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By derivation, if it is not correct, I mean a proof, something showing how the function was created.

I have found the magnus force quite easily by searching through the web and it seems to be:

fm=S(v)ω^v

As I said I need some kind of equations leading to this formula.

Also I am wondering if anyone here is familiar with solving boundary layer equations (numerically) as I have a need for doing that but am unable to.

I understand my language is not perfect and I urge you to ask if there is something that is not fully understandable.