Strong resonance in two-dimensional non-Boussinesq convection. (English) Zbl 0836.76088
The paper treats two-dimensional convection of a viscous and heat- conducting fluid in a two-dimensional rectangle heated from below. Using usual Oberbeck-Boussinesq approximation, the internal viscous heating and the temperature dependence of the viscosity (Arrhenius-type law) are taken into account. Under these assumptions, the authors investigate the solutions of the resulting \(O(2)\) symmetric bifurcation problem in dependence on the Prandtl number Pr. They found that there exist critical values of the Prandtl number \(\text{Pr}_s\) and \(\text{Pr}_c\); for \(\text{Pr}_s < \text{Pr} < \text{Pr}_c\) time-dependent solutions persist, and for values greater than \(\text{Pr}_c\) (or smaller than \(\text{Pr}_s)\) only stationary patterns are observed. If the viscosity ratio is relatively large compared to other coefficients (as in liquid metals), then there exists a stable heterocline orbit close to the onset of convection for small Prandtl numbers. A similar result with small amplitudes is shown if \(\text{Pr}_s < \text{Pr} < \text{Pr}_c\) and some non-Oberbeck-Boussinesq effects are taken into account. The authors show that for substances with small Prandtl numbers the viscous heating coefficient that appears in the equations of bifurcation is too small to affect the dynamics.
The paper is carefully prepared and presented. It is aimed at researchers having good knowledges of fluid mechanics and heat transfer.
The paper is carefully prepared and presented. It is aimed at researchers having good knowledges of fluid mechanics and heat transfer.
Reviewer: I.Pop (Cluj-Napoca)
MSC:
| 76R10 | Free convection |
| 76E15 | Absolute and convective instability and stability in hydrodynamic stability |
| 80A20 | Heat and mass transfer, heat flow (MSC2010) |
Keywords:
Arrhenius-type law; critical values of Prandtl number; \(O(2)\) symmetric bifurcation problem; two-dimensional rectangle; internal viscous heating; stable heterocline orbitReferences:
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