Nov. 23, 1871] 



NA TURE 



71 



COLDING ON THE LAWS OF CURRENTS 

 IN ORDINARY CONDUITS AND IN THE 

 SEA 



[I SEND to Nature for translation the abstract (in French) 

 appended, according to a most excellent custom, to Coining's 

 great paper in the Copenhagen Transactions. The subject is 

 one of especial interest at the present time, though, of course, 

 everything written by such a man is deserving of careful atten- 

 tion. Those in particular who met the genial Dane at the 

 British Associaiion will he glad to have in a compact form his 

 views on a question which has given rise to much discussion, and 

 which is of very great practical importance. — P. G. Tait] 



T PRESENTED in 1S63 to the Scientific Society, and some 

 -'■ months later to the Congress of Scandinavian Naturalists at 

 Stockholm, a short exposition of my researches on the motion of 

 fluid bodies, on which I had been occupied for several years, and 

 the results of which appeared to me worthy of being submitted 

 to the Society. 



The characteristic of this work is that it does not suppose, 

 like previous works of the same kind, that all the parts of a 

 current are endowed with the same rapidity ; for it owes, in fact, 

 its existence to my conviction that this mode of looking at the 

 subject can only lead in exceptional cases to exact results. My 

 researches are based on the different motions assumed by the 

 liquid threads or elements of the currents, and are supported by 

 the well-knoivn fact ihat any body, and consequetitly any portion 

 of a fluid, can only move with a constant rapidity when the 

 accelerating force is equal to the resistance. 



In the case of a fluid flowing by virtue cf its own weight over 

 a plane surface which opposes a resistance to the motion of the 

 water, it was easy to determine how this motion varies with the 

 depth, wdien the rapidity of the current is constant in all its 

 parts ; and, by pursuing this train of thought, I was led to the 

 law of the variation of the rapidity with the depth, when the 

 current moves in a cylindrical conduit with circular section, com- 

 pletely filled with the liquid. These researches are of greater 

 interest from the circumstance that the laws at which I have in 

 this manner arrived from theoretical considerations, are confirmed 

 by the experiments which have recently been carried out in 

 France by Capt. Boileau and Inspector General Darcy. These 

 laws of the motion of water may be expressed by the formula 



where V\i the rapidity of the first elements of the current, the 

 motion of which is the most rapid, v the rapidity at the depth 



■'■> 7 the fall per foot of the water, and K^ a magnitude which 



depends entirely on the nature and dimensions of the conduit, 

 on the depth of the current, &c. The theory shows besides 

 that the laws of the motion of water on a level surface are in- 

 cluded in the general law of the motion of water on a cylindri- 

 cal surface, when the radius of the cylinder is supposed infinite 

 Darcv, who has experimentally established the formula given 

 above for cyHndrical conduits, endeavoured, at the same time, to 

 determine Kff by means of certain experiments performed with 

 four different kinds of pipes, and found that K^- was inversely 

 proportional to the square of the radius of the conduit. It 

 resulted, according to the theory, that, for level conduits, A"„= 

 should be in the same manner in an inverse ratio to the square of 

 the depth of the current. But two series of e.xperiments per- 

 formed by Boileau mth level conduits led, on the contrary, to 

 the supposition that K^- was inversely proporuon.il simply to 

 the depth of the current. There was thus a want of agreement 

 between the results of the two experiments, and the point was to 

 discover which of these two hypotheses was correct. Several 

 circumstances leading me to believe that Darcy's theory could not 

 be exact, I took ai my starling point the experiments of Boileau, 

 and considered A'„- as inversely proportional to the depth of the 

 current, which I did with the less scruple since this hypothesis 

 agreed almost as well with Darcy's experiments as with his own. 

 I pursued, therefore, my researches on this basis, and, after many 

 difficulties, arrived at results which, on the whole, were so en- 

 tirely in accordance with experiments that I could not suppose 

 the possibility of Buileau's hypothesis being inexact. It was 

 only afterwards, when I approached the study of marine currents, 

 that new difficulties constantly arose, which I endeavoured at 

 first to overcome, but which became day by day more insur- 



mountable, until at last there was nothing left but to doubt the 

 correctness of my calculations, since they led to results which 

 were in obvious contradiction to facts. 



