also be achieved by rotating two rods, at the
same speed but in opposite directions, as in commercial
egg beaters [849].
A particularly efficient mixing protocol is the ‘blink-
ing vortex’ [843], where two rotors alternately spin in the
same direction. For the first half period, the first rod ro-
tates while the other one is stationary, then vice versa. A
task that could perhaps be performed by a cooking robot
[850]. This canonical and time-periodic blinking vortex
is used ubiquitously in chaotic mixing theory to com-
pare the effectiveness of different mixing protocols [851].
FIG. 26. Chaotic stirring protocols can be used for a broad
range of applications, from making macarons to cryptography.
(a) Experiment (left) and simulation (right) of the figure-of-
eight stirring protocol used to mix a blob of dye in sugar
syrup. From Thiffeault et al. [847]. (b) Chaotic advection
used to create a digital message hashing function. Image cour-
tesy of William Gilpin.
Similarly, the Arnold-Beltrami-Childress (ABC) flow is
often considered the archetypal flow for many studies on
chaotic advection in 3D [845]. Besides expedient cook-
ing, chaotic advection has numerous applications in other
disciplines [852]. For example, Gilpin [853] used a blink-
ing vortex model to create digital hash functions with
potential applications in cryptography [Fig. 26b].
By adding another pair of counter-rotating rods, for
example by using two eggbeaters, one obtains the famous
‘four-roll mill’. This concept was invented by Taylor [854]
to study the formation of emulsions [§IV D]. Oil drops
immersed in golden syrup (ideal for baking) were placed
at the center of the mill, in a stagnation-point flow, which
elongates them. The drops split when the viscous stresses
on their surface exceed the stabilising surface tension, as
described by the capillary number,
Ca ≡viscous stresses
surface tension =µU0
γ,(79)
where the characteristic velocity, U0= ˙γL0, can be writ-
ten as the local shear rate times the droplet size. The
four-roll mill laid the basis for studies of droplet break-
up and stability, but fluctuating stagnation points made
practical implementation difficult. To alleviate this is-
sue, Bentley and Leal [855] implemented an image-based
feedback loop that controlled the speed of each roller
independently. Using their invention, the same group
[856] validated theoretical limits for drop deformation
[857,858] and paired their experiments with theory for

45
studying drop dynamics [859]. The flow fields generated
by the automated four-roll mill also pioneered polymer
elongational rheometry [860,861]. More recently, Hud-
son et al. [862] introduced the microfluidic analogue of
the four-roll mill. This device been used extensively to
characterise the material properties of biomaterials and
single cells by extensional rheometry [863], and in ap-
plications of stagnation point flows in microfluidics have
been extended to include substrate patterning [864–866]
and the trapping of cells by hydrodynamic confinements,
allowing new developments in analytical chemistry and
in life sciences [867].
F. Sweetening tea with honey: mixing at low Re
and high Pe
Returning to the mixing of two liquids, we now make
a cup of tea sweetened with a drop of honey. By pure
dissolution, a viscous drop mixes slowly with the tea,
but stirring can help us again. However, since the drop
is very viscous, turbulent eddies cannot stretch the drop
into thin filaments, as was the case with much less viscous
milk drops [see §VIII E]. Instead, the sharp flow velocity
gradients around the drop increase the mass transfer by
maintaining a correspondingly sharp concentration gra-
dient [868].
We seek an estimate of the mixing time. We con-
sider a honey drop of size L0∼1 mm and diffusivity
D∼10−10 m2/s in water [869]. We also assume a very
viscous honey drop, so the mass transport is dominated
by advection due to large Schmidt numbers [Eq. (51)].
Using a stirring speed of U0∼1 mm/s, the P´eclet num-
ber [Eq. (48)] is large, Pe ∼104, but the flow close to the
drop is still laminar at an intermediate Reynolds num-
ber, Re .1. Then, as the drop dissolves, a diffusion layer
develops between the pure phases. Acrivos and Goddard
[870] showed that, in the low Reynolds number and high
P´eclet number limit, this diffusion layer has thickness
δ∼L0Pe−1/3.(80)
In our case, at high Pe, the boundary layer is rather thin,
L0/δ ∼20. By substituting δfor L0in the expression for
the diffusion time, τD∼L2
0/D, following Mossige et al.
[871], we obtain a typical mixing time
τmix ∼(L2
0/D)Pe−2/3.(81)
By inspecting this expression, we can immediately appre-
ciate the dramatic effect of fluid flow: It can reduce the
mixing time by a factor of a thousand or more. Putting
in the numbers, it takes ∼22s to stir the viscous honey
droplets (or sugar grains) into our tea. This approxima-
tion can be improved by accounting for open streamlines
and inertial effects [872,873].
Instead of stirring, we can also let the honey drop sed-
iment down. If it is sufficiently small, it will have be
spherical and sediment at low Reynolds number [§VI C].
