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ordinary path integral of the vector field represented by i–/z– = i/z–, (see the note just after
the definition of the complex path integral). Therefore, the path integral of i/z– equals 2π
Ki
It follows that the path integral of
is just K, so that the moment of the vortex
2πz–
is given by K“0, 0, 1‘ or Kk, and its strength is K.
(C) Point Vortex F(z) = (C) Combing Sources, Sinks, and Vortices
Since all of the above functions only have singularities at isolated points, we can combine
them to form velocity fields with vortices, sources and sinks as we desire. We can also
combine these things with some of the other flows we have studied above.
(D) Flow with Circulation Around a Cylinder
47 1
Start with the complex potential for basic flow around a cylinder: F(z) = z + . Then
z
add a circulation at the origin with some strength K:
1
Ki
F(z) = z +
+
lnz
z
2π
A stagnation point is a point where the velocity equals zero. Setting the speed equal to
zero and solving for z gives...
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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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