Diffuse interface models in fluid mechanics

It can be shown that this system of equations is thermodynamically consistent in the sensethat the total energy of the system is a decreasing function of time:∫ddt V[F 0 (c) + λ 2 (∇c)2 + ρ 0 v 22] ∫dV = −{κ [ ∇ ( µ 0 (c) − λ∇ 2 c )] }2+ τ : ∇v dV < 0VTo account for buoyancy effects the gravity term ρ g (where g is the acceleration of gravity)has to be added in the momentum balance equation. If ρ = ρ 0 in this force, this force has noeffect on the flow: it only modifies the pressure (the pressure P can be replaced by the pressure(P − ρ 0 g z), which does not modify the structure of the equations). For the gravity to have aneffect on the flow and in particular to account for a buoyancy effect, variations of the density mustbe accounted for. However, these variations are a priori not compatible with the incompressibilityapproximation. That is why the Boussinesq approximation is generally used: the variation of thedensity is neglected except in the gravity term where it is linearized. The momentum balanceequation therefore readswhereρ 0∂v∂t + ρ 0 v · ∇v = −∇P − ∇ · (λ∇c ⊗ ∇c) + ∇ · τ + ρ 0 (1 + β(c − c 0 )) gWe summarize the system of equations:β ˆ= 1 ( ) dρρ 0 dc0∇ · v = 0 (63)∂c+ v · ∇c = ∇ · [κ ∇˜µ] (64)∂t˜µ ˆ= µ 0 (c) − λ∇ 2 c (65)∂vρ 0∂t + ρ 0 v · ∇v = −∇P − ∇ · (λ∇c ⊗ ∇c) + ∇ · τ + ρ 0 (1 + β(c − c 0 )) g (66)(µ 0 (c) = 4 A (c − c 1 ) (c − c 2 ) c − c )1 + c 2(67)2It is worth noting that, like in the van der Waals model (cf. section 2.3), the “stress form” of themomentum balance equation can be transformed into an equivalent “potential form” in whichthe generalized chemical potential appears [Jacqmin, 1999]:ρ 0∂v∂t + ρ 0 v · ∇v = −∇ ˜P + ˜µ ∇c + ∇ · τ + ρ 0 (1 + β(c − c 0 )) g (68)˜P ˆ= P + FThis form of the momentum balance equation shows the influence of the curvature on themomemtum balance equation. Indeed, we showed in the previous section that, at equilibriumof a spherical inclusion, the value of the chemical potential is proportional to the local interfacecurvature (cf. (61)). Therefore, the term ˜µ ∇c represents a spreading (over the interface thickness)of a force proportional to the interface curvature and oriented in the direction normal to the interfaceapproximated by ∇c. This interpretation is very close to the Continuous Surface Forcecommonly used in sharp interface numerical methods [Brackbill et al., 1992].This system of equations is particularly attractive numerically. Indeed, we one has a numericalcode dedicated to the simulation of incompressible flows, it can be very easily generalized totwo-phase capillary flows. Indeed, one only needs to implement a source term in the momentum25