  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 non-zero) 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.

Fluid Dynamics;Chaotic Dynamics