Failure of energy stability in Oldroyd-B fluids at arbitrarily low Reynolds numbers

Energy theory for incompressible Newtonian fluids is, in many cases, capable of producing strong absolute stability criteria for steady flows. In those fluids the kinetic energy naturally defines a norm in which perturbations decay monotonically in time at sufficiently low (but nonzero) Reynolds numbers. There are, however, at least two obstructions to the generalization of such methods to Oldroyd-B fluids. One previously recognized problem is the fact that the natural energy does not correspond to a proper functional norm on perturbations. Another problem, original to this work, is the fact that fluctuations in Oldroyd-B fluids may be subject to non-normal amplification at arbitrarily low Reynolds numbers (albeit at sufficiently large Weissenberg numbers). Such transient growth, occuring even when the base flow is linearly stable, precludes the uniform monotonic decay of any reasonable measure of the disturbance’s amplitude.