Kutta–Joukowski theorem

First of all, the force exerted on each unit length of a cylinder of arbitrary cross section is calculated.^[6] Let this force per unit length (from now on referred to simply as force) be ${\displaystyle \mathbf {F} }$. So then the total force is:

${\displaystyle \mathbf {F} =-\oint _{C}p\mathbf {n} \,ds,}$

where C denotes the borderline of the cylinder, ${\displaystyle p}$ is the static pressure of the fluid, ${\displaystyle \mathbf {n} \,}$ is the unit vector normal to the cylinder, and ds is the arc element of the borderline of the cross section. Now let ${\displaystyle \phi }$ be the angle between the normal vector and the vertical. Then the components of the above force are:

${\displaystyle F_{x}=-\oint _{C}p\sin \phi \,ds\,,\qquad F_{y}=\oint _{C}p\cos \phi \,ds.}$

Now comes a crucial step: consider the used two-dimensional space as a complex plane. So every vector can be represented as a complex number, with its first component equal to the real part and its second component equal to the imaginary part of the complex number. Then, the force can be represented as:

${\displaystyle F=F_{x}+iF_{y}=-\oint _{C}p(\sin \phi -i\cos \phi )\,ds.}$

The next step is to take the complex conjugate of the force ${\displaystyle F}$ and do some manipulation:

${\displaystyle {\bar {F}}=-\oint _{C}p(\sin \phi +i\cos \phi )\,ds=-i\oint _{C}p(\cos \phi -i\sin \phi )\,ds=-i\oint _{C}pe^{-i\phi }\,ds.}$

Surface segments ds are related to changes dz along them by:

${\displaystyle {\begin{aligned}dz&=dx+idy=ds(\cos \phi +i\sin \phi )=ds\,e^{i\phi }\\{}\Rightarrow d{\bar {z}}&=e^{-i\phi }ds.\end{aligned}}}$

Plugging this back into the integral, the result is:

${\displaystyle {\bar {F}}=-i\oint _{C}p\,d{\bar {z}}.}$

Now the Bernoulli equation is used, in order to remove the pressure from the integral. Throughout the analysis it is assumed that there is no outer force field present. The mass density of the flow is ${\displaystyle \rho .}$ Then pressure ${\displaystyle p}$ is related to velocity ${\displaystyle v=v_{x}+iv_{y}}$ by:

${\displaystyle p=p_{0}-{\frac {\rho |v|^{2}}{2}}.}$

With this the force ${\displaystyle F}$ becomes:

${\displaystyle {\bar {F}}=-ip_{0}\oint _{C}d{\bar {z}}+i{\frac {\rho }{2}}\oint _{C}|v|^{2}\,d{\bar {z}}={\frac {i\rho }{2}}\oint _{C}|v|^{2}\,d{\bar {z}}.}$

Only one step is left to do: introduce ${\displaystyle w=f(z),}$ the complex potential of the flow. This is related to the velocity components as ${\displaystyle w'=v_{x}-iv_{y}={\bar {v}},}$ where the apostrophe denotes differentiation with respect to the complex variable z. The velocity is tangent to the borderline C, so this means that ${\displaystyle v=\pm |v|e^{i\phi }.}$ Therefore, ${\displaystyle v^{2}d{\bar {z}}=|v|^{2}dz,}$ and the desired expression for the force is obtained:

${\displaystyle {\bar {F}}={\frac {i\rho }{2}}\oint _{C}w'^{2}\,dz,}$

which is called the Blasius theorem.

To arrive at the Joukowski formula, this integral has to be evaluated. From complex analysis it is known that a holomorphic function can be presented as a Laurent series. From the physics of the problem it is deduced that the derivative of the complex potential ${\displaystyle w}$ will look thus:

${\displaystyle w'(z)=a_{0}+{\frac {a_{1}}{z}}+{\frac {a_{2}}{z^{2}}}+\cdots .}$

The function does not contain higher order terms, since the velocity stays finite at infinity. So ${\displaystyle a_{0}\,}$ represents the derivative the complex potential at infinity: ${\displaystyle a_{0}=v_{x\infty }-iv_{y\infty }\,}$. The next task is to find out the meaning of ${\displaystyle a_{1}\,}$. Using the residue theorem on the above series:

${\displaystyle a_{1}={\frac {1}{2\pi i}}\oint _{C}w'\,dz.}$

Now perform the above integration:

${\displaystyle {\begin{aligned}\oint _{C}w'(z)\,dz&=\oint _{C}(v_{x}-iv_{y})(dx+idy)\\&=\oint _{C}(v_{x}\,dx+v_{y}\,dy)+i\oint _{C}(v_{x}\,dy-v_{y}\,dx)\\&=\oint _{C}\mathbf {v} \,{ds}+i\oint _{C}(v_{x}\,dy-v_{y}\,dx).\end{aligned}}}$

The first integral is recognized as the circulation denoted by ${\displaystyle \Gamma .}$ The second integral can be evaluated after some manipulation:

${\displaystyle \oint _{C}(v_{x}\,dy-v_{y}\,dx)=\oint _{C}\left({\frac {\partial \psi }{\partial y}}dy+{\frac {\partial \psi }{\partial x}}dx\right)=\oint _{C}d\psi =0.}$

Here ${\displaystyle \psi \,}$ is the stream function. Since the C border of the cylinder is a streamline itself, the stream function does not change on it, and ${\displaystyle d\psi =0\,}$. Hence the above integral is zero. As a result:

${\displaystyle a_{1}={\frac {\Gamma }{2\pi i}}.}$

Take the square of the series:

${\displaystyle w'^{2}(z)=a_{0}^{2}+{\frac {a_{0}\Gamma }{\pi iz}}+\cdots .}$

Plugging this back into the Blasius–Chaplygin formula, and performing the integration using the residue theorem:

${\displaystyle {\bar {F}}={\frac {i\rho }{2}}\left[2\pi i{\frac {a_{0}\Gamma }{\pi i}}\right]=i\rho a_{0}\Gamma =i\rho \Gamma (v_{x\infty }-iv_{y\infty })=\rho \Gamma v_{y\infty }+i\rho \Gamma v_{x\infty }=F_{x}-iF_{y}.}$

And so the Kutta–Joukowski formula is:

${\displaystyle {\begin{aligned}F_{x}&=\rho \Gamma v_{y\infty }\,,&F_{y}&=-\rho \Gamma v_{x\infty }.\end{aligned}}}$