The theory then was shown to be inexact ; but since in so large 

 a number of cases it was evidently in agreemen' with experi- 

 ment, I attempted by a variety of means to discover the error 

 which I must tiave committed ; still all my attempts were un- 

 attended with result, and I was on the point of abandoning the 

 resolution of the problem to which I had already devoted so 

 much time, when the idea struck me of examining what woul d 

 happen if I rejected Boileau's determination of K^, and adopted 

 Darcy's hypothesis, although it still appeared to me impossible ; 

 when I found, with as much delight as surprise, that it removed 

 not only the great difficulties which I had up to that time en- 

 countered, but also all the contradictions which had occurred to 

 me as an inevitable consec|uencc of that hypothesis, and from 

 that moment the results of the calculations showed themselves to 

 be entirely in the most perfect accordance with what exists in 

 nature. 



The circumstance that the experiments of Darcy are almost as 



satisfactory whether - 



is supposed to be proportional to the 



first or to the second power of the depth of the currents, made 

 me think that the reality would be still more nearly approached 

 by expressing this magnitude by a binomial of the first and 

 second degree, and this was completely confirmed by facts. In 

 determining the constants of the binomial according to the results 

 of Darcy's experiments, I found the law of the motion of the water 

 in cylindrical pipes with a radius R, with a coefficient of resist- 

 ance ;;/, and a rapidity v„ at the surface of the conduit, may be 

 represented by the formula 



V - V = 6-8 slm 



^^^ ' s/ 62-5 -h 1177-^ 



/'being the rapidity next the axis, to which corresponds x = o. 

 This formula may be applied equally to the movement of water 

 in level conduits, if by A" is designated the depth of the current ; 



only the coefficient then becomes — :—- = 4'8, instead of 6'S. 



This formula shows, among other things, that the ratio -^ 



which corresponds to any point in a given conduit entirely filled 

 by the current, is entirely independent of the rapidity of the 

 current, a fact which Darcy's experiments confirm in a re- 

 markable manner. This relation furnishes us besides with the 

 means of determining the value of the coefficient of resistance w 

 for different kinds of pipes which were employed by Darcy, and 

 it is thus found that for 



Old pipes . . w = from o-oi20 to o-oo8o 



New pipes . . ?« = from o 0050 to 0'0033 



New varnished pipes . m = from 0.0033 to o'O025 

 values which are altogether independent of the diameter 

 of the conduit. For level wooden conduits, it is found, 

 according to the experiments of Boileau, that m = OO160 to 

 00090, while the resistance of the air, according to the same 

 author, corresponds tow = 00003 to o '0002 



Inspector-General Darcy has unfortunately died, but the 

 researches on currents which he commenced were continued by 

 the French engineer Bazin, who published in 1865 a great work 

 on the results of a considerable number of expeiiments carried on 

 with conduits of very different kinds. 



However interesting otherwise these researches may be, they 

 do not display eitherthe powers of observation or the grandeur of 

 conception which distinguish the works of Darcy. Among those 

 experiments which are of the greatest interest, there are some 

 begun by Darcy and finished by Bazin, such as researclies into 

 the motion of water in rectangular conduits, where the rapidity is 

 determined in 45 points symmetrically distributed. The result 



for these, as for circular conduits, is that the ratio -7, is indepen- 

 dent of the absolute rapidity of the current, and if the resuPs of 

 experiment on the motion of water in level conduits are compared 

 with those given by the theoretic formulae, it will be found that 

 these last agree so completely with experiment, that the difference 

 between the calculated and observed rapidities, in each of the 45 

 points mentii ned above, falls within the limit of errors of obser- 

 vation. This agreement is especia'ly remarkable in f^e case of 

 the conduit which Darcy employed in 1857 for the carrying out 



