Diffuse interface models in fluid mechanics

q = −φ dρ − k ∇T (49)dtT = (−P + ρ ∇ · φ) I − φ ⊗ ∇ρ + τ (50)where k must be positive and where τ is the dissipative stress tensor that must satisfyτ : ∇v ≥ 0A classical Newtonian fluid satisfies this condition.Using these closure relations, the system of balance equations that describes the motion of thefluid is the following:dρ= −ρ ∇ · v (51)dtρ dvdt= −∇P + ∇(ρ ∇ · φ) − ∇ · (φ ⊗ ∇ρ) + ∇ · τ (52)ρ de (dt = ∇ · φ dρdt2.2.1 Korteweg stress tensor and surface tension force)+ ∇ · (k ∇T ) + ∇ · (v · T ) (53)In the most classical case where the energy of the fluid is expressed by (14),φ = λ ∇ρandP = P 0 (ρ, T ) − λ 2 (∇ρ)2The momentum balance equation thus readsρ dvdt = −∇P 0 + ∇(λ ρ ∇ 2 ρ + λ )2 (∇ρ)2 − ∇ · (λ∇ρ ⊗ ∇ρ) + ∇ · τ (54)The stress tensor (λ∇ρ ⊗ ∇ρ) is called the Korteweg stress tensor [Korteweg, 1901]. We showin the following that this stress tensor implies a tension force in the tangential direction to theinterface, interpreted as the surface tension force.To simplify the analysis, we consider a planar interface at equilibrium. We denote z and x thecoordinates respectively normal and tangential to the interface (because of symmetry, only onetangential direction can be accounted for). Thus, all the variables depend only on z.Let us first define˜P ˆ= P 0 −(λ ρ ∇ 2 ρ + λ )2 (∇ρ)2so that, at equilibrium, the momentum balance equation (55) reduces toBy integration, one simply getsd ˜Pdz + λ d dz˜P (z) = P ∞ − λwhere P ∞ in the pressure in the bulk phases.( ) 2 dρ= 0dz( ) 2 dρdz17