Title: Single-PDE models of air-driven film flow in a vertical tube
Abstract: Films flowing over a flat surface are often modeled using long-wave asymptotics, where a small ratio of film thickness to wavelength is exploited to derive a single PDE governing the evolution of the free surface. Films which flow over a cylindrical surface such as a tube offer a second natural choice of lengthscale ratio, that of small film thickness to tube radius. Such thin-film models offer simpler nonlinearities and fewer parameters which govern the dynamics, but lose some qualitative information about the flow arising from the geometry of the problem. I will give an overview of a series of long-wave models and their thin-film counterparts developed for the particular problem of a film which coats the interior of a tube and is driven by some combination of gravity, airflow, and capillary forces. A comparison with experiments shows long-wave models are successful in quantitatively capturing the main features of these flows, while their thin-film counterparts exhibit significant differences in wave amplitude and speed, and the streamline topology of the underlying fluid flow. We will end the talk by showing recent improvements on a long-wave model for a film driven up a tube by steady airflow. This is joint work with Roberto Camassa and Jeffrey Olander.