基于广义流体密度和Reynolds输运定理的一类新守恒律
New Classes of Conservation Laws Based on Generalized Fluid Densities and Reynolds Transport Theorems
摘要：
雷诺输运定理在流体动力学中占有中心地位，它为流体中任何守恒量的输运提供了一个广义积分守恒方程，并与其相应的微分方程相联系。最近，一个新的广义框架提出了这个定理，使参数变换的位置之间的流形或广义坐标空间，利用底层的多元李对称性与守恒量。我们研究的影响，这个框架的流体流动系统，在欧拉位置速度（相空间）的描述。该分析调用了五个概率密度函数的层次结构，通过卷积来定义五个流体密度和适用于不同空间的广义密度。我们得到11个配方的广义雷诺输运定理的不同选择的坐标空间，参数空间和密度，只有第一个是已知的。这些被用来生成11个表格的积分和微分守恒定律适用于这些系统，为8个重要的守恒量（流体质量，物种质量，线性动量，角动量，能量，电荷，熵和概率）。这些大大扩展了流体流动和动力系统分析的守恒定律。
英文摘要：
The Reynolds transport theorem occupies a central place in fluid dynamics, providing a generalized integral conservation equation for the transport of any conserved quantity within a fluid, and connected to its corresponding differential equation. Recently, a new generalized framework was presented for this theorem, enabling parametric transformations between positions on a manifold or in a generalized coordinate space, exploiting the underlying multivariate Lie symmetries associated with a conserved quantity. We examine the implications of this framework for fluid flow systems, within an Eulerian position-velocity (phase space) description. The analysis invokes a hierarchy of five probability density functions, which by convolution are used to define five fluid densities and generalized densities appropriate for different spaces. We obtain 11 formulations of the generalized Reynolds transport theorem for different choices of the coordinate space, parameter space and density, only the first of which is known. These are used to generate 11 tables of integral and differential conservation laws applicable to these systems, for eight important conserved quantities (fluid mass, species mass, linear momentum, angular momentum, energy, charge, entropy and probability). These substantially expand the set of conservation laws for the analysis of fluid flow and dynamical systems.
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