A general kinematic theory of fluid-element rotation and intrinsic vorticity decompositions

The present study establishes a general theory for fluid-element rotation and intrinsic vorticity decompositions within the framework of vorticity kinematics. We propose two direction-dependent vorticity decompositions (DVDs) based on the analysis of rotation of directed material line and surface elements, with the rigid-rotation and spin modes of vorticity being explicitly defined. Intrinsic coupling relations are then derived for a pair of orthogonal line and surface elements, demonstrating their complementary roles in both kinematics and geometry. Notably, the surface-element-based spin mode is shown to coincide with the relative vorticity in the generalized Caswell formula, thereby providing a faithful representation of surface shear stress in Newtonian fluids. Correspondingly, another two DVDs are constructed based on the geometry of streamlines and streamsurfaces in the field description. Furthermore, within the characteristic algebraic description, in terms of the rotational invariants $(\psi ,\gamma )$ in the real Schur form of the velocity gradient tensor, two invariant vorticity decompositions (IVDs) are formulated. The first IVD with positive spin aligns with the Liutex-shear decomposition, which corresponds to the Klein–Kaden–Betz (KKB) mechanism by which wrapping shear layers form axial vortices. The second IVD is indispensable for understanding unidirectional swirling motion around a point on the rotation-axis-normal plane ${\mathcal{P}}$ , corresponding to an anti-KKB mechanism/phenomenon characterized by the negative spin. Importantly, it is proved that the DVD vorticity modes are rigorously bounded by the IVD vorticity modes $(R_{N}^{\pm },s_{N}^{\pm })=(2\psi ^{\pm },\gamma ^{\pm })$ on ${\mathcal{P}}$ . Finally, distinctive features and applicability of these kinematic tools are demonstrated with representative examples. The results indicate that a coupled IVD–DVD approach provides a powerful diagnostic tool for unravelling the subtle structures and fundamental physics inherent to complex flow fields.