Waves and instabilities of viscoelastic fluid film flowing down an inclined wavy bottom

Evolution of waves and hydrodynamic instabilities of a thin viscoelastic fluid film flowing down an inclined wavy bottom of moderate steepness have been analyzed analytically and numerically. The classical long-wave expansion method has been used to formulate a nonlinear evolution equation for the development of the free surface. A normal-mode approach has been adopted to discuss the linear stability analysis from the viewpoint of the spatial and temporal study. The method of multiple scales is used to derive a Ginzburg-Landau-type nonlinear equation for studying the weakly nonlinear stability solutions. Two significant wave families, viz., γ_{1} and γ_{2}, are found and discussed in detail along with the traveling wave solution of the evolution system. A time-dependent numerical study is performed with Scikit-FDif. The entire investigation is conducted primarily for a general periodic bottom, and the detailed results of a particular case study of sinusoidal topography are then discussed. The case study reveals that the bottom steepness ζ plays a dual role in the linear regime. Increasing ζ has a stabilizing effect in the uphill region, and the opposite occurs in the downhill region. While the viscoelastic parameter Γ has a destabilizing effect throughout the domain in both the linear and the nonlinear regime. Both supercritical and subcritical solutions are possible through a weakly nonlinear analysis. It is interesting to note that the unconditional zone decreases and the explosive zone increases in the downhill region rather than the uphill region for a fixed Γ and ζ. The same phenomena occur in a particular region if we increase Γ and keep ζ fixed. The traveling wave solution reveals the fact that to get the γ_{1} family of waves we need to increase the Reynolds number a bit more than the value at which the γ_{2} family of waves is found. The spatiotemporal evolution of the nonlinear surface equation indicates that different kinds of finite-amplitude permanent waves exist.