Chap. XVII.

THEORY OF THE MOTION OF FLUIDS

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cylindrical tube, when compared with that which issues through a hole in a thin plate,whatever may be the direction of the tube ; this Venturi has proved upon the principle ofvertical ascension combined with the pressure of the atmosphere, a heavy fluid moving in adescending cylindrical pipe accelerating its motion: the lower particles have a tendency toseparate themselves from the upper, and thus cause the pressure of the atmosphere toincrease the velocity of the upper ; this cannot take place in an ascending or horizontal pipe,though the pressure of the atmosphere in such cases increases the velocity of the fluidwithin the pipe. In descending cylindrical tubes, the upper ends of which have the formof the vena contracta, the velocity of the effluent water corresponds with the height of thefluid above the lower extremity of the tube. A descending tube applied to a reservoirincreases the expenditure, as was remarked by Frontinus , De Aqueduct, “ Calix devexusamplius rapit."

Let B L K O be a conical tube having the form of the vena contracta, the cylindricaltube LCQK having the same diameter as the contracted part LK ; the fluid stratum LKcontinuing to descend through the height LC, will have itsmotion accelerated in the same manner as all other bodies fallingby the force of gravity : hence, when it passes from L K to L Mit tends to detach itself from the stratum immediately above it,or, in other words, to produce a vacuum between LKand MN,and the same effect is produced through the whole length, L C,of the ttibe : the pressure of the atmosphere becomes active as faras is necessary to prevent the vacuum, and its action is the sameboth at A, the surface of the fluid, and at C, the lower extre-mity of the tube ; the atmospherical pressure at A increases thevelocity of the fluid, which issues at C Q, while at C it destroysthe sum of the accelerations, which would be produced alongL C, so that the fluid remains continuous in the tube.

Let t represent the time in which the continuous column offluid LCQK passes through the tube LC, whatever be thevelocity' at L, and the successive acceleration from L to C; ifwe suppose this column to return upwards from D to E, it willpass through the space DE, which is equal to LC in the sametime t , during which it will lose all the acceleration it ac-quired in its descent from L to C ; the pressure of the column E D continued during thetime t is therefore the force necessary to destroy the successive acceleration from L to C,and to prevent the fluid from losing the continuity in the tube LC ; hence it follows thatthe pressure of the atmosphere, which is exerted at CQ to destroy the sum of the acceler-ations along L C, is equal to the pressure of a column E D of a fluid of the same nature asthat of the reservoir from which the water flows; and since the same pressure must also beexerted on the surface A of the reservoir, if we take F A = L C, the fluid at LK willpossess the velocity which is proper to the height FL= AC, without considering the re-tardation produced by the internal inequalities of the tube LCQK. It is supposed byHachette that the cause of this increased expenditure by tubes is the adhesion of thefluid to the sides of the tubes, which results from capillary attraction.

On the Shock of Water against Plane Surfaces -The shock or impetus of a column of

water is in the compound ratio of the bases of the columns and the square of the velocityof the water, or their falls: wherefore, when the bases of the columns are equal, the im-pacts will be to each other as the squares of the velocities of the water, or as the heights offalls capable of the same velocities. When the bases of the columns are equal, and theystrike equal surfaces, the water may be considered as a mass of small spheres whoseimpact will depend on their velocity, and the number which strike at the same time : for aportion of water flowing with twice the velocity of another portion, striking a surfaceopposed to it not only with tw’ice its force, but with twice its number of particles, itsimpetus must increase in the double proportion, or as the square of its velocity, that is, ifthe velocities are as 3 to 5, the shocks will be as 9 to 25. The shocks being to each otheras the product of the orifices multiplied by the heights of the water, or as the columns B Rand F S, in fig. 1725., which are the forces causing the impacts, it follows that the impactsmay be measured by the weight of the same columns, since the effect of a force acting simplywithout modification may be taken for the force itself. As the pressure which the columnsB R and F S exercise on their base is nothing more than a tendency to motion w’hose effectwould be to produce a shock, which may be measured by the cause itself, it follows that theweight which will express the pressure of water on a surface will express the impact also.

The difference produced by friction between the natural and the effective discharge fromthe same orifice of a medium size will also be very great in the effect of impact, becausethe impacts being in the double ratio of the velocities, the impression of the effective willbe as much less than that of the natural discharge, as the square of the velocity of the firstwill be less than that of the velocity of the second.

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