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 and into an analytic function of a complex variable 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, and are real single valued continuous functions. We get from the above

and

Therefore,

and

Consider

(22.13)

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

and

which means

and

Finally we get

and

This completes the definition

(22.14)

Also,

(22.15)

Therefore and are perpendicular to each other.

Let us consider another function or complex potential

Which gives,

(22.16)

Therefore, we get

and

and

;   which means

and       is the complex velocity

Therefore,

(22.17)