Diffuse interface models in fluid mechanics

is no longer the case and some of them are now discussed.At a critical point, the properties of both phases become equal. Above the critical point, atwo-phase system cannot exist and the fluid exists only as a one-phase fluid. On the contrary, belowthe critical point, under certain conditions, the fluid can co-exist under two different phasesseparated by an interface. As the two-phase system system approaches the critical point frombelow, the thickness of the interface actually increases and becomes infinite at the critical point.Thus, just below the critical point, the typical size of the transition layer that separates the bulkphases becomes of the same order of magnitude as the typical size of the bulk phases. Therefore,the interface cannot be modeled as a surface of discontinuity and its internal structure has to bedescribed .Let us consider two air bubbles in liquid water. It is commonly observed that two bubbles cancoalesce, i.e. they merge to give rise to a single bubble. If the interfaces are modeled as a surfaceof discontinuity, just at the moment where they merge, the model is singular. In a sense, one ofthe interfaces desappears and this is not possible if the interface is discontinuous. To overcomethis singularity, one has to study the detailed interaction of the interfaces during their merging.This can be done only by accounting for the internal structure of the interfaces [Lee et al., 2002a,b].Another situation, actually similar to the previous one, is the description of the creation of asecond phase in an initially single-phase system; this is called nucleation. In this case, one hasto describe how, from a single phase, an interface is created. This continuous process can bemodeled only if the interface under construction is modeled as a continuous medium [Dell’Isolaet al., 1996].1.3.2 Numerical limitationsIn the previous section, we have shown that, in some cases, the idealization of an interface asa surface of discontinuity is physically irrelevant and the detailed structure has to be described.However, in many applications, these phenomena can be neglected or do not occur. Nevertheless,the coupled partial differential equations that describe the two-phase flow are highlynon-linear and numerical simulation is often necessary to solve them. The numerical simulationof two-phase flows is very challenging because it is a moving boundary problem. Several numericaltechniques exist to solve this kind of problems and their description is beyond the scopeof this presentation. These techniques are often difficult to implement numerically, especially inthree space dimensions, and sometimes depend on the know-how of the code developper. Thisis partially due to the lack of a clear mathematical background for some of the methods. Nevertheless,the boundary conditions that must be applied at the moving interfaces need a particulartreatment in the numerical algorithm, which is difficult and often tedious. This is why diffuseinterface methods can be numerically attractive. If one can come up with a system of partial differentialequations that is valid in the entire two-phase system, including within the continuoustransition interfacial zones, the motion of the entire two-phase system would be describe by thissingle system of equations, which thus eliminates the difficult problem of the particular treatmentof the boundary conditions at the interfaces. The programming effort would therefore be highlydecreased. Moreover, if these equations are obtained from first principles, the development ofaccurate numerical schemes can be based on a better mathematical ground [Jamet et al., 2002].2 Liquid-vapor flows with phase-change: the van der Waals modelof capillarityIn this section, we present the van der Waals model of capillarity. This model is a diffuse interfacemodel dedicated to the description of an interface that separates a liquid and a vapor phase of8

a pure fluid. Extensions to binary mixtures is possible but will not be presented here [Fouilletet al., 2002]. It is interesting to note that this model is the first diffuse interface developed byvan der Waals [van der Waals, 1894].2.1 Thermodynamic modelAny diffuse interface model is actually a thermodynamic model. Indeed, the internal structureof an interface is mainly an equilibrium feature. Dynamic effects only perturb this equilibriumstructure, which is thus important to characterize.2.1.1 A mean-field approximationThe main issue is the following: is it possible to describe the internal structure of a liquid-vaporinterface at equilibrium by considering a “classical” thermodynamic description of the fluid? By“classical”, we mean that the energy of a fluid particle depends only on local variables such as thedensity ρ and the temperature T . Van der Waals showed that it is actually impossible [Rowlinsonand Widom, 1982]: the interface would be sharp and surface tension would be null. That is whynon-local terms have to be considered. In the case of a liquid-vapor interface, van der Waalspostulated the following thermodynamic description:F = F 0 (ρ, T ) + λ 2 (∇ρ)2 (14)where F is the volumetric free energy of the fluid, F 0 is its “classical” part and λ is the capillarycoefficient. For the sake of simplicity, we will always consider that λ is constant.F 0AρFigure 2: Illustration of the graph of the classical volumetric free energy F 0 (ρ).It can be shown that this particular form is justified from a molecular point of view. We willnot proove this and the interested reader can refer to [Rocard, 1967] for instance. In particular, itcan be shown that the value of λ depends only on the intermolecular potential.2.1.2 General equilibrium conditionsFor the sake of generality, we will consider that the volumetric free energy of the fluid is givenby the general expression F (ρ, T, ∇ρ). The differential of F thus readsdF = −S dT + g dρ + φ · d∇ρ (15)which defines the entropy S, the Gibbs free enthalpy g as well as φ.The second law of thermodynamics states that a closed and isolated system at equilibrium issuch that its entropy is maximum. Mathematically, this reads∫δ [S + L 1 U(S, ρ, ∇ρ) + L 2 ρ] dV = 0 (16)V9

where δ represents the variation, V is the fluid domain S is the volumetric entropy, U is the volumetricinternal energy and where L 1 and L 2 are two constant Lagrange multipliers accountingfor the constraints of conservation respectively of the energy and mass of the system.SinceU = F + S Tone hasThusNow∫V∫[(1 + L 1 T ) δ S + (L 1 g + L 2 ) δρ + L 1 φ · δ∇ρ] dV = 0Vφ · δ∇ρ = φ · ∇(δρ)= ∇ · (φ δρ) − (∇ · φ) δρ[(1 + L 1 T ) δ S + (L 1 (g − ∇ · φ) + L 2 ) δρ] dV = 0 (17)where we have used the condition (that is discussed in section 2.4)∫∫∇ · (φ δρ) dV = n · φ δρ dS = 0Vwhere ∂V is the boundary of the domain and n its unit normal outwardly directed and wherewe have assumed that n · φ = 0 on ∂V .Since equation (17) must be satisfied for any variation δS and δρ, one has:∂VT = − 1 L 1g − ∇ · φ = − L 2L 1Since L 1 and L 2 are constants, the equilibrium conditions readT = cste (18)g − ∇ · φ = cste (19)The first condition means that the temperature of the system is uniform at equilibrium. Thesecond condition means that the generalized Gibbs free enthalpy˜g ˆ= g − ∇ · φ (20)is uniform at equilibrium. This latter condition is a generalization of the classical equilibriumcondition stating that the Gibbs free enthalpy is uniform at equilibrium.It must be emphasized that these equilibrium condition are valid in the entire two-phasesystem, including the bulk liquid and vapor phases as well as the interfacial zones.2.1.3 Internal structure of the interfaceIn this section, we discuss the consequences of the equilibrium conditions derived in the previoussection on the internal structure of a liquid-vapor interface. For that purpose, we restrictthe analysis to the case where the expression for F (ρ, T, ∇ρ) is given by (14). In this case, theequilibrium condition (19) reads∂F 0∂ρ (ρ, T 0) − λ ∇ 2 ρ = cste (21)10

