Abstract
In this paper, we discuss recent results on 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 between parallel plates located at x = ±1, and 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 steady shear strain rate that results in steady states having, in general, discontinuities in the strain rate. We explain why every solution tends to a steady state as t → ∞, and we identify steady states that are stable; more details and proofs will be presented in [8].
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).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G. Andrews and J. Ball, “Asymptotic Stability and Changes of Phase in One-Dimensional Nonlinear Viscoelasticity,” J. Diff. Eqns. 44 (1982), pp. 306–341.
C. Guillopé and J.-C. Saut, “Global Existence and One-Dimensional Nonlinear Stability of Shearing Motions of Viscoelastic Fluids of Oldroyd Type,” Math. Mod. Numer. Anal., 1990. To appear.
C. Guillopé and J.-C. Saut, “Existence Results for Flow of Viscoelastic Fluids with a Differential Constitutive Law,” Math. Mod. Numer. Anal., 1990. To appear.
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840 Springer-Verlag, New York, 1981.
J. Hunterand M. Slemrod, “Viscoelastic Fluid Flow Exhibiting Hysteretic Phase Changes,” Phys. Fluids 26 (1983), pp. 2345–2351.
D. Malkus, J. Nohel, and B. Plohr, “Dynamics of Shear Flow of a Non-Newtonian Fluid,” J. Comput. Phys., 1989. To appear.
D. Malkus, J. Nohel, and B. Plohr, “Analysis of New Phenomena In Shear Flow of Non-Newtonian Fluids,” in preparation, 1989.
J. Nohel, R. Pego, and A. Tzavaras, “Stability of Discontinuous Steady States in Shearing Motions of Non-Newtonian Fluids,” Proc. Roy. Soc. Edinburgh, Series A, 1989. submitted.
A. Novick-Cohen and R. Pego, “Stable Patterns in a Viscous Diffusion Equation,” preprint, 1989. submitted.
R. Pego, “Phase Transitions in One-Dimensional Nonlinear Viscoelasticity: Admissibility and Stability,” Arch. Rational Mech. and Anal. 97 (1987), pp. 353–394.
M. Renardy, W. Hrusa, and J. Nohel, Mathematical Problems in Viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 35, Longman Scientific & Technical, Essex, England, 1987.
A. Tzavaras, “Effect of Thermal Softening in Shearing of Strain-Rate Dependent Materials,” Arch. Rational Mech. and Anal. 99 (1987), pp. 349–374.
G. Vinogradov, A. Malkin, Yu. Yanovskii, E. Borisenkova, B. Yarlykov, and G. Berezh-Naya, “Viscoelastic Properties and Flow of Narrow Distribution Polybutadienes and Polyiso-prenes,” J. Polymer Sci., Part A-2 10 (1972), pp. 1061–1084.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer-Verlag New York, Inc.
About this paper
Cite this paper
Nohel, J.A., Pego, R.L., Tzavaras, A.E. (1991). Nonlinear Stability in Non-Newtonian Flows. In: Glimm, J., Majda, A.J. (eds) Multidimensional Hyperbolic Problems and Computations. The IMA Volumes in Mathematics and Its Applications, vol 29. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9121-0_20
Download citation
DOI: https://doi.org/10.1007/978-1-4613-9121-0_20
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4613-9123-4
Online ISBN: 978-1-4613-9121-0
eBook Packages: Springer Book Archive