We examine the effects of stationary vortices in superfluid ⁶Li atoms at zero temperature in the frame of the recently developed fluiddynamical scheme, that includes the pair density and its associated pair current and pair kinetic energy in addition to the fields appearing in the hydrodynamical description of normal fluids. In this frame, the presence of any particle velocity field gives rise to the appearance of a pair current. As an illustration, we consider a stationary vortex with cylindrical geometry in an unpolarized fluid, and examine the effects of the rotational velocity field on the spatial structure of the equilibrium gap and the profiles of the pair current. We show that the latter is intrinsically complex and its imaginary part is the source of a radial drift for the velocity field. We discuss the consequences on the stationary regime.