A consistent energy integral model for a film over a substrate featuring topographies

Abstract

A depth‐averaged model, based on energy integral method (EIM) has been employed to capture the steady‐state free surface profiles in a gravity‐driven thin film flow of a viscous, incompressible fluid over inclined substrates with topographical features. The model incorporates inertial effects, energy balance and is consistent up to O(ε), where ε≪1, is the film thickness parameter. The governing equations are solved using a meshless numerical scheme based on radial basis functions. The predictions of the effects of inertia, inclination of the substrate, and aspect ratio of the topographical features are compared with the available finite‐element solutions and experimental observations. The steady‐state profiles generated using the EIM equations are in close agreement with the experimental observations. Being a consistent model, EIM model predictions are expected to be more reliable and accurate than the results based on the depth averaged form of the momentum equations, which are known to be not consistent. This feature is the motivating factor for the present work which is also a first attempt in implementing the meshless numerical scheme to free surface problems. The analysis paves a way for similar investigations on three‐dimensional flows related to film flow over chemically patterned substrates.