Potential "ways of thinking" about the shear-banding phenomenon

TL;DR

This paper reviews various interpretations of the shear-banding phenomenon in soft matter, focusing on the diffusive Johnson-Segalman equation and its connections to thermodynamics, reaction-diffusion systems, and boundary conditions.

Contribution

It provides a comprehensive overview of different theoretical approaches to shear-banding, highlighting the unifying role of the dJS equation and introducing a new thermodynamic perspective.

Findings

dJS equation can be expressed in multiple equivalent forms

Analogy between dJS and reaction-diffusion equations

Boundary conditions significantly influence shear-banding states

Abstract

Shear-banding is a curious but ubiquitous phenomenon occurring in soft matter. The phenomenological similarities between the shear-banding transition and phase transitions has pushed some researchers to adopt a 'thermodynamical' approach, in opposition to the more classical 'mechanical' approach to fluid flows. In this heuristic review, we describe why the apparent dichotomy between those approaches has slowly faded away over the years. To support our discussion, we give an overview of different interpretations of a single equation, the diffusive Johnson-Segalman (dJS) equation, in the context of shear-banding. We restrict ourselves to dJS, but we show that the equation can be written in various equivalent forms usually associated with opposite approaches. We first review briefly the origin of the dJS model and its initial rheological interpretation in the context of shear-banding. Then we describe the analogy between dJS and reaction-diffusion equations. In the case of anisotropic diffusion, we show how the dJS governing equations for steady shear flow are analogous to the equations of the dynamics of a particle in a quartic potential. Going beyond the existing literature, we then draw on the Lagrangian formalism to describe how the boundary conditions can have a key impact on the banding state. Finally, we reinterpret the dJS equation again and we show that a rigorous effective free energy can be constructed, in the spirit of early thermodynamic interpretations or in terms of more recent approaches exploiting the language of irreversible thermodynamics.