Thermohydrodynamics of boiling in a van der Waals fluid.

Abstract

We present a modeling approach that enables numerical simulations of a boiling Van der Waals fluid based on the diffuse interface description. A boundary condition is implemented that allows in and out flux of mass at constant external pressure. In addition, a boundary condition for controlled wetting properties of the boiling surface is also proposed. We present isothermal verification cases for each element of our modeling approach. By using these two boundary conditions we are able to numerically access a system that contains the essential physics of the boiling process at microscopic scales. Evolution of bubbles under film boiling and nucleate boiling conditions are observed by varying boiling surface wettability. We observe flow patters around the three-phase contact line where the phase change is greatest. For a hydrophilic boiling surface, a complex flow pattern consistent with vapor recoil theory is observed.

Full text

Page 1

PHYSICAL REVIEW E 85, 026320 (2012)
Thermohydrodynamics of boiling in a van der Waals fluid
T. Laurila,1A. Carlson,2M. Do-Quang,2T. Ala-Nissila,1,3and G. Amberg2
1COMP CoE at the Department of Applied Physics, P.O. Box 11100, Aalto University School of Science, FI-00076 AALTO, Finland
2Linn´ e Flow Center, Department of Mechanics, The Royal Institute of Technology, Stockholm, Sweden
3Department of Physics, Brown University, Providence, Rhode Island 02912-8143, USA
(Received 29 September 2011; published 29 February 2012)
We present a modeling approach that enables numerical simulations of a boiling Van der Waals fluid based
on the diffuse interface description. A boundary condition is implemented that allows in and out flux of mass
at constant external pressure. In addition, a boundary condition for controlled wetting properties of the boiling
surface is also proposed. We present isothermal verification cases for each element of our modeling approach.
By using these two boundary conditions we are able to numerically access a system that contains the essential
physics of the boiling process at microscopic scales. Evolution of bubbles under film boiling and nucleate boiling
conditions are observed by varying boiling surface wettability. We observe flow patters around the three-phase
contact line where the phase change is greatest. For a hydrophilic boiling surface, a complex flow pattern
consistent with vapor recoil theory is observed.
DOI: 10.1103/PhysRevE.85.026320PACS number(s): 47.55.D−, 47.11.Fg, 44.35.+c, 64.70.fh
I. INTRODUCTION
Boiling and condensation are common phenomena in
everyday life. Perhaps the most obvious examples are various
processes during cooking, but an even more ubiquitous
example would be weather phenomena due to the interplay of
water,watervapor,andair.Inengineering,boilingheattransfer
is a common method of heat transfer in thermal power plants,
conventional and nuclear alike, and is thus an indispensable
part of electricity production. Efficiency of heat transfer by
boiling is limited by the creation of a vapor film at the hot
surface when heat throughput is too high [1–4]. This effect is
generally called criticalheat flux, boiling crisis,or dryout. The
insulatingeffectofthevaporfilmcausesarapidanddestructive
jump in temperature at the newly dry wall.
While macroscopic and effective properties of boiling and
condensation have been studied for centuries, and properties
such as latent heats of materials are known in great detail, the
dynamicsofhowboilingandcondensationhappens,especially
the initial stages of the formation of bubbles or droplets, is still
largely unresolved. A classical issue is heterogenous boiling
and condensation, where the phase transition is macroscop-
ically observed to occur at much smaller superheats than
classical theories predict. The boiling crisis mentioned above
is an example where understanding and thereby controlling
the dynamics of bubbles as they form would have an obvious
technological impact.
Observing in experiments the incipience of bubble nucle-
ation in boiling is challenging, as the dynamics is inherently
fast and occur on microscopic scales. The separation of time
and length scales is significantly reduced when dealing with a
liquid close to its critical point and in a microgravity environ-
ment, but this produces a host of challenges of its own [5,6].
Mathematically and computationally, thermalmultiphase flow
problemsarechallengingtodescribe[7–9].Toourknowledge,
neither computational nor experimental observations have
been reported of the microscopic flow around the three-phase
contact line where boiling on a heated surface predominantly
happens. The vapor recoil theory proposed by Nikolayev
and coworkers links the flow profile to the boiling crisis by
presenting dryout as a single-bubble spreading event [10].
In his pioneering work on liquid-gas phase transitions, Van
der Waals [11] considered the coexistence of liquid and gas
to consist of a density field that attains two different values
and varies smoothly but rapidly in between. Van der Waals
also attributed a free energy cost to the gradients of the density
field. These ideas have been extended for thermodynamics
and hydrodynamics of nonuniform phases by, among others,
Korteweg [12], Ginzburg and Landau [13], Cahn and Hilliard
[14], and Dunn and Serrin [15]. The results of this work
are in modern parlance called phase field methods or diffuse
interface methods in the materials science and fluid dynamics
communities, respectively.
Diffuse interface methods have received renewed attention
in recent years, much of which can be attributed to the in-
creasedcapabilitiesofmoderncomputersenablingsimulations
of relevant problems by using these methods. A review of
the contemporary diffuse interface methods was given by
Anderson [7]. Recently, Onuki presented a new formulation
[8]. These two have a difference in how the thermal gradient
couples to stress at the interface, although as guiding principle
both use the reversibility of capillary forces.
A number of numerical simulation studies have been
performed considering condensation, boiling, drop spreading,
bubbles, and droplets in nonuniform thermal fields [8,17–21];
however, allofthesearemadeforsystemsthatareenclosed by
solid walls. A key factor in simulating the dynamics of boiling
is that a bubble must be able to grow while the surrounding
liquidmaintainsitsmetastablestate.Inanumericalsimulation
this means that there must be an open boundary on the
computational domain that allows mass to either flow in or
out, making the average density of the system change in time.
An ideal open boundary would not reflect any information
of outgoing flow back into the computational domain [22].
Because sound waves are always present in the compressible
system, this should particularly hold for the acoustic modes.
In our work we observe the phase change between gas and
liquid to cause compressibility waves. We assume that the
026320-1
1539-3755/2012/85(2)/026320(15)©2012 American Physical Society

Page 2

T. LAURILA et al.
PHYSICAL REVIEW E 85, 026320 (2012)
opposite phenomenon (i.e., sound waves affecting the phase
change) is insignificant and can be ignored. Thus we neglect
the acoustic properties of our boundaries, which leads to a
significant simplification. The study presented here for diffuse
interface boiling properly includes both the thermodynamics
and hydrodynamics in an open system.
The aim here is to develop two separate boundary
conditions, one that will allow us to control the wetta-
bility of the solid substrate and the other to allow mass
flux through an open boundary enabling us to simulate
boiling. The proposed modeling approach for the wetting
boundary condition is adopted from the isothermal Cahn-
Hilliard method [16]. Modeling of contact lines are par-
ticularly relevant in boiling, which is the final goal of
the study presented in this article. The basis of the open
boundary condition is simple thermodynamics, but we find
it necessary to introduce pressure as an additional variable to
make it work in practice.
Figure 1 presents a sketch of all the computational cases
we study and as such presents an outline of the article. After
presenting the mathematical model (Sec. II) and numerical
formulation (Sec. III), we proceed to verify the model with
various isothermal cases in Sec. IV. First we verify the
Young-Laplace law for a bubble immersed in its liquid at
FIG. 1. Schematic view of the computational cases considered in
this paper with corresponding boundary conditions. We simulate the
Young-Laplace law for a single bubble immersed in its coexistence
liquid. We verify the open boundary condition by comparing with
an analytical solution. The proposed wetting boundary condition
is verified in simulations of isothermal systems but for different
temperatures. Numerical simulations of boiling are performed in a
domainwithatemperaturegradient,wheretheupperboundaryallows
mass flux in and out.
coexistence. The case is illustrated in the upper-left panel in
Fig. 1. To verify the open boundary condition we look at
the growth or collapse of bubbles immersed in metastable
liquid as function of their initial size. A theoretical estimate
for limiting size is obtained from the Gibbs free energy, and
numerical simulations sketched in lower-left panel of Fig. 1
are in good agreement with the prediction. A sketch of the
numerical system that is used to verify the wetting condition is
illustrated in the panel to the upper right in Fig. 1. Initially
a straight interface is connected with two walls with the
oppositeequilibriumangleimposed.Theequilibriuminterface
is straight and we measure the angle it takes between the two
walls. In Sec. V we present numerical simulations of boiling
dynamicsofaVanderWaalsfluid,whereoneoftheboundaries
isopen.Weshowinparticulartheeffectofthesolidwettability
on the rate of phase change from liquid to gas.
II. MATHEMATICAL MODEL
The Van der Waals (VdW) model of liquid-gas coexistence
is described by the Helmholtz free energy per volume [8]:
f(ρ,T) = kBT
ρ
mp
?
ln
?ρ
1 − bρ
mpT3/2
mp
?
− 1
?
− a
?ρ
mp
?2
.
(1)
The corresponding VdW equation of state is obtained as
p(ρ,T) = ρ∂ρf(ρ,T) − f(ρ,T) =
kBT
1 − bρ
ρ
mp
mp
− a
?ρ
mp
?2
,
(2)
and the internal energy is obtained as
e(ρ,T) = f(ρ,T) − T∂Tf(ρ,T) =3kB
2mTρ −
a
m2ρ2.
(3)
Above, we have the pressure p, mass density ρ, temperature
T,BoltzmannconstantkBandmolecularmassmp.∂ρ= ∂/∂ρ
is a short-hand notation for the partial derivative that we will
keep throughout this article. The VdW parameters a and b
describe the fluid in question. The scope of the present work
is the qualitative description of the general thermodynamics
and hydrodynamics of liquid-gas phase changes and, for
this purpose, the VdW model provides a well-established
benchmark. It should be noted that the VdW model might not
be the most computationally convenient and efficient among
thesimpleequationsofstate[27],butastheclassicbenchmark
of a liquid-vapor phase transition it is the most appropriate
choice for our purpose.
In addition to an equation of state with two stable phases,
the description of two-phase coexistence needs to account
for the interfaces between the phases. The central idea of
the diffuse interface method; namely, that there is an energy
cost associated with spatial change of an order parameter and
the order parameter changes rapidly but smoothly across an
interface, was considered already by Van der Waals [11]. In
the case of the liquid-gas phase transition, the density plays
the role of the order parameter. In modern guise, the diffuse
026320-2

