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  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 115, Issue 1-2
  • January 1990, pp. 39-59

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

  • John A. Nohel (a1), Robert L. Pego (a1) and Athanasios E. Tzavaras (a1)
  • DOI: http://dx.doi.org/10.1017/S0308210500024550
  • Published online: 14 November 2011
Abstract
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.

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1G. Andrews and J. Ball . Asymptotic behavior and changes of phase in one-dimensional nonlinear viscoelasticity. J. Differential Equations 44 (1982), 306341.

4D. Henry . Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics 840 (New York: Springer, 1981).

5J. Hunter and M. Slemrod . Viscoelastic fluid flow exhibiting hysteretic phase changes. Phys. Fluids 26 (1983), 23452351.

6M. Johnson and D. Segalman . A model for vrscoelastic fluid behavior which allows non-afflne deformation. J. Non-Newtonian Fluid Mech. 2 (1977), 255270.

7J. Lions and E. Magenes . Non-Homogeneous Boundary Value Problems and Applications (New York: Springer, 1972).

12J. Oldroyd . Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids. Proc. Roy. Soc. London Ser. A 245 (1958), 278297.

13A. Pazy . Semigroups of Linear Operators and Applications to Partial Differential Equations (New York: Springer, 1983).

14R. Pego . Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability. Arch. Rational Mech. Anal. 97 (1987), 353394.

15N. Phan-Thien and R. Tanner . A new constitutive equation derived from network theory. J. Non-Newtonian Fluid Mech. 2 (1977), 353365.

17A. Tzavaras . Effect of Thermal Softening in Shearing of Strain-Rate Dependent Materials. Arch. Rational Mech. Anal. 99 (1987), 349374.

18G. Vinogradov , A. Malkin , Yu. Yanovskii , E. Borisenkova , B. Yarlykov and G. Berezhnaya . Viscoelastic properties and flow of narrow distribution polybutadienes and polyisoprenes. J. Polymer Sci. Part A-2 10 (1972), 10611084.

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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