May 22, 1884] 



NA TURE 



of the two manners of internal motion of fluids so important is 

 that all those problems to which the theory fits belong to the one 

 class of internal motions. The point before us to-night is simple 

 enough, and may be well expressed by analogy. Most of us 

 have more or less familiarity with the motion of troops, and we 

 can well understand that there exists a science of military tactics 

 which treats of the best manoeuvres to meet particular circum- 

 stances. Suppose this science proceeds on the assumption that the 

 discipline of the troops is perfect, and hence takes no account of 

 such moral effects as may be produced by the presence of an 

 enemy. Such a theory would stand in the same relation to the 

 movements of troops -as that of hydrodynamics does to the move- 

 ments of water. For although only disciplined motion may be 

 recognised in military tactics, troops have another manner of 

 motion when anything disturbs their order. And this is pre- 

 cisely how it is with water : it will move in a perfectly direct, 

 disciplined manner under some circumstances, while under 

 others it becomes a mass of eddies and cross streams, which 

 may be well likened to a whirling struggling mob, where 

 each individual element is obstructing the others. Nor does 

 the analogy end here. The circumstances which determine 

 whether the motion of troops shall be a march or a scramble 

 are closely analogous to those which determine whether the 

 motion of water shall be direct or sinuous. In both cases 

 there is a certain influence necessary for order : with troops, 

 it is discipline; with water, it is viscosity or treaclyness. 

 The better the discipline of the troops, or the more treacly the 

 fluid, the less likely is steady motion to be disturbed under any 

 circumstances. On the other hand, speed and size are in both 

 influences conducive to unsteadiness. The larger the army 

 and the more rapid the evolutions the greater the chance of da- 

 nder ; so with fluid, the larger the channel and the greater the 

 velocity the more chance of eddies. With troops some evolu- 

 tions are much more difficult to effect with steadiness than others, 

 and some evolutions which would be perfectly safe on parade 

 would be sheer madness in the presence of an enemy. It is 

 much the same with water. 



One of my chief objects in introducing this analogy is to illus- 

 trate the fact that even while executing mane envies in a steady 

 manner there may be a fundamental difference in the condition 

 of the fluid. This is easily realised in the case of troop-,, difficult 

 and easy manoeuvres may be executed in equally steady manners 

 if all goes well, but the conditions of the moving troop, are 

 essentially different, for while in the one case any slight dis- 

 arrangement would be easily rectified, in the other it would 

 inevitably lead to a scramble. The source of such a change in 

 the manner of motion may be ascribed either to the delicacy of 

 the manoeuvre or to the upsetting disarrangement, but as a matter 

 of fact both these causes are necessary. In the case of extreme 

 delicacy an indefinitely small disturbance, such as is always to be 

 counted upon, will effect the change. Under these circumstances 

 we may well describe the condition of the troops in the simple 

 manoeuvre as stable, while that in the difficult manoeuvre is un- 

 stable, i.e. will break down on the smallest disarrangement. 

 The small disarrangement is the immediate cause of the break- 

 down in the same sense as the sound of a voice is sometimes the 

 cause of an avalanche, but since such disarrangement is certain 

 to occur a condition of instability is the real cause of the change. 



All this is exactly true for the motion of water. Supposing no 

 disarrangement, the water would move in the manner indicated 

 in the theory, just as if there were no disturbance an egg would 

 stand on its end, but as there is always some slight disturbance 

 it is only when the condition of steady motion is more or less 

 stable that it can exist. In addition then to the theories either 

 of military tactics or of hydrodynamics, it is necessary to know 

 under what circumstances the manoeuvres of which they treat are 

 stable or unstable. It is in definitely separating these that the 

 method of colour-bands has done good service, which will re- 

 move the discredit in which the theory of hydrodynamics has 

 been held. 



In tlie first place it lias shown that the properly of viscosity 

 or treaclyness possessed more or less by all fluids is the general 

 influence conducive to steadiness, while, on the oilier hand, 

 space and velocity have the counter influence. Also that the 

 effect of these influences is subject to a perfectly definite law. 

 which is that a particular evolution becomes unstable for a defi- 

 nite value of the viscosity divided by the product of the velocity 

 and space. This law explains a vasl number of phenomena 

 which have hitherto appeared paradoxical. One general con- 

 clusion i- that with sufficiently slow motion all manners ol motion 

 are stable. 