Page 3

THERMOHYDRODYNAMICS OF BOILING IN A VAN DER ...
PHYSICAL REVIEW E 85, 026320 (2012)
interface method starts with the free energy [16]
?
where surface tension is described in terms of the gradient
energy coefficient κ. The actual surface tension depends on κ
and the form of the free energy density f, which must be of
double tangent form as function of density [14]:
√2κ
ρg
where ρg and ρl are the gas and liquid densities at
coexistence at temperature T. ?f(ρ) is the difference
between value of f(ρ) and the double tangent line at
ρ. This is valid when temperature is uniform across
the interface. Different assumptions have been made
on how κ(ρ,T) depends on density and temperature
[7,8,18,19]. κ couples to both the surface tension and the
interface width and experimental data on κ as function of
temperature and pressure is sparse. In this work we have
made the simplest assumption that κ is constant, which can
be determined so that the surface tension of our VdW fluid
corresponds in order of magnitude to that of a real fluid.
Qualitativelythesurfacetensionbehavesproperlyasafunction
of temperature and pressure, vanishing at the critical point and
increasing below the criticalpoint as afunction of temperature
and pressure.
Coexistence between the liquid and gas phases is obtained
as a minimum of the free energy, constrained such that
densitiesfarawayfromtheinterfaceareρlandρg.Considering
thecaseofconstanttemperature,theminimaofthefreeenergy
are obtained via the Euler-Lagrange equation of the functional
ofEq.(4).Ontheotherhand,mechanicalbalanceofthesystem
is obtained from the stress tensor as zero divergence:
F =
dx
?
f(ρ(x),T(x)) +κ
2|∇ρ(x)|2
?
,
(4)
σ =
?ρl
dρ??f(ρ,T),
(5)
∂jPij= 0,
for every i.
(6)
We use the Einstein summation notation and the shorthand
notation ∂i= ∂/∂xi, and similarly for j and k. This zero stress
divergence criterion, or zero force, corresponds to the Euler-
Lagrange equation of F or minimum free energy if the stress
tensor equals
ij=?p(ρ,T) − κρ∂k2ρ −1
The above result for the stress tensor in constant temperature
and constant κ is equivalent among the existing diffuse
interface models [7,8,29].
The hydrodynamic equations for mass continuity and fluid
motion with the stress tensor above read
PT
2κ∂kρ∂kρ?δij+ κ∂iρ∂jρ.
(7)
∂tρ + ∂j(ρvj) = 0,
(8)
∂t(ρvi) + ∂j(ρvivj) = −∂jPT
where σij is the viscous stress and giincludes external bulk
forces such as gravity.
Considering full thermodynamics, that is a temperature
field as function of time and space as well, the foundation of
extendingtheisothermalmodelabovehasbeentodemandthat
interfacialforcesduetothegradientenergyworkadiabatically.
This can be obtained in two different ways, either as a flux
of internal energy proportional to temperature gradients with
ij+ ∂jσij+ gi,
(9)
Fourier’s law in the energy conservation equation [7,29], or
the pressure tensor itself is extended with terms proportional
to temperature gradient [8].
The two-phase model must also account for diffusive
transport of both heat and mass that changes between the two
phases. This means incorporating a model for the viscosity (η)
andthermaldiffusivitycoefficients(α)asafunctionofdensity.
In this work we assume these material properties to be linearly
proportional to the density; namely,
η = η0ρ,
α = α0ρ,
(10)
(11)
where η0and α0are constants.
For the two-dimensional (2D) simulations we present here,
we use the model formulated by Onuki [8], which explicitly
written out in our case takes the form
∂tρ + ∂j(ρvj) = 0,
(12)
∂tvi+ ∂j(ρvivj) = −∂jPij+ η0∂j[ρ(∂ivj+ ∂jvi)] + ρgi,
(13)
∂te(ρ,T) + ∂j(e(ρ,T)vj)
= −Pjk∂jvk+ η0[ρ(∂jvk+ ∂kvj)]∂jvk+ α0∂j(ρ∂jT),
(14)
Pij=
?
+κ∂iρ∂jρ.
p(ρ,T) − κρ∂k2ρ −1
2κ∂kρ∂kρ + κρ
T∂kρ∂kT
?
δij
(15)
Note that, above, we have applied the standard viscous stress
tensor in 2D and identical bulk and shear viscosities, which
we use in our simulations. Phase change is driven here by the
nonequilibrium state of the system, which produces a large
contribution in the stress tensor given in Eq. (15). Since this
is a system of coupled equations, this gives a contribution
into the momentum and then the mass conservation equation.
Equation (15) is the stress tensor derived by Onuki [8], where
the last term is the correction to the pressure tensor making the
interfacial force act adiabatically. The coefficient κ from the
free energy formulation is assumed here to be constant in both
densityandtemperature.Throughaconstantκ theVdWtheory
predicts the interface tension of the fluid. A more detailed fit
to more complex fluid data is beyond the scope of this work.
It is worthwhile to note here that, for test purposes, we
have also implemented the formulation proposed by Anderson
et al. [7]. Testing the two models against each other yielded
practically indistinguishable results for a boiling simulation in
a domain with a temperature gradient. It would be interesting
to study in detail if any differences between these two models
exist in a benchmark thermohydrodynamics case, but this
is beyond the scope of the present work. However, our
preliminary numerical tests indicate that the results of the two
models are nearly identical and that the results presented here
do not depend on the choice between these two models.
A. Boundary conditions
In addition to bulk dynamics boundary conditions for the
thermodynamic variables need to be prescribed for studies of
026320-3

