11
Nov
07

Schwarz-Christoffel Flow

Here’s a potential flow of a vortex near some stair steps. This was accomplished by mapping a flat wall to stair steps using a Schwarz-Christoffel transformation:

(* runtime: 0.7 second *)
Clear[z]; gamma = 1; z0 = 1 + I; z1 = I;z2 = 0; z3 = 1; zeta1 = -1; zeta2 = 0; zeta3 = 1; theta1 = 3Pi/2; theta2 = Pi/2; theta3 = 3Pi/2;
z = z[zeta] /. DSolve[{z'[zeta] == K(zeta - zeta1)^(theta1/Pi - 1)(zeta - zeta2)^(theta2/Pi - 1)(zeta -zeta3)^(theta3/Pi - 1), z[zeta1] == z1}, z[zeta], zeta][[1]];
z = z /. K -> (K /. Solve[(z /. zeta -> zeta2) == z2, K][[1]]); zeta0 = zeta /. FindRoot[z == z0, {zeta, zeta2 + I}];
Z[w_] = z /. zeta -> (zeta /. Solve[w == (I gamma/(2Pi))(Log[zeta - zeta0] - Log[zeta - Conjugate[zeta0]]), zeta][[1]]);
ToPoint[z_] := {Re[z], Im[z]}; grid = Table[ToPoint[Z[phi + I psi]], {phi, -1, 0, 0.05}, {psi, -1*^-6, -1, -0.02}];
Show[Graphics[{Map[Line, grid], Map[Line, Transpose[grid]]}, AspectRatio -> Automatic, PlotRange -> {{-1, 2}, {-1, 2}}]]

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