where T 0 is a constant corresponding to the equilibrium temperature of the system; T 0 can beviewed as a parameter and is dropped in the following developments.Mathematically, this equation is a differential equation that the density field ρ(x) must satisfyat equilibrium. To better understand the consequences of this differential equation, we nowconsider a planar interface at equilibrium at we denote z the coordinate normal to the interface.We thus seek for the function ρ(z) that defines the profile of the interfacial zone. This functionmust satisfywhere g 0 ˆ= ∂F 0 /∂ρ.g 0g 0 (ρ) − λ d2 ρ= cste (22)dz2 AρFigure 3: Illustration of the graph of g 0 (ρ).Very far from the interfacial zone, bulk liquid and vapor phases exist and therefore d 2 ρ/dz 2 =0; likewise, dρ/dz = 0. This equation shows thatg 0 (ρ v ) = cste = g 0 (ρ l ) ˆ= g eq (23)where ρ v and ρ l are the densities of the vapor and liquid phases respectively far from the interface.This equation shows that the specific free Gibbs energy of the phases are equal. This correspondsto the condition of equilibrium of the interface discussed in section 1.2.2.By multiplying equation (22) by dρ/dz and integrate, one getsF 0 (ρ) − F 0 (ρ v ) − g eq (ρ − ρ v ) = λ 2( ) 2 dρ(24)dzIf we defineone haswhich is clearly a differential equation for ρ(z).W (ρ) ˆ= F 0 (ρ) − F 0 (ρ v ) − g eq (ρ − ρ v ) (25)λ2This equation shows in particular thatwhich is equivalent to( ) 2 dρ= W (ρ)dzW (ρ v ) = W (ρ l ) (26)F 0 (ρ l ) − g eq ρ l = F 0 (ρ v ) − g eq ρ v (27)Now, using classical thermodynamic relations, it can be shown that the pressure P 0 is given byP 0 = ρ g 0 − F 0 (28)11

Thus equation (27) readsP 0 (ρ v ) = P 0 (ρ l ) (29)This relation means that the liquid and vapor phases at equilibrium are equal. We recover theclassical condition of equilibrium of a planar interface.It is worth noting that the two conditions (23) and (29) are two equations of the two unknownsρ v and ρ l that can thus be determined. Moreover, these conditions have a simple graphical interpretationon the graph of the function F 0 (ρ). Indeed, the condition (23) means that the two slopesof the tangent to this graph at ρ v and ρ l are equal. The condition (29) means that the y-interceptsof these tangents are equal. Therefore, these two tangents are the same. Thus, ρ v and ρ l are definedby the bi-tangent to the graph of the function F 0 (ρ). Moreover, this allows to show that W (ρ) (definedby (25)) is actually the height between F 0 (ρ) and its bi-tangent.For the sake of simplicity (as will be shown hereafter), the function W (ρ) is often modeled asfollows:W (ρ) = A (ρ − ρ v ) 2 (ρ − ρ l ) 2 (30)where A is a parameter characteristic of the function F 0 (ρ) (cf. figure 2). This particular formallows to simplify the differential equation (24):This differential admits the following solutionwheredρdz = √2 Aλ (ρ − ρ v) (ρ l − ρ)ρ(z) = ρ l + ρ v2where h represents the interface thickness.+ ρ l − ρ v2h ˆ= 1ρ l − ρ v√λ2 A( z)tanh2 h(31)2.1.4 Surface excess energyThis analysis also allows to determine the energy “concentrated” at the interface. This energy,denote F ex is defined as follows:whereF ex ˆ=∫ zi−∞(F (ρ, ∇ρ) − F 0 (ρ v )) dz +ρ ex ˆ=∫ zi−∞∫ +∞z i(ρ(z) − ρ v ) dz +(F 0 (ρ l ) − F (ρ, ∇ρ)) dz − g eq ρ ex∫ +∞z i(ρ l − ρ(z)) dzwhere z i is any position. Actually it is straightforward to show that F ex does not depend on z i .Using the relations (25) and (24), it can be shown thatF ex =∫ +∞−∞λ( ) 2 ∫ dρρl√ √dz = 2 λ W (ρ) dρ (32)dzρ vThis energy concentrated at the interface is interpreted as the surface tension.It is worth noting that the above expression is general and is valid for any expression for thefunction F 0 (ρ) and therefore W (ρ). In the particular case where W (ρ) is given by (30), one findsthat the expression for the surface tension is the following:σ = (ρ l − ρ v ) 36√2 A λ (33)12

2.1.5 Equilibrium of a pherical inclusionIn section 1.2, we showed that surface tension has an effect on the outer bulk phases through theLaplace relation. Is it recovered by the van der Waals model?The equilibrium condition (21) is always valid and is in particular valid for a spherical inclusion(bubble or droplet) at equilibrium. In this case, we consider a spherical system of coordinateswhose origin is the center of the inclusion. The only variations that must accounted for are alongthe radial direction. The equilibrium condition thus readswhere r is the radial coordinate.This equation shows in particular thatg − λ d2 ρdr 2 − λ 2 rdρ= cste (34)drg(ρ s v) = g(ρ s l ) ˆ= g s eq (35)This relation shows that the bulk phase specific Gibbs free enthalpies are equal. It is importantto note that, at this point, the value of geq s is unknown and is a priori different from g eq (theequilibrium value of a planar interface). This implies in particular that the densities of the bulkphases (ρ s v and ρs l) are different from those of a planar interface.By multiplying equation (34) by dρ/dr and integrating from r = 0 to r yieldsF 0 (ρ) − F 0 (ρ s i ) − g s eq(ρ − ρ s i ) = λ 2( ) 2 ∫ dρr( ) 22 dρ+dr 0 η λ dη (36)dηwhere the subscript i denotes the interior phase (i.e. vapor for a bubble or liquid for a drop).This expression shows in particular thatF 0 (ρ s e ) − F 0 (ρ s i ) − gs eq (ρs e − ρs i ) = ∫ ∞where the subscript e denotes the exterior phase.Using the definition (28) of the pressure, this relation reads:P 0 si∫ ∞( ) 2− Pe 0 s 2 dρ=0 η λ dηdη0( ) 22 dρη λ dηdηThe variations of ρ are significant only in the vicinity of the radius R of the inclusion; thus∫ ∞0( ) 22 dρη λ dη ≃ 2 dη R∫ ∞0λ( ) 2 dρdηdηMoreover, if the density profile (determined by the integro-differential equation (36)) is onlyweakly influenced by the curvature effects the last integral term can be approximated by takingthe density profile of a planar interface, in which caseThereforeThis is the Laplace relation.∫ ∞0λP 0 si( ) 2 dρdη ≃ σdη− P 0 se≃ 2 σRThe graphical determination of the condition of equilibrium of a spherical inclusion is illustratedin figure 4.13

WW2σ/Rρρ2σ/RbubbledropFigure 4: Graphical determination of the states of a spherical inclusion at equilibrium.2.1.6 Expression for the volumetric free energyThe study of of the equilibrium conditions developed in the previous sections shows the importanceof the dependence in ρ of the volumetric free energy F 0 on the determination of the internalstructure of the interface and on the value of the surface tension. At a given temperature T 0 , theexpression for F 0 (ρ) is the following (cf. relations (25) and (28)):F 0 (ρ) = W (ρ) + g eq ρ − P eqwhere P eq is the saturation pressure at the temperature considered. The equilibrium conditionsstudied in the previous sections show that the values of P eq and of g eq have no influence the resultsfor an isothermal system. Therefore, very often, one takes F 0 (ρ) = W (ρ).Because the equilibrium condition on the temperature is simply T = cste, in the previoussections, the temperature of the system has been considered as a parameter. However, the effectsof the temperature are important to account for accurately. Indeed, in many applications, phasechangeoccurs because the system is heated, which means that the temperature gradients drivethe phase transition. Moreover, the latent heat is also an important thermodynamic propertythat has to be accounted for. The general expression (15) for the differential of F shows that itsdependence in T is related to the entropy:s = − 1 ( ) ∂Fρ ∂TρIn particular, the latent heat L is such that[ ( )1 ∂FL = T (s v − s l ) = T(ρ v (T )) − 1 ( ) ]∂F(ρ l (T ))ρ v (T ) ∂Tρ l (T ) ∂Twhere ρ v (T ) and ρ l (T ) are the vapor and liquid densities at saturation.Moreover, the heat capacity at constant volume Cv (another important thermo-physical property)is given by( ) ∂sCv = T(38)∂TρThus the dependence in T of F must be consistent with the data L(T ) and Cv(T ). Without anyapproximation, it is difficult to provide an expression for F 0 (ρ, T ) that satisfies this consistency.To keep the simplicity of the polynomial expression (30) for W (ρ), we can make the followingapproximation:F 0 (ρ, T ) = W (ρ, T ) + g eq (T ) ρ − P eq (T )(37)14

