[From the Transactions of the Cambridge Philosophical Society, Vol. VIII. p. 105.] ON SOME CASES OF FLUID MOTION. [Read May 29, 1843.] THE equations of Hydrostatics are founded on the principles that the mutual action of two adjacent elements of a fluid is normal to the surface which separates them, and that the pressure is equal in all directions. The latter of these is a necessary consequence of the former, as has been shewn by Mr Airy*. An exactly similar proof may be employed in Hydrodynamics, by which it may be shewn that, if the mutual action of two adjacent elements of a fluid in motion is normal to their common surface, the pressure must be equal in all directions, in order that the accelerating force which acts on the centre of gravity of an element may not become infinite, when we suppose the dimensions of the element indefinitely diminished. In Hydrostatics, the accurate agreement of the results of our calculations with experiments, (those phenomena which depend on capillary attraction being excepted), fully justifies our fundamental assumption. The same assumption is made in Hydrodynamics, and from it are deduced the fundamental equations of fluid motion. But the verification of our fundamental law in the case of a fluid at rest, does not at all prove it to be true in the case of a fluid in motion, except in the very limited case of a fluid moving as if it were solid. Thus, oil is sufficiently fluid to obey the laws of fluid equilibrium, (at least to a great extent), yet no one would suppose that oil in motion ought to be considered a perfect fluid. It would appear from the following consideration, that the fluidity of water and other such fluids is not quite perfect. * See also Professor Miller's Hydrostatics, page 2. When a mass of water contained in a vessel of the form of a solid of revolution is stirred round, and then left to itself, it presently comes to rest. This, no doubt, is owing to the friction against the sides of the vessel. But if the fluidity of water were perfect, it does not appear how the retardation due to this friction could be transmitted through the mass. It would appear that in that case a thin film of fluid close to the sides of the vessel would remain at rest, the remaining part of the fluid being unaffected by it. And in this respect, that part of Poisson's solution of the problem of an oşcillating sphere, which relates to friction, appears to me in some degree unsatisfactory. A term enters into the equation of motion of the sphere depending on the friction of the fluid on the sphere, while no such term enters into the equations of motion of the fluid, to express the equal and opposite friction of the sphere on the fluid. In fact, as long as we regard the fluidity of the fluid as perfect, no such term can enter. The only way by which to estimate the extent to which the imperfect fluidity of fluids may modify the laws of their motion, without making any hypothesis as to the molecular constitution of fluids, appears to be, to calculate according to the hypothesis of perfect fluidity some cases of fluid motion, which are of such a nature as to be capable of being accurately compared with experiment. The cases of that nature which have hitherto been calculated, are by no means numerous. My object in the present paper which I have the honour to lay before the Society, has been partly to calculate some such cases which may be useful in determining how far we are justified in regarding fluids as perfectly fluid, and partly to give examples of the methods by which the solution of problems depending on partial differential equations may be effected. In the first seven articles, I have mentioned and explained some general principles, which are afterwards applied. Some of these are not new, but it was convenient to state them for the sake of reference. Others are I believe new, at least in their development. In the remaining articles, I have given different problems, of which I have succeeded in obtaining the solutions. As the problem 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 mass of the ball by b3 + 2a3 2 (b3 — a3) times that of the fluid displaced, a being the radius of the ball, b 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 erroneous step by which 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 gravity, and principal axes, but having different moments of inertia, these moments being given by infinite series, which converge with great rapidity. I have also in Art. 11, given some cases of progressive motion, deduced on the supposition that the 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 temperature 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 22nd Cahier of the Journal de l'Ecole Polytechnique. The cases alluded to are those of the temperature in a solid sphere, and in a rectangular parallelepiped. Since, however, the application of the formulæ in the two cases of fluid motion and of the permanent state of temperature is different, as well as the formulæ 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, without 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 from rest, which allows us to suppose that udx+vdy + wdz is an exact differential do, where u, v and w are the components, parallel to the axes of x, y, and z, 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 rotatory motion, in consequence, in all probability, of the mutual friction of the parts of the fluid; so that after a time udx + vdy+wdz 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 that quantity will cease to be an exact differential. For the same reason, in the progressive motion of a solid in a fluid, the effect of friction continually accumulating, the motion might at last be sensibly different from what it would be if there were no friction, and that, even if the friction were very small. In the case of small oscillatory motions however it would appear that the effect of friction in the forward oscillation, supposing that friction small, would be counteracted by its effect in the backward oscillation, at least if the two were symmetrical. In this case then we might expect our results to agree very nearly with experiment, so far at least as the time of oscillation is concerned. The forces which act on the fluid are supposed in the following investigations to be such that Xdx + Ydy + Zdz is the exact differential of a function of x, y and z, where X, Y, Z, are the components, parallel to the axes, of the acccelerating force acting on the particle whose co-ordinates are x, y, z. The only effect of such forces, in the case of a homogeneous, incompressible fluid, being to add the quantity pf(Xdx + Ydy+Zdz) to the pressure, the forces, as well as the pressure due to them, will for the future be omitted for the sake of simplicity. 2. It is a recognized principle, and one of great importance in these investigations, that when a problem is determinate any solution which satisfies all the requisite conditions, no matter how obtained, is the solution of the problem. In the case of fluid motion, when the initial circumstances and the conditions with respect to the boundaries of the fluid are given, the problem is determinate. If it were required to find what sort of steady motion could take place between given surfaces, the problem would not be determinate, since different kinds of steady motion might result from different initial circumstances. It may be well here to enumerate the conditions which must be satisfied in the case of a homogeneous incompressible fluid without a free surface, the case which is considered in this paper. We have first the equations, responding quantities for y and z, and omitting the forces. We have also the equation of continuity, du dv dw + + dx dy dz ...........(B); (A) and (B) hold at all times for all points of the fluid mass. If σ be the velocity of the point (x, y, z) of the surface of a solid in contact with the fluid resolved along the normal, and v the velocity, resolved along the same normal, of the fluid particle, |