Diffuse interface models in fluid mechanics

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

is no longer the case and some of them are now discussed.At a critical point, the properties of both phases become equal. Above the critical point, atwo-phase system cannot exist and the fluid exists only as a one-phase fluid. On the contrary, belowthe critical point, under certain conditions, the fluid can co-exist under two different phasesseparated by an interface. As the two-phase system system approaches the critical point frombelow, the thickness of the interface actually increases and becomes infinite at the critical point.Thus, just below the critical point, the typical size of the transition layer that separates the bulkphases becomes of the same order of magnitude as the typical size of the bulk phases. Therefore,the interface cannot be modeled as a surface of discontinuity and its internal structure has to bedescribed .Let us consider two air bubbles in liquid water. It is commonly observed that two bubbles cancoalesce, i.e. they merge to give rise to a single bubble. If the interfaces are modeled as a surfaceof discontinuity, just at the moment where they merge, the model is singular. In a sense, one ofthe interfaces desappears and this is not possible if the interface is discontinuous. To overcomethis singularity, one has to study the detailed interaction of the interfaces during their merging.This can be done only by accounting for the internal structure of the interfaces [Lee et al., 2002a,b].Another situation, actually similar to the previous one, is the description of the creation of asecond phase in an initially single-phase system; this is called nucleation. In this case, one hasto describe how, from a single phase, an interface is created. This continuous process can bemodeled only if the interface under construction is modeled as a continuous medium [Dell’Isolaet al., 1996].1.3.2 Numerical limitationsIn the previous section, we have shown that, in some cases, the idealization of an interface asa surface of discontinuity is physically irrelevant and the detailed structure has to be described.However, in many applications, these phenomena can be neglected or do not occur. Nevertheless,the coupled partial differential equations that describe the two-phase flow are highlynon-linear and numerical simulation is often necessary to solve them. The numerical simulationof two-phase flows is very challenging because it is a moving boundary problem. Several numericaltechniques exist to solve this kind of problems and their description is beyond the scopeof this presentation. These techniques are often difficult to implement numerically, especially inthree space dimensions, and sometimes depend on the know-how of the code developper. Thisis partially due to the lack of a clear mathematical background for some of the methods. Nevertheless,the boundary conditions that must be applied at the moving interfaces need a particulartreatment in the numerical algorithm, which is difficult and often tedious. This is why diffuseinterface methods can be numerically attractive. If one can come up with a system of partial differentialequations that is valid in the entire two-phase system, including within the continuoustransition interfacial zones, the motion of the entire two-phase system would be describe by thissingle system of equations, which thus eliminates the difficult problem of the particular treatmentof the boundary conditions at the interfaces. The programming effort would therefore be highlydecreased. Moreover, if these equations are obtained from first principles, the development ofaccurate numerical schemes can be based on a better mathematical ground [Jamet et al., 2002].2 Liquid-vapor flows with phase-change: the van der Waals modelof capillarityIn this section, we present the van der Waals model of capillarity. This model is a diffuse interfacemodel dedicated to the description of an interface that separates a liquid and a vapor phase of8
a pure fluid. Extensions to binary mixtures is possible but will not be presented here [Fouilletet al., 2002]. It is interesting to note that this model is the first diffuse interface developed byvan der Waals [van der Waals, 1894].2.1 Thermodynamic modelAny diffuse interface model is actually a thermodynamic model. Indeed, the internal structureof an interface is mainly an equilibrium feature. Dynamic effects only perturb this equilibriumstructure, which is thus important to characterize.2.1.1 A mean-field approximationThe main issue is the following: is it possible to describe the internal structure of a liquid-vaporinterface at equilibrium by considering a “classical” thermodynamic description of the fluid? By“classical”, we mean that the energy of a fluid particle depends only on local variables such as thedensity ρ and the temperature T . Van der Waals showed that it is actually impossible [Rowlinsonand Widom, 1982]: the interface would be sharp and surface tension would be null. That is whynon-local terms have to be considered. In the case of a liquid-vapor interface, van der Waalspostulated the following thermodynamic description:F = F 0 (ρ, T ) + λ 2 (∇ρ)2 (14)where F is the volumetric free energy of the fluid, F 0 is its “classical” part and λ is the capillarycoefficient. For the sake of simplicity, we will always consider that λ is constant.F 0AρFigure 2: Illustration of the graph of the classical volumetric free energy F 0 (ρ).It can be shown that this particular form is justified from a molecular point of view. We willnot proove this and the interested reader can refer to [Rocard, 1967] for instance. In particular, itcan be shown that the value of λ depends only on the intermolecular potential.2.1.2 General equilibrium conditionsFor the sake of generality, we will consider that the volumetric free energy of the fluid is givenby the general expression F (ρ, T, ∇ρ). The differential of F thus readsdF = −S dT + g dρ + φ · d∇ρ (15)which defines the entropy S, the Gibbs free enthalpy g as well as φ.The second law of thermodynamics states that a closed and isolated system at equilibrium issuch that its entropy is maximum. Mathematically, this reads∫δ [S + L 1 U(S, ρ, ∇ρ) + L 2 ρ] dV = 0 (16)V9
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 28 and 29: This system is thermodynamically co
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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