CHAPTER VIII. GENERAL PROPERTIES OF FLUIDS IN MOTION. (HYDRO- AND PNEUMO-DYNAMICS.) Pressure against the Sides of Vessels, 156. Theorem of Torricelli, 157. Velocity of Fluid Currents, 158. General Law of Currents, 159— lateral Reaction of, 160. Velocity of Fluid through Narrow Channels, 161. Fountains, 162. Friction between Fluids and Solids, 163— between Fluids only, 164. Properties of Gaseous Currents, 185.— Apparent Attraction of Disks by Currents of Air, 166-7.—Pumps, 168-171.-Syphons, 172-3.-Hiero's Fountain, 174.-Hydraulic Press, 175. 156. WHEN a fluid is poured into any vessel, the sides of the latter becomes acted upon by two opposed forces, one acting from without to within, and the other in the converse direction. The internal pressure arises from the weight of the column of fluid pressing against the sides (126), and the external force is the pressure of the medium in which the vessel is immersed. If an opening be made in the side of a vessel thus circumstanced, the fluid thus exposed is acted upon by the same forces as pressed against the portion of the side renewed; and accordingly, if the pressure from within to without is greater than the opposite pressure, the fluid will flow out of the opening, and obeying the force of gravitation, fall to the earth. But if the external pressure is the most powerful, the fluid will not escape: this may be illustrated B B by filling a glass-tumbler, a, with water, placing a piece of paper, BB, over its mouth, and carefully inverting it; on holding it in this direction, the fluid will not escape, for the upward pressure of the atmosphere against the paper will exceed the action of the attraction of gravitation on the water, and accordingly the glass will remain full. 157. Liquids, escaping from orifices in vessels containing them, obey the force of gravitation, and their motion becomes accelerated in a corresponding manner, providing all mechanical obstacles, arising from friction or other causes, be absent. The expression of this fact is known as the theorem of Torricelli, and may be thus stated: particles of fluid, on escaping from an orifice, possess the same degree of velocity as if they had fallen freely, in vacuo, from an height equal to the distance of the surface of the fluid above the centre of the orifice. Fluids obey this law without any relation to their density, their velocity solely depending upon the depth of the orifice from which they escape, below the level of the fluid. 158. The velocity of fluids thus escaping from orifices, is, cæteris paribus, as the square roots of the depths of the orifices below the surface of the fluid. Thus, calling the velocity of a fluid, escaping from an orifice one foot below the surface, unity; the velocity of a fluid, escaping from a similar orifice 4 feet below the level, will be 2; at 9 feet 3; at 16 feet 4, and so on. From this fact we learn that two vessels perfectly alike, being filled with fluid, and allowed to discharge a certain measure by similar orifices, one of them being kept quite full by the addition of fresh fluid, the quantity of water discharged in a given time from the latter vessel, as compared with the quantity escaping from that which was not kept full, will be as 2 to 1. 159. When fluids escape from lateral apertures, they describe parabolic curves, and obey the laws of projectiles (61); and when allowed to escape through a circular orifice pierced in the bottom of a containing vessel, providing the latter be composed of some thin material, the following phenomena are observed: (A.) The particles of fluid descend vertically, to within three inches of the bottom, and then turn towards the orifice. GENERAL LAWS OF CURRENTS. 113 (B.) The surface of the fluid gradually falls, remaining horizontal until within a certain distance of the bottom, when it forms a hollow cone, immediately above the centre of the orifice. (C.) The current of fluid having escaped from the vessel, contracts in diameter at a certain distance from the orifice. (D.) The greatest contraction of this fluid vein takes place at a distance from the orifice equal to half its diameter: the diameter of the contracted portion of the vein being to that portion nearest the orifice as 5:8. (E.) Beyond this contraction (D) the liquid vein continues to diminish in thickness, if moving from above to below, and to increase, if moving in the opposite direction. (F.) The surface of a fluid escaping by a lateral aperture does not form a hollow cone (B), but becomes depressed on the side in which the orifice exists. (G.) Every fluid vein, moving vertically downwards from a circular orifice, is composed of two well-defined portions. The portion nearest the orifice is perfectly transparent, like a rod of glass or crystal; its section is circular, and it gradually decreases in diameter, until it joins the second portion of the current, which is nearly opaque, and apparently much agitated, consisting of a multitude of drops, each produced by an annular dilatation of a portion of fluid at the orifice of the vessel, and undergoing during the time of its falling, a series of periodic vibrations, by which each drop alternately elongates and contracts. A series of pulsations thus occur at the orifice of the vessel, their number being in the direct ratio of the rapidity of the current, and in the inverse ratio of the diameter of the orifice; they are frequently sufficiently rapid to produce a distinct musical sound. (H.) In consequence of the contraction of the fluid-vein (C) (D), liquids escape with equal rapidity from a conical tube, as from a cylindrical one of equal length, providing the truncated apex of the latter corresponds in situation and section, to the point of greatest contraction of the fluid current. 160. In a vessel full of water, the downward pressure of any B D column, of fluid, as AB, pressing on the horizontal layer CD, acts with a certain degree of force on the sides of the vessel; if then an aperture be made at c, the pressure there becomes null, and fluid escapes, whilst the pressure remains active at D. As the pressure BC against the side is removed, and that against BD continues in action: the vessel, if carefully suspended, will move as if repelled in a direction opposed to that of the current escaping from c. A The movement, arising from this reaction against the sides of the vessel, is readily illustrated by means of the apparatus ABC, consisting of a large glass tube, A, closed at both ends with corks; two tubes, CB, bent twice at right angles, are fixed in the lower cork, their ends at EF being bent in opposite directions. Fill a with water, place the cork & in its place, and suspend the whole, by the thread H, from the ceiling. The apparatus will remain at rest, for no fluid can escape, as the pressure of the air against the open ends FE, is more intense than the gravitation of the fluid (156). Remove the cork G, then atmospheric pressure acts on the water in a, forces it through the tubes BC, and escaping at FE, produces a rapid rotation of the apparatus, in a direction contrary to that of the current of escaping fluid. C D Α E F 161. When fluids pass through a tube, or channel, whose section is greater at one part than another, as the same quantity must pass through every part in the same time, the velocity of the liquid is necessarily greater in the narrow, than in the wide parts: thus, if in the tube AB, water be allowed to run through in a constant stream, its velocity at CD will be much greater than at the wide parts EF. The momentum of the fluid will be equal in every part; for as this is equal to the quantity of matter multiplied by the quantity of motion (66), the quantity of fluid contained in CD, is less than in EF, yet its momentum is proportionately greater. For the same reason, when water flows through a funnel, its velocity is as much greater when passing through the tube, as the latter is narrower than the wider part of the instrument; and hence also the current of rivers is more rapid under the arches of bridges than at any other part. A B 162. Springs and fountains are formed by some concealed reservoir of water escaping through a cleft, or fissure, in the rocks containing the supply of fluid. On the water escaping, it possesses a velocity regulated according to the theorem of Torricelli, and therefore sufficient to project it upwards in the form of a jet d'eau. Artificial fountains are constructed on a similar principle; thus, if the tube AC be filled with water, it will escape from the aperture at c, in a jet rising to an elevation somewhat less than that of the column of water in a; according to the experiments of Marriotte, attaining an elevation of 5 feet, if the column of water in the reservoir be 5 feet 1 inch high. The elevation of the jet d'eau would be equal to the height of fluid in the reservoir, if all friction from angular projections, &c., as well as the resistance of the atmosphere, were removed. The greatest elevations, cæteris paribus, is obtained when the fluid escapes through an aperture pierced |