Diffuse interface models in fluid mechanics

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