we show that particular boundary conditions arise from the existence of the capillary term <strong>in</strong> theenergy functional and that these boundary conditions are related to the contact angle 4 .Let us consider a liquid-vapor system <strong>in</strong> contact with a solid wall. Let us consider an energyof <strong>in</strong>teraction between the solid and the <strong>fluid</strong> U s (per unit surface area) and let us assume thatthis energy depend only on the local density of the <strong>fluid</strong> at the boundary (this assumption can bejustified by a mean-field approximation [Gou<strong>in</strong>, 1998]). Therefore, the total <strong>in</strong>ternal energy of thesystem is∫∫U(S, ρ, ∇ρ) dV + U s (ρ) dAV∂Vwhere ∂V is the boundary of the <strong>fluid</strong> doma<strong>in</strong> V . Follow<strong>in</strong>g the developments made <strong>in</strong> section2.1.2, the application of the second law of thermodynamics to determ<strong>in</strong>e 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 rem<strong>in</strong>d that L 1 is the constant Lagrange multiplier account<strong>in</strong>g for the constra<strong>in</strong>t ofconservation of the total <strong>in</strong>ternal 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 <strong>in</strong>tegral is a term that did not appear <strong>in</strong> the study developed <strong>in</strong> section 2.1.2.S<strong>in</strong>ce the above condition must be satisfied for any variation δρ, the follow<strong>in</strong>g condition must besatisfied at the boundary ∂V :n · φ = − dU s(56)dρTo illustrate the physical mean<strong>in</strong>g of this boundary condition, let us consider the follow<strong>in</strong>g assumptions:the expression for F (ρ, ∇ρ, T ) is given by (14) so that φ = λ ∇ρ and U s (ρ) is assumedto be l<strong>in</strong>ear so that dU s /dρ = β = cste. The boundary condition therefore reads∂V∂Vn · ∇ρ = − β λ(57)S<strong>in</strong>ce β and λ are constant, this consdition imposes the value of the normal derivative of ρ. Asillustrated <strong>in</strong> figure 6, this imposes the value of the contact angle. The <strong>in</strong>terested reader can referto [Seppecher, 1996, Jacqm<strong>in</strong>, 2000] for <strong>in</strong>stance where this boundary condition has been studied.It is worth not<strong>in</strong>g 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 l<strong>in</strong>e (a variation that is observedexperimentally). This form is thermodynamically coherent s<strong>in</strong>ce it ensures that the entropy of thesystem <strong>in</strong>creases.3 Two-phase flows of non-miscible <strong>fluid</strong>s: the Cahn-Hilliard modelIn the previous section, we presented the van der Waals model of capillarity dedicated to themodel<strong>in</strong>g of an <strong><strong>in</strong>terface</strong> that separates the liquid and vapor phase of the same pure substance.Many two-phase systems are made of different substances, for <strong>in</strong>stance air and water or oil andwater. In this case, an <strong><strong>in</strong>terface</strong> 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 <strong>in</strong> contact with a solid wall, it is observed that the angle θ between the liquid-gas<strong><strong>in</strong>terface</strong> and the solid surface is characteristic of the triplet solid-liquid-gas. For <strong>in</strong>stance, 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 <strong>in</strong>teraction between the solid and the liquid. If the solid hasmore aff<strong>in</strong>ity with the liquid than with the gas, the system tends to m<strong>in</strong>imize its energy by <strong>in</strong>creas<strong>in</strong>g 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 po<strong>in</strong>t of view, why do these two species tend to be separated? At the molecularlevel, the energy of <strong>in</strong>teraction of two molecules of different species is larger than that one twomolecules of the same species. Thus, s<strong>in</strong>ce the system tends to m<strong>in</strong>imize its energy, if the speciesare separated, the energy of the system is weaker. This simple reason<strong>in</strong>g shows that the “driv<strong>in</strong>gforce” of the transition, which eventually gives rise to the existence of an <strong><strong>in</strong>terface</strong>, is the amountof one species <strong>in</strong>to the other. If not too many molecules of one species is present is the other,the energy of the system is not that <strong>in</strong>creased but if more molecules are added, the energy getsto large and the system tends to separate <strong>in</strong>to two different phases: one rich <strong>in</strong> the first speciesand the other rich <strong>in</strong> the other species. In this case, the relevant thermodynamic variable is theconcentration of one species <strong>in</strong> 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 <strong>in</strong> the mixture; it is denoted ρ 0 . Let c denote themass fraction (or concentration) of one species <strong>in</strong> 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 po<strong>in</strong>t of view.Indeed, us<strong>in</strong>g a mean-field approximation, it can be shown that the attractive energy of <strong>in</strong>teractionof molecules of different types gives rise to this form for the energy of the mixture and thatλ depends only on the <strong>in</strong>ter-molecular potentials.21