Papkovich potentials for a strained sphere ($\mathbf{\Phi }=-5a^3\mathbf{Ex}/(3r^3),\chi =a^5(\mathbf{x}\cdot \mathbf{Ex})/r^5$)

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.