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\listfiles
\documentclass[a4paper]{article}% Herbert Voss 2004-10-06
\usepackage{pstricks}
\usepackage{pstricks-add}
\usepackage{multido}
\def\func{%
const0 x sin dup mul 3 mul div neg /p exch def
p abs sqrt neg /r exch def
const1 2.0 div neg /q exch def
p 0 lt {
p dup mul p mul q dup mul add 0 le {
q r 3 exp div arccos
3 div cos r dup add neg mul
}{
q r 3 exp div arch
3 div cosh r dup add neg mul
} ifelse
}{
q r 3 exp div arsh
3 div sinh r dup add neg mul
} ifelse
}
\begin{document}
\textbf{Dynamical flow around a cylinder}
\[
\left(r^2-\frac{c_1}{r}\right)\sin^2\theta=c_0
\]
which is the same as
\[
r^3-\frac{c_0}{\sin^2\theta}\cdot r-c_1=0
\]
which can be solved by the formula of Cardano:
\[
r^3+3pr+2q=0
\]
\bigskip
\begin{pspicture*}(-5,-2.5)(5,2.5)
\pstVerb{
/e 2.718281828459045 def
/arccos { dup dup mul neg 1.0 add sqrt exch atan } def
/sinh { dup e exch exp exch neg e exch exp sub 2 div } def
/cosh { dup e exch exp exch neg e exch exp add 2 div } def
/arsh { dup dup mul 1 add sqrt add ln } def
/arch { dup dup mul 1 sub sqrt add ln } def
}
\psaxes(0,0)(-5,-2.5)(5,2.5)
\pscircle[linecolor=red]{1}
\psset{plotpoints=100,linewidth=0.05pt,polarplot=true,linecolor=blue}
\multido{\rA=0.005+0.05}{5}{
\psplot{1}{179}{%
/const0 \rA\space def
/const1 1.0 def % cylinder radius
\func
}
\psplot{181}{359}{%
/const0 \rA\space def
/const1 1.0 def % cylinder radius
\func
}%
}
\multido{\rA=0.03+0.25}{20}{
\psplot{1}{179}{%
/const0 \rA\space def
/const1 1.0 def % cylinder radius
\func
}
\psplot{181}{359}{%
/const0 \rA\space def
/const1 1.0 def % cylinder radius
\func
}%
}
\end{pspicture*}
\end{document}
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