-1CS 



HYDRODYNAMICS. 



Definitions. 



Discharge 



ai Kluidi 



from 



v ^ >/ __ - _, 1 . A vertical orifice is an orifice placed in a vertical 

 Definitions, direction, so to allow the water to issue in a horizontal 

 stream. 



2. A horizontal orifice is an orifice placed in a hori- 

 zontal direction, so as to allow the fluid to escape in a 

 vertical direction. 



3. An ajutage is a name given to any orifice, or cy- 

 linder, or cone from which water issues. 



4. An additional tube is a tube of any form, insert- 

 ed in a simple orifice made in the sides or bottom of a 

 vessel. 



5. A head of mater is a term used to denote the height 

 of the fluid above the orifice, or in general the height 

 of a spring or source of water above the lowest point 

 where it can be employed to exert a mechanical force, 

 either by its impulse or by its weight. 



6. If water issues with a velocity V, equal to that 

 which a heavy body would acquire by falling through 

 a height H, the velocity is "Said to be the velocity due 

 to the height H, and the height is said to be due to the 

 velocity V. 



PROF-. L 



If a fluid moves in an open canal, or through a 

 tube, kept constantly full, whose diameter gradually 

 varies, and if the fluid has the same velocity in every 

 point of the same section, the velocities in different sec- 

 tions will be in the inverse ratio of the areas of the 

 sections. 



Since the canal and tube are always full, the same 

 quantity of fluid must pass through every section in 

 the same time. But as the quantity of fluid which 

 passes through any section, whose area is A, is propor- 

 tional to that area, and also to the velocity V with 

 which it flows, it must be proportional to A and V 

 jointly, or A X V. In like manner the quantity of 

 fluid which runs through the area a of any other sec- 

 tion in which v is the velocity, will be proportional to 

 a x v. Hence V : v =. a : A. 



SCHOLIUM. 



The case stated in the proposition is one which 

 is purely theoretical, and can never occur in practice. 

 In every canal the velocity of the surface is always 

 greatest, and in every tube the particles in its axis al- 

 ways move most rapidly. 



PROP. II. 



If a fluid is discharged from a vertical or horizontal 

 orifice infinitely small, in a vessel where the fluid is kept 

 constantly at the same height, the velocity with which 

 the fluid issues, is equal to that which a heavy body 

 would acquire by falling through a height equal to the 

 height of the fluid above the orifice. 



PLAT* Let ABDC, Fig. 1. be a vessel in which the surface of 



CCCXVHIJ the water always stands at AB, and let m n be the very 

 Kg. 1. small orifice through which the fluid is discharged. Let 

 us suppose the fluid divided by horizontal planes into an 

 infinite number of lamina, then since the area of the ori- 

 fice m n is infinitely small compared with the area of the 

 larniniB, it will follow, from Prop. I. that the velocity 



with which the laminae descend must be infinitely small. 

 Now it is obvious that the lowest film of fluid m n is 

 pressed out by the weight of the column mnpo. 

 (See Chap. I. Sect. I. Prop. IV.) Let M be the mass 

 of the column of fluid m n h g, which is discharged 

 at every instant by the pressure of mn p o, or by the 

 force m n X m o, and let m be the mass of a lesser co- 

 lumn of fluid m efn, which would have been dis- 

 charged in the same time, solely by its own gravity, 

 which may be represented by the line E m. Then if 

 V be the velocity of the column m g h n, and a the ve- 

 locity of the column m efn, the quantity of motion of 

 the column mghn will be V X M. an d the quantity of 

 motion of the column m efn will be u X . But the 

 moving forces are m n y. m o, and mn X E m ; and as 

 they must be proportional to the quantities of motion 

 which they produce, (see DYNAMICS, p. 286',) we have 

 mn X o : wi X E m =: V X M : H X m or 



Discharge 

 of Kfoid* 



from 

 Orifices. 



But the masses M, m discharged in the same time are as 

 the area of the orifice multiplied by the velocity ; that 

 is, M : m = m n x V : m n X or M : m =r V : u, and 

 as magnitudes have the same ratio as their equimulti- 

 ples have, (Euclid, V. 15,) we have 

 M V : M u = V 1 :*; but it has already been shewn that 

 M V : M u = mo: Em, hence 

 V 1 : U 1 = mo: Em. 



Now if v is the velocity which a heavy body would 

 acquire by falling through the height m o, we have, by 

 DYNAMICS, p. 292, Case 4, 



v* : u'=m o : m E, consequently 

 V : u *= * : u 1 , 



and V 1 1' and V = v, that is, the velocity V, with 

 which the fluid issues from the orifice m n, under the 

 pressure of the column mnpo, is equal to the velocity v, 

 which a heavy body would acquire by falling through 

 the height m n. 



It is obvious, that the preceding reasoning is appli- 

 cable to a vertical orifice, or to an orifice in any posi- 

 tion, provided its depth is equal to m n, for the pressure 

 of the fluid is the same in all directions. 



Cor. 1. If the vessel ABDC, instead of being kept 

 constantly full, is allowed to empty itself by the orifice 

 m , the velocity will always diminish ; and when the 

 surface has assumed a lower level GH, the velocity will 

 be that which is due to h m. 



Cor. 2. As the velocities of heavy bodies, descend- 

 ing by the force of gravity are as the square roots of the 

 spaces or heights through which they fall, (see DYNA- 

 MICS, p. 292, Case 4,) the velocity of the issuing fluid 

 will be as the square roots of the altitude of the surface 

 of the fluid above the orifice. That is, if the water 

 stands successively at the heights o m, h m, the veloci- 

 ties will be as */mo : V mh. 



Cor. 3. As the quantities of fluid discharged are pro- 

 portional to the velocities when the orifices remain 

 the same, they will also be proportional by Cor. 2. to 

 the square roots of the height of the fluid in the vessel. 



Cor. 4. If the orifice is horizontal, but opening up- 

 wards, so as to discharge the fluid in a vertical direc- 

 tion, the water will rise in a jet to the same height as 

 the surface of the fluid in the reservoir. As all heavy 

 bodies acquire in falling a velocity which would carry 

 them upward to the same height from which they fell, 

 the same must be true of fluids. In practice, however, 

 the resistance of the air, and the friction of the fluid 

 upon the sides of the orifice, prevent this from being 

 true. 



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