ben682 said:
i was wondering if anyone knew how to derive the magnus effect equation?
Sort of. At least, in a somewhat idealized setup, which should give you a qualitative answer, at least.
Using Kutta-Joukowski theorem, lift per infinitesimal slice can be computed via circulation.
dF_L = \rho \Gamma v dl
Where gamma is the circulation. Technically, all three of these should be taken at infinity. However, if you assume a laminar flow, the circulation is going to be the same everywhere, and you know circulation at the boundary of the sphere. Circulation is given by the integral along a closed curve.
\Gamma = \int_C v\cdot ds
And the layer at the boundary of the sphere moves along with the sphere. That gives Γ=2ωπr², where ω is angular velocity component perpendicular to v. The total lift can then be computed by integrating over all slices of the sphere.
F_L = \int_{-R}^{R} 2\pi \rho r^2 (\omega \times v) dl = \frac{8}{3}\pi \rho R^3 \omega \times v
Which gives you the correct ωxv dependence of lift. How close the coefficient is going to be, however, I have no idea.
Edit: I just sort of assumed spherical shape in that last integral. If it's a cylinder, or whatever, you'll need to correct for whatever r(l) dependence you actually have.
