Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/164

§ 109 aeroplane are diverted these will react on the neighbouring layers of air, and so on, so that a stratum of some considerable thickness will be involved. Now the factor that must limit the thickness of this stratum is evidently the size and shape of the plane, for the more remote layers of the fluid only escape by the fact that a circulation takes place from the side of greatest to the side of least pressure, which circulation depends chiefly upon the size and shape, and but little upon the angle of the plane. The elasticity of the air might become sensible if the velocity were sufficient, but at ordinary velocities this factor is unimportant.

Let us then assume for our convention that the depth of the layer affected for a plane of given shape depends upon its linear dimension and is constant in respect of angle, the latter being supposed to be of small magnitude. Then, since under the pre- sent supposition the lines of flow will require to follow the surfaces of the plane (the fluid being unable to bounce off as in the previous case), we have

where $$\kappa$$ is a constant, and by (1) we obtain:—

This result for planes of certain general proportions, at small angles to the line of flight, agrees closely with experiment.

The quantity $$\kappa\ A$$ of equation (3) may be aptly termed the sweep of the aeroplane or wing. It is a measure of the effective cross-section of the horizontal column of air dealt with by the aeroplane or supporting member. It has been found, experimenting with superposed planes, that two planes fifteen inches by four inches in pterygoid aspect, and at angles less than ten degrees, do not suffer any sensible diminution of their individual sustaining power if they are separated by a vertical distance of four inches. It is therefore fair to assume that a plane of the dimensions stated is sustained by the inertia of a layer of air not more than four inches thick. That is to say, the sweep does not