Diffuse interface models in fluid mechanics
Diffuse interface models in fluid mechanics
Diffuse interface models in fluid mechanics
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which is the equation of evolution of the <strong>in</strong>ternal energy.The expression for the <strong>in</strong>ternal energy isu = F ρ + T s (44)and the pressure P is def<strong>in</strong>ed byso that the differential of u reads( ) ∂FP ˆ= ρ − F (45)∂ρρ du = ρ T ds + P ρdρ + φ · d∇ρ (46)Us<strong>in</strong>g equations (39), (42) and (43) to express dρ/dt, ds/dt and du/dt <strong>in</strong> 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 E<strong>in</strong>ste<strong>in</strong> convention on the repeated <strong>in</strong>dices has been used.Thusφ · d∇ρ (= ∇ · φ dρ )− dρ (∇ · φ) − (φ ⊗ ∇ρ) : ∇vdtdt dtSubstitut<strong>in</strong>g this relation <strong>in</strong> 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 follow<strong>in</strong>g expressionsfor q and T satisfy this condition 3 :3 These relations are not the most general. For <strong>in</strong>stance, the thermal conductivity k is <strong>in</strong> 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 <strong>in</strong> the normal andtangential directions to the <strong><strong>in</strong>terface</strong> can be different. Likewise, a Newtonian behavior is very restrictive compared to thegeneral expressions found for the dissipative stress tensor <strong>in</strong> which five different “viscosity” coefficients appear.It is worth not<strong>in</strong>g 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 po<strong>in</strong>t of view. Indeed, us<strong>in</strong>g the Hamilton’s pr<strong>in</strong>cipal, it canbe shown that the term φ dρ/dt is actually not a heat flux per say but is rather a work s<strong>in</strong>ce it has no contribution tothe entropy source and is thus a conservative contribution. This term is known as the “<strong>in</strong>tersticial work<strong>in</strong>g” [Dunn andSerr<strong>in</strong>, 1965].16