Mathematical Sciences: Applied and Computational Mathematics and Experimental Analysis

小行星9403915 该项目包括对力学的一些基本方面进行广泛的调查，包括一些相关数学技术和工具的开发。其主持范围包括各种流体流动的理论和实验研究，大型水体近海岸区的水波和砂结构形成的研究，以及液晶的理论和实验研究。其中一个较大的调查将关注的水波建模，着眼于更好地理解波在相反方向上传播的相互作用。另一个有趣的研究涉及的是耗散效应和表面张力之间存在平衡的流动。另一个子项目将重新考虑一些经典的流动，如管道中的球体流动，泰勒-库埃特流动以及稳定性和涡破裂的相关问题。描述液晶等奇异材料所需的复杂数学模型也将是议程上的重点。结盟所有这些调查将是具体的理论成果在偏微分方程和数值分析的发展。特别重要的是，在后一个领域将工作的高效率的方法来解决大型线性系统所产生的算法，解决复杂的非线性微分方程。由此产生的计算机代码将与上述所有专题的实验工作以及在相关理论框架内解释的结果结合使用。 这项工作令人兴奋的原因有几个。首先，许多正在解决的问题不仅是基本的理论重要性，但他们也有可能提供新的见解，工程和技术设计和处理。其次，由于存在的流体力学实验室设在数学系在宾夕法尼亚州，该小组有能力联合收割机结合理论，数值和实验工作，很少实现密切合作。此外，一些最佳数值技术的发展工作将在更广泛的科学和工程活动的竞技场的影响，而不仅仅是本研究的范围。

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.