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 johny September 2, 2005 01:24

Numerical solution to the rotating disk problem?

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 axis-symmetric 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,

 ramp September 2, 2005 04:20

Re: Numerical solution to the rotating disk proble

The following two refs, are for stationary disk, may be of some use to you.....

S. Nitin and R.P. Chhabra, Wall effects in two-dimensional axisymmetric flow over a circular disk oriented normal to flow in a cylindrical tube, Canadian Journal of Chemical Engineering, 83, 450-457 (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,

 johny September 2, 2005 04:37

Re: Numerical solution to the rotating disk proble

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, :)

 Tom September 2, 2005 04:38

Re: Numerical solution to the rotating disk proble

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 Falkner-Skan problem. You could also try Schlicting or any other book on boundary-layers (your problem is usually referred to as the von Karman similarity solution).

 johny September 2, 2005 05:24

Re: Numerical solution to the rotating disk proble

Yes, I have consulted that book. However, in Falkner-Skan 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 boundary-value problem of ODE; but i got no reply. :(

 Tom September 2, 2005 07:44

Re: Numerical solution to the rotating disk proble

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 e-mail because he's retired (he's about 80 years old now!)

 johny September 4, 2005 22:30

Re: Numerical solution to the rotating disk proble

Hi, Tom:

Sorry I didn't pay attention to the Finite-Difference 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.

 zxaar September 5, 2005 05:53

Re: Numerical solution to the rotating disk proble

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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