Onset of Taylor vortices and chaos in viscoelastic fluids

Abstract

The influence of fluid elasticity on the onset and stability of axisymmetric Taylor vortices is examined for the Taylor–Couette flow of an Oldroyd-B fluid. A truncated Fourier representation of the flow field and stress leads to a six-dimensional dynamical system that generalizes the three-dimensional system for a Newtonian fluid. The coherence of the model is established through comparison with existing linear stability analyses and finite-element calculations of the nonlinear dynamics of the transition to time-periodic (finite-amplitude) flow. The stability picture and flow are drastically altered by the presence of the nonlinear (upper convective) terms in the constitutive equation. It is found that the critical Reynolds number Rec at the onset of Taylor vortices decreases with increasing fluid elasticity or normal stress effects, and is strongly influenced by fluid retardation. For weakly elastic flows, there is an exchange of stability at Re=Rec through a supercritical bifurcation, similar to the one predicted by the Newtonian model. As the elasticity number exceeds a critical value, a subcritical bifurcation emerges at Rec similar to the one predicted by the Landau–Ginzburg equation. More importantly, it is shown that, if fluid elasticity is adequately accounted for, any small but nonvanishing amount of fluid elasticity can lead to the onset of chaos usually observed in experiments on the Taylor–Couette flow of supposedly Newtonian fluids. This is in sharp contrast to the Newtonian model, which does not predict the destabilization of the Taylor vortices, and therefore cannot account for the onset of periodic and chaotic motion.