Diffuse interface models in fluid mechanics

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