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Stability of steady shearing flows subject to large shearing perturbations in a non-linear viscoelastic fluid*†

Published online by Cambridge University Press:  14 November 2011

M. Slemrod
M. Slemrod
Affiliation:
Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, N.Y. 12181, U.S.A.
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Synopsis

This paper studies stability of steady state solutions in a non-linear viscoelastic fluid. The main technique is to imbed the equation of motion in singularly perturbed equations and apply an energy method and the parabolic maximum principle.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 85 , Issue 3-4 , 1980 , pp. 251 - 262
Copyright
Copyright © Royal Society of Edinburgh 1980

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References

1Slemrod, M.. Instability of steady shearing flows in a non-linear viscoelastic fluid. Arch. Rational Mech. Anal. 68 (1978), 211–225.CrossRefGoogle Scholar
2Coleman, B. D. and Noll, W.. Foundations of linear viscoelasticity. Reu. Modem Phys. 33 (1961), 239–249.CrossRefGoogle Scholar
3Green, A. E. and Rivlin, R. S.. Mechanics of nonlinear materials with memory. Arch. Rational Mech. Anal. 1 (1957), 1–21.CrossRefGoogle Scholar
4Chacon, R. V. S. and Rivlin, R. S.. Representation theorems in the mechanics of materials with memory. Z. Angew. Math. Phys. 15 (1964), 444–447.CrossRefGoogle Scholar
5Coleman, B. D. and Noll, W.. Recent results in the continuum theory of viscoelastic fluids. Ann. N.Y. Acad. Sci. 89 (1961), 672–714.CrossRefGoogle Scholar
6Coleman, B. D. and Noll, W.. Simple fluids with fading memory. Second-order Effects in Elasticity, Plasticity, and Fluid Dynamics (Internat. Sympos., Haifa, 1962), pp. 530–552 (Israel: Jerusalem Academic Press; Oxford: Pergamon, 1964).Google Scholar
7Coleman, B. D. and Gurtin, M. E.. On the stability against shear waves of steady flows of non-linear viscoelastic fluids. J. Fluid Mech. 33 (1968), 165–181.CrossRefGoogle Scholar
8Hopf, E.. The partial differential equation u t, + uu x = μu xx. Comm. Pure Appl. Math. 3 (1950), 201–230.CrossRefGoogle Scholar
9Greenberg, J. M., MacCamy, R. C. and Mizel, V. J.. On the existence, uniqueness and stability of solutions of the equation σ' (u x)u xx + λu xtx = p ou u. J. Math. Mech. 17 (1968), 707–728.Google Scholar
10MacCamy, R. C.. Existence, uniqueness, and stability of u n = ə/əx(σ(u x) + λ(u x)u xt). Indiana Univ. Math. J. 20 (1970), 231–238.CrossRefGoogle Scholar
11Pritchard, W. G.. Measurement of the viscoemetric functions for a fluid in steady shear flows. Philos. Trans. Roy. Soc. London Ser. A 270 (1971), 507–556.Google Scholar
12Henry, D.. Geometric theory of semilinear parabolic equations. Lecture Notes, Univ. of Kentucky (Lexington, KY, 1972).Google Scholar
13Chueh, K. N., Conley, C. C. and Smoller, J. A.. Positively invariant regions for systems of nonlinear diffusion equations. Indiana Univ. Math. J. 26 (1977), 373–392.CrossRefGoogle Scholar