Let P1, P2, be the values of p at the ends of the times t1, t2: then Clog P2 or the interval between the density being P1, P2, is equal to If = 1 P2 0, t2- t1 = ∞, whence we conclude that the air would not all escape from the vessel in any finite time. Bossut Traité d'Hydrodynamique, tom. 1. p. 386. 3. To determine the velocity with which air will escape from a vessel through a small orifice into circumambient air of indefinite extent, the density of which is not so great as that of the air in the vessel. Let o represent the density of the circumambient air and F its elastic force; p the initial density and accordingly F the initial elastic force of the confined air; p' the density of the ρ σ confined air after a certain time, and Fits corresponding σ elastic force; M the small mass of air which issues initially in a certain small time with a velocity V; M' the small mass of air which issues in an equal portion of time some time afterwards, with a velocity V'. The initial expulsive force will therefore be equal to and the expulsive force after a time t will be equal to Now the expulsive forces vary as the quantities of motion which they generate in a given time: hence But M, M', are equal to pk V, p'k V', respectively, k denoting the area of the orifice; hence Bossut: Traité d'Hydrodynamique, tom. 1. p. 388. 4. Under the circumstances of the preceding problem, to find how long the air in the vessel will be decreasing in density from p to p'. C denoting the capacity of the vessel and Q the mass of the contained air at any time t, we have 5. A vessel being supposed to contain air more rare than that of the atmosphere, to find the velocity with which the surrounding air will flow into the vessel through a small orifice, and to determine the relation between the time and the density of the air in the vessel. Leto denote the density of the circumambient air, p the initial density of the air in the vessel, and p' its density at the end of any time t. Then, V denoting the initial velocity of the influx and V' its velocity at the time t, C the capacity of the vessel and k the area of the orifice, 6. Two closed vessels containing air of unequal condensation, to determine the velocity of the efflux of the air from one vessel into the other through a small orifice, and to find the equation between the time and the corresponding density of the air in either vessel. Let ρ denote the initial density of the air in the vessel which contains the denser air, and p' its density at the end of a time t: let σ, o', denote analogous quantities in regard to the other vessel. Then, A, B, denoting the capacities of the vessels of denser and rarer air respectively, k the area of the orifice, V the initial velocity of efflux and V' after a time t, we shall have CHAPTER II. OSCILLATION OF FLUIDS IN PIPES. THE general problem of the motion and pressure of fluids in pipes with either uniform or variable bores is discussed at length, with many interesting applications, by Euler, in a Memoir in the Novi Commentarii Academic Scientiarum Petropolitanæ, tom. XV., pro anno 1770. This memoir consists of five chapters: Caput 1. De principiis motus linearis fluidorum. Cap. II. De motu aquæ in tubis æqualiter ubique amplis. Cap. III. De motu aquæ in tubis inæqualiter amplis. Cap. IV. De elevatione aquæ antliarum ope. Cap. v. De motu aquæ per tubos diverso caloris gradu infectos. The reader may consult also a memoir by M. Coriolis, entitled, "Sur une manière simple de calculer la pression produite contre les parois d'un canal dans lequel se meut un fluide incompressible: in Liouville's Journal de Mathématiques, année 1837; p. 130. 1. An incompressible fluid oscillates in a siphon ABCD of uniform bore, (fig. 47) which consists of two vertical branches and one horizontal: to ascertain the period of an oscillation. Let x, y, be the altitudes of the two free surfaces of the column above the horizontal portion of the siphon. Let c represent the whole length of the column of fluid, and a the length of the horizontal branch of the siphon. Then, k denoting the area of a section of the siphon and p the density of the fluid, the mass of fluid in motion will be equal to kpc, and the force producing motion, being the difference of the weights of the two vertical portions of the fluid, will be equal to But, a denoting the length of the horizontal branch of the which shews that the period of an oscillation is equal to π ( and is therefore the same as that of an oscillation of a perfect pendulum the length of which is half that of the column of fluid. Newton compares the undulatory motion of the waves of an indefinite mass of fluid to the oscillation of the fluid in a siphon. Newton: Principia, lib. II. Sect. vii. Prop. 44, 45, 46. 2. An incompressible fluid oscillates under the action of gravity in a smooth continuous tube of variable bore to determine the motion of the fluid. We shall conceive the fluid to be divided into an infinite number of parallel sections, each of them pierced perpendicularly by the axis of the tube. We shall also suppose the motion of the particles in each slice to move at right angles to either face of the slice, in accordance with the hypothesis of parallel sections. Let L (fig. 48) be the area of any section of the fluid, L' and L" being the values of L at the extremities of the column of fluid. Let M be the area of a section of the fluid at any other point. Let u, v, be the velocities of the fluid at L, M. Let s, s', s", be the lengths of the axis of the tube reckoned from a fixed point A in it to the sections M, L, L", respectively, and let z be the depth of the point of the axis at M below the horizontal plane through A. |