This gives i 1 i 1 1 fz are e ar cos i ar

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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) such that 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 of ∞: 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 disguise. Examples (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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This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.

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