Dec. 30, 1875] 



NATURE 



171 



turn are necessarily balanced by equal forces in the opposite 

 direction required to reinstate the former momentum. 



It ^vilI be useful to consider more in detail the action of all the 

 forces operating 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 throw a forward stress on the pipe, tending to force 

 it in the direction AC. But similarly velocity has to be given 

 to the fluid in 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 direction AC, 

 which had to be destroyed, it follows that the forces of reaction 

 upon the pipe in the directions AC and BN are equal. These 

 forces can be met in two ways, either by securing the bent part 

 of the pipe DEF so that it ^v^ll in each part resist the stresses 

 that come on it, or by letting the forces be resisted by the ten- 

 slonal strength of the straight parts of the pipe AD 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 

 force acting along BN, so that In fact the forces brought into 



Fig. h. 



play by the right-angled bend produce a longitudinal tension on 

 the pipe at either end of the bend equal to tlie force required to 

 destroy the forward momentum of the fluid. 



Proceeding to the case cf the non-right-angled bend, as shown 

 in Fig. 31 : in this case, as we have seen, a portion only of the 

 forward momentum of the fluid in the line AC has to be de- 

 stroyed, also a ceitain amoimt of sideways momentum has to be 

 created in a direction which we may consider parallel to the line 

 QP ; and the composition of the remaining forward momentum 

 in the line AC with the created sideways momentum in the line 

 QP, results in the progress of the fluid along the path FB ; this 

 partial destruction of forward momentum and establishment of 

 some sideways momentum are essential to the onward progress 

 of the fluid along FB. The bend DEF will be subject to the 

 reaction of the forces necessarj- to produce these changes ; and 

 cither the bend may be locally secured, or the stress upon it may 

 be met, as in the case of the right-angled bend we have just 

 been considering, by a tensional drag on the pipe at either end 

 «f the bend. There is, however, this difference between the 



cases, that the force required to establish sideways momentum 

 parallel to QP cannot be directly met by the reaction of tension 

 along the line BF of the second part of the pipe ; but this 

 force may be met by the obliquely acting tension of the 

 pipe BF combined with the induced tension along the pipe 

 AD. It is well known that in the case of a given force, such 

 as that we are supposing parallel to PQ, resisted by two ob- 

 liquely placed forces such as the tension iong the lines DA and 

 FB, the nearer the lines DA and FB are to one straight line, the 

 greater mxist be the tension along those lines to balance a given 

 force acting on the line PQ. Now the less the line FB diverges 

 from the line AC, the less will be the sideways momentum 

 parallel to QP that has to be imparted to the fluid ; but at the 

 same time and to precisely the same extent will the proportionate 

 tension put upon the limbs DA and FB of the pipe be aggravated 

 by the greater obliquity of their action. The sideways pull is 

 greatest when the bend is a right angle ; and then it amounts to 

 a force that will take up or give out the entire momentum of the 

 fluid, and it is supplied directly by the tension of the limb of the 

 pipe at FB. If the bend is made less than a right angle, the 

 less the bend is made, the less is the sideways pull, but the 

 greater by the same degree is the disadvantage of the angle at 

 which the tension on the pipe resists the pull ; and it results 

 from this that in the case of a bend other than a right angle. 



Fig. 35. 



the tension on the pipe is the same as in the case of a right- 

 angled bend. A geometrical proof of this is given in Fig. 33. It 

 is evident that the radius of curvattire of the bend does not enter 

 into this consideration, and that the forces acting are not affected 

 by the rate of curvature of the pipe, the simple measure of the 

 forces being the increase or decrease in the momentum of the 

 fluid in each direction. It results from this that if a fluid be 

 flowing along a pipe with a bend in it, no matter what may be 

 the angle of the bend, or the radius of its curvature, the reac- 

 tions necessary to deflect the path of the fluid will be met by a 

 tensional resistance along the pipe ; and this tension is equal to 

 the force that would be required to entirely destroy the momen- 

 tum of the fluid. 



If we now assume any number of bends, of any angle or curva- 

 ture, to be connected together (see Fig. 3), the equilibrium of 

 each bend is satisfied by a longitudinal tension which is in every 

 case the same ; and this tension is therefore uniform throughout 

 the pipe ; for the tension at any intermediate point in a bend is 

 clearly the same as at the ends of the bend, as we may suppose 



