Mathematical Sciences: Research in Nonlinear Problems Arising in Mechanics

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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. ***