Free Vortex

The velocity components of the free vortex are given by

     v_r = 0   and    v_θ = K/r   (assumed v_z = 0)

The vorticity can be determined by the curl of the velocity as follows:

Since the vorticity is zero, Tom has learned that the given free vortex is indeed irrotational.

The velocity potential associated with the free vortex can then be determined as follows:

     

Since V_r = ∂φ/∂r, the velocity potential is not a function of r, and f(r) reduces to an arbitrary constant. For simplicity, set f(r) = 0 (the value of an individual velocity potential is irrelevant) to yield

     φ = Kθ

Note that when a flow is referred to as irrotational, it simply implies that the individual fluid elements do not rotate. Although the fluid elements of the given free vortex follow a circular path, it is still irrotational flow.