Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/182

§ 120 decided by aerodynamic considerations alone, and that the question of equilibrium is involved.

It is evident that we are not bound to our assumption of uniform load distribution, and that if we suppose the pressure difference (between the under and upper surfaces) to be less towards the extremities, the latter may be made proportionately fuller without seriously disturbing the relative distribution of the stray field; we might thus take an elliptical form as a standard, with a pressure distribution appropriately proportioned. In general, the wing-plan of a bird has ordinates that approximate more or less closely to those of an ellipse. The discussion of the practical aspect of this question will be resumed in a subsequent chapter.

§ 121. Hydrodynamic Interpretation and Development.—We may recognise in the foregoing investigation (§§ 115 and 116) an elaboration of the theory initially put forward in § 90 (Chap. III), where the forces acting on the fluid were dealt with in bulk, instead of as in the present instance being studied in detail.

In § 90 it was shown that the disturbance peculiar to the neighbourhood of the aerofoil possesses angular momentum, and it was inferred that this being the case, the disturbance comprises a cyclic motion, for otherwise it must involve rotation, which is excluded by the nature of the hypothesis. We are consequently confined, in an inviscid atmosphere, strictly to the case where the aerofoil is of infinite lateral extent, for a cyclic motion is only possible in a multiply connected region.

The problem, then, from the hydrodynamic standpoint, resolves itself into the study of cyclic motion superposed on a translation. We have already devoted some attention to such a combination, and we have traced the field in a simple case for values of the functions $$\psi$$ and $$\phi ,$$ Fig. 48. In Fig. 70 we have the stream lines for this particular case plotted over a considerably greater area, the internal system of flow being replaced by a solid of