Rayleigh–Bénard and interfacial instabilities in two immiscible liquid layers

Publisher Website
Google Scholar
Abstract
The linear stability of two immiscible liquid layers heated from below through a rigid perfectly conducting boundary with a free deformable upper surface and a deformable liquid–liquid interface is examined. Three modes of instability are allowed simultaneously in the analysis: surface tension driven at each of the two interfaces, buoyancy driven because of the presence of adverse density gradients in each liquid, and an interfacial mode (the Rayleigh–Taylor instability), related to the density difference and interfacial tension at the interfaces. For purely surface tension driven convection, the presence of a middle interface that can suppress normal deformations makes the two liquid layers more stable than a single layer of the same total depth. The interaction of the buoyancy and interfacial modes leads to overstability when the physical properties of the two liquids are only slightly different from each other. For certain Rayleigh numbers, both stationary and oscillatory modes display positive growth constants over a certain range of wavenumbers. As the Marangoni number is increased, the coupling between the surface tension and buoyancy mechanisms makes the system more unstable but removes the oscillatory eigenmodes. The addition of trace amounts of insoluble surface active agents at the two interfaces has a very strong stabilizing influence by introducing the expected hydrodynamic rigidity to the surfaces. However, their more interesting effect is their ability to change the nature of the most unstable eigenmode from stationary to oscillatory.