Matched asymptotic solutions for the steady banded flow of the diffusive Johnson-Segalman model in various geometries

TL;DR

This paper derives analytical solutions for steady banded flows in the Johnson-Segalman model with diffusion, revealing how diffusion influences the structure and stability of flow bands across different geometries.

Contribution

It introduces matched asymptotic expansions to solve the diffusive Johnson-Segalman model, elucidating the effect of diffusion on band formation in various geometries.

Findings

Stable steady flows always have two bands in studied geometries.

Two distinct band configurations are possible in Couette flow with small curvature.

Diffusion lifts degeneracy, selecting unique steady states.

Abstract

We present analytic solutions for steady flow of the Johnson-Segalman (JS) model with a diffusion term in various geometries and under controlled strain rate conditions, using matched asymptotic expansions. The diffusion term represents a singular perturbation that lifts the continuous degeneracy of stable, banded, steady states present in the absence of diffusion. We show that the stable steady flow solutions in Poiseuille and cylindrical Couette geometries always have two bands. For Couette flow and small curvature, two different banded solutions are possible, differing by the spatial sequence of the two bands.