In the past several years it has come to be appreciated that in low Reynolds number flow the nonlinearities provided by non-Newtonian stresses of a complex fluid can provide a richness of dynamical behaviors more commonly associated with high Reynolds number Newtonian flow. For example, experiments by V. Steinberg and collaborators have shown that dilute polymer suspensions being sheared in simple flow geometries can exhibit highly time dependent dynamics and show efficient mixing. The corresponding experiments using Newtonian fluids do not, and indeed cannot, show such nontrivial dynamics. To better understand these phenomena we study the Oldroyd-B viscoelastic model. We first explain the derivation of this system and its relation to more familiar systems of Newtonian fluids and solids and give some analytical results for small data perturbations. Next we study this and related models numerically for low-Reynolds number flows in two dimensions. For low Weissenberg number (an elasticity parameter), flows are "slaved" to the four-roll mill geometry of the fluid forcing. For sufficiently large Weissenberg number, such slaved solutions are unstable and under perturbation transit in time to a structurally dissimilar flow state dominated by a single large vortex, rather than four vortices of the four-roll mill state. The transition to this new state also leads to regions of well-mixed fluid and can show persistent oscillatory behavior with continued destruction and generation of smaller-scale vortices.
