Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/25

Rh 14°C, and 760 m.m. pressure, that is to say, let one cubic foot contain $$\tfrac{1}{13}$$th lb. mass. Let $$P$$ represent the pressure per square foot, that is, the total force $$F$$ divided by the area of the plane. Then the mass dealt with per second to develop a force $$P$$ will be $$\frac{V}{13} ,$$ and the velocity $$v$$ being equal to $$V ,$$ we have:—

Momentum per second $$= P = \frac{V^2}{13} .$$ But $$P$$ here is in absolute units, poundals. Reducing to pounds, we have:—

Pressure $$= \frac{V^2}{13\ g} = \frac{V^2}{13 \times 32.2} = \frac{V^2}{420}$$ approximately. If the velocity be expressed in miles per hour, this becomes $$\frac{V^2}{200}$$ (nearly). This may be recognised at once as a result often given in text books as the pressure-velocity equation for air, and is tacitly put forward as if founded on experiment. It is approximately 50 per cent. higher than the true value.

If, instead of introducing a value for the density, we denote this by $$p ,$$ the expression (absolute units) is: $$P = \rho\ V^2 ;$$ the experimental value is, in the case of air, $$P = .7\ \rho\ V^2$$, or, in the case of water, $$P =.55\ \rho\ V^2 ,$$ as ascertained for flat plates of compact outline. (See Chap. V.)

§ 5. Deficiency of the Newtonian Method.—It is evident from the foregoing that the theory of the Newtonian medium is capable of giving results within measure of the truth, when applied to real fluids. The degree of accuracy varies with the circumstances, and the author will now endeavour to point out the reasons why, and the manner in which, the method fails, and indicate the circumstances under which the Newtonian theory is applicable and those under which it is not.

At the outset it may be set down that any defect in the theory is due, not to any want of exactitude in the fundamental theory—this rests definitely on the third law of motion and is absolute—but rather to the difficulty and uncertainty as to its manner of application in the case of real fluids.