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Multiple buoyancy-driven flows in a vertical cylinder heated from below. (English) Zbl 0573.76079

The authors have calculated the form and stability of the steady two- dimensional flows in a cylinder with rigid boundaries and sidewalls insulated. By combining finite-element methods for solving the Boussinesq equations with efficient computer- aided schemes for tracking evolution and multiplicity of the flow field with changes in parameters like Rayleigh number, these flows are computed. The numerical techniques for calculating solution multiplicity and stability developed by H. B. Keller [in: P. H. Rabinowitz (ed.)., Application of bifurcation theory (1977; Zbl 0456.00014) on pp. 339-384] and by R. A. Brown and L. E. Scriven [Phil. Trans. R. Soc. Lond., A 297, 51-79 (1980; Zbl 0438.76040)] have been applied to the finite-dimensional equation set that results from the finite element approximation to the Boussinesq equations. The authors have been able to extend back calculations for the flows evolving from the rest. Based on a linear analysis of the stability of the finite-element solutions to small perturbations in the field variables, the stability results are presented well. The general techniques are applied here for all the disturbances except that of time- periodic bifurcation. They are much more efficient than the eigenvalues calculations used previously by Brown and Scriven.
Reviewer: K.K.Srivastava

MSC:

76R10 Free convection
76E15 Absolute and convective instability and stability in hydrodynamic stability
76M99 Basic methods in fluid mechanics
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