Nonlinear Waves in Continuum Mechanics

这项研究的变形和流动的材料的数学是由应用程序的地下流动的流体，如地下水，石油和天然气，雪崩动力学，包括处理的粒状材料在滚筒和斜槽，微观液滴的运动，和冲击波的作用，在信号的传播，例如在光纤。连接这些应用的数学涉及用非线性方程对连续过程进行建模，以及对方程的波动解的研究。最近的实验和新模型为这些复杂系统的新见解开辟了机会。该研究项目将包括数学和物理方面的研究生培训，重点是在科学会议上清楚地介绍结果。这个研究项目的连续介质力学的非线性模型集中在四个领域，最近的发展提供了进步的机会。分散和耗散冲击波的调查将发生在磁化和多孔介质流动模型的背景下。新的两相流模型具有奇异的非线性，激发了分析和数值模拟的创新。软弹性基底上的液滴接触线的研究涉及非对称液滴模型的制定，可以通过变换和对偶积分方程的方法来解决。一种用于运动和控制微观液滴的固定机制是诱导基底弹性模量的空间变化，从而在液滴上提供驱动力。 颗粒流模型，特别是多分散粒径，将使用非线性偏微分方程理论进行分析。这些方程将结合最近对颗粒材料中应力分布的描述，再加上一个已建立的动力学机制，该机制量化了不同尺寸的颗粒如何在不同的层中积累。

This research into the mathematics of deformation and flow of materials is motivated by applications to subsurface flows of fluids such as groundwater, oil and gas, avalanche dynamics, including the processing of granular materials in tumblers and chutes, the motion of microscopic fluid droplets, and the role of shock waves in the propagation of signals, for example in optical fibers. The mathematics linking these applications concerns the modeling of continuous processes with nonlinear equations, and the study of wave-like solutions of the equations. Recent experiments and new models have opened up opportunities for fresh insights into these complex systems. The research project will include graduate student training in mathematics and physics, with an emphasis on clear presentation of results at scientific conferences. This research project on nonlinear models of continuum mechanics focuses on four areas where recent developments provide opportunities for progress. The investigation of dispersive and dissipative shock waves will take place in the context of models of magnetization and porous media flow. New models of two phase flow possess singular nonlinearities that stimulate innovations in analysis and numerical simulation. The study of contact lines of droplets on soft elastic substrates involves the formulation of models for asymmetric drops that can be solved by the method of transforms and dual integral equations. A conjectured mechanism for the motion and control of microscopic droplets is to induce spatial variations in the elastic modulus of the substrate, thereby providing a driving force on the droplet. Granular flow models, specifically with poly-disperse grain sizes, will be analyzed using the theory of nonlinear partial differential equations. The equations will incorporate recent descriptions of stress distribution in granular materials, coupled to an established dynamic mechanism that quantifies how grains of different sizes accumulate in distinct layers.