Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/163

§ 109. Dynamic Support.—Endeavours have been made in the past to apply the principles of the conservation of momentum—that is, the doctrine of the continuous communication of momentum (§ 3)—to estimate directly the efficiency of an aeroplane sustaining a load and the expenditure of power necessary. If the air were a fluid discontinuous after the manner of the Newtonian medium, then such methods would lead to immediate and reliable results, for we know that if $W$ be the weight supported, and $v$ the velocity of downward discharge, and $m$ the mass per second of the projected particles,

$W=mv\quad$ or $\quad v=W/m$

(1)

and if $E=$ energy expended per second,

$E={\frac {1}{2}}\ mv^{2}={\frac {W^{2}}{2m}}$

or for any given weight to be sustained ($W=$ constant) the energy is inversely as the mass of fluid dealt with per second.

Something by way of convention is necessary to connect the above quantities with the size and velocity of the wing member. Thus if we suppose the latter to be an elastic aeroplane of area $A$ and angle $\beta ,$ travelling at a velocity $V,$ we shall have:—

$v=V\times 2\sin \beta$ (where $v$ is the velocity imparted at right angles to the plane), and $m=VA\ \rho \ \sin \beta ,$ and (1) becomes

$W=\rho \ A\ V^{2}\ 2\ sin^{2}\beta .$

If we had taken for our convention that the surface of the aeroplane is inelastic,^[1] then, since the particles on impact would not bounce off, $v=V\ \sin \beta$ and

$W=\rho \ A\ V^{2}\ \sin ^{2}\beta .$

(2)

The above results are not altogether in harmony with experience. The weight sustained does vary approximately with the area of the plane and density of the fluid, and as the square of the velocity, but the relationship in respect of angle does not hold good.

Let us introduce an elementary notion of continuity into the fluid. It is evident that when the layers of air adjacent to the

1. Compare “Principia,” prop, xxx., Book II.

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