This system is thermodynamically coherent and has been used to study p<strong>in</strong>ch-off and reconnectionof <strong><strong>in</strong>terface</strong>s for <strong>in</strong>stance [Lee et al., 2002a,b]. However, this sytem is much more coupledthan the <strong>in</strong>compressible model (or the thermodynamically <strong>in</strong>coherent model (69)-(71)). Indeed,the density ρ(c) appears <strong>in</strong> many terms and <strong>in</strong> particular <strong>in</strong> expression for the Korteweg stresstensor and <strong>in</strong> the Laplacian part of ˜µ. Moreover (and certa<strong>in</strong>ly most importantly), the pressureP appears <strong>in</strong> the expression for ˜µ, which <strong>in</strong>duces a complex coupl<strong>in</strong>g between the Cahn-Hilliardand momentum balance equation.4 <strong>Diffuse</strong> <strong><strong>in</strong>terface</strong> <strong>models</strong> and numerical simulation of mesoscopicproblems4.1 Numerical vs physical <strong><strong>in</strong>terface</strong> thicknessIn section 1.3, we showed that, <strong>in</strong> <strong>fluid</strong> <strong>mechanics</strong>, diffuse <strong><strong>in</strong>terface</strong> <strong>models</strong> are attractive fortwo ma<strong>in</strong> reasons: (i) they allow to study peculiar physical phenomena where sharp <strong><strong>in</strong>terface</strong><strong>models</strong> fail and (ii) they can be easily implemented numerically, which virtually elim<strong>in</strong>ates thedifficult and tedious numerical treatment of the mov<strong>in</strong>g boundaries. Whatever the reason, whenwe have to solve the equations of motion numerically, these equations are discretized <strong>in</strong> time andspace. In particular, if we consider the discretization <strong>in</strong> space, the <strong><strong>in</strong>terface</strong>s have to be capturedby the mesh. To illustrate this po<strong>in</strong>t, let us take a particular example of a regular mesh wherethe size of a mesh cell is ∆x <strong>in</strong> all directions. S<strong>in</strong>ce the cont<strong>in</strong>uous partial differential equationsare discretized on this mesh, all the spacial variations must be captured, so that the numericaltruncation errors do not make the system degenerate. In particular, the <strong><strong>in</strong>terface</strong> structure mustbe captured. This means that ∆x must be smaller than the <strong><strong>in</strong>terface</strong> thickness h. Typically, forstandard discretization schemes, one has ∆x ≃ h/4 so that the bi-Laplacian of the Cahn-Hilliardequation is well approximated. Even for higher order schemes, ∆x is of the order of h.The diffuse <strong><strong>in</strong>terface</strong> <strong>models</strong> presented <strong>in</strong> the previous sections (either for a liquid-vapor systemof for non-miscible phases) have been derived based on physical arguments: the ma<strong>in</strong> thermodynamicvariable are physical variables (mass density ρ or mass concentration c), the energyfunctional is physical and the equations of motion are based on physical first pr<strong>in</strong>ciples of conservation.Therefore, the <strong>in</strong>ternal structure of the <strong><strong>in</strong>terface</strong> is physical and <strong>in</strong> particular the <strong><strong>in</strong>terface</strong>thickness is also physical. This approach is relevant for any study where the <strong><strong>in</strong>terface</strong>s must bemodeled as cont<strong>in</strong>uous transition zones: coalescence and rupture of <strong><strong>in</strong>terface</strong>s, nucleation, etc. Inthese cases, the typical <strong><strong>in</strong>terface</strong> thickness is of the order of 10 −9 m and ∆x ≃ 10 −9 m. Therefore,assum<strong>in</strong>g that the maximum size of the mesh is of the order 10 6 mesh cells, the typical size of the3D doma<strong>in</strong> studied is about ( 10 −7) 3m 3 for a regular mesh. This is extremely small, even thoughit is relevant for this k<strong>in</strong>d of physical processes occur<strong>in</strong>g at the scale of an <strong><strong>in</strong>terface</strong>.However, when diffuse <strong><strong>in</strong>terface</strong> <strong>models</strong> are used for numerical convenience, if the physicalmodel is used as it is, one must restrict the studies to doma<strong>in</strong>s whose typical size is 10 −7 m. Now,<strong>in</strong> many applications, the typical size of the <strong>in</strong>clusions (bubbles or droplets) of the two-phase systemis much larger than this, by at least one order of magnitude. At this po<strong>in</strong>t, different strategiesare possible: (i) diffuse <strong><strong>in</strong>terface</strong> <strong>models</strong> are abandonned as well as their potential ease of use, (ii)adaptative mesh ref<strong>in</strong>ement techniques may be developed to capture the <strong>in</strong>terfacial zones or (iii)diffuse <strong><strong>in</strong>terface</strong> <strong>models</strong> are adapted to the numerical simulation of mesoscopic problems. In thelatter case, the adaptation of the model must be such that the