# Kutta-Joukowski theorem

### From Physics wiki

The **Kutta-Joukowski theorem** applies to 2-dimensional flow around an aerofoil, and gives the lift on the aerofoil as

- ,

under the following assumptions:

- incompressible flow
- the circulation is evaluated in an inviscid region (outside the boundary layer)
- flow is described by a potential

## Derivation

The force per unit length on the aerofoil is

where is the line element along the contour of the aerofoil and is the unit vector normal to it. Introduce the complex function . Then

- ,

where is the angle that makes with the direction. So

, ,

Note that if then (we take the contour to be counterclockwise in the complex plane). For later use, note that . Then

,

Using Bernoulli's principle,

- ,

so that

- ,

Outside of the boundary layer, we can encode the velocity by introducing a velocity potential . Immediately outside the boundary layer the flow is assumed to be tangential to the contour, although it might have opposite sense. In other words, or along various segments of the contour. Thus , and

- ,

or

,

which is the Blasius–Chaplygin formula. The function may be expanded as a Laurent series,

- ,

where we exclude positive powers since the velocity is to remain finite at infinity. Therefore

- ,

, by the residue theorem, , , introducing the stream function , , , using the assumption that is a stream line, , where is the vorticity.

Therefore,

- ,

and

- .

We can now evaluate , or its complex conjugate, using the residue theorem:

- .

Thus

,

.