HYDRAULICS.] HYDKOMECHANICS 469 of a centrifugal pump or turbine into a chamber, it forms a free vor tex of this kind. The water flows spirally outwards, its velocity diminishing and its pressure increasing according to the law stated above, and the head along each spiral stream line is constant. 31. Forced Vortex. If the law of motion in a rotating current is different from that in a free vortex, some force must be applied to cause the variation of velocity. The simplest case is that of a ro tating current in which all the particles have equal angular velocity, as for instance when they are driven round by radiating paddles revolving uniformly. Then in equation (2), 29, considering two cir cular stream lines of radii r and r + dr (fig. 40), we have p = r, ds = dr. If the angular velocity is o, then v = ar and dv=adr. Hence g 99 Comparing this with (1), 29, and putting dz = 0, because the motion is horizontal, dp o?rdr 2oV , dp o 2 r , j4-= dr G g 11 - = 1- constant G (9). Let p lt be the pressure, radius, and velocity of one cylin drical section, p.^ r. 2 , r. 2 those of another ; then PA. _ ?!? i 2 = P* _ g2r 2 2 . G 2g Pa-Pi_ 2 - G (10). That is, the pressure increases from within outwards in a curve which in radial sections is a parabola, and surfaces of equal pressure are paraboloids of revolution (tig. 40). / I I DISSIPATION OF HEAD IN SHOCK. 32. Relation of Pressure and Velocity in a Stream in Steady Motion when (he Changes of Section of the Stream arc Abrupt. When a stream changes section abruptly, rotating eddies are ^d which dissipate energy. The energy absorbed in producing rotation is at once abstracted from that effective in causing the flow, and sooner or later it is wasted by frictional resistances due to the rapid relative motion of the eddying parts of the fluid. In such cases the work thus expended internally in the fluid is too important to be neglected, f and the energy / i thus lost is com monly termed en ergy lost in shock. Suppose fig. 41 to - represent a stream having such an *? abrupt change of section. Let AB, CD be normal sec tions at points where ordinary streamline motion has not been dis- DD Fig, 41. turbed and where it has been re-established. Let u,p, v be the area of section, pressure, and velocity at AB, and u 1 ,p } , v, corresponding quantities at CD. Then if no work were expended internally, and assuming the stream horizontal, we should have _ G ^ 20 -_ 2g (1). But if work is expended in producing irregular eddying motion, the head at the section CD will be diminished. Suppose the mass ABCD comes in a short time t to A B C D . The resultant force parallel to the axis of the stream is P& + Po( w i - u ) -Pii> where p is put for the unknown pressure on the annular space be-t tween AB and EF. The impulse of that force is The horizontal change of momentum in the same time is the differ ence of the momenta of CDC D and ABA B , because the amount of momentum between A B and CD remains unchanged if the motion is steady. The volume of ABA B or CDC D , being the inflow and outflow in the time t, is Qt = (avt = u,v l t, and the momen- P C turn of these masses is Q,rt and - Oit . The change of momen- g ff G turn is therefore Qt^ Equating this to the impulse, Assume that p =p, the pressure at AB extending unchanged through the portions of fluid in contact with AE, BF which lie out of the path of the stream. Then (since Q = 01^^) = . , ^ = ^ 2g (2); (3). This differs from the expression (1), 26, obtained for cases where no sensible internal work is done, by the last term on the right. That is/- M has to be added to the total head at CD, which is 2ff * - + -^1 . to make it equal to the total head at AB, or Jf_J51ll is G* 2g 2g the head lost in shock at the abrupt change of section. But v - t j is the relative velocity of the two parts of the stream. Hence, when an abrupt change of section occurs, the head due to the relative velocity is lost in shock, or L I_l2. foot-pounds of energy is wasted for eacli pound of fluid. Experiment verifies this result, so that the assumption that/> = ?> appears to be admissible. If there is no shock, = G G If there is shock, 2g _ G G g Hence the pressure head at CD in the second case is less than in the former by the quantity or, puttin 2*7
= o>r, by the quantity
