Numerical observations on the continuous spectrum of the linearized viscoelastic operator in shear dominated complex flows

The eigenspectrum of the viscoelastic operator of the upper convected Maxwell (UCM) model in two-dimensional mixed-kinematic flows without internal stagnation points is contained on a line with real part given by - 1/De, where De denotes the Deborah number [9]. We examine the manifestation of this continuous spectrum in numerical simulations of viscoelastic journal bearing and eccentric Dean flows using a pseudo-spectral Chebyshev-Fourier collocation technique. Our numerical results show that for a given set of geometric parameters the maximum imaginary part of the continuous spectrum increases with the largest wavenumber that can be accomodated by the mesh. Hence, increasing azimuthal resolution for a given eccentricity introduces higher wavenumbers leading to an overall convergence that is poorer than that obtained for a coarser mesh. The eigenfunctions exhibit singular behavior in both the wall-normal and azimuthal directions. The locations of these singularities depend on the imaginary part of the eigenvalue. (C) 2000 Elsevier Science B.V. All rights reserved.