Given a uniform velocity of a fluid at any position in space:
This flow is incompressible because the velocity is constant, the first derivatives of the velocity components are zero, and the total divergence is zero: 
Given the circulation is always zero the flow is also irrotational, we can derive this from the Kelvin's circulation theorem and from the explicit computation of the vorticity:
Being incompressible and two-dimensional, this flow is constructed from a stream function:
from which
and in cylindrical coordinates:
from which
As usual the stream function is defined up to a constant value which here we take as zero. We can also confirm that the flow is irrotational from:
Being irrotational, the potential function is instead:
and therefore
and in cylindrical coordinates