777 



HYDRODYNAMICS. 



HYDHODYNAMICS. 



778 



which the same quantity must flow in the game time ; since otherwise 

 the equilibrium of the river would not be permanent. It follows that 

 the products of the areas of the sections multiplied by the velocities 

 in each must be equal to each other, and that the velocities in different 

 sections must be inversely proportional to the areas of those sections. 



If the difference of level between any two points on the surface of a 

 river or canal, in a longitudinal section, be equal to one inch, and if I, 

 in inches, be the distance of those points on the surface, the slope of 



the river may be represented by -. Then, since the accelerative power 

 of gravity vertically, is to the accelerative power on any plane, as the 

 length of the plane is to its vertical height ; we shall have | for the 



accelerative power in a river whose slope is -. Again, if the resistances 



I 



to the motion of the fluid were, as is sometimes the case, nearly pro- 

 portional to the squares of the velocities, so that the resistance might 



T' 2 



be represented by (m being constant, and v representing the mean 



velocity) ; then (because when water in a river moves uniformly, the 

 resistance is, as in all like cases, equal to the accelerative force) we 



** i mg 



should have = i ; whence = V~r- But the resistances m canals 



ml l 



and rivers are not strictly proportional to the squares of the velocities ; 

 and it is found by experiment that, in one and the same bed, 



v { V' hyp. lg- vT+T6} may be considered aa constant, and may 



be represented by /itvj. Also, in beds whose transverse sections 

 differ both in area and figure, when the mean radius is represented 

 by r (where 



area A c B of the section 

 " 



V 



it is found by experiment that 



is constant and equal to 



of the velocity which the water should have at the bottom, of canals, 

 according to their nature, without wearing them away : 



Vr-0-1 



307 inches; hence -\A^=307 (Vr O'l) and m = 244 (Vr O'l) 3 . 

 Consequently we obtain 



r [VJ-hyp. log. vT+Tej =307 (Vr-0-1), 



307 (Vr-0-1) 



or r= -- rr:. But further investigation leads to 



Vf hyp. log. V + 1'6 



the conclusion that this expression for > must be diminished by 

 0'3 (Vr O'l) on account of the resistance by which the particles of 

 water oppose a separation from each other. (Du IJu.it, ' Traitc! 

 d'Hydrodynamique.') 



As an approximation we may make */ my = 307 / r, and 

 r=307 /-. And by experiment it has been found that if i'' = the 



velocity at the surface of a river, >" the velocity at the bottom, and 

 the mean velocity (all being expressed in inches per second), we shall 

 have 



t," = (-/i-'_l)' and t-=J 



The mean velocity in any one section may be practically found, 

 tolerably near the truth, by placing in it a rod of wood loaded at one 

 end with a weight sufficient to allow it to float upright in still water. 

 The greater velocity at the upper surface will make the rod incline 

 towards the direction of the stream ; and, consequently, when it has 

 acquired a state of equilibrium, it will float in an oblique position : the 

 top of the rod will move slower than the water at the upper surface of 

 the river, and the bottom will move faster than that in the lower part. 

 Hence the mean velocity of the water in that part of the breadth of 

 the river may be considered as equal to '8 of the observed velocity of 

 the rod. Often a Woltmann's drum is used, in which is a turning 

 shaft, communicating by a screw-channel with a meter, and carrying 

 four wings like a windmill. The experiment must be tried in different 

 parts of the breadth of the river ; and, in order to find the quantity of 

 water which flows through the section in a given time, the area of the 

 section must be obtained by measuring the breadth and sounding the 

 depths at intervals across the river. 



A knowledge of the velocity at the bottom of a river is of consider- 

 able use in enabling the hydraulic engineer to judge of the action of 

 the stream on its bed ; and it is evident that, to ensure permanency, 

 the accelerative force of the water should be in equilibrio with the 

 tenacity of the channel. The following table shows the superior limits 



Weakened earth . . 



Light soft clays . . 



Sand 



Gravel 



Pebbles 



Broken stones, flints 

 Agglomerated pebbles and schist 

 Rocks in layers . 

 Hard rocks .... 



2-5 inches per second. 



