tion of water above the body which was exposed to the Restaaer

stream.

Resistance even rebound, as if frequently observed. This actual of Fluids. rebounding is here prevented by the surrounding water, which is moving with the same velocity: but the pressure may be almost annihilated by the tendency to rebound of the inner filaments.

72 Substance of Buat's theory.

73 Experi

which it is

confirmed.

Part (and perhaps a considerable part) of this apparent non pression is undoubtedly produced by the tenacity of the water, which licks off with it the water lying in the hole. But at any rate, this is an important fact, and gives great value to these experiments. It gives a key to many curious phenomena in the resistance of fluids; and the theory of M. Buat deserves a very serious consideration. It is all contained in the two following propositions.

1,"hf, by any cause whatever, a column of fluid, whether making part of an indefinite fluid, or contained in solid canals, come to move with a given velocity, the pressure which it exerted laterally before its motion, either on the adjoining fluid or on the sides of the canal, is diminished by the weight of a column having the height necessary for communicating the velocity of the motion.

over it.

2. "The pressure on the centre of a plane surface perpendicular to the stream, and wholly immersed in it, is of the weight of a column having the height necessary for communicating the velocity. For 33 is of 214." Ile attempted to ascertain the medium pressure on ments by the whole surface, by opening 625 holes dispersed all With the same velocity of current, he found the height in the tube to be 29 lines, or 74 more than the height necessary for producing the velocity. But he justly concluded this to be too great a measure, because the holes were of an inch from the edge: had there been holes at the very edge, they would have sustained a non-pression, which would have diminished the height in the tube very considerably. He exposed to the same stream a conical funnel, which raised the water to 34 lines. But this could not be considered as a measure of the pressure on a plane solid surface; for the central water was undoubtedly scooped out, as it were, and the filaments much more deflected than they would have been by a plane surface. Perhaps something of this happened even in every small hole in the former experiments. And this suggests some doubt as to the accuracy of the measurement of the pressure and of the velocity of a current by Mr Pitot's tube. It surely Lenders some corrections absolutely necessary. It is a fact, that when exposed to a vein of fluid coming through a short passage, the water in the tube stands on a level with that in the reservoir. Now we know that the velocity of this stream does not exceed what would be produced by a fall equal to of the head of wa, ter in the reservoir. Mr Buat made many valuable observations and improvements on this most useful instrument, which will be taken notice of in the articles RIVERS and WATER-WORKS.

82

100

Mr Buat, by a scrupulous attention to all the circumstances, concludes that the medium of pressure on the whole surface is equal to 25.5 of the weight of a co21.5 lumn, having the surface for its base, and the produetive fall for its height. But we think that there is an uncertainty in this conclusion; because the height of the water in the vertical tube was undoubtedly augmented by an hydrostatical pressure arising from the accumula

21.5

Since the pressures are as the squares of the velocities, or as the heights 4 whieh produce the velocities, 25.5 we may express this pressure by the symbol h, or 1.186 h, ar mh, the value of being 1.186. This exceeds considerably the result of the experiments of the French academy. In these it does not appear that m sensibly exceeds unity. Note, that in these experiments the body was moved through still water; here it is exposed to a stream. These are generally supposed to be equivalent, on the authority of the third law of motion, which makes every action depend on the relative motions. We shall by and by see some causes of difference.

of Fluids

74

The writers on this subject seem to think their task The act completed when they have considered the action of the the im fluid on the anterior part of the body, or that part of der part af body it which is before the broadest section, and have paid ship tqunbe little or no attention to the hinder part. Yet those who important are most interested in the subject, the naval architects, with that seem convinced that it is of no less importance to at- on the fore tend to the form of the hinder part of a ship. And pirt

