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# An analytic approach short of seeing directly what

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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 2 F = Aw = Az as our complex potential. Therefore, 2 2 ∞ = A(x - y ) and § = 2Axy Equipotentials: These are the curves ∞ = constant, or 2 2 A(x - y ) = const giving radial lines emanating from the origin. Streamlines: These are the curves 2Axy = const giving hyperbolas. 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 compute A. 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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## This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.

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