a1 Center for the Mathematical Sciences, University of Wisconsin-Madison, Madison, WI 53705, U.S.A.
We study the nonlinear stability of discontinuous steady states of a model initial-boundary value problem in one space dimension for incompressible, isothermal shear flow of a non-Newtonian fluid driven by a constant pressure gradient. The non-Newtonian contribution to the shear stress is assumed to satisfy a simple differential constitutive law. The key feature is a non-monotone relation between the total steady shear stress and shear strain-rate that results in steady states having, in general, discontinuities in the strain rate. We show that every solution tends to a steady state as t → ∞, and we identify steady states that are stable.
(Received July 28 1989)
1 Also Department of Mathematics
2 Department of Mathematics, University of Michigan
3 Also Department of Mathematics