Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/94

§ 55 general that the experimental determinations, except for velocities quite small in comparison with that of wave motion, will be in excess of those indicated by the theory. It is evident that in elasticity we have a factor foreign to the dimensional theory as given, and the existence of such a factor invalidates the hypothesis upon which the theory is founded.

An apparent discrepancy occurs in the case of some experiments made by Newton (Book II., Section VI.), who found, from the motion of a pendulum whose spherical "bob" was immersed in water, that the resistance was augmented in more than the duplicate ratio of the velocity.

Newton supposed this to be an error due to the narrowness of the trough employed, but this in the light of dimensional theory is insufficient.

The probable explanation is that for a large arc the discontinuous type of motion has time to establish itself on each swing, whereas for small arcs of motion the flow has not had time to fully develop discontinuity; for very small arcs the flow will approximate to the Eulerian form (compare Chap. III.). Consequently the resistance for small amplitude is far less than is the case for continuous motion, and thus factors are introduced outside the dimensional hypothesis, which presumes a steady state.

§ 56. General Conclusions.—The importance of the results attained in the present chapter, in relation to aerial flight, is to some extent an unknown quantity.

It is evident that under ordinary conditions the law of viscosity does not apply, and it would appear further that the tangential resistance does not follow the normal law of skin friction, but that the conditions commonly involve turbulence, and $$R$$ varies as some higher power of $$V$$ between $$V^{1.5}$$ and $$V^2 .$$ It is highly probable that the conditions may be different in the case of the smaller flying insects, such as flies, mosquitoes, etc., and it may be the relatively greater importance of viscosity in such cases that is primarily responsible for the peculiarities of insect flight.