Editing Lift (force) (section)

===Integrated force/momentum balance in lifting flows===
[[File:Airfoil control volumes.jpg|thumb|400px|right|Control volumes of different shapes that have been used in analyzing the momentum balance in the 2D flow around a lifting airfoil. The airfoil is assumed to exert a downward force −L' per unit span on the air, and the proportions in which that force is manifested as momentum fluxes and pressure differences at the outer boundary are indicated for each different shape of control volume.]]

The flow around a lifting airfoil must satisfy Newton's second law regarding conservation of momentum, both locally at every point in the flow field, and in an integrated sense over any extended region of the flow. For an extended region, Newton's second law takes the form of the ''momentum theorem for a control volume'', where a [[control volume]] can be any region of the flow chosen for analysis. The momentum theorem states that the integrated force exerted at the boundaries of the control volume (a [[surface integral]]), is equal to the integrated time rate of change ([[material derivative]]) of the momentum of fluid parcels passing through the interior of the control volume. For a steady flow, this can be expressed in the form of the net surface integral of the flux of momentum through the boundary.<ref>Shapiro (1953), Section 1.5, equation 1.15</ref>

The lifting flow around a 2D airfoil is usually analyzed in a control volume that completely surrounds the airfoil, so that the inner boundary of the control volume is the airfoil surface, where the downward force per unit span <math>-L'</math> is exerted on the fluid by the airfoil. The outer boundary is usually either a large circle or a large rectangle. At this outer boundary distant from the airfoil, the velocity and pressure are well represented by the velocity and pressure associated with a uniform flow plus a vortex, and viscous stress is negligible, so that the only force that must be integrated over the outer boundary is the pressure.<ref name="Lissaman 1996">Lissaman (1996), "Lift in thin slices: the two dimensional case"</ref><ref name="Durand 1932">Durand (1932), Sections B.V.6, B.V.7</ref><ref name="Batchelor 1967 p. 407">Batchelor (1967), Section 6.4, p. 407</ref> The free-stream velocity is usually assumed to be horizontal, with lift vertically upward, so that the vertical momentum is the component of interest.

For the free-air case (no ground plane), the force <math>-L'</math> exerted by the airfoil on the fluid is manifested partly as momentum fluxes and partly as pressure differences at the outer boundary, in proportions that depend on the shape of the outer boundary, as shown in the diagram at right. For a flat horizontal rectangle that is much longer than it is tall, the fluxes of vertical momentum through the front and back are negligible, and the lift is accounted for entirely by the integrated pressure differences on the top and bottom.<ref name="Lissaman 1996"/> For a square or circle, the momentum fluxes and pressure differences account for half the lift each.<ref name="Lissaman 1996"/><ref name="Durand 1932"/><ref name="Batchelor 1967 p. 407"/> For a vertical rectangle that is much taller than it is wide, the unbalanced pressure forces on the top and bottom are negligible, and lift is accounted for entirely by momentum fluxes, with a flux of upward momentum that enters the control volume through the front accounting for half the lift, and a flux of downward momentum that exits the control volume through the back accounting for the other half.<ref name="Lissaman 1996"/>

The results of all of the control-volume analyses described above are consistent with the Kutta–Joukowski theorem described above. Both the tall rectangle and circle control volumes have been used in derivations of the theorem.<ref name="Durand 1932"/><ref name="Batchelor 1967 p. 407"/>