Mathematical Sciences: Program in Computational and Applied Mathematics

9306720卡夫利施旋流和对流流体流动对于各种现象都很重要，例如流体混合、绕翼型和障碍物的流动以及湍流。(1)通过分叉分析和数值计算对轴对称旋转涡片的动力学、不稳定性和奇异性形成进行了研究，并将其应用于涡流破裂。(2)研究了具有随机性和时间依赖性的对流流动的有效扩散速率。(3)采用偏微分方程组理论、代数几何(突变理论)、符号计算和数值模拟相结合的方法研究偏微分方程组的奇异性。该方案的研究是为了对复杂的流体流动进行分析和数值计算，如在许多具有科学和技术意义的问题中出现的。对这些流动的有效理解和应用需要对其微观和宏观特征进行描述。这项研究首先将奇点作为理想流动中的一种细尺度现象加以研究。通过对微观变化的均匀化，得到了流动的宏观描述。这项研究的结果将对评估复杂流动的质量特征以及开发有效的计算方法具有重要意义。*9306720卡夫利施旋流和对流流体流动对于各种现象都很重要，例如流体混合、绕翼型和障碍物的流动以及湍流。(1)通过分叉分析和数值计算对轴对称旋转涡片的动力学、不稳定性和奇异性形成进行了研究，并将其应用于涡流破裂。(2)研究了具有随机性和时间依赖性的对流流动的有效扩散速率。(3)采用偏微分方程组理论、代数几何(突变理论)、符号计算和数值模拟相结合的方法研究偏微分方程组的奇异性。该方案的研究是为了对复杂的流体流动进行分析和数值计算，如在许多具有科学和技术意义的问题中出现的。对这些流动的有效理解和应用需要对其微观和宏观特征进行描述。这项研究首先将奇点作为理想流动中的一种细尺度现象加以研究。通过对微观变化的均匀化，得到了流动的宏观描述。这项研究的结果将对评估复杂流动的质量特征以及开发有效的计算方法具有重要意义。***

9306720 Caflisch Vortical and convective fluid flows are important for a wide range of phenomena, such as fluid mixing, flow around airfoils and obstacles, and turbulence. In this project the investigator undertakes research in three related topics: (1) Dynamics, instabilities and singularity formation for axisymmetric, swirling vortex sheets are studied through bifurcation analysis and numerical computation, with applications to vortex breakdown. (2) The effective diffusion rate for a convective flow is studied for flows with randomness and time-dependence. (3) Singularities are investigated for systems of PDE's through a combination of PDE theory, algebraic geometry (catastrophe theory), symbolic computation, and numerical simulation. The research in this proposal is for analysis and numerical computation of complex fluid flows, such as occur in many problems of scientific and technological importance. Effective understanding and application of these flows requires a description of both their microscopic and their macroscopic features. This investigation first focuses on singularities as a fine-scale phenomena in idealized flows. A macroscopic description of flows is also developed through homogenization of the microscopic variations. The results from this study will be important in assessing the qualitative features of complex flows,as well as in the development of effective computational methods. *** 9306720 Caflisch Vortical and convective fluid flows are important for a wide range of phenomena, such as fluid mixing, flow around airfoils and obstacles, and turbulence. In this project the investigator undertakes research in three related topics: (1) Dynamics, instabilities and singularity formation for axisymmetric, swirling vortex sheets are studied through bifurcation analysis and numerical computation, with applications to vortex breakdown. (2) The effective diffusion rate for a convective flow is studied for flows with randomness and time-dependence. (3) Singularities are investigated for systems of PDE's through a combination of PDE theory, algebraic geometry (catastrophe theory), symbolic computation, and numerical simulation. The research in this proposal is for analysis and numerical computation of complex fluid flows, such as occur in many problems of scientific and technological importance. Effective understanding and application of these flows requires a description of both their microscopic and their macroscopic features. This investigation first focuses on singularities as a fine-scale phenomena in idealized flows. A macroscopic description of flows is also developed through homogenization of the microscopic variations. The results from this study will be important in assessing the qualitative features of complex flows, as well as in the development of effective computational methods. ***