solnsheet8 - Fluid Dynamics 3 Solutions to Sheet 8 1(i If |...

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Unformatted text preview: Fluid Dynamics 3 - Solutions to Sheet 8 1. (i) If | z | = R (on the boundary) then z z = R 2 and thus R 2 / z = z . But this means than w ( z ) = f ( z ) + f ( z ) for z on the boundary, and thus w ( z )- w ( z ) = f ( z ) + f ( z )- ( f ( z ) + f ( z )) = 0 . (ii) The complex potential of the vortex at z is f ( z ) =- i 2 ln( z- z ), and thus w ( z ) =- i 2 ln( z- z ) + i 2 ln R 2 z- z . Then the complex velocity which moves the vortex is u- iv | z = z =- i 2 R 2 z 2 ( R 2 /z- z ) =- i 2 h 2 R 2 z R 2- h 2 (iii) Complex conjugation leads to u + iv = 2 h 2 R 2 h 2- R 2 ( y- ix ) . In vector notation,- y + ix (- y,x ) = h , and thus u =- 2 h R 2 h 2- R 2 Thus the vortex rotates around the cylinder clock- wise at an angular speed = 2 h 2 R 2 h 2- R 2 . 2. (i) The complex potential of the two vortices in the upper half plane is w = i 2 (- ln( z- z ) + ln( z + z )) f ( z ) , with the two vortices placed at z = s + ih and- z =- s + ih . Then the total potential in the presence of the wall at y = 0 is w = f ( z ) + f ( z ) = i 2 (- ln( z- z ) + ln( z + z ) + ln( z- z )- ln( z + z )) ....
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This note was uploaded on 01/20/2012 for the course MATH 33200 taught by Professor Eggers during the Fall '11 term at University of Bristol.

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solnsheet8 - Fluid Dynamics 3 Solutions to Sheet 8 1(i If |...

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