  The Oldroyd-B model has been used extensively to predict a host of
instabilities in shearing flows of viscoelastic fluids, often realized
experimentally using polymer solutions. The present review, written on the
occasion of the birth centenary of James Oldroyd, provides an overview of
instabilities found across major classes of shearing flows. These comprise (i)
the canonical rectilinear shearing flows including plane Couette, plane and
pipe Poiseuille flows; (ii) viscometric shearing flows with curved streamlines
such as those in the Taylor-Couette, cone-and-plate and parallel-plate
geometries; (iii) non-viscometric shearing flows with an underlying extensional
flow topology such as the flow in a cross-slot device; and (iv) multilayer
shearing flows. While the underlying focus in all these cases is on results
obtained using the Oldroyd-B model, we also discuss their relation to the
actual instability, and as to how the shortcomings of the Oldroyd-B model may
be overcome by the use of more realistic constitutive models. All the three
commonly used tools of stability analysis, viz., modal linear stability,
nonmodal stability, and weakly nonlinear stability analyses are discussed, with
supporting evidence from experiments and numerical simulations as appropriate.
Despite only accounting for a shear-rate-independent viscosity and first normal
stress coefficient, the Oldroyd-B model is able to qualitatively predict the
majority of instabilities in the aforementioned shearing flows. The review also
highlights, where appropriate, open questions in the area of viscoelastic
stability.
