= Sphere in a uniform straining Stokes flow
Let the ambient velocity be $\mathbf E\mathbf x$, with $\mathbf E$ symmetric and trace free, and centre a rigid sphere of radius $a$ at the origin. It is <force-free> and <torque-free>, with zero translation and rotation. Its disturbance is
$$
\mathbf u'=-\frac{a^5}{r^5}\mathbf E\mathbf x
-\frac{5a^3}{2r^5}\left(1-\frac{a^2}{r^2}\right)
(\mathbf E:\mathbf x\mathbf x)\mathbf x,
\qquad p'=-\frac{5\mu a^3}{r^5}(\mathbf E:\mathbf x\mathbf x).
$$
The total velocity vanishes at $r=a$, and the leading disturbance is a <stresslet>. Harmonic dipole and quadrupole potentials in the <Papkovich–Neuber representation> give the complete field.
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