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Traditional Cauchy-Stokes decomposition of velocity gradient tensor gives asymmetric and an anti-symmetric subtensors which are called the strain-rate andvorticity tensors. There are two problems with Cauchy-Stokes decomposition. Thefirst one is that the anti-symmetric or vorticity tensor cannot represent thefluid rotation or vortex. The second one is that the symmetric (strain-rate)tensor cannot distinguish the stretching (compression) and shear. Thestretching and shear are dependent on the coordinate or are not Galileaninvariant. Since vorticity cannot distinguish between the non-rotational shearand the rigid rotation, vorticity has been decomposed to a rigid rotationcalled Liutex and anti-symmetric shear in our previous work. A Liutex-basedprincipal coordinate was developed and the velocity gradient tensor wasdecomposed in the principal coordinate as a rigid rotation (Liutex tensor), apure shear tensor and a stretching (compression) tensor, which is called theprincipal decomposition. However, the principal decomposition is made at eachpoint which has own principal coordinate different from other points. Thispaper derives the principal decomposition in the original xyz coordinatesystem, and, therefore, provides a new tool for fluid kinematics to conduct thevelocity gradient tensor decomposition to rigid rotation, pure shear, andstretching (compression) which is Galilean invariant and has clear physicalmeanings. The new velocity gradient tensor decomposition could become afoundation for new fluid kinematics.

Principal Decomposition of Velocity Gradient Tensor in the Cartesian Coordinates