Page 4

T. LAURILA et al.
PHYSICAL REVIEW E 85, 026320 (2012)
different physical phenomena. In addition to assigning values
forthetemperatureandvelocityatsolidwalls,weneedanopen
boundary allowing a flow of mass in and out. For instance,
in a liquid-gas system fully enclosed by walls, two stable
coexistence phases can appear after domain decomposition.
Carefully engineering the initial condition in a wall-enclosed
system can make the decomposition mimic the dynamics of
phase change under constant pressure, but it is difficult to say
a priori how well. Also a boundary condition for the contact
line, here imposed via the density variable, is required to
adequately describe boiling phenomena near a hot wall.
A solid wall has a different surface tension when in contact
with liquid or gas. We denote these by σsland σsgrespectively,
where the subscript l denotes liquid, s solid, and g gas.
Togetherwiththeliquid-gas(σ)interfacetensionthesegivethe
equilibrium contact angle ?edescribed by Young’s equation
cos(?e) =σsg− σsl
In addition to the wetting boundary condition on the wall we
prescribe a no-slip velocity, constant temperature (or a fully
insulating wall), and no normal forces.
We follow the methodology by Jacqmin [16] and add a
surface contribution to the free energy at the wall:
?
+
S
In the same spirit as the isothermal Cahn-Hilliard model,
we represent g(ρ,T) as a polynomial that gives the values
g(ρg(T),T) = 0 and g(ρl(T),T) = 1 for the coexistence
densities of gas and liquid at a given temperature. We also
require g to have a minimum and maximum in ρ at the gas
and liquid densities [i.e., ∂ρg(ρ,T)|ρ=ρg/l= 0]. This ensures
that the boundary condition does not generate any artificial
contributiontothe(δF/δρ)foranyofthecoexistencedensities
at the wall. Furthermore, we assume local equilibrium at the
wall, implying that the contact angle will immediately relax
to its equilibrium angle. The wetting boundary condition here
reads
∇nρ =σ cos?
κ
where ∇nis the normal gradient. This boundary condition is
well defined on a wall with constant temperature, where a
third-order polynomial describes g in analogy to the case of
the Cahn-Hilliard freeenergy [28].This makes thevariation in
g with respect to density to take the form of a regularized delta
pulse, which is only nonzero in the interfacial region. Since no
simple analytic expression exists for gas and liquid densities
as a function of temperature for the VdW model, extending
this to a wall with varying temperature is complicated.
For the open boundary condition of constant temperature
and bulk pressure we propose to use vanishing normal gradi-
ents of the velocity fields, and constant density, temperature,
and (bulk) pressure. The equation of state must be fulfilled
by the values we impose to the latter three, meaning that the
density at the open boundary must be either the gas or liquid
density at the given temperature and pressure. Note that one
σ
.
(16)
F =
V
dx
?
?
dS[σsg+ (σsl− σsg)g(ρ,T)].
f (ρ(x),T(x)) +κ
2|∇ρ(x)|2
?
(17)
∂ρg(ρ,T),
(18)
of these phases is in general stable and the other is metastable,
unless the wall temperature and pressure is set exactly at the
boiling point.
To summarize our boundary conditions, we have for the
solid wall
vi= 0,
for every i,
(19)
∇nρ =σ cos?
T = Twall,
κ
∂ρgT(ρ),
∇nρ = 0,
(20)
∇nT = 0,
(21)
(22)
∇nPij= 0.
Here the density and temperature conditions on the left-hand
side are for a constant-temperature wall, and the right-hand
sideisforafullyinsulatingwall.Theno-slipandzero-pressure
gradient conditions apply for both cases.
The open boundary of a bulk fluid is given by
∇nvi= 0,
for every i,
(23)
(24)
(25)
(26)
ρ = ρwall,
T = Twall,
Pij= pwallδij,
consistency where
p(ρwall,Twall) = pwall.
for physical itmust holdthat
III. NUMERICAL METHODOLOGY
The numerical simulations were carried out using the finite
element toolbox FEMLEGO [25]. FEMLEGO is a symbolic tool
that defines the differential equations, boundary conditions,
initial conditions, and the method of solving each equation
in a single MAPLE worksheet. It also inherits adaptive mesh
refinement capabilities [26], which are used in these sim-
ulations. This enables us to have a high resolution of the
interface without spending excessive computational time. The
implementation of the mesh adaptivity can be described as
follows: At each mesh refinement step an element is marked
for refinement if the element size is still larger than the
minimummeshsizeallowed, and itdoes notmeetaprescribed
error criterion. In the case that an element meets the error
criterion, it is marked for derefinement unless it is an original
element. At the next refinement step, elements containing
hanging nodes are marked for refinement. The refinement
or derefinement stops if and only if no element is marked
for refinement or de-refinement (see [26]). All variables are
discretized in space using piecewise linear functions. A modi-
fied version of the characterized-splitting-based scheme [24],
originally developed for single-phase compressible flow, is
developedandtailoredfortheVdWphasechangesimulations.
Details concerning the numerical scheme are presented in
Appendix A.
A. Dimensionless units
Thegoverningequationshavebeensolvedindimensionless
form. Since the dimensions of the systems in this study are at
microscopic scales (i.e., <1 μm), gravitational effects can be
neglected. The equations are scaled in such a way that the
026320-4

Page 5

THERMOHYDRODYNAMICS OF BOILING IN A VAN DER ...
PHYSICAL REVIEW E 85, 026320 (2012)
fluid’s critical properties are all of size unity. This means the
equation of state (2) is turned into
˜ p( ˜ ρ,˜T) =
8˜T ˜ ρ
3 − ˜ ρ− 3 ˜ ρ2,
(27)
where the tilde denotes dimensionless variables. The corre-
sponding dimensionless internal energy is
˜ e( ˜ ρ,˜T) = 4˜T ˜ ρ − 3ρ2.
(28)
In dimensionless units the critical point is at˜Tc= ˜ ρc= ˜ pc=
1. This gives us three equations that set our three physical
dimensions of distance, time, and mass. Note that in physical
dimensions the critical temperature is determined by the
energy scale through the Boltzmann constant kB. This is
achieved by scaling the density and temperature by
ρ =mp
3b˜ ρ,
8a
27b
(29)
kbT =
˜T,
(30)
and by 27b2/a = x0t2
of length (x0), mass (m0), and time (t0):
0/m0. This leads to the following scaling
x0= 2b1/3,
m0=8
(31)
3mp,
?mp
(32)
t0= 6b5/6
a.
(33)
The gradient energy coefficient κ is related to the surface
tensioncoefficientthroughEq.(5),whichistheexperimentally
measurable quantity to which we can fit our model. The
kinematic viscosity (η0) and heat conductivity (α0) per density
scale by their physical units. This leads to the three free
variables that determine our VdW system:
˜ η0=t0
˜ α0=m0t0
x2
0
η0,
(34)
x2
t4
0kb
0σ2
α0,
(35)
˜ κ =
2m2
0
??˜ ρl
˜ ρgd ˜ ρ
?
?˜ f( ˜ ρ)?.
(36)
One should note that the model predicts all the properties of
the fluid once these three constants are set. This is, however,
not enough to quantitatively model a real two-phase fluid,
but the essential physical phenomena are captured with these
parameters.
In the numerical simulations we choose the three param-
eters above based on material properties similar to those
of water. The dimensionless units are determined from the
critical properties of water: Tc= 647 K, pc= 22.1 MPa,
ρc= 322 kg/m3, with the characteristic length x0= 0.74 nm,
time t0= 2.8 ps, and mass m0= 1.3 × 10−25kg. Note that
the characteristic length scale is similar to the interface width
and not the bubble size in the simulations. Choosing the
dimensionless coefficients
˜ η0= 1,
˜ α0= 30,
˜ κ = 1,
(37)
(38)
(39)
leads to the following dimensional properties for our VdW
fluid at 0.5Tc: kinematic viscosity η0= 2 × 10−7m2/s, heat
conductivity per density α0= 6 × 10−4 Jm2
tension σ = 0.07 N/m. The lambda value for the liquid is
then λl= 0.7W
kgKs, and surface
Km.
IV. MODEL VERIFICATION AT
ISOTHERMAL CONDITIONS
Toverifyourproposedmodelandthenumericalschemewe
study isothermal systems of a single vapor bubble immersed
in its liquid at coexistence. At constant temperature we have
coexistence with well-defined values for the gas and liquid
densities. This allows us to numerically measure the surface
tension via the Laplace pressure. We also use this as a measure
of the accuracy of the numerical scheme. For simulations
with an open boundary the pressure is fixed and allows us
to observe the effect of surface tension on the metastability
of the gas bubble and the liquid surrounding it. We extract
from simulations the threshold for the critical bubble size that
will either shrink or grow when placed in a superheated liquid.
This critical bubble size is compared against the theoretical
estimate. Finally, we verify the wetting boundary condition by
measuring the contact angles from simulations after reaching
steadystate.Theinitialconditionfortheinterfaceisfarfromits
equilibrium shape and thus the interface undergoes significant
evolution before reaching equilibrium.
A. Young-Laplace law
Sincethecoexistencevaluesforgasandliquiddensitiesata
given temperature for the VdW fluid can be calculated and the
formoftheHelmholtzfreeenergyisknown,thesurfacetension
σ can be computed from Eq. (5) [18]. The interface tension
causes a pressure increase in a circular domain of radius r
compared to the surroundings. This pressure difference is
given by the Young-Laplace law, which in 2D is given by
?p =σ
r.
(40)
We compute the numerical steady state of a bubble in
coexistence with a liquid in a closed system (i.e., one where
all edges of a square 2D computational domain are solid
walls). Initially a bubble is placed at the center of the domain
surrounded by the liquid phase. Both the densities of the gas
and liquid are at the coexistence density, and we set the initial
interfaceprofiletointerpolatebetweentheseusingatanhfunc-
tion. Even though small spurious velocity currents persist at
the interface after reaching what we interpret as the numerical
steadystate,thebulkvaluesforbothliquidandgasareconstant
to the fifth decimal. These density values tell us how well our
numerical simulation reproduce the VdW coexistence and by
using the equation of state (27) we obtain the pressure inside
026320-5

