Instability of a viscoelastic Oldroyd-B fluid flowing over a slippery inclined plane with an imposed shear

We investigate both linear and weakly nonlinear stability analyses for the viscoelastic Oldroyd-B fluid falling down an impermeable slippery inclined plane, where a constant shear stress is applied at the fluid surface to take into account the dynamic effect of surrounding airflow. A linearized boundary value problem in terms of the amplitudes of perturbation variables is derived to execute the linear stability analysis. In the linear regime, Deborah number De, slip length α, and applied shear stress τs destabilize surface mode instability by decreasing the critical Reynolds number, while the solvent viscosity ratio β stabilizes it by increasing the critical Reynolds number. However, surface mode instability is stabilized if the external shear stress is applied in the counter-flow direction. Interestingly, we find a critical value of τs(<0) for a given slip length; below this value, the Newtonian fluid is more unstable, while above it, the viscoelastic upper-convected Maxwell fluid is more unstable compared to other fluids. Moreover, we observe the emergence of a purely elastic instability induced by elastic stress, and this instability can be made faster by introducing the slip effect at the solid impermeable plane. On the other hand, the weakly nonlinear stability analysis is performed based on the Benney-type nonlinear surface evolution equation. We find a filtered wave equilibrium solution in both the subcritical and supercritical zones. However, in the case of a purely elastic instability, we observe only a supercritical equilibrium solution. Finally, the nonlinear traveling wave solution exhibits that its amplitude amplifies as the Deborah number, the slip length, and the imposed shear stress rise.