Perturbation analysis of steady and oscillatory onset in a Bénard problem with two similar liquids

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Abstract
In a recent paper, Renardy and Joseph [Phys. Fluids 2 8, 788 (1985)] studied the Bénard problem for two layers of different fluids lying on top of each other and bounded by walls. Their study shows that, in contrast to the Bénard problem for one fluid, the onset of instability can be oscillatory. The number of parameters involved in the problem is large, and there is as yet no comprehensive picture of when the instability is oscillatory and when it is not. The study of limiting cases, accessible by perturbation methods, may be helpful in this respect. In this paper, an analysis is given for the case when the properties of the two fluids are nearly equal and the fluids are allowed to slip at the boundaries.