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Numerical solution to the rotating disk problem? 

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September 2, 2005, 01:24 
Numerical solution to the rotating disk problem?

#1 
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Hi, all:
I am long harassed by finding a numerical method to the problem of rotating disk. Here, the name "rotating disk" problem refer to a set of four ordinary differential equations modelled from the axissymmetric flow from a rotating disk (see P163 ~ 168 of the book <<Viscous Fluid Flow>> by Frank M. White): H' = 2F F'' = G^2 + F^2 + F'H G'' = 2FG + HG' P' = 2FH  2F' with boundary conditions F(0)=H(0)=P(0)=0, G(0)=1, and F(infinity)=G(infinity)=0. I know that to solve the equations numerically, we must find out the needed inital value of F'(0), and G'(0), to let F and G vanish at the infinite far end. But for me, it seems a stiff system, and shooting just can't converge. Any suggestions? Btw, is there any paper that is on this topic? After all, it is a textbook problem, and should be solved numrically long before. Thanks, 

September 2, 2005, 04:20 
Re: Numerical solution to the rotating disk proble

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The following two refs, are for stationary disk, may be of some use to you.....
S. Nitin and R.P. Chhabra, Wall effects in twodimensional axisymmetric flow over a circular disk oriented normal to flow in a cylindrical tube, Canadian Journal of Chemical Engineering, 83, 450457 (2005). S. Nitin and R.P. Chhabra, Sedimentation of a circular disk in power law fluids, Journal of Colloid and Interface Science, In Press, Available online 1 September 2005, 

September 2, 2005, 04:37 
Re: Numerical solution to the rotating disk proble

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I don't know why the book's name can't be displayed.
The problem is described in Frank M. White's book "Viscous Fluid Flow" in Page 163 ~ 168. It can be reduced to solve the four ordinary differential equations numerically: H' = 2F F'' = G^2 + F^2 + F'H G'' = 2FG + HG' P' = 2FH  2F' subject to F(0)=H(0)=P(0)=0, G(0)=1, and F(infinity)=G(infinity)=0 The key point is to find the two initial values F'(0) and G'(0) to let F(inf) and G(inf) vanish. Up to this point, it is a pure math problem. And maybe it can be solved with shooting method. The difficulty is: there are two valued to be shooted, and this may be quite difficult due to the system's stiffness. Since this is quite an old textbook problem, I don't think these 2 new papers can be of help. If it is solved, it should be long before. Anyway, thanks for your attention, 

September 2, 2005, 04:38 
Re: Numerical solution to the rotating disk proble

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The easiest way to solve this equation is to write the equations in finite difference form and Newton iterate. Otherwise try the book "Two point boundary value problems" by H.B. Keller which I vaguely recall discusses the related FalknerSkan problem. You could also try Schlicting or any other book on boundarylayers (your problem is usually referred to as the von Karman similarity solution).


September 2, 2005, 05:24 
Re: Numerical solution to the rotating disk proble

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Yes, I have consulted that book. However, in FalknerSkan problem, there is only one value to be shooted, so that's relatively easy.
I even solved Blasius equation numerically only by linear shooting. But for this rotating disk problem, two values are to be determined. Schlicting's classical book also didn't talk of how to numerically get F'(0) and G'(0), it mentioned that those two valued were obtained by the method of series expansion and match. But today this method is clearly two complicated and outdated. With the power of modern computer, it should not be that difficult to solve it numerically. Actally, I have tried to mail Prof Keller about this problem, coz he is an established expert in boundaryvalue problem of ODE; but i got no reply. 

September 2, 2005, 07:44 
Re: Numerical solution to the rotating disk proble

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Have you tried my ealier suggestion of Newton iteration on the finite difference version of the equations  there is no need to know either F'(0) or G'(0) in this case?
I think you're being a bit unfair on the series solution method  have a look in either the Journal of Fluid Mechanics or the European Journal of Mechanics B/Fluids to see that these methods/ideas are still used. Also have a look at http://anziamj.austms.org.au/V42/CTAC99/Kels/home.html I suspect Keller didn't reply to your email because he's retired (he's about 80 years old now!) 

September 4, 2005, 22:30 
Re: Numerical solution to the rotating disk proble

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Hi, Tom:
Sorry I didn't pay attention to the FiniteDifference method you methioned in last post. And it didn't occure to me that series method is still in large use these days. Thanks for your opinions and that URL  it helps. 

September 5, 2005, 05:53 
Re: Numerical solution to the rotating disk proble

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I am not sure how much it is relevant to your work, but if you have not, have a look at numerical recipe book, might find something useful , here is the link:
http://www.library.cornell.edu/nr/cbookcpdf.html 

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