Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 77 / 2 / c / Solution

Taking the curl of the Brinkman equation eliminates pressure and the constant matrix velocity. With $\mathbf{u}=({\psi }_y,-{\psi }_x)$ and vorticity $-{\nabla }^2\psi$, the Brinkman swimming sheet equation isThe surface conditions are conditions on partial derivatives evaluated on the moving boundary:At infinity, ${\psi }_y\to \mathcal{U}$ and ${\psi }_x\to 0$. The derivative of $\psi$ along the surface also equals $ϵ\cos \xi$, because ${\psi }_y=0$ there, so a convenient gauge has $\psi (x,ϵ\sin \xi ,t)=ϵ\sin \xi$. Decaying perturbations and the force-free condition determine the swimming solution.