Diffuse interface models in fluid mechanics

which is the equation of evolution of the internal energy.The expression for the internal energy isu = F ρ + T s (44)and the pressure P is defined byso that the differential of u reads( ) ∂FP ˆ= ρ − F (45)∂ρρ du = ρ T ds + P ρdρ + φ · d∇ρ (46)Using equations (39), (42) and (43) to express dρ/dt, ds/dt and du/dt in the above relation, onehas−∇ · q + T : ∇v = T (−∇ · q s + ∆ s ) − P ∇ · v + φ · d∇ρdtThe last term of this equation is transformed as follows(47)φ · d∇ρdt( )∂∇ρ= φ · + v · ∇∇ρ∂t( ) ∂ρ= φ · ∇ + φ∂t i v j ρ ,ij(= ∇ · φ ∂ρ )∂t(= ∇ · φ ∂ρ )∂t(= ∇ · φ ∂ρ∂t− ∂ρ∂t (∇ · φ) + (φ i v j ρ ,j ) ,i− (φ i v j ) ,iρ ,j− ∂ρ∂t (∇ · φ) + ∇ · (φ v · ∇ρ) − φ i,i v j ρ ,j − φ i v j,i ρ ,j)− ∂ρ (∇ · φ) + ∇ · (φ v · ∇ρ) − (v · ∇ρ) ∇ · φ − (φ ⊗ ∇ρ) : ∇v∂twhere ψ ,i ≡ ∂ψ/∂x i and where the Einstein convention on the repeated indices has been used.Thusφ · d∇ρ (= ∇ · φ dρ )− dρ (∇ · φ) − (φ ⊗ ∇ρ) : ∇vdtdt dtSubstituting this relation in equation (47) allows to express the entropy source ∆ s as follows[∆ s = ∇ · q s − 1 (q + φ dρ )]− 1 (T dt T 2 q + φ dρ )· ∇T + 1 [T + (P − ρ ∇ · φ) I + φ ⊗ ∇ρ] : ∇vdt T(48)The second law of thermodynamics states that ∆ s ≥ 0 for any motion. The following expressionsfor q and T satisfy this condition 3 :3 These relations are not the most general. For instance, the thermal conductivity k is in general a tensor of order twoof the form k = k 1 I + k 2 ∇ρ ⊗ ∇ρ/(∇ρ) 2 . This relation expresses that the thermal conductivity in the normal andtangential directions to the interface can be different. Likewise, a Newtonian behavior is very restrictive compared to thegeneral expressions found for the dissipative stress tensor in which five different “viscosity” coefficients appear.It is worth noting that, despite its simplicity, the method used to derive these expressions, and especially the expressionfor q, might not be the most rigorous from a fundamental point of view. Indeed, using the Hamilton’s principal, it canbe shown that the term φ dρ/dt is actually not a heat flux per say but is rather a work since it has no contribution tothe entropy source and is thus a conservative contribution. This term is known as the “intersticial working” [Dunn andSerrin, 1965].16