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The potential flow which is symmetric to the axis of a plane flow is known as axisymmetric potential flow. It can be determined using superposition technique of plane flows. Some of the examples are as follows:

Table1: Plane flow and counterpart axisymmetric flow

Spherical Polar Coordinates[edit]

Spherical polar coordinates are used to express axisymmetric potential flows. Only two coordinates (r,θ) are used and the flow properties are constant on a circle of radius about the x-axis.

The equation of continuity for incompressible flow in spherical polar coordinates is:

+                                (1)

where and are radial and tangential velocities. Therefore a spherical polar stream function exists such that

=                     =                    (2)

Similarly a velocity potential exists such that

=                     =                                               (3)

Spherical polar coordinates for axisymmetric flow

These formulas help to deduce the and functions for various elementary axisymmetric potential flows.

Uniform stream in the x Direction[edit]

The components of a stream in x direction are:


Substituting in eqs. 2 and and 3 and integrating them gives

The arbitrary constants of integration have been neglected.

Point Source or Sink[edit]

Consider a volume flux Q emanating from a point source. The flow will spread out radially and at radius r, it will be equal to .  Thus


With for convenience. Integrating Eqs. 2 and 3 gives


For a point sink, change m to –m in Eq. 6.

Point Doublet[edit]

A source can be placed at and an equal sink at . On taking a limit when tends to 'zero' with product being constant


The velocity potential of point doublet can be given by:


Uniform Stream plus a point source[edit]

On combination of Eqns. (5) and (7) we get the stream function for a uniform stream and a point source at the origin.


From Eqn. (2), the velocity components can be written after differentiation as:

=                                         (11)

[[File:Fig 2 Streamlines and potential lines due to a point doublet at the origin, from Eqns. (8) and (9).JPG|thumb|center|800x1200px|upright = 1.5|Fig 2:Streamlines and potential line due to a point doublet at the origin]]

Equating these equations with zero gives a stagnation point at and at , as shown in the Fig. Suppose m = , we can write the stream function as:


The value of stream surface passing through the stagnation point is which forms a half body of revolution enclosing a point source, as shown in Fig. Using this half body, a pitot tube can be simulated. The half body approaches the constant radius about the x-axis far down the stream.

Fig 3:Streamlines for Rankine half-body of revolution.

At ,, , there occurs the maximum velocity and minimum pressure along the half body surface. There exists an adverse gradient downstream of this point because Vs slowly decelerates to , but no flow separation is indicated by boundary layer theory. Thus for a real half body flow, Eqn. (12) proves to be a realistic simulation. But if one adds the uniform stream to a sink to form a half body rear surface, the separation will be predictable and inviscid pattern would not be realistic.

Uniform Stream plus Point Doublet[edit]

From Eqns. (5) and (8), if we combine a uniform steam and a point doublet at the origin, we get


On examining this relation, the steam surface corresponds to the sphere of radius:


Taking for convenience, we rewrite Eqn. (13) as

=                                       (15)

Below is the plot of streamlines for this sphere. Differentiating Eqn. (2) , we get the velocity components as



The radial velocity vanishes at the surface of sphere r = a, as expected. A stagnation point exists at the front and the rear of the sphere.

Fig 4:Streamlines and potential lines for inviscid flow past a sphere

At the shoulder , there is maximum velocity where and . The surface velocity distribution is



Fluid Mechanics - Frank M. White
Fluid Mechanics and Hydraulic mechanics by R.K. Bansal.

Category:Mechanical engineering