Stability and transitions of the second grade Poiseuille flow

Abstract

In this study we consider the stability and transitions for the Poiseuille flow of a second grade fluid which is a model for non-Newtonian fluids. We restrict our attention to perturbation flows in an infinite pipe with circular cross section that are independent of the axial coordinate. We show that unlike the Newtonian (epsilon = 0) case, in the second grade model (epsilon > 0 case), the time independent base flow exhibits transitions as the Reynolds number R exceeds the critical threshold R-c = 8.505 epsilon(-1/2) where epsilon is a material constant measuring the relative strength of second order viscous effects compared to inertial effects. At R = R-c, we find that the transition is either continuous or catastrophic and a small amplitude, time periodic flow with 3-fold azimuthal symmetry bifurcates. The time period of the bifurcated solution tends to infinity as R tends to R-c. Our numerical calculations suggest that for low epsilon values, the system prefers a catastrophic transition where the bifurcation is subcritical. We also show that there is a Reynolds number R-E with R-E < R-c such that for R < R-E, the base flow is globally stable and attracts any initial disturbance with at least exponential speed. We show that the gap between R-E and R-c, vanishes quickly as epsilon increases. (C) 2016 Elsevier B.V. All rights reserved.