FIG. 27. Waves and splashes. (a) Protecting a ship by calm-
ing the waves with oil. The Dutch fisheman Isak Kalisvaar
reported to have conducted this experiment, in a letter to
Frans van Lelyveld in 1776, after his ship got into a violent
storm. From Mertens [876]. (b) Diagram of capillary wave
dampening by surfactants. (c) Representative image of coffee
spilling. From Mayer and Krechetnikov [877]. (d) Schematic
of sloshing dynamics in an oscillated container. From Sauret
et al. [878].
When we substitute the terminal velocity U∞[Eq. (62)]
for U0in Eq. (81), we obtain a characteristic time scale
of mixing for the sinking drop,
τmix, sink ∼ µ2
(∆ρg)2D!1/3
.(82)
This timescale also applies to the inverted system of a
water drop rising in another viscous, miscible liquid like
corn syrup [871].
IX. WASHING THE DISHES: INTERFACIAL
FLOWS
Even the best restaurant cannot perform without its
dish washers. This job is somewhat strenuous and bor-
ing, but much alleviated by the mesmerizing colors of
soap bubbles and the startling wave dynamics. Fun, you
might think, but interfacial phenomena have led to ex-
ceptional scientific discoveries ranging from cell biology
to nanotechnology [see e.g. 874,875]. In this penulti-
mate section, we will pop the bubble of some old miscon-
ceptions and catch the wave of the latest developments
concerning interfacial flows.
A. Greasy galleys smooth the waves
Benjamin Franklin (1706-1790) noticed a remarkable
phenomenon during one of his journeys at sea, sailing
in a fleet of 96 ships. “I observed the wakes of two of
the ships to be remarkably smooth, while all the others

46
were ruffled by the wind, which blew fresh” [879]. Being
puzzled with the differing appearance, Franklin at last
pointed it out to the captain, and asked him the mean-
ing of it. The captain’s answer may come as a surprise:
“The cooks, says he, have, I suppose, been just emptying
their greasy water.” The calming effect of oil on water
was common knowledge to seamen at the time, and had
indeed been described since ancient Greeks. However,
legends circulated about a ship that miraculously sur-
vived a storm by taming it with olive oil [Fig. 27a], so
Franklin decided to initiate a series of systematic exper-
iments [879]. The amusing details of these stories, and
the scientific interest that emerged since, are described
eloquently by Mertens [876] and Tanford [880].
While the dampening of surface waves was known
for millennia, its precise cause was a mystery until re-
cently, as described by Henderson and Miles [881], Nico-
las and Vega [882], Behroozi et al. [883], Kidambi [884]
and references therein. Franklin thought that the oil film
stopped the wind from catching the water, but more than
a century passed before more progress was made. In
her kitchen, Agnes Pockels performed pioneering experi-
ments on the surface tension of oil films [see §IX C]. We
now know that this surface tension increases when the
oil film is stretched thin, for example by the wind. Be-
cause of the Marangoni effect [see §III B], the resulting
gradients in surface tension then induce flows that oppose
the film deformation, thus dampening the surface waves
[Fig. 27b]. This interfacial restoring force is referred to
as the Gibbs surface elasticity, or the Marangoni elastic-
ity [885], which is a multiphase flow effect that occurs
in many other applications, as reviewed extensively by
Brennen [164].
B. Splashing and sloshing
No culinary achievement happens without a little mess
left behind, be it an accidental spill, or the usual drop
of wine from the cook’s glass on the kitchen table [see
§VII E]. The question of sloshing, why liquids spill out
of a container under acceleration, has received prior at-
tention in the context of space vehicles and ballistics:
Depending on the size of the container, and the type of
agitation, large-scale oscillations of the encased fluid can
be enhanced to the point of spilling [886,887]. In the
academic context, it is known to everyone trying to walk
to seminars with their coffee cup [Fig. 27c]. It turns out
that spilling results from a combination of excess acceler-
ation for a given coffee level when we start walking, and a
complex enhancement of vibrations present in the range
of common coffee cups sizes [877]. With some relief came
the realisation that beer does not slosh so easily, since the
presence of even a few layers of foam bubbles on the free
surface introduces strong damping of surface oscillations
[878].
We generally want to avoid or control splashing or
spreading, especially when mixing and pouring liquids.
The impact and breakdown of droplets on a solid or liq-
uid surface is mainly controlled by the Reynolds number
[Eq. (5)] and the Weber number [Eq. (14)]. Another im-
portant factor determining the splashing behaviour is the
type of substrate, which regulates the contact angle dy-
namics of impinging droplets [888]. The elasticity of the
substrate also plays an important role [620], as soft solids
noticeably reduce or eliminate splashing. Estimates and
experiments show that the droplet kinetic energy needed
to splash on a very soft substrate can be almost twice
as large as in a rigid case [889]. Droplet spreading and
recoil can result in a number of complex fluid dynamics
phenomena, when the elongating and stretching drops
form jets and sheets which further destabilise into smaller
droplets via the Rayleigh-Plateau instability [§II H]. The
possible outcomes of a collision of a droplet with a solid
substrate involve deposition, a fervent splash, so-called
corona splash in which the liquid forms a circular layer
which detaches from the wall, and retraction in which the
droplet can de-stabilise and break up or rebound (par-
tially or entirely) [537,539,890]. The process is con-
trolled by the wettability of the surface, the parameters
of the droplet, and its impact speed.
