Analysis and numerical computations of moving boundaries in fluid dynamics and materials science

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0104350SiegelThis proposal is concerned with the analysis and numerical computation of moving boundaries in problems from fluid dynamics and materials science. The proposed problems are motivated by different applications but are remarkably similar from a mathematical point of view. One class of problems concerns the deformation and breakup via tip streaming of drops and bubbles in extensional flows. Tip streaming refers to the phenomenon where a bubble develops a deformed shape featuring cusplike ends, from which fine filaments or small bubbles are emitted into the exterior fluid. Experiments suggest that interfacial tension gradients such as those produced by surfactants are important for the onset of this instability. The PI shall continue his development of an analytical theory for the deformation and evolution of slender bubbles with surfactant, focusing on the effects of soluble surfactant and its influence in tip streaming. Mathematical difficulties in the analysis are associated with a jump in boundary conditions at "stagnant caps" of surfactant on the free surface. Techniques employed to address these difficulties include singular perturbation theory, theory of singular integral equations and Riemann-Hilbert problems, and complex variable theory in moving boundary problems. The theory will be complemented by accurate numerical computations. The PI also proposes to investigate tip streaming and air entrainment in simple models of fiber coating, as well as cusp formation and a possible analogue of tip streaming which may occur during the diffusion controlled evolution of voids in a stressed solid. The mathematical statements of these free boundary problems are similar, and the PI expects that there will be a certain synergism between them.Moving boundary problems, such as the evolution of waves on water, the propagation of flame fronts, or the growth of crystals, continue to challenge applied scientists and engineers. The PI proposes to study a class of fundamental moving boundary problems that are important in technological applications. A common feature among the proposed problems is that the moving interface develops very small-scale features, such as cusps or very fine filaments, which can then greatly influence the properties of the material or fluid. One application of the proposed work is in the coating of materials (such as optical fibers) by pulling at high speed through a liquid bath. Air filaments produced during the coating process can snap off, leaving voids or other blemishes that adulterate the coating. A second application is in the failure of materials. Small pores present in materials evolve (via atomic diffusion) when the material is stressed. Cusp development in the pores can initiate cracks or dislocations (misalignment of atoms); this has been implicated as a prominent cause of failure in microelectronic circuits. Other applications include emulsion formation and mixing in multi-component fluid systems.