Diffuse interface models in fluid mechanics

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$G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport ...$

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
Page 8 and 9: is no longer the case and some of t
Page 10 and 11: where δ represents the variation,
Page 12 and 13: Thus equation (27) readsP 0 (ρ v )
Page 14: WW2σ/Rρρ2σ/RbubbledropFigure 4:
Page 17: q = −φ dρ − k ∇T (49)dtT =
Page 22 and 23: 3.1.2 Equilibrium conditionsSince w
Page 24 and 25: φ ˆ= ∂F∂∇cThe following exp
Page 26 and 27: alance equation and a convection-di
Page 28 and 29: This system is thermodynamically co
Page 30 and 31: P 0modifiedP eqρ vρ lρFigure 10:
Page 32 and 33: Figure 11: Numerical simulation of
Page 34 and 35: in the Cahn-Hilliard model than in

Diffuse interface models in fluid mechanics

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