478 



NATURE 



[Sept. 28, 1876 



strictly to hypothetical fluids without viscosity, and its results as 

 applied to water fall very far short of experimental verification. 

 Thus, as was so beautifully illustrated before this Section last 

 year by Mr. Froude, a solid should move through a frictionless 

 liquid in a rigid inclosure without resistance ; the liquid moving 

 out of its way, from front to rear, in filaments or streams which, 

 closing together behind, cause pressure which exactly balances 

 the pressure in front. In fact, however, water opposes very 

 great resistance to the rapid motion of solids through it. If a 

 ball which will just float be allowed to fall from a great height 

 into water it will only descend a very short distance ; and when 

 we come to speeds like the speed of a shot, a foot of water 

 opposes nearly as much resistance as an inch of iron. Opposite 

 as these facts are to what the stream line theory might lead us to 

 expect, they do not disprove the truth of this theory because it 

 does not take into account the viscosity of water. But it is clear 

 that before we can make much practical use of this theory in 

 dealing with the only fluids with which we have to deal, we 

 must ascertain in what way it is that viscosity affects the 

 behaviour of these fluids, so that it may be taken into account in 

 applying the theory. This is a point on which I think some 

 light is thrown by rendering the motion of the water visible. 



The Stream Line Theory applies to the Vortex Ring. — The idea of 

 colouring the water to render its motion apparent was doubtless 

 suggested by the effect of smoke and the beautiful phenomena of 

 the smoke-ring. In the smoke-ring we have an instance of a most 

 important form of fluid motion accidentally rendered invisible, 

 of which we should otherwise have most certainly been in igno- 

 rance ; as it is, however, it has caught the attention of mathe- 

 maticians, and in the hands of Sir William Thomson, Prof. Tait, 

 Helmholtz, and others, has led to most important researches. 



That which is most striking in the smoke-ring is the regularity 

 and extreme beauty of its internal structure. Our familiarity 

 with objects moving rapidly through the air tends to diminish 

 our surprise at the ease with which these rings move. But when 

 we see these rings in water, this rapid motion and the small dis- 

 turbance which they cause, although only a few inches below the 

 surface, are, I think, the most striking points of the phenomenon. 



Vortex rings in water were exhibited at Edinburgh, in 1871, 

 by Mr, H. Deacon, but only on a very small scale, being formed 

 of a single drop. About three years ago I tried a method of form- 

 ing them, very similar to that used by Prof. Tait for smoke-rings. 

 This method succeeded perfectly. From an orifice f" in diame- 

 ter, I could send rings the full length of my trough (20 feet), 

 and with velocities so great, that during the first part of their 

 course the eye could not follow them. It would appear, from 

 the absence of all disturbance either behind the ring or at the 

 surface, that these rings must move without resistance ; and yet 

 this appears at first sight to be inconsistent with the way in 

 which the speed of the rings diminishes as they proceed, either 

 in water or air. There is, however, a cause for this diminution 

 of speed, which cannot properly be called resistance. The 

 rings grow in size as they proceed, and consequently they are 

 continually adding to their bulk water taken up from that which 

 surrounds them, and with which this forward momentum has to 

 be shared. A loss of velocity must result from this growth in 

 size, and the only question with regard to resistance is whether 

 the one of these is sufiicient to account for the other, whether, 

 notwithstanding the loss of velocity, the momentum of the 

 moving mass remains constant. 



To determine this I measured (by the best means 1 could de- 

 vise) the momentum of a series of equal rings at different dis- 

 tances from this origin ; the result was that (within the limits of 

 accuracy of the experiments) there was the same momentum in 

 the rings after they had travelled 15 feet, and were not moving 

 more than 3 inches per second, as when at 2 feet from the origin, 

 and moving more than 5 feet per second. I conclude, therefore, 

 that these rings do move without any appreciable resistance. 



When this freedom from resistance is considered along with 

 the internal motion of the fluid in and around these rings, it 

 shows that in them we have an instance, and I believe it is 

 the only one, in which the stream line theory applies accurately 

 to motion in a viscous fluid. The form of the mass of fluid 

 moving forward is not nearly that of the ring, but is an oblate 

 spheroid a good deal longer than this ring which it encloses. 



This spheroid, like the ring, is continually growing, but at any 

 instant it has a definite shape, and the motion of the water 

 which surrounds it, is at that instant exactly the same as it would 

 be according to the stream line theory if the spheroid were solid 

 and the water were frictionless. 



