Effect of imposed shear on the stability of a viscoelastic liquid flowing down a slippery plane

We investigate a linear stability analysis for the Oldroyd-B liquid draining down a slippery inclined plane where a constant shear stress is applied at the liquid surface in the streamwise direction. The aim is to expand the previous study (Shaqfeh et al., 1989) in the presence of imposed shear stress when the bounding plane is slippery. The stability analysis is executed analytically as well as numerically for low to high values of the Reynolds number, which detects the existence of surface and shear modes. In the low Reynolds number regime, we use the analytical long-wave asymptotic expansion to find the onset of primary instability for the surface mode. On the other hand, we implement the numerical Chebyshev spectral collocation technique to determine the short-wave instability when the Reynolds number changes from a moderate to a high value. It is found that the critical Reynolds number for the surface mode decreases with the increasing values of the slip length and imposed shear stress, and thereby, the surface instability occurs at a lower Reynolds number than that when the slip length and the imposed shear stress are excluded. More specifically, the imposed shear stress has a destabilizing effect on the surface mode when acting in the co-flow direction but a stabilizing effect when acting in the counter-flow direction. Furthermore, the viscosity ratio exhibits a stabilizing effect on the surface mode near the onset of instability but exhibits a destabilizing effect far away from the onset of instability. In the high Reynolds number regime, the primary instability induced by the shear mode can be stabilized by the viscosity ratio and slip length, but can be destabilized by applying a constant shear stress in the co-flow direction. In a special case, the result for the surface mode corresponding to Walters' liquid B '' can be recovered from that of the Oldroyd-B liquid if the viscoelastic parameter is kept at a very low value.