Diffuse interface models in fluid mechanics

DT (K)12108642mixturepure fluidh (J/m^2/s/K)16000014000012000010000080000600004000020000mixturepure fluid00 0.5 1 1.5 2 2.5 3t (s)00 0.5 1 1.5 2 2.5 3t (s)Figure 13: Comparison of the mean temperature wall of of the instantaneous mean heat flux withand without the presence of a dilute substance [Jamet and Fouillet, 2005].4.4 Two-phase flows of non-miscible fluidsIn the case of non-miscible phases, the issue is the same: how to make the interface a free parameterwithout modifying the flow at the mesoscopic scale? The solution is the same as in theliquid-vapor case: increase λ and decrease A proportionally. However, in this case, the consequencesare different. The variations of the classical chemical potential µ 0 (c) must be modified.We showed in the previous section that this modification might not be critical as long as noexternal scale of variation of the chemical potential is imposed. In common two-phase flow applications,it is rare that any external scale of variation of the chemical potential is imposed andthe situation is thus easier that in the liquid-vapor case. However, there does exist a difficulty. Indeed,the mobility κ is related to the time scale at which a system goes back to equilibrium. Sincethe interface thickness is modified, if κ is not modified, the time scale at which the interfacial zonegoes back to equilibrium is modified. Therefore, κ must be increased to ensure that the interfacekeeps close to equilibrium. Finding the optimal value for κ is not straightforward. Now, in section3.1.2 we showed that the chemical potential of equilibrium of a spherical inclusion dependson the radius of curvature of the system. Therefore, if two spherical inclusion of different radiiat put close to each other, because of this difference of chemical potentials, a diffusion mass fluxdevelops in between the inclusions, making the smaller inclusion shrink and the bigger expand(cf. figure 14). This diffusion mass flux is proportional to the difference of curvature of the inclusionsand to the mobility κ, the latter being modified. Thus the time scale at which the processoccurs is decreased. This problem is generally overcome by making κ vary so that its value inthe bulk phases vanishes, thus eliminating this “parasitic coarsening”. However, it can be shownthat, because of numerical truncation errors (and also other fundamental reasons that cannot bedeveloped here) this “parasitic coarsening” cannot be totally eliminated.Figure 14: Mass diffusion between inclusions of different sizes. The color field represents thegeneralized chemical potential and the interfaces are represents by iso-contours of the mass fraction.Even though the issue of the numerical increase of the interface thickness might be less critical33