]06 Mb. stokes, on SOME CASES OF FLUID MOTION. 



blem to be solved is usually stated at the head of each article, I shall here only mention some 



of the results. As a particular case of the problem given in Art. 8, I find that, when a cylinder 



oscillates in an infinitely extended fluid, the effect of the inertia of the fluid is to increase the mass of 



the cylinder by that of the fluid displaced. In part of Art. 9, I find that when a ball pendulum 



oscillates in a concentric spherical envelope, the effect of the inertia of the fluid is to increase the 



¥ + Za' 



mass of the ball bv times that of the fluid displaced, a beina; the radius of the ball, and 



•' 2(1/- a-') 



6 that of the envelope. Poisson, in his solution of the problem of the sphere, arrives at the strange 

 result that the envelope does not at all retard the oscillating sphere. I have pointed out the errone- 

 ous step by wliich he was led to this conclusion, which I am clearly called upon to do, in venturing 

 to differ from so high an authority. Of the different cases of fluid motion which I have given, that 

 which appears to be capable of the most accurate and varied comparison with experiment, is the 

 motion of fluid in a rectangular box which is closed on all sides, given in Art. 13. The experiment 

 consists in comparing the calculated and observed times of oscillation. I find that when the motion 

 is small, the effect of the fluid on the motion of the box is the same as that of a solid having the 

 same mass, centre of gravit}', and principal axes, but having different moments of inertia, these 

 moments being given by infinite series, whicli converge with great rapidity. I have also in Art. II, 

 given some cases of progressive motion, deduced on the supposition that tlie same particles of fluid 

 remain in contact with the solid, which do not at all agree with experiment. 



In almost all the cases given in this paper, the problem of finding the permanent state of tem- 

 perature in the several solids considered, supposing the surfaces of those solids kept up to constant 

 temperatures varying from point to point, may be solved by a similar analysis. I find that some of 

 these cases have been already solved by M. Duhamel in a paper inserted in the 2'2nd Cahier of the 

 Journal de rEcole Polyfechnique. The cases alluded to are those of the temperature in a solid 

 sphere, and in a rectangular parallelopiped. Since, however, the application of the formulae in the 

 two cases of fluid motion and of the permanent state of temperature is different, as well as the 

 formulae themselves to a certain extent, I thought it might be worth while to give them. 



1. The investigations in this paper apply directly to incompressible fluids, as the fluids spoken 

 of will be supposed to be, unless the contrary is stated. The motions of elastic fluids may in most 

 cases be divided into two classes, one consisting of those condensations on which sound depends, the 

 other, of those motions which the fluid takes in consequence of the motion of solid bodies in it. 

 Those motions of the fluid, which take place in consequence of very rapid motions of solids, (such 

 as those of bullets), form a connecting link between these two classes. The motions of the second 

 class are, it is true, accompanied by condensations, and propagated with the velocity of sound, but 

 if the motions of the solids are not great we may, witliout sensible error, suppose the motions of 

 the fluid propagated instantaneously to distances where they cease to be sensible, and may neglect the 

 condensation. The investigations in this paper will apply without sensible error to this kind of 

 motion of elastic fluids. 



In all cases also the motion will be supposed to begin fioin rest, whicii allows us to suppose that 

 udx + vdy + wdz is an exact differential d(p, where u, v and w are the components, parallel to the 

 axes of X, y, and ss, of the whole velocity of any particle. In applying our investigations however 

 to fluids such as they exist in nature, this principle must not be strained too far. When a body is 

 made to revolve continually in a fluid, the parts of the fluid near the body will soon acquire a rota- 

 tory motion, in consequence, in all probability, of the mutual friction of the parts of the fluid ; so 

 that after a time udx + vdy + wdx could no longer be taken an exact differential. It is true that 

 in motion in two dimensions there is one sort of rotatory motion for which that quantity is an exact 

 differential; but if a close vessel, filled with fluid at first at rest, be made to revolve uniformly round 

 a fixed axis, the fluid will soon do so too, and therefore liiat quantity will cease to be an exact dif- 



