vorticity-and-angular-momentum-conservation

In fluid dynamics, vorticity is defined as the curl of a 3D velocity field $\mathbf{u}$, represented by the vector $\boldsymbol{\omega} = \nabla \times \mathbf{u}$ Verified Answer #1. This vector serves as the fundamental descriptor of the local rotational state within a fluid Verified Answer #1. The relationship between curl, angular momentum, and energy dissipation involves a complex interplay of kinematics, local conservation laws, and global thermodynamics Verified Answer #1.

Kinematics of Curl and Local Rotation

The local rotational state of a fluid is analyzed by decomposing the velocity gradient tensor $\nabla \mathbf{u}$ into two distinct components Verified Answer #1:

• The symmetric strain-rate tensor $\mathbf{S}$, which represents fluid deformation Verified Answer #1.
• The anti-symmetric rotation tensor $\boldsymbol{\Omega}_{rot}$, which represents pure rigid-body-like rotation Verified Answer #1.

The vorticity vector $\boldsymbol{\omega}$ is the dual vector of the anti-symmetric rotation tensor Verified Answer #1. It is mathematically related to the local angular velocity vector $\boldsymbol{\Omega}$ of an infinitesimal fluid parcel by the equation $\boldsymbol{\omega} = 2\boldsymbol{\Omega}$ Verified Answer #1. Consequently, any point with a non-vanishing curl forces a fluid element to undergo a local spin equal to half the magnitude of that curl, aligned with the curl's axis Verified Answer #1.

Conservation of Angular Momentum

In a continuum fluid, the conservation of angular momentum is manifested through dynamic transport equations and local stress symmetries Verified Answer #1. For classical Newtonian fluids that lack internal couple forces, the conservation of angular momentum specifically requires that the Cauchy stress tensor remains symmetric Verified Answer #1.