Translating sphere in Stokes flow

A sphere of radius $a$ translating with velocity $\mathbf{U}$ through otherwise stationary fluid hasIts surface traction integrates to the Stokes drag law.

A fixed sphere in a slowly varying incident Stokes flow requires an external force $\mathbf{F}=-6\pi \mu a{\mathbf{u}}_{\infty }$ and external couple $\mathbf{G}=-4\pi \mu a^3{\mathbf{\omega }}_{\infty }$ to leading order. These follow from Faxén translation law and Faxén rotation law; the incident Stokeslet supplies ${\mathbf{u}}_{\infty }$ and ${\mathbf{\omega }}_{\infty }$. Finite-radius Laplacian terms enter the translational force at the next nonzero order. The signs here refer to applied holding loads, not hydrodynamic loads.

For a sphere of radius $a$ in a slowly varying ambient Stokes flow ${\mathbf{u}}_{\infty }$, the translational force isevaluated at the sphere centre.

For two equal, well-separated spheres, the Rotne--Prager cross-mobility through order $a^3/R^3$ is proportional to

A passing forced sphere advects a distant force-free sphere along a curved trajectory. Its longitudinal displacement can diverge logarithmically even when its transverse displacement approaches a finite value.