Convective instabilities in concurrent two‐phase flow: Part II. Global stability

Abstract

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

Onset of Rayleigh−Bénard−Marangoni Convection in Gas−Liquid Mass Transfer with Two-Phase Flow: Theory

“…convection and Marangoni convection, the long interfacial waves and short Tollmien-Schliching waves discussed by Gumerman and Homsy 16,17 are not considered here. In addition, it is assumed that the onset of Rayleigh convection and Marangoni convection is controlled by the local velocity disturbances and concentration disturbances just upstream of the onset.…”

Crack Patterns in Thin Polymer Layers

“…A heat transfer from below through the fluid layer amplifies the effect of the diffusion-controlled instability. [22] To produce a strong temperature gradient perpendicular to the layer at the beginning of the layer drying, the polymer solutions were cast on hot glass plates. The temperatures of the glass plates were 60 8C or 70 8C, see Table 1.…”

“…Furthermore, a weak air stream was directed over the layer surface to enhance the interfacial instability, see also. [22] Figure 2 shows the surface of such a structured PMMA film measured with the interferometer microscope. In this example the substrate temperature was 60 8C and the coating thickness was 0.7 mm.…”

Linear stability of a draining film squeezed between two approaching droplets

“…In similar problems this term has been shown to be a "sink" with respect to the growth of h [20]. Thus, we anticipate that our solution will provide an upper bound for the stability constraint, i.e., including interface flow should further stabilize the system [20,31,32].…”