1126

THEORY AND PRACTICE OF ENGINEERING.

Book II

ciiap. xvii.

THE THEORY OF THE MOTION OF FLUIDS.

The Theory of the Motion of Fluids necessarily follows the consideration of the lawswhich relate to their equilibrium: both Daniel Bernouilli and Huygens supposed that roforce is lost in the communication of motion between different bodies considered as belongingto any system; and that they always acquire such velocity in descending through anvspace, that the centre of gravity of the system is capable of ascending to a height equal tothat from which it descended, notwithstanding a mutual action between the bodies: anelastic ball, for instance, weighing 10 ounces, descending from the height of a foot uponanother of 1 ounce, in such a manner as to lose the whole of its motion, the smaller ballwill acquire a velocity capable of carrying it to the height of 10 feet. These laws withcertain restrictions are applicable to the motion of fluids.

We are indebted to Archimedes for the first theory upon floating bodies; he establishedthe principles that the parts of a liquid least pressed are impelled by those which are moreso, and that each part is always pressed by the weight of its corresponding column ; also,that a body impelled by a fluid is driven in a vertical direction through its centre ofgravity : such were the established laws with regard to fluids upwards of two thousandyears ago, since which time a third principle has been discovered, viz. equality of pressurein every direction, that is, if a pressure be applied to any point on the surface of a fliAd, itis equally transmitted to all other points in that liquid.

On the Measurement of Waters which flow through Tubes or Reservoirs -To establish the

principles which are the object of this proposition, wc must remark that the portions of waterenclosed in the vessel press mutually in every direction with equal forces in each horizontalbed ; and that if any escape by an opening made in the bottom, all other portions with whichit is surrounded hasten to flow on this side with a certain gradation of velocity, dependenton the force of those which follow, or on the weight with which they are loaded. Now asthe force which presses on the surface of the water to make it descend may be regarded asnothing, with respect to the pressure sustained by the film which serves as the base of acolumn of water answering to the orifice, we cannot say that it is this column which escapes,and is continually renewed by the surface, but that generally all contained in the vesselrushes to the orifice.

To make this clear, suppose a vessel A B C D filled with water to the* height I K, if weplunge a tube E F G H open at both ends to the bottom, its surface will be pressed inhorizontal directions with equal and oppositeforces, intersecting according to the order of theterms of an arithmetical progression. Fortracing on the sides E F, G H of the tube theright-angled and isosceles triangles FES andGHT, their elements will represent the actionof the water against the outer surface. As thepoint X will be pressed with a force expressed bythe element VX, and the point Y opposite to thepreceding with a force expressed by the heightF Y equal to V X ; and it will be the same withall the other points taken in the same horizontalbed L I\I; we find that the water of the vessel willmake as much effort to enter the tube, as that inthe tube will to escape.

As the extent of the base A D is indifferent tothe pressure of which we have been speaking, itwill be seen that if the vessel were itself a tube, OPQR, a little larger than that in whichit is plunged, the surface of the latter will always be pressed with the same force, howeversmall the difference between the two circles O R and EH, provided their circumferencesdo not join.

If the middle tube be suppressed, so as only to regard the column enclosed therein, thesurface will be pressed by the water with which it is surrounded with the same force asthat of the tube was : to know the ratio of this force relatively to the action of the" c ^©bt of the column on the bottom of the vessel, let r be the radius, N H, of the circle

c

: x i v,

Fig. 1708.