Vorticity and angular momentum conservation

Kinematics of Curl and Local Rotation

In fluid dynamics, the curl of a velocity field $\mathbf{u}$ is defined as the vorticity vector $\boldsymbol{\omega} = \nabla \times \mathbf{u}$ Verified Answer #2Verified Answer #3. This vector serves as the fundamental descriptor of a fluid's local rotational state Verified Answer #1. A non-vanishing curl at a specific point indicates that a fluid parcel is undergoing local rotation Verified Answer #2. The fluid parcel rotates at an instantaneous angular velocity $\boldsymbol{\Omega}$ equal to half the vorticity, expressed as $\boldsymbol{\Omega} = \frac{1}{2}\boldsymbol{\omega}$ Verified Answer #2Verified Answer #3.

The velocity gradient tensor $\nabla \mathbf{u}$ can be decomposed into a symmetric strain-rate tensor $\mathbf{S}$, representing deformation, and an anti-symmetric rotation tensor $\boldsymbol{\Omega}_{rot}$, representing rigid-body-like rotation Verified Answer #1. The vorticity vector corresponds directly to the dual vector of this anti-symmetric rotation tensor Verified Answer #1.

Conservation of Angular Momentum

The conservation of angular momentum in a fluid system is governed by the vorticity transport equation Verified Answer #2. This equation, derived from the Navier-Stokes equations, describes how vorticity is transported, tilted, and stretched by flow velocity gradients Verified Answer #2Verified Answer #3.

Vortex Stretching and Tilting

The term $(\boldsymbol{\omega} \cdot \nabla)\mathbf{u}$ represents the stretching and tilting of vorticity by velocity gradients Verified Answer #2. This mechanism is the fluid-dynamical analogy of a figure skater conserving angular momentum Verified Answer #2. When a fluid filament is stretched, its moment of inertia decreases, which forces its local spin or vorticity to increase to maintain angular momentum Verified Answer #2Verified Answer #3.

The Lamb Vector

The Lamb vector, or vortex force, is defined as $\mathbf{L} = \mathbf{u} \times \boldsymbol{\omega}$ Verified Answer #2Verified Answer #3. By rewriting the inertial term of the Navier-Stokes equations, the Lamb vector is revealed to act as a local force density that accelerates the fluid Verified Answer #2. This force governs the interaction between rotational structures, such as vortex rings or wake flows, and the surrounding mean flow Verified Answer #2Verified Answer #3.

Dynamics and Boundary Interactions

Vorticity is not only redistributed through transport but is also created at solid boundaries Verified Answer #2. In Newtonian fluids without internal couple forces, the conservation of angular momentum requires the Cauchy stress tensor to be symmetric Verified Answer #1. The evolution of the curl within a system provides a basis for understanding the mechanics of energy dissipation and the development of angular momentum over time Verified Answer #2Verified Answer #3.