HYDRAULICS,] HYDROMECHANICS 467 is the sum of the elevation and pressure head at that point, and it falls below a horizontal line A"B" drawn at H feet above XX by 2 the quantities a= v and 6 = ^-, when friction is absent. 20 20 28. Illustrations of the Theorem of Bernoulli. In a lecture to the mechanical section of the British Association in 1875, the late Mr W. Froude gave some experimental illustrations of the principle of Bernoulli. Mr . Fronde remarked that it was a common but erroneous impression that a fluid exercises in a contracting pipe A (tig. 32) an excess of pressure against the entire converging - surface which it meets, and that, conversely, as it enters an enlarge- ment B, a relief of pressure is experienced by the entire diverging surface of the pipe. Further it is commonly assumed that when passing through a con traction C, there is in the narrow neck an excess of pressure due to the squeezing together of the liquid at that point. These im pressions are in no respect correct ; the pressure is smaller as the section of the pipe is smaller and conversely. Fig. 33 shows a pine so formed that a contraction is followed by an enlargement, and fig. 34 one in which an enlargement is followed by a contraction. The vertical pressure columns show the decrease of pressure at the contraction and increase of pressure at the en- Fig. 34. largement. The line abc in both figures shows the variation of free surface level, supposing the pipe frictionless. In actual pipes, however, work is expended in friction against the pipe ; the total head diminishes in proceeding along the pipe, and the free surface level is a line such as ab^, falling below abc. Mr Froude further points out that, if a pipe contracts and enlarges again to the same size, the resultant pressure on the converging part Similarly the pressures on EC, CD balance those on GH, EG. In the same way, in any combination of enlargements and contrac tions, a balance of pressures, due to the How of liquid parallel to the axis of the pipe, will be found, provided the sectional area and direction of the ends are the same. The following experiment is interesting. Two cisterns provided with converging pipes were placed so that the jet from one was ex actly opposite the entrance to the other. The cisterns being filled very nearly to the same level, the jet from the left hand cistern A entered the right hand cistern B (fig. 36), shooting across the free space between them without any waste, except that due to indirect ness of aim and want of exact correspondence in the form of the orifices. In the actual experiment there was 18 inches of head in the right and 20J inches of head in the left hand cistern, so that about 2 inches were wasted in friction. It will be seen that in the open space between the orifices there was no pressure, ex cept the atmospheric pressure acting uniformly throughout the system. 29. Pressure, Velocity, and Energy in Different Stream Lines. The equation of Bernouilli gives the variation of pressure and velocity from point to point along a stream line, and shows that the total energy of the flow across any two sections is the same. Two other directions may be defined, one normal to the stream line and in the plane containing its radius of curvature at any point, the other normal to the stream line and the radius of curvature. For the problems most practically useful it will be sufficient to consider the stream lines as parallel to a vertical or horizontal plane. If the motion, is in a vertical plane, the action of gravity must be p+df Fig. 35. ?u a u ba l ances thc resultant pressure on the diverging part so
- there is no tendency to move the pipe bodily when u atc-r flows
unongb.it Thus the conical part AB (fig. 35) presents the same projected surface as HI, and the pressures parallel to the axis of ie pipe, normal to these projected surfaces, balance each other. Fig. 37. taken into the reckoning ; if the motion is in a horizontal plane, the terms expressing variation of elevation of the filament will dis appear. l Let AB, CD (fig. 37) be two consecutive stream lines, at present assumed to be in a vertical plane, and PQ a normal to these lines making an angle </> with the vertical. Let P, Q be two particles mov ing along these lines at a 1 distance PQ = eL?, and let z be the height of Q above the horizontal plane with reference to which The energy is measured, r its velocity, and p it 1 The following theorem is taken from a paper by Professor Cotterill, "Oa the Distribution of Energy in a Mass of Fluid in Steady Motion," Phil. May.,
February 187(i.
