Stability of compressible flow in real media

项目负责人Kevin Zumbrun，印地安那大学dms - 0300487摘要首席研究员建议研究“真实”介质中可压缩流动的稳定性，这些流动具有粘度、热传导、电磁动力学、相变、非热平衡和化学反应等经常被忽视的影响，在物理上有趣（通常是大振幅）的状态下，可能会发生向不稳定的转变。例如，强激波和爆震波或经典剪切流的多维稳定性。这涉及奇异摄动理论、动力系统和分岔、线性算子的谱理论和非线性偏微分方程中有趣的和非标准的问题，并且应该导致新的通用数学工具的发展。最终的物理目标是对稳定性现象的理解，这种理解比在简化模型中获得的更完整、更精确：一方面在数学基础层面解决哲学难题，另一方面在实际应用层面产生定量预测。在粘性激波锋稳定性研究中，以埃文斯函数为中心的攻击计划和相关的谱技术是近年来发展起来的。规则流型的稳定性是流体、气体和等离子体动力学中一个古老而核心的话题，它决定了哪些（稳定的）流型将被典型地观察到，哪些（不稳定的）流型只是数学上而不是物理上可观察到的解决方案。从稳定到不稳定的转变是特别重要的，因为它通常标志着替代的出现，更复杂的流动模式接近原始的（现在不稳定的）——这是通过研究更简单和更好理解的流动来理解复杂流动的一种方法。尽管在这个问题上有大量著名的理论，可以追溯到19世纪后期，但仍然有许多方面没有得到很好的理解，特别是压缩流、粘性流或反应流。在这里，我们建议研究在可压缩气体和等离子体动力学中出现的几个问题，以及在燃烧应用中，这些通常被忽视的效应具有相当大的实际重要性。我们的目标是，通过包括这些数学上有问题的术语，将现有的理论从定性转移到定量制度，在我们推进数学理论的同时获得对实践者有用的新信息。计划中的活动既有分析的部分，也有数值的部分，并涉及与国内外同事、现任和前任研究生和博士后的合作。这可能会加强和扩展跨领域和机构的现有合作网络，并有助于培养研究生和博士后。这些研究的最终目的，即定量预测过渡到不稳定的过程，如果能够实现，将在工程、化学、制造和其他过程中直接和实际地使用。

PI: Kevin Zumbrun, Indiana UniversityDMS-0300487ABSTRACTThe principal investigator proposes to study stability of compressible flows in ``real'' media featuring often-neglected effects such as viscosity, heat conduction, electromagneticdynamics, phase-transition, non-thermoequilibrium, and chemical reaction, in the physically interesting (usually large-amplitude) regime where transition to instability may be expected to occur: for example, multidimensional stability of strong shock and detonation waves, or of classical shear flows. This involves interesting and nonstandard issues in singular perturbation theory, dynamical systems and bifurcation, spectral theory of linear operators, and nonlinear partial differential equations, and should result in the development of new mathematical tools of general application. The ultimate physical goal is an understanding of stability phenomena that is both more complete and more precise than can be obtained within simplified models: on the one hand resolving philosophical puzzles at the level of mathematical foundations and on the other yielding quantitative predictions at the level of practical application. The plan of attack centers around Evans function and related spectral techniques developed recently in the study of stability of viscous shock fronts.The stability of regular flow patterns is an old and central topic in fluid, gas, and plasma dynamics, deciding which (stable) patterns will typically be observed, and which (unstable) are only mathematical and not physically observable solutions. The transition from stability to instability is of particular importance, since it usually signals the arising of alternative, more complicated flow patterns close to the original (now unstable) one- this is a way to understand complicated flows by the study of simpler and better-understood ones. Despite a large and well-known body of theory on this subject, dating back to the late 1800's, there are still many aspects that are poorly understood, particularly for compressive, viscous, or reacting flows. Here, we propose to study several of these issues arising in compressible gas and plasma dynamics, and in combustion, applications in which such usually neglected effects are of considerable practical importance. Our goal is, by including these mathematically problematic terms, to move existing theory from the qualitative to the quantitative regime, obtaining new information of use to practitioners at the same time that we advance the mathematical theory. The planned activities have both analytic and numerical components, and involve collaboration with domestic and foreign colleagues and with current and former graduate students and post doctorates. This may be expected to strengthen and extend existing networks of cooperation across field and institution, and to aid in training of graduate and postdoctoral students. The ultimate aim of these investigations, of quantitative predictions of transition to instability, would, if achieved, be of direct and practical use at the level of engineering, in chemical, manufacturing, and other processes.