Kutta–Joukowski theorem

Remarks

The proof uses the Blasius theorem. Far from the body the complex potential has the expansion $w(z) = Uz - \frac{i\Gamma}{2\pi}\log z + c + \frac{m}{z} + \cdots$, since the flow is a uniform stream plus a vortex plus terms that decay. Then $(w')^2 = U^2 - \frac{iU\Gamma}{\pi z} + O(z^{-2})$, and the residue theorem on a large circle gives $F_x - i F_y = \frac{i\rho}{2} \cdot 2\pi i \cdot \left(-\frac{iU\Gamma}{\pi}\right) = i\rho U\Gamma$. For a flat plate of chord $c$ at angle of attack $\alpha$, the Kutta condition fixes $\abs{\Gamma} = \pi U c \sin\alpha$, which gives a lift $\pi \rho U^2 c \sin\alpha$ per unit span. Without circulation the force is zero, which is the two-dimensional form of d'Alembert's paradox.

Going deeper

For a student

The theorem reduces the lift of any two-dimensional profile in potential flow to a single number, the circulation. The shape of the body enters only through the value of $\Gamma$, which potential flow alone does not determine. In airfoil theory the Kutta condition at the sharp trailing edge selects the physical value of $\Gamma$. The same formula, applied to each section of a wing, is the basis of lifting-line theory, and it also explains the Magnus force on a spinning cylinder.