75

this subject

the universal practice of all nations has been to make the hinder part more acute than the fore-part. This has undoubtedly been deduced from experience; for it is in direct opposition to any notions which a person would naturally form on this subject. Mr Buat therefore thought it very necessary to examine the action of the water on the hinder part of the body by the same method. And, previous to this examination, in order to Experacquire some scientific notions of the subject, he made tent on the following very curious and instructive experiment. by Buat, Two little conical pipes AB (fig. 24.) were inserted Fig. into the upright side of a prismatic vessel. They were an inch long, and their diameters at the inner and outer ends were five and four lines. A was 57 lines under the surface, and B was 73. A glass syphon was made of the shape represented in the figure, and its internal diameter was 14 lines. It was placed with its mouth in the axis, and even with the base of the conical pipe. The pipes being shut, the vessel was filled with water, and it was made to stand on a level in the two legs of the syphon, the upper part being full of air. When this syphon was applied to the pipe A, and the water running freely, it rose 32 lines in the short leg, and sunk as much in the other. When it was applied to the pipe B, the water rose 41 lines in the one leg of the syphon, and sunk as much in the other.

it.

He reasons in this manner from the experiment. The and his rearing comprehended between the end of the syphon and soning upon the sides of the conical tube being the narrowest part of the orifice, the water issued with the velocity corresponding to the height of the water in the vessel above the orifice, diminished for the contraction. If therefore the cylinder of water immediately before the mouth of the syphon issued with the same velocity, the tube would be emptied through a height equal to this HEAD OF WATER (charge). If, on the contrary, this cylinder of water, immediately before the mouth of the syphon, were stagnant, the water in it would exert its full pressure on the mouth of the syphon, and the water in the syphon would be level with the water in the vessel.

Between

y

We see that in both experiments it bears an accurate proportion to the depth under the surface. For 57: 73 32 41 very nearly. He therefore estimates the non-pressure to be of the height of the water above the orifice.

56

100

We are disposed to think that the ingenious author te. has not reasoned accurately from the experiment. In the first place, the force indicated by the experiment, whatever be its origin, is certainly double of what he supposes; for it must be measured by the sum of the rise of the water in one leg, and its depression in the other, the weight of the air in the bend of the syphon being neglected. It is precisely analogous to the force acting on the water oscillating in a syphon, which is acknowledged to be the sum of the elevation and depression. The force indicated by the experiment therefore is of the height of the water above the orifice. The force exhibited in this experiment bears a still greater proportion to the productive height; for it is certain that the water did not issue with the velocity acquired by the fall from the surface, and probably did not exceed of it. The effect of contraction must have been considerable and uncertain. The velocity should have been measured both by the amplitude of the jet and by the quantity of water discharged. In the next place, we apprehend that much of the effect is produced by the tenacity of the water, which drags along with it the water which would have slowly issued from the syphon, had the other end not dipped into the water of the vessel. We know, that if the borizontal part of the syphon had been continued far enough, and if no retardation were occasioned by friction, the column of water in the upright leg would have accelerated like any heavy body; and when the last of it had arrived at the bottom of that leg, the whole in the horizontal part would be moving with the velocity acquired by falling from the surface. The water of the vessel which issues through the surrounding ring very quickly acquires a much greater velocity than what the water descending in the syphon would acquire in the same time, and it drags this last water along with it both by tenacity and friction, and it drags it out till its action is opposed by the want of equilibrium produced in the syphon, by the elevation in the one leg and the depression in the other. We imagine that little can be concluded from the experiment with respect to the real nonpressure. Nay, if the sides of the syphon be supposed infinitely thin, so that there would be no curvature of the filaments of the surrounding water at the mouth of the syphon, we do not very distinctly see any source of nonpressure: For we are not altogether satisfied with the proof which Mr Buat offers for this measure of the pressure of a stream of fluid gliding along a surface, and obstructed by friction or any other cause. We imagine that passing water in the present experiment would be a little retarded by accelerating continually the water descending in the syphon, and renewed a-top, supposing the upper end open; because this water would not of itself acquire more than half this velocity. It however drags it out, till it not only resists with a force equal to the weight of the whole vertical column, but even exceeds it by. This it is able to de, because the

757 whole pressure by which the water issues from an orifice Resistanc has been shown (by Daniel Bernoulli) to be equal to of Fluids. twice this weight. We therefore consider this beantiful experiment as chiefly valuable, by giving us a measure of the tenacity of the water; and we wish that it were repeated in a variety of depths, in order to discover what relation the force exerted bears to the depth. It would scem that the tenacity, being a certain determinate thing, the proportion of 100 to 112 would not be constant, and that the observed ratio would be made up of two parts, one of them constant, and the other proportional to the depth under the surface.