which is the equation of evolution of the internal energy.The expression for the internal energy isu = F ρ + T s (44)and the pressure P is defined byso that the differential of u reads( ) ∂FP ˆ= ρ − F (45)∂ρρ du = ρ T ds + P ρdρ + φ · d∇ρ (46)Using equations (39), (42) and (43) to express dρ/dt, ds/dt and du/dt in the above relation, onehas−∇ · q + T : ∇v = T (−∇ · q s + ∆ s ) − P ∇ · v + φ · d∇ρdtThe last term of this equation is transformed as follows(47)φ · d∇ρdt( )∂∇ρ= φ · + v · ∇∇ρ∂t( ) ∂ρ= φ · ∇ + φ∂t i v j ρ ,ij(= ∇ · φ ∂ρ )∂t(= ∇ · φ ∂ρ )∂t(= ∇ · φ ∂ρ∂t− ∂ρ∂t (∇ · φ) + (φ i v j ρ ,j ) ,i− (φ i v j ) ,iρ ,j− ∂ρ∂t (∇ · φ) + ∇ · (φ v · ∇ρ) − φ i,i v j ρ ,j − φ i v j,i ρ ,j)− ∂ρ (∇ · φ) + ∇ · (φ v · ∇ρ) − (v · ∇ρ) ∇ · φ − (φ ⊗ ∇ρ) : ∇v∂twhere ψ ,i ≡ ∂ψ/∂x i and where the Einstein convention on the repeated indices has been used.Thusφ · d∇ρ (= ∇ · φ dρ )− dρ (∇ · φ) − (φ ⊗ ∇ρ) : ∇vdtdt dtSubstituting this relation in equation (47) allows to express the entropy source ∆ s as follows[∆ s = ∇ · q s − 1 (q + φ dρ )]− 1 (T dt T 2 q + φ dρ )· ∇T + 1 [T + (P − ρ ∇ · φ) I + φ ⊗ ∇ρ] : ∇vdt T(48)The second law of thermodynamics states that ∆ s ≥ 0 for any motion. The following expressionsfor q and T satisfy this condition 3 :3 These relations are not the most general. For instance, the thermal conductivity k is in general a tensor of order twoof the form k = k 1 I + k 2 ∇ρ ⊗ ∇ρ/(∇ρ) 2 . This relation expresses that the thermal conductivity in the normal andtangential directions to the interface can be different. Likewise, a Newtonian behavior is very restrictive compared to thegeneral expressions found for the dissipative stress tensor in which five different “viscosity” coefficients appear.It is worth noting that, despite its simplicity, the method used to derive these expressions, and especially the expressionfor q, might not be the most rigorous from a fundamental point of view. Indeed, using the Hamilton’s principal, it canbe shown that the term φ dρ/dt is actually not a heat flux per say but is rather a work since it has no contribution tothe entropy source and is thus a conservative contribution. This term is known as the “intersticial working” [Dunn andSerrin, 1965].16

q = −φ dρ − k ∇T (49)dtT = (−P + ρ ∇ · φ) I − φ ⊗ ∇ρ + τ (50)where k must be positive and where τ is the dissipative stress tensor that must satisfyτ : ∇v ≥ 0A classical Newtonian fluid satisfies this condition.Using these closure relations, the system of balance equations that describes the motion of thefluid is the following:dρ= −ρ ∇ · v (51)dtρ dvdt= −∇P + ∇(ρ ∇ · φ) − ∇ · (φ ⊗ ∇ρ) + ∇ · τ (52)ρ de (dt = ∇ · φ dρdt2.2.1 Korteweg stress tensor and surface tension force)+ ∇ · (k ∇T ) + ∇ · (v · T ) (53)In the most classical case where the energy of the fluid is expressed by (14),φ = λ ∇ρandP = P 0 (ρ, T ) − λ 2 (∇ρ)2The momentum balance equation thus readsρ dvdt = −∇P 0 + ∇(λ ρ ∇ 2 ρ + λ )2 (∇ρ)2 − ∇ · (λ∇ρ ⊗ ∇ρ) + ∇ · τ (54)The stress tensor (λ∇ρ ⊗ ∇ρ) is called the Korteweg stress tensor [Korteweg, 1901]. We showin the following that this stress tensor implies a tension force in the tangential direction to theinterface, interpreted as the surface tension force.To simplify the analysis, we consider a planar interface at equilibrium. We denote z and x thecoordinates respectively normal and tangential to the interface (because of symmetry, only onetangential direction can be accounted for). Thus, all the variables depend only on z.Let us first define˜P ˆ= P 0 −(λ ρ ∇ 2 ρ + λ )2 (∇ρ)2so that, at equilibrium, the momentum balance equation (55) reduces toBy integration, one simply getsd ˜Pdz + λ d dz˜P (z) = P ∞ − λwhere P ∞ in the pressure in the bulk phases.( ) 2 dρ= 0dz( ) 2 dρdz17

Since λ is constant, s = s 0 . Therefore( ) ∂sρ T ds = ρ Cv 0 0dT + ρ T∂ρTdρThusρ Cv 0 dT( ) ∂s0dt = ∇ · (k ∇T ) − ρ T ∂ρTdρ+ τ : ∇v (55)dtThis equation is very similar to the classical equation of evolution of the temperature in asingle-phase fluid. However, the interpretation of the term in dρ/dt is very different. Indeed, for asingle-phase incompressible fluid, this term vanishes (by definition). In the case of a liquid-vaporinterfacial zone, (dρ/dt) represents the rate of vaporization denoted γ c . Indeed, let us consideran interface where vaporization occurs. If we follow a liquid fluid particle in its motion, as itcrosses the interface to become vapor, its density drastically decreases, therefore dρ/dt < 0. Now,ρ T (∂s 0 /∂ρ) T is analogous to the latent heat (cf. (37)). Therefore, this term can be approximatedby γ c L ρ/(ρ l − ρ v ). This term is thus a spreading of the latent heat source over the interfacialzone.2.3 Different forms of the momentum balance equationIn the form (52) of the momemtum balance equation, the Korteweg tensor might not be the mostappropriate. Indeed, from a numerical point of view for instance, the discretization of this termis not straightforward. Moreover, we will see that there exists an equivalent from in which onlyLaplacian and gradient operators appear, which is generally much easier to implement.The following identity holds:∇ · (φ ⊗ ∇ρ) = ∇(ρ ∇ · φ) − ρ ∇(∇ · φ) + φ · ∇∇ρGiven the expression (45) for P and the differential of F (15), one hasThus, equation (52) readsρ dvdt∇P = ρ ∇g + ρ s ∇T − φ · ∇∇ρ= −ρ ∇ (g − ∇ · φ) − ρ s ∇T + ∇ · τThis form of the momemtum balance equation makes clearly appears the two conditions ofequilibrium (18)-(19). This shows in particular that if any of these thermodynamic equilibriumconditions is not satisfied, it triggers a fluid motion. Moreover, since the thermodynamic equilibriumconditions appear in this equation, it shows its thermodynamic consistency. Moreover, froma numerical point of view, Jamet et al. [2002] showed that, through a detailed analysis of the discretizedenergy exchanges, this form allows to get rid of the so-called parasitic currents [Brackbillet al., 1992]. These non-physical currents, concentrated in the close vicinity to the interface, areinduced by numerical truncation that are very difficult to eliminate without a detailed analysisof the energy exchanges. The thermodynamic consistency of this model gives a framework todevelop accurate numerical schemes.2.4 Boundary conditions and contact angleSo far, in the determination of the equilibrium conditions or of the equations of motion, we havenot studied the boundary conditions that must be applied on the density field. In this section,19

