# Modeling a 3D Wing — Joukowsky Airfoils and Prandtl Lifting-Line Theory

I am trying to calculate the distribution of lift over the span of a rectangular, non-swept wing with a (constant) Joukowsky airfoil cross-section. The wing is rectangular in the sense that the chord length is constant. The hope is to combine results from Joukowsky airfoils and Lifting-Line theory. I have followed the derivations from the following sources:

http://brennen.caltech.edu/fluidbook/basicfluiddynamics/potentialflow/complexvariables/joukowskiairfoils.pdf

https://en.wikipedia.org/wiki/Lifting-line_theory

For the Joukowsky airfoil, the circulation is fully determined by the airfoil geometry and angle of attack by imposing the Kutta condition (finite fluid velocity at the trailing edge):

$$\Gamma = -4\pi UR\sin (\alpha + \beta)$$

where $$\alpha$$ is the angle of attack and $$R$$ and $$\beta$$ determine airfoil geometry.

Now, in Lifting-Line theory, we require knowledge of the slope of the airfoil lift coefficient $$C_{\ell \alpha}$$, which I assume is $$\frac{dC_{\ell}}{d\alpha}$$. However, we come up with a solution for both the induced angle of attack $$\alpha_i$$ and the circulation $$\Gamma$$ which have nothing to do with Joukowsky airfoils or the Kutta condition. It would appear that both circulation and the effective angle of attack (geometric angle of attack $$-$$ $$\alpha_i$$) are specified now, and the above relation should not be true in general.

My question is: how is the circulation found from Lifting-Line theory consistent with results of Joukowksy airfoils, and the requirements of the Kutta condition? Can the two methods be used in tandem?