But still this experiment is intimately connected with the matter in hand; and this apparent non-pressure on the hinder part of a body exposed to a stream, from whatever causes it proceeds, does operate in the action of water on this hinder part, and must be taken into the

account.

of De Buat.

78 We must therefore follow the chevalier de Buat in Further his discussions on this subject. A prismatic body, ha- discussions ving its prow and poop equal and parallel surfaces, and plunged horizontally into a fluid, will require a force to keep it firm in the direction of its axis precisely equal to the difference between the real pressures exerted on its prow and poop. If the fluid is at rest, this difference will be nothing, because the opposite dead pressures of the fluid will be equal: but in a stream, there is superadded to the dead pressure on the prow the active pressure arising from the deflections of the filaments of this fluid.

If the dead pressure on the poop remained in its full intensity by the perfect stagnation of the water behind it, the whole sensible pressure on the body would be the active pressure only on the prow, represented by m h. If, on the other hand, we could suppose that the water behind the body moved continually away from it (being renewed laterally) with the velocity of the stream, the dead pressure would be entirely removed from its poop, and the whole sensible pressure, or what must be opposed by some external force, would be mh+h. Neither of these can happen; and the real state of the case must be between these extremes.

79.

The following experiments were tried: The perfo- Experirated box with its vertical tube was exposed to the ments.. stream, the brass plate being turned down the stream. The velocity was again 36 inches per second.

The central hole A alone being opened, gave a non-
pressure of
13 lines.

A hole B, of an inch from the edge,

Here it appears that there is a very considerable non-pressure, increasing from the centre to the border. This increase undoubtedly proceeds from the greater lateral velocity with which the water is gliding in from the sides. The water behind was by no means stagnant, although moving off with a much smaller velocity than that of the passing streara, and it was visibly removed from the sides, and gradually licked away at its further extremity.

Another box, having a great number of holes, all open, indicated a medium of non-pressure equal to 13. lines. Another

Resistance

Another of larger dimensions, but having fewer holes, of Fluids. indicated a non-pressure of 12%.

80

Great uti

in ship

building.

But the most remarkable, and the most important phenomena were the following.

The first box was fixed to the side of another box, so that, when all was made smooth, it made a perfect cube, of which the perforated brass plate made the poop.

The apparatus being now exposed on the stream, with the perforated plate looking down the stream,

The hole A indicated a non-pression

B C

=7.2

6

Those are most valuable experiments. They plainly lity of them show how important it is to consider the action on the hinder part of the body. For the whole impulse or resistance, which must be withstood or overcome by the external force, is the sum of the active pressure on the fore-part, and of the non-pressure on the hinder-part; and they show that this does not depend solely on the form of the prow and poop, but also, and perhaps chiefly, on the length of the body. We see that the nonpressure on the hinder-part was prodigiously diminished (reduced to one-fourth) by making the length of the body triple of the breadth. And hence it appears, that merely lengthening a ship, without making any change in the form either of her prow or her poop, will greatly diminish the resistance to her motion through the water; and this increase of length may be made by continuing the form of the midship frame in several timbers along the keel, by which the capacity of the ship, and her power of carrying sail, will be greatly increased, and her other qualities improved, while her speed is aug

SI Physical cause of it explained.

mented.

It is surely of importance to consider a little the physical cause of this change. The motions are extremely complicated, and we must be contented if we can but perceive a few leading circumstances.