we show that particular boundary conditions arise from the existence of the capillary term in theenergy functional and that these boundary conditions are related to the contact angle 4 .Let us consider a liquid-vapor system in contact with a solid wall. Let us consider an energyof interaction between the solid and the fluid U s (per unit surface area) and let us assume thatthis energy depend only on the local density of the fluid at the boundary (this assumption can bejustified by a mean-field approximation [Gouin, 1998]). Therefore, the total internal energy of thesystem is∫∫U(S, ρ, ∇ρ) dV + U s (ρ) dAV∂Vwhere ∂V is the boundary of the fluid domain V . Following the developments made in section2.1.2, the application of the second law of thermodynamics to determine the equilibriumconditions of the system yields∫∫δ [S + L 1 U(S, ρ, ∇ρ) + L 2 ρ] dV + δ L 1 U s (ρ) dA = 0Vwhere we remind that L 1 is the constant Lagrange multiplier accounting for the constraint ofconservation of the total internal energy. This yields (cf. section 2.1.2):∫∫ ( )dUs[(1 + L 1 T ) δ S + (L 1 (g − ∇ · φ) + L 2 ) δρ] dV +dρ + n · φ δρ dS = 0VThe last surface integral is a term that did not appear in the study developed in section 2.1.2.Since the above condition must be satisfied for any variation δρ, the following condition must besatisfied at the boundary ∂V :n · φ = − dU s(56)dρTo illustrate the physical meaning of this boundary condition, let us consider the following assumptions:the expression for F (ρ, ∇ρ, T ) is given by (14) so that φ = λ ∇ρ and U s (ρ) is assumedto be linear so that dU s /dρ = β = cste. The boundary condition therefore reads∂V∂Vn · ∇ρ = − β λ(57)Since β and λ are constant, this consdition imposes the value of the normal derivative of ρ. Asillustrated in figure 6, this imposes the value of the contact angle. The interested reader can referto [Seppecher, 1996, Jacqmin, 2000] for instance where this boundary condition has been studied.It is worth noting that the boundary condition (56) is an equilibrium boundary condition.However, this condition can be extended to out-of-equilibrium conditions to recover a variationof the contact angle with the speed of displacement of the contact line (a variation that is observedexperimentally). This form is thermodynamically coherent since it ensures that the entropy of thesystem increases.3 Two-phase flows of non-miscible fluids: the Cahn-Hilliard modelIn the previous section, we presented the van der Waals model of capillarity dedicated to themodeling of an interface that separates the liquid and vapor phase of the same pure substance.Many two-phase systems are made of different substances, for instance air and water or oil andwater. In this case, an interface separates two phases that are made of different species. Nonmisciblesphases are phases for which no mass transfer from one phase to the other exist.4 When a drop of liquid is put in contact with a solid wall, it is observed that the angle θ between the liquid-gasinterface and the solid surface is characteristic of the triplet solid-liquid-gas. For instance, for the same liquid and gas,if we change the nature of the solid, the same volume of liquid either tends to spread on the surface (θ < 90 ◦ ) or toretract (θ > 90 ◦ ). This tendency is related to the energy of interaction between the solid and the liquid. If the solid hasmore affinity with the liquid than with the gas, the system tends to minimize its energy by increasing the area of contactbetween the liquid and the solid (θ < 90 ◦ ).20

vaporliquid¤¡¤¡¤¡¤ §¡§¡§¡§ ¨¡¨¡¨¡¨ ©¡©¡©¡© ¡¡¡ ¡¡¡ ¡¡¡ ¡¡¡ ¡¡¡ £¡£¡£¡£¤¡¤¡¤¡¤ §¡§¡§¡§ ¨¡¨¡¨¡¨ ©¡©¡©¡© ¡¡¡ ¡¡¡ ¡¡¡ ¡¡¡ ¡¡¡ £¡£¡£¡£¤¡¤¡¤¡¤ §¡§¡§¡§ ¨¡¨¡¨¡¨ ©¡©¡©¡© ¡¡¡ ¡¡¡ ¡¡¡ ¡¡¡ ¡¡¡ £¡£¡£¡£¦¡¦¡¦ ¥¡¥¡¥¤¡¤¡¤¡¤ §¡§¡§¡§ ¨¡¨¡¨¡¨ ©¡©¡©¡© ¡¡¡ ¡¡¡ ¡¡¡ ¡¡¡ ¡¡¡ £¡£¡£¡£¦¡¦¡¦ ¥¡¥¡¥θ∇ρ¤¡¤¡¤¡¤¥¡¥¡¥ ¦¡¦¡¦§¡§¡§¡§ ¨¡¨¡¨¡¨ ©¡©¡©¡© ¡¡¡ ¡¡¡ ¡¡¡ ¡¡¡ ¡¡¡ £¡£¡£¡£solidn¤¡¤¡¤¡¤ §¡§¡§¡§ ¨¡¨¡¨¡¨ ©¡©¡©¡© ¡¡¡ ¡¡¡ ¡¡¡ ¡¡¡ ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡£¡£¡£¡£¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡Figure 6: Illustration of the contact angle boundary condition.3.1 Thermodynamic modelFrom a physical point of view, why do these two species tend to be separated? At the molecularlevel, the energy of interaction of two molecules of different species is larger than that one twomolecules of the same species. Thus, since the system tends to minimize its energy, if the speciesare separated, the energy of the system is weaker. This simple reasoning shows that the “drivingforce” of the transition, which eventually gives rise to the existence of an interface, is the amountof one species into the other. If not too many molecules of one species is present is the other,the energy of the system is not that increased but if more molecules are added, the energy getsto large and the system tends to separate into two different phases: one rich in the first speciesand the other rich in the other species. In this case, the relevant thermodynamic variable is theconcentration of one species in the mixture.3.1.1 A mean-field approximationFor the sake of simplicity, we will first consider that the density of the mixture is constant forany value of the concentration of one species in the mixture; it is denoted ρ 0 . Let c denote themass fraction (or concentration) of one species in the mixture. By analogy with the van der Waalsmodel, Cahn and Hilliard [Cahn and Hilliard, 1958, 1959a,b] postulated that the free energy ofthe system F is given byF = F 0 (c) + λ 2 (∇c)2 (58)where F 0 (c) is the “classical” part of the energy and λ is the capillary coefficient.F 0Aµ 0 AccFigure 7: Illustration of the graph of the functions F 0 (c) and µ 0 (c).However, it can be shown that this particular form is justified from a molecular point of view.Indeed, using a mean-field approximation, it can be shown that the attractive energy of interactionof molecules of different types gives rise to this form for the energy of the mixture and thatλ depends only on the inter-molecular potentials.21

