xvi ᎪᎡᎢ. Contents Notes on two-dimensional problems Viscosity in gases; dissipation function Damping of plane waves of sound by viscosity Combined influence of viscosity and thermal conduction Effect of viscosity on diverging waves Effect on the scattering of sound-waves by a spherical obstacle, fixed Damping of sound-waves in a spherical vessel Turbulent motion. Reynold's experiments; critical velocities of wat in a pipe; law of resistance. Inferences from theory of dimensio References to theoretical investigations of Rayleigh and Kelvin. CHAPTER XII ROTATING MASSES OF LIQUID Maclaurin's ellipsoids. Relations between eccentricity, angular velocity and angular momentum; numerical tables Jacobi's ellipsoids. Linear series of ellipsoidal forms of equilibrium Other special forms of relative equilibrium. Rotating annulus General problem of relative equilibrium; Poincaré's investigation. Linear series of equilibrium forms; limiting forms and forms of 378-380. Application to a rotating system. Secular stability of Maclaurin's and Jacobi's ellipsoids. The pear-shaped figure of equilibrium Small oscillations of a rotating ellipsoidal mass; Poincaré's method. Dirichlet's investigations; references. Finite gravitational oscillations of a liquid ellipsoid without rotation. Oscillations of a rotating Dedekind's ellipsoid. The irrotational ellipsoid. Rotating elliptic HYDRODYNAMICS CHAPTER I THE EQUATIONS OF MOTION 1. THE following investigations proceed on the assumption that the matter with which we deal may be treated as practically continuous and homogeneous in structure; i.e. we assume that the properties of the smallest portions into which we can conceive it to be divided are the same as those of the substance in bulk. The fundamental property of a fluid is that it cannot be in equilibrium in a state of stress such that the mutual action between two adjacent parts is oblique to the common surface. This property is the basis of Hydrostatics, and is verified by the complete agreement of the deductions of that science with experiment. Very slight observation is enough, however, to convince us that oblique stresses may exist in fluids in motion. Let us suppose for instance that a vessel in the form of a circular cylinder, containing water (or other liquid), is made to rotate about its axis, which is vertical. If the angular velocity of the vessel be constant, the fluid is soon found to be rotating with the vessel as one solid body. If the vessel be now brought to rest, the motion of the fluid continues for some time, but gradually subsides, and at length ceases altogether; and it is found that during this process the portions of fluid which are further from the axis lag behind those which are nearer, and have their motion more rapidly checked. These phenomena point to the existence of mutual actions between contiguous elements which are partly tangential to the common surface. For if the mutual action were everywhere wholly normal, it is obvious that the moment of momentum, about the axis of the vessel, of any portion of fluid bounded by a surface of revolution about this axis, would be constant. We infer, moreover, that these tangential stresses are not called into play so long as the fluid moves as a solid body, but only whilst a change of shape of some portion of the mass is going on, and that their tendency is to oppose this change of shape. L. H. 1 2. It is usual, however, in the first instance to neglect the tangential stresses altogether. Their effect is in many practical cases small, and independently of this, it is convenient to divide the not inconsiderable difficulties of our subject by investigating first the effects of purely normal stress. The further consideration of the laws of tangential stress is accordingly deferred till Chapter XI. If the stress exerted across any small plane area situate at a point P of the fluid be wholly normal, its intensity (per unit area) is the same for all aspects of the plane. The following proof of this theorem. is given here for purposes of reference. Through P draw three straight lines PA, PB, PC mutually at right angles, and let a plane whose direction-cosines relatively to these lines are l, m, n, passing infinitely close to P, meet them in A, B, C. Let P. P1, P2, P3 denote the intensities of the B stresses* across the faces ABC, PBC, PCA, PAB, respectively, of the tetrahedron PABC. If A be the area of the first-mentioned face, the areas of the others are, in order, lA, mA, nA. Hence if we form the equation of motion of the tetrahedron parallel to PA we have p1. lA = pl. A, where we have omitted the terms which express the rate of change of momentum, and the component of the extraneous forces, because they are ultimately proportional to the mass of the tetrahedron, and therefore of the third order of small linear quantities, whilst the terms retained are of the second. have then, ultimately, p = P1, and similarly p = p2 = P3, which proves the P2 theorem. 