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Unformatted text preview: ation...] This condition implies that
Px = -Qy.
These two equations look like the C-R equations with the wrong signs. In fact, they show
that the pair P, -Q satisfy the C-R equations, whence they are the real and imaginary
parts of a complex analytic function: Write
v(z) = velocity field in complex form = P + iQ
Then v(z) = P - iQ is analytic, and is therefore has an (analytic) antiderivative, F(z)
F'(z) = v(z)
In other words,
F'(z) = v(z),
just as in the case of the electrostatic field. F is called, as usual, the complex potential of
the flow. If we write F as ∞ + i§ as usual, then we see that the velocity of the gradient
v = Ô∞
so ∞ is called the velocity potential and § is called the stream function, since it gives
the streamlines of v . In other words, we just have the electric potential situation in
(A) Flow around a corner
We want to model flow as follows: Since the above picture is one of streamlines, § = const, we can set up the Dirichlet
problem as one for § (rather than ∞) as follows: † Viscous fluids are not irrotational....
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