Shape calculus for vortex patch equilibria and its application to lattice configurations

A new shape derivative formula for singular contour integrals with logarithmic kernels is proposed. This formula yields a simple numerical scheme for computing vortex patch equilibria. Owing to its simplicity, any steady configuration of point vortices can be extended to that of vortex patches. As a test problem, a doubly periodic array of vortex patches is considered to show the efficiency of the new formula. Several non-trivial families of stationary vortex patch lattices are also presented.