Diffuse interface models in fluid mechanics
Diffuse interface models in fluid mechanics
Diffuse interface models in fluid mechanics
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where T 0 is a constant correspond<strong>in</strong>g to the equilibrium temperature of the system; T 0 can beviewed as a parameter and is dropped <strong>in</strong> the follow<strong>in</strong>g developments.Mathematically, this equation is a differential equation that the density field ρ(x) must satisfyat equilibrium. To better understand the consequences of this differential equation, we nowconsider a planar <strong><strong>in</strong>terface</strong> at equilibrium at we denote z the coord<strong>in</strong>ate normal to the <strong><strong>in</strong>terface</strong>.We thus seek for the function ρ(z) that def<strong>in</strong>es the profile of the <strong>in</strong>terfacial zone. This functionmust satisfywhere g 0 ˆ= ∂F 0 /∂ρ.g 0g 0 (ρ) − λ d2 ρ= cste (22)dz2 AρFigure 3: Illustration of the graph of g 0 (ρ).Very far from the <strong>in</strong>terfacial zone, bulk liquid and vapor phases exist and therefore d 2 ρ/dz 2 =0; likewise, dρ/dz = 0. This equation shows thatg 0 (ρ v ) = cste = g 0 (ρ l ) ˆ= g eq (23)where ρ v and ρ l are the densities of the vapor and liquid phases respectively far from the <strong><strong>in</strong>terface</strong>.This equation shows that the specific free Gibbs energy of the phases are equal. This correspondsto the condition of equilibrium of the <strong><strong>in</strong>terface</strong> discussed <strong>in</strong> section 1.2.2.By multiply<strong>in</strong>g equation (22) by dρ/dz and <strong>in</strong>tegrate, one getsF 0 (ρ) − F 0 (ρ v ) − g eq (ρ − ρ v ) = λ 2( ) 2 dρ(24)dzIf we def<strong>in</strong>eone haswhich is clearly a differential equation for ρ(z).W (ρ) ˆ= F 0 (ρ) − F 0 (ρ v ) − g eq (ρ − ρ v ) (25)λ2This equation shows <strong>in</strong> particular thatwhich is equivalent to( ) 2 dρ= W (ρ)dzW (ρ v ) = W (ρ l ) (26)F 0 (ρ l ) − g eq ρ l = F 0 (ρ v ) − g eq ρ v (27)Now, us<strong>in</strong>g classical thermodynamic relations, it can be shown that the pressure P 0 is given byP 0 = ρ g 0 − F 0 (28)11