Dynamics of a thin viscoelastic film on an inclined plane

Abstract

A theory for two-dimensional long and stationary waves of finite-amplitude on a thin viscoelastic fluid (weakly elastic) layer flowing down an inclined plane is investigated. A set of exact averaged equations for the viscoelastic film flow system is described and linearised stability analysis of the uniform flow is performed using normal-mode formulation and the critical condition for linear instability is obtained. The linearised instability for the permanent wave equation, consistent to the second order in ϵ ( ϵ = h ¯ 0 / L , h ¯ 0 – unperturbed film thickness, L – characteristic length) is examined and the eigenvalue properties of the fixed points are classified in various parametric regimes. The possible domains of heteroclinic orbits and the regions of possible nonlinear bifurcations are analysed for different values of viscoelastic parameter Γ. Numerical integration of the permanent wave equation as a third order dynamical system is carried out. While wave transitions in real life involve complex spatio-temporal dynamics and many of these transitions lead to chaotic waves that are not stationary traveling waves, bifurcation of stationary traveling waves has been examined as a preliminary study of the more complex transitions. Different bifurcation scenarios leading to multiple hump solitary waves or leading to chaos are exhibited in the parametric space. The results are compared and contrasted with the Newtonian results. A summary of the bifurcation scenarios in the We versus cot θ/ Re plane is obtained for different values of viscoelastic parameter Γ, when Re ≈ 13.33 and Re = 100.