Analysis of Multidimensional Problems for Nonclassical Shocks and Classical Shock Reflection

Cuny College Of Staten Island, Staten Island NY

Investigators

Abstract

A shock wave is a propagating disturbance in a liquid, gas, or plasma (a fluid in physics terminology) that is characterized by an abrupt change in fluid properties, such as density or pressure, across the wave front. Shock waves are ubiquitous in nature, from quantum levels to cosmological scales. They occur in a great variety of physically important phenomena, such as the air flow around an airliner's wing, flows of dense granular materials, fluids undergoing liquid-solid or vapor-liquid phase transitions, magneto-hydrodynamics, nonlinear elasticity, and many other important situations. The mathematical theory describing realistic multi-dimensional dynamics for these phenomena is under-developed. This project will focus on several important problems whose solution holds the promise of elucidating general phenomena. In particular, the project will consider the glancing shock reflection problem (that originated from the problem initially proposed by John von Neumann), formation of shocks in supersonic regimes, and others. The project will bring high-performance computing to bear on the understanding of these problems; the numerical solutions obtained have the potential to lead to advances in the theory. This project addresses the mathematical modeling and analysis of a number of multi-dimensional Riemann problems whose solutions contain classical or nonclassical shocks. A common theme of much of the work in this project is the formulation of problems using asymptotic model equations that retain the essential physics present in the underlying governing equations. These asymptotic models can then be solved numerically, enabling vastly greater numerical resolution to be achieved in regions where the solutions exhibit small-scale features. The project addresses fundamental questions in compressible fluid flow and nonlinear hyperbolic systems, including those systems that admit small-scale dependent shocks (examples include models for magneto-hydrodynamics, phase transitions, complex fluids, etc.). The mathematical theory of the Euler equations of gas dynamics, and more generally of hyperbolic conservation laws, is largely incomplete in more than one space dimension, and the project will attack a number of multidimensional problems that are likely to be amenable to solution. Numerical solutions of these problems have the potential to lead to the development of the theory and resolve various paradoxes in the existing theories.