Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 37A (Fluid Dynamics)

Let $\mathbf{u}$ be the Stokes flow and let $\mathbf{v}$ be any other admissible incompressible flow with the same no-slip boundary condition on the sphere and the same decay at infinity. PutThen $\nabla \cdot \mathbf{w}=0$ and $\mathbf{w}=0$ on the sphere. The viscous dissipation satisfiesFor the Stokes stressincompressibility of $\mathbf{w}$ givesThe divergence theorem, the homogeneous boundary data, decay at infinity, and $\nabla \cdot \mathbf{\sigma }=0$ giveThereforeThis proves the Minimum-dissipation theorem for Stokes flow.

The componentwise product rule for the Laplacian and the harmonicity of $\mathbf{\Phi }$ giveHenceThus $\boxed{\nabla \cdot \mathbf{u}=0}$.

Applying the Laplacian to $\mathbf{u}$ and using ${\nabla }^2\mathbf{\Phi }=0$ givesForone therefore hasTogether with part (i), these are the Stokes equations, so the fields form the Unscaled Papkovich–Neuber representation.

Let the sphere have radius $a$; set $a=1$ for the unit sphere implicit in the question. Sinceone findsOn $r=a$, the no-slip boundary condition requires $\mathbf{u}=\mathbf{U}$ for every surface normal. The coefficient of
$(\mathbf{U}\cdot \mathbf{n})\mathbf{n}$ must vanish, while the coefficient of $\mathbf{U}$ must equal one:Solving,For a unit sphere,Substitution reproduces the standard translating sphere in Stokes flow.