Convection in a fluid layer with asymmetric boundary conditions

Abstract
Steady convection rolls in a horizontal fluid layer heated from below are described numerically with a Galerkin method. A rigid lower and stress-free upper boundary are assumed, while the temperature is fixed at both boundaries. The stability of the steady solutions with respect to arbitrary three-dimensional infinitesimal disturbances is analyzed and the stability boundaries in the Rayleigh number–wave-number plane are determined for selected Prandtl numbers. It is found that results of the analysis correspond more closely to the case of two rigid boundaries than to the case of two stress-free boundaries. The domains of stability in the case of asymmetric boundaries are larger at high Prandtl numbers than in the case of two rigid boundaries, but smaller for low Prandtl numbers. Some of the asymmetric properties of convection rolls are discussed.

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