Asymptotic Analysis of Thin Film Flows on Curved Surfaces

Abstract

This paper presents an in-depth asymptotic analysis of thin film flows on curved surfaces, focusing on both spherical and cylindrical geometries. The study begins with a review of the mathematical preliminaries, including differential geometry, the governing Navier-Stokes equations for thin films, and asymptotic analysis techniques. The thin film flow problem is formulated on general curved surfaces, with assumptions and simplifications such as the lubrication approximation. The governing equations are then non-dimensionalized and analyzed using perturbation methods to obtain leading-order solutions. The paper includes case studies on thin film flows on spherical and cylindrical surfaces, using hypothetical data to demonstrate the application of the theoretical models. The leading-order solutions reveal significant insights into the effects of curvature on thin film dynamics, with numerical methods used to validate and compare the results. The findings highlight the critical role of curvature in determining film thickness and pressure distribution, with practical implications for applications in coating processes, biological systems, and environmental engineering. The study concludes with a discussion on the contributions to the field of thin film flows, including the development of analytical and numerical frameworks and the practical insights gained from the case studies. Future research directions are suggested, focusing on higher-order asymptotic analysis, complex geometries, non-Newtonian fluids, experimental validation, and application-specific studies.