Mathematical Sciences: Computational Two-Phase Viscous Drop Spreading

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9623092 Braun Motivated by the complex transport associated with spreading process such as the spreading of solder droplets and coating processes, a research program is proposed on the spreading of a droplet that is covered by a film. Often in soldering and other processes involving spreading, the drop may become very thin which indicates the presence of multiple length scales. Based on the author's previous research, the first approach proposed is a multi-scale asymptotic analysis (lubrication theory) to derive nonlinear diffusion equations for the free surfaces; those nonlinear diffusion equations will be solved using a spectrally-accurate implementation of the method of lines. All quantities of interest may be computed in this approximation once the required free surfaces have been found. A second approach is also proposed, via boundary integral equations, under the approximation of Stokes flow in the drop and film. This approach relaxes the requirement that the drop and film be thin, but still requires that the flow be slow. We propose making a direct comparison between the two approaches in order to evaluate the utility of each in various situations. The work proposed will develop models only for the fluid dynamics, and inclusion of reactive effects will be proposed in a subsequent project. Important processes in manufacturing involve the spreading of a fluid droplet in the presence of another fluid; one example is soldering, where, in many instances, the presence of a thin layer of a second fluid, the flux, is required for the process to work at all. Because the use of lead-based solder is no longer allowed, understanding of this droplet spreading process for other kind of solder materials would be very helpful in understanding alternative solder materials. The work proposed in the project is developing some preliminary models necessary for the development of a complete model for a spreading solder drop. The proposed program is to develop theory for the spreading of a droplet on a flat plate that is covered by a thin film of a different fluid. Computer codes will be developed that predict how fast the drop spreads, and the shapes of the drop and the film that covers the drop. In summary, sophisticated mathematical techniques will be applied to a problem of technological importance. The computer codes resulting from this project are expected to be applicable to related wetting problems in different technological areas (e.g. coating processes) subsequent to the completion of this project.