Resonance in Thermal Convection

Abstract

Resonance in thermal convection has many applications since interactions between the fluid layers may greatly increase the critical Rayleigh number threshold for the onset of convection, which in turn is of interest to the energy industry. In particular, with modern heat transfer devices being increasingly employed on a microscale there is much need to understand penetrative and resonant convection on a nanoscale. MEMS (micro-electro-mechanical-systems) will play an important part in future heat transfer technology, and we believe an analysis of thermal convection influenced by resonance is important. There have been some very interesting recent analyses of resonance in thermal convection. For clarification, by resonance we mean where instability in one part of a fluid layer may simultaneously occur with instability in another part of the layer. Such resonance can lead to unusually high Rayleigh numbers at the onset of thermal convection and so this is very much of interest to the heat transfer industry. Typically, a high Rayleigh number may be synonymous with delaying or prohibiting heat transfer and is thus important in insulation, while low Rayleigh numbers may be desirable when one requires rapid heat transfer such as in cooling pipes used in many modern devices such as computers. The implications of resonance in the energy sector may be important, especially with nano devices, or with application in building design and heat loss. This chapter studies a model for resonance in thermal convection in a fluid where the resonance effect is produced by having a density in the buoyancy force which is quadratic in the temperature, and additionally having a constant heat source. The analogous problem of resonant penetrative convection in a fluid saturated porous layer is also analysed. We also describe analysis of a model for resonance in thermal convection where the density in the buoyancy force is linear in temperature but the heat source is linear in the vertical coordinate. This allows for the possibility of strong resonance between fluid layers. A further model is investigated where the density in the buoyancy force is quadratic in temperature but the heat source is linear in the vertical coordinate. This allows for a situation of thermal convection in a fluid layer where resonance may possibly occur simultaneously in three distinct sublayers. This chapter is completed by considering another possibility which may possibily lead to convective fluid motion in sublayers and so possible resonance and we do this by taking a density quadratic in the temperature field, but we allow the gravity field to vary linearly in $z$. This situation may possibly arise in an experimental set-up, or even possibly in stellar convection.