Affine velocity field ($\mathbf{u}(\mathbf{x},t)=K(t)\mathbf{x}+\mathbf{b}(t)$)

A velocity field with spatially constant velocity gradient $K$. Differences of fluid velocities satisfy $\mathbf{u}(\mathbf{x}+\mathbf{R})-\mathbf{u}(\mathbf{x})=K\mathbf{R}$, even when the uniform translation $\mathbf{b}$ is nonzero. Incompressibility requires $\mathrm{tr}K=0$.

An affine velocity field with zero uniform translation. It describes the relative motion about a comoving reference point even when the original velocity field has a nonzero spatially uniform component.