Stationary waves on an inclined sheet of viscous fluid at high Reynolds and moderate Weber numbers

Abstract

A theory is described for the nonlinear waves on the surface of a thin film flowing down an inclined plane. Attention is focused on stationary waves of finite amplitude and long wavelength at high Reynolds numbers and moderate Weber numbers. Based on asymptotic equations accurate to the second order in the depth-to-wavelength ratio, a third-order dynamical system is obtained after changing to the frame of reference moving at the wave propagation speed. By examining the fixed-point stability of the dynamical system, parametric regimes of heteroclinc orbits and Hopf bifurcations are delineated. Extensive numerical experiments guided by the linear analyses reveal a variety of bifurcation scenarios as the phase speed deviates from the Hopf-bifurcation thresholds. These include homoclinic bifurcations which lead to homoclinic orbits corresponding to well separated solitary waves with one or several humps, some of which occur after passing through chaotic zones generated by period-doublings. There are also cases where chaos is the ultimate state following cascades of period-doublings, as well as cases where only limit cycles prevail. The dependence of bifurcation scenarios on the inclination angle, and Weber and Reynolds numbers is summarized.