Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

A nonlinear singularly perturbed Volterra integrodifferential equation occurring in polymer rheology

A. S. Lodgea112, J. B. McLeoda11 and J. A. Nohela1134

a1 Mathematics Research Center, University of Wisconsin, Madison, U.S.A.

Synopsis

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

S0308210500010167_eqnU1

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.

(Received March 29 1977)

Footnotes

1 sponsored by The U.S. Army under Contract No. DAAG29-75-C-0024

2 sponsored by The National Science Foundation under Grant No. ENG 75-18397

3 sponsored by The U.S. Army under Grant No. DAHC04-74-G-0012

4 sponsored by The National Science Foundation under Grant No. MCS 75-21868.