Stress modes in linear stability of viscoelastic flows

Although parallel shear flows of the upper convected Maxwell fluid are stable, recent articles [R. Kupferman, On the linear stability of plane Couette flow for an Oldroyd-B fluid and its numerical approximation, J. Non-Newt. Fluid Mech. 127 (2005) 169-190; C.R. Doering, B. Eckhardt,J. Schumacher, Failure of energy stability in Oldroyd-B fluids at arbitrarily low Reynolds numbers, J. Non-Newt. Fluid Mech. 135 (2006) 92-96] have exhibited transient growth at high Weissenberg number which is analogous to that observed in high Reynolds number Newtonian shear flows. The perturbations involved are pure stress perturbations with zero velocities. In this paper, we extend the analysis of such stress modes in plane Couette How to the three-dimensional case. We also exhibit a class of "stress pressure modes" in plane Poiseuille flow. Finally, we discuss pure stress modes in elongational flow. (C) 2009 Elsevier B.V. All rights reserved.