In deRham's theory, a closed 1form, A, consists of two parts: a perfect
differential and a harmonic part. In three dimensions, such covariant vector
fields consist of a pure gradient (with zero curl, but perhaps finite
divergence), and a nongradient harmonic part (that has zero divergence and zero
curl). Such flows have zero vorticity. The harmonic part cannot exist on a
topological domain which is simply connected.
The two dimensional theory of Lift (such as that championed by Joukowski)
considers the topologically nonsimply connected domain exterior to an airfoil
of arbitrary shape. The Lift force is orthogonal to the velocity field (similar
to a Lorentz or so called Magnus force) such that the power (force times
velocity) is zero. Joukowski showed that the Lift is due to the line integral
of the velocity along any closed path in the exterior domain that encloses the
wing. The value of such an integral is defined as the net circulation, and as
Joukowski proved, depends entirely on the harmonic component of the vector
field, and is
If the harmonic component of the 1form vanishes so does the Lift. In two
dimensions, the shape of the airfoil does nothing except to insure that a
translation from an initial condition at rest to a constant translational
velocty will induce a harmonic component of the flow vector field. Any
mechanism that will produce the same harmonic component will produce the same
Lift.
In two dimensions such flows have zero vorticity and zero drag. In three
dimensions, a wing of finite aspect ratio will induce a flow component which has
finite curl, or vorticity, which leads to the generation of tip vortices and
induced drag. High performance sail planes have very high aspect ratios just to
minimize this topological effect. For more info click here.
