Resonant Penetrative Convection with an Internal Heat Source/Sink

Abstract

We develop a detailed linear instability and nonlinear stability analysis for the situation of convection in a horizontal plane layer of fluid when there is a heat sink / source which is linear in the vertical coordinate which is in the opposite direction to gravity. This can give rise to a scenario where the layer effectively splits into three sublayers. In the lowest one the fluid has a tendency to be convectively unstable while in the intermediate layer it will be gravitationally stable. In the top layer there is again the possibility for the layer to be unstable. This results in a problem where convection may initiate in either the lowest layer, the upmost layer, or perhaps in both sublayers simultaneously. In the last case there is the possibility of resonance between the upmost and lowest layers. In all cases penetrative convection may occur where convective movement in one layer induces motion in an adjacent sublayer. In certain cases the critical Rayleigh number for thermal convection may display a very rapid increase which is much greater than normal. Such behaviour may have application in energy research such as in thermal insulation. * Research supported by a "Tipping Points, Mathematics, Metaphors and Meanings" grant of the Leverhulme Trust. I should like to thank an anonymous referee for pointing out some errors and for several helpful remarks.

Convection with Local Thermal Non-Equilibrium and Microfluidic Effects

“…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, cf. [18,118,133,300,303,402,420,425,428,433]. MEMS (microelectro-mechanical-systems) will play an important part in future heat transfer technology, cf.…”

“…In this section we describe work of [433] who develops a model for penetrative convection in a fluid involving a heat source which varies linearly with vertical height across the layer. This allows one to obtain very strong resonance between sublayers.…”

“…This allows one to obtain very strong resonance between sublayers. The mathematical analysis in [433] involves a linear instability technique which yields a definite instability boundary coupled with a global nonlinear energy stability analysis which yields a definite stability threshold. [433] points out that such analyses for a constant heat source have been performed previously, cf.…”

“…The mathematical analysis in [433] involves a linear instability technique which yields a definite instability boundary coupled with a global nonlinear energy stability analysis which yields a definite stability threshold. [433] points out that such analyses for a constant heat source have been performed previously, cf. [198,377,380], although the richness of the results for resonance obtained in [433] are not found with a constant heat source or sink.…”

Magnetic field and throughflow effects on double-diffusive convection in internally heated anisotropic porous media

“…Hence, quantifying the discrepancy between these two thresholds makes it possible to provide an assessment of the suitability of linear theory to predict the de-stabilisation of the double diffusive convection in anisotropic porous media with throughflow, magnetic and variable gravity effects. Recent contributions on the study of convective instabilities in fluid and porous media include [19][20][21][22][23][24][25][26][27][28][29][30][31].…”