datum line XX. Hence we see that in stream line motion, under the restrictions named above, the total ener y per lb of fluid is uniformly distributed along the stream line. if the free surface of the fluid OO is taken as the datum, and -h, -hi are the depths of A and B measured down from the free surface, the equation takes the form V2/2§ “f'P/G -h =vi2/2g+pi/G -hi; (3) or generally sv'/2 g ~i-p/G - h = constant. (311) § 3o Second Form of the Theorem of Bernoulli Su ose at the pp “ two sections A, B (fig. 262 of an elementary stream small vertical Q pipes are introduced, which may be termed pressure columns; ? f1f q s Bl

'i

~l" ~1:;;: '—~ ”-”—~-~ —i7—5§ 1 'f j, f B' T " if I 6

rf fj Ag *B g

I ' | o ~ - ' it i' ';, f, »f ' -'lf rg C V C 1 El s t FIG. 26. 1£. (§ 8), having their lower ends accurately parallel to the direction of flow. In such tubes the water will rise to heights corresponding to the pressures at A and B. Hence b=p/G, and b'=p-/G. Consequently the tops of the pressure columns A' and B' will be at total heights bl c=p/G+z and b'+c'=p1/G+z1 above the datum line XX. The difference of level of the pressure column tops, or the fall of free surface level between A and B, is therefore E = (St-P 1)/G+(z-Zi); and this by equation (1), 29 is (1112-112)/2g. That is, the fall of free surface level between two sections is equal to the difference of the heights due to the velocities at the sections. The line A'B is sometimes called the line of h draulic gradient, though this term is also used in cases where f>i”iction needs to be taken into account. It is the line the height of which above datum is the sum of the elevation and pressure head at that point, and it falls below a horizontal line A”B" drawn at H ft. above XX by the quantities a =t»2/2g and a' =v,2/2g, when friction is absent. § 31. Illustrations of the Theorem of Bernoulli. In a lecture to the mechanical section of the British Association in 1875, W. Froude gave some experimental illustrations of the principle of Bernoulli. He remarked that it was a common but erroneous impression that a fluid exercises in a contracting pipe A (fig. 27) an excess of pressure against the entire converging surface projected surface as HI, and the pressures parallel to the axis of the pipe, normal to these lgrojected surfaces, balance each other. Similarly the pressures on C, 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 flow of liquid parallel to the 1 F = —-1*—r-, ;—°-""* —" I; I I lr I II I.. v.” 32 €=s I i*.9, §

. -'%»-“

Q ={ 1:2 /// C

• "|"— 2 E ' / /"'T'E

gf" IN, 5. . ”"r"' 5 -9% f /'§ I " ' Z Z 5' . H 1 - ! 1 Y, /- ! -~1—~ -~~'—}}i})}T?*' "'”

FIG.28.

l 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 exactly opposite the entrance to the other. The cisterns being filled I li . 2, . . @ g 1 ' -5 -5- 'f 5 T !::T-: I -e- ~~' S | / " —1- i '/':b. . 15 — -. | / - -~ '3 ", l “Z/, ' I —|— I ' -}> . I G E E Q H L I G I 5; ff ~ ""*"°"'§ T°>——"°%'T>- '— -, — K-.75-> FIG. 29. very nearly to the same level, the 'et from the left-hand cistern A entered the right-hand cistern B (fig. 31), shooting across the free space between them without any waste, except that due to indirectness of aim and want of exact correspondence in the form of the orifices. In the actual experiment there was 18 in. of head in the right and 20% in. of head in the left-hand cistern, so that about which it meets, and that, conversely, A , ' ' A I as it enters an enlargement B, a relief { 3 ' ' ' ";E;' ' ' ' 21"jj2 ' ' ' ' ' ' ' ' ' ' "Pg '°" ' “"'1fI' Q r of pressure is experienced bv the, mg | i -D "“ "'“"L v entire diverrin ~ f f ' ' id -~-'- —- 'J »»»""" g g sur ace o the pipe. | 9 E » Further it is commonly assumed that 9 #aw —- — 5 — .when passing through a contraction A C, there is in the narrow neck an excess of pressure due to the squeezing together of the llquld at that point. These impressions are in no rCSp€Ct Correct; the pressure is smaller as the section of the pipe is smaller and conversely. Fig. 28 shows a pipe so formed that a contraction is followed by an enlargement, and fig. 29 one in which an enlargement is followed by a contraction. The A B Vertical pressure columns

  - show the decrease of

~ pressure at the contraction and increase of pressure at the enlargement. The line abc in both figures shows the C variation of free surface level, supposing the pipe frictionless. In actual pipes, however, work is FIG 27 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 abici, falling below abc. Froude further pointed out that, if a pipe contracts and enlarges again to the same size, the resultant pressure on the converging part exactly balances the resultant pressure on the diverging part so that there is no tendency to move the pipe bodily when water flows through it. Thus the conical part AB (fig. 30) presents the same if » 1 FIG. 30. 2% in. were wasted in friction. It will be seen that in the open space between the orifices there was no pressure, except the atmospheric § pressure acting uniformly throughout the system. § 32. Venturi Meter.-An ingenious application of the variation of pressure and velocity in a converging and diverging pipe has been

Fig. 31. made by Clemens Herschel in the construction of what he terms a I Venturi Meter for measuring the flow: in water mains. Suppose that, as in fig. 32, a contraction is made in a water main, the change of section bein radual to avoid the reduction of eddies. The ratio p

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