Nonlinear waves and conservative transport in soluble-surfactant falling films on compliant substrates

Abstract

We investigate finite-amplitude waves in a liquid film containing a soluble surfactant as it flows down an inclined compliant substrate. The coupled motion of the liquid and a spring-backed plate is described by a conservative weighted-residual model, which also accounts for surfactant transport in the bulk and at the free surface. The reduction retains viscous terms arising from wall motion, and its pressure closure recovers fluid added mass under uniform translation. Comparison with the coupled Orr-Sommerfeld problem gives a difference of approximately $1\%$ in the peak growth rate of the film mode at $Re=25$. Nonlinear computations are then used to examine travelling states and their dependence on substrate properties and surfactant coupling. Along the prescribed stiffness-damping path, substrate displacement decreases by more than four orders of magnitude, while wave speed changes by less than $1\%$. The extrema of free-surface elevation and liquid thickness vary differently, so surface height alone does not reliably indicate film thinning over a deformable support. In the cases studied, Marangoni coupling reduces surfactant localisation and weakens the wave and substrate response. Moving-frame reconstruction reveals recirculation near the wave crest and distinguishes propagation of the deformation pattern from transverse substrate motion. The load-work budget identifies capillary pressure as the principal net source of substrate work for the two compliant supports considered. A control calculation in which stiffness is varied at fixed damping shows a substantial change in damping loss but a comparatively small change in deformation.