# 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.

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**ABSTRACT:**Using a continuum model capable of describing the one-component liquid-gas hydrodynamics down to the contact line scale, we carry out numerical simulation and physical analysis for the droplet motion driven by thermal singularity. For liquid droplets in one-component fluids on heated or cooled substrates, the liquid-gas interface is nearly isothermal. Consequently, a thermal singularity occurs at the contact line and the Marangoni effect due to temperature gradient is suppressed. Through evaporation or condensation in the vicinity of the contact line, the thermal singularity makes the contact angle increase with the increasing substrate temperature. This effect on the contact angle can be used to move the droplets on substrates with thermal gradients. Our numerical results for this kind of droplet motion are explained by a simple fluid dynamical model at the droplet length scale. Since the mechanism for droplet motion is based on the change of contact angle, a separation of length scales is exhibited through a comparison between the droplet motion induced by a wettability gradient and that by a thermal gradient. It is shown that the flow field at the droplet length scale is independent of the statics or dynamics at the contact line scale.Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics 06/2012; 85(6). - [Show abstract] [Hide abstract]

**ABSTRACT:**Droplet motion on solid substrates has been widely studied not only because of its importance in fundamental research but also because of its promising potentials in droplet-based devices developed for various applications in chemistry, biology, and industry. In this paper, we investigate the motion of an evaporating droplet in one-component fluids on a solid substrate with a wettability gradient. As is well known, there are two major difficulties in the continuum description of fluid flows and heat fluxes near the contact line of droplets on solid substrates, namely, the hydrodynamic (stress) singularity and thermal singularity. To model the droplet motion, we use the dynamic van der Waals theory [ Phys. Rev. E 75 036304 (2007)] for the hydrodynamic equations in the bulk region, supplemented with the boundary conditions at the fluid-solid interface. In this continuum hydrodynamic model, various physical processes involved in the droplet motion can be taken into account simultaneously, e.g., phase transitions (evaporation or condensation), capillary flows, fluid velocity slip, and substrate cooling or heating. Due to the use of the phase field method (diffuse interface method), the hydrodynamic and thermal singularities are resolved automatically. Furthermore, in the dynamic van der Waals theory, the evaporation or condensation rate at the liquid-gas interface is an outcome of the calculation rather than a prerequisite as in most of the other models proposed for evaporating droplets. Numerical results show that the droplet migrates in the direction of increasing wettability on the solid substrates. The migration velocity of the droplet is found to be proportional to the wettability gradients as predicted by Brochard [ Langmuir 5 432 (1989)]. The proportionality coefficient is found to be linearly dependent on the ratio of slip length to initial droplet radius. These results indicate that the steady migration of the droplets results from the balance between the (conservative) driving force due to the wettability gradient and the (dissipative) viscous drag force. In addition, we study the motion of droplets on cooled or heated solid substrates with wettability gradients. The fast temperature variations from the solid to the fluid can be accurately described in the present approach. It is observed that accompanying the droplet migration, the contact lines move through phase transition and boundary velocity slip with their relative contributions mostly determined by the slip length. The results presented in this paper may lead to a more complete understanding of the droplet motion driven by wettability gradients with a detailed picture of the fluid flows and phase transitions in the vicinity of the moving contact line.Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics 05/2012; 85(5). - [Show abstract] [Hide abstract]

**ABSTRACT:**Using the dynamic van der Waals theory [Phys. Rev. E 75, 036304 (2007)], we numerically investigate the hydrodynamics of Leidenfrost droplets under gravity in two dimensions. Some recent theoretical predictions and experimental observations are confirmed in our simulations. A Leidenfrost droplet larger than a critical size is shown to be unstable and break up into smaller droplets due to the Rayleigh-Taylor instability of the bottom surface of the droplet. Our simulations demonstrate that an evaporating Leidenfrost droplet changes continuously from a puddle to a circular droplet, with the droplet shape controlled by its size in comparison with a few characteristic length scales. The geometry of the vapor layer under the droplet is found to mainly depend on the droplet size and is nearly independent of the substrate temperature, as reported in a recent experimental study [Phys. Rev. Lett. 109, 074301 (2012)]. Finally, our simulations demonstrate that a Leidenfrost droplet smaller than a characteristic size takes off from the hot substrate because the levitating force due to evaporation can no longer be balanced by the weight of the droplet, as observed in a recent experimental study [Phys. Rev. Lett. 109, 034501 (2012)].Physical Review E 04/2013; 87(4-1):043013. · 2.31 Impact Factor

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

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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

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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

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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

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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

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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.

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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

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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.

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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

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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.

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(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.

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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

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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

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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.

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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)

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