有限变形理论的若干进展及其在流体力学中的相关应用

Some Developments of Finite Deformation Theories with Some Applications in Fluid Mechanics

概要性地叙述了作者新近提出的"当前物理构型对应之曲线坐标系显含时间的有限变形理论"、"几何形态为曲面的连续介质的有限变形理论",前者针对介质几何形态为Euclid流形(体积形态),后者针对Riemann流形(曲面形态).类比于一般有限变形理论,上述理论均包括物理及参数构型构造,变形梯度定义及其基本性质,变形刻画,输运定理以及守恒律方程.基于上述理论提出对应曲线坐标系显含时间的流函数涡量解法,固定曲面上二维不可压缩流动的流函数涡量解法以及海面油污扩散控制方程,并给出了相关数值研究结果.

Some recent developments of finite deformation theories termed as ＂finite deformation theory with respect to curvilinear coordinates corresponding to current physical configurations including time explicitly＂ and ＂finite deformation theory with respect to continuous mediums whose geometrical configurations are two dimensional surfaces＂ are narrated. The former and the latter theories correspond to continuous mediums whose geometrical configurations are Euclidian manifolds （bulk status） and Riemannian manifolds （surface status）, respectively. As compared to general theories, both newly developed theories include constructions of physical and parametric configurations, definition of deformation gradient tensor with its primary properties, deformation descriptions, transport theories and governing equations of conservation laws. As applications, stream function vorticity algorithm with respect to curvilinear coordinates including time explicitly, stream function ga vorticity algorithm for two dimensional incompressible flows on fixed surfaces, and governing equations for oil diffusion on sea surfaces are presented with some results of tentative numerical studies.