基于曲面主方向的正交系的非完整基理论及其在流体力学中的相关应用

Theory of nonholonomic basis of orthogonal coordinate system based on principal directions of surface and related applications in fluid mechanics

本文将微分几何中光滑曲面上局部存在的基于主方向的正交系联系于张量分析中的非完整基理论,以此为曲面与其邻域上的张量场场论提供了一种新方法,称为基于曲面主方向的正交系的非完整基理论。这种场论方法,基于基面的沿主方向的参数坐标,然后沿基面的法方向进行空间延拓,以此获得基面邻域内的完整的正交系。对此引入非完整基理论,可得所有的非零Christoffel符号直接对应为基面或者当地曲面的主曲率或者测地曲率,因而张量场的各种微分算子相对于曲面的主方向与法方向展开,所得分量表达式仅含物理量与曲面曲率。由此不仅可以清晰地展现曲面几何特征与物理量/物理过程之间的关系,而且所获得的分量表达式形式上最为简单。另一方面,经典的诸如柱坐标系、球坐标系等正交系也隶属基于曲面主方向的正交系,从而本文方法可以统一相关正交系下的张量场场论。作为应用,本文推导了可变形曲面上涡量、涡量法向梯度与变形率张量的表达式,曲面上流体边界层方程的分量方程,曲面介质相关守恒律方程等。

The locally existed orthogonal parameter coordinates of smooth surface in differential geometry has been related to nonholonomic basis in tensor analysis.This provides a new method for tensor field theory on surfaces and their neighborhoods termed as the nonholonomic basis theory based on the orthogonal system of the principal directions of the surface.This field theory method is based on the parameter coordinates of the base surface with respect to the principle directions,and then performs spatial extension in the normal direction of the base surface to obtain a complete orthogonal system in the base surface neighborhood.Introducing the nonholonomic basis theory,it can be obtained that all non-zero Christoffel symbols directly correspond to the principal curvatures or geodesic curvatures of the base surface or the local surface.Consequently,the resulting components of all kinds of differential operators of any tensor field with respect to the principle directions and the normal direction contains only the physical quantities and the curvatures of the surface.This not only clearly shows the relationship between the geometric characteristics of the surface and the physical quantity/physical process,but also the component expression obtained is the simplest in form.On the other hand,classical orthogonal systems such as cylindrical coordinate system and spherical coordinate system also belong to the orthogonal system based on the principal directions of the surface,so the method in this paper can unify the tensor field analysis of the related orthogonal systems.As an application,the expressions of vorticity,normal gradient of vorticity and deformation rate tensor on deformable surfaces,component equations of fluid boundary layer equations on surfaces,and related conservation law equations for curved media are derived.