Leap-Frogging Bubble Rings

Here is a simple technique for modeling vortices assuming inviscid incompressible potential flow (irrotational). This technique was inspired from Kerry Mitchell’s paper based on Ultra Fractal. You can also see this here on Dynamical Systems’ Picture Gallery.

Here is some Mathematica code to plot the entrained fluid:

(* runtime: 25 seconds, increase n for better resolution *)
tmax = 0.85; rcore = 0.1; Klist = {1, -1, 1, -1}; zlist0 = {-1 - 0.5I, -1 + 0.5I, -0.5 - 0.5I, -0.5 + 0.5I}; m = Length[zlist0];
v[K_, z_, z0_] := Module[{r2 = Abs[z - z0]^2}, I K(z0 - z)/r2(1 - Exp[-r2/rcore^2])];
zlist = NDSolve[Flatten[Table[{Subscript[z,i]'[t] == Sum[If[i == j, 0, v[Klist[[j]], Subscript[z, i][t], Subscript[z, j][t]]], {j, 1, m}], Subscript[z, i][0] ==zlist0[[i]]}, {i, 1, m}]], Table[Subscript[z, i][t], {i, 1, m}], {t, 0, tmax}][[1, All, 2]];
n = 23; image = Table[NDSolve[{z'[t] == Sum[v[Klist[[i]], z[t], zlist[[i]]], {i, 1, m}], z[tmax] == x + I y}, z[t], {t, 0, tmax}, MaxSteps -> 5000][[1, 1, 2]] /. t -> 0, {y, -1.125, 1.125, 2.25/n}, {x, -0.35, 1.9, 2.25/n}];
ListDensityPlot[Map[Sign[Im[#]]Arg[#] &, image, {2}], Mesh -> False, Frame -> False, AspectRatio -> Automatic]