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