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The method of energy is used to study the stability of uniformly accelerated flows, i.e. those flows characterized by an impulsive change in boundary temperature or velocity. Two stability criteria are considered: strong stability, in which there is exponential decay of the disturbance energy, and marginal stability, in which the disturbance energy is less than or equal to its initial value. For the important case in which the critical stability parameter (measured by the Marangoni, Rayleigh or Reynolds number) decreases with time, it is proved that an onset time exists. Furthermore, it is shown that the experimental onset time is bounded below by the marginal stability limit, which in turn is bounded below by the strong stability limit.The method is then applied to the problem of an impulsively cooled liquid layer susceptible to instabilities driven by interfacial-tension gradients. The strong stability and marginal stability boundaries are calculated and bounds on the onset time are given. These results represent the first rigorous bounds for convective instability problems of this class. Comparison with the limited available experimental data shows the calculated results to be lower bounds on the experimental onset times, and hence the theory is in agreement with available experimental results.

The linear stability of thermally stratified horizontal two-phase Couette flow is analyzed for the case of a constant vertical temperature gradient. Instabilities driven by buoyancy, surface tension gradients, or shear are allowed for. It is shown that the instability can take three possible forms: streamwise oriented roll vortices, long interfacial waves, and short TollmienSchlichting waves. It is shown that the stability limits for rolls are identical to those for plane, stagnant layers. A long wave expansion is presented and the stability limits for this mode are given algebraically. The nonexistence of a Squire's Theorem is demonstrated and some numerical experiments at moderate Reynolds numbers are described. Detailed comparisons with previous work are possible for only one fluid pair, but it is shown that reasonably accurate statements may be made to determine which mode may manifest itself in any given experimental situation. SCOPEUnder certain conditions, diffusive heat or mass transfer across a fluid-fluid interface can result in a convective instability. The secondary convection is driven by the temperature or concentration gradients. Because rates of transport are enhanced severalfold over that occurring by the diffusion mechanism alone, an understanding of thz phenomena is of great practical interest. Study of this instability of diffusive transport has attracted considerable attention since it was first observed by B6nard (1900). It is now well known to be due to either surface tension variations or buoyantly unstable density variations.Many theoretical analyses of instabilities driven by buoyancy and surface tension gradients exist for the case of stagnant plane fluid layers, (Berg et al., 1966;Sawistowski, 1971). The results of these studies have been substantiated by experiments, but it has been difficult to apply these results to cases of practical importance because of several restrictive assumptions. One of these is the assumption of initially stagnant phases. The objective in this work is to examine the effect of a rectilinear flow on some of the predictions made in stagnant systems. The work is motivated by the pervasiveness of shear in situations of practical interest such as liquid-liquid extraction, film evaporators, and wetted wall columns. All of these are situations where convective instabilities would be expected. In this paper, the first of a three part series, we employ the techniques of linear stability theory to (1) categorize the possible modes of instability in two-phase concurrent flow and (2) determine quantitative instability criteria which would allow one to predict which of these modes would occur. CONCLUSIONS AND SIGNIFICANCEThe model system used here is the simplest incorporating the important physics of the problem. Thermally stratified two-phase Couette flow is considered. We take the gravity vector to be perpendicular to the boundar'es generating the shear field and the thermal stratification to be linear. The treatment for mass transfer ij identical except for the sta...

The method of energy is used to determine global stability limits for thermally stratified two-phase plane Couette flow. Instabilities due to surface tension variations, buoyancy effects, and shear are allowed for, but surface waves are specifically excluded from consideration. Under these assumptions, the Marangoni, Rayleigh, and Reynolds numbers completely describe the system. Stability plots, valid for disturbances of any magnitude, are presented for both streamwise oriented roll vortices and two-dimensional transversely oriented disturbances. It is shown that rolls are the most dangerous disturbances in the sense that they cannot be shown to be stable relative to transverse disturbances at any nonzero Reynolds numbers. Comparisons are made with existing linear limits, and these are seen to be close only for moderate Reynolds numbers.This study is the second in a series which focuses on convective instabilities due to surface tension, density, or velocity gradients in two phase flow. The need for a unified stability theory for these phenomena is clear: in the cases in which they occur, they significantly affect the rates of heat, mass, or momentum transport between the phases. These increases have yet to be exploited in any systematic manner in applications for a variety of reasons. The foremost of these pertains to the relative uncertainty in being able to predict when instabilities will and will not occur. As we have noted (Gumerman and Homsy, 1974), most theoretical analyses have been for stagnant plane layers, while most practical applications involve the presence of shear, for example, liquid-liquid extraction and film evaporation. In that work, we employed the techniques of linear theory to discuss the various modes of instability and gave quantitative stability SCOPE CONCLUSIONS AND SIGNIFICANCEThe model system employed here is that of thermally stratified plane Couette flow. The model is specifically restricted to fluid pairs with large density differences and large interfacial tension. These restrictions exclude the existence of interfacial waves discussed extensively in Part I (Gumerman and Homsy, 1974). Thus, instabilities due to surface tension, density, and velocity gradients are allowed for.It is shown that of the possible disturbance modes, streamwise oriented roll vortices are the most dangerous, that is, they yield the lowest global stability boundaries. These results are valid for disturbances of arbitrary magnitude. For the longitudinal roll mode, the linear limits are