Translating sphere in Stokes flow
= Translating sphere in Stokes flow
A sphere of radius $a$ translating with velocity $\mathbf U$ through otherwise stationary fluid has
$$
\mathbf u=\frac{3a}{4r}\left(\mathbf I+\frac{\mathbf x\mathbf x}{r^2}\right)\mathbf U
+\frac{a^3}{4r^3}\left(\mathbf I-3\frac{\mathbf x\mathbf x}{r^2}\right)\mathbf U,
\qquad
p=\frac{3\mu a}{2r^3}\mathbf U\mathbin\cdot\mathbf x.
$$
Its surface traction integrates to the <Stokes drag law>.