Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 66 / 1 / a

Work in the frame in which the distant fluid is stationary, and take the pressure at infinity as zero. In the Unscaled Papkovich–Neuber representation, choose the harmonic potentialsBoth are harmonic away from the sphere's centre: $1/r$ is harmonic there and $\mathbf{V}\cdot \mathbf{x}/r^3=-\mathbf{V}\cdot \nabla (1/r)$. Substitution into $\mathbf{u}=\nabla (\mathbf{x}\cdot \mathbf{\Phi }+\chi )-2\mathbf{\Phi }$, $p=2\mu \nabla \cdot \mathbf{\Phi }$, gives the translating sphere in Stokes flow:Here $\mathbf{n}=\mathbf{x}/r$. At $r=a$ the coefficients of $\mathbf{I}$ sum to one and those of $\mathbf{nn}$ sum to zero, so the no-slip boundary condition is satisfied. The field decays at infinity. The Papkovich–Neuber representation enforces the Stokes equation and incompressibility, and Uniqueness of Stokes flow identifies the physical solution.

Differentiate radially at fixed direction $\mathbf{n}$:On the surface, multiplying the specified Newtonian fluid stress tensor by the outward normal from the solid givesThus the surface traction in translating-sphere Stokes flow is uniform and the fluid's force on the sphere isThis is the Stokes drag law; an external force of the opposite sign maintains the motion.