An analytic approach short of seeing directly what

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.

Ask a homework question - tutors are online