On the generation of discontinuous shearing motions of a non-Newtonian fluid
type: Item
instance of: scholarly article
title: On the generation of discontinuous shearing motions of a non-Newtonian fluid
zbMATH Open document ID: 0906.76006
author: John A. Nohel
author: Robert L. Pego
published in: Archive for Rational Mechanics and Analysis
publication date: 1997-12-10T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 76A10
Mathematics Subject Classification ID: 76E30
Mathematics Subject Classification ID: 76A05
Mathematics Subject Classification ID: 35Q35
zbMATH DE Number: 1096004
zbMATH Keywords: fading memory
zbMATH Keywords: isothermal conditions
zbMATH Keywords: channel flow
zbMATH Keywords: constant pressure gradient
zbMATH Keywords: degenerate singularly perturbed system of nonlinear reaction-diffusion equations
zbMATH Keywords: spurt phenomena
MaRDI profile type: Publication
full work available at URL: s002050050056
OpenAlex ID: W2036869439
DOI: 10.1007/S002050050056
description: scientific article; zbMATH DE number 1096004