Before a stream separates into impacting droplets, liq-
uid jets are frequently seen and used in the kitchen [106].
When plating a gourmet meal, the way sauces are spread
on a plate is carefully engineered to achieve a variety of
shapes and textures. The same questions appear when
glazing a cake, where various edible jets and streams are
placed on surfaces in a skillful way that manages buck-
ling instabilities in such elongated filaments to produce
the desired visual effect. In art, the understanding of
hydrodynamics was crucial to Jackson Pollock, for one,
who used a stick to drizzle paint on his canvas in a va-
riety of ways [99]. The complex fluid dynamics behind
different painting effects has only recently been analysed
and reviewed by Herczy´nski et al. [891] and Zenit [892].
C. Dishwashing and soap film dynamics
The interference patterns on soap bubbles have fasci-
nated physicists for centuries [894], which has resulted
in pioneering discoveries in optics, statistical mechanics,
and in fluid mechanics by Newton [895], Plateau [896],
and De Gennes et al. [84]. An even more remarkable
story is how the self-taught chemist Agnes Pockels (1862-
1935) was inspired to study surface tension while doing
the dishes. Women were not allowed to enter universi-
ties, so she did not have a scientific training and could
not publish her work in scientific journals [897]. Ten years
after her first experiments, she was encouraged to write
a letter explaining her findings to Lord Rayleigh, who
then forwarded it to Nature [898]. Along with her subse-
quent papers [899–902], all in top-level journals, she con-
tributed to establishing the field of surface science. With-
out formal training and without access to a lab, Pockels
also used simple kitchen tools to develop the precursor

47
FIG. 28. Thin-film interferograms showing the evolution of
surfactant driven flows in soap bubbles. Top: When the soap
bubble is brought through an air-liquid interface in one single
step, an unstable dimple forms at the apex. The dimple is
quickly washed away by surfactant plumes rising from the
periphery. Bottom: When the bubble is instead elevated in
multiple, small steps, controllable Marangoni instabilities can
be utilized to stabilise the bubble and to prolong its life span.
From Bhamla and Fuller [893].
to the now widely celebrated Langmuir trough, which is
now used widely to measure the surface pressure of soap
molecules and other surfactants upon compression [903].
Soap bubbles are comprised of a thin aqueous film that
is sandwiched between two surfactant layer, where each
color corresponds to a different film thickness. This film
starts to drain immediately due to gravity [893]. In turn,
this drainage causes a small deficit in soap concentration
at the bubble apex and the formation of a small dimple
[Fig. 28, top left panel]. This gradient in surfactant con-
centration sets up a Marangoni flow towards the apex.
By replenishing interfacial material, these flows stabilise
the bubble against rupture. However, these Marangoni
flows are short-lived and are quickly destroyed by chaotic
flows which do not stabilise the bubble [Fig. 28, top right
panel]. A simple trick can solve this issue. Bhamla et al.
[904] showed that by elevating the soap bubble in multi-
ple small steps through a soap solution (instead of in one
huge step), it is possible to induce a cascade of Marangoni
instabilities. Each Marangoni instability arrests the pre-
vious one, and this prevents chaotic flows from develop-
ing. This method produces beautiful flow patterns, as
displayed in the bottom panel in Fig. 28.
The rate of draining depends on the viscosity of the
soap film: Adding glycerol, a natural ingredient in soap,
effectively extends the life span of a bubble. Adding corn
syrup or honey does the same job, but it might not help
to clean your dishes. However, it will help to make giant
soap bubbles. By retarding film drainage and by reduc-
ing the evaporation rate, a bubble stabilized by viscosity
has sufficient time to grow before it eventually pops. An-
other approach was taken by Frazier et al. [905]. They
appreciated the central role of viscoelasticity in stabi-
lizing thin liquid films, and utilized polyethylene glycol
(PEG), a long-chained polymer commonly found in hand
sanitizers, to create bubbles with surface areas close to
100 m2, the area of a badminton field.
To conclude this section, we note that modern video
games use computer-generated imagery (CGI) techniques
coupled to mathematical models that vary the soap film
thickness to render bubbles realistically [906]. Moreover,
the nature of the boundary between water and oil is cru-
cial to many nanometre-scale assembly processes, includ-
ing biological functions such as protein folding and liquid-
liquid phase separation [907,908].
