Vorticity and angular momentum conservation

Vorticity is defined as the curl of a velocity field $\mathbf{u}$, expressed as $\boldsymbol{\omega} = \nabla \times \mathbf{u}$ Verified Answer #2. This vector characterizes the local rotational state of fluid elements Verified Answer #2. A non-vanishing curl at a specific point indicates that a fluid parcel is undergoing local rotation Verified Answer #2. The vorticity vector is mathematically related to the local angular velocity $\boldsymbol{\Omega}$ of an infinitesimal fluid parcel by the relationship $\boldsymbol{\omega} = 2\boldsymbol{\Omega}$ Verified Answer #1.

Kinematics of Curl

The local rotational state of a fluid can be understood by decomposing the velocity gradient tensor $\nabla \mathbf{u}$ into symmetric and anti-symmetric components Verified Answer #1.

The curl of the velocity field corresponds directly to the dual vector of the anti-symmetric rotation tensor Verified Answer #1. This relationship dictates that a fluid element undergoes a local spin equal to half the magnitude of the curl, aligned with the axis of the curl Verified Answer #1.

Conservation of Angular Momentum

In a continuum fluid, the conservation of angular momentum is expressed through local stress symmetries and dynamic transport equations Verified Answer #1. For classical Newtonian fluids without internal couple forces, this conservation requires the Cauchy stress tensor to be symmetric Verified Answer #1.

At the local level, the conservation of angular momentum is described by the vorticity transport equation: $$\frac{D\boldsymbol{\omega}}{Dt} = \frac{\partial \boldsymbol{\omega}}{\partial t} + (\mathbf{u} \cdot \nabla)\boldsymbol{\omega} = (\boldsymbol{\omega} \cdot \nabla)\mathbf{u} + \nu \nabla^2 \boldsymbol{\omega}$$ Verified Answer #2.

Vortex Stretching and Tilting

The term $(\boldsymbol{\omega} \cdot \nabla)\mathbf{u}$ represents vortex stretching and tilting, which is central to the conservation of angular momentum Verified Answer #2. In three-dimensional flows, stretching a fluid filament via velocity gradients decreases its moment of inertia Verified Answer #2. To conserve angular momentum, the local spin magnitude $|\boldsymbol{\omega}|$ must increase Verified Answer #2. This mechanism is the fluid-dynamical analog of an ice skater pulling in their arms to spin faster Verified Answer #2.

The Lamb Vector

The Lamb vector is defined as $\mathbf{L} = \mathbf{u} \times \boldsymbol{\omega}$ Verified Answer #2. It acts as a "vortex force" density within the fluid Verified Answer #2. In various flow configurations, such as wake flows or vortex rings, the Lamb vector governs the interaction between rotational structures and the surrounding mean flow Verified Answer #2. This provides a link between the geometry of the curl and the dynamical forces in the system Verified Answer #2.