Diffuse interface models in fluid mechanics

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$G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport ...$

This system is thermodynamically coherent and has been used to study pinch-off and reconnectionof interfaces for instance [Lee et al., 2002a,b]. However, this sytem is much more coupledthan the incompressible model (or the thermodynamically incoherent model (69)-(71)). Indeed,the density ρ(c) appears in many terms and in particular in expression for the Korteweg stresstensor and in the Laplacian part of ˜µ. Moreover (and certainly most importantly), the pressureP appears in the expression for ˜µ, which induces a complex coupling between the Cahn-Hilliardand momentum balance equation.4 Diffuse interface models and numerical simulation of mesoscopicproblems4.1 Numerical vs physical interface thicknessIn section 1.3, we showed that, in fluid mechanics, diffuse interface models are attractive fortwo main reasons: (i) they allow to study peculiar physical phenomena where sharp interfacemodels fail and (ii) they can be easily implemented numerically, which virtually eliminates thedifficult and tedious numerical treatment of the moving boundaries. Whatever the reason, whenwe have to solve the equations of motion numerically, these equations are discretized in time andspace. In particular, if we consider the discretization in space, the interfaces have to be capturedby the mesh. To illustrate this point, let us take a particular example of a regular mesh wherethe size of a mesh cell is ∆x in all directions. Since the continuous 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 interface structure mustbe captured. This means that ∆x must be smaller than the interface 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 interface models presented in the previous sections (either for a liquid-vapor systemof for non-miscible phases) have been derived based on physical arguments: the main 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 principles of conservation.Therefore, the internal structure of the interface is physical and in particular the interfacethickness is also physical. This approach is relevant for any study where the interfaces must bemodeled as continuous transition zones: coalescence and rupture of interfaces, nucleation, etc. Inthese cases, the typical interface thickness is of the order of 10 −9 m and ∆x ≃ 10 −9 m. Therefore,assuming that the maximum size of the mesh is of the order 10 6 mesh cells, the typical size of the3D domain studied is about ( 10 −7) 3m 3 for a regular mesh. This is extremely small, even thoughit is relevant for this kind of physical processes occuring at the scale of an interface.However, when diffuse interface models are used for numerical convenience, if the physicalmodel is used as it is, one must restrict the studies to domains whose typical size is 10 −7 m. Now,in many applications, the typical size of the inclusions (bubbles or droplets) of the two-phase systemis much larger than this, by at least one order of magnitude. At this point, different strategiesare possible: (i) diffuse interface models are abandonned as well as their potential ease of use, (ii)adaptative mesh refinement techniques may be developed to capture the interfacial zones or (iii)diffuse interface models 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 interface thickness isno longer dictated by physical arguments but rather by numerical arguments.The third solution will be studied in the subsequent sections. Nevertheless, the second solutiondeserves to be discussed. Indeed, we believe that it is a promising strategy because manyphysical phenoma occur close to the interface and a fine discretization is therefore necessary tocapture them. However, using mesh refinement to capture the internal structure of an interfacemight not be the most relevant solution. Indeed, if one is interested in problems whose typicalsize is that of a bubble or droplet (mesocopic scale), the sharp interface approximation is the most28
elevant. This means that all the physical processes of interest occur at the scale of the inclusion(bubble or droplet) and not of the interface structure. Thus, there is a clear and justified scaleseparation. Now, using a mesh refinement technique for these problems means that a complexnumerical technique is used to capture phenomena of no physical interest that have been introducedto simplify the numerical implementation. Thus, seeking for a way to use diffuse interfacemodels with an artificial interface thickness appears as the most relevant solution.4.2 Necessary modification of the diffuse interface modelsIn the previous section we showed that, for many problems where the relevant scale is the radiusof an inclusion, the scale separation with the interface thickness is justified. In this case, it isalmost impossible to use physical diffuse interface models on a regular mesh: much too manymesh points would be necessary only to capture the interfacial zones. Therefore, the typical sizeof the interfaces 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 interface thickness h must be adaptedso that the interface structure can be captured by the mesh. The interface 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 begin 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 interface 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 fluid F (ρ) (see equation (25)).The goal is that h can be chosen arbitrarily. Now, the interface thickness is a consequence ofthe diffuse interface 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 involved inthe properties of the bulk phases. In particular, the parameter A is characteristic not only of thethermodynamic behavior of the fluid within the interface but also within the bulk phases: thefunction F 0 (ρ) (in which the parameter A appears) is valid for any value of ρ and in particular forthe values of ρ reached within 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 interface is λ and it appears as the mostobvious parameter that can be modified to increase the interface thickness. Equation (73) showsthat λ should be increased to increase h.Now, we have also shown that the expression for the surface tension is the following:σ = (ρ l − ρ v ) 36√2 A λ (76)29
Page 8 and 9: is no longer the case and some of t
Page 10 and 11: where δ represents the variation,
Page 12 and 13: Thus equation (27) readsP 0 (ρ v )
Page 14: WW2σ/Rρρ2σ/RbubbledropFigure 4:
Page 17: q = −φ dρ − k ∇T (49)dtT =
Page 20 and 21: we show that particular boundary co
Page 22 and 23: 3.1.2 Equilibrium conditionsSince w
Page 24 and 25: φ ˆ= ∂F∂∇cThe following exp
Page 26 and 27: alance equation and a convection-di
Page 30 and 31: P 0modifiedP eqρ vρ lρFigure 10:
Page 32 and 33: Figure 11: Numerical simulation of
Page 34 and 35: in the Cahn-Hilliard model than in

Diffuse interface models in fluid mechanics

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