D. Ripples and waves
Whenever an interface between two fluids is disturbed,
ripples and waves emerge and propagate along the sur-
face [also see §III A]. When a group of waves moves across
a pond, we see waves of different wave lengths λprop-
agating at different speeds and, importantly, groups of
waves travelling at different speeds than the crests and
troughs of individual sinusoidal perturbations. The rea-
son for this is the dispersity of water waves, that is the
dependence of the wave propagation speed on the wave-
length, with longer waves generally travelling faster. For
a wave of frequency ω, the relationship between the wave
speed cand the wavenumber kis c=ω/k, where the de-
pendence of the frequency ω=ω(k) on the wavenumber
is called the dispersion relation. In a wave packet, where
each crests travels at the speed c, the velocity of travel
of the whole group is cg= dω/dkand is called the group
velocity.
On deep water, where the dispersion relation reads
ω2=gk, the wave speed c=pg/k is twice as large
as the group propagation speed, which is the reason why
in a travelling wave packet individual crests will seem to
continuously appear at the back of the packet, propa-
gate through it towards the front, and eventually vanish
there. Deep water here means that the depth of the layer
is much larger than the wavelength, λ= 2π/k, in which
case the dispersion relation above is obtained from the as-
sumption of a potential flow with a linearised boundary
condition at the free surface, which is appropriate when
the wave amplitude is small compared to the wavelength
[135]. Such waves are referred to as gravity waves.
However, in many small-scale flows, the surface tension
forces at the interface cannot be neglected. Account-
ing for them leads to a dispersion relation for capillary-
gravity waves,ω2=gk +γk3/ρ, where the importance of
the surface tension parameter is measured by the dimen-
sionless number S = γk2/ρg.
For very short waves, the capillary term dominates,
so S 1 and the dispersion relation simplifies to ω2=
σk3/ρ. Such waves are termed capillary waves, and for

48
water typical cross-over wavelength when S = 1 is about
1.7 cm. Notably, for capillary waves, the group velocity
exceeds the wave (or phase) velocity (cg= 3c/2) and
so crests move backwards in propagating wave packet.
In most small-scale kitchen flows surface tension has a
pronounced effect on the appearance and propagation of
waves. Such are the waves created by a dripping faucet
in a filled sink.
Moreover, in various food science circumstances we
might have to do with waves of wavelength comparable
to the depth of the vessel in which they propagate. For
such shallow-water waves the propagation speed depends
on the local depth with larger speeds at deeper water.
In particular, for gravity waves the dispersion relation
becomes in this case ω2=gk tanh kh, with hbeing the
water depth. This again holds for wavelengths small com-
pared to h. The general case is much more complex and
nonlinear in nature, yet the linear wave theory is often
enough to grasp the dominant behaviour. We considered
here only free-surface flows but the reasoning is easily
generalised to any fluid-fluid interface [909].
E. Rinsing flows: thin film instabilities
Thin fluid films are a remarkable example of a kitchen
flow which has already received a considerable attention
exactly in this context. The stability of falling films was
the subject of investigation of a father-son team of the
Kapitza family, led by the elder Nobel prize winner Py-
otr Leonidovich Kapitza, in the 1940s [910]. After World
War II, Kapitza was removed from all his positions, in-
cluding the directorship of his own Institute for Physical
Problems, for refusing to work on nuclear weapons. He
was ordered to stay at his country house and, deprived
of advanced equipment, devised experiments to work on
there, including a famous set of experiments on falling
films of liquid [911]. Kapitza and Kapitza [912] were
the first to experimentally investigate traveling waves on
the free surface of a liquid film falling down a smooth
plate. The emerging Kapitza instability takes form of
roll waves [913], and evolves from a two-dimensional dis-
turbance (i.e., invariant in the spanwise direction) into
a fully developed three-dimensional flow [914]. Since the
early works of Kapitza, the dynamics of waves in viscous
films over the flat substrates were reviewed extensively
[583,598,915]. We often encounter such waves after a
rainfall, on an inclined asphalt road, or even in flowing
mud [916]. Film and rivulet flows at solid surfaces bear
importance for gas exchange also in industrial applica-
tions, including distillation columns [917]. In the kitchen
context, they emerge predominantly in rinsing flows or
spreading flows, where the thin film dynamics may be
governed either by capillarity or external diving, such as
gravity, or centrifugal forces [918]. We discussed viscous
spreading in §VI G, thus here we focus purely on waving
instability.
The phenomenon of roll waves formation is governed
by two dimensionless parameters: the Reynolds number
describing the flow character, and the Kapitza number
Ka = surface tension
inertial forces =σ
ρg1/3ν1/4,(83)
where gis the gravitational acceleration driving the flow.
In the context of thin films flows down an inclined slope,
the formation of roll waves can also be discussed in terms
of the Froude number Fr , defined in Eq. (13) in §II I. For
moderate Reynolds numbers, the value of Fr ≈2 marks
the onset of instability in the thin film flow equations
[919]. However, Benjamin has shown in his seminal pa-
per [920] that such flow is unstable for all values of Re,
but the rates of amplification of unstable waves become
very small when Re is made fairly small, while their wave-
lengths tend to increase greatly. He proposed a criterion
that for an observable instability of flow down a slope
with inclination angle βthe critical Reynolds number is
Re = 5
6cot β, as later corroborated by Yih [921].