49 



9'8 



19-7 



19-9 



. 59-1 

 . 98 



Irregularities in the sides and beds of rivers, whether arising from 

 natural causes, or produced by artificial obstructions, are the causes of 

 currents setting obliquely across and of eddies being formed. These 

 not only diminish the velocity of the water by creating impediments to 

 its motion, but are sometimes seriously detrimental to the navigation, 

 and to the stability of the structures which are founded in the bed of 

 the river. When walls are made to project into the stream, the water 

 striking them is forced to rise above its general level, on account of the 

 obstruction ; and is afterwards reflected towards the middle of the 

 channel, with a velocity due to the rise thus produced. This current 

 carries with it, by a lateral communication of motion, some of the 

 water from the parts beyond the obstruction ; the surface of the river 

 being here, consequently, depressed, a portion of the water from the 

 oblique current falls by gravity into the lower part, and thus a sort of 

 whirlpool is formed at the place where the obstruction terminates. 

 This process goes on continually ; and the pressure upon the bed of 

 the river under the whirlpool being diminished in consequence of the 

 centrifugal force arising from the spiral motion, the water under the 

 bed forces its way upwards, removing the gravel and sand, and fre- 

 quently displacing the materials which form the foundation of the 

 work there constructed. 



When a body moves in a fluid at rest, its anterior surface being per- 

 pendicular to the direction of the motion; if an indefinitely thin 

 lamina of fluid be supposed at every successive instant of time to be 

 displaced, the resistance experienced by the moving surface may be 

 considered equal to the weight of a column of the fluid whose base is 

 the surface pressed, and whose height is that which is due to the 

 velocity : that is to say, the resistance may be supposed to be equal to 

 the pressure which would produce the same velocity at an orifice in 

 the base or side of a vessel. A difference of opinion has however 

 existed respecting the amount of the pressure sustained by the moving 

 surface. For a vein of water issuing from a vessel and striking a plane 

 surface at rest is shown by Newton (' Principia,' lib. ii., prop. 36), (and 

 the fact seems to be confirmed by the experiments of Krafft and 

 Bossut), to exert a pressure upon that surface equal to the weight of a 

 column of water whose height is twice that which is due to the velocity. 

 Du 1 !u.it , however, has proved that, even if such should be the case with 

 respect to the central part of the impinging column of fluid, the mean 

 pressure is less, on account of the lateral deviations of the exterior 

 filaments, and the amount first stated above is that which is generally 

 assumed. 



If the velocity be represented by r, the height due to that velocity 



is equal to ; then a representing the area of the moving surface, 

 and D the specific gravity of the fluid, we shall have -2-11 D for the 



pressure against, or the resistance experienced by that surface in 

 moving through the fluid. 



But when the anterior surface of the moving body is oblique to the 

 direction of the motion, the resistance above found must be diminished 

 on account of the inclination. Thus, let I be that inclination ; the 

 number of parallel filaments which act against a plane perpendicularly 

 is, to the number which can act upon it in an oblique position, as 

 radius ( = 1) is to sin. I. And by mechanics, the intensity of any force 

 acting obliquely on any plane is a decomposed part of the whole force, 

 and is to the latter in the ratio of sin. 2 1 to rad. z ( = 1). Therefore the 

 effective pressure against an oblique plane varies, as sin. 3 i; conse- 

 quently when the moving plane is oblique to the direction of its 



motion, the resistance which it experiences is to be expressed by av ~ 



20 

 D sm. 3 r. 



If a cylindrical body, terminated in front by an equilateral cone, 

 move through a fluid in the direction of its axis ; it can easily be 

 shown that the resistance experienced is one-fourth, and if the body be 

 terminated in front by a hemisphere, the resistance is one-half of that 

 which would be experienced by the same cylinder if it were termi- 

 nated in front by a plane perpendicular to its axis. 



When a prismatical body is placed in a stream of water the effort 

 necessary to keep it immovable in the fluid is equal to the difference 

 between the pressures in front and .behind. The pressure in front is 

 equal to the sum of the pressure produced by the moving water and 

 of the dead pressure, as it is called, which takes place when the body 

 is at rest in still water ; and the pressure on the rear face is merely 

 equal to this last. When a body of that kind is made to move in a 

 fluid at rest, its progress is retarded by the same difference of tho 



