Linear stability analysis of a shear-imposed Oldroyd-B liquid flowing down a slippery inclined plane

A linear stability analysis is performed by using the Orr-Sommerfeld type boundary value problem (OS BVP) for an Oldroyd-B liquid flowing down a slippery plane in the presence of an imposed shear stress. The OS BVP is solved analytically and numerically by using the long-wave asymptotic expansion at low Reynolds number and the Chebyshev spectral collocation technique for low to high values of the Reynolds number, respectively. The long-wave approach shows the existence of surface mode, while the finite wavelength shear mode appears in the high Reynolds number regime. Expression of critical Reynolds number for the surface mode indicates a destabilizing nature of slip and imposed shear stress (co-flow direction). But when the imposed shear stress acts in the counter-flow direction, the critical Reynolds number for the surface mode increases, thus a stabilizing effect is seen. While using the numerical approach in the moderate Reynolds number regime, the Weissenberg number and the viscosity ratio have a dual nature on the primary instability. Finally, at the high Reynolds number, the shear mode is stabilized by the viscosity ratio, slip length, and imposed shear stress (counter-flow direction), and destabilized by the Weissenberg number and imposed shear stress (co-flow direction).