Conditions of convective instability in a vertical circular microchannel with slippage effects

Abstract

The paper outlines a novel mathematical model of convective instability in a vertical cylindrical microchannel. The model yields a transcendental equation, whose solution represents a dependence of the critical Rayleigh number on the harmonic number of the perturbation function, the ratio of thermal conductivities of the fluid and the solid wall, as well as the Prandtl and Knudsen numbers. The most dangerous are harmonics with n = 1, which corresponds to diametrically antisymmetric motions. An increase in the Knudsen number causes weakening of the system stability and a decrease in the critical Rayleigh number. This tendency is observed in the entire range of the ratio of thermal conductivities (0 … ∞) and Prandtl numbers (0.01 … 100) studied in the paper. A decrease in the thermal conductivity of the solid wall and the Prandtl numbers causes loss of the system stability, which can be explained by an increase in the amount of heat spent to arouse instability.