Traveling waves on a falling weakly viscoelastic fluid film

The weighted residual integral method is employed to investigate the flow of a thin layer of Walters-type B″ viscoelastic fluid flowing down an inclined plane. A simplified second-order two-equation model is derived; the model is analogous to the simplified model proposed by Ruyer-Quil and Manneville [Ruyer-Quil, C., & Manneville, P. (2000). Improved modeling of flows down inclined planes. European Physical Journal B: Condensed Matter and Complex Systems, 15, 357–369] for Newtonian fluid. The normal mode analysis is used to investigate the linear stability of the Nusselt’s flow and the correct critical condition for linear stability was found. The results of linear analysis indicate that the viscoelastic parameter, Γ, destabilizes the film flow as its magnitude increases. The two-equation model is used to investigate the particular case of traveling waves. The result is that the model exhibits bifurcation scenarios such heteroclinic, homoclinic, Hopf and period-doubling bifurcations. The influence of viscoelastic parameter on the nonlinear development of these traveling waves is discussed.