Instability of channel flow of a shear-thinning White–Metzner fluid

Abstract

We consider the inertialess planar channel flow of a White–Metzner (WM) fluid having a power-law viscosity with exponent n. The case n = 1 corresponds to an upper-convected Maxwell (UCM) fluid. We explore the linear stability of such a flow to perturbations of wavelength k −1. We find numerically that if n < n c ≈ 0.3 there is an instability to disturbances having wavelength comparable with the channel width. For n close to n c, this is the only unstable disturbance. For even smaller n, several unstable modes appear, and very short waves become unstable and have the largest growth rate. If n exceeds n c, all disturbances are linearly stable. We consider asymptotically both the long-wave limit which is stable for all n, and the short-wave limit for which waves grow or decay at a finite rate independent of k for each n. The mechanism of this elastic shear-thinning instability is discussed.