Pattern formations in heat convection problems

Abstract

After Bénard’s experiment in 1900, Rayleigh formulated heat convection problems by the Oberbeck-Boussinesq approximation in the horizontal strip domain in 1916. The pattern formations have been investigated by the bifurcation theory, weakly nonlinear theories and computational approaches. The boundary conditions for the velocity on the upper and lower boundaries are usually assumed as stress-free or no-slip. In the first part of this paper, some bifurcation pictures for the case of the stress-free on the upper boundary and the no-slip on the lower boundary are obtained. In the second part of this paper, the bifurcation pictures for the case of the stress-free on both boundaries by a computer assisted proof are verified. At last, Bénard-Marangoni heat convections for the case of the free surface of the upper boundary are considered.

Keywords

Oberbeck-Boussinesq equation / Heat convection / Pattern formation / Computer assisted proof

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