Page 6

T. LAURILA et al.
PHYSICAL REVIEW E 85, 026320 (2012)
0 0.51 1.52 2.5
0.6
0.7
0.8
0.9
1
ρ
T
0 1000 20003000 40005000
1
2
3
4x 10−3
Δ p
t
(a)
(b)
FIG. 2. (a) Numerical coexistence densities denoted by hollow markers plotted against the analytical VdW coexistence lines (dashed lines).
Uncertainty due to Laplace pressure caused by inaccuracy in the numerical densities are all within the marker size. (b) Time series of the
pressure difference inside and outside the gas bubble. The dashed line is the analytical Laplace pressure.
and outside the bubble. The difference in these pressure values
gives a numerical value for the interface tension via Eq. (40).
We study the Laplace pressure for isothermal coexistence
ateightdifferenttemperaturesbetweenT = 0.95andT = 0.6
and for five different bubble sizes between 50 ? r ? 150. The
width of the diffuse interface depends on temperature, but
stays well within the range of 1 to 10, ensuring that there is
still a length-scale separation between the bubble size and the
interface width. At temperatures below T < 0.6, or density
contrast beyond ρl/ρg> 50 our numerical results become
inaccurate. In Fig. 2(a) we compare the numerical density
coexistence values against the analytical prediction for dif-
ferent temperatures. For each temperature we plot the largest
and smallest density values taking into account the predicted
Laplace pressure. Thus the smallest bulk density of the gas is
the smallest value observed minus the Laplace pressure. Since
in a finite system the Laplace pressure could also manifest
itself as decreased pressure in the liquid domain, the largest
bulk density of the liquid is the largest value observed plus the
Laplace pressure. As seen from Fig. 2, the largest and smallest
value overlap in most cases and agree well with the VdW
result. Some differences are observable at low temperatures in
the gas branch due to the large compressibility of the gas.
InFig.2(b),weshowatimeseriesofthepressuredifference
inside and outside the bubble as it approaches equilibrium.
The system is here at T = 0.95 and the bubble has a radius
of r = 75. As an initial condition we prescribe coexistence
values with the same pressure. As the bubble equilibrates,
soundwavesareproducedthatpropagateacrossthesimulation
domain.Whenthedensityprofilehasstabilizedwithonlyvery
minor variations, we measure the pressure difference between
the gas and liquid phases.
The gas-liquid interface tension provides a sensitive check
oftheaccuracyofthesolutionmethod,sincechangesindensity
due to Laplace pressure are typically in the third decimal in
the density. This means only fractions of a percent on the
liquid side and a few percent on the gas side. By observing
densities of the gas and liquid domains we obtain the interface
0.6 0.7 0.80.91
0.95
1
1.05
1.1
1.15
T
σ
σpred
0.60.70.8
T
0.91
0
0.5
1
1.5
2
2.5
σ
0 0.01 0.020.03 0.04
−2
−1.5
−1
−0.5
0
0.5
1
Δ p
p−pcx
Δ p
(a)
(b)
FIG. 3. (Color online) (a) Numerically measured surface tension of bubbles with different sizes for eight temperatures at isothermal
conditions. The surface tension is measured by extracting the pressure difference between the gas and liquid domains, which is scaled with the
predicted values from Eq. (5). At temperatures above T = 0.65 we observe interface tensions within 5% of the prediction. The inset shows the
unscaled data, where the dashed line is the theoretical value. (b) Data for the pressure difference, where the gas data are denoted by stars and
the liquid data as diamonds. The solid line is a fit to the gas data and intended as a guide to the eye. The dashed line is a shift by unity in the
y axis that is equivalent to the Laplace pressure. This dataset shows that Laplace pressure difference between the gas and liquid sides is well
characterized numerically and obtained to a much higher degree of accuracy than the coexistence pressure in the bulk phases.
026320-6

Page 7

THERMOHYDRODYNAMICS OF BOILING IN A VAN DER ...
PHYSICAL REVIEW E 85, 026320 (2012)
tension data shown in Fig. 3(a). The results agree with the
interface tension values predicted by Eq. (5) to an accuracy of
about 5%. Our numerical accuracy for the surface tension is
in the same range as presented in the recent study by Pecenko
et al. [18]; however, they evaluated the numerical error in a
different manner. The error bars show the smallest and largest
values we obtain for the surface tension using different bubble
sizes at the same temperature.
Observations at temperature T = 0.6Tcshow that our sim-
ulations become inaccurate when we decrease the temperature
further from Tcthan this. The density contrast increases the
further the temperature is from the critical temperature, which
makestheproblemhardertosolvenumerically.Anotheraspect
is that the equations have a singularity at zero density, which
adds another numerical complication. At lower temperatures
the interface thickness reduces significantly, requiring also a
much finer grid resolution. Exploring these different effects
is, however, beyond the scope of the current study and we
note that the range of validity where we verify our method
(T > 0.7) compares favorably with earlier studies [8,17,19].
One reason for this is the adaptive mesh refinement employed
on our work, which makes it computationally easier to resolve
the interface as it becomes thinner and thinner as temperature
is quenched deeper below the critical point.
In Fig. 3(b) we show the data for the pressure difference
in the gas and liquid sides as a function of the predicted
pressure difference from Eqs. (5) and (40). The simple picture
of Laplace pressure is to have the coexistence pressure in
the external liquid domain and the coexistence pressure plus
the Laplace pressure in the internal gas domain. This would
correspond to stars at 1 and diamonds at 0 in the plot.
Figure 3(b) shows that, surprisingly, the Laplace pressure is
recovered by our method better than the coexistence pressure
itself. That is, the pressure difference between the bubble and
liquid bulk is obtained to a much higher degree of accuracy
than the coexistence bulk pressure itself. Nevertheless, the
VdW bulk density values are reproduced to within a few
percent for the gas and a fraction of a percent for the liquid.
The unexpected result is that we consistently observe Laplace
pressure differences corresponding to density differences less
than the accuracy of the coexistence densities.
B. Growing or shrinking of a bubble
ToverifyouropenboundaryconditioninEqs.(23)–(26)we
study an isothermal system in a square 2D domain with three
physical walls and one open boundary with a fixed pressure.
If the pressure is slightly below the coexistence value for the
given temperature, the gas phase is stable and the liquid is
metastable. An initial condition set as a gas bubble immersed
in the metastable liquid will then force the gas bubble either
to grow or shrink depending on the initial size of the bubble.
Referring to the details in Appendix B, we predict the
critical radius of the bubble that will neither grow nor shrink
in 2D to be
?
+ρext− ρg
rpred
crit = −σ
2
[f(ρg,T) − f(ρext,T)]
ρl− ρext[f(ρl,T) − f(ρext,T)]
?
.
(41)
204060 80
20
30
40
50
60
70
80
rcrit
pred
rcrit
FIG. 4. Numerically measured critical bubble radius of a bubble
in a superheated liquid plotted against the prediction from Eq. (41),
here represented by the dashed line. Stars are lower bound observa-
tions and diamonds are upper bound observations.
In order to determine whether the bubbles grow or shrink,
we start with an initial condition that is close to the theoretical
estimate. Exactly at the threshold for the critical bubble radius
acousticwavesandspuriousvelocitycurrentsmighttriggerthe
bubbletoeithergroworshrink.Asthisthresholdisapproached
we notice that the mass fluxes become increasingly slow,
resultinginanunfeasiblesimulationtimetodeterminewhether
the bubble shrinks or grows. Instead, we successively increase
and decrease the bubble size to a point where shrinking or
growing is well defined in the simulations. This gives us upper
and lower bounds for the critical bubble size.
InFig.4weshowthenumericallyobtainedupperandlower
bounds for the critical bubble radius. The markers denote the
numerical result, diamonds illustrate the upper bound, and
squares show the lower bound. The dashed line shows the an-
alytical prediction from Eq. (23). Simulations were preformed
at three different temperatures, and three different superheats.
At temperature T = 0.8 with the coexistence pressure pcoex=
0.383, we use external pressures p = {0.37,0.36,0.35}; at
T = 0.85, pcoex= 0.504, and p = {0.495,0.485,0.475}; and
at T = 0.9, pcoex= 0.647, and p = {0.64,0.635,0.63}. We
note that our method captures the critical bubble sizes well,
considering the sensitivity of the test, but the numerical values
are consistently about 5% to 10% higher than what we predict.
As Appendix B shows, the estimate is also approximate
since, numerically, the pressure in the metastable liquid varies
smoothly between the coexistence and external pressures.
C. Wetting contact angle
In addition to the open boundary condition there is also
another crucial ingredient needed to model boiling; namely
the boundary condition for the contact line. The contact line
is the point where the three phases meet. We adopt here a
methodology similar to what has been commonly employed
026320-7

