= Rotne--Prager mobility
{c}
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 $\widehat{\mathbf R}=\mathbf R/R$. The cross-mobility, defined by the contribution $\mathbf U_1=\mathsf M_{12}\mathbf F_2$ to sphere one's translation, is
$$
\mathsf M_{12}=\frac{1}{6\pi\mu a}\left[
\frac{3a}{4R}(I+\widehat{\mathbf R}\widehat{\mathbf R})
+\frac{a^3}{2R^3}(I-3\widehat{\mathbf R}\widehat{\mathbf R})\right].
$$
The 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.
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