The effect of viscosity is well shown by introducing a band o 

 coloured water across a beaker filled with clear water at rest. 

 Then, when all is quite still, turn the beaker about its axis. The 

 glass turns, but not the water, except that which is quite close to 

 the glass. The coloured water which is close to the glass is 

 drawn out into what looks like a long smear, but it is not a 

 smear. It is simply a colour-band extending from the point in 

 which the colour touched the glass in a spiral manner inwards ; 

 showing that the viscosity is slowly communicating the motion of 

 the glass to the water within. To show this it is only necessary 

 to turn the beaker back, and the smear closes up • until the 

 colour-band assumes its radial position. Throughout this evolu- 

 tion the motion has been quite steady — quite according to the 

 theory. 



When water flows steadily, it flows in streams. Water flowing 

 along a pipe is such a stream. This is bounded by the solid 

 surface of the pipe, but if the water is flowing steadily we can 

 imagine the water to be divided by ideal tubes into a faggot of 

 indefinitely small streams, any one of which may be coloured 

 without altering its motion, just as one column of infantry may- 

 be distinguished from another by colour. 



If there is internal motion, it is clear that we cannot consider 

 the whole stream bounded by the pipe as a faggot of elementary 

 streams, as the water is continually crossing the pipe from one 

 side to another, any more than we can distinguish the streaks of 

 colour in a human stream in the corridor of a theatre. 



Solid walls are not necessary to form a stream. The jets from 

 a fountain or the cascade in Niagara are streams bounded by 

 free surfaces. A river is a stream half bounded by a solid surface. 

 Streams may be parallel, as in a pipe ; converging or diverging, 

 as in conical pipes ; or they may be straight and curved. All 

 these circumstances have their influence on stability in the manner 

 indicated in the accompanying diagram. 



ClRCUMSTAN'CES CONDUCIVE TO 

 Direct or Steady Motion Sinuous or Unsteady Motion 



(1) Viscosity or fluid friction (5) Particular variation of 



winch continually destroys dis- velocity across the stream, as 



turbance. Thus treacle is when a stream flows through 



steadier than water. still water. 



{2) A free bounding surface. 



(6) Solid bounding walls. 



(3) Converging solid boun- (7) Diverging solid bounding 

 daries. walls. 



(4) Curvature of the streams (8) Curvature with the velo- 

 with the velocity greatest on city greatest on the inside. 



the outside. 



It has for a long time been noticed that a stream of fluid 

 through fluid otherwise at rest is in an unstable condition. It is 

 this instability wdiich renders flames and jets sensitive to the 

 slight disarrangement caused by sound. 



I have here a glass vessel of clean water in front of the lantern, 

 so that any colour-bands will be projected on to the screen. 

 You see the ends of two vertical tubes facing each other : 

 nothing is flowing through these tubes, and the water in the 

 vessel is at rest. I now open two taps, so as to allow a steady 

 stream of coloured water to enter at the lower pipe, water 

 flowing out at the upper. The water enters quite steadily, forms 

 a sort of vortex ring at the end, which proceeds across the 

 vessel, and passes out at the lower pipe. The coloured stream 

 then extends straight across the vessel, and fills both pipes : you 

 see no motion ; it looks like a red glass rod. The red water is, 

 however, flowing slowly, so slowly that viscosity is paramount, 

 and hence the stream is steady. As the speed is increased, a 

 certain wriggling, sinuous motion appears in the column ; a little 

 faster and the column breaks up into beautiful and well-defined 

 eddies, and spreads into the surrounding water, which, becoming 

 opaque with colour, gradually draws a veil over the experiment. 

 The final breaking up of the column was doubtless determined 

 by some slight vibration in the apparatus, but such vibration, 

 which is always going on. will not affect the stream until it is in 

 a sufficiently unstable condition. The same is true of all streams 

 bounded by standing water. 



If the motion is sufficiently slow, . o 01 ling to the size of the 

 stream and the viscosity, the stream is steady and stable. Then 

 at a certain critical velocity, determined by the ratio of the 

 viscosity of the water to the diameter of" ihe stream, the stream 

 becomes unstable. So that under any conditions which involve 

 a stream through surrounding water, the motion becomes un- 

 stable at sufficiently great velocities. 



