Diffuse interface models in fluid mechanics

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

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
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 20 and 21: we show that particular boundary co
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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