Beyond Vorticity: An Angular Momentum Perspective on Fluid Flow

TL;DR

This paper proposes an angular momentum density framework for fluid flow analysis, offering new insights and advantages over traditional vorticity-based methods, including better understanding of lift, torque, and inertial effects.

Contribution

It introduces a novel angular momentum perspective that unifies various fluid dynamics phenomena and provides new theoretical tools and decompositions.

Findings

Decomposes viscous torque into diffusive and spin dissipative parts.

Explains lift generation via angular momentum in boundary layers.

Enables direct calculation of viscous added mass force.

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.