Chap. XVII.
THEORY OF THE MOTION OF FLUIDS.
1 HI
infinitum , but soon arrives at a state of uniformity, which is not afterwards increased ; newdegrees of velocity are not obtained in consequence of being retarded by the friction of itschannel.
Depth of theupper edge of
Cubic feet ofwater dischargedin a minute byevery inch of the
Cubic feet ofwater dischargedin a minute by
the waste-boardbelow the sur-
every inch of thewaste-board, ac-
waste-board, ac-cording to DuBuat's formula.
cording to ex-periment made inScotland .
I
-
-
0-403
-
-
0*428
2
.
-
1 140
.
1*211
3
-
-
2 095
-
-
2*226
4
.
-
3-225
.
_
3*427
5
-
-
4*507
4*789
6
.
-
5*925
.
6*295
7
-
-
7*466
.
-
7*933
8
-
-
9*122
-
.
9*692
9
.
-
10*884
.
.
11 *564
10
-
-
12*748
_
13*535
11
-
.
14*707
_
15*632
12
-
-
16*758
.
-
17-805
13
-
-
18*895
.
_
20076
14
-
-
21-117
-
-
22*437
15
-
-
23*419
-
-
24*883
16
-
-
25*800
.
.
27*413
17
-
-
28*258
.
.
30024
18
-
-
30-786
-
-
32*710
When the quantity of water discharged over a weir is
known, the depth of the e
the wasteboard, or H, may be found from the following formula, -
V 11-4172//
Michelotti’s experiments give 0*2703 \/H for the number of cubical inches dischargedin a second over a weir, when the height H is one inch ; and the real discharge to the theo-retical discharge is 9536 to 1000. These numbers suppose the length of the weir to beinfinite; his formula includes only the contractions produced by the upper edge of thewaste-board.
M. Eytelwein, in calculating the discharge of rectangular orifices reaching to the surfacerepresents the velocity, which varies as the square root of the height, by the ordinates andthe quantity of water discharged by the area of a parabola, two-thirds that of the circum-scribing rectangle. This may also be found by taking two-thirds of the velocity due to themean height, and allowing for the contraction of the vein. He purposes, for example, alake, in which there is a rectangular opening without any oblique lateral walls, 3 feet wide,and extending 2 feet below the surface of the water : here the co-efficient of the velocitycorrected for contraction is 5*1, and the corrected mean velocity \*/ 2 x5*l=4*8; therefore,the area being 6, the discharge of water in a second is 28-8 cubic feet: C being the co-efficient corrected for contraction, W the width of the aperture, and H its depth belowthe surface, we have the general formula,
Q = |/HxCxHx\V,for the quantity of water in cubic feet.
As the same co-efficient serves to determine the discharge over a weir of considerablebreadth, it is easy to deduce the depth or breadth requisite for the discharge of a givenquantity of water : for example, a lake has a weir 3 feet in breadth, and the surface of thewater stands at a height of 5 feet above it; it is required how much the weir must bewidened that the water may be a foot lower; we have the velocity = | 5 x 5T, and the
quantity of water 5*1 x 3 x 5, but the velocity must be reduced to \ 4 x 5*1, and
then the section will be
£ */5 x 5*1 x 3 x 5 a/5x3x5 „ „
- TT7—n—=- T~a — = 7 ‘ 5 *
|v4x5 , l V 4
7*5
and the height being 4 feet, the breadth must be 5 = 4-19 feet
On the Motion of Water in Rivers .—~The theorem by which the velocity of a river isdetermined, is one of the most valuable of M. Eytelwein’s calculations : the friction isnearly as the square of the velocity, not because a number of particles proportionate to thevelocity are torn asunder in a time proportionately short, (for according to the analogy ofsolid bodies no more force is destroyed by friction when the motion is rapid than when
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