Therefore, we can find different flow deformations by calculating the velocity gradient tensor and then decomposing it into symmetric and antisymmetric parts, then we have the following for flow deformations

• Rotation: Off-diagonal elements of the antisymmetric part \( \Omega_{ij} \). Note that in 2D, antisymmetric tensor has only one independent component (the off-diagonal elements are just negative of each other), which corresponds to the vorticity in 2D. In 3D there are 3 independent components corresponding to the three components of the vorticity \( \omega_x \), \( \omega_y \), and \( \omega_z \)
• Dilation/Compression: Sum of the diagonal elements of the symmetric part \( E_{ij} \)
• Angular deformation: Off-diagonal elements of the symmetric part \( E_{ij} \)