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

Abstract

In contrast to the Newtonian case, linear stability problems for viscoelastic flows involve continuous as well as discrete spectra, even if the flow domain is bounded. Numerical methods approximate these continuous spectra poorly, and incorrect claims of instability have been published as a result of this on more than one occasion. In this paper, we shall derive some analytical results on the location of the continuous spectrum for linear stability 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 shear wave propagation, there are only three possible contributions to the continuous spectrum: 1. A part on the line Re λ=−1/W , where W is the relaxation time of the fluid. 2. A part associated with the short wave limit of wall modes which has real parts confined between −1/ W and −1/(2 W). 3. A part associated with the integration of stresses in a given velocity field. If the flow is two-dimensional and has no stagnation points, then the latter part also has real part on the line Re λ=−1/W , and hence the continuous spectrum is always stable.