Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/71

and their distance from the material plane, shall be in the constant ratio $n$. And let us denote the distance between adjacent planes by the symbol $\triangle y$, and the corresponding velocity difference in the axis of $x$ by $\triangle u$. Then

$\triangle y\propto n$ and $\triangle u\propto V.$

(1)

Let $F=$ the tangential force.

We have $F$ as measured by viscosity varies as the area (which is constant) $\times$ velocity gradient, or—

$F\propto {\frac {du}{dy}}\propto {\frac {\triangle u}{\triangle y}}\propto {\frac {V}{n}}.$

(2)

And $F$ as dependent on dynamic considerations $=$ momentum imparted per second to the fluid. For unit width of any stratum we have mass $=\rho \triangle y\quad V,$ and velocity varies as $V$ or $F=\Sigma \rho \ \triangle y\ V^{2}.$

∴	$F\propto nV^{2}.$	(3)
By (2) and (3) we have—
	$nV^{2}\propto {\frac {V}{n}},$
or	$n\propto V^{-{\frac {1}{2}}},$
substituting in either (2) or (3)—
	$F\propto V^{1.5}.$	(4)

This may be taken as the normal law of skin-friction.^[1]

§ 36. Kinematical Relations.—In dealing with problems relating to fluid resistance it is found to lead to simplification to eliminate the density of the fluid by introducing two new quantities, kinematic resistance and kinematic viscosity.

Kinematic resistance, which we will denote by the symbol $R$, may be defined as the resistance per unit density, or $R=F/\rho$, and is consequently of the dimensions ${\frac {L^{4}}{T^{2}}}$.

1. The foregoing demonstration is here presented for the first time by the author; the experimental fact was discovered by Mr. H. S. Allen (compare § 50). The relation $F/V^{1.5}=$ const. may appropriately be termed Allen's law. It is evident in the above investigation that the balance of viscous and dynamic forces is demonstrated for all corresponding layers of the region each to each, for any number of cases of $V$ variation, and consequently the method is comprehensive, and includes both the plus and minus momentum of the wake and counterwake currents.