3.1.2 Equilibrium conditionsSince we deal with a mixture, we can assume that thermal effects are negligible. We can thusassume that the temperature of the system is imposed to a constant. Since the system is assumedto be isothermal, the equilibrium is characterized by a minimum of a minimum of its free energy.Mathematically, this reads: ∫δ (F (c, ∇c) + L ρ 0 c) dV = 0Vwhere L is a constant Lagrange multiplier accounting for the fact that the system is closed andthat the total mass of each species is constant, which reads∫ρ 0 c dV = csteVFollowing the same developments as those presented in section 2.1.2, one finds that the equilibriumcondition is the following:( )∂F ∂F∂c − ∇ · = cste∂∇cThis condition is very similar to the condition (19) found for the van der Waals model.In the particular case where F (c, ∇c) is given by the expression (58), this equilibrium conditionsimply readswhereµ 0 (c) − λ∇ 2 c = cste (59)µ 0 (c) ˆ= dF 0(60)dcThis equation is the same as that obtained for the van der Waals model and the same conclusionshold.In particular, for a planar interface at equilibrium, it is found that the equilibrium conditionscorrespond to the double-tangent to the graph of the function F 0 (c). This condition defines themass fraction of the phases at equilibrium of a planar interface, c 1 and c 2 , as well as the chemicalpotential of a planar interface µ eq . It is worth emphasizing that c 1 and c 2 are not equal to 0or 1; actually, from a physical point of view c 1 and c 2 cannot be exactly equal to 0 or 1. Theirvalue actually depend on the physical system considered. It is then convenient to introduce thedouble-well function W (c) defined as the difference between F (c) and its double-tangent:W (c) ˆ= F (c) − (F (c 1 ) + µ eq (c − c 1 ))This function is very often approximated by a polynomial of degree 4:W (c) = A (c − c 1 ) 2 (c − c 2 ) 2It is worth noting that this particular form can be justified from a mean-field approximation closeto a critical point. It must be emphasized that this approximation is valid when c 1 ≃ c 2 , whichmeans in particular that it is not valid for c 1 ≃ 0 and c 2 ≃ 1. This is important because, often,Cahn-Hilliard models are used with c 1 = 0 and c 2 = 1, which actually only corresponds to arenormalization of the “true” mass fraction.Moreover, the study of the equilibrium of a spherical inclusion implies that same results as inthe van der Waals model: (i) the chemical potentials of the bulk phases surrounding the curvedinterface are equalµ 0 (c i ) = µ 0 (c e ) ˆ= µ s e22

and (ii) there is an equivalent of the Laplace relationwhere the “pressure” P 0 is defined byP 0 (c i ) − P 0 (c i ) = 2 σRP 0 (c) ˆ= c dF 0dc − F 0 (c)In this case, the “pressure” P 0 does not have the interpretation of a physical pressure.A Taylor expansion of these equilibrium conditions of a spherical inclusion around the equilibriumstate of a planar interface allow to show thatµ s e ≃ µ eq + 1 2 σc i − c e R(61)It is worth noting that the form of this equilibrium condition is actually similar to the Gibbs-Thompson condition (12).This equilibrium condition shows in particular that the value of the chemical potential is proportionalto the curvature of the interface. This explains how an interface tends to get spherical.Indeed, let us consider an closed interface whose shape is initially irregular. If, at each point ofthe interface, the interfacial zone is at local thermodynamic equilibrium, the above equilibriumcondition is satisfied locally. These means in particular that, along the interface, the chemicalpotential is not uniform. According to the Cahn-Hilliard equation, this yields a diffusion massflux and therefore a mass diffusion. This mass diffusion makes the overall system evolve and,since the Cahn-Hilliard equation is thermodynamically coherent, this evolution tends to makethe system get closer to an equilibrium state. If we consider a spherical inclusion at equilibrium,the chemical potential along the interface is constant, therefore no mass flux exist and the systemkeeps at rest.3.2 The Cahn-Hilliard equationIn the previous section, we derived the equilibrium condition for a Cahn-Hilliard fluid. The issueis to determine how the system behaves out of equilibrium. To start this analysis, we will firstsimplify the system by assuming that the velocity of the mixture is null. In this case, the evolutionof the concentration evolves through a mass diffusion equation that reads∂c∂t = −∇ · jwhere j is the diffusion mass flux, whose expression must be determined.The evolution of the free energy of the mixture is given by∂F∂t = −∇ · q − ∆ fwhere q is the heat flux and ∆ f is the energy dissipation. The second law of thermodynamicsimposes that ∆ f ≥ 0.Using the same developments as those presented in section 2.2 for the van der Waals model,one finds that[∆ f = −j · ∇ (µ − ∇ · φ) − ∇ · q + φ ∂c]∂t − j (µ − ∇ · φ)whereµ ˆ= ∂F∂c23

φ ˆ= ∂F∂∇cThe following expression for j ensures that ∆ f ≥ 0j = −κ ∇ (µ − ∇ · φ)where κ is called the mobility. It is worth noting that κ ≥ 0 and is any function of the thermostaticparameter of the system (i.e. c and ∇c) 5 .In the particular case where F (c, ∇c) is given by (58), the mass diffusion equation reads∂c∂t = ∇ · [κ∇ ( µ 0 (c) − λ∇ 2 c )] (62)This equation is called the Cahn-Hilliard equation. It is worth noting that it is a generalizedmass diffusion equation that is valid in the entire two-phase system.The Cahn-Hilliard equation is a generalized mass diffusion equation. In particular, it shows thatthe local diffusion mass flux is proportional to the gradient of the generalized chemical potential(µ 0 (c) − λ∇ 2 c).3.3 Equations of motionIn the previous section, we derived the Cahn-Hilliard equation. This equation is the only equationthat has to be accounted for if the velocity of the mixture is null. In this section, we derivethe equations of motion of the mixture when this velocity is not null.3.3.1 Boussinesq approximationIn this section, we assume that the density of the mixture does not depend on the compositionof the mixture: ρ(c) = ρ 0 . This the density is constant, the general mass balance equation of themixture∂ρ+ ∇ · (ρ v) = 0∂tdegenerates to the solenoidal condition∇ · v = 0In this case, we know that, in the momemtum balance equation, the pressure is the Lagrangemultiplier associated to this solenoidal constraint. Using a similar approach as that developed insection 2.2 for the van der Waals model, it can be shown that the momemtum balance equationreadsρ 0∂v∂t + ρ 0 v · ∇v = −∇P − ∇ · (λ∇c ⊗ ∇c) + ∇ · τwhere τ can be expressed as the classical viscous stress tensor.The Cahn-Hilliard equation (62) has to be modified to account for the convection term asfollows:∂c∂t + v · ∇c = ∇ · [κ∇ ( µ 0 (c) − λ∇ 2 c )]5 In the general case, the mobility is a tensor of the formκ = κ 1 I + κ 2 ∇c ⊗ ∇c/(∇c) 2andj = −κ · (µ − ∇ · φ)24

It can be shown that this system of equations is thermodynamically consistent in the sensethat the total energy of the system is a decreasing function of time:∫ddt V[F 0 (c) + λ 2 (∇c)2 + ρ 0 v 22] ∫dV = −{κ [ ∇ ( µ 0 (c) − λ∇ 2 c )] }2+ τ : ∇v dV < 0VTo account for buoyancy effects the gravity term ρ g (where g is the acceleration of gravity)has to be added in the momentum balance equation. If ρ = ρ 0 in this force, this force has noeffect on the flow: it only modifies the pressure (the pressure P can be replaced by the pressure(P − ρ 0 g z), which does not modify the structure of the equations). For the gravity to have aneffect on the flow and in particular to account for a buoyancy effect, variations of the density mustbe accounted for. However, these variations are a priori not compatible with the incompressibilityapproximation. That is why the Boussinesq approximation is generally used: the variation of thedensity is neglected except in the gravity term where it is linearized. The momentum balanceequation therefore readswhereρ 0∂v∂t + ρ 0 v · ∇v = −∇P − ∇ · (λ∇c ⊗ ∇c) + ∇ · τ + ρ 0 (1 + β(c − c 0 )) gWe summarize the system of equations:β ˆ= 1 ( ) dρρ 0 dc0∇ · v = 0 (63)∂c+ v · ∇c = ∇ · [κ ∇˜µ] (64)∂t˜µ ˆ= µ 0 (c) − λ∇ 2 c (65)∂vρ 0∂t + ρ 0 v · ∇v = −∇P − ∇ · (λ∇c ⊗ ∇c) + ∇ · τ + ρ 0 (1 + β(c − c 0 )) g (66)(µ 0 (c) = 4 A (c − c 1 ) (c − c 2 ) c − c )1 + c 2(67)2It is worth noting that, like in the van der Waals model (cf. section 2.3), the “stress form” of themomentum balance equation can be transformed into an equivalent “potential form” in whichthe generalized chemical potential appears [Jacqmin, 1999]:ρ 0∂v∂t + ρ 0 v · ∇v = −∇ ˜P + ˜µ ∇c + ∇ · τ + ρ 0 (1 + β(c − c 0 )) g (68)˜P ˆ= P + FThis form of the momentum balance equation shows the influence of the curvature on themomemtum balance equation. Indeed, we showed in the previous section that, at equilibriumof a spherical inclusion, the value of the chemical potential is proportional to the local interfacecurvature (cf. (61)). Therefore, the term ˜µ ∇c represents a spreading (over the interface thickness)of a force proportional to the interface curvature and oriented in the direction normal to the interfaceapproximated by ∇c. This interpretation is very close to the Continuous Surface Forcecommonly used in sharp interface numerical methods [Brackbill et al., 1992].This system of equations is particularly attractive numerically. Indeed, we one has a numericalcode dedicated to the simulation of incompressible flows, it can be very easily generalized totwo-phase capillary flows. Indeed, one only needs to implement a source term in the momentum25

