Two-dimensional perturbations in a scalar model for shear banding

TL;DR

This paper analytically investigates a simplified shear banding model, revealing that the absence of normal stresses leads to linear stability against 2D perturbations, unlike models with normal stresses.

Contribution

It demonstrates that normal stresses are responsible for linear instabilities in shear banding models, using a piecewise linear approximation for the flow curve.

Findings

The toy model is linearly stable against 2D perturbations.

Normal stresses induce linear instabilities in shear banding.

The model provides insight into the role of normal stresses in shear banding stability.

Abstract

We present an analytical study of a toy model for shear banding, without normal stresses, which uses a piecewise linear approximation to the flow curve (shear stress as a function of shear rate). This model exhibits multiple stationary states, one of which is linearly stable against general two-dimensional perturbations. This is in contrast to analogous results for the Johnson-Segalman model, which includes normal stresses, and which has been reported to be linearly unstable for general two-dimensional perturbations. This strongly suggests that the linear instabilities found in the Johnson-Segalman can be attributed to normal stress effects.