Beyond Vorticity: An Angular Momentum Perspective on Fluid Flow

Abstract: While vorticity is the classical tool for analyzing rotational fluid kinematics, it inherently focuses on local, differential spin. This paper introduces a complementary framework based on the angular momentum density field, $\mathbf{L} = \mathbf{r} \times \mathbf{u}$, deriving generalized transport equations that explicitly balance macroscopic torque and rotational momentum. This $\mathbf{L}$ perspective offers several distinct theoretical advantages over traditional velocity/vorticity formulations. Specifically, this approach: (i) provides a novel decomposition of the viscous torque into a diffusive component and a local spin dissipative term; (ii) shows the mechanism by which lift is generated in viscous boundary layers by vorticity acting as a source of angular momentum; it also explains stall (iii) reformulates the hydrodynamic impulse to yield a remarkably clean separation of terms into dilatational, volumetric, and rotational flux components; The $\mathbf{L}$ formalism provides the kinematic closure necessary to unify non-circulatory added mass and circulatory lift within a single, dimensionally consistent budget. (iv) enables the direct calculation of the viscous added mass force, accounting for the inertial resistance of boundary layers and separated wakes; (v) simplifies geophysical fluid dynamics by absorbing the planet's rotation, traditionally treated as an artificial virtual vorticity term which directly gets absorbed into the conserved axial angular momentum $m$, revealing the fundamental physics of global circulation through explicit torque balances; (vi) identifies the rotlet as a fundamental Green's function for the $\mathbf{L}$ transport equation in the Stokes regime; and (vii) demonstrates that both oblique shocks and vortex sheets act as singular sources of $\mathbf{L}$ that turn the macroscopic flow.