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

0104350Siegel本建议涉及流体力学和材料科学问题中移动边界的分析和数值计算。提出的问题是由不同的应用程序驱动的，但从数学角度来看却非常相似。一类问题涉及通过伸展流动中液滴和气泡的尖端流动而变形和破裂。尖端流动是指气泡形成具有尖端的变形形状，从中将细丝或小气泡排放到外部流体中。实验表明，表面活性剂产生的界面张力梯度对于这种不稳定性的开始是很重要的。PI应继续发展含有表面活性剂的细长气泡变形和演化的分析理论，重点研究可溶表面活性剂的作用及其在尖端流动中的影响。分析中的数学困难与自由表面上表面活性物质“停滞帽”处边界条件的跳跃有关。用来解决这些困难的技术包括奇异摄动理论、奇异积分方程组理论和Riemann-Hilbert问题，以及运动边界问题的复变量理论。这一理论将得到精确的数值计算的补充。PI还建议研究纤维涂层的简单模型中的尖端流动和空气夹带，以及尖端流动的形成和可能的尖端流动的模拟，这可能发生在应力固体中空洞的扩散控制演化过程中。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，如水中波的演化、火焰锋面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具有重要意义的基本移动边界问题。提出的问题中的一个共同特征是，运动界面会产生非常小的特征，如尖端或非常细的细丝，这些特征会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在液浴中高速拉动来对材料(如光纤)进行涂层。涂布过程中产生的空气长丝可能会折断，留下空洞或其他污点，使涂布掺杂。第二个应用是材料的失效。当材料受到压力时，材料中存在的小孔会演化(通过原子扩散)。气孔中的尖端发展可能引发裂纹或位错(原子的错位)；这已被认为是微电子电路故障的一个重要原因。其他应用包括在多组分流体系统中形成和混合乳状液。

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.