alance equation and a convection-diffusion-like equation for c (the Cahn-Hilliard equation). Anexample is given in figure 8. The system simulated is the impact of a heavier droplet on a solidgrid. This figure shows in particular that topological changes are automatically accounted for.Using standard second order schemes for the discretization in space, the interface is captured byabout 4 mesh cells. However, Jacqmin [1999] shows that it is possible to reduce it to about 2 byusing a more complex scheme.Figure 8: Numerical simulation of the impact of a droplet on a grid using the Cahn-Hilliardmodel with a Boussinesq approximation.Because of its simplicity, simulating three dimensional systems is particularly easy, even onparallel computers. An illustration of the complex three dimensional systems that can be simulatedis shown in figure 9.Figure 9: Numerical simulation of a complex Rayleigh-Taylor instability in a I-shape reservoirusing a Cahn-Hilliard diffuse interface model.3.3.2 Finite density contrastThe case where the bulk phases have very different densities (for instance air and water at roomtemperature ρ a ≃ 1 kg/m 3 and ρ w ≃ 1000 kg/m 3 ) is rather different. From a physical point of26

view, in this case, the velocity is not solenoidal [Galdi et al., 1991]. Therefore, a priori, the pressurecan no longer be introduced as a Lagrange multiplier accounting for this constraint. However,from a practical point of view, the solenoidal constraint on the velocity field is attractive: it allowsto use standard numerical methods such as the projection method. The natural choice is thus touse the following equations∇ · v = 0 (69)dc= ∇ · [κ ∇˜µ]dt(70)ρ(c) dvdt = −∇ ˜P + ˜µ ∇c + ∇ · τ + ρ(c) g (71)where ρ(c) is generally a linear functionρ(c) = ρ 1 + c − c 1c 2 − c 1(ρ 2 − ρ 1 ) (72)It must be emphasized that this system does not satisfy the mass balance equation (1). Indeed,from equations (69) and (70), it is straightforward to show that∂ρdρ+ ∇ · (ρ v) = ∇ · [κ ∇˜µ] ≠ 0∂t dcHowever, if dρ/dc = cste, this equation can be written in a conservative form, which shows that(by integration over the entire volume), the total mass of the system is conserved (an importantproperty that many sharp interface numerical methods do not satisfy).Despite its simplicity, it can be shown that this system is ill-posed, in the sense that no monotonicallydecreasing energy is associated to this system of equations; it thus violates the secondlaw of thermodynamics. This may be the price to pay to conserve the easiness of the numericalimplementation. . .However, it is possible to develop a diffuse interface model with a finite density contrast thatis thermodynamically coherent. One of the main issue is to define incompressibility flow whenthe velocity field is not solenoidal. Lowengrub and Truskinovsky [1998] showed that, in this case,it is relevant to use the thermodynamic incompressibility: the density is independent of the pressure.The main thermodynamic variables are the mass fraction c and the thermodynamic pressure Pand the relevant thermodynamic potential is not the free energy F but the specific Gibbs freeenthalpy g(c, P, ∇c) whose expression is:g(c, P, ∇c) = f 0 (c) + α 2 (∇c)2 +where the linearity in P is due to the incompressibility assumption. The coefficient α is equivalentto a capillary coefficient but its dimension is not that of λ ([α] = [λ]/[ρ]).Lowengrub and Truskinovsky [1998] show that the corresponding system of balance equationsis the following:Pρ(c)whereρ dvdt∂ρ+ ∇ · (ρ v) = 0∂tρ dc = ∇ · (κ ∇˜µ)dt= −∇P − ∇ · (α ρ ∇c ⊗ ∇c) + ∇ · τ˜µ = df 0dc − P dρρ 2 dc − 1 ∇ · (ρ α ∇c)ρ27

This system is thermodynamically coherent and has been used to study pinch-off and reconnectionof interfaces for instance [Lee et al., 2002a,b]. However, this sytem is much more coupledthan the incompressible model (or the thermodynamically incoherent model (69)-(71)). Indeed,the density ρ(c) appears in many terms and in particular in expression for the Korteweg stresstensor and in the Laplacian part of ˜µ. Moreover (and certainly most importantly), the pressureP appears in the expression for ˜µ, which induces a complex coupling between the Cahn-Hilliardand momentum balance equation.4 Diffuse interface models and numerical simulation of mesoscopicproblems4.1 Numerical vs physical interface thicknessIn section 1.3, we showed that, in fluid mechanics, diffuse interface models are attractive fortwo main reasons: (i) they allow to study peculiar physical phenomena where sharp interfacemodels fail and (ii) they can be easily implemented numerically, which virtually eliminates thedifficult and tedious numerical treatment of the moving boundaries. Whatever the reason, whenwe have to solve the equations of motion numerically, these equations are discretized in time andspace. In particular, if we consider the discretization in space, the interfaces have to be capturedby the mesh. To illustrate this point, let us take a particular example of a regular mesh wherethe size of a mesh cell is ∆x in all directions. Since the continuous partial differential equationsare discretized on this mesh, all the spacial variations must be captured, so that the numericaltruncation errors do not make the system degenerate. In particular, the interface structure mustbe captured. This means that ∆x must be smaller than the interface thickness h. Typically, forstandard discretization schemes, one has ∆x ≃ h/4 so that the bi-Laplacian of the Cahn-Hilliardequation is well approximated. Even for higher order schemes, ∆x is of the order of h.The diffuse interface models presented in the previous sections (either for a liquid-vapor systemof for non-miscible phases) have been derived based on physical arguments: the main thermodynamicvariable are physical variables (mass density ρ or mass concentration c), the energyfunctional is physical and the equations of motion are based on physical first principles of conservation.Therefore, the internal structure of the interface is physical and in particular the interfacethickness is also physical. This approach is relevant for any study where the interfaces must bemodeled as continuous transition zones: coalescence and rupture of interfaces, nucleation, etc. Inthese cases, the typical interface thickness is of the order of 10 −9 m and ∆x ≃ 10 −9 m. Therefore,assuming that the maximum size of the mesh is of the order 10 6 mesh cells, the typical size of the3D domain studied is about ( 10 −7) 3m 3 for a regular mesh. This is extremely small, even thoughit is relevant for this kind of physical processes occuring at the scale of an interface.However, when diffuse interface models are used for numerical convenience, if the physicalmodel is used as it is, one must restrict the studies to domains whose typical size is 10 −7 m. Now,in many applications, the typical size of the inclusions (bubbles or droplets) of the two-phase systemis much larger than this, by at least one order of magnitude. At this point, different strategiesare possible: (i) diffuse interface models are abandonned as well as their potential ease of use, (ii)adaptative mesh refinement techniques may be developed to capture the interfacial zones or (iii)diffuse interface models are adapted to the numerical simulation of mesoscopic problems. In thelatter case, the adaptation of the model must be such that the value of the interface thickness isno longer dictated by physical arguments but rather by numerical arguments.The third solution will be studied in the subsequent sections. Nevertheless, the second solutiondeserves to be discussed. Indeed, we believe that it is a promising strategy because manyphysical phenoma occur close to the interface and a fine discretization is therefore necessary tocapture them. However, using mesh refinement to capture the internal structure of an interfacemight not be the most relevant solution. Indeed, if one is interested in problems whose typicalsize is that of a bubble or droplet (mesocopic scale), the sharp interface approximation is the most28

