The Objective Deformation Component of a Velocity Field

For an arbitrary velocity field $\mathbf{v}$ defined on a finite, fixed spatial domain, we find the closest rigid-body velocity field ${\mathbf{v}}_{RB}$ to $\mathbf{v}$ in the $L^2$ norm. The resulting deformation velocity component, ${\mathbf{v}}_d={\mathbf{v}-\mathbf{v}}_{RB}$, turns out to be frame-indifferent and physically observable. Specifically, if ${\mathbf{Q}}_{\text{RB}}(t)$ is the rotation tensor describing the motion of the closest rigid body frame, then $\mathbf{v}$ is seen as ${\mathbf{Q}}_{\text{RB}}^T{\mathbf{v}}_d$ by an observer in that frame. As a consequence, the momentum, energy, vorticity, enstrophy, and helicity of the flow all become frame-indifferent when computed from the deformation velocity component ${\mathbf{v}}_d$.