628 



NA TURE 



{Oct. 25, 1883 



In their practical aspect they relate to the law of resistance to 

 the motion of water in pipes, which appears in a new form, the 

 law for all velocities and all diameters being represented by an 

 equation of two terms. 



In their philosophical aspect these results relate to the funda- 

 mental principles of fluid motion ; inasmuch as they afford for 

 the case of pipes a definite verification of two principles, which 

 art that the general character of the motion of fluids in contact 

 with solid surfaces depends on the relation (1) between the dimen- 

 sions of the space occupied by the fluid and a linear physical con- 

 stant of the fluid ; (2) between the velocity and a physical velocity 

 constant of the fluid. 



The results as viewed in their philosophical aspect were the 

 primary object of the investigation. 



As regards the practical aspects of the results it is not reces- 

 sary to say anything by way of introduction ; but in order to 

 render the philosophical scope and purpose of the investigation 

 intelligible it is necessary to describe shortly the line of reason- 

 ing which determined the order of investigation. 



2. The Leading Features of the Motion of Actual Fluids. — 

 Although in rnost ways the exact manner in which water moves 

 is difficult to perceive, and still more difficult to define, as are 

 also the forces attending such motion, certain general features 

 both of the forces and motions stand prominently forth as if to 

 invite or defy theoretical treatment. 



The relations between the resistance encountered by, and the 

 velocity of a solid body moving steadily through, a fluid in which 

 it is completely immersed, or of water moving through a tube, 

 present themselves mostly in one or other of two simple forms. 

 The resistance is generally proportional to the square of the 

 velocity, and when this is not the case it takes a simpler form, 

 and is proportional to the velocity. 



Again, the internal motion of water assumes one or other of 

 two broadly distinguishable forms — either the elements of the 

 fluid follow one another along lines of motion which lead in the 

 most direct manner to their destination, or they eddy about in 

 sinuous paths, the most indirect possible. 



3. Connection between the Leading Features of Fluid Motion. — 

 These leading features of fluid motion are well known, and are 

 supposed to be more or less connected, but it does not appear 

 that hitherto any very determined efforts have been made to 

 trace a definite connection between them, or to trace the 

 characteristics of the circumstances under which they are usually 

 presented. 



Certain circumstances have been definitely associated with the 

 particular 1 iws of force. Resistance as the square of the velocity 

 is associated with motion in tubes of more than capillary dimen- 

 sions, and with the motion of the bodies through the water at 

 more than insensibly small velocities, while resistance as the 

 velocity is associated with capillary tubes and small velocities. 



The equation; of hydrodynamics, although they are applicable 

 to direct motion, i.e. without eddies, and show that then the 

 resistance is as ihe velocity, have hitherto thrown no light on 

 the circumstances on which such motion depends. And although 

 of late years these equations have been applied to the theory of 

 the eddy, they have not been in the least applied to the motion 

 of water, which is a mass of eddies, i.e. in sinuous motion, nor 

 have they yielded a clue to the cause of resistance varying as the 

 square 01 the velocity. Thus, while as applied to waves and the 

 motion of water in capillary tubes the theoretical results agree 

 with the experimental, the theory of hydrodynamics has so far 

 failed to afford the slightest hint why it should explain these 

 phenomena, anl signally failed to explain the law of resistance 

 encountered by large bodies moving at sensibly high velocities 

 through water, or that of water in sensibly large pipes. 



This acciden'al fitness of the theory to explain certain of the 

 phenomena, while entirely failing to explain others, affords strong 

 presumption that there are some fundame tal principles of fluid 

 motion of which due account has not been taken in the theory ; 

 and several years ago it seemed to me that a careful examination 

 as to the connection between these four leading feature-, together 

 with the circumstances on which they severally depend, was 

 the most likely means of finding the clue to the principles over- 

 looked. 



4. Space and Velocity. — The definite association of resistance 

 as the square of the velocity with sensibly large tubes and high velo- 

 cities, and of resistance as the velocity with capillary tubes and 

 slow velocities, seemed to be evidence of the very general and 

 important influence of some properties of fluids not recognised 

 in the theory of hydrodynamics. 



