Mathematical Sciences: Research in Nonlinear Problems Arising in Mechanics

这个项目需要对表现相变化的“类气”系统的动力学模型和宏观流体动力学进行研究。这项研究将有助于理解液-气相变的基本原理，分析液-气界面动力学的计算方法，以及颗粒材料的“类气体”建模。研究中使用的工具将是稠密气体动力学理论中的数学思想、渐近分析和单复变量理论中的基本思想（如Yang-Lee相变理论所需要的）。然后，该项目将使用这些工具开发包括亚稳态在内的液-汽系统的相关宏观流动方程，并分析这种流动。此外，在颗粒流的研究中，该项目将发展理解快速颗粒流中剪切带形成和弹性崩塌问题所需的渐近思想。该项目将提供几个重要技术领域的知识。例如，颗粒物料快速流动的建模在散装物料（煤、谷物、药品）的转移中很重要。液-气相变动力学的新成果将为研究核反应堆中气溶胶、水下爆炸和两相流体流动的行为提供新的数学和计算技术。另一个可能的好处是，在研究液体-蒸汽流动中获得的想法也可以转移到固体的相变中，例如，现代“形状记忆”材料的建模。＊＊＊

9531845 Slemrod The project entails investigation into kinetic models and macroscopic hydrodynamics of "gas like" systems which exhibits changes of phase. The investigation will yield understanding of fundamental principles of liquid-vapor phase transitions, computational methods for analysis of liquid-vapor interface dynamics, and with respect to "gas like" modeling of granular materials. The tools used in the investigation will be mathematical ideas in the kinetic theory of dense gases, asymptotic analysis, and fundamental ideas in the theory of one complex variable (as needed for the Yang-Lee theory of phase transitions). The project will then use these tools to develop the relevant macroscopic flow equations for liquid-vapor systems including metastable states and analyze such flows. In addition, in the study of the granular flows the project will develop the asymptotic ideas necessary to understand the issues of shear band formation and elastic collapse in rapid granular flow. %%% The project will provide knowledge in several technologically important fields. For example, modeling of rapid flow of granular materials is important in the transfer of bulk materials (coal, cereals, pharmaceuticals). New results in dynamics of liquid-vapor phase transitions will provide new mathematical and computation techniques for the studying the behavior aerosols, underwater detonations, and two phase fluid flow in nuclear reactors. Another possible benefit is that ideas obtained in studying liquid-vapor flow may be carried over to phase transitions in solids as well, e.g. modeling of modern "shape memory" materials. ***