Elliptic curves and three-dimensional flow patterns

This paper is concerned with the geometry of flow patterns in the classical problem of an impulsely-started, incompressible, axisymmetric, laminar jet generated by a point force. The second and third invariants of the velocity gradient tensor, evaluated at critical points in the jet, describe the fundamental dependence of the flow on the jet Reynolds number. As the Reynolds number is increased from zero to infinity, the critical points follow elliptic curves in the space of invariants and rational roots occur at bifurcation points in this space. The corresponding invariants of the acceleration gradient tensor trace out a trajectory with infinitely many, densely spaced rational roots. The results provide new insight into the viscous and pressure forces which act in the jet and the balance between strain and rotation which leads to the onset of a starting vortex.