Subcritical instabilities in plane Poiseuille flow of an Oldroyd-B fluid

TL;DR

This paper investigates subcritical instabilities in visco-elastic Poiseuille flow using the Oldroyd-B model, confirming similar nonlinear transition mechanisms as in Couette flow and providing detailed analysis of the instability's structure.

Contribution

It extends previous Couette flow analyses to Poiseuille flow, demonstrating the universality of the subcritical instability scenario in visco-elastic channel flows.

Findings

Subcritical instability occurs at Weissenberg numbers greater than one.

The spatial structure of the unstable mode is characterized.

Results suggest similarities with Newtonian pipe flow transition mechanisms.

Abstract

Recently, detailed experiments on visco-elastic channel flow have provided convincing evidence for a nonlinear instability scenario which we had argued for based on calculations for visco-elastic Couette flow. Motivated by these experiments we extend the previous calculations to the case of visco-elastic Poiseuille flow, using the Oldroyd-B constitutive model. Our results confirm that the subcritical instability scenario is similar for both types of flow, and that the nonlinear transition occurs for Weissenberg numbers somewhat larger than one. We provide detailed results for the convergence of our expansion and for the spatial structure of the mode that drives the instability. This also gives insight into possible similarities with the mechanism of the transition to turbulence in Newtonian pipe flow.