Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 36E / ii / Solution

Put $Q=F\cdot x$. Differentiate the given Stokeslet velocity:Symmetrizing and subtracting the pressure term cancels the diagonal terms, leaving the Cauchy stress tensorOn the radius-$R$ surface with outward sphere normal $n=x/R$, the force exerted by the exterior fluid on the sphere is the tractionUse $dS=R^{2}d\Omega$ and ${\int }_{S^2}{n}_i{n}_{j}d\Omega =(4\pi /3){\delta }_{ij}$ to findThe $R^2$ factors cancel, proving independence of radius. This calculation uses the printed far-field Stokeslet; higher multipoles of the exact sphere flow have zero net force through enclosing surfaces.