Meniscus solutions and shock dynamics for Marangoni-driven flows Submitted for Presentation in ECS2005 The European Coating Symposium 2005

We consider a situation where a thin liquid film is driven from a reservoir up a substrate by a thermally induced Marangoni shear stress and investigate the meniscus that connects the film with the reservoir, as well as the wave dynamics near the contact-line. We identify two types of meniscus solutions, and show, via a phase space investigation of the third order ODE that governs the profile, when these solutions appear. The first type fixes the film thickness, hence the flow rate, and exists only below a critical inclination angle (measured with respect to the vertical position). The second yields thicker films and does not meter the flow. The rich wave dynamics near the contact-line involve non-classical (undercompressive) shocks and have themselves been the focus of intensive investigations in recent years. We summarise these results to demonstrate how the interplay of meniscus solutions and front dynamics determines the profile of the liquid surface. The discussion of the meniscus solutions carries over to the drag-out problem, for which in particular the dependence on the inclination angle has, to the best of our knowledge, not yet been systematically investigated.