On the Resistance of Fluids.

One of the most important but at the same time most difficult practical questions connected with the subject of this book is the determination. of the resistance experienced by a solid moving through a fluid, e.g. by a ship moving through the water, or by a projectile through the air.

The effect of a liquid on the motion of a solid through it has been discussed, on certain assumptions, in Arts. 105-119. It was there found that the whole effect was simply equivalent to an increase in the inertia of the solid. The latter yields more sluggishly to the action of force than it would do if the fluid were removed, whether the tendency of the force be to increase or to diminish the motion which the body already has; but there is no tendency to a total transfer of energy of motion from the solid to the fluid, or in any other way to reduce the solid to rest. Thus a sphere immersed in a liquid will move under the action of any forces exactly as if its inertia were increased by half that of the fluid displaced, and the fluid then annihilated. For instance, if set in motion and then left to itself, it will describe a straight path with uniform velocity.

These theoretical conclusions do not at all correspond with what is observed in actual cases. As a matter of fact a body moving through a fluid requires a continual application of force to maintain the motion, and if this be not supplied the body is quickly brought to rest.

This discrepancy between theory and observation must be due to unreality in one or more of the fundamental assumptions on which the investigations referred to were based. These assumptions are

A. That the fluid is 'perfect,' i.e. that there is no tangential action between adjacent portions of fluid, or between the fluid and the solid;

B. That the motion is continuous, i.e. that the velocities u, v, w are everywhere continuous functions of the co-ordinates; and

C. That the fluid is unlimited except by the surface of the solid.

Let us take these in order.

A. The effects of imperfect fluidity, or viscosity, have been treated of in Chapter IX. The resistance due to this cause is proportional to the velocity of the solid, and, for a body of given shape, to its linear dimensions. This has been proved in the text for the case of the sphere, and it is easy, by the method of 'dimensions,' to extend the result to the general case. Thus if 1, L, t, T, u, U, p, P be corresponding lengths, times, velocities, and pressures in two geometrically similar cases of motion, it appears from equations (14) of Art. 180 that we must have

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That is, the resultants of the pressures (normal and tangential) on any corresponding areas are proportional to the products of corresponding lines and velocities.

It must be remembered however that the investigations of Chapter IX. proceed on the assumption that the motion is slow. Thus it is very doubtful whether the equations of Art. 180, or the boundary conditions of Art. 181, would be applicable to the motion of the stratum of water in contact with the side of a ship in rapid motion.

B. It appears from Art. 30 that there is a certain limit to the velocity of a solid of given shape if the motion be continuous. When this limit is reached, the pressure at some point of the surface of the solid sinks to zero, and if the limit be exceeded, a surface of discontinuity is formed. See Art. 94. If the surface of the solid have a sharp projecting edge or angle, this limiting velocity is very small (zero if the edge be of perfect geometrical sharpness), whilst for bodies of 'fair' easy shape it may be considerable.

When the motion is continuous a certain amount of momentum is expended in each unit of time in starting the elementary streams which diverge from the body in front, but this momentum is restored again to the body by the streams as they close in behind. But when a surface of discontinuity is formed the streams do not close in again; on the contrary, we have a mass of 'dead water' following the body and pressing on its rear with merely the general pressure which obtains in

the parts of the fluid which are at rest. Hence the restoration of momentum no longer takes place, and a force equal to the loss per unit time must be applied to the solid in order to maintain its motion.

The only case of this kind which has as yet been mathematically worked out is that in which the solid is a long plane lamina, with parallel straight edges. If the lamina move broadside-on, the motion is obtained from Art. 98 by considering a velocity q, parallel to y impressed on everything. The resistance to the motion of the lamina with velocity is then pq2l, where l is the breadth of the lamina. The resistance

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experienced by a lamina moving obliquely has been calculated by Lord Rayleigh (1. c. Art. 96). The result is

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where a is the angle which the direction of motion of the lamina makes with the normal to its plane.

Generally the resistance due to this cause is proportional to the square of the velocity, so long as we can assume that the motions of the fluid corresponding to different velocities of the solid are geometrically similar.

The formation of the 'wake' which is observed to follow a vessel in motion has been attributed by some writers to the friction between the surface of the ship and the water. It is supposed that the tendency of this friction is to drag a mass of water bodily after the ship. This explanation, if correct, shews that the laws of fluid friction laid down in Chapter IX. do not hold for such rapid motions as are here in question.

It seems possible however that the wake may be in a great measure due to the cause at present under consideration. It is at all events admitted that the wake is greatly increased by discontinuity in the lines of the vessel, a state of things favourable as we have seen to the formation of a surface of discontinuity in the fluid. Again it is possible that the bottom of a ship (especially when foul) is to be regarded not as a geometrical surface, but as a mass of projections each of which establishes a surface of discontinuity, with dead water in its rear, on its own account. The aggregate of these masses of dead water would then build up the wake. We should thus also have an explanation of the law, laid down by some writers on this subject, that the 'skinresistance' is proportional to the square of the velocity.

* Cf. Stokes, Camb. Trans., vol. vIII, p. 301.

Of course there is not, practically, in any of these cases absolute discontinuity of motion. It was shewn in Art. 132 that a surface of discontinuity may be regarded as made up of a system of vortices distributed in a certain way. Owing to fluid friction the vortex-motion does not remain concentrated in this surface, but is diffused through the fluid on each side. Thus the boundary of the dead water is marked by a band of eddies.

C. The effect of a rigid boundary to the fluid may be estimated from Art. 125 (a). It appears that if the dimensions of the solid be small compared with its distance from the boundary, the influence of the latter may be neglected.

A free surface at a great distance from it also has little effect on a solid. The case is otherwise however when the solid is only partially submerged, e. g. a ship. One effect of the motion is to make the pressure deviate from its statical value; to increase it, for instance, at the bows and at the stern, and to diminish it amidships*. Hence the level of the fluid is disturbed, there is an elevation of the water at each of the former points, and a depression between. A wave like this, accompanying the ship, would of course cause no loss of energy beyond what is necessary to maintain it against viscosity. But we have also waves produced which travel over the surface, and carry off energy to the distant parts of the fluid. The energy thus dispersed must of course come directly or indirectly from the ship; and the loss from this cause constitutes a special form of resistance, called 'wave-resistancet.'

A body moving through air will in like manner experience a resistance due to the dispersion of energy by air-waves. It appears, however, that if the motion be steady, and the velocity of the body small compared with that of sound, the resistance due to this cause is inappreciable. For we have, with the same notation as in Art. 169,

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*This is easily seen by impressing on everything a velocity equal and opposite to that of the ship, and so reducing the case to one of steady motion.

For a discussion of the various kinds of waves produced by a ship, and of the probable laws of wave-resistance, we must refer to a paper by Rankine, in the Phil. Trans. for 1871. The student may also consult, on the general question, Rankine "On Stream-Lines in connection with Naval Architecture," Nature vol. II, and Froude "On Stream-Lines in Relation to the Resistance of Ships," Nature, vol. III.

and writing = ay, where a is the velocity of the solid, we have, as in

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differentiated are referred to axes fixed in the solid. On this supposition

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i.e. v is of the second order of small quantities. Hence to a first approximation

v2 = 0,

i.e. the fluid moves sensibly as if it were incompressible.

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