  We study theoretically pressure driven planar channel flow of shear thinning
viscoelastic fluids. Combining linear stability analysis and full nonlinear
simulation, we study the instability of an initially one-dimensional base state
to the growth of two-dimensional perturbations with wavevector in the flow
direction. We do so within three widely used constitutive models: the
microscopically motivated Rolie-Poly model, and the phenomenological
Johnson-Segalman and White-Metzner models. In each model, we find instability
when the degree of shear thinning exceeds some level characterised by the
logarithmic slope of the flow curve at its shallowest point. Specifically, we
find instability for $n<n^*$, with $n^*\approx 0.21, 0.11$ and $0.30$ in the
Rolie-Poly, Johnson-Segalman and White-Metzner models respectively. Within each
model, we show that the critical pressure drop for the onset of instability
obeys a criterion expressed in terms of this degree of shear thinning, $n$,
together with the derivative of the first normal stress with respect to shear
stress. Both shear thinning and rapid variations in first normal stress across
the channel are therefore key ingredients driving the instability. In the
Rolie-Poly and Johnson-Segalman models, the underlying mechanism appears to
involve the destabilisation of a quasi-interface that exists in each half of
the channel, across which the normal stress varies rapidly. In the
White-Metzner model, no such quasi-interface exists, but the criterion for
instability nonetheless appears to follow the same form as in the Rolie-Poly
and Johnson-Segalman models. This presents an outstanding puzzle concerning any
possibly generic nature of the instability mechanism. We finally make some
briefly comments on the Giesekus model, which is rather different in its
predictions from the other three.