F. Dynamics of falling and rising drops
1. Immiscible drops
The literature describing buoyant immiscible drops is
vast, and we try not to give a comprehensive review here;
for that, we refer the reader to the many excellent reviews
and books previously published, see e.g. Harper [922]
and Leal [868]. Nonetheless, due to its central position
in the field of fluid dynamics and its omnipresence in the
kitchen, we mention a few seminal works regarding freely
suspended drops.
When a drop of water is released in cooking oil, it
starts to fall immediately due to gravity. During its de-
scent, surface traction from the outer liquid mobilise the
fluid-fluid interface and the degree of surface mobility is
given by the viscosity ratio between the ambient and drop
fluid, ˆµ/µ. For very viscous drops translating through
low viscosity liquids (such as a drop of oil rising through
water, ˆµ/µ →0), the small surface traction is insuffi-
cient to mobilise the interface: this results in the drop
translating at the velocity of a rigid Stokes’ sphere of
the same size and volume. The opposite mobility limit
is reached when the viscosity ratio is reversed such that
ˆµ/µ → ∞): the interface is then expected to be com-
pletely mobile, which causes a vortex ring to develop
within the drop. A completely mobile interface is not
able to resist viscous stresses, and this leads to the ter-
minal drop velocity being one and a half times as high
as that of a Stokes’ sphere of the same size and density.
The solution to the flow field within a translating, rigid
drop at low Reynolds number was worked out simulta-
neously and independently by the French mathematician
Jacques Salomon Hadamard (1865-1963) [923] and the
Polish physicist and mathematician Witold Rybczynski
(1881-1949) [924] as early as in 1911.

49
In reality, most small droplets rise or descend at ve-
locities that lie between the theoretical prediction by
Hadamard and Rybczynski and the Stokes prediction for
rigid spheres [290], and this is true even in pure liquids
with no surfactants added [Fig. 29a,b]. The terminal ve-
locity generally depends strongly on size as reported by
Bond [925], who found small water droplets to descend
through castor oil at only 1.16 times the Stokes’ velocity,
while drops exceeding a critical radius of about 0.6 cm de-
scended at 1.4 times the Stokes’ velocity. To explain the
sudden jump in velocity with drop size, Bond and New-
ton [926] postulated that a ratio of buoyancy to surface
tension determines the mobility of the interface. Boussi-
nesq [927] instead suggested that an increased viscosity
at the drop’s surface is responsible for slowing the drop.
However, without experimental evidence of the flow field
within the drop, it is impossible to judge the correctness
of these models.
Aiming to obtain a better description, Savic [928] pub-
lished photographic evidence of the flow streamlines in-
side water droplets descending through castor oils. Savic
[928] visualizations show that the streamline patterns of
drops exceeding 1 cm in radius are almost indistinguish-
able from the Hadamard-Rybczynski-solution and that
the terminal velocities for large drops are in good agree-
ment with theory as well. However, for smaller drops,
the vortex rings are shifted forward, and this occurs as
a stagnant cap emerges in the rear of the drop. As the
drop size is further reduced, the stagnant region covers
a larger and larger portion until it envelops the entire
drop, with the result of the drop sedimenting as a Stokes
sphere.
To explain his observations, Savic [928] proposed that
the interface is immobilized by surface active molecules,
which are in turn de-stabilized by viscous stresses from
the outer fluid. For the smallest drops, the viscous stress
is insufficient to distort the surface layer: this leads to
a complete immobilization. However, as the drop size
increases, the shear stress increases as well, and this
leads to a gradual removal of the surface layer until the
Hadamard-Rybczynski-theory is fully recovered for the
largest drops.
Savic [928] developed a theory to calculate the drag of
a drop from the degree of surface coverage, which he ex-
tracts from the flow visualizations. He also attempted to
calculate the critical drop size of the transition between
a mobile and an immobile no-slip boundary, however the
transition occurred at larger radii than predicted, and he
suggests this discrepancy to be due to a finite solubility
between water and castor oil not accounted for in the the-
oretical model. Later, Davis and Acrivos [929] improved
Savic’s analysis to obtain better agreement with exper-
iments, and Sadhal and Johnson [930] extended these
results to obtain an exact solution of the drag force on
the drop for a given surface coverage. For a droplet sed-
imenting at a given rate, Sadhal and Johnson [930] also
obtained an analytical expression for the total amount
of surfactant adsorbed to the interface. However, the so-
FIG. 29. Interfacial phenomena in rising bubbles and drops.
(a) The drag force on a rising bubble is affected by ‘hidden’
surfactant transport variables (b) including (i) Interfacial vis-
cosity can resist the surface flow. (ii) Surfactant concentration
gradients generate Marangoni stresses. (iii) Marangoni forces
weakened by surface diffusion against the gradient. (iv) Dif-
fusive transport of surfactants in the bulk. (v) Adsorption
and desorption kinetics of soluble surfactants. (a-b) From
Manikantan and Squires [290]. (c) Water drops ascending
through corn syrup and glycerol undergo shape transforma-
tions from prolate to oblate spheroids. The travel time tis
rescaled by the characteristic mixing time τfrom Eq. (81).
