The year has been divided into three segments, with a total of nine workshops. In each case, we aim to bring together researchers with overlapping interests who may move in disjoint scientific circles, and expose applied mathematicians to activity in the selected area of the workshop. The overall focus will be on identifying situations where an infusion of existing mathematical technology can lead to rapid progress, as well as recognizing areas where the existing theoretical framework needs to be.
Fall Quarter (September - December, 1999)
Although combustion has a long history and the subject is of great economical and technological importance, its emergence as a science is of relatively recent origin. Applied mathematicians "discovered" Combustion only about thirty years ago, and the scientific, analytical and computational challenges of the field have now made combustion an important application area of applied mathematics.
Fluid mechanics, chemical reaction, and thermodynamics are the essential ingredients of combustion. The governing equations are conservation laws of compressible flow, augmented by chemical kinetics. Nonlinearities and disparate scales abound. Even in the simplest of combustible mixtures, there is a bewildering array of chemical reactions among myriad species. When multiple phases are encountered, there is a lack of consensus even on the governing equations. It is clear, then, that the field continues to Offer major opportunities to the modeller, analyst, and numericist alike.
The aim of the quarter of concentration on combustion is to focus on three specific contexts in the field of combustion (each a workshop topic), to review recent successes, and to acquaint the participant with the challenges that remain.
Winter Quarter(January - March, 2000)
Natural Resources and Environment
This term will focus on the increasingly important role that reaction-transport processes play in the recovery of natural resources and in the confinement and remediation of environmental hazards. Geochemical phenomena which are essential to the understanding of the processes involved in enhanced oil recovery are also critical to developing effective strategies for the bioremediation of petroleum wastes. In both cases the effects of the chemical reactions enter the mathematical model not only as lower order terms in the reaction-transport equation but also can be nonlinearly coupled to the transport through concentration-dependent diffusion and permeability coefficients. Mechanical effects are also becoming important in these processes. During this term we will study the phenomena which arise from the full coupling of these reaction, transport and mechanical effects. We shall also be interested in studying them in the context of "real life", strategically important processes such as enhanced oil recovery, location of mineral deposits, chemical and nuclear waste repository dynamics, bioremediation of petroleum wastes, spread of pollutants in the atmosphere and lakes, etc. Experts in the modeling, analysis and numerical simulation of such processes will be brought together with experimentalists and participants from industry during the period.
Spring Quarter (April - June, 2000)
Multiscale and Transition Regimes
This term will focus on modeling processes for which transport is one of the most complicated components. This includes processes that include length scales that range from the order of mean-free paths to many times over different spatial-temporal regions of the problem, thereby requiring different transport models in each region for effective modeling. In some cases the underlying kinetic description is understood, such as the Boltzmann equation for rarified gases, or the transport equation for radiation. In such cases the main issue is one of economy, a fully resolved kinetic simulation being impractical, and one therefore develops homogenization, stochastic, or moment based subgrid models. Such is the focus of two of the workshops: "Model Hierarchies for the Evolution of Surfaces under Chemically Reacting Flows" and "Tr nsport Phenomena in Transition Regimes." In other cases there is considerable disagreement about the underlying kinetic description, especially when dispersive effects become macroscopic, for example due to quantum effects in semiconductors and superfluids. These disagreements are the focus of the workshop: "Dispersive Corrections to Transport Equations".