Mathematical Sciences: Computations in Fluids and Materials

小行星9707494 这些项目涉及复杂流体中的动力学和图案形成，以及牛顿流体中的奇异性形成和拓扑转变。 第一个项目考虑细长弹性细丝的流体动力学，如液晶流动中出现的，磷脂双层管的动力学，以及生物聚合物的动力学。 建立易处理的计算模型，占流体动力学相互作用的细丝本身，依赖于有区别地利用细长。 这仍然给出了一个计算密集的问题，高阶时间步长的弹性约束，奇异核的相互作用积分，积分方程在每个时间步长求解。 第二个项目继续对不混溶液体之间的流体/流体界面的拓扑转变的理解。 基本问题是：表面张力如何引起或介导转变？ 奇点的特征是什么？ 需要添加什么物理来跟踪过渡？ 奇点之后还剩下什么？ 基于以前的工作，在不混溶流体之间的开尔文-亥姆霍兹不稳定性的这种奇异性，建议研究，计算和分析，奇异性和过渡射流分离不混溶的流体，无论是通过使用尖锐的界面模型，和流体模型，具有粘度，并允许一定的粘性。 第三个项目研究剪切稀化的影响，这是许多非牛顿流体和液晶流共有的一个特性，对Saffman-Taylor不稳定性的发展。 一些建模工作已经完成，产生了一个自然的非牛顿版本的达西定律，有关流体速度的非线性椭圆问题的解决方案。 建议现在模拟这种气泡膨胀成剪切稀化液体的完全非线性动力学。 这是一个非常具有挑战性的计算问题，因为它涉及到一个不断发展的区域上的非线性椭圆问题的解决方案。 流体和材料的许多基本动力学-奇异性和图案形成是两个主要例子-将通过从数学建模到发展计算方法和相关数学理解，再到通过高性能计算进行大规模模拟和数据分析的过程来理解。 这里要进行的三个项目都位于流体动力学和材料科学的交叉点，都说明了上述声明。 在第一个项目中，它被提议理解和模拟晶体结构的动力学，如液晶流体的相变，磷酸脂双层管的动力学，以及生物聚合物的动力学。 在第一个实例中，这种长丝在高强度长丝的制造中具有潜在的技术重要性。 第二个项目继续从理论上理解是什么驱使射流分解成液滴，进入第二种不同的流体（比如油和水）。 虽然观察起来很容易且常见，但这种行为与表面张力密切相关，这种效应仍然不为人所知，但却是许多基本流体现象的核心。 这将通过建模，分析和大规模计算相结合的方式进行研究。 最后一个项目涉及剪切稀化液体在薄间隙中流动的动力学。 这种流动对于显示装置设计和注射成型是重要的。 特别令人感兴趣的是与气/液界面相关的不稳定性和图案形成，气/液界面由表面张力驱动但介导。 这是一个极具挑战性的计算问题，需要开发新的模拟方法。

9707494 Michael Shelley These projects concern dynamics and pattern formation in complex fluids, and singularity formation and topological transitions in Newtonian fluids. The first project considers the hydrodynamics of slender elastic filaments, such as arise in liquid crystal flows, the dynamics of phospho-lipid bilayer tubes, and in the dynamics of biological polymers. Building tractable computational models, that account for hydrodynamic interactions of the filament with itself, relies on discriminating exploitation of slenderness. This still gives a computationally intensive problem with high-order time-step constraints from elasticity, interaction integrals with singular kernels, and integral equations to be solved at every time-step. The second project continues towards an understanding of topological transitions of fluid/fluid interfaces between immiscible liquids. The fundamental questions are: How does surface tension provoke or mediate transitions? What characterizes the singularity? What physics needs to be added to follow the transition? And what is left of the singularity in its aftermath? Building upon previous work on such singularities in the Kelvin-Helmholtz instability between immiscible fluids, it is proposed to study, computationally and analytically, singularities and transitions in jets that separate immiscible fluids, both by using sharp interface models, and fluid models that have viscosity and allow some miscibility. The third project studies the effect of shear-thinning, a property shared with many non-Newtonian fluids and liquid crystal flows, on the development of the Saffman-Taylor instability. Some of the modelling work has already been done, yielding a natural non-Newtonian version of Darcy's law, relating the fluid velocity to the solution of a nonlinear elliptic problem. It is proposed to now simulate the full nonlinear dynamics of such a bubble expanding into a shear-thinning liquid. This is a very challenging comp utational problem as it involves the solution of nonlinear elliptic problems on an evolving domain. Much of the fundamental dynamics of fluids and materials -- singularity and pattern formation are two central examples -- will be understood by a progression from mathematical modelling, to developing computational methods and relevant mathematical understanding, and thence to large-scale simulation and data analysis through high-performance computing. The three projects to be pursued here all lie at the intersection of fluid dynamics and materials science, and all illustrate the above statement. In the first project, it is proposed to understand and simulate the dynamics of filamentary structures, as arise in phase transitions of liquid crystalline fluids, the dynamics of phospho-lipid bilayer tubes, and in the dynamics of biological polymers. In first example, such filaments are of potential technological importance in the manufacture of high-strength filaments. The second project continues towards a theoretical understanding of what drives the break-up into droplets of a jet of fluid into a second, different fluid (say, oil and water). While easy and common to observe, such behavior is strongly associated with surface tension, an effect that is still ill-understood, and yet lies at the heart of much basic fluid phenomena. This will be studied by a combination of modelling, analysis, and large-scale computation. The final project concerns the dynamics of shear-thinning liquids flowing in thin gaps. Such flows are important to display device design, and to injection molding. Of particular interest is the instability and pattern formation associated with a gas/liquid interface which is driven but mediated by surface tension. This is an extremely challenging computational problem, requiring the development of new simulational methods.