Diffuse interface models in fluid mechanics

The modification of A does not modify any property at saturation. In particular, the saturationpressure and the densities at saturation are not modified. However, it modifies the variations ofthe pressure with the density around the states at saturation ρ v and ρ l . In particular, it modifiesthe compressibility of the bulk phases. Compressibility has an effect on the motion of the fluidonly if the typical velocity of the fluid is of the order of magnitude of the speed of sound (thatdepends on the compressibility of the fluid). In many applications, the bulk velocities are verysmall compared to the speed of sound and, even though the latter is decreased by decreasing A,this remains valid.However, thermal expansion effects are important to account for because they are at the originof natural convection. The expression for the thermal expansion coefficient isκ T = 1 ρ(∂P/∂ρ) T(∂P/∂T ) ρThis expression shows that, if (∂P/∂ρ) T is decreased, (∂P/∂T ) ρ must be decreased as well to conservethe value of κ T . Now, the decrease of (∂P/∂T ) ρ is equivalent to the decrease of (dP sat /dT ).Because of the Clapeyron relationdP satdT = LT (v v − v l )in order the keep the value of L constant, it is necessary to modify the value of T . Actually, athorough analysis of this issue [Fouillet, 2003] shows that only the value of the temperature ofreference has to be modified. Since only temperature differences are important, the increase ofthe reference temperature is not very important. Moreover, it has been shown that this modificationmainly influences the time scale at which the interfacial zone gets back to the saturationtemperature. This time is physically extremely small and an increase of the time generally barelyhas any influence.This short analysis shows that the issue of the numerical increase of the interface thickness isnot trivial. It also shows that, even though the initial goal was to modify only a property of theinternal structure of the interface, it lead to modify important properties of the bulk phases aswell; it actually lead to modify all the variation of the equation of state P (ρ, T ). This is mainlybecause the graph of this function (or equivalently the function F 0 (ρ, T )) influences bulk propertiesas well as the internal structure of the interface.However, using these modifications, the van der Waals model can be used to simulate complexproblems such as nucleate boiling on a heated surface [Jamet and Fouillet, 2005] as shownin figure 11.This model can be extended to dilute binary mixtures as shown in figure 12. The influenceof the addition of a small amount of an extra component has been shown to have an importantinfluence on the wall heat exchange coefficient (cf. figure 13), which is commonly observed experimentally.Nevertheless, it is very difficult to use this model to get quantitative results. One of themain issues is gravity. Indeed, it has been shown that all the pressure scales had to be modified:(∂P/∂ρ) and (∂P/∂T ). These modifications are acceptable as long as no external pressurescale is imposed. But gravity imposes an external scale of pressure variation. To illustrate theissue, let us consider a vapor bubble on a heated wall. Because of gravity, the pressure at the topof the bubble is lower than the pressure at the bottom of the bubble. The local thermodynamicequilibrium of the interface is determined by the Clapeyron relation which determines the equilibrepressure as a function of the local temperature. So that the interface remains at equilibrium,the temperature must vary in the direction of the gravity field: lower at the top of the bubble thanat its bottom:ρ g∇T =(dP sat /dT ) ≃ ρ g(∂P/∂T ) ρ31