= Solution
The <Magnus effect> is the transverse <force> produced when a spinning body moves relative to a <fluid>. Rotation changes the flow and <circulation> around the body; the resulting asymmetric <pressure> and wake deflect the <fluid>, with the opposite transverse <force> on the body. Examples include the curved flight of spinning balls and the lift of rotating-cylinder sails.
A useful quantitative model is steady two-dimensional <incompressible flow> past a cylinder of radius $a$, with <fluid> <velocity> $\mathbf U=U\hat{\mathbf x}$ far away. Let <circulation> $\Gamma$ be positive anticlockwise as viewed from $+z$. The potential-flow <velocity> at the cylinder surface is
$$
v_\theta(a,\theta)=-2U\sin\theta+\frac{\Gamma}{2\pi a}.
$$
The <Bernoulli equation> outside the <boundary layer> gives
$$
p(\theta)=p_\infty+\frac\rho2\left(U^2-v_\theta^2\right).
$$
Integrating the <pressure> <force> around the cylinder, per unit span, yields
$$
F_y'=-a\int_0^{2\pi}p(\theta)\sin\theta\,d\theta=-\rho U\Gamma.
$$
Thus the <Kutta–Joukowski theorem> gives
$$
\boxed{\mathbf F'=\rho\mathbf U\times(\Gamma\hat{\mathbf z}),\qquad |\mathbf F'|=\rho U|\Gamma|.}
$$
The sign here refers to the <fluid> <velocity> relative to the cylinder. For a body moving through otherwise stationary <fluid>, that relative <fluid> <velocity> points opposite to the body's translational <velocity>.
Real <viscosity> and the <no-slip boundary condition> transmit the surface rotation to the <fluid> and help set the <circulation> and separation pattern. Pure inviscid flow with only an impermeability condition does not determine $\Gamma$ from the cylinder's <angular speed>. Therefore the <circulation> must not be assigned a universal spin-rate formula without additional assumptions. The cylinder result illustrates the lift mechanism; quantitative <forces> on a finite spinning ball also depend on its geometry, <Reynolds number>, rotation rate, surface roughness and wake.
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