Stability of vorticity defects in viscoelastic shear flow

We analyze the stability of certain kinds of two-dimensional, incompressible, viscoelastic shear flows. These particular flows consist of a strong background linear shear, with a superposed ``defect;'' i.e., a localized region over which the vorticity varies rapidly. A matched asymptotic expansion recasts the problem in a simpler form for which one can derive explicit dispersion relations for the normal modes. This technique is used to explore the stability properties of some constitutive models including the Oldroyd-B, Johnson-Segalman and Phan-Thien-Tanner models. We derive sufficient stability conditions based on Nyquist methods, and consider the inviscid limits of the visco-elastic system. Thus this provides an analytic method for investigating stability in regimes when numerical approaches are cumbersome. Finally, for large Weissenberg number, the defect approximation allows us to study the continuous spectrum associated with the elasticity of the constitutive models; this is a considerable advance over previous analyses and verifies analytically the presence of such a continuous spectrum. The dynamics associated with this continuous spectrum contains features such as transient amplification and ultimate decay, phenomena familiar for Newtonian fluids in the inviscid limit.

J Non-Newt. Fluid Mech. 72, 281-304, 1997 Co-authored with Dr N.J. Balmforth.