Thin-Film Flows: Classical Examples, Marangoni Motions, and Viscous Membranes

These notes summarize five 45 min lectures the author presented at CISM in June 2023. The topics are centered around the theme of “thin fluid films,” which constitutes an area of (mostly low-Reynolds-number) fluid dynamics with wide applicability. It is also a set of topics where nonlinearity is common, yet analytical results, either in the form of scaling laws and/or the reduction of partial differential equations to ordinary differential equations, are possible. The lectures seek to highlight this intersection of physical problems, scaling laws, analyses, including similarity solutions and detailed results, spanning traditional coating flows and surfactant-mediated dynamics, as well as thin-film descriptions common to dynamics of cellular membranes, which gives a link to biophysics. To start the article, we survey a few problems where surface tension is important and where dimensional analysis yields insights and quantitative results. Then, in turn, we analyze the differential equations and boundary conditions that describe physically common thin-film flows, with emphasis on analytical insights and the steps towards developing, where possible, similarity solutions. The motion of a particle in a viscous membrane constitutes the last lecture. In preparing the notes the author filled in various steps and other explanations that time did not allow for during the actual lectures.