Page 8

T. LAURILA et al.
PHYSICAL REVIEW E 85, 026320 (2012)


t=0
t=200
t=500
t=1000
t=4000
FIG. 5. Interface evolution of an initially straight interface that
spreads on two walls until it finds its equilibrium shape. The
temperature of the system is T = 0.8Tc. The lower wall has an
equilibrium angle ?e= 60◦and the upper wall has ?e= 120◦.
for the isothermal Cahn-Hilliard method [16], and for now
assume that the temperature at the boundary is fixed. This
impliesthatwecanapriorideterminethecoexistencedensities
for gas and liquid. The wetting boundary condition is obtained
from Eq. (18) and it appears by taking the variation of the free
energy with respect to density.
In Eq. (18), gT(ρ) is an interpolating function between the
two stable phases acting as a switch between having a dry
or wet solid. It is convenient to use higher-order polynomials
for gT(ρ), where we require that, at the coexistence, densities
gT(ρl) = 1 and gT(ρg) = 0. In addition, we require that the
variation of ∂ρgT(ρ) = 0 for ρ = ρland ρ = ρg, making the
boundary condition to be effective only at the interface area.
Often a third-order polynomial is used since it fulfills these
requirements. The third-order polynomial is exact in the case
of the Ginzburg-Landau free energy. For the VdW, the third-
order polynomial is an approximation. Note that a constant
temperature at the wall is needed to set g as above.
As a verification to the wetting boundary condition we
measure the apparent contact angle after the solution has
reached equilibrium. The simulation setup is as follows:
A rectangular domain with constant temperature is applied,
where the domain is separated in the middle with a straight
interface with gas on the left and liquid on the right side of
the interface. The upper wall has an equilibrium angle of
π − ?e and the lower wall has an angle ?e, which makes
the interface a straight line in the steady state. We define the
contact angle on the liquid side of the interface. The interface
goes through nontrivial relaxation dynamics before reaching
equilibrium. In Fig. 5 we show the initial condition and the
interface shape as four snapshots in time. Initially the contact
line moves rapidly over the solid walls due to the capillary
force generated at the wetting foot region of the contact line.
Astheinterfaceapproachesitsequilibriumshape,theinterface
motion decelerates. Finally, the interface finds its equilibrium
shape as seen in Fig. 5.
Weevaluatethenumericalpredictionbasedontheboundary
conditions (19)–(22) for the contact angle for three differ-
ent temperatures and for several surface wettabilities [see
Fig. 6(a)]. The contact angle is measured along the density
contour (ρg+ ρl)/2, where the angle is defined between the
straight line along the interface and the wall on the liquid
side. We notice in Fig. 6(a) that the deviation between the
numerically measured contact angle and the imposed angle is
within about three degrees. The deviation seems to be slightly
dependent on the equilibrium angle, which might imply that
our method for extracting the angle could be refined. Overall,
we find the results in good agreement with expected values. It
should be noted that the constants appearing in the third-order
polynomial gT(ρ) are adjusted according to the coexistence
densities at the respective temperatures and that the change in
temperature did not influence the results significantly.
Figure 6(b) shows the evolution of the apparent contact
angle ? for an equilibrium angle ?e= 60◦at T = 0.9,
where ? − ?eis plotted on the y axis. Initially the interface
is perpendicular to the wall and it quickly approaches the
imposed value, where it eventually takes a steady-state value
0200040006000 8000 10000
0
5
10
15
20
25
30
t
Θ−Θe
40 60 80
(a)
100 120140
−5
−3
−1
1
3
5
Θe
Θ−Θe
T=0.7
T=0.8
T=0.9
(b)
FIG. 6. Contact angle measured in the system with a straight interface, see Fig. 5. (a) Measured difference of the contact angle at numerical
equilibrium and the imposed one as a function of the set contact angle in degrees. Six different ?evalues were measured at three temperatures
each. Note that the numerical contact angle is a few degrees toward the perpendicular compared to what it is set. (b) Time series of the
contact angle as measured along the isocontour for the density (ρg+ ρl)/2 where it meets the top and bottom walls. The initial condition
at perpendicular contact relaxes quickly and obtains a value about two degrees toward the perpendicular compared to what we theoretically
impose.
026320-8

Page 9

THERMOHYDRODYNAMICS OF BOILING IN A VAN DER ...
PHYSICAL REVIEW E 85, 026320 (2012)
withadifferenceinangleof<2◦.Bycomparingthenumerical
results in Fig. 6 to the Laplace test in Fig. 2, we notice
that any oscillations from acoustic waves are not significantly
influencing the results seen in Fig. 6. Acoustic waves are also
present in the simulations presented in Fig. 6, but they give a
very small contribution compared to the flow generated by the
spreading motion.
V. DYNAMICS OF BOILING
We numerically investigate a bubble near a hot (de)wetting
wall in a closed and an open system. The computational
domain is again a 2D square. The lower boundary is the
hot wall and has constant temperature. The boundaries to the
left and right are insulated walls with an equilibrium angle
?e= 90◦,whereallwallsareimposedwithano-slipcondition
for the velocity. A temperature gradient is imposed between
the upper and lower boundaries corresponding to a heated
system. The density of the liquid phase changes from top to
bottom as a function of the initial temperature profile such that
the pressure in the liquid is constant. A coexistence gas bubble
is initiated close to the wall to mimic the first nucleate. This
bubble is at first not in direct contact with the wall, but slowly
drifts toward the hot wall by the unbalanced surface tension at
its front and rear, thus generating a type of Marangoni flow. A
hyperbolic tangent interpolation is used to set initial interface
width close to the predicted numerical solution. In the closed
system this bubble has a radius r = 50 that is slightly larger
than for the open system. The reason for this choice is that the
mass fractions of gas and liquid phases in the closed system
remain constant and we want to study the evolution of a larger
bubble. In the open system the initial bubble radius is r = 30,
but the bubble grows similar to a boiling process contrary to
the closed system.
The objective is to first observe the difference between
the closed and the open systems emphasizing the role of
the constant pressure boundary in describing the physics of
boiling. Then we verify that we obtain both film and nucleate
boiling asfunctions ofthewettabilityofthehotwall.Weshow
simulations with two different equilibrium angles ?e= 45◦
and?e= 177◦.OntheupperwallthetemperatureisT = 0.88
and the lower hotter wall is at a temperature T = 0.9. The
pressureattheopenboundaryissettocorrespondtotheboiling
temperature of T = 0.89.
Figure 7 shows the simulation result for the spreading in a
closed system with a thermal gradient as the bubble comes in
contact with the lower wall, which has an equilibrium angle of
?e= 177◦.Thewallfavorsthegasphaseandthebubblestarts
to spread rapidly along the wall. The temperature decreases
at the foot region of the bubble, where it has the smallest
radius of curvature, as shown in Fig. 7(a). In this region the
mass flux is also the greatest; see Fig. 7(d). As the bubble
continuestospread,thetemperatureinsidethebubblebecomes
more and more uniform, as shown in Fig. 7(b). Figure 7(e)
shows two vortices are generated, showing the evaporation
close to the contact line and the condensation at the top of the
bubble.
Later in time the bubble has almost spread across the
domain and formed a gas film at the hot solid surface; see
Fig. 7(c). As shown in Fig. 7(e) two evaporation-condensation
vortices persist in the steady state. The vortices resemble
Rayleigh-B´ enard convection rolls, but they are created by the
(a) t=250(b) t=750(c) t=5500
(d) t=250(e) t=750(f) t=5500
FIG. 7. (Color online) Closed system with ?e= 177◦on the hot wall where the interface is illustrated by the contour line drawn at the
mean between the liquid and gas coexistence densities ρ = (ρl+ ρg)/2. Panels (a), (b), and (c) show the temperature profile with equidistant
isotherms between T = 0.9 and T = 0.88. Panels (d), (e), and (f) show mass flux vectors on the right-hand side of the vertical symmetry axis
at the middle of the system.
026320-9

