Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/74

if the mean velocity exceeds ${\frac {1000\ \nu }{\alpha }}$ where $\alpha$ is the radius of the tube^[1] (c.g.s. units).

§ 38. General Expression.—Homomorphous Motion.—Let us examine generally the relations of geometrically similar systems possessed of homomorphous motion—that is, under circumstances when the theory of dimensions is strictly applicable, then the quantities upon which the motion depends are comprised by—velocity $=V,$ kinematic viscosity $\nu$, and a linear (scale) dimension $l.$

Let us write

$l=c\ V^{p}\ \nu ^{q},$

or, in terms of dimensions

$L={\frac {L^{p}}{T^{p}}}\times {\frac {L^{2q}}{T^{q}}},$

and we have the equations

	$p+q=0,$
	$p+2q=1,$
∴	$q=1\quad \quad p={\boldsymbol {-}}1,$
∴	$l=c\ {\frac {\nu }{V}}\quad$ or $\quad V=c\ {\frac {\nu }{l}},$	(7)

which may be taken as the general equation connecting all similar systems of flow in viscous fluids.

In a number of tubes, such as may be supposed employed for experimentally investigating the phenomenon of turbulence, we have a number of such similar systems, and it will be noted that the expression is identical with that arrived at by Mr. Osborne Reynolds, in whose equation we have $c=1000$ as expressing a particular state of motion.

§ 39. Corresponding Speed.—The above expression enables us to formulate at once a law of corresponding speed for motion in any viscous fluid, for, if the physical properties of the fluid do not vary in any way $\nu$ will be constant and we have $V\propto {\frac {1}{l}},$ that

1. Poynting and Thomson, “Properties of Matter,” Chap. XVIII.

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