Dr. Andreas Muench
Technische Universität München
The formation of non-classical shock profiles in Marangoni-driven thin film flow, and their interaction with the meniscus
Abstract: We consider an experiment (carried out by AM Cazabat, X. Fanton, et. al., at the Colége de France), where a thin film of silicon oil climbs out of a reservoir up a tilted wafer, due to the action of a thermally induced Marangoni shear stress. The film is connected to the reservoir by a capillary meniscus. At the moving contact-line, one observes, for vertical or fairly vertical positions of the wafer, a fingering instability, which is absent if the wafer is tilted to more horizontal positions.We first investigate a lubrication model describing the evolution of the thin film above the meniscus, which turns out to be a conservation law with high order smoothing terms, and a non-convex flux function. We show, by a combination of numerical simulations and direct investigation of the third order ODE for the traveling wave profiles, that this equation permits classical shock profiles (consistent with Lax' criterion), as well as non-classical undercompressive profiles, for slightly thicker films. We can then explain the unusual transition in the dynamical behaviour of the thin film, using e.g. our results on the different stability properties of the two types of waves at the front of the film.
The film thickness, which plays an important role in determining which shocks form, is typically fixed by the capillary meniscus. If time permits, we will derive an asymptotic formula for the film thickness, via matched asymptotics, and also discuss other types of interaction between the waves in the thin film and the meniscus.
| Zeit: | Freitag, 25. Januar 2002, 14.15 Uhr |
| Ort: | FU Berlin, Arnimallee 2-6, Raum 032 im EG |