Uniform decomposition of velocity gradient tensor

In this paper, the principal decomposition of the velocity gradient tensor [∇v] is discussed in 3 cases based on the discriminant ∆: ∆ < 0 with 1 real eigen value and a pair of conjugate complex eigen values, ∆ > 0 with 3 distinct real eigen values, and ∆ = 0 with 1 or 2 distinct real eigen values. The velocity gradient tensor can also be classified as rotation point, which can be decomposed into three parts, i.e., rotation [R], shear [S] and stretching/compression [SC], and non-rotation point, we defined a new resistance term [L], and the tensor can be decomposed into three parts, i.e., resistance [L], shear [S] and stretching/compression [SC]. Example matric are also displayed to demonstrate the new decomposition. Connections of principal decomposition between 3 different cases, and between Resistance and Liutex will also be discussed.