Mathematical Sciences: Nonlinear Waves, Nonlinear Materials and Chaotic Mixing

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9500568 Grove This project seeks to develop a fundamental approach to the study of mixing layers, nonlinear materials, and fluid chaos. Nonlinear waves in hyperbolic conservation laws commonly display sensitive dependence to the problem formulation and to numerical solution algorithms. A completed theory of a hyperbolic wave should contain the following ingredients: (1) jump conditions describing the influence of the wave on the flow in which it is embedded, (2) the width or growth rate of the layer, (3) a complete theory of the internal structure, and (4) a systematic unfolding of all possible sensitive dependencies for the wave. The mathematical tools and theories used in the analysis are varied, and include: partial differential equations (hyperbolic conservation laws), random fields, perturbation theory (ordinary and renormalized), renormalization group methods, traveling wave analysis, bifurcation theory, and the geometric theory of ordinary differential equations. A representative mixing layer problem is acceleration driven layers, arising in instabilities of a fluid interface. Such mixing layers arise in many areas of basic science and technology, including supernovae, inertial confinement fusion, and injection jets in carburetors for internal combustion engines. Detailed mathematical modeling and analysis will seek to modify and refine the defining mathematical equations describing such flows and develop an improved understanding of the structure of these equations. This analysis will in turn be used to develop high resolution numerical methods for the solution of these equations. The nonlinear material wave patterns we study describe important metal forming processes such as punching and cutting (shear bands) and flow instabilities in forming plastic components by injection molding (viscoelastic materials). The analysis of waves in nonlinear materials uses a similar integrated approach: modeling, theor y, computations, and applications. Interactions with collaborators will allow transfer of this technology to appropriate applied physics and engineering communities.