Mathematical Sciences: Applied and Computational Mathematics and Experimental Analysis

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9403915 Bona This project comprises a broadly-based investigation of some fundamental aspects of mechanics, including the development of some of the relevant mathematical techniques and tools. Included within its auspices are theoretical and experimental studies of various fluid flows, studies of water waves and formation of sand structures in the near- shore zone of large bodies of water, and theoretical and experimental studies of liquid crystals. One of the larger investigations will be concerned with the modelling of water waves with an eye toward a better understanding of the interaction of waves travelling in opposite directions. Another interesting study is concerned with flows in which there is a balance between dissipative effects and surface tension. Still another sub-project will reconsider some classical flows such as flow past a sphere in a pipe, Taylor-Couette flow and the related issues of stability and vortex breakdown. The complex mathematical modelling needed to describe exotic materials like liquid crystals will also be high on the agenda. Allied to all of these investigations will be the development of specific theoretical results in partial differential equations and in numerical analysis. Especially important in the latter area will be work on highly efficient methods for the solution of large linear systems arising in the algorithms for the solution of complicated nonlinear differential equations. The resulting computer codes will be used in conjunction with experimental work on all the forementioned topics and the outcomes interpreted within the relevant theoretical framework. This work is exciting for several reasons. First, many of the issues being addressed are not only of fundamental theoretical importance, but they also have potential to provide new insights into engineering and technological design and processing. Secondly, because of the presence of the fluid mechanics laboratory sited within the mathematics department at Pe nn State, the group has the ability to combine theoretical, numerical and experimental work in a rarely realized close collaboration. In addition, some of the work on development of optimal numerical techniques will have impact in a much broader arena of science and engineering activity than just the confines of the present study.