Weakly viscoelastic film flowing down a rotating inclined plane

We investigate the nonlinear stability of a thin viscoelastic film flowing under the effects of gravity and Coriolis and centrifugal forces. We assume that the viscoelastic liquid satisfies the rheological property of Walters' liquid B & DPRIME;. We may consider this case as a viscoelastic flow down a rotating cone and far from the apex. Using the classical long wave expansion technique, we derive a nonlinear evolution equation describing the shape of the liquid interface as a function of space and time and also derive its stability characteristics. We solve the physical system in a two-step procedure. In the first step, we use the normal mode method to characterize the linear nature. The linear study reveals that the linear growth rate is invariant with the Coriolis effect but is significantly affected by the viscoelastic parameter & UGamma; as well as the Taylor number Ta. It is found that both & UGamma; and Ta destabilize the flow. In the second step, we solve an elaborated nonlinear film flow model based on the method of multiple scales and demarcate different instability zones. The weakly nonlinear study shows that with an increase in & UGamma; and Ta, the supercritical stable region and the explosion area increase whereas the unconditional stable and the subcritical unstable region shrink. Finally, on validating our analytical predictions by performing a direct numerical simulation, a good agreement between the results of the linear stability analysis, weakly nonlinear stability analysis, and the numerical simulations is found.