90 



NATURE 



{Dec. 2, 1875 



fluid ia a pipe exercise,^, in the case of its meeting a contraction 

 (see Fig. 9), an excess of pressure against the entire converging 

 surface which it meets, and that, conversely, as it enters an enlarge- 

 ment (see Fig. 10), a relief of pressure is experienced by the entire 



Fig. 10. 



Fig. 9. 



diverging surface of the pipe. Further, it is commonly assumed 

 that, wlien passing through a contraction (see Fig. 11), there is 



Fig. It. 



in'the narrow neck an excess of pressure due to the squeezing 

 together of the fluid at that point. 



These impressions are in no respect correct ; the pressure at 

 the smallest part of the pipe is, in fact, less than that at any 

 other point, and vice versd. 



If a fluid be flowing along a pipe which has a contraction in 

 it (see Fig. 12), the forward velocity of the fluid at B must be 



Fig. 12. 



greater than that at A, in the proportion in whiqh the sectional 

 area of the pipe at B is less than that at A ; and therefore while 

 passing from A to B the forward velocity of the fluid is being in- 

 creased. This increase of velocity implies the existence of a force 

 acting in the direction of the motion ; that is to say, each particle 

 which is receiving an increase of forward velocity must have a 

 greater fluid pressure behind it than in front of it ; for no other 

 condition will cause that increase of forward velocity. Hence a 

 particle of fluid, at each stage of its progress along the tapering 

 contraction, is passing from a region of higher pressure to a region 

 of lower pressure, so that there must be a greater pressure in the 

 larger part of the pipe than in the smaller, and a diminution of 

 pressure at each point corresponding with the diminution of 

 sectional area ; and this difference of pressure must be such as to 

 supply the force necessary to establish the additional forward 

 velocity required at each point of the passage of the fluid through 

 the contraction. Consequently, differences of pressure at different 

 points in the pipe depend simply upon the velocities at those 

 points, or, in other words, on the relative sectional areas of the 

 pipe at those points. * 



It is simple to apply the same line of reasoning to the converse 

 case of an enlargement. Here the velocity of the particles is 

 being reduced through precisely the same series of changes, but 

 in an opposite order. The fluid in the larger part of the pipe 

 moves more slowly than that in the smaller, so that, as it ad- 

 vances into the enlargement, its forward velocity is being checked; 

 and this check implies the existence of a force acting in a direc- 

 tion opposite to the motion of the fluid, and each particle being 

 thus retarded must therefore have a greater fluid pressure in front 

 of it than behind it ; thus a particle of fluid at each stage of its 

 progress along a tapering enlargement of a pipe is passing from 

 a region of lower pressure to a region of higher pressure. As is 

 well known, the force required to produce a given change of 

 velocity is the same, whether the change be an increase or a de- 

 crease. Hence, in the case of an enlargement of a pipe, as in 

 the case of a contraction, the changes of velocity can be satisfied 

 only by changes of pressure, and the law for such change of pres- 

 sure will be the same, mutatis mutandis. 



In a pipe in which there is a contraction and a subsequent en- 

 largement to the same diameter as before (see Fig. 11), since 

 the differences of pressure at different points depend on the differ- 

 ences of sectional area at those points, by a law which is exactly 

 the same in an "enlarging as in a contracting pipe, any points 

 which have the same sectional area will have the same pressures, 



• See Supplementary Note B. 



the pressures at the larger areas being larger, and those at the 

 smaller areas smaller. 



Precisely the same result will follow in the case of an enlarge- 

 ment followed by a contraction (see Fig. 13).* 



Fig. 13. 



This proposition can} be illustrated by experiments performed 

 with water. 



Figs. 14, 15 show certain pipes, the one a contraction followed 

 by an enlargement, and the other an enlargement followed bj 



Fig. 14. 



a contraction. At certain points in each pipe, vertical gauge- 

 glasses are connected, the water-levels in which severally indicate ] 

 the pressures in the pipe at the points of attachment. 



In Fig. 14 the sectional areas at P and E are equal to one! 

 another. Those at C and K are likewise equal to one another,] 

 but are smaller than those at P and E. The area at I is the 

 smallest of all. Now, if the water were a perfect fluid, thej 

 pressures P Q and E D would be equal, and would be greate 



Fig. 15. 



than C H and K N. C H and K N would also be equal to one 

 another, and would be themselves greater than I J. 



The results shown in Fig. 15 are similar in kind, equal pres- 

 sures corresponding to equal sectional areas. 



Fig. 16. 



As water is not a perfect fluid, some of the pressure at each 

 successive point is lost in friction, and this growing defect in 

 pressure is indicated in the successive gauge-glasses in the maim< 

 shown in Figs. 16, 17. 



* In a perfect fluid, we may say in a sense, the vis viva of each partial 

 remains constant. If the particle is stationary, therzi viva is entirely n 

 presented by the pressure ; if it be under no pressure, the vis viva is entire!; 

 represented by the velocity ; if it be moving at some intermediate velocity," 

 the vis vi-^'i is partly represented by the pressure and partly by the velocity 



i 



