Page:Encyclopædia Britannica, Ninth Edition, v. 12.djvu/463

447 HYDRODYNAMICS.] HYDROMECHANICS 4c-(c~ - a 2 ) As an example of the use of moving axes in hydrodynamics, con- sider the liquid filling the ellipsoidal case where ~2 7^ iT == 1 I and first suppose the liquid to be frozen, and to have componenl angular velocities f, 77, about the axes, then If the liquid now be suddenly melted, and additional component singular velocities n 1( n s, n 3 communicated to the ellipsoid about the axes, then (vide infra) and if U, V, W denote the component velocities of the liquid relative to the axes, V = v + SUl - ar 3 = r-^z - J&x , We see that so that a liquid particle always remains on a similar ellipsoid. The hydrodynamical equations with moving axes, taking into account the mutual gravitation of the liquid, are p dx 1 dp P dy where dt dv A = ~ dy (1), - = (2), 2 + )P / i /&quot; _i_ i TJ * v o ^ i **/* 3jj(J T&amp;gt;2 / 2 r V72 / 2  With the above values of u, v , -w, U, V, W, the hydrodynamical equations are of the form 1 dp 7 dx + A * + x + hy + gz=0, 1 dp !^r The component accelerations in space of the liquid particle at xyz parallel to the axes are therefore ax + hy + gz, hx + p, J+ f Z) gx+fy + yz; and by the dynamical equations the rates of change of angular momentum about the coordinate axes are zero, and therefore 2w { (gx +fy + yz)y - (hx + /3y +fz)z } = ; - = ; and therefore y^O. and similarly g and h vanish. Therefore the hydrodynamical equations become 1 dp 7 1 = 7 4 2 (c 2 - a 2 _ 2 &quot;3 -sft- 02- Therefore, integrating, c- + a- = constant ; and therefore the surfaces of equal pressure are the similar and co-axial quadrics (A + a)* 2 + (B + j8)?/ 2 + (C + 7)s 2 = constant. If we can make a, ft, y constant, and (A + a)a 2 = (B + j8)& 2 = C + y)c the surfaces of equal pressure are similar to the external case, which can therefore be removed without affecting the motion. This is the case when the axis of revolution is a principal axis; and, supposing it the axis of z, then O 1 = 0, fi 2 = 0, | = 0, 77 = 0. If in addition we put fl 3 = 0, or w 3 =, we obtain the solution of the particular case considered by Jacobi, of a liquid ellipsoid of three unequal axes, rotating about its least axis in relative equi librium ; or, putting a = b, we obtain Maclaurin s solution of the equilibrium of a rotating spheroid (Cam. Phil. Soc. Proc., iii.). Equation (11) is called Bernoulli s equation, and for homogeneous liquids under gravity is a very useful principle in hydraulics ; the equation may be established from first principles by considering the energy which enters and leaves a certain portion of a tube of flow. (Lamb, Motion of Fluids, p. 23). If homogeneous liquid be drawn off from a vessel, so large that the motion of the free surface may be neglected, then Bernoulli s equation becomes, P being the atmospheric pressure and h the height of the free surface, r&amp;gt; P + gz+ % q&quot; = -- + gh ; and in particular, for a jet issuing into the atmosphere, where _p = F, &-?(*-); or the velocity is due to the depth below the free surface. This is Tonicelli s theorem (Do Motu gravium Projectorum, 1643). If we suppose fluid to escape according to this law from a large closed vessel in which the pressure is p where the motion is insen sible, and neglect the variations of velocity due to variations of level, p being sufficiently great, then If A be the sectional area of the jet (at the vena contracta), the quantity of fluid which escapes per unit of time is the momentum per unit of time is Ap7 2 = 2A and the energy per unit of time is Suppose, for instance, two equal pipes leading one from the steam space and the other from the water space of a steam boiler at a pressure p, and suppose Torricelli s theorem to hold for the rate of efflux of the]steam and water, then, if &amp;lt;r denote the density of steam, and p the density of water, Q ^ The velocity of steam jet_ / p The velocity of water jet a The quantity of steam jet_ f a_ The quantity of water jet V p /o The momentum of steam jet _ ^ The momentum of water jet / j&amp;gt;_ The energy of steamjet The energy of water jet For instance, with steam at 8 atmospheres, or 120 Ib to the square inch, . / -P- =15 nearly. (T (Rankine, Steam Engine, appendix).
 * &amp;gt;i, w.,, a&amp;gt; 3 being the component angular velocities of the axes.