Page 10

T. LAURILA et al.
PHYSICAL REVIEW E 85, 026320 (2012)
(a) t=1000 (b) t=4500(c) t=13000
(d) t=1000(e) t=4500(f) t=13000
FIG. 8. (Color online) Closed system with ?e= 45◦on the hot wall where the interface is illustrated by the contour line drawn at the mean
between the liquid and gas coexistence densities ρ = (ρl+ ρg)/2. Panels (a), (b), and (c) show temperature profiles with equidistant isotherms
between T = 0.9 and T = 0.88. Panels (d), (e), and (f) show mass flux vectors on the right-hand side of the vertical symmetry axis in the
middle of the system.
two-phase structure and temperature gradient—no gravity is
present here.
Bymakingthelowerwallmoreattractedtotheliquidphase,
the dynamics observed in the simulations change (see Fig. 8).
The lower wall here has an equilibrium contact angle of ?e=
45◦andallotherparametersarethesameaspresentedinFig.7.
Thebubblespreadshereinamuchslowerfashion,butthesame
evaporation to condensation flow is observed, [see Figs. 8(b)
and 8(d)]. At the final stage of the spreading and close to
steady state the mass flux at the contact line changes direction
as shown in Figs. 7(c) and 7(e). Two vortices are present in the
steadystateasforthecaseofthehydrophobicwallinFig.7,but
thevortices areintheopposite direction.Asimilarsystemwas
experimentally investigated by Nikolayev et al. [6]. However,
the length scales of observation of their cryogenic hydrogen
experiment are larger by orders of magnitude than our current
numerical system, making direct comparison impossible.
We note that the bubble evolution in the closed system
as shown in Figs. 7 and 8 strongly resembles an isothermal
spreading event, and that the mass fraction with gas seems
fairly constant. The observed dynamics is also consistent with
observations made by Ref. [8] for a similar VdW system.
In Figs. 9 and 10 we show the cases equivalent to those
of Figs. 7 and 8, with the important difference that we now
have the constant pressure boundary condition at the top.
This allows mass flux in and out of the domain and thus this
corresponds to boiling at a constant pressure.
Figure 9 shows boiling at a wall with an equilibrium
angle ?e= 177◦. As the gas bubble comes in contact with
the hot solid wall the temperature profile and the mass flux,
shown in Figs. 9(a) and 9(d), are indeed similar to those in
the corresponding closed system. But as the bubble starts to
spread, a rapid mass transfer from liquid to gas takes place
and leads quickly to the formation of a thin gas film on the hot
wall, see Figs. 9(b) and 9(c). At time t = 2000 the surface is
entirely covered by gas and, in analogy to the dryout process,
a sharp temperature gradient is created as the gas insulates the
hot wall from the interface where boiling occurs [3]. After the
interface comes in contact with the insulated walls at the two
sidesofthedomain,therateofphasechangedecreasesrapidly,
corresponding to film boiling. Notice that the interface has a
nearly constant temperature at its boiling point T = 0.89 [see
Fig. 9(e)] and a linear temperature profile in accordance to
Fourier’s law while a drastically higher temperature gradient
in the gas than in the liquid is about to be established.
By making the hot boiling surface more hydrophilic, the
qualitative features of boiling dynamics change dramatically
(see Fig. 10). Initially, the bubble starts to wet the hot wall
as shown in Figs. 10(a) and 10(c), although the phase change
is much less than for the hydrophobic wall. The temperature
takes a nearly linear profile inside the gas, where it is distorted
close to the contact line, as shown in Fig. 10(b). The majority
of the phase change takes place at the contact line. As the
bubble evolves past the initial contact with the wall a steady
boiling regime is established, characterized by a semicircular
interface at a temperature close to the boiling point, as shown
in Figs. 10(b) and 10(c). This boiling regime is characteristic
of nucleate boiling, where the bubble would grow to orders
of magnitude larger than our numerical system until gravity
detaches it.
026320-10

Page 11

THERMOHYDRODYNAMICS OF BOILING IN A VAN DER ...
PHYSICAL REVIEW E 85, 026320 (2012)
(a) t=250(b) t=750 (c) t=2000
(d) t=250(e) t=750 (f) t=2000
FIG. 9. (Color online) Open system with ?e= 177◦on the hot wall where the interface is illustrated by the contour line drawn at the mean
between the liquid and gas coexistence densities ρ = (ρl+ ρg)/2. Panels (a), (b), and (c) show temperature profiles with equidistant isotherms
between T = 0.9 and T = 0.88. Panels (d), (e), and (f) show mass flux vectors on the right-hand side of the vertical symmetry axis in the
middle of the system.
The latent heat of boiling keeps the temperature at the gas-
liquid interface close to the boiling point even when the inter-
facereachesdowntothehotwallatthecontactline.Thismeans
that most of the thermal energy flowing into the fluid from the
hot wall, and thus most of the boiling, happens at the contact
line. Notice that there is a strong peak in mass flux going into
(a) t=750(b) t=4500(c) t=13000
(d) t=750
(e) t=4500(f) t=13000
FIG. 10. (Color online) Open system with ?e= 45◦on the hot wall where the interface is illustrated by the contour line drawn at mean
between the liquid and gas coexistence densities ρ = (ρl+ ρg)/2. Panels (a), (b), and (c) show temperature profiles with equidistant isotherms
between T = 0.9 and T = 0.88. Panels (d), (e), and (f) show mass flux vectors on the right-hand side of the vertical symmetry axis in the
middle of the system.
026320-11

Page 12

T. LAURILA et al.
PHYSICAL REVIEW E 85, 026320 (2012)
(a) Mass flux(b) Velocity
FIG. 11. Closer look at the contact line region of Fig. 10(f). The
mass flux magnitude profile (a) and the velocity magnitude profile
(b) are shown near the contact line in the steady nucleate boiling
regime. The black line is the interface as in Fig. 10. The arrows are
schematic guides to the eye of the direction of the flow. The flow is
into the bubble at the foot and out of the bubble further away from
the contact line. This prominent effect is interpreted as vapor recoil.
the bubble at the contact line in the steady-boiling regime.
While average mass flux must be away from the growing
bubbleduetomassconservation,weobservetheoppositeatthe
contactlinefootofthebubble,andaflowvortexisgeneratedat
the foot. This flow is consistent with the mechanism of vapor
recoil and we interpret our results as verification that vapor
recoilispresentatthemicroscaleflowsofnucleateboilingand
leadstoanapparentcontactanglethatappearsmorehydropho-
bicthanthemicroscopicequilibriumcontactangleweimpose.
The flow at the contact line in the steady nucleate boiling
regime is shown in more detail in Fig. 11, where we plot the
magnitudes of the mass flux and the velocity. A prominent jet
into the bubble from the contact angle is seen in the velocity
profile. The steep gradient in the mass flux at the contact line
causes a force that distorts the interface leading to an apparent
contact angle. The interface profile meets the boiling surface
at the equilibrium contact angle if observed at the scale of
the interface width, which is the smallest physical scale. This
we have imposed via the wetting boundary condition. If the
contact line is observed further away, distortion of the angle is
seen clearly beyond what is expected due to the finite radius
of the bubble. This is due to dynamical effects at the contact
line in the steady nucleate boiling regime. Qualitatively the
full thermal hydrodynamics are thus consistent with the vapor
recoil picture [10]. Quantification of this effect is beyond the
scope of this paper.
Figure 12 shows the time evolution of total mass in the four
computational systems we considered here. m0is the initial
mass of the system and m(t) =?ρ(t,x,y)dV is the mass at a
is indeed conserved. For the open systems the hydrophobic
wall causes a rapid initial phase change as the gas spreads
on the wall and film boiling sets on. The inefficiency of film
boilingisapparentasarapiddecreaseinthephase-changerate
at later times. As the insulating gas layer grows, so does the
insulation, and the boiling rate continuously decreases.
Figure 12 shows a linear boiling regime for the hydrophilic
boiling surface. The linear rate is consistent with the boiling
happening dominantly at the contact line point. Since the
interface adjusts to the boiling temperature, the heat flux in
the contact line region remains nearly constant in time as the
given time. The closed systems are included to show that mass
02000 40006000 800010000
−0.06
−0.05
−0.04
−0.03
−0.02
−0.01
0
(m(t)−m0)/m0
t
Closed:Θe=177
Closed:Θe=45
Open:Θe=177
Open:Θe=45
FIG. 12. Rate of mass change in systems with temperature
gradients.Boilingisseenaschange intotalmassintheopensystems.
contact line moves as the bubble grows. In fact, all the flow
properties in the contact line region remain the same as the
contact line advances in the steady-boiling regime.
VI. SUMMARY AND DISCUSSION
We have presented a numerical model for VdW fluids that
enables study of boiling phenomena under constant, tunable
external pressure and at a boiling surface of tunable wetting
properties. The main development is here the implementation
of an open boundary under constant external pressure that
allows a flux of mass in and out of the domain. We propose
also a boundary condition for the contact line, which lets us
prescribe the equilibrium contact angle for the solid substrate.
The numerical scheme is based on a modification of
the characteristic-based-splitting scheme, first developed for
single-phase compressible flow by Ref. [24]. We verify our
methodology by several isothermal validation cases. First, we
study the Young-Laplace law of a static bubble submerged in
its coexistence liquid. This allows us to measure the surface
tension coefficient, where the results were found in to be
good agreement with theory. Notice that the surface tension
coefficient was reproduced with a similar level of accuracy as
in the recent work [18].
An open boundary condition based on thermodynamics
has been proposed to set a constant external pressure for
the system, allowing mass flux into or out of the system.
We verify this boundary condition by considering bubbles in
metastable, superheated liquid. The Gibbs free energy yields
an estimate for the limiting size of initial gas bubble that,
absent fluctuations, will overcome the interface tension and
grow toward the stable gas state as opposed to shrinking
toward the metastable liquid state. We find our numerical test
cases agreeing reasonably well with the estimate at different
temperatures and superheats.
Contact lines and the solid-surface energy are believed to
be important elements in a boiling process. It is therefore
important to allow for modeling of solid walls with different
equilibrium contact angles. A model for the wetting is pro-
posedhereforVdWfluids,whichisbasedonthemethodology
026320-12