elevant. This means that all the physical processes of interest occur at the scale of the inclusion(bubble or droplet) and not of the interface structure. Thus, there is a clear and justified scaleseparation. Now, using a mesh refinement technique for these problems means that a complexnumerical technique is used to capture phenomena of no physical interest that have been introducedto simplify the numerical implementation. Thus, seeking for a way to use diffuse interfacemodels with an artificial interface thickness appears as the most relevant solution.4.2 Necessary modification of the diffuse interface modelsIn the previous section we showed that, for many problems where the relevant scale is the radiusof an inclusion, the scale separation with the interface thickness is justified. In this case, it isalmost impossible to use physical diffuse interface models on a regular mesh: much too manymesh points would be necessary only to capture the interfacial zones. Therefore, the typical sizeof the interfaces has to be adapted so that a reasonnable mesh can be used. In this case, the meshis more or less given, and therefore ∆x is given. Thus, the interface thickness h must be adaptedso that the interface structure can be captured by the mesh. The interface thickness h shouldtherefore be a free parameter whose value can be chosen arbitrarily. In the subsequent sections,we study if and how this is possible. We begin with the van der Waals model and then we studythe Cahn-Hilliard model.4.3 Liquid-vapor flows with phase-change4.3.1 Modification of the parametersIn section 2.1.2, we showed that, with the van der Waals model, the interface thickness is givenby√1 λh =(73)ρ l − ρ v 2 Awhere λ is the capillary coefficient and A is a coefficient that characterizes the function W (ρ) andtherefore the free energy of the fluid F (ρ) (see equation (25)).The goal is that h can be chosen arbitrarily. Now, the interface thickness is a consequence ofthe diffuse interface model; it is not a primary parameter of the model but rather a secondaryparameter. The primary parameters of the model are ρ l , ρ v , λ and A. Therefore, these are theparameters on which one might have a degree of freedom to fix the value of h arbitrarily. Amongthe primary parameters, λ is the only “non-classical” parameter: all the others are involved inthe properties of the bulk phases. In particular, the parameter A is characteristic not only of thethermodynamic behavior of the fluid within the interface but also within the bulk phases: thefunction F 0 (ρ) (in which the parameter A appears) is valid for any value of ρ and in particular forthe values of ρ reached within the bulk phases. In particular, it can be shown that the isothermalcompressibility of the bulk phases at saturation are given by( ) ∂P= 2 A ρ v (ρ l − ρ v ) 2 (74)∂ρv( ) ∂P= 2 A ρ l (ρ l − ρ v ) 2 (75)∂ρlThus, the only parameter clearly associated to the interface is λ and it appears as the mostobvious parameter that can be modified to increase the interface thickness. Equation (73) showsthat λ should be increased to increase h.Now, we have also shown that the expression for the surface tension is the following:σ = (ρ l − ρ v ) 36√2 A λ (76)29

P 0modifiedP eqρ vρ lρFigure 10: Modification of the equation of state P 0 (ρ) induced by the increase of the interfacethickness.This expression shows that if λ is increased while all the other parameters of the models are keptconstant, the value of σ is increased by the same factor of h. Surface tension is an importantphysical parameter. Indeed, through the Laplace and Gibbs-Thompson relations (cf. equations(11) and (13)), this interfacial property has an influence on bulk properties such as the pressure,the temperature and thus all the other bulk properties. Therefore, modifying σ implies that theoverall flow at the mesoscopic scale (of interest) is modified as well. This cannot be acceptable.To overcome this issue, at least one of the other parameters of the model has to be modifiedas well. The parameter A is chosen to be modified. Equations (31) and (33) show that, in order toincrease h and to keep σ constant, λ must be increased and A must be decreased proportionally. It can beshown that, for given values of σ (physical) and h (numerical), λ and A are given byλ =A =32 (ρ l − ρ v ) 2 σ h12(ρ l − ρ v ) 4 σhThis analysis shows that the overall thermodynamic behavior of the fluid must be modifiedand not only its “non-classical” part in (∇ρ) 2 . This means in particular that the classical part ofthe equation of state F 0 (ρ) must be modified. The main consequences of this modification areanalyzed in the following section.4.3.2 Consequences of this modificationIn the previous section, we showed that to make the interface thickness a free parameter of thesystem, it is necessary to modify the thermodynamic behavior of the fluid. In particular, it isnecessary to modify the “classical” part of the free energy F 0 (ρ). This is an unexpected consequenceof the procedure. Indeed, the initial goal was to modify only the interface thickness.By doing this, we implicitly accepted to violate the detailed physics of the interface layer. Butwe wanted to modify only interface structure without modifying the outer bulk properties. Theabove analysis shows that it is not possible. Now, the issue is to study whether the modificationof the free energy F 0 (ρ) has important consequences on the behavior of the two-phase system atthe mesoscopic scale.This study is rather difficult. Very often, this is done through an asymptotic analysis of thesystem. The goal is to determine to which sharp interface model the diffuse interface model isequivalent. In the particular case where a diffuse interface model is used mainly for numericalconvenience, the sharp interface model is known (cf. section 1.2) and we want the diffuse interfacemodel to mimic this given sharp interface model as well as possible. In particular, we aim atrecovering the same boundary conditions made of interfacial mass, momentum, energy balanceequations and, at first approximation, local thermodynamic equilibrium of the interface. The issueis thus to determine whether the modification of the parameter A modifies these boundaryconditions.30

The modification of A does not modify any property at saturation. In particular, the saturationpressure and the densities at saturation are not modified. However, it modifies the variations ofthe pressure with the density around the states at saturation ρ v and ρ l . In particular, it modifiesthe compressibility of the bulk phases. Compressibility has an effect on the motion of the fluidonly if the typical velocity of the fluid is of the order of magnitude of the speed of sound (thatdepends on the compressibility of the fluid). In many applications, the bulk velocities are verysmall compared to the speed of sound and, even though the latter is decreased by decreasing A,this remains valid.However, thermal expansion effects are important to account for because they are at the originof natural convection. The expression for the thermal expansion coefficient isκ T = 1 ρ(∂P/∂ρ) T(∂P/∂T ) ρThis expression shows that, if (∂P/∂ρ) T is decreased, (∂P/∂T ) ρ must be decreased as well to conservethe value of κ T . Now, the decrease of (∂P/∂T ) ρ is equivalent to the decrease of (dP sat /dT ).Because of the Clapeyron relationdP satdT = LT (v v − v l )in order the keep the value of L constant, it is necessary to modify the value of T . Actually, athorough analysis of this issue [Fouillet, 2003] shows that only the value of the temperature ofreference has to be modified. Since only temperature differences are important, the increase ofthe reference temperature is not very important. Moreover, it has been shown that this modificationmainly influences the time scale at which the interfacial zone gets back to the saturationtemperature. This time is physically extremely small and an increase of the time generally barelyhas any influence.This short analysis shows that the issue of the numerical increase of the interface thickness isnot trivial. It also shows that, even though the initial goal was to modify only a property of theinternal structure of the interface, it lead to modify important properties of the bulk phases aswell; it actually lead to modify all the variation of the equation of state P (ρ, T ). This is mainlybecause the graph of this function (or equivalently the function F 0 (ρ, T )) influences bulk propertiesas well as the internal structure of the interface.However, using these modifications, the van der Waals model can be used to simulate complexproblems such as nucleate boiling on a heated surface [Jamet and Fouillet, 2005] as shownin figure 11.This model can be extended to dilute binary mixtures as shown in figure 12. The influenceof the addition of a small amount of an extra component has been shown to have an importantinfluence on the wall heat exchange coefficient (cf. figure 13), which is commonly observed experimentally.Nevertheless, it is very difficult to use this model to get quantitative results. One of themain issues is gravity. Indeed, it has been shown that all the pressure scales had to be modified:(∂P/∂ρ) and (∂P/∂T ). These modifications are acceptable as long as no external pressurescale is imposed. But gravity imposes an external scale of pressure variation. To illustrate theissue, let us consider a vapor bubble on a heated wall. Because of gravity, the pressure at the topof the bubble is lower than the pressure at the bottom of the bubble. The local thermodynamicequilibrium of the interface is determined by the Clapeyron relation which determines the equilibrepressure as a function of the local temperature. So that the interface remains at equilibrium,the temperature must vary in the direction of the gravity field: lower at the top of the bubble thanat its bottom:ρ g∇T =(dP sat /dT ) ≃ ρ g(∂P/∂T ) ρ31

