Potential flow around a circular cylinder with circulation

Adding circulation $\kappa$ gives the possibly multivalued velocity potentialThe velocity remains single-valued, and on the cylinder its tangential component is $-2U\sin \theta +\kappa /(2\pi a)$.

For steady two-dimensional inviscid flow of density $\rho$, far-field velocity $\mathbf{U}$, and circulation $\kappa$, the force per unit span is perpendicular to the flow. With counterclockwise circulation taken positive and $\mathbf{U}=U{\mathbf{e}}_x$, it is $-\rho \kappa U{\mathbf{e}}_y$.