By Analytical Means

For a given flow pattern in the physical plane, each streamline of the flow can be represented by a separate stream function. Transferring these stream functions, using the transformation function, Z = f (z), the corresponding streamlines in the transformed plane can be obtained. For example, the streamlines in the physical plane given by the stream function:

^ = f (x, y) = constant

can be expanded, using the transformation function:

Z = f (z),

where z = x + iy, to obtain the following three equations:

Z = constant = f (x, iy)

= f1(x, y) + if2(x, y).

But f = § + in, therefore:

§ + in = fi(x, y) + if2(x, y).

Comparing the real and imaginary parts, we have:

§ = f1 (x, y) n = h{x, y).

From these equations, § and n can be isolated, by eliminating x and y. The resulting expressions for § and n will represent the transformed line in the f-plane.

Example 4.2

Transform the straight lines, parallel to the x-axis in the physical plane, with the transformation function f = 1/z.

Solution

From the transformation function f = /z, we have:

1

Z = –

z

_ 1

x + iy

Multiplying and dividing the numerator and denominator of the right-hand side by (x — iy), we get:

x iy

f = ——————–

(x + iy)(x — iy)

x — iy

x2 + y2

x iy

x2 + y2 x2 + y2

But

Z = § + in.

Therefore:

t,-= у

^ x2 + y2 x2 + y2