Steady Motion of a Non-Circular Vortex.

The problem considered is that of the steady motion of an infinite, two-dimensional, incompressible fluid with zero vorticity everywhere except within a closed curve, where the vorticity is uniform and non-zero. The objective is to find an analytical expression for the shape of such a curve. A non-linear integro-differential equation is presented whose solution is the curve expressed in polar coordinates. The curve is assumed to be expressible as a power series in the cosine function; substitution of the series in the equation yields the co-efficients of the series. The coefficients are found to be of two types: those exhibiting a regular behavior allowing the inference of a general term, and those showing an unpredictable behavior. The regular part is shown to consist of generalized hypergeometric functions with known convergence properties, and, in some cases, with a representation in terms of elementary functions.