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
sa ^ '
r°° hus, if V=\ Vnein*dn, ........................... (30)
dn ....... (31)
o (\ sa/ sa sa
Of the four integrals in (31),
the first = \ in Vn eint dn = i V ; '
the fourth = ^ f Vneint dn = _9z/, V. Ait ,' d JjOj"
Also t-he Hccoud and third together give
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
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.