Page 13

THERMOHYDRODYNAMICS OF BOILING IN A VAN DER ...
PHYSICAL REVIEW E 85, 026320 (2012)
usedintheisothermalCahn-Hilliardmodelingofwetting[16].
We verify in a numerical test case, where the initial interface
shape is much different from the equilibrium shape, that we
indeed capture the imposed equilibrium contact angle.
Using the VdW fluid as benchmark, parametrized to have
properties of the correct order of magnitude compared to
water across the coexistence region, we study a numerical
system that we believe contains the relevant physics of boiling
at the microscopic scales. Initially we seed the simulation
with a bubble nucleate close to the hot wall. We study both
the dynamics in a closed and open system with a thermal
gradient. The results indicate that the open constant-pressure
boundary brings out the physics instrumental to boiling, and
we find both film and nucleate boiling regimes as function
of surface wettability. In particular, the flow profile close the
the three-phase contact line at the hot wall shows the vapor
recoil mechanism [6,10,30] as it arises from the full thermal
hydrodynamics.Inthecaseofboilingonthehydrophobicsolid
we observed a gas layer that is rapidly formed and covers the
hot surface and then acts to insulate the gas-liquid interface
from the hot wall, drastically reducing the boiling rate.
Whileourmodelcapturesmuchoftheboilingphysics,lim-
itationsstillexistinnumericallysimulatingpracticalboilingas
ithappensinindustriallyrelevantconditionsanddevices.First,
no thermal fluctuations exist in the model and as such nucle-
ationcannotbemodeled.Thismeansthattheveryinitialstages
of bubbles as they are formed cannot be predictively captured.
Given that nucleation sites in heterogenous boiling are deter-
ministically activated or deactivated as functions of superheat,
thelackofnucleationinthemodelisnotexpectedtobecrucial.
The more important question is whether a site remains active
and emits a steady trail of bubbles by pinch-off once seeded,
which the model could rather directly be applied to. Second,
the diffuse interface methodology is also limited in how large
systems can be studied, since the interface width, which
must be numerically resolved, is limited to physical scales if
physical surface tension is desired. This means that we cannot
reachscaleswherebubbledeparturebypinch-offduetogravity
happens or where coalescence of several bubbles is important.
However, if the vapor recoil picture of dryout is correct, the
transition happens at the microscale of single-bubble growth,
and this regime can be reached by our numerical methodology
whereas it is notoriously difficult by experiments.
Two crucial extensions for the methodology presented
here are desirable, but beyond the scope of this paper.
They represent the third and fourth limiting elements of the
method. Unlike the physical scales, these should be relatively
straightforward to address in future work. Third is extending
the numerical solution of the model to cylindrical coordinates,
as done in Ref. [17]. In cylindrical coordinates the numerical
load is not much larger than here, but the boiling at the contact
line is expected to be much better described. In particular,
the extension of the length of the contact line as the bubble
grows is crucial. As the contact line becomes longer the
total heat transfer from the hot wall, and thus the boiling
rate increases and so does the force of the interface tension
attempting to keep the bubble spherical. Balance of these is
expected to be crucial in how vapor recoil is suggested to
cause dryout by single-bubble spreading. Fourth and finally,
a hot wall of constant heat flux as opposed to constant
temperaturewouldbeinterestinganduseful,especiallyiffinite
heat conductivity along the wall could also be included. This
calls for a significantly more complex boundary condition if
wetting properties are also to be included, however.
Another interesting possibility for future studies is compar-
isonwithmoleculardynamicssimulations,whichwithmodern
computers should be feasible in the microscopic scales here
considered.Recently,itwasshownbyRef.[31]thatmolecular
dynamics simulations capture the molecular scale density
fluctuations in bulk phases of the square gradient energy
model. We note that our contact angle boundary condition
assumes local equilibrium by minimizing surface energy. As
such it enforces the equilibrium contact angle at the smallest
scale. The results we get appear reasonable, but detailed
comparison with experiments might require a finite relaxation
timetowardequilibriumatthecontactlinefollowingRef.[32].
ACKNOWLEDGMENTS
This work has been supported in part by the Academy of
Finland through its COMP Center of Excellence grant and by
the Finnish Funding Agency for Technology and Innovation
(TEKES) via its NanoFluid Consortium Grant.
APPENDIX A: NUMERICAL SCHEME
We apply a modification of the characteristic-based split
(CBS) method of Nithiarasu et al. [23,24] to solve the
governingequations.InitsoriginalformtheCBSmethoddoes
not consider the energy equation as part of the flow solution;
rather, it is treated as a passive scalar. We find that the latent
heat strongly couples the energy and density equations and,
thus, under certain circumstances, a significant improvement
in numerical efficiency is obtained by treating the energy
equation in the same manner as the density equation in the
correction step of the CBS method.
Here we will present the method for Onuki’s set of the Van
der Waals–Navier-Stokes equations (12)–(15). The CBS time
integrationstepfromtimestepnton + 1startswithanexplicit
predictormassfluxU∗
equation
?
+?t
?
i(Ui= ρvi)solvedfromthemomentum
?
??
+?t
U∗
i= Un
i+ ?t
?
−∂j
?Un
ivn
j
?+ ∂jηn
p − κ
∂i
?U∗
j
ρn
?
+ ∂j
?U∗
??
i
ρn
???
Fi− ∂j
?
ρ∂k∂kρ +1
2∂kρ∂kρδij
?n
+κ∂iρ∂jρ
2vk∂k[∂j(vjUi) + ∂jPij− ρgi]
.
(A1)
Here, U is the mass flux and superscripts denote the time step.
The viscosity term is calculated implicitly for U∗to increase
the stable time step also for viscosity-dominated flows.
Next, the correction is made for the mass flux. It involves
calculating the density and energy fields at time step n + 1
using the predictor mass flux U∗. Since the pressure is deter-
minedbydensityandtemperaturefields,wecouldconsiderthe
pressure in Eq. (15) merely a shorthand and solve for density
and temperature equations. This straightforward approach is
not convenient, however, for two reasons. First, in our finite
026320-13

