  When a shallow layer of inviscid fluid flows over a substrate, the fluid
particle trajectories are, to leading order in the layer thickness, geodesics
on the two-dimensional curved space of the substrate. Since the two-dimensional
geodesic equation is a two degree-of-freedom autonomous Hamiltonian system, it
can exhibit chaos, depending on the shape of the substrate. We find chaotic
behaviour for a range of substrates.
