Diffuse interface models in fluid mechanics
Diffuse interface models in fluid mechanics
Diffuse interface models in fluid mechanics
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DT (K)12108642mixturepure <strong>fluid</strong>h (J/m^2/s/K)16000014000012000010000080000600004000020000mixturepure <strong>fluid</strong>00 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 <strong>in</strong>stantaneous mean heat flux withand without the presence of a dilute substance [Jamet and Fouillet, 2005].4.4 Two-phase flows of non-miscible <strong>fluid</strong>sIn the case of non-miscible phases, the issue is the same: how to make the <strong><strong>in</strong>terface</strong> a free parameterwithout modify<strong>in</strong>g the flow at the mesoscopic scale? The solution is the same as <strong>in</strong> theliquid-vapor case: <strong>in</strong>crease λ and decrease A proportionally. However, <strong>in</strong> this case, the consequencesare different. The variations of the classical chemical potential µ 0 (c) must be modified.We showed <strong>in</strong> 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 <strong>in</strong> 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. S<strong>in</strong>cethe <strong><strong>in</strong>terface</strong> thickness is modified, if κ is not modified, the time scale at which the <strong>in</strong>terfacial zonegoes back to equilibrium is modified. Therefore, κ must be <strong>in</strong>creased to ensure that the <strong><strong>in</strong>terface</strong>keeps close to equilibrium. F<strong>in</strong>d<strong>in</strong>g the optimal value for κ is not straightforward. Now, <strong>in</strong> section3.1.2 we showed that the chemical potential of equilibrium of a spherical <strong>in</strong>clusion dependson the radius of curvature of the system. Therefore, if two spherical <strong>in</strong>clusion of different radiiat put close to each other, because of this difference of chemical potentials, a diffusion mass fluxdevelops <strong>in</strong> between the <strong>in</strong>clusions, mak<strong>in</strong>g the smaller <strong>in</strong>clusion shr<strong>in</strong>k and the bigger expand(cf. figure 14). This diffusion mass flux is proportional to the difference of curvature of the <strong>in</strong>clusionsand to the mobility κ, the latter be<strong>in</strong>g modified. Thus the time scale at which the processoccurs is decreased. This problem is generally overcome by mak<strong>in</strong>g κ vary so that its value <strong>in</strong>the bulk phases vanishes, thus elim<strong>in</strong>at<strong>in</strong>g this “parasitic coarsen<strong>in</strong>g”. However, it can be shownthat, because of numerical truncation errors (and also other fundamental reasons that cannot bedeveloped here) this “parasitic coarsen<strong>in</strong>g” cannot be totally elim<strong>in</strong>ated.Figure 14: Mass diffusion between <strong>in</strong>clusions of different sizes. The color field represents thegeneralized chemical potential and the <strong><strong>in</strong>terface</strong>s are represents by iso-contours of the mass fraction.Even though the issue of the numerical <strong>in</strong>crease of the <strong><strong>in</strong>terface</strong> thickness might be less critical33