 Rotne--Prager mobility

The Rotne--Prager mobility approximates the hydrodynamic coupling between two equal spheres of radius $a$ in an unbounded fluid of dynamic viscosity $\mu$ at negligible Reynolds number. For centre separation $R$ large compared with $a$, put $\hat{\mathbf{R}}=\mathbf{R}/R$. The cross-mobility, defined by the contribution ${\mathbf{U}}_1={\mathsf{M}}_{12}{\mathbf{F}}_2$ to sphere one's translation, isThe prefactor is the isolated-sphere translational mobility. The leading term is the Stokeslet interaction; the second accounts for finite sphere size. Combining the source-sphere correction with Faxén's first law for the receiving sphere gives this correction. The hydrodynamic mobility matrix maps applied force to velocity, so omitting the prefactor loses both its physical normalization and units. This far-separated pair approximation needs further treatment for near-contact or overlapping particle models.