Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/207

Rh clear; the existence of 3uch changes shows the form of the expression to be inexact, for, according to Allen (§§ 35 and 42), under these circumstances the $$V^2$$ law cannot strictly apply. It might, without departing from the form of the expression, be possible to establish an empirical relationship, and it is in any ease of interest to endeavour to ascertain the probable magnitude of the constant for the particular case when viscosity becomes vanishingly small.

There is no fluid known of which it can be said that viscosity is a negligible quantity; neither is it possible to deduce from the data of known fluids what the behaviour of such a fluid would be. We have consequently to fall back on pure theory.

§ 136. Normal Plane Theory Summarised.—Several methods of computing the pressure on a normal plane have been proposed; up to the present none of these can be considered entirely satisfactory.

1. The Method of the Neutonian Medium.—The theory of the Newtonian medium has been already discussed (§ 4); it has been shown that on this hypothesis we have two possible results: (a) if the particles are elastic, $$P = 2\ \rho\ V^2 ;$$ (b) if the particles are inelastic, $$P = \rho\ V^2 .$$

Both these results are higher than that given by experiment for a viscous fluid, a defect that is due to the faulty hypothesis, the Newtonian medium possessing no continuity. Newton was fully conscious of this fact.

2. The Neutonian Method (Book II., Section VII. prop. xxxvii.).—In this proposition, Newton arrives at a result for a fluid possessing continuity the equivalent of which is:—

3. The Torricellian Method is here so named merely as a matter of convenience as being based on the Torricellian principle, and not as due to Torricelli himself.

In a continuous fluid, the theorem of Torricelli, which is