**Interfacial instability of pressure-driven channel
flow for a two-species model of entangled wormlike micellar
solutions**

By Michael Cromer, L. Pamela Cook, Gareth H. McKinley

We examine the linear stability of the one dimensional
inhomogeneous (shear banded) pressure-driven flow through a rectilinear microchannel predicted by the VCM model (Vasquez et al., A
network scission model for wormlike micellar
solutions I: model formulation and homogeneous flow
predictions. *J. Non-Newtonian
Fluid Mech*. 144:122-139, 2007). The VCM
model is a microstructural network model that
incorporates the breakage and reforming of two elastically-active species (a
long species ‘A’ and a shorter species ‘B’). The model consists of a set of coupled
nonlinear partial differential

equations describing the two micellar
species, which relax due to reptative and Rousian stress-relaxation mechanisms as well as breakage
events. The model includes nonlocal effects arising from stress-microstructure
diffusion and we investigate the effect of these nonlocal terms on the linear
stability of the pressure-driven flow. Calculation of the full eigenspectrum shows that the mode of instability is a
sinuous (odd) interfacial mode, in agreement with previous calculations for the
shear-banded Johnson-Segalman model (Fielding and
Wilson, Shear banding and interfacial instability in planar Poiseuille
flow. *J. Non-Newtonian Fluid Mech*. 165:196-202, 2010). Increased diffusion, or
smaller characteristic channel dimensions, smoothes the kink in the velocity
profile that develops at the shear band and progressively reduces spectrum of
unstable modes. For sufficiently large diffusion this smoothing effect
eliminates the instability entirely and restabilizes
the base (shear-banded) velocity profile.

*Keywords: *Viscoelasticity, Non-Newtonian Fluids, Rheology, Constitutive Modeling,
Wormlike Micelles, Linear Stability