Mechanics of Fluids

Then
dw
dz
=−U =−u + iv
Hence u = U and v = 0, so eqn 9.36 represents uniform flow parallel to the
x-axis.
Next consider
which on differentiation yields
dw
dz
w = iVz (9.37)
= iV =−u + iv
Hence u = 0 and v = V, so eqn 9.37 represents uniform flow parallel to the
y-axis.
Finally let
w =−qze −iα = z(−q cos α + iq sin α) (9.38)
From the preceding results, we see that u = q cos α and v = q sin α, showing
that eqn 9.38 represents a uniform flow with resultant velocity q inclined at
an angle α to the x-axis.
9.9.2
Let
Flow from a line source
w =− m
ln z =−m
2π 2π ln(reiθ ) =− m m
ln r − i θ
2π 2π
(9.39)
Hence
φ =− m
ln r
2π
and ψ =−m
2π θ
showing that eqn 9.39 represents the complex potential of a line source.
9.9.3 Free vortex
Let
Then
w = i Ɣ Ɣ Ɣ
ln z =− θ + i ln r (9.40)
2π 2π 2π
φ =− Ɣ
Ɣ
θ and ψ = ln r
2π 2π
which are respectively the velocity potential and stream function of a free
vortex.
Functions of a complex variable 401