STRUCTURE OF THE SPECTRUM IN ZERO REYNOLDS-NUMBER SHEAR-FLOW OF THE UCM AND OLDROYD-B LIQUIDS

We provide a mathematical analysis of the spectrum of the linear stabi lity problem for one and two layer channel hows of the upper-convected Maxwell (UCM) and Oldroyd-B fluids at zero Reynolds number. For plane Couette flow of the UCM fluid, it has long been known (Gorodstov and Leonov, J. Appl. Math. Mech. (PMM) 31 (1967) 310) that, for any given streamwise wave number, there are two eigenvalues in addition to a con tinuous spectrum. In the presence of an interface, there are seven dis crete eigenvalues. In this paper, we investigate how this structure of the spectrum changes when the how is changed to include a Poiseuille component, and as the model is changed from the UCM to the more genera l Oldroyd-B. For a single layer UCM fluid, we find that the number of discrete eigenvalues changes from two in Couette flow to six in Poiseu ille flow. The six modes are given in closed form in the long wave lim it. For plane Couette flow of the Oldroyd-B fluid, we solve the differ ential equations in closed form. There is an additional continuous spe ctrum and a family of discrete modes. The number of these discrete mod es increases indefinitely as the retardation time approaches zero. We analyze the behavior of the eigenvalues in this limit. (C) 1999 Elsevi er Science B.V. All rights reserved.