A Minimalist Model for the Capture and Investigation of Solitary Waves in Gravity–Driven Liquid Films

ABSTRACT

A potential–based, first–integral formulation of the Navier–Stokes equations is used as a platform for modelling solitary waves in gravity‐driven film flow, giving rise to a nonlinear partial differential equation in complex form. Its subsequent modal decomposition—leading to an infinite set of ordinary differential equations (ODEs), followed by a bimodal approximation and elimination of unknowns—finally results in just two coupled ODEs to be solved. The latter is achieved both analytically and numerically using a Ritz–Galerkin methodology. Illustrative solitary wave results are provided, which exhibit the expected internal recirculating flow pattern and corresponding pronounced, “hump”–like, free–surface deviation—typically observed experimentally and consistent with other analytical/numerical results in the literature. Despite the numerous simplifying assumptions involved in arriving at such a minimalist model, the parameter study undertaken shows that it is capable—based on the number of test functions employed—of providing valid predictions apropos solitary waves; namely, regarding the existence of multiple solutions, having different size and propagation speed, for a given film thickness and effective inclination angle.