Principal coordinates and principal velocity gradient tensor decomposition

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.