From Mossige et al. [871].
lution to the internal flow field and the corresponding
sedimentation rate for a drop of a given size remains an
open question.
2. Miscible drops
As compared to immiscible drops covered in the last
section, transport problems involving miscible drops have
enjoyed far less attention, which is surprising given their
omnipresence in our daily lives and their rich dynamics.
In section §VIII F we looked at how fluid motion can
accelerate the mixing rate between a viscous drop and
its surroundings in the low Reynolds number case. In
this section, we discuss how finite inertia may influence
the shape of falling drops, and we discuss the stabilizing
effects of transient tensions between miscible liquids.
When a miscible drop descends in another liquid, it
changes shape in response to the viscous drag acting on
it, and when it reaches a critical velocity, inertial effects
also start to play a role. A simple way to visualise the
effect of inertia on the drop shape is to produce a drop

50
of food dye in air and let it fall into a glass of water.
Upon impact with the water surface, the central part of
the drop gets accelerated upward in a Rayleigh-Taylor
instability, and this causes the drop to evolve into an
open torus. For drops made of honey or corn syrup, this
shape transformation is delayed by the high viscosity, but
on a long time scale, even the most viscous drop deform
into oblate spheroids or donuts.
Kojima et al. [931] developed a theory to explain the
shape transitions of miscible drops and validated their
theory against experiments of corn syrup drops falling
through diluted corn syrup solutions. When the drop is
created in air, immediately above a water surface, they
showed that the descending drop does not experience
inertia in the early stages of its descent: In this case,
it is solely deformed by viscous traction forces, causing
the drop to develop into an oblate spheroid. However,
by pairing theory with experiments, Kojima et al. [931]
showed that later in the drop’s descent, inertia does play
a key role in its shape evolution, and this causes the drop
to develop into an open torus. The fact that inertia is
relevant at long time scales is intuitive; however, they
also had to incorporate a small, but finite tension across
the miscible interface to fully explain the material de-
formation. The inverted system concerning water drops
ascending through corn syrup was recently examined by
Mossige et al. [871] [Fig. 29c].
The tension existing between miscible liquids is not a
surface tension as defined in the classical sense between
immiscible phases [see §II F]. Instead, it is caused by
sharp gradients in composition between the pure phases
by giving rise to so-called Korteweg stresses [932] that
mimic the effect of a surface tension. These tensions
are typically at least two orders of magnitude smaller
than in immiscible systems (for example, 0.43 mN/m be-
tween glycerol and water [933] as compared to 73 mN/m
between water and air) and diminish in time as dif-
fusion smears out the miscible interface; as a result,
they are inherently difficult to measure and usually ne-
glected. However, in many situations including miscible
displacements in capillary tubes [934] and in Hele-Shaw
geometries [935], effective interfacial tensions must be ac-
counted for to accurately describe a deforming, miscible
interface, and theoretical and experimental evidence for
this is given in Refs. [936–941]. Non-equilibrium stresses
are not only of academic interest, but can be tuned to
control the morphology of miscible interfaces in modern
industrial processes. Notably, Brouzet et al. [942] uti-
lized transient tensions to align nanofibrils in microfluidic
flow focusing geometries, with implications in the paper
production industry and in the development of new, sus-
tainable alternatives to plastics, and Wylock et al. [943]
used it to control gravitational instabilities in carbon se-
questration plants.
X. DISCUSSION
A. Summary
In this Review, we have presented a overview of culi-
nary fluid mechanics and other currents in food sci-
ence. Starting from ancient times, the connection be-
tween cooking and fluid mechanics has led to innovations
that benefit both. We have explored how this connection
grows stronger every day, to the frontier of modern re-
search and gastronomy. Culinary fluid mechanics brings
people together from across society, from chefs to food
scientists, physicists and chemical engineers, medical and
nutrition specialists, and students in any discipline. To
make this article accessible to this broad audience, we
started our discussion with an overview of kitchen sink
fundamentals [§II], where we summarised the basics of
fluid mechanics in the context of food science. Starting
the meal with drinks [§III], we reviewed hydrodynamic
instabilities in cocktails, Marangoni flows, bubble effer-
vescence and culinary foams. Getting into the thick of
it with a soup for starters [§IV], we discussed the rheo-
logical properties of viscoelastic food, non-linear sauces,
suspensions and emulsions. Moving on to a hot main
course [§V], we analysed the role of heat in cooking, in-
cluding the Leidenfrost effect, Rayleigh-B´enard convec-
tion, double-diffusive convection, flames and smoke. Go-
ing for a sticky desert [§VI], we described flows at low
Reynolds numbers, from Stokes’ law to lubrication the-
ory, viscous gravity currents, ice cream and microbial
fluid dynamics. Eager for a postprandial espresso [§VII],
we examined the physics of granular matter and porous
media flows, different brewing methods, and the coffee
ring effect. Thirsty for another cup of tea [§VIII], we de-
lineated the tea leaf paradox and other non-linear flows,
succeeded by turbulence and chaos. Finally, when doing
the dishes [§IX], we explore interfacial phenomena in-
cluding the Gibbs surface elasticity, soap film dynamics,
waves and jets, miscible drops and roll wave instabilities.
