Movement of a line vortex pair downstream of a circular cylinder in potential flow

Extract

Foppl’s stability analysis (1913) of the equilibrium positions of a line vortex pair in the downstream neighbourhood of a circular cylinder in the two-dimensional potential flow of an incompressible fluid moving with uniform speed at infinity has been corrected recently by Smith (1973). Smith shows that it is only for small symmetric displacements from any one pair of points on the locus of possible rest positions which is known as the Foppl curve there is neutral spatial stability, all other displacements being unstable. This correction now makes the results of the stability analysis consistent with the work of Dolaptchiev and Sendov (1959) who determined the trajectories followed by such a vortex pair when constrained so as to move in a wholly symmetric motion. Dolaptchiev and Sendov obtained the analytical solution of the differential equations for the completely symmetric motion, but for motion other than symmetric no simple solutions of this sort appear to exist although the vortex trajectories for any case can be deduced numerically without difficulty.