TKANSACTIONS OF THE SECTIONS. 337 



produce a longitudinal tension on the pipe at either end of the bend equal to the 

 foice required to destroy the forward moiuentuni of the fluid. 



Proceeding to the case of 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 destroyed, also a certain amount of sideways momentum has 

 to be created in a direction which we may consider parallel to the Hue 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 progi-ess of the fluid along 

 the path FB ; this partial destruction of forward momentum and establishment of 

 some sideways momentum are essential to tlie onward progress of the fluid along FB. 

 The bend DEF will be subject to the reaction of the forces necessary to produce 

 these changes ; and either 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 of 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 additional 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 obliquely placed forces such as the 

 tension along the lines DxV and FB, the nearer the lines DA and FB are to one 

 straight line, the greater must 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 pro- 

 portionate 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, the ten- 

 sion on the pipe is the same as in the case of a right-angled beud. A geometrical 

 proof of this is given in fig. 33. It is evident that the radius of curvature of tha bend 

 does not enter into this consideration, and that the forces acting are not afiected 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 reactions necessaiy 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 momentum of the fluid. 



If we now assume any number of bends, of any angle or curvature, to be con- 

 nected together (see Plate IX. 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 

 imiform 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 the bend to be divided 

 at that point into two bends, and there joined together by an infinitely short piece 

 of straight pipe. 



If then the tortuous pipe 1 have above refeiTed to has its ends at A and B parallel 

 to one another, as shown in fig. 4, it is clear that the tensional forces at its ends 

 balance one another, and the pipe, as a whole, does not tend to move endways. 



Note B. 



The law regulating these changes of pressure due to changes of velocity can be 

 best understood by considering the case of a stream of perfect fluid flowing from a gra- 

 dually tapeerd pipe or nozzle placed horizontally and connected with the bottom of 

 a cistern, as shown in Plate Xll. fig. 34. Let us suppose that at the points B and 

 the sectional areas of the pipe are severally twice and foiu- times that at the point 

 of exit A. 



1875. 18 