Quite a bit to digest, but a place worth coming back to.
B. Learning from kitchen experiments
Humans are naturally curious. From an infant age,
we explore by actively interacting with our surroundings
[944]. Through touch, smell and taste, we learn about the
natural world. Becoming a scientist starts with asking
questions like “what?”, “why?” or “how?”.
In physics education, we try to answer these ques-
tions the by comparing observations with theoretical de-
scriptions. Traditionally, this knowledge is transferred
from teacher to students through in-class lecturing and
instructor-made assignments [945], but this linear learn-
ing protocol is not necessarily compatible with curiosity-
driven exploration and observation [946]. As a result,
students often feel alien to the physics topics taught in
class [39] and lose the natural intuition and curiosity that

51
FIG. 30. Kitchen-based learning is an affordable and accessi-
ble strategy to foster curiosity and intuition for a wide range
of physics topics. (a-c) In a science class accessible to non-
science majors, cake making was used to demonstrate heat
transfer and elasticity. From Rowat et al. [39]. (a) Ther-
mocouples are used to measure the rise in temperature at
different points inside a molten chocolate cake as it bakes in
an oven. (b) The thickness of the solid crust of the cake L
increases over time. (c) Results of experiments. (d) Students
pour pancake batter to learn about viscous gravity currents.
They used cell phones to video-record the spreading rate and
fit their data into a theoretical model to back-calculate the
viscosity. Image courtesy of Roberto Zenit.
is so important for learning [947,948]. Inevitably, physics
has a reputation for being difficult and abstract and with
little relevance to students’ daily lives, and this discon-
nect is largely responsible for the relatively poor recruit-
ment to science and education disciplines in higher edu-
cation [949]. To address this issue, it is vitally important
to develop effective teaching strategies that fosters both
intuition, engagement and curiosity. This is best achieved
through an hands-on active learning strategy [950,951],
without creating the perception of learning by ineffective
engagement [952], where experiments that relate to our
daily lives have a prominent role.
The kitchen is an accessible learning environment
where simple physics experiments can be performed at
home with humble ingredients; For example, Benjamin
[40] showed that students in elementary physics educa-
tion can learn about surface tension, mixing and gravity
by studying Rayleigh-Taylor instabilities [see §III A] in
their own kitchen. The simple experimental design al-
lowed for a high degree of flexibility and were designed in
such a way that they could be performed either individu-
ally, or in groups to foster collective accomplishment and
collaborative learning. As compared to experiments con-
ducted in school laboratories, kitchen experiments have
a higher potential for engagement as we encounter them
every single day, and since they require very little equip-
ment. They offer a low-cost ‘frugal science’ alternative
that is less susceptible to budget cuts, and more acces-
sible to students from underrepresented socioeconomic
backgrounds [953,954].
Affordable and accessible kitchen experiments can also
be utilized to develop intuition for advanced mathemat-
ical concepts. Notably, a famous class at Harvard and
UCLA teaches general physics concepts such as heat
transfer and phase transformations to non-science ma-
jors through the lens of cooking [Fig. 30a-c] [39]. In this
popular course, top chefs give weekly seminars for fur-
ther engagement. Kitchen experiments can also be used
to learn about more specialized topics in fluid mechanics;
For example, take-home experiments such as measuring
the flow rate from a hose and estimating the density and
the viscosity of household fluids has been used to en-
hance learning in an introductory fluid mechanics class
[955]. In addition, the kitchen can be a gateway to learn
about the intrinsic fluid properties that govern these
flows. Notably, in a special session on Kitchen Flows
at the 73rd Annual Meeting of the APS Division of Fluid
Dynamics (APS-DFD), Zenit et al. [956] demonstrated
how pancake-making can be used to teach students about
fluid viscosity. Instead of extracting the viscosity from
a classical sedimenting-sphere experiment [555], which is
less common in our daily lives, the students were asked to
pour pancake batter and other viscous fluids like honey
and syrup into frying pans and measure the spreading
rate [Fig. 30d]. By fitting their data to a theoretical pre-
diction [596], which is described in §VI G, the students
were able to back-calculate the viscosity. Such in-situ
kitchen measurements can be used for numerous other
scientific concepts, as discussed throughout this Review,
thereby creating direct links between physics and every-
day experiences.
In addition, many canonical flows can be generated
with simple kitchen tools, including the circular hydraulic
jumps [see §II I], and Poiseuille flows [§II C], which can
be used to validate theoretical predictions taught in class
as a means to develop intuition for advanced mathemat-
ical concepts. Finally, to further accelerate the learning
in fluid mechanics, e-learning tools can be implemented
[957] such as the extremely extensive Multimedia Fluid
Mechanics Online [958].
