Unformatted text preview: Think of the a viscous fluid moving down a pipe, and choose a closed
path going down the center, to the edge, and up the edge. The path integral will not be zero, so that there is
a net circulation. 39 2 We now use w = z to map this onto H, and use
§ = Ay
for the associated potential in H . Remembering that this is the i maginary part of a
complex potential in H, we simply use
F = Aw = Az
as our complex potential. Therefore,
∞ = A(x - y )
§ = 2Axy
Equipotentials: These are the curves ∞ = constant, or
A(x - y ) = const
giving radial lines emanating from the origin.
Streamlines: These are the curves
2Axy = const
Velocity: v(z) = F'(z) = 2Az, so F'(z) = 2Az–. In other words,
v = 2A“x, -y‘.
This gives an interpretation of A:
2 2 speed = |v| = 2A x + y
So by knowing the speed of the flow at any particular point away from the wall, we can
Note The speed is not constant along a streamline (hyperbola) but varies as the distance
from the origin. The particle slow dow...
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