Neo-Newtonianism and the “Sweep” of a Wing
G. H. Bryan described the approach to lift adopted by the practical men as neo-Newtonian.64 The label accurately identified two salient features of their work. First, like everyone else, the practical men operated within the framework of Newton’s mechanics. Ultimately the wing must act on a mass of air, accelerating it downward, thus ensuring, in accordance with Newton’s third law of motion, that the wing suffered an equal and opposite reaction. This reaction was the ultimate source of the lift. Second, the practical men adopted a line of reasoning that was, in some respects, analogous to one that Newton used in the Principia when he compared the forces exerted by a flowing fluid on a sphere and a cylinder “described on equal diameters.”65 Recall that Newton assumed that his fluid, or “rare medium,” consisted of a number of independent particles which would hit the sphere and cylinder and give up their momentum. (This was the model I previously likened to a shower of hailstones). The practical men greatly simplified the analysis by addressing the case of a flat plate exposed to the uniform flow of this rare medium. While Newton was no doubt conscious of the distinction between his hypothetical fluid and real air, this difference tended to be blurred in some of the later aerodynamic discussions. By applying the reasoning to a simplified wing moving in air, the following argument was constructed.
Suppose that a flat plate has area A and is at an angle 0 to a uniform, horizontal flow of a fluid that was, like Newton’s, composed of independent particles. Suppose, further, that the collisions are inelastic so that the particles simply slide along the plane after impact. Let the particles in the main flow move at speed V units of distance per second. Then the volume of fluid striking the plate each second is given by multiplying the vertical projection of the plate (A sin0) with the velocity V. The projection A sin0 was the “sweep” of the wing and sin0 was the “sweep factor.” Now multiply the volume AVsin0 by the density p (presumed to be the same as that of the air) to give the mass, and then multiply the mass by the velocity component normal to the plate (V sin0) to give the momentum exchanged per second. This is the source of the pressure P whose vertical component, P cos0, is the lift and whose horizontal component, P sin0, is the drag. This neo-Newtonian argument gave the formula for the resultant aerodynamic force P on the plate as
P = pAV2 sin2 0.
The formula was often called Newton’s sin2 law, although it is not to be found, in an explicit form, in the text of the Principia. There are two reasons for its absence. First, Newton was dealing with a curved surface not a flat plate, and second, his reasoning was geometrical in form, so that the trigonometric terms appear as geometrical ratios.66 The label is, however, a reasonable one. All of the subsequent work of the practical men involved versions of, and variations on, this formula.
For the range of angles relevant to aeronautics, sin0 is a small quantity, so its square is very small indeed. The Newtonian formula condemns any predicted lift to be small, except where the magnitude of A and V2 can offset the smallness of the squared sin0 term. On this analysis, lift would demand enormous velocities or unrealistic wing areas. Had the formula been true it would have rendered artificial flight a practical impossibility. It is little wonder that in his 1876 paper Rayleigh had expressed satisfaction that his own formula made pressure proportional to sin0 rather than, as in Newton’s for mula, to sin2 0. In following the Newtonian tradition the practical men inherited a serious problem and resorted to a variety of expedients in an attempt to overcome it. A number of examples will show how comprehensively they failed to meet this challenge.