From these examples, it is evident that easy-to-do
kitchen experiments can be implemented for enhanced
learning and engagement across all ages. They are highly
scalable, and can even can be taught on an online plat-

52
form to make learning available for large groups of stu-
dents, including students who can not afford enrollment
in an educational program. Therefore, kitchen-based
learning represents a viable strategy to increase the num-
ber of competent scientists and engineers in the world,
which is necessary to address immediate threats to hu-
mankind and ensure a sustainable future [959].
C. Curiosity-driven research
As well as being a vehicle for accessible and affordable
science education, culinary fluid mechanics is a hotspot
for curiosity-driven research [960]. Indeed, Agnes Pock-
els found inspiration for her breakthrough discoveries in
surface science and hydrodynamic instabilities from dish
washing [§IX]. Valuable data can be extracted relatively
quickly from a kitchen-based laboratory, in the spirit of
a ‘Friday afternoon experiment’ [961]. A minimum of in-
vestments of time, training and equipment are needed,
which makes this field approachable not only to experi-
mentalists, but also to theorists and researchers in other
fields. Since many kitchen flows can be described by scal-
ing theories and other analytical techniques, they can
serve to validate theoretical models in fluid mechanics
and materials science [469]. As such, kitchen experiments
are attractive to theorists, and by lowering the activa-
tion barrier to start a new experiment, they can be com-
bined with mathematical models to solve a large class of
problems in science and in engineering. Curiosity-driven
learning is foundational to human cognition [962], and
sometimes the best discoveries are made in a few hours.
Perhaps the most influential fluid mechanician of all
time, Sir. G. I. Taylor, was known for his special ability
to make groundbreaking discoveries from humble ingre-
dients and to design simple experiments that could be
described theoretically [963]. Instead of following the
hypes in science and ‘going with the flow’, Taylor was
merely driven by his own interest and curiosity, without
thinking about specific applications. Outstanding con-
tributions in fundamental science always find useful ap-
plications, which is immediately evident when we look at
the enormous implication of Taylor’s contributions to sci-
ence and engineering. However, today’s funding schemes
often require that research should preferentially address
a particular problem and have immediate impact [964],
which leaves little room for curiosity-driven research and
scientific investigations for its own sake [965]. But, since
curiosity is a prerequisite for exploration and discovery,
the scientific philosophy of Taylor and his predecessors
could serve as inspiration for the modern physicist.
D. Conclusion
Culinary fluid mechanics is the study of everything
that flows in the food supply chain, covering a wide range
of surprising phenomena that can be harnessed for the
benefit of gastronomy, food science, and for our planet
as a whole. This field naturally connects practical tech-
nologies with basic research, just how fluid mechanics
once started. Culinary flows are accessible to experi-
mentalists and theorists alike: Their intuitive geometry
and well-defined conditions are suitable for mathematical
modelling, while the relatively low equipment costs re-
duces the activation energy for pilot investigations, thus
catalysing curiosity-driven education [§X B] and research
[§X C].
Where ‘kitchen flow’ papers may initially have been
considered occasional or incidental, their breath and
depth now constitute a rapidly growing field. A field
that this Review can cover only partially because it is
so interconnected. Yet, culinary fluid mechanics is uni-
fied by a number of well-defined research directions and
goals. Firstly, it aims to establish a sustainable and fair
global food supply. Secondly, it has the potential to de-
velop reliable food technologies with a strong fundamen-
tal backbone. Thirdly, it can facilitate new discoveries
far beyond gastronomy by making science and engineer-
ing more accessible. Finally, it can advise policy makers
on important decisions for our future generations, such
as the announced EU ban on PFAS non-stick coatings
by 2030 [531] and help the reduction of climate change
[966,967]. To achieve these goals, scientists from related
fields must become even more interconnected.
Indeed, as we discussed, culinary fluid mechanics di-
rectly links to other disciplines across the sciences, from
molecular gastronomy to biological tissue mechanics and
rheology. Furthermore, it has extensive engineering ap-
plications ranging from the stream engine to 3D print-
ing and nanotechnology. Not least, there are immediate
connections with food safety, microbiology and medicine.
However, unlike many fields in science, kitchen flows cre-
ate a bond with people who couldn’t have a scientific
training. People who want to learn more, and people
who want to contribute themselves. People like Agnes
writing to Lord Rayleigh. So much talent is lost in this
world full of inequality, and we have a responsibility to
make science more inclusive and accessible to people from
under-represented backgrounds. Through science com-
munication, through education, and through research it-
self. We hope that more scientists will stand up to this
challenge.
ACKNOWLEDGEMENTS
Comments or suggestions on this arXiv submission are
most welcome. Please contact us, and we will try to in-
corporate them. A.J.T.M.M. acknowledges funding from
the United States Department of Agriculture (USDA-
NIFA AFRI grants 2020-67017-30776 and 2020-67015-
32330).

53
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