Natural convection and thermal drift

Abstract

An analysis of natural convection in a horizontal, geometrically non-uniform slot exposed to spatially non-uniform heating has been carried out. The upper plate is smooth and isothermal, and the lower plate has sinusoidal corrugations with a sinusoidal temperature distribution. The distributions of the non-uniformities are characterized in terms of the wavenumber$\unicode[STIX]{x1D6FC}$and their relative position is expressed in terms of the phase difference$\unicode[STIX]{x1D6FA}_{TL}$. The analysis is limited to heating conditions which do not give rise to secondary motions in the absence of the non-uniformities. The heating creates horizontal temperature gradients which lead to the formation of vertical and horizontal pressure gradients which drive the motion regardless of the intensity of the heating. When the hot spots (points of maximum temperature) overlap either with the corrugation tips or with the corrugation bottoms, convection assumes the form of pairs of counter-rotating rolls whose size is dictated by the heating/corrugation wavelengths. The formation of a net horizontal flow, referred to as thermal drift, is observed for all other relative positions of the hot spots and corrugation tips. Both periodic heating as well as periodic corrugations are required for the formation of this drift, which can be directed in the positive as well as in the negative horizontal directions depending on the phase difference between the heating and corrugation patterns. The most intense convection and the largest drift occur for wavelengths comparable to the slot height, and their intensities increase proportionally to the heating intensity as well as proportionally to the corrugation amplitude, with the drift being a very strong function of the phase difference. Convection creates forces at the plates which would cause horizontal displacement of the corrugated plate and deform the corrugations if such effects were allowed. Tangential forces generated by the uniform heating always contribute to the corrugation buildup while similar forces generated by the periodic heating contribute to the buildup only when the hot spots overlap with the upper part of the corrugation. The processes described above are qualitatively similar for all Prandtl numbers$Pr$, with the intensity of convection and the magnitude of the drift increasing with a reduction in$Pr$.