Hydrodynamic centre of a two-sphere dumbbell (${\mathbf{r}}_c=(a_1{\mathbf{r}}_1+a_2{\mathbf{r}}_2)/(a_1+a_2)$)

Two widely separated spheres rigidly joined by a hydrodynamically negligible connector obey the free-draining approximation. Their translational hydrodynamic resistance matrix is $6\pi \mu (a_1+a_2)I$. The displayed point is the centre about which every pure translation has zero hydrodynamic torque: ${\sum }_i{\zeta }_i({\mathbf{r}}_i-{\mathbf{r}}_c)=0$. A pure rotation about that point gives no resultant force at this order. This is an exact cancellation within the approximation, rather than an exact formula at arbitrary finite separation. It does not assert that every chiral rigid body has an origin eliminating its translation-rotation coupling.