A vortex is a region where the fluid flows around an imaginary axis.
For an irrotational vortex, the flow at every point is such that a small particle placed there undergoes pure translation and does not rotate.
Velocity varies inversely with radius in this case. Velocity will tend to  at  that is the reason for center being a singular point. The velocity is mathematically expressed as –
Since the fluid flows around an axis,
The stream function for irrotational vortices is given by –
While the velocity potential is expressed as –
For the closed curve enclosing origin, circulation (line integral of velocity field)  and for any other closed curves,