Papkovich–Neuber representation

Every homogeneous Stokes flow can be represented using a harmonic vector field $\mathbf{\Phi }$ and harmonic scalar $\chi$ as

After rescaling the harmonic potentials, the Papkovich–Neuber representation may be written

For $\Phi =U+\alpha aU/r$ and $\chi =\beta a^{3}U\cdot \nabla (1/r)$ in the Unscaled Papkovich–Neuber representation, the Newtonian fluid stress tensor gives the displayed traction at $r=a$. The $\alpha$ contribution is the Stokeslet stress; the $\beta$ contribution is twice the viscosity times the Hessian of its harmonic scalar potential. The prefactor is $6\mu /a$, not $12\mu /a$. The rigid translating-sphere limit is a direct normalization check.