1843.] 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



30 



the column. Our correspondent disputes this position; but instead 

 of adducing arguments to prove that it is an " important error," he 

 takes for granted the very question in dispute, and thereupon founds a 

 formula for determining another question, with which the present has 

 no immediate connexion. We affirmed, that the velocity of water 

 flowing through vertical pipes differs from the velocity of water is- 

 suing from an orifice, in the bottom of a large column of eqnal 

 height; we stated, also, the cause of this difference, and its amount. 

 Our correspondent, by way of refuting this opinion, assumes thai there 

 is no difference ; and then proceeds, on the ordinary data for calcu- 

 lating the velocities of falling bodies, to estimate the height from 

 which a body must fall, before an equilibrium is established between 

 the accelerating force of gravitation and the resistance of the air. 

 We shall not imitate this summary process of disposing of the sub- 

 ject in dispute, but shall endeavour to show that, according to the 

 generally recognized laws of motion, the velocity of water issuing 

 from a long vertical pipe, cannot be the velocity which is due to the 

 height ; and we shall then show the cause of this apparent deviation 

 from the usual law. 



In the first place, it must be borne in mind, that it is admitted, that 

 a pipe having the same diameter throughout, continues full during the 

 flow of water through it; therefore, as water is incompressible, the 

 velocity of the water must be the same at the top of the pipe as at 

 the bottom. Suppose the pipe to be 10 feet vertical, and to be 

 covered with water just sufficiently to keep it constantly full. Then, 

 as the velocity due to a height of 16 ft. is, in round numbers, 32 ft. 

 per second, if the water issue from the pipe with that velocity, the 

 same velocity must be communicated to the fluid in all parts of the 

 pipe, as it forms part of the hypothesis that the velocity is uniform. 

 We should, therefore, be obliged to assume the existence of some 

 force, which could communicate to the water flowing into a tube from 

 a state of rest, a velocity equal to that it would acquire after falling 

 freely through 10 ft. It can scarcely be asserted, that the pressure of 

 the atmosphere would communicate this addition \\ velocity, for the 

 upward pressure on the fluid at the bottom of the pipe must always 

 counterbalance the downward pressure on the top ; and were the pipe 

 a very long one, the upward pressure would be the greater, owing to 

 the increasing density of air at lower elevations. There is, indeed, 

 no rationally conceivable force called into action but gravitation ; and 

 if the whole column of water instantly acquire a velocity, which is 

 due only to a fall through its whole length, the force of gravitation 

 must, iu some unaccountable manner, be doubled ; for the momentum 

 of a column of water, moving with a uniform velocity of 32 feet per 

 second, is equal to the mean momentum of the same weight, were its 

 motion to increase progressively from a state of rest to a velocity of 

 64 feet. There is not, however, the slightest ground for assuming 

 that the force of gravitation produces any such effect. The final ve- 

 locity of a body falling freely through 16 feet is, within a fraction, 

 32 feet per second, the mean velocity of the fill will therefore be one 

 half, or 16 feet per second ; and that, we contend, is the velocity with 

 which a continuous and equal column of water would fall through a 

 vertical pipe 10 feet long; putting out of consideration the friction 

 of the pipe and the resistance of the air. In the case of water is- 

 suing through an orifice, the velocity due to a height of 10 feet is 32 

 feet per second, when the areas of the column and of the orifice are 

 greatly disproportioned ; but it appears from the preceding reasoning, 

 that the whole column of fluid would issue with only half that velocity, 

 which was the point to be proved. 



Having, therefore, shown that t lie conclusion at which we have ar- 

 rived may be deduced as a necessary consequence of the continuous uni- 

 form flow of water in vertical pipes, we shall next proceed to consider 

 the conditions of water when flowing down vertical pipes, and when 

 issuing from an orifice ; and we shall endeavour to arrive at the same 

 conclusion by a different procpss of reasoning. 



