CHAPTER VII. MOTION OF FLUIDS. General Equation of the Motion of Fluids. 248. THE mass of a fluid particle being p dx dy dz, its momentum in the axis x arising from the accelerating forces is, by article 144, Consequently the equation of the motion of a fluid mass in the axis or, when free, is {x - dr} p - dp P = 0. dx In the same manner its motions in the axes y and z are (56) And by the principle of virtual velocities the general equation of fluids This equation is not rigorously true, because it is formed in the hypothesis of the pressures being equal on all sides of a particle in motion, which Poisson has proved not to be the case; but, as far as concerns the following analysis, the effect of the inequality of pressure is insensible. 249. The preceding equation, however, does not express all the circumstances of the motion of a fluid. Another equation is requisite. A solid always preserves the same form whatever its motion may be, which is by no means the case with fluids; for a mass ABCD, fig. 57, formed of particles possessing perfect mobility, changes its form by the action of the forces, so that it always continues to fit into the intervals of the surrounding molecules without leaving any void. In this consists the continuity of fluids, a property which furnishes the other equation necessary for the determination of their motions. Equation of Continuity. 250. Suppose at any given time the form of a very small fluid mass to be that of a rectangular parallelopiped ABCD, fig. 57. The action of the forces will change it into an oblique angled figure N E FK, during the indefinitely small time that it moves from its first to its second position. Now N EFG may differ from A B C D both in form and density, but not in mass; for if the density depends on the pressure, the same forces that change the form may also produce a fig. 57. B D N G F K change in the pressure, and, consequently, in the density; but it is evident that the mass must always remain the same, for the number of molecules in A B C D can neither be increased nor diminished by the action of the forces; hence the volume of A B C D into its primitive density must still be equal to volume of N E F G into the new density; hence, if p dx dy dz, be the mass of A B C D, the equation of continuity will be 251. This equation, together with equations (56), will determine the four unknown quantities x, y, z, and p, in functions of the time, and consequently the motion of the fluid. Developement of the Equation of Continuity. 252. The sides of the small parallelopiped, after the time dt, become drd.dx, dy + d.dy, dz + d.dz; or, observing that the variation of dx only arises from the increase of a, the co-ordinates y and z remaining the same, and that the variations of dy, dz, arise only from the similar increments of y and z; If the angles GNF and FNE, fig. 58, be represented by and, the volume of the parallelopiped NK will be NE.NG sin 0. NF sin ; F R K Fa, Nb being at right angles to NE and RG; but as and were right angles in the primitive volume, they could only vary by indefinitely small arcs in the time dt; hence in the new volume and the volume becomes NE. NG. NF; substituting for the three edges their preceding values, and omitting indefinitely small quantities of the fifth order, the volume after the time dt is The density varies both with the time and space; hence the primi tive density, is a function of t, x, y and z, and after the time dt, it is consequently, the mass, being the product of the volume and density, as will readily appear by developing this quantity, which is the general equation of continuity. 253. The equations (56) and (59) determine the motions both of incompressible and elastic fluids. 254. When the fluid is incompressible, both the volume and density remain the same during the whole motion; therefore the increments of these quantities are zero; hence, with regard to the volume 255. By means of these two equations and the three equations (56), the five unknown quantities p, p, x, y and z, may be determined in functions of t, which remains arbitrary; and therefore all the circumstances of the motion of the fluid mass may be known for any assumed time. 256. If the fluid be both incompressible and homogeneous, the density is constant, therefore de = 0, and as the last equation becomes identical, the motion of the fluid is obtained from the other four. Second form of the Equation of the Motions of Fluids. 257. It is occasionally more convenient to regard x, y, z, the co-ordinates of the fluid particle dm, as known quantities, and dx dy dz dt' dť' dt' its velocities in the direction of the co-ordinates, as unknown. In order to transform the equations (56) and (59) to suit this case, let these quantities being functions of x, y, z, and t. The differentials of these equations when x, y, z, and t, vary all at once; and when the equations (56) become, by the substitution of the preceding and by the same substitution, the equation (59) of continuity becomes The equations (63) and (64) will determine s, u, and v, in func tions of x, y, x, t, and then the equations dx sdt dy = udt dz = vdt will give x, y, z, in functions of the time. of the fluid mass will therefore be known. The whole circumstances Integration of the Equations of the Motions of Fluids. 258. The great difficulty in the theory of the motion of fluids, consists in the integration of the partial equations (63) and (64), which has not yet been surmounted, even in the most simple problems. It may, however, be effected in a very extensive case, in which sdx+udy + vdx |
