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 ,
- >i, w.,, a> 3 being the component angular velocities of the axes.
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 /" _i_ i TJ * v o ^ i **/* 3jj(J T>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 "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> P + gz+ % q" = -- + 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 <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>_ 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).
