Purely elastic linear instabilities in parallel shear flows with free-slip boundary conditions

TL;DR

This study conducts a linear stability analysis of viscoelastic plane Couette and Poiseuille flows with free-slip boundaries, revealing instabilities linked to elastic turbulence phenomena.

Contribution

It demonstrates that free-slip boundary conditions induce linear instabilities in viscoelastic shear flows, connecting them to no-slip boundary mode stability.

Findings

Both flows become linearly unstable under free-slip conditions.

Unstable modes are related to the least stable no-slip modes.

Boundary condition homotopy links free-slip and no-slip instabilities.

Abstract

We perform a linear stability analysis of viscoelastic plane Couette and plane Poiseuille flows with free-slip boundary conditions. The fluid is described by the Oldroyd-B constitutive model, and the flows are driven by a suitable body force. We find that both types of flow become linearly unstable, and we characterise the spatial structure of the unstable modes. By performing a boundary condition homotopy from the free-slip to no-slip boundaries, we demonstrate that the unstable modes are directly related to the least stable modes of the no-slip problem, destabilised under the free-slip situation. We discuss how our observations can be used to study recently discovered purely elastic turbulence in parallel shear flows.