The effect of shear and stratification on the stability of a rotating fluid layer

Abstract

The stability of a layer of Newtonian fluid confined between two horizontal disks which rotate with different angular velocities is studied. Both isothermal and adversely stratified fluids are considered for small shear rates at low to moderate Taylor numbers. The linearized formulation of the stability problem is given a finite-difference representation, and the resulting algebraic eigenvalue problem is solved using efficient numerical techniques. The critical parameters and disturbance orientations are determined as a function of the Taylor number for the isothermal flow, and for the stratified flow for Prandtl numbers of 0·025, 1·0 and 6·0.At high Taylor numbers, the unstratified fluid flows in Ekman-like layers near the disks, and two modes of instability are noted: the viscous-type ‘class A’ travelling wave, whose existence depends on Coriolis forces, and the inflexional ‘class B’ mode, which is nearly stationary with respect to the nearer bounding disk. As the Taylor number is decreased, the Ekman layers coalesce to form a fully developed flow. In this regime there is a Taylor number below which the class A waves are always damped. The critical Reynolds number for the class B waves increases rapidly as the Taylor number approaches zero.For Prandtl numbers of 1·0 and 6·0, the adversely stratified flow exhibits two distinct types of instability: convective and dynamical. At low Reynolds numbers, a stationary mode associated with Bénard convection in a rotating fluid is critical. It is stabilized and given orientation by the shear. At higher Reynolds numbers, the critical mode is a travelling wave of the nature of either the class A or class B waves, depending upon the Taylor number. For a Prandtl number of 0·025, the critical mode resembles oscillatory convection at small Reynolds numbers and a class A wave at larger shear rates.