The spheroidal form of the bounding surface, which, of course 



being fluid, is perfectly flexible, is maintained against the 

 unequally distributed pressure of the surrounding water by the 

 motion of the water within it, the motion being such that at each 

 point in the boimding surface it causes the same pressure as that 

 arising from the motion of the external water. 



The Effect of Viscosity on the Motion of the Ring. — The motion 

 of the internal water, besides maintaining the shape of the 

 bounding surface, is such that at each point of this surface the 

 motion of the water in contact with it on the inside is identical 

 with the motion of that in contact with the outside. So that not 

 only is the bounding surface at each instant definite in form, but 

 every point of the surface is in motion in exactly the same 

 manner as the water in contact with it. 



The action of the viscosity of the water in causing the gradual 

 growth of the ring and its attendant mass is not confined to the 

 bounding surface, but extends throughout the moving water both 

 internal and external to this surface. There is a gradual diminu- 

 tion in the velocity with which the water moves along the stream 

 lines from the centres outwards in all directions as far as the 

 motion extends into the surrounding water, and, as is well 

 known, when the velocity of a stream varies from point to point 

 in a section across the direction of motion, the effect of viscosity 

 is towards equalising the velocity. Hence, in the case of the 

 vortex ring, the effect of viscosity will be to diffuse the motion 

 outwards, to diminish the whirling velocity at the centre where 

 greatest, and to extend the space through which the water is 

 in motion, that is, to diminish the velocity and extend the size ot 

 each element of the ring. 



And it appears that this effect to diminish the velocity 

 and extend the size of the ring is the only effect of vis- 

 cosity. That is to say, if the water at any instant were to lose 

 its viscosity, then the ring would proceed onwards in exactly the 

 same manner as it was proceeding at that instant, for the internal 

 motion would be just as necessary to balance the external pres- 

 sures and preserve the form of the bounding surface in friction- 

 less fluid, as in water, and hence the same law must hold 

 between the internal and external motions. 



The author then supposes that at a certain instant one of the 

 vortex rings is converted into ice, and further, that there is no 

 friction between the surface of the ice and the surrounding water. 

 He proceeds : — 



Whatever might be the resistance of the ideal smooth ice, it is 

 clear that its actual resistance must exceed this by what ship- 

 builders call its skin resistance, the drag of the water moving 

 past the surface. This may be estimated from the resistance of 

 a plane surface of equal extent when moving edgewise through 

 the water, and this is not much. 



This, however, is only one of the effects of the surface friction. 

 Whatever drag the friction may cause on the surface, there is an 

 equal drag on the water moving past it, and thus the surface 

 friction aids the diffusion in diminishing the velocity with which 

 the water would otherwise move in the stream lines near the 

 surface, and so tends to increase the disturbance of the stream 

 lines in the rear of the solid. 



Actually we find that the resistance resulting from the disturb- 

 ance of the stream lines is ten times as great as the mere skin 

 resistance, and so far is a solid, of the shape of the bounding 

 surface of the vortex^ring, from moving freely, that if it be set in 

 motion, it stops at once and is altogether dead in the water. 



That the disturbance of the stream lines as described above 

 really takes place, is shown when we colour the water. When 

 we first start the solid we see a somewhat irregular vortex ring 

 behind it, which grows rapidly and then breaks up j after this 

 the water behind is all confused, and follows the solid. Whereas 

 with the vortex ring, if there are streaks of colour in the water 

 through which the ring passes, it leaves them so nearly as they 

 were before, that there is scarcely a trace of its path. Thus this 

 disturbance of the stream line appears to be the cause of the 

 resistance encountered by a solid over and above the skin- 

 resistance. 



The magnitude of the effect depends on the curvature of the 

 streams, and hence we see why a body having a fine after- 

 part like a fish encounters so much less resistance than a full 

 body like a spheroid. Whereas if the stream lines were com- 

 plete according to the theory, the extra surface of the fish 

 should cause it much greater resistance. 



Relation between the Vortex Ri?ig and the Stream Lines of a Disc. 

 — Another matter on which I have been able to throw some 

 light by colouring the water, relates to the form of the stream 

 lines of a thin surface, such as a disc. It is, I believe, generally 

 assumed that the theory of stream lines shows all bodies would 



