TT1 



HYDROCYANIC ACID. 



HYDRODYNAMICS. 



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HYDRODYNAMICS. Under this word we usually compreheucled 

 the condition* of equilibrium and of motion in non-clastic fluid*, with 

 the resistances which they oppose to bodies moving in them. When a 

 fluid it in a state of rest, the investigation of iU equilibrium and that of 

 bodies immened in it, with the investigation of the pressure exerted by 

 the fluid on bodies immened in it, or containing it, form the subject* of 

 hydrostatic*. Hydrodynamics, which was formerly included uu 

 term kydrauKa, i* concerned chiefly in investigating by mathematical 

 reasoning, or in showing from olservatiou and experiment, the laws 

 relating to the discharge of fluids through orifices and tubes in vessels 

 or reservoirs, and to their motion* in canal* r rivers. 



Concerning the laws of the motions of fluids as they were known to 

 the ancients, little can be said ; the only notice of this branch of 

 science, even in the time of the Roman empire, is contained in the 

 treatise ' De Aqtueductibus,' which was composed by Frontinus, in the 

 reign of Nerva or Trajan. This writer shows that the quantity of 

 water issuing from an orifice depends on its magnitude, and on the 

 height of the water in the reservoir above the orifice ; and he states 

 that a short tube applied to an orifice permitted a greater discharge of 

 fluid than could be obtained from a simple perforation of equal dia- 

 meter. He appears, however, to be unacquainted with the manner of 

 determining the velocity when the height or head of the water is 

 Kiven ; and it is not certain that this elementary proposition wa* 

 solved till the time of Torricellius, who, in 1643, assigned the law 

 correctly, for that case only, however, in which the aperture is very 

 small compared with the height of the water in the vessel or reservoir. 

 It appears that, even at the end of the ItSth century, the cause of 

 the ascent of water in pumps was little known ; for Galileo, having 

 occasion to make some observations on the phenomenon, could give no 

 better reason for it than that it was caused by an attraction which he 

 supposed the piston exercised on the water ; and not being able to 

 make the column of water follow the piston when the latter was about 

 34 feet above the surface of that in the well, he attempted to explain 

 the circumstance by saying thiit the weight of the column was then so 

 great as to overcome the attraction of the piston. Wo .-ire indebted to 

 Torricellius for the discovery that the rise of the water ia owing to the 

 pressure of the atmosphere on that which, in the well, sumran< 

 pump, and which is thus form! into the barrel, in consequence of 

 the removal of the internal air, till the weight of the column raised is 

 in equilibrio with that pressure. [BAROMETER.] 



Uastelli, a disciple of Galileo, in his treatise 'Delia Mesura dell' 

 Acque Correnti ' (1628), appears to have been the first who applied 

 himself to the investigation of the motion of fluids in rivers; and, 

 together with several other circumstances relating to such motion, he 

 shows that when the bed of a canal whose transverse section is variable 

 has taken a permanent form, the velocities at different sections are 

 inversely proportional to the areas of those sections. This branch of 

 the science was subsequently much cultivated in Italy, probably on 

 account of its connection with the important operations then in pro- 

 gress for improving the navigation of the Po ami draining the marshes 

 in the northern part of that country. The Marquis Poleni wrote, in 

 1695, a work entitled ' De Motu Aqutc mixto ; ' and, in 1718, another 

 concerning the flow of water through orifices and short tubes ; and 

 numerous works containing the results of their investigations, and 

 experiments relating to the same subjects, have been made public by 

 other distinguished Italian mathematicians. 



The ' Principia' of Newton contains (lib. ii., sect. 7) a series of pro- 

 positions concerning the motions of fluids. In the first edition (1687) 

 the law of the velocity of water flowing from vessels, being founded on 

 experiments made with orifices of considerable magnitude, appeared to 

 differ from that which had been observed by Torrieellius ; but the 

 discovery of the vena contraeia, which was introduced in the second 

 edition (1713), explained the reason of the apparent diwri>|Kincy. 

 Newton also investigated the resistance of fluids to bodies moving in 

 them ; and it may be said that his theory forms the groundwork of all 

 our knowledge concerning that subject. 



