(9.3) |

According to Equation (6.172), the circulation, , of air about the airfoil is determined by performing the integral

(9.4) |

around a loop that lies just above the airfoil surface. However, as discussed in Section 6.10, the value of this integral is unchanged if it is performed around any loop that can be continuously deformed onto , while not passing through the airfoil surface, or crossing a singularity of the complex velocity, (i.e., a line source or a -directed vortex filament). Because (in the high Reynolds number limit in which the boundary layer and the wake are infinitely thin) there are no line sources or -directed vortex filaments external to the airfoil, we can evaluate the integral around a large circle of radius , centered on the origin. It follows that and . Hence,

(9.5) |

which implies that

at large .

As discussed in Section 6.11, the net force (per unit length) acting on the airfoil, , is determined by performing the Blasius integral,

(9.7) |

around a loop that lies just above the airfoil surface. However, as before, the value of the integral is unchanged if instead we perform it around a large circle of radius , centered on the origin. Far from the airfoil,

(9.8) |

So, we obtain

(9.9) |

or

In other words, the resultant force (per unit length) acting on the airfoil is of magnitude , and has the direction obtained by rotating the wind vector through a right-angle in the sense opposite to that of the circulation. This type of force is known as

As discussed in Section 6.11, the net moment per unit length (about the origin), , acting on the airfoil is determined by performing the integral

(9.11) |

around a loop that lies just above the airfoil surface. As before, we can deform into a circle of radius , centered on the origin, without changing the value of the integral. Hence, we obtain

(9.12) |

or