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Stability of discontinuous steady states in shearing motions of a non-Newtonian fluid

Published online by Cambridge University Press:  14 November 2011

John A. Nohel ,
Robert L. Pego  and
Athanasios E. Tzavaras
John A. Nohel
Affiliation:
Center for the Mathematical Sciences, University of Wisconsin-Madison, Madison, WI 53705, U.S.A.
Robert L. Pego
Affiliation:
Center for the Mathematical Sciences, University of Wisconsin-Madison, Madison, WI 53705, U.S.A.
Athanasios E. Tzavaras
Affiliation:
Center for the Mathematical Sciences, University of Wisconsin-Madison, Madison, WI 53705, U.S.A.
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Synopsis

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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 115 , Issue 1-2 , 1990 , pp. 39 - 59
Copyright
Copyright © Royal Society of Edinburgh 1990

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