On the existence of local formulae of the transfer velocity of local tubes that conserve their strengths

In the paper we discuss different approaches to express Fridman velocity, which is the transfer velocity of vortex tubes that conserves their strengths in a viscous fluid. It is known that Fridman velocity exists for any elementary vortex fragment, though it is not unique. The existence of such expressions (which we call local) of Fridman velocities depending only on the velocity components and their derivatives, has been an open question for nonstationary incompressible flows where the scalar product of vorticity and its curl is nonzero. This question is of importance for the development of numerical vortex methods. In this paper, in terms of the examples of cylindrical flows, we demonstrate the existence of flows with nonzero scalar product of vorticity and its curl, which have local Fridman velocity; as well as the existence of the flows where only nonlocal Fridman velocity is possible.