The water is turned aside by the anterior part of the body, and the velocity of the filaments is increased, and they acquire a divergent motion, by which they also push aside the surrounding water. On each side of the body, therefore, they are moving in a divergent direction, and

gra.

with an increased velocity. But as they are on all sides Resist pressed by the fluid without them, their motions of Flus dually approach parallelism, and their velocities to an equality with the stream. The progressive velocity, or that in the direction of the stream, is checked, at least at first. But since we observe the filaments constipated round the body, and that they are not deflected at right angles to their former direction, it is plain that the real velocity of a filament in its oblique path is augmented. We always observe, that a stone lying in the sand, and exposed to the wash of the sea, is laid bare at the bottom, and the sand is generally washed away to some distance all round. This is owing to the increased velocity of the water which comes into contact with the stone. It takes up more sand than it can keep floating, and it de posits it at a little distance all around, forming a little bank, which surrounds the stone at a small distance. When the filaments of water have passed the body, they are pressed by the ambient fluid into the place which it has quitted, and they glide round its stern, and fill up the space behind. The more divergent and the more rapid they are, when about to fall in behind, the more them into the trough behind the body, and less of it will of the circumambient pressure must be employed to turn remain to press them to the body itself. The extreme of this must obtain when the stream is obstructed by a thin plane only. But when there is some distance between the prow and the poop, the divergency of the filaments which had been turned aside by the prow, is diminished by the time that they have come abreast of the stern, and should turn in behind it. They are therefore more readily made to converge behind the body, and a more considerable part of the surrounding pressure remaius unexpended, and therefore presses the water against the stern; and it is evident that this advantage must be so much the greater as the body is longer. But the advantage will soon be susceptible of no very considerable increase for the lateral and divergent, and accelerated filaments, will soon become so nearly parallel and equally rapid with the rest of the stream, that a great increase of length will not make any considerable change in these particulars; and it must be accompa nied with an increase of friction.

These are very obvious reflections. And if we attend minutely to the way in which the almost stagnant fluid behind the body is expended and renewed, we shall see all these effects confirmed and augmented. But as we cannot say any thing on this subject that is precise, or that can be made the subject of computation, it is needless to enter into a more minute discussion. The diminu tion of the non-pressure towards the centre most probably arises from the smaller force which is necessary to be expended in the inflection of the lateral filaments, already inflected in some degree, and having their velocity dimi nished. But it is a subject highly deserving the attention of the mathematicians; and we presume to invite them to the study of the motions of these lateral filaments. passing the body, and pressed into its wake by forces which are susceptible of no difficult investigation. It seems highly probable, that if a prismatic box, with a square stern, were fitted with an addition precisely shaped like the water which would (abstracting tenacity and friction) have been stagnant behind it, the quantity of non-pression would be the smallest possible. The mathematician would surely discover circumstances which

would

de

18

is

-OJ

In the mean time, let us attend to the deductions which Mr de Buat has made from his few experiments.

When the velocity is three feet per second, requiring the productive height 21.5 lines, the heights corresponding to the non-pressure on the poop of a thin plane is 14.41 lines (taking in several circumstances of judicious correction, which we have not mentioned), that of a foot cube is 5.83, and that of a box of triple length is 3.31.

Let q express the variable ratio of these to the height producing the velocity, so that q h may express the nonpressure in every case; we have,

For a thin plane

a cube

a box 3 cubes

9

served near the edges of the surface. The general fac- Resistance tor of the pressure of a stream on the anterior surface of Fluids. was m 1.186; but that on a moving body through a still fluid is only m≈ 1. He observed no non-pressure even at the very edge of the prow, but even a sensible pressure. The pressure, therefore, or resistance is more equably diffused over the surface of the prow than the impulse is. He also found that the resistances diminished in a less ratio than the squares of the velocities, especially in small velocities.

