Stability of narrow-gap Taylor–Dean flow with radial heating: Stationary critical modes

Abstract

A linear stability analysis for the Taylor–Dean flow, a viscous flow between concentric horizontal cylinders with a constant azimuthal pressure gradient, keeping the cylinders at different temperatures, when the inner cylinder is rotating and outer one is stationary has been implemented. The analysis is made under the assumption that the gap spacing between the cylinders is small compared to the mean radius. A parametric study covering wide ranges of β, a parameter characterizing the ratio of representative pumping and rotation velocities and N, the parameter characterizing the direction of temperature gradient is conducted. The eigenvalue problem is solved by differential transform method using unit disturbance scheme along with shooting technique. Emphasis is given to the occurrence of critical stability for the onset of instability by finding the intersection of the two neutral curves for the inner and outer part in a range of values of the radial temperature gradient parameter N,−1.25<N<0.75. We find that the existence of such critical point of stability is realizable near N=−0.995. It is found that with the introduction of radial heating, the discontinuity in critical Taylor number is not seen for N (=−0.75,−1.25). The most stable state occurs for β=βmax (=−3.62,−3.66,−3.95,−4.2) at which maximum of critical Taylor number occurs corresponding to N (=0.25,0,−0.75,−1.25) and can be concluded that for N<0, the most stable state occurs, when βmax<−3.666 (the value of βmax near which the maximum Taylor number occurs, when N=0).