Principal Decomposition of Velocity Gradient Tensor in the Cartesian Coordinates

TL;DR

This paper introduces a Galilean invariant principal decomposition of the velocity gradient tensor directly in Cartesian coordinates, distinguishing rigid rotation, pure shear, and stretching with clear physical meanings, improving fluid kinematics analysis.

Contribution

It derives a new principal decomposition method in the original Cartesian coordinates, overcoming limitations of traditional methods and providing a more physically meaningful analysis tool.

Findings

Decomposition in original xyz coordinates is Galilean invariant.

The method clearly separates rotation, shear, and stretching.

Provides a foundation for advanced fluid kinematics analysis.

Abstract

Traditional Cauchy-Stokes decomposition of velocity gradient tensor gives a symmetric and an anti-symmetric subtensors which are called the strain-rate and vorticity tensors. There are two problems with Cauchy-Stokes decomposition. The first one is that the anti-symmetric or vorticity tensor cannot represent the fluid rotation or vortex. The second one is that the symmetric (strain-rate) tensor cannot distinguish the stretching (compression) and shear. The stretching and shear are dependent on the coordinate or are not Galilean invariant. Since vorticity cannot distinguish between the non-rotational shear and the rigid rotation, vorticity has been decomposed to a rigid rotation called Liutex and anti-symmetric shear in our previous work. A Liutex-based principal coordinate was developed and the velocity gradient tensor was decomposed in the principal coordinate as a rigid rotation (Liutex tensor), a pure shear tensor and a stretching (compression) tensor, which is called the principal decomposition. However, the principal decomposition is made at each point which has own principal coordinate different from other points. This paper derives the principal decomposition in the original xyz coordinate system, and, therefore, provides a new tool for fluid kinematics to conduct the velocity gradient tensor decomposition to rigid rotation, pure shear, and stretching (compression) which is Galilean invariant and has clear physical meanings. The new velocity gradient tensor decomposition could become a foundation for new fluid kinematics.