value of the <strong><strong>in</strong>terface</strong> thickness isno longer dictated by physical arguments but rather by numerical arguments.The third solution will be studied <strong>in</strong> the subsequent sections. Nevertheless, the second solutiondeserves to be discussed. Indeed, we believe that it is a promis<strong>in</strong>g strategy because manyphysical phenoma occur close to the <strong><strong>in</strong>terface</strong> and a f<strong>in</strong>e discretization is therefore necessary tocapture them. However, us<strong>in</strong>g mesh ref<strong>in</strong>ement to capture the <strong>in</strong>ternal structure of an <strong><strong>in</strong>terface</strong>might not be the most relevant solution. Indeed, if one is <strong>in</strong>terested <strong>in</strong> problems whose typicalsize is that of a bubble or droplet (mesocopic scale), the sharp <strong><strong>in</strong>terface</strong> approximation is the most28
elevant. This means that all the physical processes of <strong>in</strong>terest occur at the scale of the <strong>in</strong>clusion(bubble or droplet) and not of the <strong><strong>in</strong>terface</strong> structure. Thus, there is a clear and justified scaleseparation. Now, us<strong>in</strong>g a mesh ref<strong>in</strong>ement technique for these problems means that a complexnumerical technique is used to capture phenomena of no physical <strong>in</strong>terest that have been <strong>in</strong>troducedto simplify the numerical implementation. Thus, seek<strong>in</strong>g for a way to use diffuse <strong><strong>in</strong>terface</strong><strong>models</strong> with an artificial <strong><strong>in</strong>terface</strong> thickness appears as the most relevant solution.4.2 Necessary modification of the diffuse <strong><strong>in</strong>terface</strong> <strong>models</strong>In the previous section we showed that, for many problems where the relevant scale is the radiusof an <strong>in</strong>clusion, the scale separation with the <strong><strong>in</strong>terface</strong> thickness is justified. In this case, it isalmost impossible to use physical diffuse <strong><strong>in</strong>terface</strong> <strong>models</strong> on a regular mesh: much too manymesh po<strong>in</strong>ts would be necessary only to capture the <strong>in</strong>terfacial zones. Therefore, the typical sizeof the <strong><strong>in</strong>terface</strong>s has to be adapted so that a reasonnable mesh can be used. In this case, the meshis more or less given, and therefore ∆x is given. Thus, the <strong><strong>in</strong>terface</strong> thickness h must be adaptedso that the <strong><strong>in</strong>terface</strong> structure can be captured by the mesh. The <strong><strong>in</strong>terface</strong> thickness h shouldtherefore be a free parameter whose value can be chosen arbitrarily. In the subsequent sections,we study if and how this is possible. We beg<strong>in</strong> with the van der Waals model and then we studythe Cahn-Hilliard model.4.3 Liquid-vapor flows with phase-change4.3.1 Modification of the parametersIn section 2.1.2, we showed that, with the van der Waals model, the <strong><strong>in</strong>terface</strong> thickness is givenby√1 λh =(73)ρ l − ρ v 2 Awhere λ is the capillary coefficient and A is a coefficient that characterizes the function W (ρ) andtherefore the free energy of the <strong>fluid</strong> F (ρ) (see equation (25)).The goal is that h can be chosen arbitrarily. Now, the <strong><strong>in</strong>terface</strong> thickness is a consequence ofthe diffuse <strong><strong>in</strong>terface</strong> model; it is not a primary parameter of the model but rather a secondaryparameter. The primary parameters of the model are ρ l , ρ v , λ and A. Therefore, these are theparameters on which one might have a degree of freedom to fix the value of h arbitrarily. Amongthe primary parameters, λ is the only “non-classical” parameter: all the others are <strong>in</strong>volved <strong>in</strong>the properties of the bulk phases. In particular, the parameter A is characteristic not only of thethermodynamic behavior of the <strong>fluid</strong> with<strong>in</strong> the <strong><strong>in</strong>terface</strong> but also with<strong>in</strong> the bulk phases: thefunction F 0 (ρ) (<strong>in</strong> which the parameter A appears) is valid for any value of ρ and <strong>in</strong> particular forthe values of ρ reached with<strong>in</strong> the bulk phases. In particular, it can be shown that the isothermalcompressibility of the bulk phases at saturation are given by( ) ∂P= 2 A ρ v (ρ l − ρ v ) 2 (74)∂ρv( ) ∂P= 2 A ρ l (ρ l − ρ v ) 2 (75)∂ρlThus, the only parameter clearly associated to the <strong><strong>in</strong>terface</strong> is λ and it appears as the mostobvious parameter that can be modified to <strong>in</strong>crease the <strong><strong>in</strong>terface</strong> thickness. Equation (73) showsthat λ should be <strong>in</strong>creased to <strong>in</strong>crease h.Now, we have also shown that the expression for the surface tension is the follow<strong>in</strong>g:σ = (ρ l − ρ v ) 36√2 A λ (76)29