弯管中均匀流动的流线

hbghlyj

从mathcurve.com看到Streamlines of a uniform flow in a bent tube的公式
$$
\left\{\begin{array}{l}
x=\operatorname{arcsinh} e^u-\frac{\pi}{2} \\
y=\operatorname{arcsinh} e^{-u}+\frac{\pi}{2}
\end{array}\right.
$$
这是怎样得到的呢

hbghlyj

用Asympotote作图
size(8cm);
import graph;

pair acos(pair z){
  return -I*log(z + sqrt(z^2 - 1));
}
pair conformalMap(pair z) {
    return I*acos(exp(z))-acos(exp(-z));
}

real xmin = -2;
real xmax = 2;
real ymin = .1;
real ymax = .9;

xaxis("$\mathrm{Re}$",gray);
yaxis("$\mathrm{Im}$",gray);

for (real y:uniform(ymin,ymax,10)) {
    guide l;
    for (real x : uniform(xmin, xmax,10)) {
        l=l..conformalMap(x+I*y);
    }
    draw(l);
}
for (real x:uniform(xmin,xmax,10)) {
    guide l;
    for (real y : uniform(ymin, ymax,10)) {
        l=l..conformalMap(x+I*y);
    }
    draw(l,red);
}
缩小一下
size(8cm);
import graph;

pair acos(pair z){
  return -I*log(z + sqrt(z^2 - 1));
}
pair conformalMap(pair z) {
    return I*acos(exp(z))-acos(exp(-z));
}

real xmin = -2;
real xmax = 2;
real ymin = .1;
real ymax = .5;

xaxis("$\mathrm{Re}$",gray);
yaxis("$\mathrm{Im}$",gray);

for (real y:uniform(ymin,ymax,10)) {
    guide l;
    for (real x : uniform(xmin, xmax,10)) {
        l=l..conformalMap(x+I*y);
    }
    draw(l);
}
for (real x:uniform(xmin,xmax,10)) {
    guide l;
    for (real y : uniform(ymin, ymax,10)) {
        l=l..conformalMap(x+I*y);
    }
    draw(l,red);
}

hbghlyj

wolfram.com/language/12/complex-visualization/2d-ideal-fluid-flow.html

1. ComplexPlot[(-I z)^(2/3) - (-z)^(2/3), {z, -3 - 3 I, 3 + 3 I},
2. Mesh -> {Range[0, 3, .2]}, MeshFunctions -> {Im[#2] &}, Sequence[
3. RegionFunction -> Function[{z},
4. Or[Arg[z] < (-Pi)/2, Arg[z] > 0]], BoundaryStyle -> None]]

复制代码

这个是拐角的外部（全平面去掉拐角的内部）。和上面不一样。