Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 74 / 2 / a

One normalization of the Papkovich–Neuber representation uses a harmonic vector $\mathbf{\Phi }$ and harmonic scalar $\psi$:These formulae give $\nabla \cdot \mathbf{u}=0$ because ${\nabla }^2(\mathbf{x}\cdot \mathbf{\Phi })=2\nabla \cdot \mathbf{\Phi }$, and give $\mu {\nabla }^2\mathbf{u}=\nabla p$. Rescaling the potentials gives the other common normalizations of the representation.

A point force requires a decaying, rotationally covariant field linear in $\mathbf{F}$, with no length scale. Thus choose the harmonic monopole $\mathbf{\Phi }=c\mathbf{F}/r$, $\psi =0$, for $r>0$. Differentiating givesThe Newtonian fluid stress tensor is ${\sigma }_{ij}=-p{\delta }_{ij}+\mu ({\partial }_i{u}_j+{\partial }_j{u}_i)$, which simplifies to ${\sigma }_{ij}=-3\mu c(\mathbf{F}\cdot \mathbf{x})x_i{x}_j/r^5$. Its traction resultant on a sphere, using an outward normal from the origin, is $-4\pi \mu c\mathbf{F}$. Momentum balance with the force $\mathbf{F}\delta (\mathbf{x})$ therefore fixes $c=1/(4\pi \mu )$. The resulting Stokeslet isAn arbitrary constant pressure may be added. The stress has purely radial traction on concentric spheres, and its integrated traction $-\mathbf{F}$ supplies the required distributional point-force normalization.