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System of second-order nonlinear ordinary differential equations

  • To: mathgroup at smc.vnet.net
  • Subject: [mg128657] System of second-order nonlinear ordinary differential equations
  • From: killer_katschinski <jan.hildenbeutel at gmail.com>
  • Date: Wed, 14 Nov 2012 01:29:46 -0500 (EST)
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Hello,

I'm new to this forum and also to Mathematica, so please treat me gentle :)

I would solve numerically the velocity profiles of a rotating disk. Thefore following system must be solved for the 3 (nondimensional) unknown velocity-functions F(eta), G(eta) and H(eta), with eta=nondimensional wall distance (see "Boundary Layer Theory" by H. Schlichting):

2F+H'=0
F^2+F'*H-G^2-F''=0
2F*G+H*G'-G''=0

with the follwing boundary conditions:

F(x=0)=0
G(x=0)=1
H(x=0)=0
F(x->\inf)=0
G(x->\inf)=0

I tried to implement the problem into Mathematica with the numerical solver :

NDsolve[{2*F[eta] + H'[eta] == 0, (F[eta])^2 + F'[eta]*H[eta] - (G[eta]^2 - F''[eta] == 0, 2*F[eta]*G[eta] + H[eta]*G'[eta] - G''[eta] == 0}, {F[0] == 0, G[0] == 1, H[0] == 0, F[1000000] == 0, G[1000000] == 0}, {eta, 1000000}]

Plot[Evaluate[{F[eta], G[eta], H[eta]} /. s], {eta, 0, 5}, PlotStyle -> Automatic]

I did not arrive to any solution but to many many errors. If you could give me any hint I would be really thankful!

Best regards

killer_katschinski



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