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

John A. Nohel; Robert L. Pego; Athanasios E. Tzavaras

doi:10.1017/S0308210500024550

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

John A. Nohel, Robert L. Pego, Athanasios E. Tzavaras

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Abstract

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.

Original language	English (US)
Pages (from-to)	39-59
Number of pages	21
Journal	Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Volume	115
Issue number	1-2
DOIs	https://doi.org/10.1017/S0308210500024550
State	Published - 1990
Externally published	Yes

Bibliographical note

Funding Information:
Supported by the U.S. Army Research Office under Grant DAAL03-87-K-0036 and DAAL03-88-K-0185, the Air Force Office of Scientific Research under Grant AFOSR-87-0191; the National Science Foundation under Grants DMS-8712058, DMS-8620303, DMS-8716132, and a NSF Post Doctoral Fellowship (Pego).

ASJC Scopus subject areas

General Mathematics

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title = "Stability of discontinuous steady states in shearing motions of a non-Newtonian fluid",

abstract = "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.",

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doi = "10.1017/S0308210500024550",

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T1 - Stability of discontinuous steady states in shearing motions of a non-Newtonian fluid

AU - Nohel, John A.

AU - Pego, Robert L.

AU - Tzavaras, Athanasios E.

N1 - Funding Information: Supported by the U.S. Army Research Office under Grant DAAL03-87-K-0036 and DAAL03-88-K-0185, the Air Force Office of Scientific Research under Grant AFOSR-87-0191; the National Science Foundation under Grants DMS-8712058, DMS-8620303, DMS-8716132, and a NSF Post Doctoral Fellowship (Pego).

PY - 1990

Y1 - 1990

N2 - 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.

AB - 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.

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

Abstract

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