Location of the continuous spectrum in complex flows of the UCM fluid

In contrast to the Newtonian case, linear stability problems for viscoelast ic flows involve continuous as well as discrete spectra, even if the flow d omain is bounded. Numerical methods approximate these continuous spectra po orly, and incorrect claims of instability have been published as a result o f this on more than one occasion. In this paper, we shall derive some analy tical results on the location of the continuous spectrum for linear stabili ty of flows of the upper convected Maxwell fluid. In general, we shall show that in 'subsonic' flows, where the fluid speed is always slower than the speed of sheer wave propagation, there are only three possible contribution s to the continuous spectrum: 1. A part on the line Re lambda = -1 W, where W is the relaxation time of t he fluid. 2. A part associated with the short wave limit of wall modes which has real parts confined between -1/W and 3. A part associated with the integration of stresses in a given velocity f ield. If the flow is two-dimensional and has no stagnation points, then the latte r part also has real part on the line Re lambda = -1/W, and hence the conti nuous spectrum is always stable. (C) 2000 Elsevier Science B.V. All rights reserved.