Diffuse interface models in fluid mechanics

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$G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport ...$

alance equation and a convection-diffusion-like equation for c (the Cahn-Hilliard equation). Anexample is given in figure 8. The system simulated is the impact of a heavier droplet on a solidgrid. This figure shows in particular that topological changes are automatically accounted for.Using standard second order schemes for the discretization in space, the interface is captured byabout 4 mesh cells. However, Jacqmin [1999] shows that it is possible to reduce it to about 2 byusing a more complex scheme.Figure 8: Numerical simulation of the impact of a droplet on a grid using the Cahn-Hilliardmodel with a Boussinesq approximation.Because of its simplicity, simulating three dimensional systems is particularly easy, even onparallel computers. An illustration of the complex three dimensional systems that can be simulatedis shown in figure 9.Figure 9: Numerical simulation of a complex Rayleigh-Taylor instability in a I-shape reservoirusing a Cahn-Hilliard diffuse interface model.3.3.2 Finite density contrastThe case where the bulk phases have very different densities (for instance air and water at roomtemperature ρ a ≃ 1 kg/m 3 and ρ w ≃ 1000 kg/m 3 ) is rather different. From a physical point of26
view, in this case, the velocity is not solenoidal [Galdi et al., 1991]. Therefore, a priori, the pressurecan no longer be introduced as a Lagrange multiplier accounting for this constraint. However,from a practical point of view, the solenoidal constraint on the velocity field is attractive: it allowsto use standard numerical methods such as the projection method. The natural choice is thus touse the following equations∇ · v = 0 (69)dc= ∇ · [κ ∇˜µ]dt(70)ρ(c) dvdt = −∇ ˜P + ˜µ ∇c + ∇ · τ + ρ(c) g (71)where ρ(c) is generally a linear functionρ(c) = ρ 1 + c − c 1c 2 − c 1(ρ 2 − ρ 1 ) (72)It must be emphasized that this system does not satisfy the mass balance equation (1). Indeed,from equations (69) and (70), it is straightforward to show that∂ρdρ+ ∇ · (ρ v) = ∇ · [κ ∇˜µ] ≠ 0∂t dcHowever, if dρ/dc = cste, this equation can be written in a conservative form, which shows that(by integration over the entire volume), the total mass of the system is conserved (an importantproperty that many sharp interface numerical methods do not satisfy).Despite its simplicity, it can be shown that this system is ill-posed, in the sense that no monotonicallydecreasing energy is associated to this system of equations; it thus violates the secondlaw of thermodynamics. This may be the price to pay to conserve the easiness of the numericalimplementation. . .However, it is possible to develop a diffuse interface model with a finite density contrast thatis thermodynamically coherent. One of the main issue is to define incompressibility flow whenthe velocity field is not solenoidal. Lowengrub and Truskinovsky [1998] showed that, in this case,it is relevant to use the thermodynamic incompressibility: the density is independent of the pressure.The main thermodynamic variables are the mass fraction c and the thermodynamic pressure Pand the relevant thermodynamic potential is not the free energy F but the specific Gibbs freeenthalpy g(c, P, ∇c) whose expression is:g(c, P, ∇c) = f 0 (c) + α 2 (∇c)2 +where the linearity in P is due to the incompressibility assumption. The coefficient α is equivalentto a capillary coefficient but its dimension is not that of λ ([α] = [λ]/[ρ]).Lowengrub and Truskinovsky [1998] show that the corresponding system of balance equationsis the following:Pρ(c)whereρ dvdt∂ρ+ ∇ · (ρ v) = 0∂tρ dc = ∇ · (κ ∇˜µ)dt= −∇P − ∇ · (α ρ ∇c ⊗ ∇c) + ∇ · τ˜µ = df 0dc − P dρρ 2 dc − 1 ∇ · (ρ α ∇c)ρ27
Page 8 and 9: is no longer the case and some of t
Page 10 and 11: where δ represents the variation,
Page 12 and 13: Thus equation (27) readsP 0 (ρ v )
Page 14: WW2σ/Rρρ2σ/RbubbledropFigure 4:
Page 17: q = −φ dρ − k ∇T (49)dtT =
Page 20 and 21: we show that particular boundary co
Page 22 and 23: 3.1.2 Equilibrium conditionsSince w
Page 24 and 25: φ ˆ= ∂F∂∇cThe following exp
Page 28 and 29: This system is thermodynamically co
Page 30 and 31: P 0modifiedP eqρ vρ lρFigure 10:
Page 32 and 33: Figure 11: Numerical simulation of
Page 34 and 35: in the Cahn-Hilliard model than in

Diffuse interface models in fluid mechanics

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