  We perform a linear stability analysis of viscoelastic plane Couette and
plane Poiseuille flows with free-slip boundary conditions. The fluid is
described by the Oldroyd-B constitutive model, and the flows are driven by a
suitable body force. We find that both types of flow become linearly unstable,
and we characterise the spatial structure of the unstable modes. By performing
a boundary condition homotopy from the free-slip to no-slip boundaries, we
demonstrate that the unstable modes are directly related to the least stable
modes of the no-slip problem, destabilised under the free-slip situation. We
discuss how our observations can be used to study recently discovered purely
elastic turbulence in parallel shear flows.
