  The linear stability of a shear-thinning, viscoelastic fluid undergoing any
of the canonical rectilinear shear flows, viz., plane Couette flow and
pressure-driven flow through a channel or a tube is analyzed in the
creeping-flow limit using the White--Metzner model with a power-law variation
of the viscosity with shear rate. While two-dimensional disturbances are
considered for plane Couette and channel flows, axisymmetric disturbances are
considered for pressure-driven flow in a tube. For all these flows, when the
shear-thinning exponent is less than $0.3$, there exists an identical
instability at wavelengths much smaller than the relevant geometric length
scale (gap between the plates or tube radius). There is also a
finite-wavelength instability in these configurations governed by the details
of the geometry and boundary conditions at the centerline of the channel or
tube. The most unstable mode could be either of the short-wave or
finite-wavelength instabilities depending on model parameters. For
pressure-driven channel flow, it is possible to have sinuous or varicose
unstable modes depending on the symmetry of the normal velocity eigenfunction
about the channel centerline. This difference in symmetry is relevant only for
the finite wavelength instability, in which case sinuous modes turn out to be
more unstable, in accordance with experimental observations. In all the three
configurations, the short wavelength unstable modes are localized near the
walls, and are insensitive to symmetry conditions at the centerline. It is
argued that this instability should be a generic feature in any wall-bounded
shear flow of strongly shear-thinning viscoelastic fluids. Our predictions for
the finite-wavelength instability in pressure-driven channel and pipe flows are
in good agreement with experimental observations for the flow of concentrated
polymer solutions in these geometries.