3. The equations of motion of a fluid have been obtained in two different forms, corresponding to the two ways in which the problem of determining the motion of a fluid mass, acted on by given forces and subject to given conditions, may be viewed. We may either regard as the object of our investigations a knowledge of the velocity, the pressure, and the density, at all points of space occupied by the fluid, for all instants; or we may seek to determine the history of every particle. The equations obtained on these two plans are conveniently designated, as by German mathematicians, the 'Eulerian' and the 'Lagrangian' forms of the hydrokinetic equations, although both forms are in reality due to Eulert. * Reckoned positive when pressures, negative when tensions. Most fluids are, however, incapable under ordinary conditions of supporting more than an exceedingly slight degree of tension, so that p is nearly always positive. † "Principes généraux du mouvement des fluides," Hist. de l'Acad. de Berlin, 1755. "De principiis motus fluidorum," Novi Comm. Acad. Petrop. t. xiv. p. 1 (1759). Lagrange gave three investigations of the equations of motion; first, incidentally, in The Eulerian Equations. These quantities are then 4. Let u, v, w be the components, parallel to the co-ordinate axes, of the velocity at the point (x, y, z) at the time t. functions of the independent variables x, y, z, t. t they express the motion at that instant at all points of space occupied by the fluid; whilst for particular values of x, y, z they give the history of what goes on at a particular place. We shall suppose, for the most part, not only that u, v, w are finite and continuous functions of x, y, z, but that their space-derivatives the first order (ĉu/dx, dv/dx, dwsdx, &c.) are everywhere finite*; we shall understand by the term 'continuous motion,' a motion subject to these restrictions. Cases of exception, if they present themselves, will require separate examination. In continuous motion, as thus defined, the relative velocity of any two neighbouring particles P, P' will always be infinitely small, so that the line PP' will always remain of the same order of magnitude. It follows that if we imagine a small closed surface to be drawn, surrounding P, and suppose it to move with the fluid, it will always enclose the same matter. And any surface whatever, which moves with the fluid, completely and permanently separates the matter on the two sides of it. 5. The values of u, v, w for successive values of t give as it were a series of pictures of consecutive stages of the motion, in which however there is no immediate means of tracing the identity of any one particle. To calculate the rate at which any function F (x, y, z, t) varies for a moving particle, we remark that at the time t + dt the particle which was originally in the position (x, y, z) is in the position (x + udt, y + vdt, z + wồt), so that the corresponding value of F is If, after Stokes, we introduce the symbol D/Dt to denote a differentiation following the motion of the fluid, the new value of F is also expressed by FDF/Dt. St, whence connection with the principle of Least Action, in the Miscellanea Taurinensia, t. ii. (1760) [Oeuvres, Paris, 1867-92, t. i.]; secondly in his "Mémoire sur la Théorie du Mouvement des Fluides," Nouv. mém. de l'Acad. de Berlin, 1781 [Oeuvres, t. iv.]; and thirdly in the Mécanique Analytique. In this last exposition he starts with the second form of the equations (Art. 14, below), but translates them at once into the 'Eulerian' notation. It is important to bear in mind, with a view to some later developments under the head of Vortex Motion, that these derivatives need not be assumed to be continuous. 6. To form the dynamical equations, let p be the pressure, p the density, X, Y, Z the components of the extraneous forces per unit mass, at the point (x, y, z) at the time t. Let us take an element having its centre at (x, y, z), and its edges dx, dy, Sz parallel to the rectangular co-ordinate axes. The rate at which the x-component of the momentum of this element is increasing is. Sx Sy Sz Du/Dt; and this must be equal to the x-component of the forces acting on the element. Of these the extraneous forces give pdx dysz X. The pressure on the yz-face which is nearest the origin will be ultimately ρ The difference of these gives a resultant - ap/dx.dx dy dz in the direction of x-positive. The pressures on the remaining faces are perpendicular to x. We have then Substituting the value of Du/Dt from (1), and writing down the symmetrical equations, we have 7. To these dynamical equations we must join, in the first place, a certain kinematical relation between u, v, w, p, obtained as follows. If v be the volume of a moving element, we have, on account of the constancy of mass, D.pv To calculate the value of 1/v. Dv/Dt, let the element in question be that which at time t fills the rectangular space dx dy Sz having one corner P at (x, y, z), and the edges PL, PM, PN (say) parallel to the co-ordinate axes. At time t + St the same element will form an oblique parallelepiped, and since * It is easily seen, by Taylor's theorem, that the mean pressure over any face of the element dx dy dz may be taken to be equal to the pressure at the centre of that face. |