is the difference of the pressures on the faces A B and C D. In the same manner it may be proved that are the differences of the pressures on the faces BF, AG, and on ED, AF. 236. But if X, Y, Z, be the accelerating forces in the direction of the axes, when multiplied by the volume dx dy dz, and by p its density, they become the momenta p.X dx dy dz, p. Y dx dy dz, p. Z dx dy dz. But these momenta must balance the pressures in the same directions when the fluid mass is in equilibrio; hence, by the principle of virtual velocities As the variations are arbitrary, they may be made equal to the differentials, and then dp = p {Xdr + Ydy + Zdz} (52) is the general equation of the equilibrium of fluids, whether elastic or incompressible. It shows, that the indefinitely small increment of the pressure is equal to the density of the fluid mass multiplied by the sum of the products of each force by the element of its direction. 237. This equation will not give the equilibrium of a fluid under all circumstances, for it is evident that in many cases equilibrium is impossible; but when the accelerating forces are attractive forces directed to fixed centres, it furnishes another equation, which shows the relation that must exist among the component forces, in order that equilibrium may be possible at all. It is called an equation of condition, because it expresses the general condition requisite for the existence of equilibrium. Equations of Condition. 238. Assuming the forces X, Y, Z, to be functions of the distance, by article 75. The second member of the preceding equation is an exact differential; and as p is a function of x, y, z, it gives the par but the differential of the first, according to y, is and the differential of the second, according to r, is These three equations of condition are necessary, in order that the equation (52) may be an exact differential, and consequently integrable. If the differentials of these three equations be taken, the sum of the first multiplied by Z, of the second multiplied by X, of the third multiplied by Y, will be and an equation expressing the relation that must exist among the forces X, Y, Z, in order that equilibrium may be possible. Equilibrium will always be possible when these conditions are fulfilled; but the exterior figure of the mass must also be determined. Equilibrium of homogeneous Fluids. 239. If the fluid be free at its surface, the pressure must be zero in every point of the surface when the mass is in equilibrio; so that p = 0, and whence p{Xdr + Ydy + Zdz } = 0, f(Xdx + Ydy + Zdz) = constant, supposing it an exact differential, the density being constant. The resulting force on each particle must be directed to the inte I rior of the fluid mass, and must be perpendicular to the surface; for were it not, it might be resolved into two others, one perpendicular, and one horizontal; and in consequence of the latter, the particle would slide along the surface. If u0 be the equation of the surface, by article 69 the equation of equilibrium at the surface will be Xdx + Ydy + Zdz = λdu, A being a function of x, y, z; and by the same article, the resultant of the forces X, Y, Z, must be perpendicular to those parts of the surface where the fluid is free, and the first member must be an exact differential. Equilibrium of heterogeneous Fluids. 240. When the fluid mass is heterogeneous, and when the forces are attractive, and their intensities functions of the distances of the points of application from their origin, then the density depends on the pressure; and all the strata or layers of a fluid mass in which the pressure is the same, have the same density throughout their whole extent. Demonstration. Let the function Xdc + Ydy + Zdz be an exact difference, which by article 75 will always be the case when the forces X, Y, Z, are attractive, and their intensities functions of the mutual distances of the particles. Assume p = f(Xdr + Ydy + Zdz), being a function of x, y, z; then equation (52) becomes dp = p.dp. (53) (54) The first member of this equation is an exact differential, and in order that the second member may also be an exact differential, the density P must be a function of p. The pressure p will then be a function of also; and the equation of the free surface of the fluid will be constant quantity, as in the case of homogeneity. Thus the pressure and the density are the same for all the points of the same layer. The law of the variation of the density in passing from one layer to another depends on the function in which expresses it. And when that function is given, the pressure will be obtained by integrating the equation dp = pdp. 241. It appears from the preceding investigation, that a homogeneous liquid will remain in equilibrio, if all its particles act on each other, and are attracted towards any number of fixed centres; but in that case, the resulting force must be perpendicular to the surface of the liquid, and must tend to its interior. If there be but one force or attraction directed to a fixed point, the mass would become a sphere, having that point in its centre, whatever the law of the force might be. 242. When the centre of the attractive force is at an infinite distance, its direction becomes parallel throughout the whole extent of the fluid mass; and the surface, when in equilibrio, is a plane perpendicular to the direction of the force. The surface of a small extent of stagnant water may be estimated plane, but when it is of great extent, its surface exhibits the curvature of the earth. 243. A fluid mass that is not homogeneous but free at its surface will be in equilibrio, if the density be uniform throughout each indefinitely small layer or stratum of the mass, and if the resultant of all the accelerating forces acting on the surface be perpendicular to it, and tending towards the interior. If the upper strata of the fluid be most dense, the equilibrium will be unstable; if the heaviest is undermost, it will be stable. 244. If a fixed solid of any form be covered by fluid as the earth is by the atmosphere, it is requisite for the equilibrium of the fluid that the intensity of the attractive forces should depend on their distances from fixed centres, and that the resulting force of all the forces which act at the exterior surface should be perpendicular to it, and directed towards the interior. 245. If the surface of an elastic fluid be free, the pressure cannot be zero till the density be zero; hence an elastic fluid cannot be in equilibrio unless it be either shut up in a close vessel, or, like the atmosphere, it extend in space till its density becomes insensible. Equilibrium of Fluids in Rotation. 246. Hitherto the fluid mass has been considered to be at rest; but suppose it to have a uniform motion of rotation about a fixed axis, as for example the axis oz. Let w be the velocity of rotation common to all the particles of the fluid, and r the distance of a par ticle dm from the axis of rotation, the co-ordinates of dm being x, y, z. Then wr will be the velocity of dm, and its centrifugal force resulting from rotation, will be wr, which must therefore be added to the accelerating forces which urge the particle; hence equation (53) will become аф = Xdx + Ydy + Zdz + w2 rdr. And the differential equation of the strata, and of the free surface of the fluid, will be Xdx + Ydy + Zdz + w2. rdr = 0. (55) The centrifugal force, therefore, does not prevent the function from being an exact differential, consequently equilibrium will be possible, provided the condition of article 238 be fulfilled. 247. The regularity of gravitation at the surface of the earth; the increase of density towards its centre; and, above all, the correspondence of the form of the earth and planets with that of a fluid mass in rotation, have led to the supposition that these bodies may have been originally fluid, and that their parts, in consolidating, have retained nearly the form they would have acquired from their mutual attractions, together with the centrifugal force induced by rotation when fluid. In this case, the laws expressed by the preceding equations must have regulated their formation. |