The non-pressures increased in a greater ratio than the squares of the velocities. The ratio of the velocities to a small velocity of 23 inches per second increased geometrically, the value of q increased arithmetically; and we may determine q for any velocity V by this proportion

9=0.67 0.271 0.153

22

It is evident that the value of q has a dependence on the proportion of the length, and the transverse section of the body. A series of experiments on prismatic bodies showed Mr de Buat that the deviation of the filaments was similar in similar bodies, and that this obtained even in dissimilar prisms, when the lengths were as the square-roots of the transverse sections. Although therefore the experiments were not sufficiently numerous for deducing the precise law, it seemed not impossible to derive from them a very useful approximation. By a dexterous comparison he found, that if / expresses the length of the prism, and s the area of the transverse section, and L expresses the common logarithm of the quantity to which it is prefixed, we shall express the non-pressure pretty accurately by the formula = น

I

Hence arises an important remark, that when the height corresponding to the non-pression is greater than √s, and the body is little immersed in the fluid, there will be a void behind it. Thus a surface of a square inch, just immersed in a current of three feet per second, will have a void behind it. A foot square will be in a similar condition when the velocity is 12 feet.

We must be careful to distinguish this non-pressure from the other causes of resistance, which are always necessarily combined with it. It is superadditive to the active impression on the prow, to the statical pressure of the accumulation a-head of the body, the statical pressure arising from the depression behind it, the effects of friction, and the effects of tenacity. It is indeed next to impossible to estimate them separately, and many of them are actually combined in the measures now given. Nothing can determine the pure non-pressures till we can ascertain the motions of the filaments.

M. de Buat here takes occasion to controvert the universally adopted maxim, that the pressure occasioned by a stream of fluid on a fixed body is the same with that on a body moving with equal velocity in a quiescent fluid. He repeated all these experiments with the perforated box in still water. The general distinction was, that both the pressures and the non-pressure in this case were less, and that the odds were chiefly to be ob

V

When Pitot's tube was exposed to the stream, we had m= 1; but when it is carried through still water, m is 1.22. When it was turned from the stream, we had is =

$4

sup

0.1573 but when carried through still water, q 0.138. A remarkable experiment. When the tube was moved laterally through the wa- and ter, so that the motion was in the direction of the plane ports his opinion by of its mouth, the non-pressure was 1. This is one a remarkof his chief arguments for bis theory of non-pression. able expeHe does not give the detail of the experiment, and only riment. inserts the result in his table.

As a body exposed to a stream deflects the fluid, heaps it up, and increases its velocity; so a body moved through a still fluid turns it aside, causes it to swell up before it, and gives it a real motion alongside of it in the opposite direction. And as the body exposed to a stream has a quantity of fluid almost stagnant both before and behind; so a body moved through a still fluid carries before it and drags after it a quantity of fluid, which accompanies it with nearly an equal velocity. This addition to the quantity of matter in a motion must make a diminution of its velocity; and this forms a very considerable part of the observed resistance.

$5

well found

We cannot, however, help remarking that it would The objecrequire very distinct and strong proof indeed to over- tion not turn the common opinion, which is founded on our most ed. certain and simple conceptions of motion, and on a law of nature to which we have never observed an exception. M. de Buat's experiments, though most judiciously contrived, and executed with scrupulous care, are by no means of this kind. They were, of absolute necessity, very complicated; and many circumstances, impossible to avoid or to appreciate, rendered the observation, or at least the comparison, of the velocities, very uncertain.

Resistance have a stream except in consequence of a sloping surof Fuids. face. Suppose a body floating on this stream. It will not only sail down along with the stream, but it will sail down the stream, and will therefore go faster along the canal than the stream does: for it is floating on an inclined place; and if we examine it by the laws of bydrostatics, we shall find, that besides its own tendency to slide down this inclined plane, there is an odds of hydrostatical pressure, which pushes it down this plane. It will therefore go along the canal faster than the stream. For this acceleration depends on the difference of pressure at the two ends, and will be more remarkable as the body is larger, and especially as it is longer. This may be distinctly observed. All floating bodies go into the stream of the river, because there they find the smallest obstruction to the acquisition of this motion along the inclined plane; and when a number of bodies are thus floating down the stream, the largest and longest outstrip the rest. A log of wood floating down in this manner may be observed to make its way very fast among the chips and saw-dust which float alongside of it.

