Problems Involving a Fluid Interface

The value of the boundary-integral method is particularly evident if we consider problems in which one or more of the boundaries is a fluid interface. Here, for simplicity, we consider the generic problem of a drop in an unbounded fluid that is undergoing some mean motion that causes the drop to deform in shape. This type of problem is particularly difficult because the shape of the interface is unknown and is often changing with time. We shall see that the boundary-integral formulation provides a powerful basis to attack this class of problems, and in fact, is largely responsible for much of the considerable theoretical progress that has 

For convenience, we begin by restating the governing equations and boundary conditions in nondimensional form  

We denote the characteristic length and velocity scales as tc and uc, the characteristic time scale as lc/uc and the characteristic stress and pressure scales as fj,uc/ic and X luc/Ic for the fluid outside and inside the drop, respectively. We express the interfacial tension in the form 

The function F is unknown and must be calculated as part of the solution of the problem. 

To approach this problem by means of the boundary-integral technique, we first note the general expression for the velocity in the exterior fluid  

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Introductory note Most transport and/or fluids problems are not amenable to analysis by classical methods for linear differential equations, either because the equations are nonlinear (or simply too comphcated in the case of the thermal energy equation, which is linear in temperature if natural convection effects can be neglected), or because the solution domain is complicated in shape (or in the case of problems involving a fluid interface having a shape that is a priori unknown). Analytic results can then be achieved only by means of approximations. One approach is to simply discretize the equations in some way and turn on the computer. Another is to use the family of approximations methods known as asymptotic approximations that lead to useful concepts such as boundary layers, etc. This course is about the latter approach. However, it is not just a... 

For motions of a single fluid involving sohd boundaries, we have already noted that the no-slip and kinematic boundary conditions are sufficient to determine completely a solution of the equations of motion, provided the motion of the boundaries is specified. In problems involving two fluids separated by an interface, however, these conditions are not sufficient because they provide relationships only between the velocity components in the fluids and the interface shape, all of which are unknowns. The additional conditions necessary to completely determine the velocity fields and the interface shape come from a force equilibrium condition on the interface. In particular, because the interface is viewed as a surface of zero thickness, the volume associated with any arbitrary segment of the interface is zero, and the sum of all forces acting on this interface segment must be identically zero (to avoid infinite acceleration). 

The object of the present review, a case study of evaporative convection, has the purpose of bringing together those aspects of the broader field of free convection which are of primary interest to chemical engineers. The specific problem of evaporative convection provides a particularly appropriate vehicle for such a study first, because it typifies free convection occurring during and as a result of heat or mass transfer in a fluid layer second, because it involves a fluid-fluid interface which gives rise to important surface phenomena in fluid mechanics and third, because it provides a needed backdrop for the many experimental studies where spontaneous convection complicates any attempt to elucidate the molecular mechanism of the evaporation process per se (Kl, P2). 

An important class of fluid flow problems involves free surface flows where only the flow phenomena in a single phase are of importance despite the fact that two or more phases are present in the system. Such situations are encountered, for example, in gas-liquid two-phase flows where, due to the large density difference, from a hydrodynamic point of view the flow can be treated as a liquid flow in a vacuum. The analysis of such free surface flows is rather complex due to the fact that in addition to the fluid flow problem the position of the interface and enforcement of appropriate boundary conditions have to be dealt with. 

Various flow problems involving evaporation and condensation phenomena are quite common in ordinary circumstances and have aroused an interest of scientists not only in the field of fluid dynamics but also of kinetic theory. The reason for this is that the ordinary continuum-based fluid dynamics cannot describe qualitatively correctly the process of evaporation and condensation occurring at the interface even in ihe continuum limit because of the existence of a nonequilibrium region, the thickness of which is of the order of the molecular mean free path, in the close vicinity of the interface between the condensed phase and the gas phase. Such a nonequilibrium region is called the Knudsen layer, in which collisions between molecules are not so frequent that the momentum and energy exchanges between the molecules leaving the interface... 

Membrane contactors provide a novel approach to the solution of many such problems (especially of the second and third kind) of contacting two different phases, one of which must be a fluid. Essentially, a porous membrane, most often in hollow-fiber form, is the basic element in such a device. Any membrane in flat or spiral-wound or hollow-fiber or any other form has two interfaces since it has two sides. However, conventional separation processes involve usually one interface in a two-phase system, for example, gas-liquid, vapor-liquid, liquid-liquid, hquid-supercritical fluid, gas-solid, liquid-solid, and the like. Membrane contactors allow the creation of one immobilized phase interface between two phases participating in separation via the porous membrane. Three types of immobilized phase interfaces in two-phase configurations are relevant ... 

Nested wells can also be used to analyze multilayer aquifer flow. There are many situations involving interaquifer transport owing to leaky boundaries between the aquifers. The primary case of interest involves the vertical transport of fluid across a horizontal semipermeable boundary between two or more aquifers. Figure 4 sets out the details of this type of problem. Unit 1 is a phraetic aquifer, bound from below by two confined aquifers, having semipermeable formations at each interface. 

The evaluation of catalyst effectiveness requires a knowledge of the intrinsic chemical reaction rates at various reaction conditions and compositions. These data have to be used for catalyst improvement and for the design and operation of many reactors. The determination of the real reaction rates presents many problems because of the speed, complexity and high exo- or endothermicity of the reactions involved. The measured conversion rate may not represent the true reaction kinetics due to interface and intraparticle heat and mass transfer resistances and nonuniformities in the temperature and concentration profiles in the fluid and catalyst phases in the experimental reactor. Therefore, for the interpretation of experimental data the experiments should preferably be done under reaction conditions, where transport effects can be either eliminated or easily taken into account. In particular, the concentration and temperature distributions in the experimental reactor should preferably be described by plug flow or ideal mixing models. 

Polymer melts are complex fluids. Their viscoelastic properties during flow depend not only on their molecular structure but also on the interactions they are likely to develop at the walls, depending on the physical and chemical features of the interface and the flow conditions. In addition, not all their properties can be determined and the constitutive equations used are in practice often limited to considerations on the shear viscosity. From a theoretical point of view, considerable difficulties are involved and the problem to be studied here has not been solved. In particular, even though the boundary conditions considered in... 

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