Daniel Bernoulli (in 1738) was the first who applied the higher 

 branches of mathematical analysis in the investigation of general equa- 

 tions relating to the problems of hydrodynamics ; and though objections 

 were made to the principle which lie adopted, yet the independent 

 investigations of succeeding mathematicians have only confirmed the 

 remits at which the former had arrived. The subject was taken up in 

 1744 by D'Alembert, who, assuming that the motion of each hori- 

 zontal lamina of fluid in a vessel, during its descent in consequence of 

 the efflux from the orifice, is compounded of two motionx, namely. 

 that which it had at the mi "d that 



which is subsequently lost, arrived at equations containing, in the 

 implicit form, all the circumstances attending the efllu\ 

 And, subsequently, he investigated corresponding equations from the 

 assumption, first, that a rectangular canal supposed to exist in a fluid 

 mass which is in equilibrio is/itself in equilibrio ; and secondly, that a 

 molecule of fluid supposed 1 to lie incompressible retains the seme volume 

 under a different form in passing from one place to another. 



The researches of Kuler, La Orange, La Place, and other Continental 

 mathematicians have, since, contributed greatly to establish th< 

 cipies of the science on an analytical basts. The laws of the m 

 of fluids in canals and rivers were, with every possible precaut 

 ensure accuracy, determined experimentally by the Ablx! Bossut in 

 1771, and by the Chevalier Du Buat in 1786. 



In investigating the circumstances attending the discharge of fluids 

 through orifices it is usual, according to the theory of ParalMitm of 

 Strata, to suppose the fluid to be divided into an infinite number of 

 indefinitely thin lamimc perpendicular to the axis A B (fy. 1) of the 



Fig. 1. 



vessel in which it is contained, ami that in i i the fluid 



these lamina: preserve their parallelism till they come near the orifice, 

 when they assume the shape of a funnel, about which the fluid in 

 stagnant. In the process immediately following, let the vessel be 

 cylindrical or prismatical, in a vertical position, and have an 

 at B. Now if pp'l'ii be such a lamina, its A i i \ U-injt 



expressed by a;, and its area (perpendicularly to the axis* by a, w 

 have a.rf-r for the volume of the lamina ; also n Iwini; its Aen> 

 have aTtds for its mass. The force with which this lamina tends 

 downwards is its gravity, and the resistance exp o the descent 



is the excess of the pressure from below upwards over the do\M 

 pressure of that above the plate, If .</ represent the force of p 

 then D ag dx is equal to the action of gravity on the plate in its descent, 

 that is, the iceight of such a lamina. 



l.rt /< represent the pressure exerted downward" by the water above, 

 against any elementary portion of the lamina ; then, the pres- 

 the water at the upper and under surfaces of the lamina bein 

 pth of those surfaces below \,p + dp acting up 

 present that of the water below : hence the resistance of tin- 

 water below the lamina will be nit/i; and therefore the motive 

 force by which the lamina descends will be <;n n </.r a </y>. This 

 being divided by the mass of the latter will (by mechanics) give 

 gavte-atp a* Ax-dp for the Ke<)}entive foroe o{ dMcent 

 a D ax i ' i/. < 



But in variable motions the accelerntive force is expressed 



(r being the velocity and t the time>; or, since r= 57, the accelerativc 



,/-./ <P.r gndx dp 



force is expressed by -^j : therefore ^, = 



But the quantity of water Honing through tin- orifice at n in ;my 

 given tune being evidently equal to that which would pass through 

 the space pp' <fq, whose depth is its, in the same time ; if r be the 

 velocity of a particle in its descent through the depth ;/ "r </._-. ami // 

 that of a particle in the orifice, we shall have, in the element ill of 



time (' being the area of the orifice), n' n dt = adr, or dx- - 



a' i' 

 whence, considering a, a' and dt as constant, we get fx 



which, being substituted in the above equation for ^5, we obtain 



,, [i,/.,- .'/, a'dlirll 



. .r nil n il.ra an n a rf -j-. 



I) ll.r a <lt 



il '' a' u 

 But the equation '(fr = arf.r gives -^ = -^-; thereto) 



D a'-' 

 equation becomes gDdxdp= r , , whose integral in go x- j, 



D0"li 5 



= , + constant; whence 



a"- 



n x \ j * D + constant . 



h (- .\n) and n - a', '' 



]>=ifDh 4 " * * constant : 



