236 REPORT— 1875, 



momentum of the fluid -which is destroyed or created ; and from this it will be 

 inferred that the forces will be the same, no matter what is the radius of the curve 

 of the pipe, inasmuch as the curvature of the pipe does not affect the amount of 

 the forward momentum of the fluid that has to be destroyed or replaced. 



Let lis next take the case of a bend in a pipe that is not a right angle, as shown 

 in Plate XII. lig. .31 ; and here, as before, I only propose to deal with the forces 

 that come into play in the direction A C of the original motion of the fluid. Now 

 in this case the forward motion of the fluid is not, as in the instance of the right- 

 angled bend, entirely destroyed in its progress from A to B ; only a portion of the 

 motion is checked, and a portion of the momentum destroyed ; and the magnitude 

 of the force required to destroy the momentum is in proportion to the amount 

 by which the forward velocity of the fluid in the line A C is destroyed. This 

 force is administered to the fluid by the curved portion of the pipe at the bend 

 DEF, and, as in the former case, exercises by reaction on the pipe a forward 

 stress, which will be in proportion to the extent by which the forward motion of 

 the fluid is checked by the divergence of the pipe from its original line. 



Suppose to this bend we attach rigidly another bend BG of the same angle, 

 as shown in fig. 32, so that the termination of this second bend at G is parallel to 

 the commencement of the first bend at A. Here, in the portion of the pipe B G, 

 that part of the forward velocity which was taken away has to be again given to 

 the fluid; this requires force, which is administered to the fluid by the curved part 

 UK, of the pipe. There is thus thrown on the pipe a rearward stress represented 

 by M. The force required in the bend between B and G to reinstate completely 

 the forward velocity, is evidently the same in amount as the force required in the 

 bend between A and B to destroy in part the forward velocitJ^ 



It follows, therefore, that the two stresses on the pipe, represented by the 

 arrows L and M, which indicate the forces acting on the pipe, are equal and 

 opposite to one another ; and these are the only forces acting on the rigid pip in 

 the line A of the original motion of the fluid at A. It follows, therefore, that in 

 the case of two right-angled bends rigidly attached, or in the case of two con- 

 nected equal-angled bends of any other angle, the stresses brought on the pipe by 

 the flow of the fluid will not tend to move the pipe bodily endwaj's. 



It will be seen also by this reasoning that the forces we have referred to do not 

 depend on the cm-vature of the pipes, but are simply measured bj' the amount of 

 the forward momentum of the fluid and the extent to which that momentum is 

 modified in the total amount of deflection of the course of the fluid at the bend, 

 or, in other words, by the angle of the bend. And from this reasoning it becomes 

 apparent that by whatever bends or combinations of bends we divert tlie course of 

 a stream of fluid in the pipe, provided the combination be such as to restore the 

 stream to its original direction, the aggregate of the forces in one direction required 

 to destroy forward momentum are necessarily balanced by equal forces in the 

 opposite direction required to reinstate the former momentum. 



It will be useful to consider more in detail the action of all the forces acting on a 

 fluid in a bend of the pipe ; and I will return to the case of a single right-angled 

 bend, as shown in fig. 29. I before spoke merely of the forces acting parallel to 

 the line AC, and said that the forward momentum of the fluid in that line had to be 

 destroyed in its passage round the bend DEF, and that this must be effected by a 

 force acting parallel to AC, which would by its reaction throw a forward stress on 

 the pipe, tending to force it in the dirgction AC. But similarly velocity has to be 

 given to the fluid iu the direction NB ; and to do this a force must be administered 

 to the fluid which will cause a reaction on the pipe in the direction BN ; and as the 

 momentum to be established in the direction NB, has to be equal to that in the di- 

 rection AC, which had to be destroj'ed, it follows that the forces of reaction upon the 

 pipe in the directions AC andBN are equal. These forces can be met in two waj's, 

 either by securing the bent part of the pipe DEF so that it will in each part resist the 

 stresses that come on it, or by letting the forces be resisted by the tensional strength 

 of the straight parts of the pipe Al) and BF, operating in the direction of their 

 length ; and in this case we see that the tension on AD must be equal to the force 

 acting along AC, and the tension on BF must be equal to the equal force acting 

 along BN, so that iu fact the forces brought into play by the right-angled bend 



