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

## TL;DR

This paper develops a comprehensive kinematic theory for fluid vorticity and rotation, introducing new decompositions and relations that improve understanding of vortex dynamics in various flows.

## Contribution

It introduces a unified framework combining algebraic and geometric approaches to decompose vorticity and velocity gradient tensors, advancing vortex analysis.

## Key findings

- Derived intrinsic coupling relations for orthogonal line-surface elements.
- Proved the spin mode equals the relative vorticity in the generalized Caswell formula.
- Validated the theory across diverse flow scenarios.

## Abstract

The present study proposes a general kinematic theory for fluid-element rotation and intrinsic vorticity decompositions within the context of vorticity and vortex dynamics. Both the angular velocities of material line and surface elements comprise a classical contribution driven by volume-element rotation (equal to half the local vorticity), and a strain-rate-induced specific angular velocity. Then, two direction-dependent vorticity decompositions (DVDs) are constructed, revealing the rigid rotation and spin modes of vorticity. We derive intrinsic coupling relations for orthogonal line-surface element pair, elucidating their complementary kinematic and geometric roles. Notably, we rigorously prove that the spin mode (in the surface-element-based DVD) is identical to the relative vorticity in the generalized Caswell formula, thereby faithfully accounting for surface shear stress in Newtonian fluids. Next, within a field-theoretic framework, vorticity decompositions are proposed based on streamline and streamsurface using differential geometry. The physical roles of the six rotational invariants in the characteristic algebraic description (including the the normal-nilpotent decomposition (NND) of the velocity gradient tensor (VGT) and the resulting invariant vorticity decomposition (IVD)) are clarified through unified analysis with the DVD, Caswell formula, and Helmholtz-Hodge decomposition. ... It is found that physical admissible DVD vorticity modes must be bounded by IVD modes in phase space, whereas enforcing a minimization principle naturally yields the Liutex formula. Finally, the effectiveness of theory is validated across diverse flows, from simple to complex. Results show that a coupled IVD-DVD analysis could enhance physical understanding of complex vortical flows under both algebraic and field-theoretic frameworks.