170 



NATURE 



[Dec. 30, 1875 



along the pipe at A towards the bend. This force is administered 

 to the fluid by the curved portion of the pipe at the bend DEF ; 

 and as the pipe is assumed to be rigid, the work of arresting the 

 forward velocity of the fluid throws a forward stress on the pipe 

 in a direction parallel to the line AC. 



Let us now assume that to the right-angled bend AB we 

 attach rigidly a second right-angled bend, BG, as shown in Fig. 

 30, in such a manner that the termination of this second bend 



Fig. 30. 



at G is parallel to the commencement of the first bend at A. 

 Here I will again, for the present, deal only with the forces in 

 a direction parallel to the line AC. 



The fluid at B has no velocity in the direction of the line AC, 

 and at G it has a velocity in that direction equal to the velocity 

 which it had at A. To give it this velocity in a forward direc- 

 tion (I mean forward in its original direction of motion), to 

 establish this forward momentum, requires the application of a 

 force in the direction HG ; and this force is administered to the 

 fluid by the curved portion of the pipe at the bend IJK ; and as 

 the pipe is assumed to be rigid, the duty of establishing the for- 

 ward velocity of the fluid throws a rearward stress on the pipe 

 in the direction GH. Now as the forward momentum given to 

 the fluid between B and G in the line GH is exactly the same as 

 the momentum destroyed between A and B in the line AC, it 

 follows that the rearward stress thrown on the pipe at the bend 

 IJK is exactly equal to the torward stress thrown on the pipe 

 at the bend DEF. Hence it will be seen that the forces acting 

 on the rigid pipe AG, treated as a whole, balance, so far as relates 

 to the forces parallel to the line AC, the original line of motion 

 of the fluid — the forward stress acting on the pipe at the bend 

 DEF being balanced by the equal rearward stress acting on the 

 pipe at the bend IJK. These two of the forces acting on the 

 pipe are shown by the arrows L and M, which, it must be 

 remembered, are the only forces which act in a direction parallel 

 to the line AC. 



It will have been seen that the measure of these forces is the 

 amount of forward 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 aff'ect the 

 amount of the forward momentum that has to be destroyed or 

 replaced in the fluid. 



Let us next take the case of a bend in a pipe that is not a 

 right angle, as shown in Fig. 31 ; and here, as before, I only 



propose to deal with the forces that operate in a direction parallel 

 to the line AC, that is, of the original motion of the fluid. Now 

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

 stance ofJ:he right-angled bend, entirely destroyed in its progress 

 from A to A ; only a portion of the forward motion is checked, 

 and the same portion of the forward momentum destroyed ; and 

 the force by which it is destroyed is administered to the fluid by 

 the curved portion of the pipe at the bend DEF, and, as in the 

 former case, constitutes a forward stress on the pipe in the direc- 



tion of the line AC, which will bear the same ratio to the stress 

 which would follow from the destruction of the whole, as the 

 portion destroyed bears to the whole forward momentum. 



Suppose to this bend we attach rigidl/ anotlier bend BG, of 

 same angle, as shown in Fig. 32, so that the termination of this 



^6.> 



Fig. 32. 



second bend at G is parallel to the commencement of the first bend 

 at A. Here, in the portion of the pipe BG, that part ot 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 IJK 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 

 velocity. 



It follows, therefore, that the two stresses oa the pipe, repre- 

 sented 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 pipe in a direction parallel 

 to the line AC or the original motion of the fluid at A. It 

 follows, therefore, that in case of two right-angled bends rigidly 

 connected, or in the case of two connected 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 endways. 



It will be seen also by this reasoning that the forces we have 

 referred to do not depend on the curvature of the pipes, but are 

 simply measured by the amount of the forward momentum of 

 the fluid and the extent to which that momentum is modified by 

 the total of the deflection which the course of the fluid experi- 

 ences in passing the bend, or, in other words, by the angle of 

 the bend. And from this reasoning it becomes apparent that by 



I 



Fig. 33. 

 Let AGB = angle of bend. 



Let GC = force required to destroy the whole momentum of fluid in line 

 AC. 

 ,, = tension which would be put on pipe AD by a right-angled 

 bend. 

 Then HC = force required to destroy momentum lost at the bend in the 



line AC. 

 And HB = force required to establish momentum acquired at bend in line 



.'. BC = total force acting on pipe. 



This force must be in equilibrium with the tensions of pipe along I3G and 



AC. 

 .'. the tension of pipe = GC cr GB. 



t e. = the tension of pipe when the bend is right angled. 

 Therefore the tension of the bent pipe is constant for a given velocity of 



flow, whatever be the angle of the bend. 



whatever bends or combinations of bends we divert the course 

 of a stream of fluid in a 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 momen« 



