Mathematical Sciences: Hydrodynamic Interface Motion

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9409484 Bertozzi The dynamics of a thin viscous film on a solid surface are modeled by a fourth order nonlinear diffusion equation for the film thickness (Greenspan, J. of Fluid Mech. 1978, vol. 84, p. 125). The project involves three aspects of this problem, singularity formation, weak solutions, and pattern formation. A singularity occurs when the thickness of the film goes to zero. Physically the singularity corresponds to a topological transition in the fluid. The study will be made via a combination of numerics and analysis the onset of such singularity formation forth order degenerate diffusion equations. This work may include collaborations with M. Brenner, L. Kadanoff, T. Dupont, and A. Bernoff. The investigator is working with Mary Pugh at the Courant Institute on the weak solution problem. This collaboration recently established sharper existence and long time behavior results for weak solutions to the thin film equation. The existence is in a regularity class that just includes a family of unique `source type' solutions analogous to the `Barenblatt' solutions of the porous media equation. Preliminary numerical computations show that in general weak solutions converge rapidly onto the `source type' solution. It is planned to complete the numerics as well as to relate these weak solutions to slip models' suggested for the spreading drop problem. Lastly, in a joint project with M. Brenner, problems in pattern formation such a gravity and temperature driven fingering instabilities in thin films will be studied. ***