The effects of gap width and dilute solution properties on the viscoelastic Taylor-Couette instability

Abstract

The effects of finite gap and various dilute solution properties on the previously studied purely elastic Taylor-Couette instability reported by Muller et al. (1989) and Larson et al. (1990) are investigated. The dilute solution properties which we consider are the ratio of the second to the first normal stress coefficient, ψ2/ψ1, and the ratio of the solvent to the polymer contribution to the shear viscosity, S. Linear stability predictions for the flow of an Oldroyd-B fluid are presented over a wide range of Deborah number, De, gap ratio, ε, and S. In addition, the Oldroyd-B model is modified to include second normal stress differences, and new stability predictions are presented for small negative and small positive ψ2/ψ1. Both the critical conditions and changes in the flow structure are presented. It is demonstrated that finite-gap effects are stabilizing even for relatively small gap ratios (0 < ε < 0.35). Furthermore, it is shown that there are two possible flow structures which can be chosen near the onset of instability: a standing wave structure (i.e. radially propagating vortices) or a travelling wave (i.e. vortices propagating up or down the coaxial cylinders). However, the strength and both the axial and radial dimensions of these vortices depend markedly on the gap, with both dimensions decreasing as the gap ratio increases. Thus, the number of vortices filling the gap increases with the gap ratio.In a second study, we show that the instability is sensitive to the presence of second normal stress differences. Positive second normal stress differences are shown to be destabilizing, while negative differences are strongly stabilizing. Furthermore, when both finite-gap effects and small negative second normal stress differences are included, the predicted gap dependence of the critical De is in good agreement with previous measurements on the flow of a dilute polyisobutylene solution. Finally, we present new measurements of the critical values of the De for a series of dilute, viscous polystyrene solutions, for which ψ2 was found to be near zero. We find that as the polymer concentration increases (and therefore S decreases) the critical Deborah number decreases, in qualitative agreement with the theoretical predictions.