Figure 11: Numerical simulation of nucleate boiling using the van der Waals’ diffuse interfacemodel [Jamet and Fouillet, 2005]. The color field represents the temperature and the interfacecorresponds to iso-contours of the density field.Figure 12: Numerical simulation of nucleate boiling using an extension of the van der Waals’diffuse interface model to binary mixtures [Jamet and Fouillet, 2005].The color field represents themass fraction of a dilute substance and the interface corresponds to iso-contours of the densityfield.Since (∂P/∂T ) ρ is decreased because of the increase of the interface thickness, ∇T is increased.If the thermal conditions of the system more or less impose the temperature gradient (if the heatflux is imposed at the heated wall for instance), the disequilibrium of the bubble interface ismodified and the dynamics of the bubble as well.32

DT (K)12108642mixturepure fluidh (J/m^2/s/K)16000014000012000010000080000600004000020000mixturepure fluid00 0.5 1 1.5 2 2.5 3t (s)00 0.5 1 1.5 2 2.5 3t (s)Figure 13: Comparison of the mean temperature wall of of the instantaneous mean heat flux withand without the presence of a dilute substance [Jamet and Fouillet, 2005].4.4 Two-phase flows of non-miscible fluidsIn the case of non-miscible phases, the issue is the same: how to make the interface a free parameterwithout modifying the flow at the mesoscopic scale? The solution is the same as in theliquid-vapor case: increase λ and decrease A proportionally. However, in this case, the consequencesare different. The variations of the classical chemical potential µ 0 (c) must be modified.We showed in the previous section that this modification might not be critical as long as noexternal scale of variation of the chemical potential is imposed. In common two-phase flow applications,it is rare that any external scale of variation of the chemical potential is imposed andthe situation is thus easier that in the liquid-vapor case. However, there does exist a difficulty. Indeed,the mobility κ is related to the time scale at which a system goes back to equilibrium. Sincethe interface thickness is modified, if κ is not modified, the time scale at which the interfacial zonegoes back to equilibrium is modified. Therefore, κ must be increased to ensure that the interfacekeeps close to equilibrium. Finding the optimal value for κ is not straightforward. Now, in section3.1.2 we showed that the chemical potential of equilibrium of a spherical inclusion dependson the radius of curvature of the system. Therefore, if two spherical inclusion of different radiiat put close to each other, because of this difference of chemical potentials, a diffusion mass fluxdevelops in between the inclusions, making the smaller inclusion shrink and the bigger expand(cf. figure 14). This diffusion mass flux is proportional to the difference of curvature of the inclusionsand to the mobility κ, the latter being modified. Thus the time scale at which the processoccurs is decreased. This problem is generally overcome by making κ vary so that its value inthe bulk phases vanishes, thus eliminating this “parasitic coarsening”. However, it can be shownthat, because of numerical truncation errors (and also other fundamental reasons that cannot bedeveloped here) this “parasitic coarsening” cannot be totally eliminated.Figure 14: Mass diffusion between inclusions of different sizes. The color field represents thegeneralized chemical potential and the interfaces are represents by iso-contours of the mass fraction.Even though the issue of the numerical increase of the interface thickness might be less critical33

in the Cahn-Hilliard model than in the van der Waals model, it is nevertheless a real difficulty toensure that these modifications do not involve modifications on the mesoscopic characteristics ofthe flow.This analysis shows that the adaptation of physical diffuse interface models to mesoscopicproblems is rather difficult and tricky. We showed in particular that it is important to know thesharp interface model that the diffuse interface model must mimic. It is indeed a reference thathelps to develop the “equivalent” diffuse interface model. Moreover, we showed that the difficultycomes from the fact that we use only physical variables (mass density ρ or mass fractionc) and that these physical variables are important to characterize (i) the interface structure (thatis aimed at being modified) and (ii) bulk phases properties (pressure, chemical potential, etc).We showed that modifying one feature without modifying the other is difficult and tricky. Thesystem lacks degrees of freedom. This degree of freedom can come from the introduction of anotherparameter whose main goal would be to characterize only the interface structure and notthe bulk properties. The phase-field variable ϕ often used in other applications of diffuse interfacemodels can be interpreted as such a variable. It has recently been shown [Jamet and Ruyer,2004] that such a phase-field modeling is possible for liquid-vapor flows. The introduction of thephase-field indeed allows to easily decouple the interface properties from the bulk properties.ReferencesJ. U. Brackbill, D. B. Kothe, and C. Zemach. A continuum method for modeling surface tension.J. Comp. Phys., 100:335–354, 1992.J. W. Cahn and J. E. Hilliard. Free energy of a nonuniform system. I. interfacial free energy. J.Chem. Physics, 28(2):258–267, 1958.J. W. Cahn and J. E. Hilliard. Free energy of a nonuniform system. II. thermodynamic basis. J.Chem. Physics, 30(5):1121–1124, 1959a.J. W. Cahn and J. E. Hilliard. Free energy of a nonuniform system. III. nucleation in a twocomponentincompressible fluid. J. Chem. Physics, 31(3):688–699, 1959b.J.-M. Delhaye, M. Giot, and M. L. Riethmuller. Thermohydraulics of two-phase systems for industrialdesign and nuclear engineering. Hemisphere Publishing Corporation, 1981.F. Dell’Isola, H. Gouin, and G. Rotoli. Nucleation of spherical shell-like interfaces by secondgradient theory: Numerical simulations. Eur. J. Mech. B/Fluids, 15(4):545–568, 1996.J. E. Dunn and J. Serrin. On the thermodynamics of intersticial working. Arch. Rational Mech.Anal., 88:88–133, 1965.C. Fouillet. Généralisation à des mélanges binaires de la méthode du second gradient et application à lasimulation numérique directe de l’ébullition nucléée. Thèse de doctorat, Université Paris VI, 2003.C. Fouillet, D. Jamet, and D. Lhuillier. A continuous interface model for the direct numerical simulationof phase-change in two-component liquid-vapor flows. In 2002 Joint ASME/EuropeanFluid Engineering Division Summer Conference, Montreal, Canada, July 14-18, 2002.G. P. Galdi, D. D. Joseph, L. Preziosi, and S. Rionero. Mathematical problems for miscible andcompressible fluids with korteweg stresses. European Journal of Mechanics B Fluids, 10(3):253–267, 1991.H. Gouin. Energy of interaction between solid surfaces and liquids. J. Phys. Chem. B, 102:1212–1218, 1998. doi: 10.1021/jp9723426.34

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