As there is no such thing as absolute space or absolute time 



recognised in mechanical philosophy, to suppose that the cha- 

 racter of motion of fluids in any way depended on absolute size 

 or absolute velocity would be to suppose such motion c utside 

 the pale of the laws of motion. If, then, fluids in their motions, 

 are suhject to these laws, what appears to be the dependence of 

 the character of the motion on the absolute size of the tube and 

 on the absolute velocity of the immersed body must in reality be 

 a dependence on the size of the tube as compared with the size 

 of some other object, and on the velocity of the body as com- 

 pared with some other velojity. What is the standard object 

 and what the standard velocity which come into comparison with 

 the size of the tube and the velocity of an immersed body are 

 questions to which the answers were not obvious. Answers, 

 however, were found in the discovery of a circumstance on which 

 sinuous motion depends. 



5. The Effect of Viscosity on the Character of Fluid Motion. 

 —The small evidence which clear water shows as to the exist- 

 ence of internal eddies, not less than the difficulty of estimating 

 the viscous nature of the fluid, appears to have hitherto obscured 

 the very important circumstance that the more viscous a fluid is 

 the less prone is it to eddying or sinuous motion. To express this 

 definitely, if p. is the viscosity and the density of the fluid, for 



water - diminishes rapidly as the temperature rises ; thus at 



5° C ' p is double what it is at 45 C. What I observed was 



that the tendency of water to eddy becomes much greater as the 

 temperature rises. 



Hence, connecting the change in the law of resistance with 

 the birth and development of eddies, this discovery limited 

 further search for the standard distance and standard vel icity to 

 the physical properties of the fluid. 



To follow the line of this search would be to enter upon a 

 molecular theory of liqui Is, and this is beyond my present pur- 

 pose. It is sufficient here to notice the well-known fact 



that — is a quantity of the nature of a product of a distance and 

 a velocity. 



6. Evidence from the Equations of Motion. — In this article it 

 is pcinted out that the equations of motion afford definite evi- 

 dence of a dependence of the dynamical equilibrium of a fluid on 



the value of ■£ — , c being the diameter of the pipe and (J the 



mean velocity of the fluid. 



7. The Cause of Eddies. — There appeared to be two possible 

 causes for the change of direct motion into sinuous. These are 

 best discussed in the language of hydrodynamics ; but as the 

 results of this investigation relate to both ihese causes, which, 

 although the distinction is subtile, are fundamentally distinct and 

 lead to distinct results, it is necessary that they should I e indi- 

 cated. 



The general cause of the change from steady to eddying motion 

 was, in 1843, pointed out by Prof. Stokes as being that, under 

 certain circumstances, the steady motion becomes unstable, so 

 that an indefinitely small disturbance may lead to a change to 

 sinuous motion. Both the causes above referred to are of this 

 kind, and yet they are distinct ; the distinction lying in the part 

 taken in the instability by vi-cosity. If we imagine a fluid free 

 from viscosity and absolutely free to glide over solid surfaces, 

 then comparing such a fluid with a viscous fluid in exactly the 

 same motion — 



(1.) The frictionless fluid might be unstable and thevi-cous 

 stable. Under these circumstances the cause of eddies is the 

 instability as a perfect fluid, the effect of viscosity being in the 

 direction of stability. 



(2.) The frictionless fluid might be stable, and the viscous 

 fluid unstable ; under which circumstances the cause of instabi- 

 lity would be the viscosity. 



It was clear to me that the conclusion I had drawn from the 

 equations of motion immediately related only to the first cause. 

 Nor could I then perceive any possible way in which instability 

 could result from viscosity. All the same I felt a certain amount 

 of uncertainty in assuming the first cause of instat ility to be 

 general. This uncertainty was the result of various considera- 

 tions, but particularly from my having observed that eddies 

 apparently come on in very different ways, according to a very 

 definite circumstance of motion, which may be illustrated. 



When in a channel the water is all moving in the same direc- 

 tion, the velocity being greatest in the middle, and diminishing 

 to zero at the sic' , as indicated by the curve in Fig. I, eddies 



