The onset of cellular convection in a shallow two-dimensional container of fluid heated non-uniformly from below

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Abstract
A theoretical study is made of the onset of cellular convection in a shallow two-dimensional container of fluid when the temperature difference between the horizontal boundaries is a monotonic function of horizontal distance. The typical lengthscale of the horizontal variation in the temperature difference is taken to be of the same order of magnitude as the length of the container, and both are much longer than the depth of the container. The resulting flow may be regarded as consisting of two parts, a steady base flow and a disturbed flow. It is found that a weak disturbance taking the form of transverse rolls is first set up near the endwalls, but as the temperature difference between the horizontal boundaries is uniformly increased, an instability in the form of longitudinal rolls takes place near the hotter end of the container. This description is in good qualitative agreement with experiment.

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