Page 14

T. LAURILA et al.
PHYSICAL REVIEW E 85, 026320 (2012)
element formulation all variables are approximated with a
linear approximation function. Since higher-order derivatives
appear in the derivative of stresses we need to introduce
additional variables in order to evaluate the derivatives of
stresses. Second, for our boundary conditions we need to
impose a Dirichlet boundary condition for the pressure at
the open boundary. This is greatly simplified by having the
pressure as an additional variable in our system of equations.
Thus, we split the equations such that the pressure tensor is an
additional variable.
Inserting the correction of the mass flux to the density
and temperature equations gives the correction step. The flux
correction step is
i= ?t?∂jPn+1
where Pn+1needs to be determined and depends on ρn+1and
Tn+1. The correction step involving the pressure, density, and
temperature equations is
?
+ρm
?U∗
e(ρn+1,Tn+1) = e(ρn,Tn) + ?t
?
?
Un+1
i
− U∗
ij
− ∂jPn
ij
?,
(A2)
Pn+1
ij
=
p(ρm,Tm) − κρm∂k2ρm−1
2κ∂kρm∂kρm
Tm∂kρm∂kTm
?
δij+ κ∂iρm∂jρm,
j− ?t?∂kPn+1
−∂j
?U∗
?U∗
(A3)
ρn+1= ρn− ?t∂j
jk
?
− ∂kPn
e(ρm,Tm)U∗
jk
??,
j
ρm
(A4)
?
?
−
Pn+1
jk∂k
j
ρm
?
??
+ α0∂j(ρm∂jTm)
?U∗
+η0
∂j
k
ρm
+ ∂k
j
ρm
??
∂j
?U∗
k
ρm
??
(A5)
.
The above correction step can be taken in two different ways
depending on the choice of the time step m used. If we use the
previous time step in the correction step (m = n) then each of
the above equations can be straightforwardly solved. If we in-
steaduseanimplicitmethod(m = n + 1)thentheaboveequa-
tions need to be solved together as a coupled set and each de-
pends on the solution of the others. The implicit method leads
to a significant increase in the maximal stable time step of the
method,butthecoupledsolutioniscomputationallyexpensive.
By numerical trial we find that the implicit method is
advantageous when the liquid-gas interface is far from the
wallsofthecomputationaldomain.Thelargertimestepgained
by the implicit method is lost when a contact line is present
in the system, however, and the explicit method is found to be
computationally more efficient.
To summarize, the CBS solution method starts by taking a
predictor step (A1) for the intermediate mass flux U∗. Then a
corrector step (A3)–(A5) gives the density and temperature at
the new time step. Finally, the mass flux, and thus velocity, at
the new time step is obtained by the flux correction (A2).
By numerical trial we put the limit of the feasibility of the
present implementation at temperatures T ? 0.7Tc. The lower
below the critical point we quench the system the higher all
the contrasts between the phases become, and the harder is the
numerical task. Consider the CFL condition with the highest
speedinthesystem[i.e.,thespeedofsoundintheliquidphase,
which at the limit is vc(T = 0.7Tc) ≈ 3.15], and the smallest
element mesh size when adapted to the interface, which is
?x ≈ 0.1. Our implicit solution method works with time step
?t = 0.5 under these conditions, whereas the explicit method
needs ?t = 0.01. Referred to these the CFL number of the
implicit method is CFL ≈ 16, and the explicit CFL ≈ 0.3.
TheseCFLnumbersofferaguidelinetothenumericalmethod,
but recall that we are not attempting to correctly resolve the
time evolution of the compressibility waves in this study.
However, we do desire that the acoustic modes present in
the compressible hydrodynamic equations do not make our
numerical solver diverge.
APPENDIX B: ESTIMATE OF CRITICAL
SHRINKING OR GROWING BUBBLE
The change in Gibbs free energy of the system upon a local
increase of bubble radius can be used to predict whether a
bubbleofgivensizeimmersedinsuperheatedliquidwillshrink
or grow, assuming the system will evolve in the direction of
locally decreasing Gibbs free energy. The Gibbs free energy
density is the Helmholtz free energy plus the pressure, where
wecanshiftzeroofGibbstoremoveambientpressure,leaving
only the Laplace pressure:
g = f + ?p.
Considering an isothermal system at temperature T we have
the coexistence gas and liquid densities ρgand ρl. Assuming
an initial condition of a bubble of radius r immersed in
metastable liquid at density ρext< ρl, the system can decrease
its Helmholtz free energy by increasing the bubble size,
thereby phase separating some of the metastable liquid to
the coexistence densities. However, this costs free energy in
interface tension as the bubble circumference grows, which is
(B1)
ext
ρ
ρ
ρ
ext
liq
gas
y
r
dr
y
r
dr
ρgas
ρliq
ρ
FIG. 13. Schematic to estimate whether a bubble immersed in
metastable liquid will locally gain or lose Gibbs free energy by
growing. On the left-hand side initially the bubble density profile is
thought to be the solid line, with proper interface tension associated
with the sharp interface. Growing the bubble by dr causes the
metastable liquid to phase separate to liquid and gas (as per
coexistence at this temperature), thereby gaining free energy. The
result of the bubble growing is the dashed line with a square
hump at the interface. Shaded boxes are of equal area for mass
conservation. Curved dash-dotted line schematically shows how the
density profile actually looks numerically with a bubble and an
open bound, showing the approximate nature of the estimate. The
right-hand side is the potentially growing bubble shown from above,
with the one-dimensional profile cut shown.
026320-14

Page 15

THERMOHYDRODYNAMICS OF BOILING IN A VAN DER ...
PHYSICAL REVIEW E 85, 026320 (2012)
taken into account by integrating ?p ∝ r−1over the bubble
area ∝r2. Local mass conservation determines how much
liquid is created for every unit of gas. Increasing the size of the
gasbubblefromr tor + ?r,theGibbsfreeenergychangesby
?G = πσ?r + 2πr?r
+ρext− ρg
?
[f(ρg,T) − f(ρext,T)]
ρl− ρext[f(ρl,T) − f(ρext,T)]
?
,
(B2)
where (ρext− ρg)/(ρl− ρext) is the amount of metastable
liquid turned to coexistence liquid for every unit of
metastable liquid turned to coexistence gas according to
mass conservation, which is obeyed locally at the bubble
circumference even though the system can exchange mass at
an open boundary. Considering where ?G changes sign upon
positive ?r, we predict the critical radius of the bubble that
will neither grow nor shrink to be
?
+ρext− ρg
This estimate and how it approximates the numerical density
profiles is illustrated in the schematic in Fig. 13.
rpred
crit = −σ
2
[f(ρg,T) − f(ρext,T)]
ρl− ρext[f(ρl,T) − f(ρext,T)]
?
.
(B3)
[1] V. K. Dhir, Annu. Rev. Fluid Mech. 30, 365 (1998).
[2] T. G. Theofanous, T. N. Dinh, J. P. Tu, and A. T. Dinh, Exp.
Therm. Fluid Sci. 26, 775 (2002).
[3] T. G. Theofanous, J. P. Tu, A. T. Dinh, and T. N. Dinh, Exp.
Therm. Fluid Sci. 26, 775 (2002).
[4] H. T. Phan, N. Caney, P. Marty, S. Colasson, and J. Gavillet,
C. R. Mecanique 337, 251 (2009).
[5] V. S. Nikolayev, D. Beysens, Y. Garrabos, C. Lecoutre, and
D. Chatain, Microgravity Sci. Technol. 18, 34 (2006).
[6] V. S. Nikolayev, D. Chatain, Y. Garrabos, and D. Beysens, Phys.
Rev. Lett. 97, 184503 (2006).
[7] D. M. Anderson, G. B. McFadden, and A. A. Wheeler, Annu.
Rev. Fluid Mech. 30, 139 (1998).
[8] A. Onuki, Phys. Rev. E 75, 036304 (2007).
[9] G. Tryggvason, B. Bunner, A. Esmaeeli, and N. Al-Rawahi,
Adv. Appl. Mech. 39, 81 (2003).
[10] V. S. Nikolayev and D. A. Beysens, Europhys. Lett. 47, 345
(1999).
[11] J. D. Van der Waals, Verh.-K. Ned. Akad. Wet., Afd. Natuurkd.,
Eerste Reeks 1, 56 (1893); J. S. Rowlinson, J. Stat. Phys. 20,
197 (1979).
[12] D. Korteweg, Arch. Neerl. Sci. Exactes Nat. Ser II 6, 1 (1901).
[13] V. L. Ginzburg and L. D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064
(1950).
[14] J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 28, 258 (1958).
[15] J. E. Dunn and J. Serrin, Arch. Ration. Mech. Anal. 88, 95
(1985).
[16] D. Jacqmin, J. Comput. Phys. 155, 96 (1999).
[17] R. Teshigawara and A. Onuki, Phys. Rev. E 82, 021603
(2010).
[18] A. Pecenko, J. G. M. Kuerten, and C. W. M. van der Geld, Int.
J. Multiphase Flow 36, 558 (2010).
[19] A. Pecenko, L. G. M. van Deurzen, J. G. M. Kuerten, and C. W.
M. van der Geld, Int. J. Multiphase Flow 37, 149 (2010).
[20] S. Merabia and I. Pagonabarraga, Eur. Phys. E 20, 209
(2006).
[21] R. Teshigawara and A. Onuki, e-print arXiv:1108.0213v1
(2011).
[22] T. Colonius, Annu. Rev. Fluid Mech. 36, 315 (2004).
[23] P. Nithiarasu, R. Codina, and O. C. Zienkiewicz, Int. J. Numer.
Meth. Engng. 66, 1514 (2006).
[24] O. C. Zienkiewicz et al., The Finite Element Method for Fluid
Dynamics, 6th ed. (Elsevier, 2005).
[25] G.Amberg,R.Tonhardt,andC.Winkler,Math.Comput.Simul.
49, 149 (1999).
[26] M. Do-Quang, W. Villanueva, I. Singer-Loginova, and
G. Amberg, Bull. Pol. Acad. Sci-Te 55, 229 (2007).
[27] P. Yuan and L. Schaefer, Phys. Fluids 18, 042101 (2006).
[28] W. Villanuava and G. Amberg, Int. J. Multiphase Flow 32, 1072
(2006).
[29] P. Papatzacos, Phys. Scr. 61, 349 (2000).
[30] H. J. Palmer, J. Fluid Mech. 75, 487 (1976).
[31] L.-T. Kong, D. Vriesinga, and C. Denniston, Europhys. Lett. 93,
50004 (2011).
[32] A. Carlson, D.-Q. Minh, and G. Amberg, Phys. Fluids 21,
121701 (2009).
026320-15