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

0104350 Siegel这一建议涉及流体动力学和材料科学问题中移动边界的分析和数值计算。所提出的问题是出于不同的应用程序，但从数学的角度来看是非常相似的。一类问题涉及拉伸流中液滴和气泡通过尖端流的变形和破裂。 尖端流是指气泡形成以尖状末端为特征的变形形状的现象，细丝或小气泡从尖状末端发射到外部流体中。实验表明，界面张力梯度，如那些由表面活性剂产生的是重要的，这种不稳定性的发病。PI应继续开发表面活性剂作用下细长气泡变形和演化的分析理论，重点关注可溶性表面活性剂的作用及其对尖端流的影响。分析中的数学困难与自由表面上的表面活性剂的“停滞帽”处的边界条件的跳跃有关。 解决这些困难的技术包括奇异摄动理论、奇异积分方程理论和Riemann-Hilbert问题，以及移动边界问题中的复变量理论。这一理论将得到精确的数值计算的补充。PI还建议调查尖端流和空气夹带在简单的模型中的纤维涂层，以及尖点的形成和尖端流的可能模拟过程中可能发生的扩散控制的演变过程中的空隙在一个强调固体。这些自由边界问题的数学表述是相似的，PI预计它们之间会有一定的协同作用。移动边界问题，如水面上波浪的演化、火焰前缘的传播或晶体的生长，继续挑战着应用科学家和工程师。PI建议研究一类在技术应用中非常重要的基本移动边界问题。 所提出的问题中的一个共同特征是移动界面发展出非常小尺度的特征，例如尖点或非常细的细丝，这可以极大地影响材料或流体的性质。所提出的工作的一个应用是通过高速拉过液浴来涂覆材料（如光纤）。在涂层过程中产生的空气丝可能会折断，留下空隙或其他瑕疵，使涂层掺假。第二个应用是材料的失效。 当材料受到应力时，材料中存在的小孔会演变（通过原子扩散）。孔隙中的尖端发展可以引发裂纹或位错（原子的错位）;这已经被认为是微电子电路故障的主要原因。其它应用包括在多组分流体系统中的乳液形成和混合。

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.