773 



HYDRODYNAMICS. 



HYDRODYNAMICS. 



774 



and subtracting this equation from the preceding, we have 



a' 2 

 o=gx ^-5 



2 



This expresses, according to the theory of fluid threads, the relation 

 between the velocity u and the difference between the weights of two 

 filaments of the fluid having unity for the base of each, and whose 



a'- 

 hejghts are A and x. When x=o, the equation becomes o= J u' 



gh + J " s ; or considering the orifice as infinitely small so that a' 

 and the whole first term of the second member vanishes, we have 

 o= yh + 4 u-; whence u = VZyli. 



Now gh expresses the weight of a prism of fluid having unity for 

 the area of its base and whose height is h, h being the vertical distance 

 of the surface of the fluid from the centre of gravity of the orifice, 

 which is called the " charge of water " on the orifice ; and this is the 

 pressure of the fluid against a small orifice at the bottom of the vessel : 

 but, while the height h is the same, the pressure is the same whatever 

 be the position or inclination of the orifice : therefore V2 gh will ex- 

 press the velocity at the same depth, whether the orifice be at the 

 bottom or side of the vessel. By the theory of dynamics this is equal 

 to the velocity acquired by a body in descending by gravity through a 

 height h, equal to that of the column of fluid, the orifice being infi- 

 nitely small. 



It may be concluded from the above theorem, that the velocity of a 

 fluid spouting upwards through an orifice in a vessel would cause it to 

 ascend to the level of the upper surface of that in the vessel, if the 

 resistance of the air and of friction were abstracted. Hence we see 

 that if Q = waste of water per second, m = a constant coefficient, d the 

 surface of the orifice; then <t = mud = md V'2,gh. Now m is found to 

 bo always = '62 in orifices with thin walls; hence q = i-98Vk~. In 

 orifices with c. cylindrical spout, 3 or 4 times the size of the orifice, 

 we shall have u = -82 */2 gh, and the mutc = Q = '82dy/2 #A = 6-58V2A. 



It follows, also, that the velocities of spouting fluids, at different 

 depths below the upper surface, are proportional to the square roots of 

 the depths ; that the quantities of fluids discharged in equal times at 

 different depths in the vessel, the latter briny constantly full, are to one 

 another in a ratio compounded of the areas of the orifices and the 

 square roota of the depths ; and the quantity of water which would 

 be discharged in a given time t, through an orifice a in a vessel kept 

 constantly full at the height h, is expressed by a' t V fy/i- 



The velocity or V 2?A expresses the length of a cylinder of water 

 which would flow through the orifice iu one second ; consequently the 

 time of discharging, from a cylindrical or prismatical vessel, the area 

 of whose base is a and whose height is h, a quantity of water equal 

 to that which the vessel will contain, the latter being however kept 

 full during all the time that the water is flowing, will be found by 

 making ah equal to a't V 2yA ; whence t (the time required) = 

 a It 

 f'Vff The value of g is 32-19 feet, or 386-28 inches; and in these 



values of a and t it is evident that the areas and height must be of the 

 same denomination aa g. 



When a vessel is suffered to discharge itself gradually, the velocity of 

 the effluent water diminishes continually. Now if x be the depth to 

 which the water has descended at the end of the time t, A being the 

 whole height when the vessel is full, h x will be the height of the 

 fluid at that time ; and we shall have V 2jr (A x) for the velocity in 

 the orifice. This may be supposed constant during the time d t, and 

 then the quantity of fluid discharged in that element of time would be 

 equal to a' d t V 20 (A x). In the time of this discharge the upper 

 surface of the fluid will descend through the depth d x ; therefore the 



area of the upper surface being a, we have a dx = a' dt 

 adx 



and dt = 



u'Sl'J 



. If the vessel in an upright cylinder or prism. 



a is constant, and the integral of the expression is 1= Vhx 



' 



+ constant. But, when xo, we have t = o; therefore o 



a'Vlg 



^ 

 VA + constant; whence t= ^=~ { V7t - -/A - x) ; which, when 



a 2A 



Incomes <= -, V ; and comparing this with the time in 



which an equal quantity would run off, the vessel being kept full, it will 

 be found to be double the latter. 



Next, if it were required to determine the quantity of water which 

 would flow through an orifice of finite magnitude when cut in the 

 vertical side of a vessel which is kept constantly full, it must be 

 observed that the velocity of the effluent fluid at different points in the 

 depth of the orifice varies as the square root of the distance of the point 

 from the upper surface. Now let A B ( = A) be the vertical height of 

 the water in a vessel in one side of which is formed the orifice whose 

 axis is c u, and imagine the horizontal ordiuates at m and n to be drawn 



indefinitely near each other. Let c B = A', c m = x, the ordiuate at m = y, 

 mn = dx : then ydx is equal to the elementary area of the orifice ; 

 and the water flowing through the area in the time t, being that which 



is due to the height A m, is expressed by t y dx ^/ 2 g (h 1 + x) ; which, 

 being integrated between x = o and x = A A', would give the 



Quantity of water discharged through the whole orifice in the time t. 

