Research on Mathematical Problems in the dynamics of thin films of viscous liquids and the evolution of fluid interfaces

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UCLA-IPAM-NSF workshop on Thin Films and Fluid Interfaces: Jan 30-Feb 2, 2006

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Project Summary

We are pursuing a combined experimental, analytical, and computational study of fundamental problems in the dynamics of thin viscous films and fluid interfaces. This research program includes related studies of solid-liquid-vapor interfaces, moving contact lines, and surface tension effects. Experiments will be directly integrated with analytical and computational studies in this project. of behavior in moving contact lines. Our models for these phenomena are given by nonlinear partial differential equations, whose analysis and numerical computation present a rich variety of challenging problems. Several specific problems arising from earlier work on undercompressive waves and rupturing films will also be pursued. The research program will drive curriculum developments in the Departments of Mathematics, Physics and the Center for Nonlinear and Complex Systems at Duke University.

Thin films research

The contact line, or triple juncture, is the point on a fluid-fluid interface where the interface meets a solid boundary. Properties of the behavior of the contact line play an important role in the evolution of the entire fluid. When the system is at rest the local interfacial energies determine the preferred state of the system according to Young's law. However, when the contact line is in motion, as in the case of a wetting fluid, energy from the bulk of the fluid dissipates at the contact line, in a fashion that is not very well understood. The microscopic physics at the contact line greatly influences large scale properties of the flow and plays a leading role in the dynamics of front propagation and fingering of driven films.

For thin film flows, the lubrication approximation simplifies bulk flow fluid dynamics to a single equation, relating the depth-averaged horizontal fluid velocity to the shape of the gas-liquid interface. The lubrication approximation can be derived from an asymptotic expansion of the Navier Stokes equations with small Capillary number Ca and Reynolds number Re. The resulting equations take the general form of a nonlinear fourth-order degenerate PDE for the film height h as a function of space and time.  
 \begin{displaymath} h_t + (f(h))_x = -\nabla\cdot(m(h)\nabla[\nabla^2 h + P(h)]).\end{displaymath} (1)
In the formulation above, the convective term f(h) includes any directed driving forces such as gravity or Marangoni stress. The right hand side includes fourth-order diffusion from surface tension and second-order diffusion from physical effects like the normal component of gravity to the solid surface. A recent review article in the Notices of the AMS discusses some of the basic mathematical aspects of (1) in the context of modeling moving contact lines.

Our research group has produced many papers related to this project including papers on the mathematical analysis of thin film equations, numerical analysis of thin film equations, nonlinear dynamics of undercompressive waves, experiments involving undercompressive waves in thin films, self-similarity and singularities and rupture of films, and undergraduate reports and papers.

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