Diffuse interface models in fluid mechanics

Since λ is constant, s = s 0 . Therefore( ) ∂sρ T ds = ρ Cv 0 0dT + ρ T∂ρTdρThusρ Cv 0 dT( ) ∂s0dt = ∇ · (k ∇T ) − ρ T ∂ρTdρ+ τ : ∇v (55)dtThis equation is very similar to the classical equation of evolution of the temperature in asingle-phase fluid. However, the interpretation of the term in dρ/dt is very different. Indeed, for asingle-phase incompressible fluid, this term vanishes (by definition). In the case of a liquid-vaporinterfacial zone, (dρ/dt) represents the rate of vaporization denoted γ c . Indeed, let us consideran interface where vaporization occurs. If we follow a liquid fluid particle in its motion, as itcrosses the interface to become vapor, its density drastically decreases, therefore dρ/dt < 0. Now,ρ T (∂s 0 /∂ρ) T is analogous to the latent heat (cf. (37)). Therefore, this term can be approximatedby γ c L ρ/(ρ l − ρ v ). This term is thus a spreading of the latent heat source over the interfacialzone.2.3 Different forms of the momentum balance equationIn the form (52) of the momemtum balance equation, the Korteweg tensor might not be the mostappropriate. Indeed, from a numerical point of view for instance, the discretization of this termis not straightforward. Moreover, we will see that there exists an equivalent from in which onlyLaplacian and gradient operators appear, which is generally much easier to implement.The following identity holds:∇ · (φ ⊗ ∇ρ) = ∇(ρ ∇ · φ) − ρ ∇(∇ · φ) + φ · ∇∇ρGiven the expression (45) for P and the differential of F (15), one hasThus, equation (52) readsρ dvdt∇P = ρ ∇g + ρ s ∇T − φ · ∇∇ρ= −ρ ∇ (g − ∇ · φ) − ρ s ∇T + ∇ · τThis form of the momemtum balance equation makes clearly appears the two conditions ofequilibrium (18)-(19). This shows in particular that if any of these thermodynamic equilibriumconditions is not satisfied, it triggers a fluid motion. Moreover, since the thermodynamic equilibriumconditions appear in this equation, it shows its thermodynamic consistency. Moreover, froma numerical point of view, Jamet et al. [2002] showed that, through a detailed analysis of the discretizedenergy exchanges, this form allows to get rid of the so-called parasitic currents [Brackbillet al., 1992]. These non-physical currents, concentrated in the close vicinity to the interface, areinduced by numerical truncation that are very difficult to eliminate without a detailed analysisof the energy exchanges. The thermodynamic consistency of this model gives a framework todevelop accurate numerical schemes.2.4 Boundary conditions and contact angleSo far, in the determination of the equilibrium conditions or of the equations of motion, we havenot studied the boundary conditions that must be applied on the density field. In this section,19