Helmholtz velocity decomposition and Cauchy-Stokes tensor decomposition have been widely accepted as the foundation of fluid kinematics for a long time. However, there are some problems with these decompositions which cannot be ignored. Firstly, Cauchy-Stokes decomposition itself is not Galilean invariant which means under different coordinates, the stretching (compression) and deformation are quite different. Another problem is that the anti-symmetric part of the velocity gradient tensor is not the proper quantity to represent fluid rotation. To show these two drawbacks, two counterexamples are given in this paper. Then “principal coordinate” and “principal decomposition” are introduced to solve the problems of Helmholtz decomposition. An easy way is given to find the Principal decomposition which has the property of Galilean invariance.

中文翻译：

---

主坐标和主速度梯度张量分解

长期以来，亥姆霍兹速度分解和柯西-斯托克斯张量分解已被广泛接受为流体运动学的基础。但是，这些分解存在一些不可忽视的问题。首先，柯西-斯托克斯分解本身不是伽利略不变的，这意味着在不同坐标下，拉伸（压缩）和变形是完全不同的。另一个问题是速度梯度张量的反对称部分不是代表流体旋转的适当量。为了显示这两个缺点，本文给出了两个反例。然后引入“主坐标”和“主分解”来解决亥姆霍兹分解的问题。给出了一种简单的方法来找到具有伽利略不变性的主分解。