i8, 1875] 



NA TURE 



51 



that its figure is perfectly symmetric on either side of C, the 

 middle point of its length. 



I-et us now assume that the pipe has a stream of perfe«t fluid 



Fig. 1. 



running through it from A towards E, and that the pipe is free 

 to move bodily endways. 



It is not unnatural to assume at first sight that the tendency of 

 the fluid would be to push the pipe forward, in rirtue of the 

 opposing surfaces offered by the beads in it — that both the 

 divergence between A and C from the original line at A, and the 

 return between C and E to that line at E, would place parts of 

 the interior surface of the pipe in some manner in opposition to 

 the stream or flow, and that the flov7 thus obstructed would drive 

 the pipe forward ; but if we endeavour to build up these supposed 

 causes in detail, we find the reasoning to be illusory. 



I will now trace the results which can be established by correct 

 reasoning. 



The surface being assumed to be smooth, the fluid, being a 

 perfect fluid, can exercise no drag by friction or other^vise on the 

 side of the pipe in the direction of its length, and in fact can 

 exercise no force on the side of the pipe, except at right angles to it. 

 Now the fluid flowing round the curve from A to B will, no doubt, 

 have to. be deflected from its course, and, by what is commonly 

 known as centrifugal action, will press against the outer side of 

 the curve, and this with a determinable force. The magnitude 

 and direction of this force at each portion of the curve of the 

 pipe between A and B are represented by the small anows 

 markedy ; and the aggregate of these forces between A and B is 

 represented by the larger arrow marked G. In the same way 

 the forces acting on the parts BC, CD, and D E are indicated by 

 the arrows H, I, and J ; and as the conditions under which the 

 fluid passes along each of the successive parts of the pipe are 

 precisely alike, it follows that the four forces are exactly equal, 

 and, as shown by the arrows in the diagram, they exactly 

 neutralise one another in virtue of their respective directions ; 

 and therefore the whole pipe from A to E, considered as a rigid 

 single structure, is subject to no disturbing force by reason of 

 the fluid running through it. 



Though this conclusion that the pipe is not pushed endways 

 may appear on reflection so obvious as to have scarcely needed 

 elaborate proof, I hope that it has not seemed needless, even 

 though tedious, to follow somewhat in detail the forces that act, 

 and which are, under the assumed conditions, the only forces 

 that act, on a symmetrical pipe such as I have supposed. 



Having shown that in the case of this special symmetrically 

 curved pipe the flow of a perfect fluid through it does not tend 

 to push it endways, I will now proceed to show that this is also 

 the case whatever may be the outline of the pipe, provided that 

 its beginning and end are in the same straight line. 



Assume a pipe bent, and its ends joined so as to form a com- 

 plete circular ring, and the fluid within it running with velocity 

 round the circle. This fluid, by centrifugal force, exercises a 

 uniform outward pressure on every part of the uniform curve ; 

 and this is the only force the fluid can exert. This pressure 

 tends to tear the ring asunder, and causes a uniform longitudinal 

 tension on each part of the ring, in the same manner as the 

 pressure within a cylindrical boiler makes a uniform tension on 

 the shell of the boiler. 



Now, in the case of fluid running round within rings of various 

 diameter, just as in the case of railway trains running round 

 curves of various diameter, if the velocity along the curve remain 

 the same, the outward pressure on each part of the circumference 

 is less, in proportion as the diameter becomes greater ; but the 

 circumferential tension of the pipe is in direct proportion to the 

 pressure and to the diameter ; and since the pressure has been 

 shown to be inversely as the diameter, the tension for a given 

 'ocity will be the same, whatever be the diameter. 

 Thus, if we take a ling of doubled diameter, if the velocity is 

 unchanged, the outward pressure per Ihieal inch will be halved ; 

 but this halved pressure, acting with the doubled diameter, will 

 give the same circumferential tension. 



Now thi^i longitudinal tension is the same at every part of the 



ring ; and if we cut out a piece of the ring, and supply the longi- 

 tudinal tension at the ends of the piece, by attaching two straight 

 pipes to it tangentially (see Fig. 2), and if we maintain the flow 

 of the fluid through it, the curved portion of the pipe will be 

 under just the same strains as when it formed part of the com- 



plete ring. It will be subject merely to a longitudinal tension ; 

 and if the pipe thus formed be flexible, and fastened at the ends, 

 the flow of fluid through it will not tend to disturb it in anyway. 

 Whatever be the diameter of the ring out of which the piece is 

 assumed to be cut, and whatever be the length of the segment 

 cut out of it, we have seen that the longitudinal tension will be 

 the same if the fluid be moving at the same velocity ; so that, if 

 we piece together any number of such bends of any lengths and 

 any curvatures to form a pipe of any shape, such pipe, if flexible 

 and fastened at the ends (see Fig. 3), will not be disturbed by the 



Fig. 



flow of fluid through it ; and the equilibrium of each portion and 

 of the whole of the combined pipe will be satisfied by a imiform 

 tension along it. 



Further, if the two ends of the pipe are in the same straight 

 line, pointing away from one another (see Fig. 4), since the 



Fig. 4. 



tensions on the ends of the pipe are equal and opposite, the flow 

 of the fluid .through it does not tend to push it bodily end- 

 ways. * 



This is the point which it was my object to prove ; but in the 

 course of this proof there has incidentally appeared the further 

 proposition, that a flexible tortuous pipe, if fastened at the ends, 

 will not tend to be disturbed in any way by the flow of fluid 

 through it. This proposition may to some persons seem at first sight 

 to be so paradoxical as to cast some doubt on the validity of the 

 reasoning which has been used ; but the proposition is neverth:?- 

 less true, as can be proved by a closely analogous experiment, as 

 follows : — 



Imagine the ends of the flexible tortuous pipe to be joined so 

 as to form a closed figure (see Fig. 5), there will then be no need 

 for the imaginary fastenings at the ends, since each end will 

 * Se« Supplementary Note A.' 



