<< Graphics`PlotField`
<< Graphics`ImplicitPlot`
<< Graphics`Graphics`

example 9.4 円柱まわりの流れ　p. 214

step1 ContourPlot

fz = U (z + a^2/z )

w = 
      Together[Expand[Factor[fz /. z ->  x +  I y]]] /. 
        x + I y ->  (x^2 + y^2)/(x - 
                I y)

para = {a ->  1, U ->  1}

{ComplexExpand[Re[w]], ComplexExpand[Im[w]]}
{ϕ, ψ} = % /. para;

{Simplify[D[ϕ, x] - D[ψ, y]], 
      Simplify[D[ϕ, y] + D[ψ, x]]}

Plot[ϕ /. 
        y ->  1.1, {x, -3, 3}]

zu11 = 
      ContourPlot[ψ, {x, -3, -0.001}, {y, -3, 3}, 
        AspectRatio -> Automatic, ContourShading -> False, 
        Contours ->  25, 
        Epilog -> {Disk[{0, 0}, 
              1, {0, 2  Pi}]}]
zu12 = 
      ContourPlot[ψ, {x, 0.001, 3}, {y, -3, 3}, 
        AspectRatio -> Automatic, ContourShading -> False, 
        Contours ->  25, 
        Epilog -> {Disk[{0, 0}, 1, {0, 2  Pi}]}]

zu1 = 
      Show[zu11, zu12, 
        PlotRange ->  {{-3, 3}, {-3, 3}}]

zu21 = 
      ContourPlot[ϕ, {x, -3, -0.001}, {y, -3, 3}, 
        AspectRatio -> Automatic, ContourShading -> False, 
        Contours ->  20]
zu22 = 
      ContourPlot[ϕ, {x, 0.001, 3}, {y, -3, 3}, 
        AspectRatio -> Automatic, ContourShading -> False, 
        Contours ->  20]

zu2 = 
      Show[zu21, zu22, 
        PlotRange ->  {{-3, 3}, {-3, 3}}]

zu1zu2 = Show[zu1, zu2]

zu3 = 
      PlotGradientField[ϕ, {x, -3, 3, 2}, {y, -3, 3, 0.25}, 
        ScaleFactor ->  0.5]

zu1zu3 = Show[zu1, zu3]

step 1 summary

Show[
      GraphicsArray[{zu1zu2, zu1zu3}]]

step2  別解 NDSolve

eq = z'[t] == Conjugate[D[fz, z] /. z -> z[t]] /. 
        para

ne = 25;
z0 = Table[-10 - 3  I + 0.25 I i, {i, 0, ne}];

zuall21 = 
      Table[{ans1 = NDSolve[{eq, z[0] == z0[[i]]}, 
                z[t], {t, 0, 100}][[1]]; 
          ParametricPlot[{Re[z[t]] /. ans1, Im[z[t]] /. ans1}, {t, 0, 100}, 
            PlotRange ->  {{-3, 3}, {-3, 3}}]}, {i, 
          ne}]

zuall22 = 
      Show[zuall21, Graphics[Disk[{0, 0}, 1]], 
        DisplayFunction ->  $DisplayFunction, 
        AspectRatio ->  Automatic]

Show[zuall22, zu3]

p215 Fig 9.4 memo

eq1 = 1 - Exp[2 I θ]

eq2 = -2 I Exp[I θ] Sin[θ]

Simplify[eq1 - eq2]