It was demonstrated by Daniel Bernoulli, that the impulse of a 



"vein" of fluid falling perpendicularly, is equal to the weight of a 

 column whose base is the area of the vein, and whose height is twice 

 the fall producing the velocity. For example ; if r be taken as the 

 final velocity of the efflux acquired by falling freely from a height /;, 

 then it is well known that a body falling with the final velocity, during 

 the time of the fall, will pass through a space equal to 2 h, or twice, 

 the height. As the water commences and continues to flow through 

 an orifice with the final velocity due to the height, the quantity of 

 water falling through the aperture in a given time is double the quan- 

 tity that would flow through it if the flow commenced with the initial 

 velocity of a falling body, and progressively increased to its final ve- 

 locity. 



Bernoulli's hypothetical vein of fluid was without any tan- 

 gible boundaries, and the particles of the fluid in the vein were sup- 

 posed to be pressed against, and changing places with, all the other 

 particles in the containing vessel. It is this transmission of the pres- 

 sure through the fluid, that causes the difference between the imagi- 

 nary vein of fluid and a re.il pipe passing from the orifice to the sur- 

 face. When the communication between the orifice and all other 

 parts of the vessel is free, the water near the orifice is forced out, net 

 only by the weight of the particles immediately above it, but all the 

 particles of fluid are pressing towards the aperture and contributing 

 towards the effect. The space occupied by the particles of fluid 

 forced through the aperture, is immediately filled by other particles 

 sustaining equal pressure. The continuity and equality of the pres- 

 sure are thus preserved, which consequently ma : ntains an equal and 

 continuous flow, the. height of the fluid being supposed constant. 

 The velocity at the first moment of efflux is the same as would be 

 acquired by a body falling freely from the surface, because the whole 

 gravitating effects of the perpendicular vein of fluid instantly act on 

 the portion of water above the orifice, and this action is continued, 

 because the pressure remains free and constant. When a vertical 

 pipe passes from the orifice to the surface of the water, so as to ex- 

 clude the action of the surrounding fluid, the conditions are essentially 

 changed. Suppose such a pipe to be filled with water, the base of 

 the vein of water within, when at res/, would sustain the same pres- 

 sure as another equal area on the bottom of the vessel, the heights 

 being equal. But as the force then acting on the lowest lamina of 

 the fluid is produced solely by the pressure of the laminae of fluid 

 above, were the lowest one to separate from the upppr by the impulse 

 of this pressure, the force would instantly cease, for the lamina im- 

 mediately above the lowest not being impelled with equal force, 

 would not have the same velocity. The adhesion of the particles of 

 water would, however, prevent the falling vein of fluid from being 

 divided, because the difference of the force acting on one minute 

 lamina, and that acting on the. fluid particles immediately above, 

 would not equal the cohesive attraction which holds them together. 

 The vein of fluid would, therefore, cohere ami fall through the ver- 

 tical length of pipe as a solid mass. Again ; as the upper part of 

 the vertical vein of fluid in the pipe would be as free to move under 

 the influence of gravitation, when the supporting base was re- 

 moved, as the lower portions of the vein, and as the force would 

 be exerted in the same time, the velocities they would re^po lively 

 acquire, would be the same ; and they would fall through equal spaces 

 in equal times. The length of pipe we have assumed to be 16 feet, 

 therefore, it would be emptied by the fall of water in one second, 

 the final velocity on issuing fro:-; the pipe would be 32 feet per 

 second, and the mean of the initial and final velocities woul 1 be 10 

 feet per second. 



If we suppose the water just to cover the top of the pipe, so as to 

 keep it constantly full, the flow of water would then, it is admitted, 

 be uniform instead of being accelerated, as in the preceding illustra- 

 tion. The water at the lower portion of the pipe would be retarded 

 in its fall, by the continuity of cohesion between the particles 

 fluid in the failing vien; or, in other Word*, the velor I 



