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Analysis of Potential Flows through Complex Variables

The properties exhibited by the velocity potential and stream function of two dimensional irrotational flow of an inviscid fluid are identical to those exhibited by the real and imaginary part of an analytic function of a complex variable. It is natural to combine   into an analytic function of a complex variable z = x + ly in the region of z plane occupied by the flow. Here,    is called imaginary unit.

An analytic function,                             (22.11)

and                                                                     (22.12)

These are known as Cauchy-Riemann condition. Also, u and v are real single valued continuous functions. We get from the above

Therefore  

Consider                                                                          (22.13)

where  is velocity potential function and ψ  is stream function. This leads to 

which means                

Finally we get              

This completes the definition

                                                            ( 22.14)

Also                                                                                                               (22.15)

Therefore  are perpendicular to each other.

Let us consider another function or complex potential

                                                (22.16)

Therefore, we get

Therefore, 

                                                                           (22.17)

...contd...Analysis of Potential Flows through Complex Variables

Now consider another situation, where the complex potential is given by

                                                                             (22.18) 

and 

Therefore     A z = F(z) signifies uniform flow. The flow was earlier represented via Figure 20.2(a)

We may choose yet another complex potential, given by

                                                                                                                             (22.19)

                          (22.20)
 

This signifies

                                                                                                                   (22.21)

                                                                                                                 (22.22)

The flow is basically elementary uniform flow at an angle as represented by  Figure 20.2(b)

Consider another complex potential given by

                                                                                                   (22.23)

We obtain     

If A is positive, ψ is in the outward direction and it is a source flow Figure 20.3. If A is negative, ψ is in the inward direction and it is sink flow.

The radial and tangential component of velocities are given by

Let

The quantity K is,