Sphere in a uniform straining Stokes flow

Let the ambient velocity be $\mathbf{Ex}$, 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 isThe 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.

With symmetric traceless strain $\mathbf{E}$, these harmonic potentials solve the decaying disturbance around a stationary sphere in the convention $\mathbf{u}=\mathbf{\Phi }-\frac{1}{2}\nabla (\mathbf{x}\cdot \mathbf{\Phi }+\chi )$ and $p=-\mu \nabla \cdot \mathbf{\Phi }$. The vector potential supplies the leading stresslet; the scalar degree-two potential corrects no slip at finite radius. Matching the independent $\mathbf{Ex}$ and $(\mathbf{x}\cdot \mathbf{Ex})\mathbf{x}$ structures fixes both coefficients.