Diffuse interface models in fluid mechanics

where T 0 is a constant corresponding to the equilibrium temperature of the system; T 0 can beviewed as a parameter and is dropped in the following 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 interface at equilibrium at we denote z the coordinate normal to the interface.We thus seek for the function ρ(z) that defines the profile of the interfacial 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 interfacial 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 interface.This equation shows that the specific free Gibbs energy of the phases are equal. This correspondsto the condition of equilibrium of the interface discussed in section 1.2.2.By multiplying equation (22) by dρ/dz and integrate, one getsF 0 (ρ) − F 0 (ρ v ) − g eq (ρ − ρ v ) = λ 2( ) 2 dρ(24)dzIf we defineone 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 in 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, using classical thermodynamic relations, it can be shown that the pressure P 0 is given byP 0 = ρ g 0 − F 0 (28)11