is no longer the case and some of them are now discussed.At a critical po<strong>in</strong>t, the properties of both phases become equal. Above the critical po<strong>in</strong>t, atwo-phase system cannot exist and the <strong>fluid</strong> exists only as a one-phase <strong>fluid</strong>. On the contrary, belowthe critical po<strong>in</strong>t, under certa<strong>in</strong> conditions, the <strong>fluid</strong> can co-exist under two different phasesseparated by an <strong><strong>in</strong>terface</strong>. As the two-phase system system approaches the critical po<strong>in</strong>t frombelow, the thickness of the <strong><strong>in</strong>terface</strong> actually <strong>in</strong>creases and becomes <strong>in</strong>f<strong>in</strong>ite at the critical po<strong>in</strong>t.Thus, just below the critical po<strong>in</strong>t, 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 <strong><strong>in</strong>terface</strong> cannot be modeled as a surface of discont<strong>in</strong>uity and its <strong>in</strong>ternal structure has to bedescribed .Let us consider two air bubbles <strong>in</strong> liquid water. It is commonly observed that two bubbles cancoalesce, i.e. they merge to give rise to a s<strong>in</strong>gle bubble. If the <strong><strong>in</strong>terface</strong>s are modeled as a surfaceof discont<strong>in</strong>uity, just at the moment where they merge, the model is s<strong>in</strong>gular. In a sense, one ofthe <strong><strong>in</strong>terface</strong>s desappears and this is not possible if the <strong><strong>in</strong>terface</strong> is discont<strong>in</strong>uous. To overcomethis s<strong>in</strong>gularity, one has to study the detailed <strong>in</strong>teraction of the <strong><strong>in</strong>terface</strong>s dur<strong>in</strong>g their merg<strong>in</strong>g.This can be done only by account<strong>in</strong>g for the <strong>in</strong>ternal structure of the <strong><strong>in</strong>terface</strong>s [Lee et al., 2002a,b].Another situation, actually similar to the previous one, is the description of the creation of asecond phase <strong>in</strong> an <strong>in</strong>itially s<strong>in</strong>gle-phase system; this is called nucleation. In this case, one hasto describe how, from a s<strong>in</strong>gle phase, an <strong><strong>in</strong>terface</strong> is created. This cont<strong>in</strong>uous process can bemodeled only if the <strong><strong>in</strong>terface</strong> under construction is modeled as a cont<strong>in</strong>uous medium [Dell’Isolaet al., 1996].1.3.2 Numerical limitationsIn the previous section, we have shown that, <strong>in</strong> some cases, the idealization of an <strong><strong>in</strong>terface</strong> asa surface of discont<strong>in</strong>uity is physically irrelevant and the detailed structure has to be described.However, <strong>in</strong> 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-l<strong>in</strong>ear and numerical simulation is often necessary to solve them. The numerical simulationof two-phase flows is very challeng<strong>in</strong>g because it is a mov<strong>in</strong>g boundary problem. Several numericaltechniques exist to solve this k<strong>in</strong>d of problems and their description is beyond the scopeof this presentation. These techniques are often difficult to implement numerically, especially <strong>in</strong>three 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 mov<strong>in</strong>g <strong><strong>in</strong>terface</strong>s need a particulartreatment <strong>in</strong> the numerical algorithm, which is difficult and often tedious. This is why diffuse<strong><strong>in</strong>terface</strong> methods can be numerically attractive. If one can come up with a system of partial differentialequations that is valid <strong>in</strong> the entire two-phase system, <strong>in</strong>clud<strong>in</strong>g with<strong>in</strong> the cont<strong>in</strong>uoustransition <strong>in</strong>terfacial zones, the motion of the entire two-phase system would be describe by thiss<strong>in</strong>gle system of equations, which thus elim<strong>in</strong>ates the difficult problem of the particular treatmentof the boundary conditions at the <strong><strong>in</strong>terface</strong>s. The programm<strong>in</strong>g effort would therefore be highlydecreased. Moreover, if these equations are obta<strong>in</strong>ed from first pr<strong>in</strong>ciples, 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 <strong><strong>in</strong>terface</strong>model dedicated to the description of an <strong><strong>in</strong>terface</strong> that separates a liquid and a vapor phase of8
a pure <strong>fluid</strong>. Extensions to b<strong>in</strong>ary mixtures is possible but will not be presented here [Fouilletet al., 2002]. It is <strong>in</strong>terest<strong>in</strong>g to note that this model is the first diffuse <strong><strong>in</strong>terface</strong> developed byvan der Waals [van der Waals, 1894].2.1 Thermodynamic modelAny diffuse <strong><strong>in</strong>terface</strong> model is actually a thermodynamic model. Indeed, the <strong>in</strong>ternal structureof an <strong><strong>in</strong>terface</strong> is ma<strong>in</strong>ly an equilibrium feature. Dynamic effects only perturb this equilibriumstructure, which is thus important to characterize.2.1.1 A mean-field approximationThe ma<strong>in</strong> issue is the follow<strong>in</strong>g: is it possible to describe the <strong>in</strong>ternal structure of a liquid-vapor<strong><strong>in</strong>terface</strong> at equilibrium by consider<strong>in</strong>g a “classical” thermodynamic description of the <strong>fluid</strong>? By“classical”, we mean that the energy of a <strong>fluid</strong> particle depends only on local variables such as thedensity ρ and the temperature T . Van der Waals showed that it is actually impossible [Rowl<strong>in</strong>sonand Widom, 1982]: the <strong><strong>in</strong>terface</strong> 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 <strong><strong>in</strong>terface</strong>, van der Waalspostulated the follow<strong>in</strong>g thermodynamic description:F = F 0 (ρ, T ) + λ 2 (∇ρ)2 (14)where F is the volumetric free energy of the <strong>fluid</strong>, 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 po<strong>in</strong>t of view. We willnot proove this and the <strong>in</strong>terested reader can refer to [Rocard, 1967] for <strong>in</strong>stance. In particular, itcan be shown that the value of λ depends only on the <strong>in</strong>termolecular potential.2.1.2 General equilibrium conditionsFor the sake of generality, we will consider that the volumetric free energy of the <strong>fluid</strong> is givenby the general expression F (ρ, T, ∇ρ). The differential of F thus readsdF = −S dT + g dρ + φ · d∇ρ (15)which def<strong>in</strong>es 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