228 REPORT— 1875. 



without introduciiig endways pressure. Now tortuosity of flow is but anotlier 

 word for sideways deviation of flow. 



This leads us* up to the case of more sudden contractions or enlargements in 

 pipes, where the particles next the sides of the pipes have to follow their surfaces 

 and must therefore be moved rapidly sideways in their course. 



We will, for simplicity, consider the case of a contraction (see Plate XI. f)g. 21), 

 and one in which the pipe resumes the same diameter beyond the contraction. 



The particles along the central line pua-sue a straight course, and are subject 

 only to the changes of pressiu-o necessary to induce the changes of velocity. 



lo consider the behaviour of the other particles, let us assume that we insert 

 anumber of perfectly thin partitions (see fig. 22), which we lay in sach a manner that 

 they exactly follow the paths of the particles of fluid at each point, so as not in 

 any way to affect tlieii' motion ; these partitions are quite imaginary, and merely 

 assist us in looking upon the entire fluid in question as divided into a number of 

 small streams. These streams are generally curvilinear, and -vary in sectional 

 area ; and at the point beyond the contraction where the pipe resimies its former 

 sectional area, we shall natiu-ally find these minor streams occupying the same 

 sectional area as before, and moving with the same velocity as before. 



Now each of these small streams is exactly represented by a stream of fluid 

 flowing within a pipe, that pipe being curvilinear and gradually varying in sectional 

 area, and its two ends being of the same sectional area and in the same straight 

 line. We have seen that in the case of such a stream the sum total of all 

 the forces due to its motion has no resultant longitudinally ; and this will be 

 equally the case, whether the envelope of the stream be an actual pipe or the 

 mutual pressure of adjacent streams ; this envelope will not be moved endways by 

 the flow of the fluid. What is true of each stream is true of all put together; and 

 thus it follows that the whole bodj^ of fluid which these separate streams constitute, 

 does not exert any endways force ; or, in other words, there will be equilibrium of 

 fluid forces throughout tlie passage of the fluid through a local contraction in a 

 pipe, such as we have been considering. The same line of argument evidently 

 holds good in the case of an enlargement, where the pipe beyond the enlargement 

 regains the same diameter as before. 



In illustration of the conclusions which have been thus far established, if we 

 had a perfect fluid with which to try the experiment, we luight exhibit a very 

 instructive and striking result. 



Assume a stream of perfect fluid flowing through a pipe of very large diameter, A B 

 C, with a contraction in it, at B, as shown in fig. 23, and that the equal pressures at 

 A and C on either side of the contraction are indicated by the head of fluid in pres- 

 sure-gauges A D, C E — the pressure at B, which will be less, being represented by 

 the height B F. Now, the condition of the pipe at A will be just the same if we sup- 

 pose the pipe supplied from a large cistern G, as shown in fig. 24; and the appro- 

 priate pressure at A will be maintained if the fluid stands in the cistern G at a 

 height H, equal to the head A D in the pressure-gauge. So, again, the condition of 

 the pipe at C will be the same if the pipe discharges into a cistern, I ; and the appro- 

 priate pressure at C will be maintained, and can only be maintained, if the water 

 m the cistern stands at a height J, equal to the head E in the pressure-gauge, 

 which is, in fact, the same level as TI in the cistern G ; so that if we once esta- 

 blish the motion through the pipe ABC, and maintain the supply of fluid, we 

 shaU have the fluid running rapidly, and continuing to nin •wdth unabated 

 rapidity, from one cistern into another, though both are at the same head. 



If we take such a condition of things that the pressure at B is zero, or, in other 

 words, if the velocity at B is that due to the head A D, then we might cut the 

 pipe at B and separate the two cisterns, and we shoidd find the fluid issuing at 

 B in a jet, and re-entering the pipe again at K, and rising as before in the cistern 

 I to the same level with a perpetual flow. 



The experiment here suggested is, if rightly understood, only a specialized 

 instance of the properties of what in the previous experiment was termed a con- 

 traction followed by an enlargement ; it is, in fact, as if in that experiment the 

 diameter of the contracted part had been so far reduced that the pressure within it 

 would have sunk apparently to zero — that is lo say, in reality, to the pressure of 



