1911] ON THE MOTION OF SOLID BODIES THEOUGH VISCOUS LIQUID 35
§ 5. The problem of the sphere moving with arbitrary velocity through a viscous fluid is of course more difificult than the corresponding problem of the plane, lamina, but 'it has been satisfactorily solved by Boussinesq* and by Basse b f. The easiest road to the result is by the application of Fourier's theorem to the periodic solution investigated by Stokes. If the velocity of the .sphere at time t be V- Vneint, a the radius, M' the mass of the liquid displaced by the sphere, and s = vW2l/)> v being as before the kinematic viscosity, Stokes finds as the total force at time t
(29)
sa ^ '
r°° hus, if V=\ Vnein*dn, ........................... (30)
J o
dn ....... (31)
^ '
o (\ sa/ sa sa
Of the four integrals in (31),
f°°
the first = \ in Vn eint dn = i V ; '
.o
the fourth = ^ f Vneint dn = _9z/, V. Ait ,' d JjOj"
Also t-he Hccoud and third together give
4>a
and fclii.s is fcho only part which could present any difficulty. We have, however, already conaiderecl this integral in connexion with the motion of a piano antl its value ia expressed by (10). Thus
' (32)
( ^
Tho firafc tt»nu depends upon the inertia of the fluid, and is the same as would be obtained by ordinary hydrodynamics when v = 0. If there is no acceleration at the moment, this term vanishes. If, further, there has been no acceleration for a long time, the third term also vanishes, and we obtain the result appropriate to a uniform motion.
an in (i). The general result (32) is that of Bouseinesq and Basset.
* 6'. R. t. c. p. 985 (1883) ; Theorie Analytiftue de la Chaleur, t. n. Paris, 1903. t Phil, Tram. 1888 ; Hydrodynamics', Vol. rt. chap. xxn. 1888.
3—2d with Boggio's work.