 f the orifice were rectangular, y would be constant : suppose it = 6 ; 



then the indefinite integral would be b t V 2</ f (A' + x) * dx, or 

 |6 1 +/%g (h' + x) *, which (between the said limits) becomes | bt *J2g 



(A* A' 5 ) : and if the orifice extended from the bottom to the top of 

 the vessel, having then xh, or A' = o, the expression would be 



| bt \/2</.A' 2 . If a rectangular orifice of the same form and magnitude 

 were situated at the bottom B, with its longer side ( = A) horizontal, 

 the breadth 5 being very small in this, and also in the preceding case, 

 the quantity discharged in the same time t, the velocity of the effluent 

 water being now equal in every part of the orifice, and being that 



3 



which is due to the whole height A, would be expressed by 4 t V %g A 5 . 

 The discharge found above is manifestly equal to two-thirds of this 

 quantity. 



In the second book of the ' Principia,' Newton shows that all the 

 particles of water issuing from an orifice in a vessel do not pass per- 

 pendicularly to the side or bottom in which it is formed, many of 

 them converging towards the orifice in every direction ; so that after 

 passing it they form a stream of diminished breadth, which he called 

 the rena coniracta. The section of the vena contracta may be taken as 

 equal to 5-8ths of the actual orifice, as has been shown by Mr. Renuie, 

 in his ' Report to th British Association,' for 1834. 



M. Savart has demonstrated the existence of certain eddies formed at 

 the orifice by the issuing jet, caused by some water above the orifice 

 trying to get out and coming into contact with the resisting walls, and 

 by some water below it being moved by the falling mass above it, thus 

 producing a set of forces acting by couples. Hence the issuing water, 

 although it always has its motion of translation perpendicular to the 

 resisting surface, has, besides, a rotatory motion caused by these 

 eddies. The irregularities of this rotatory motion then tend to cause 

 a disintegration of each successive section ; and hence, between the 

 thicker or normal drops there come out smaller ones, thus forming an 

 irregular stream of varying width. Savart has shown that each drop is 

 formed by an annular enlargement at the orifice, which is propagated 

 along the jet and causes this disintegration by a succession of such pulsa- 

 tions. The number of these is probably directly as the velocity of the 

 jet, and inversely as the size of the orifice. .It is remarkable that these 

 pulsations are continuous enough to cause a clear musical note ; and if 

 with any instrument we produce the same note near, the pulsations 

 become very regular, but cause no change in the amount or velocity of 

 emission. When the orifices are not circular, curious variations in the 

 geometrical figures, representing the sections at different distances from 

 the orifice, are produced. 



When, again, a rising column of water impinges against a horizontal 

 plate, we have a remarkable appearance, namely, a sort of disc of 

 water, of which the interior is a transparent sheet, and the outer rim 

 is a streaked space, along which lines of fluid stretch out and fall back 

 in a very fine spray. The pulsations here, also, are very regular, and 

 produce a musical sound. The relative size of the striated part to the 

 whole varies with the position of the intercepting solid. When a 

 certain distance is reached this part vanishes, and we have a wholly 

 transparent sheet. The forms also vary very beautifully, according as 

 the plate is perpendicular or oblique to the issuing jet. M. Savart has 

 also given some curious results on the subject of the clashing together 

 of two liquid veins, which we have not room here to describe, but 

 which will be found in the ' Annales de Chimie et de Physique,' 

 vol. liv. ; and a brief summary in Pouillet's ' Traite" de Physique.' 



We have been recently enlightened as to the phenomena of issuing 

 jets by the researches of MM. Savary and Magnus. Bidoue asserted 

 that the spiral form of the jet was illusory ; but Savary, when account- 

 ing for the curious dilations which he calls ventral segments, showed 

 that the efflux itself gives a vibratory motion to the liquid vein, thus 

 causing the protuberances in question. Prof. Magnus, iu the ' Phil. 

 Mag.' for February and March, 1856, regards every jet as composed of 

 an indefinite number of united jets. He then examines the effect pro- 

 duced by the collision of two equal jets, coming centrally iu opposite 

 directions. These, as we should suppose from Savart's experiments, 

 spread out at the confluence into a flat plate perpendicular to the axes 

 of the jets. When they meet obliquely, but centrally, they form a flat 

 plate, not circular, but oval. This also we should imagine d priori ; 

 for the force of each jet may be resolved into two others, one parallel 

 to the plane bisecting the angle between the jets, and the other per- 

 ] peudicular to it. The latter causes the elongation of the plate in a 

 ! plane perpendicular to the direction of the force, as in the former case. 

 I When the water thus spreads out laterally, its motion does not cease, 

 but the plate, by its cohesion, contracts in width, and collects into two 

 new jets converging to one another. These then throw out a new 

 plate perpendicular to the first, and the same process is repeated, 

 forming thus a succession of elliptic plates in perpendicular planes, like 

 the links of ,1 chain. 



