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Publication Detail
Linear instability of planar shear banded flow of both diffusive and non-diffusive Johnson-Segalman fluids
We consider the linear stability of shear banded planar Couette flow ofthe Johnson-Segalman fluid, with and without the addition of stressdiffusion to regularise the equations. In particular, we investigate thelinear stability of an initially one-dimensional "base" flow, with a flatinterface between the bands, to two-dimensional perturbations representingundulations along the interface. We demonstrate analytically that, for thelinear stability problem, the limit in which diffusion tends to zero ismathematically equivalent to a pure (non-diffusive) Johnson-Segalman modelwith a material interface between the shear bands, provided the wavelengthof perturbations being considered is long relative to the (short)diffusion lengthscale.For no diffusion, we find that the flow is unstable to long waves foralmost all arrangements of the two shear bands. In particular, for any setof fluid parameters and shear stress there is some arrangement of shearbands that shows this instability. Typically the stable arrangements ofbands are those in which one of the two bands is very thin. Weak diffusionprovides a small stabilising effect, rendering extremely long wavesmarginally stable. However, the basic long-wave instability mechanism isnot affected by this, and where there would be instability as wavenumber ktends to 0 in the absence of diffusion, we observe instability formoderate to long waves even with diffusion.This paper is the first full analytical investigation into an instabilityfirst documented in the numerical study of [Fielding, PRL95, 134501(2005)]. Authors prior to that work have either happened to chooseparameters where long waves are stable or used slightly differentconstitutive equations and Poiseuille flow, for which the parameters forinstability appear to be much more restricted.We identify two driving terms that can cause instability: one, a jump inN1, as reported previously by Hinch et al. [Hinch, Harris & Rallison,JNNFM 43, 311-324 (1992)]; and the second, a discontinuity in shear rate.The mechanism for instability from the second of these is not thoroughlyunderstood.We discuss the relevance of this work to recent experimental observationsof complex dynamics seen in shear-banded flows.
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