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Unformatted text preview: n the most nearest the origin, where the width of
the flow channels is widest: A typical flow channel
[The above potential also gives a model of the flow along any flow channel such as the
one above.] also, the flow speeds up as the flow channel gets narrower and narrower.
This is how water pistols work.
(B)Flow around a cylinder
This leads to a description of § again: 40 This region D maps into H via w = z + 1/z, and, on H, § can again be taken to be
§ = Ay
Giving us a complex potential
È
1˘
F(z) = AÍz + ˙
Î
z˚
iø To see the real and imaginary parts, use polar form: z = re . This gives
È iø 1 iø˘
È
È
1˘
1˘
F(z) = AÍre + e ˙ = AÍr + ˙ cos ø + i AÍr  ˙ sin ø
Î
˚
Î
˚
Î
r
r
r˚
So we can now get the potentials and streamlines in polar form:
Equipotentials:
È
˘
Ír + 1˙ cos ø = const
Î
r˚
Kind of complicated to draw these piggies  wait until we do things parametrically in the
next section.
Streamlines:
È
˘
Ír  1˙ sin ø = const
Î
r˚
Again, these are not standard curves. However, at large distances, 1/r ‡ 0...
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This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.
 Fall '03
 StefanWaner
 Math, Algebra, Geometry, Complex Numbers

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