Spatio-temporal instabilities in viscoelastic channel flows: The centre mode

Abstract

We study in this work the spatio-temporal characteristics of the centre-mode linear instability in viscoelastic channel flows of Oldroyd-B and FENE-P fluids. The linear complex Ginzburg–Landau equation derived using an amplitude expansion method is adopted to determine whether the flow is convectively unstable or absolutely unstable. Comparison of the obtained results with those from the conventional saddle-point searching method shows a favourable agreement. This demonstrates the good predictability of the linear complex Ginzburg–Landau equation which is more suitable for a parametric study in a large space than the conventional method. We found that the centre-mode instability in the viscoelastic channel flow is of a convective nature, i.e., the trailing edge of the wavepacket travels downstream. This could be attributed to the relatively high phase speed, low spreading rate and low growth rate of the instability, collectively preventing a disturbance from travelling upstream. Our results show that stronger polymer elasticity (a greater elastic number or larger polymer concentration) marginally affects the trailing-edge velocity of the centre mode. Decreasing the maximum polymer extensibility in the FENE-P model reduces the spreading rate of the wavepacket and makes the centre-mode disturbance more convective. Theoretical derivations at asymptotically high Reynolds numbers confirm the convective nature of the instabilities from numerical observations. Our results may be conducive to a better understanding of the spatio-temporal instability in viscoelastic experiments.