Stability of viscoelastic shear flows subjected to parallel flow superposition

Abstract

Steady viscoelastic shear flows in curved geometries are susceptible to instabilities due to the radial force associated with elastic stresses along curved streamlines. Recent work has shown that the addition of steady or oscillatory shear flow in a transverse direction (orthogonal superposition) can suppress these instabilities. The present work instead investigates the effect of oscillatory parallel superposition, for the particular case of circular Couette flow. For flow of an Oldroyd-B fluid, the oscillation has a weak stabilizing effect if the oscillation amplitude is not too large. If on the other hand, the oscillation amplitude is such that the angular velocity of the moving cylinder changes sign over part of the cycle, the flow is destabilized. However, in the limit where the motion is purely oscillatory (i.e., large amplitude oscillatory shear), the flow is stabilized relative to steady circular Couette flow with the same maximum shear rate. Finally, in the limit where curvature goes to zero—plane Couette flow—parallel superposition makes the already stable flow slightly more stable, increasing the decay rate of fluctuations in the range of parameters studied. Furthermore, two-dimensional disturbances are the most slowly decaying; we have extended Squire’s theorem in this case to encompass an arbitrary time dependence.