The motion of a vortex near two circular cylinders

The motion of a vortex near two circular cylinders of arbitrary radii-a problem of geophysical significance-is studied. The fluid motion is governed by the two-dimensional Euler equations and the flow is irrotational exterior to the vortex. Two models are considered. First, the trajectories of a line vortex are obtained using conformal mapping techniques to construct the vortex Hamiltonian which respects the zero normal flow boundary condition on both cylinders. The vortex paths reveal a critical trajectory (i.e. separatrix) that divides trajectories into those that orbit both cylinders and those that orbit just one cylinder. Second, the motion of a patch of constant vorticity is computed using a combination of conformal mapping and the numerical method of contour surgery. Although the patch can deform, the results show that when the islands have comparable radii the patch remains remarkably coherent. Moreover, it is demonstrated that the trajectory of the centroid of the patch is well modelled by a line vortex. For the limiting case when one of the cylinders has infinite curvature (i.e. it becomes a straight line or wall) it is shown that the vortex patch, which propagates under the influence of its image in the wall, may undergo severe deformation as it collides with the smaller cylinder, with portions of the vortex passing around different sides of the cylinder.