Free Vortex Flow
• Fluid particles move in circles about a point.
• The only non-trivial velocity component is tangential.
• This tangential speed varies with radius r so that same circulation is maintained.
• Thus, all the streamlines are concentric circles about a given point where the velocity along each streamline is inversely proportional to the distance from the centre. This flow is necessarily irrotational.
Velocity components
In a purely circulatory (free vortex flow) motion, the tangential velocity can be written as
For purely circulatory motion we can also write
Stream Function
Using the definition of stream function, we can write
Combining Eqs (1) and (2) with the above said relations for stream function, it is possible to write
Velocity Potential Function
Because of Irrotationality, it should satisfy
Eqs (21.1) and (21.2) and the above solution of Laplace's equation yields
Since, the integration constants C1 and C2 have no effect on the structure of velocities or pressures in the flow. We can ignore the integration constants without any loss of generality.
It is clear that the streamlines for vortex flow are circles while the potential lines are radial. These are given by
In Fig. 1, point 0 can be imagined as a point vortex that induces the circulatory flow around it.
• The point vortex is a singularity in the flow field (vθ becomes infinite).
• Point 0 is simply a point formed by the intersection of the plane of a paper and a line perpendicular to the plane.
This line is called vortex filament of strength 
where is the circulation around the vortex filament .
Circulation is defined as
This circulation constant denotes the algebraic strength of the vortex filament contained within the closed curve.
From Eq. (6) we can write
For a two-dimensional flow
Consider a fluid element as shown in Fig. 2. Circulation is positive in the anticlockwise direction (not a mandatory but general convention).
After simplification
Physically, circulation per unit area is the vorticity of the flow.
Now, for a free vortex flow, the tangential velocity is given by Eq. (1) as
For a circular path (refer Fig.2)
Therefore
G = 2πC (9)
It may be noted that although free vortex is basically an irrotational motion, the circulation for a given path containing a singular point (including the origin) is constant (2πC) and independent of the radius of a circular streamline.
However, circulation calculated in a free vortex flow along any closed contour excluding the singular point (the origin), should be zero.
Considering Fig 21.3 (a) and taking a closed contour ABCD in order to obtain circulation about the point, P around ABCD it may be shown that
Forced Vortex Flow
If there exists a solid body rotation at constant ω (induced by some external mechanism), the flow should be called a forced vortex motion (Fig. 3 (b).
We can write
Equation (10) predicts that
1. The circulation is zero at the origin
2. It increases with increasing radius.
3. The variation is parabolic.
It may be mentioned that the free vortex (irrotational) flow at the origin is impossible because of mathematical singularity. However, physically there should exist a rotational (forced vortex) core which is shown by the dotted line (in Fig. 3a).
Below are given two statements which are related to Kelvin's circulation theorem (stated in 1869) and Cauchy's theorem on irrotational motion (stated in 1815) respectively
1. The circulation around any closed contour is invariant with time in an inviscid fluid.--- Kelvin's Theorem
2. A body of inviscid fluid in irrotational motion continues to move irrotationally.------------ Cauchy's Theorem
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