The stability of plane Couette flow with viscosity stratification

Abstract

The stability of plane Couette flow is examined for liquids in which the viscosity varies with depth. Under suitable conditions, the flow may be stable or unstable at small Reynolds numbers for disturbances with wavelengths long compared with the liquid depth.A mechanism which may be stabilizing or destabilizing is found to derive from the role of diffusion in the neighbourhood of a ‘critical layer’, where the liquid velocity equals the phase velocity of a wave-like disturbance. This mechanism, which requires a viscosity gradient, is quite distinct from that found by Yih (1967) for the case of a viscosity discontinuity. An example is considered, for which these two mechanisms may be of comparable importance.