Question

2) Let F (Z) = f ilog z, 6 is real, be the complex - potential of a flow. Find and sketch the streamlines of the flow

Answer 1

We start by dividing the function log(z) into real and imaginary parts.

Let,
2 = x +iy = relt
, where
r= r2 + y2
and
8 = tan-19

Then,

log(z) = log(re) = log(r) + idlog(e) = log(x2 + y2) + itan-19

Now the potential function is given by,

F(z) = silog(z) = sillog( Vx2 + y2)+itan-1%) = -stan-19+ilog(V x2 + y2)

Stream function is given by the imaginary part of the complex potential. Hence, the stream function is,

V = Slog(Vx2 + y2) = -log(x2 + y2) (Using, log(ak) = klog(a))

The streamlines are where the stream function is constant. Hence, the streamlines are given by,

V = -log(x² + y2) =D (Where, D is a real constant)

x2 + y2 = e =C (Where, C is a non- negative constant)

Hence, the streamlines are given by ,

x + y2 = C (Where, C is a positive constant)

This for of equation represents circle centered at the origin. For different values of C we get the streamlines. I have added a graph showing the streamlines for C = 0, 1, 4, 9, 16.