Stability of plane Poiseuille-Couette flows of a piezo-viscous fluid

We examine stability of fully developed isothermal unidirectional plane Poiseuille-Couette flows of an incompressible fluid whose viscosity depends linearly on the pressure as previously considered in Hron et al. [J. Hron, J. Málek, K.R. Rajagopal, Simple flows of fluids with pressure-dependent viscosities, Proc. R. Soc. Lond. A 457 (2001) 1603-1622] and Suslov and Tran [S.A. Suslov, T.D. Tran, Revisiting plane Couette-Poiseuille flows of a piezo-viscous fluid, J. Non-Newtonian Fluid Mech. 154 (2008) 170-178]. Stability results for a piezo-viscous fluid are compared with those for a Newtonian fluid with constant viscosity. We show that piezo-viscous effects generally lead to stabilisation of a primary flow when the applied pressure gradient is increased. We also show that the flow becomes less stable as the pressure and therefore the fluid viscosity decrease downstream. These features drastically distinguish flows of a piezo-viscous fluid from those of its constant-viscosity counterpart. At the same time the increase in the boundary velocity results in a flow stabilisation which is similar to that observed in Newtonian fluids with constant viscosity.