Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/87

law, resistance varies as the 1.5th power of the velocity, for, where the force overcoming the resistance is supplied by the difference of specific gravity of the fluid and the sphere, we have—

	$R\propto l^{3},$
and	$R\propto V^{r}\ l^{r}\ \nu ^{2-r},$
∴	$l^{3}\propto V^{r}\ l^{r}\ \nu ^{2-r},$
∴	$l^{3-r}\propto V^{r}\ \nu ^{2-r},$
that is,	$V\propto {\frac {l^{\frac {3-r}{r}}}{\nu ^{\frac {2-r}{r}}}}.$
But for a given fluid $\nu$ is constant, and we have—
	$V\propto {\frac {l^{\frac {3-r}{r}}}{,}}$
or if	$V\propto l^{n}\quad n={\frac {3-r}{r}},$  or $\ r={\frac {3}{n+1}}.$

Now, if in stage (2) (Fig. 29) we ignore the small constant, we have $V\propto l$ or $n=1,$ and $r={\frac {3}{2}},$ that is to say, the general expression in this case becomes:—

$R=c\ \nu ^{\frac {1}{2}}\ l^{1{\frac {1}{2}}}\ V^{1{\frac {1}{2}}},$

that is, during this stage the resistance follows the normal law of skin-friction, or Allen's law.

 

§ 51. Characteristic Curve, Spherical Body.—The form of the experimental curve, as plotted by Mr. Allen, is given in Fig. 29, in which ordinates $=V,$ and abscissae = values of linear dimension, i.e., radius of sphere. The first stage or Stokes portion of this curve is a parabola, $V\propto l^{2};$ this corresponds to an $r$ value = unity; the second stage is approximately a straight line, the value of $r$ here being as shown 1.5; the third (or Newtonian) stage of the curve, not shown on this plotting, has an $r$ value equal 2, that is $n=.5$ or $V\propto l^{.5}.$ This form of plotting is the outcome of the method of experiment, i.e., measuring the limiting velocity acquired under the influence of gravity; if we re-plot as a resistance-velocity diagram (Fig. 30), the size of the body being supposed constant, we are able to obtain a general idea of the “characteristic curve of resistance” for a spherical