404 ON THE FLOW OF COMPRESSIBLE FLUID PAST AN OBSTACLE [407
If we assume that <£ varies as rmcosn0, we see that m=p + 2, and that the complete solution is
p+2 •
............(12)
A and B being arbitrary constants. In (10) we have to deal with n = l associated with p = — o and —7, and with ?z=3 associated with p — — 3. The complete solution as regards terms in cos 0 and cos 30 is accordingly 0 = (Ar + Br-1) cos 0 + (<7r3 + Dr~s) cos 30
2U3o2 f „ / c2 c4 \ cos301 /10.
-I------^— cos 0 (-ri+ 94-5)-----o— .......(13)
The conditions to be satisfied at infinity require that, as in (7), A — U, and that C = 0. We have also to make d<p/dr vanish when r = c. This leads to
..................(14)
iiU, JLJ-lLU
Thus
, rr 1 - °2 <6 = U 4
r 12a2r __
4- ^ Ion, fi(-*- + JL\ COS3^
satisfies all the conditions and is the value of </> complete to the second approximation.
That the motion determined by (15) gives rise to no resultant force in the direction of the stream is easily verified. The pressure at any point is a function of g2, and on the surface of the cylinder q2 = c~2 (d<p/d0)*. Now (dfyjdQy* involves 0 in the forms sin20, sin2 30, sin 0 sin 30, and none of these are changed by the substitution of TT - 0 for 0; the pressures on the cylinder accordingly constitute a balancing system.
There is no particular difficulty in pursuing the approximation so as to include terms involving the square and higher powers of' ?72/a2. The right-hand member of (6) will continue to include only terms in the cosines of odd multiples of 0 with coefficients which are simple powers of r, so that the integration can be effected as in (11), (12). And the general conclusion that there is no resultant force upon the cylinder remains undisturbed.
The corresponding problem for the sphere is a little more complicated, but it may be treated upon the same lines with use of Legendre's functions Pn (cos 0) in place of cosines of multiples of 0. In terms of the usual polar coordinates (r, 6, ««), the last of which does not appear, the first approximation, as for an incompressible fluid, is
,, ...............(16) is room for a supplementary volume which should have regard more especially to the practical side of the subject. Perhaps the time for this has not yet come. During the last few years much work has been done in connexion with artificial flight. We may hope that before long this may be coordinated and brought into closer relation with theoretical hydrodynamics. In the meantime one can hardly deny that much of the latter science is out of touch with reality.