A novel analytical formulation for obtaining solitary wave solutions with a focus on gravity-driven liquid films

A potential-based, first-integral formulation of the Navier–Stokes equations is used as the platform for modelling solitary waves in the context of gravity-driven film flow on planar surfaces, resulting in a nonlinear partial differential equation (PDE), 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, 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 the corresponding pronounced hump-shaped, free-surface displacement typically observed in related experiments and consistent with other analytical/numerical results in the literature. Despite the numerous approximations involved in arriving at such a minimalist model, the same is capable of providing valid predictions apropos solitary waves; namely, regarding the existence of multiple solutions for a given film thickness and effective inclination angle, with different size and propagation speed, as demonstrated by the parameter study undertaken. Another remarkable feature is the paired occurrence of solitary wave and anti-wave solutions.