DOI: 10.1051/jp2:1996197
J. Phys. II France 6 (1996) 551-571

Nonmonotonic Constitutive Laws and the Formation of Shear-Banded Flows

N.A. Spenley^{1, 2}, X.F. Yuan^{1, 3} and M.E. Cates<sup>1, 4

1</sup>  Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, U.K.
²  Max-Planck-institut für Kolloid- und Grenzflächenforschung, Kantstrasse 55, 14513 Teltow, Germany
³  H. H. Wills Physics Laboratory, University of Bristol, Royal Fort, Tyndall Avenue, Bristol BS8 1TL, U.K.
⁴  Department of Physics, University of Edinburgh, King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, U.K.

(Received 18 September 1995, received in final form 29 November 1995, accepted 3 January 1996)

Abstract
We consider constitutive models of viscoelastic behaviour which predict a shear stress which is a nonmonotonic function of the shear rate. It is known that a homogeneous shear flow is unstable when the shear stress decreases with shear rate. We use a novel simulation technique (the Lagrangian-Eulerian method for the fluid dynamics combined with Öttinger's stochastic method for the constitutive equation) to solve one- and two-dimensional models of plane Couette flow for an integral constitutive equation describing entangled wormlike micelles. The results are compared with those of a `toy' model (with a differential constitutive equation). We show that the steady state actually consists of bands of different shear rate. Such a flow is strongly inhomogeneous, and our preliminary results indicate that the constitutive equation must be modified to allow for spatial variations in the viscoelastic stress.

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