57

Mr Buat's calculation of resistance,

Now when, in the course of our experiments, a body is supported against the action of the stream, and the impulse is measured by the force employed to support it, it is plain that part of this force is employed to act against that tendency which the body has to outstrip the stream. This does not appear in our experiment, when we move a body with the velocity of this stream through still water having a horizontal surface.

The other distinguishing circumstance is, that the retardations of a stream arising from friction are found to be nearly as the velocities. When, therefore, a stream moving in a limited canal is checked by a body put in its way, the diminution of velocity occasioned by the friction of the stream having already produced its effect, the impulse is not affected by it; but when the body puts the still water in motion, the friction of the bottom produces some effect, by retarding the recess of the water. This, however, must be next to nothing.

The chief difference will arise from its being almost impossible to make an exact comparison of the velocities for when a body is moved against the stream, the relative velocity is the same in all the filaments. But when we expose a body to a stream, the velocity of the different filaments is not the same; because it decreases from the middle of the stream to the sides.

M. Buat found the total sensible resistance of a plate 12 inches square, and measured, not by the height of water in the tube of the perforated box, but by weights acting on the arm of a balance, having its centre 15 inches under the surface of a stream moving three feet per second, to be 19.46 pounds; that of a cube of the same dimensions was 15.22; and that of a prism three feet long was 13.87; that of a prism six feet long was 14.27. The three first agree extremely well with the determination of m and q, by the experiments with the perforated box. The total resistance of the last was undoubtedly much increased by friction, and by the retrograde force of so long a prism floating in an inclined stream. This last by computation is 0.223 pounds; this added to h (m+q), which is 13.39, gives 13.81, leaving 0.46 for the effect of friction.

If the same resistances be computed on the supposi

5

body mo

M. Buat next endeavours to ascertain the quantity ving in stil of water which is made to adhere in some degree to a water, de body which is carried along through still water, or which remains nearly stagnant in the midst of a stream. He takes the sum of the motions in the direction of the stream, viz. the sum of the actual motions of all those particles which have lost part of their motion, and he divides this sum by the general velocity of the stream. The quotient is equivalent to a certain quantity of water perfectly stagnant round the body. Without being able to determine this with precision, he observes, that it augments as the resistance diminishes; for in the case of a longer body, the filaments are observed to converge to a greater distance behind the body. The stagnant mass a-head of the body is more constant; for the deflection and resistance at the prow are observed not to be affected at the length of the body. M. Buat, by a very nice analysis of many circumstances, comes to this conclusion, that the whole quantity of fluid, which in this manner accompanies the solid body, re mains the same, whatever is the velocity. He might have deduced it at once, from the consideration that the curves described by the filaments are the same in all velocities.

He then relates a number of experiments made to ascertain the absolute quantity thus made to accompany the body. These were made by causing pendulums to oscillate in fluids. Newton had determined the resistances to such oscillation by the diminution of the arches of vibration. M. Buat determines the quantity of dragged fluid by the increase of their duration; for the stagnation or dragging is in fact adding a quantity of matter to be moved, without any addition to the moving force. It was ingeniously observed by Newton, that the time of oscillation was not sensibly affected by the resistance of the fluid: a compensation, almost complete, being made by the diminution of the arches of vibration; and experiment confirmed this. If, therefore, a great augmentation of the time of vibration be observed, it must be ascribed to the additional quantity of matter which is thus dragged into motion, and it may be employed for its measurement. Thus, let a be the length of a pendulum swinging seconds in vacuo, and 7 the length of a second's pendulum swinging in a fluid. Let p be the weight of the body in the fluid, and P the weight of the body displaced by it; P+p will exP+P press its weight in vacuo, and will be the ratio

of these weights. We shall therefore have

a

P+P =

P

7

P+P

Let n express the sum of the fluid displaced, and the fluid dragged along, n being a greater number than unity,