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
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.