TRANSACTIONS OF THE SECTIONS. 235 



APPENDIX. 



Note A. 



Tlie proposition, that the flow of fluid through a tortuous pipe having its ends 

 in the same straight line does not tend to push the pipe endways, can be treated in 

 several ways, of which only one is given in the accompanying address; but it may 

 be interesting to some readers to trace some of the other ways of viewing the 

 question. 



First let us take the case of a right-angled bend in a pipe (that is to say, where 

 the direction of a pipe is altered through a right angle by a curve of greater or lesa 

 radius; a bend of this sort is shown in Plate XII. fig. 29), and assume that the 

 fluid in it at A is flowing from A towards 0. I propose at present to deal only 

 with those forces or tendencies which act more or less powerfully in the direction 

 of the original motion of the fluid, namely along the line AC. 



I must here remind you that I am dealing with this matter entirely indepen- 

 dently of hydrostatic pressure. Perhaps to some it will be difticult to disassociate 

 the idea of hydrostatic pressure from a fluid in a pipe. This difficulty might be 

 got over by assuming that the pipe is immersed in a fluid of the same density 

 and head as the fluid within it. There will thus be hydrostatic equilibrium 

 between the fluid within and withoiit the pipe, the only difterence being that 

 the fluid inside the pipe is assumed to be in rapid motion, and thus subjects the 

 pipe to any stresses properly incidental to that motion of the fluid within it. 



The sole work that has to be done in the present case is that of deflecting the 

 current of fluid to a course at right angles to its original course AC ; and, regard- 

 ing the forces employed in this work as resolvable throughout into two sets of com- 

 fionents, the one at right angles to the line AC, the other parallel to it, it is of the 

 atter alone that account is to be taken. Manifestly the sum of these latter compo- 

 nents is measured by the circumstance that it is precisely sufticient to entirely de- 

 stroy the forward momentum of the fluid that flows 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 the 

 line AC. 



Let us now assunre 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 termina- 

 tion of this second bend 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 Gr it has 

 n velocity in that direction equal to the velocity which it had at A. To give it 

 this velocity in a forward direction (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 I.JK ; and as the pipe is assumed to be 

 rigid, the duty of establishing the forward velocity of the fluid, throws a rear- 

 ward stress on the pipe in the direction .GH. ?fow as the forward momentum 

 given to the fluid between B and G, in the line GIT, is exactly the same as the 

 momentum destroyed between A and B in the line AC, it follows that the rear- 

 ward stress thrown on the pipe at the bend UK is exactly equal to the forward 

 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 each other, so far as 

 relates to the forces in the line AC, the original line of motion of the fluid ; that is 

 to say, the forward stress acting on the pipe at the bend DEF is balanced bj-^ the 

 equal rearward stress acting on the pipe at the bend UK. These two of the forces 

 acting on the pipe are shown by the arrows L and M, which, it must be remem- 

 bered, are the only forces which act in a line parallel to AC. 



It will have been seen that the measure of these forces is the amount of forwnr(J 



