38 ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID [354
Proceeding in this way we see that the cosine factor may properly be identified with unity, and that the value of the integral for the first quadrant may be equated to I/log t'. And for a similar reason the quadrants after the first contribute nothing of this order of magnitude. Accordingly we may take
7 ,7 ,.n.
k cos nt an = —, - ; ......................... (43)
For the other part of (38), we get in like manner
8v ["sint'x.dec Sv
r -,, . _ 8v [
k smntdn = -- -I o a- j o
; - TIT/—/,-
a) log x a? JQ a log (t /as)
In the denominator of (44) it appears that ultimately we may replace log (t'/x) by log t' simply. Thus
k' sin nt dn — — : - , . ........................ (45)
a2 log t v '
so that the two integrals (43), (45) are equal. We conclude that when t is great enough,
ZvlW _ W
' o? log (4n8 ................... V >
But a better discussion of these integrals is certainly a desideratum.
§ 7. Whatever interest the solution of the approximate equations may possess, we must never forget that the conditions under which they are applicable are very restricted, and as far as possible from being observed in many practical problems. Dynamical similarity in viscous motion requires that Va/v be unchanged, a being the linear dimension. Thus the general form for the resistance to the uniform motion of a sphere will be
F = pvVa.f(Va/v'), ........................... (47)
where / is an unknown function. In Stokes' solution (I)/ is constant, and its validity requires that Vajv be small*. When V is rather large, experiment shows that F is nearly proportional to V\ In this case v disappears. " The second power of the velocity and independence of viscosity are thus inseparably connected "f.
The general investigation for the sphere moving in any manner (in a straight line) shows that the departure from Stokes' law when the velocity is not very small must be due to the operation of the neglected terms involving the squares of the velocities ; but the manner in which these act has not yet been traced. Observation shows that an essential feature in rapid fluid motion past an obstacle is the formation of a woke in the rear of the obstacle ; but of this the solutions of the approximate equations give no hint.
* Phil. Mag. Vol. xxxvi. p. 354 (1893) ; Scientific Papers, Vol. iv. p. 87. t Phil. Mag. Vol. xxxrv. p. 59 (1892) ; Scientific Papers, Vol. in. p. 576.