On the generation of discontinuous shearing motions of a non-Newtonian fluid
The authors study on unsteady one-dimensional shear flow of a highly elastic and viscous incompressible non-Newtonian fluid with fading memory under isothermal conditions. The channel flow is symmetric about the centre line and satisfies no slip condition at the wall. It is driven by a constant pressure gradient.The non-Newtonian contribution to the stress is assumed to follow a differential constitutive law (due to Oldroyd, Johnson and Segalman), its key feature being a non-monotone relation between the total steady shear stress and the rate of strain. The analysis leads to a degenerate, singularly perturbed system of nonlinear reaction-diffusion equations in a regime where the Reynolds number is much smaller than the Deborah (or Weissenberg) number. It has been shown that if the driving pressure gradient exceeds a critical value (the local shear stress maximum of the steady stress vs. strain rate relation) the solution starting initially from rest ultimately tends to a particular discontinuous - top-jumping - steady state solution, except in a small neighbourhood of the discontinuity. This discontinuous steady state is found to be nonlinearly stable in a precise sense with respect to perturbations yielding smooth initial data. Such steady states have been proposed to explain spurting flows, which exhibit a large increase in mean flow rate when the driving pressure exceeds a critical value. This paper is the continuation of the study initiated by the authors [SIAM J. Math. Anal. 24, 911-942 (1993; Zbl 0783.76011)] about the nonlinear stability and generation of certain discontinuous states which model the experimentally observed spurt phenomena.
degenerate singularly perturbed system of nonlinear reaction-diffusion equations
scientific article; zbMATH DE number 1096004