Rigid-body velocity in a deforming swimmer frame ($\mathbf{u}=\mathbf{U}+\mathbf{\Omega }\times \mathbf{r}+{\partial }_t\mathbf{r}$)

A material point at body-frame position $\mathbf{r}$ has velocity relative to stationary ambient fluid $\mathbf{U}+\mathbf{\Omega }\times \mathbf{r}+({\partial }_t\mathbf{r}{)}_{\mathrm{body}}$. The translational, rotational and deformation terms must all be retained before assigning perturbation orders.