Instability of a two-dimensional viscous flow in an annulus with permeable walls to two-dimensional perturbations

Abstract

The stability of a two-dimensional viscous flow in an annulus with permeable walls with respect to small two-dimensional perturbations is studied. The basic steady flow is the most general rotationally invariant solution of the Navier-Stokes equations in which the velocity has both radial and azimuthal components, and the azimuthal velocity profile depends on the radial Reynolds number. It is shown that for a wide range of parameters of the problem, the basic flow is unstable to small two-dimensional perturbations. Neutral curves in the space of parameters of the problem are computed. Calculations show that the stability properties of this flow are determined by the azimuthal velocity at the inner cylinder when the direction of the radial flow is from the inner cylinder to the outer one and by the azimuthal velocity at the outer cylinder when the direction of the radial flow is reversed. This work is a continuation of our previous study of an inviscid instability in flows between rotating porous cylinders [K. Ilin and A. Morgulis, “Instability of an inviscid flow between porous cylinders with radial flow,” J. Fluid Mech. 730, 364–378 (2013)].