Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-25T17:42:03.540Z Has data issue: false hasContentIssue false
  • English
  • Français

A nonlinear singularly perturbed Volterra integrodifferential equation occurring in polymer rheology

Published online by Cambridge University Press:  14 November 2011

A. S. Lodge ,
J. B. McLeod  and
J. A. Nohel
A. S. Lodge
Affiliation:
Mathematics Research Center, University of Wisconsin, Madison, U.S.A.
J. B. McLeod
Affiliation:
Mathematics Research Center, University of Wisconsin, Madison, U.S.A.
J. A. Nohel
Affiliation:
Mathematics Research Center, University of Wisconsin, Madison, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

We study the initial value problem for the nonlinear Volterra integrodifferential equation

where μ > 0 is a small parameter, a is a given real kernel, and F, g are given real functions; (+) models the elongation ratio of a homogeneous filament of a certain polyethylene which is stretched on the time interval (— ∞ 0], then released and allowed to undergo elastic recovery for t > 0. Under assumptions which include physically interesting cases of the given functions a, F, g, we discuss qualitative properties of the solution of (+) and of the corresponding reduced problem when μ = 0, and the relation between them as μ → 0+, both for t near zero (where a boundary layer occurs) and for large t. In particular, we show that in general the filament does not recover its original length, and that the Newtonian term —μy′ in (+) has little effect on the ultimate recovery but significant effect during the early part of the recovery.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 80 , Issue 1-2 , 1978 , pp. 99 - 137
Copyright
Copyright © Royal Society of Edinburgh 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bernstein, B., Kearsley, E. A. and Zapas, L. J.A Study of Stress Relation with Finite Strain. Trans. Soc. Rheol. 7 (1963), 391–410.CrossRefGoogle Scholar
2Chang, Hui Elongational Flow and Spinnability of Viscoelastic Fluids (Wisconsin Univ. Ph.D. Thesis, 1973).Google Scholar
3Eisenhart, L. P.Riemannian Geometry (Princeton: Univ. Press, 1926).Google Scholar
4Huang, T. A. Time-dependent First Normal Stress Difference and Shear Stress Generated by Polymer Melts in Steady Shear Flow (Wisconsin Univ. Ph.D. Thesis, 1976).Google Scholar
5Kaye, A.Non-Newtonian Flow in Incompressible Fluids. Cranfield College Aeronaut. Note 134 (1962).Google Scholar
6Lakshmikantham, V. and Leela, S.Differential and Integral Inequalities (London: Academic Press, 1969).Google Scholar
7Levin, J. J. and Nohel, J. A.Perturbations of a Nonlinear Volterra Equation. Michigan Math. J. 12 (1965), 431–447.CrossRefGoogle Scholar
8Lodge, A. S.Body Tensor Fields in Continuum Mechanics (London: Academic Press, 1974).Google Scholar
9Lodge, A. S. Concentrated Polymer Solutions. In Proc. Fifth Internat. Congr. Rheol. 4 (Tokyo: Univ. Press, 1970).Google Scholar
10Lodge, A. S.Constitutive Equations from Molecular Network Theories for Polymer Solutions. Rheol. Acta 7 (1968), 379–392.CrossRefGoogle Scholar
11Lodge, A. S.The Compatibility Conditions for Large Strains. Quart. J. Mech. Appl Math. 4 (1951), 85–93.CrossRefGoogle Scholar
12Lodge, A. S. and Meissner, J.On the Use of Instantaneous Strains Superposed on Shear and Elongational Flows of Polymeric Liquids, to Test the Gaussian Network Hypothesis and to Estimate the Segment Concentration and its Variation during Flow. Rheol. Acta 11 (1972), 351–352.CrossRefGoogle Scholar
13Lodge, A. S. and Meissner, J.Comparison of Network Theory Predictions with Stress/Time Data in Shear and Elongation for a Low-density Polyethylene Melt. Rheol. Acta 12 (1973), 41–47.CrossRefGoogle Scholar
14Marrucci, G. and Acierno, D. Non-affine Deformation in Impermanent Networks of Polymer Chains. In Proc. Seventh Internat. Congr. Rheol. (Gothenburg: Swedish Soc. Rheol., 1976).Google Scholar
15Meissner, J.Dehnungsverhalten von Polyäthylen-Schmelzen. Rheol. Acta 10 (1971), 230–242.Google Scholar
16Miller, R. K.Nonlinear Volterra Integral Equations (New York: Benjamin, 1971).Google Scholar
17Nohel, J. A.Some Problems in Nonlinear Volterra Integral Equations. Bull. Amer. Math. Soc. 68 (1962), 323–329.CrossRefGoogle Scholar
18Nohel, J. A. and Shea, D. F.Frequency Domain Methods for Volterra Equations. Advances in Math. 22 (1976), 278–304.CrossRefGoogle Scholar
19Walter, W.Differential and Integral Inequalities (Berlin: Springer, 1970).CrossRefGoogle Scholar
20Widder, D. V.The Laplace Transform (Princeton: Univ. Press, 1941).Google Scholar