Find the mapping of circle using confirmal mapping ( joukowski tansformation method) Assume w= z+1/z 〖z=re〗^iθ = 〖r(cosθ+isinθ)〗^1 ( a Circle with radius r and center (0,0))
1 Answer
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6
We can use Manipulate
to find such suitable Circle
.The circle through {1,0}
and center by w
(one possible w={-.2, .4}
). The Joukowsky tansformation mapping such circle to a closed curve like the wings of an airplane.
Manipulate[
Block[{w = {1, I} . pt, r = Abs[w - 1], z = w + r*Exp[I*t]},
GraphicsRow@{Graphics[{Circle[pt, r], Text[pt, pt],
AbsolutePointSize[5], {Red, Point[{1, 0}]}}, PlotRange -> 3,
Axes -> True],
ParametricPlot[ReIm[z + 1/z], {t, 0, 2 π},
PlotRange -> 3]}], {{pt, {-.2, .4}}, {-1, -1}, {1, 1}},
ControlPlacement -> Top]
*
Manipulate[
Block[{w1, r1, z1, pt2, w2, r2, z2, t, c}, w1 = {1, I} . pt1;
r1 = Abs[w1 - 1]; z1 = w1 + r1*Exp[I*t];
pt2 = (1 - c)*{1, 0} + c*pt1;
w2 = {1, I} . pt2; r2 = Abs[w2 - 1]; z2 = w2 + r2*Exp[I*t];
λ0 =
SolveValues[First[(1 - c)*{1, 0} + c*pt1] == 0 /. pt1 -> {-.2, .4},
c][[1]];
GraphicsRow@{ParametricPlot[{ReIm[z1], ReIm[z2]}, {t, 0,
2 π}, {c, λ, 1}, MeshFunctions -> {#4 &},
Mesh -> {{λ0}}, MeshStyle -> White, PlotRange -> 2,
PerformanceGoal -> "Quality",
Epilog -> {Point[pt1], Point[pt2 /. c -> λ],
Line[{pt1, pt2 /. c -> λ}], Dotted,
Line[{pt2 /. c -> λ, {1, 0}}]}, PlotStyle -> Magenta],
ParametricPlot[{ReIm[(z1 + 1/z1)/2], ReIm[(z2 + 1/z2)/2]}, {t, 0,
2 π}, {c, λ, 1}, MeshFunctions -> {#4 &},
Mesh -> {{λ0}}, MeshStyle -> White, PlotRange -> 2,
PerformanceGoal -> "Quality",
PlotStyle -> Magenta]}], {{pt1, {-.2, .4}}, {-1, -1}, {1,
1}}, {{λ, λ0}, 0, 1 - $MachineEpsilon},
ControlPlacement -> Top]
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$\begingroup$ @bmf Thanks your upvote! I don't know how to make the text and the graphics row looks good. $\endgroup$– cvgmtApr 17, 2022 at 22:42
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$\begingroup$ yes, it's a bit tricky as it would seem. but I have to say they don't look bad and things are very clear. if I manage to come up with something, I will leave a comment :) $\endgroup$– bmfApr 17, 2022 at 22:45
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$\begingroup$ update: some partial progress -not very satisfactory- can be made as follows:
Manipulate[Block[{w = pt[[1]] + pt[[2]]*I, r = Abs[pt[[1]] + pt[[2]]*I - 1], z = w + r*Exp[I*t]}, GraphicsRow@{Graphics[{Circle[pt, r], Text[pt, pt, {-2, -10}, {1, 0}], AbsolutePointSize[5], {Red, Point[{1, 0}]}}, PlotRange -> 3, Axes -> True], ParametricPlot[ReIm[z + 1/z], {t, 0, 2 \[Pi]}, PlotRange -> 3]}], {{pt, {-.2, .4}}, {-1, -1}, {1, 1}}, ControlPlacement -> Top]
Did you have in mind something like that or did I misunderstand? $\endgroup$– bmfApr 17, 2022 at 23:48 -
2$\begingroup$ Upvoted, begrudgingly. I do not much like having responses to do-my-homework when no effort on the part of the poster is shown. But this is a nice answer all the same. $\endgroup$ Apr 18, 2022 at 17:45
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