ON THE GENERATION OF DISCONTINUOUS SHEARING MOTIONS OF A NON-NEWTONIAN FLUID

The system under study models unsteady, one-dimensional shear flow of a highly elastic and viscous incompressible non-Newtonian fluid with f ading memory under isothermal conditions. The flow, in a channel, is d riven by a constant pressure gradient, is symmetric about the center l ine, and satisfies a no-slip boundary condition at the wall. The non-N ewtonian contribution to the stress is assumed to obey a differential constitutive law (due to OLDROYD, JOHNSON & SEGALMAN), the key feature of which is a non-monotone relation between the total steady shear st ress and strain rate. In a regime in which the Reynolds number is much smaller than the Deborah (or Weissenberg) number, one obtains a degen erate, singularly perturbed system of nonlinear reaction-diffusion equ ations. It is shown that if the driving pressure gradient exceeds a cr itical value (the local shear stress maximum of the steady stress vs. strain rate relation), then the solution to the governing system, star ting from rest at t = 0, tends as t --> infinity to a particular disco ntinuous steady state solution (the ''top-jumping'' steady state), exc ept in a small neighborhood of the discontinuity. This discontinuous s teady state is shown to be nonlinearly stable in a precise sense with respect to perturbations yielding smooth initial data. Such discontinu ous steady states have been proposed to explain ''spurting'' flows, wh ich exhibit a large increase in mean flow rate when the driving pressu re is raised above a critical value.