Diffuse interface models in fluid mechanics
Diffuse interface models in fluid mechanics
Diffuse interface models in fluid mechanics
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and (ii) there is an equivalent of the Laplace relationwhere the “pressure” P 0 is def<strong>in</strong>ed 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 <strong>in</strong>terpretation of a physical pressure.A Taylor expansion of these equilibrium conditions of a spherical <strong>in</strong>clusion around the equilibriumstate of a planar <strong><strong>in</strong>terface</strong> allow to show thatµ s e ≃ µ eq + 1 2 σc i − c e R(61)It is worth not<strong>in</strong>g that the form of this equilibrium condition is actually similar to the Gibbs-Thompson condition (12).This equilibrium condition shows <strong>in</strong> particular that the value of the chemical potential is proportionalto the curvature of the <strong><strong>in</strong>terface</strong>. This expla<strong>in</strong>s how an <strong><strong>in</strong>terface</strong> tends to get spherical.Indeed, let us consider an closed <strong><strong>in</strong>terface</strong> whose shape is <strong>in</strong>itially irregular. If, at each po<strong>in</strong>t ofthe <strong><strong>in</strong>terface</strong>, the <strong>in</strong>terfacial zone is at local thermodynamic equilibrium, the above equilibriumcondition is satisfied locally. These means <strong>in</strong> particular that, along the <strong><strong>in</strong>terface</strong>, the chemicalpotential is not uniform. Accord<strong>in</strong>g to the Cahn-Hilliard equation, this yields a diffusion massflux and therefore a mass diffusion. This mass diffusion makes the overall system evolve and,s<strong>in</strong>ce the Cahn-Hilliard equation is thermodynamically coherent, this evolution tends to makethe system get closer to an equilibrium state. If we consider a spherical <strong>in</strong>clusion at equilibrium,the chemical potential along the <strong><strong>in</strong>terface</strong> 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 <strong>fluid</strong>. The issueis to determ<strong>in</strong>e how the system behaves out of equilibrium. To start this analysis, we will firstsimplify the system by assum<strong>in</strong>g 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 determ<strong>in</strong>ed.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.Us<strong>in</strong>g the same developments as those presented <strong>in</strong> section 2.2 for the van der Waals model,one f<strong>in</strong>ds that[∆ f = −j · ∇ (µ − ∇ · φ) − ∇ · q + φ ∂c]∂t − j (µ − ∇ · φ)whereµ ˆ= ∂F∂c23