Shear-banding: When can we ignore diffusion?

It is well-known that the addition of a small amount of diffusion of some material property (stress, rate of strain, structure factor, etc.) can regularise a constitutive equation which permits shear-banding. Typically, in the absence of diffusion, these models are ill-posed and permit steady banded states of a sort not observed in experiment, with many sharp-edged, closely-spaced bands. In an earlier paper [1] we looked at the two-dimensional linear stability properties of banded planar shear flow for two possible diffusive modifications to the Johnson-Segalman model. We were able to show that, for perturbations whose wavelength is not as short as the diffusion lengthscale, the diffusive stability calculation is a regular perturbation to the equivalent calculation for a non-diffusive fluid with a material interface between shear-bands. In this paper we show another, similar result using the same kind of analysis: this time for a phase-separating binary blend. We believe that in general, for the purposes of the linear stability properties, the effects of weak diffusion can be neglected and the stability or otherwise of any diffuse interface can be assessed using the simpler diffusionless model. The extensive literature on coextrusion instabilities is therefore relevant to the mechanisms for breakdown and fluctuation of banded systems, phase-separating systems and, indeed, any physical system containing interface-like regions in the presence of weak diffusion.