Fluid velocity

“ The fluid which is falling from a higher to a lower level, carries with it the velocity due to its revolution with the earth at a greater distance from its centre. It will therefore accelerate, although to an almost infinitesimal extent, the earth's daily rotation. The sum of all these increments of velocity, arising from the descent of all the falling waters on the earth's surface, would in time become perceptible, did not nature, by the process of evaporation, convey the waters back to their sources ”

“ The validity of the above reasoning is unquestionable if we are dealing with a fluid whose properties are those of the Newtonian medium; the energy of turbulence being represented by a variation in the individual velocities of the particles such as will not affect their mean velocity. ”

“ In all actual streams the different fluid filaments have different velocities, those near the surface and centre moving faster than those near the bottom and sides. The ordinary formulae for the flow of streams rest on a hypothesis that this variation of velocity may be neglected, and that all the filaments may be treated as having a common velocity equal to the mean velocity of the stream. On this hypothesis, a plane layer abab (fig. 102) between sections normal to the direction of motion is treated as sliding down the channel to a′a′b′b′ without deformation. ”

“ The propagation of this velocity is only bounded by the limits of the fluid itself. If these limits consist of a solid plane parallel to the strata, then the propagation of the velocity will cease only at that point, i. e., between the last liquid stratum and the first solid stratum. Among the results found, the author lays particular emphasis on two: "In the first place, the steady motion arising in the interior of an unlimited stratum of water from an unvarying surface velocity makes itself felt with linearly decreasing velocity down to the bottom. ”

“ Besides, the more swiftly the globes move, the less are they pressed by the fluid at their hinder parts; and if the velocity be perpetually increased, they will at list leave an empty space behind them, unless the compression of the fluid be increased at the same time. For the compression of the fluid ought to be increased (by proposition 32 and 33.) in the duplicate ratio os the velocity, in order to preserve the resistance in the same duplicate ratio. ”

“ The more gross and less fluid pairs will tend to the circumference, unless they are heavy towards the centre. And tho', for the sake of demonstration, I proposed, at the beginning of this Section, an hypothesis that the resistance is proportional to the velocity, nevertheless, 'tis in truth probable that the resistance is in a less ratio than that of the velocity. Which granted, the periodic times of the parts of the vortex will be in a greater than the duplicate, ratio of the distances from its centre. ”

“ Then these filaments may have any law of variation of velocity assigned to them. If the motion is steady motion these fluid filaments (or as they are then termed stream lines) will have fixed positions in space.
Periodic Unsteady Motion.—In ordinary streams with rough boundaries, it is observed that at any given point the velocity varies from moment to moment in magnitude and direction, but that the average velocity for a sensible period (say for 5 or 10 minutes) varies very little either in magnitude or velocity. ”

“ The kinetic energy in the field of flow round a body in motion is imparted to the fluid when the body is started from rest, and is given up when the motion is arrested. The effect of the fluid motion is thus to add to the apparent inertia of the body, so that a given force requires to act through a greater distance to impart a given velocity than for the same body in vacuo. Not only has a force to act for a greater distance, but also for a longer time, which means that the body possesses in effect a greater store of momentum for a given velocity. ”

“ If the positions of these partitions are so adjusted that the volumes of flow in all the elementary streams are equal, they represent to the mind the velocity as well as the direction of motion of the particles in different parts of the current, for the velocities are inversely proportional to the cross sections of the elementary streams. No actual fluid is devoid of viscosity, and the effect of viscosity is to render the motion of a fluid sinuous, or rotational or eddying under most ordinary conditions. ”

“ A physical conception of velocity potential under these circumstances is somewhat difficult, but if we revert to the dynamical hypothesis and regard the velocity potential system as the pressure system by which the motion is generated, we encounter at once the same difficulty in another form. Before we are able to interfere either to start or to stop the fluid in cyclic motion, we must introduce some imaginary barrier in its path. ”

“ This is done by considering the angular momentum of any small spherical portion of the liquid relatively to an164 axis through the centre of the sphere, and proving that in order that it may vanish, for every axis, the component velocities of the fluid at the centre must be derivable from a velocity-potential. The angular momentum of a particle about an axis is the product of the component of the particle's momentum, at right angles to the plane through the particle and the axis, by the distance of the particle from the axis. ”

“ For the liquid in the interspace between α and η,
= 1/th 2 (η−α) th 2η;
(8) and the effective k2 of the liquid is reduced to
1/4 c2/th 2 (η−α) sh 2η,
(9) which becomes 1/4 c2/sh 2η = 1/8 (a2 − b2) /ab, when α = ∞, and the liquid surrounds the ellipse η to infinity.
An angular velocity R, which gives components −Ry, Rx of velocity to a body, can be resolved into two shearing velocities, −R parallel to Ox, and R parallel to Oy; and then ψ is resolved into ψ1 + ψ2, such that ψ1 + 1/2Rx2 and ψ2 + 1/2Ry 2 is constant over the boundary. ”

“ The slowness with which powders subside, depends partly on the specific gravity of the substance, and partly on the magnitude of the particles themselves. Bodies, in falling through a resisting medium, after a certain time acquire a uniform velocity, which is called their terminal velocity, with which they continue to descend: when the particles are very small, and the medium dense, as water, this terminal velocity is soon arrived at. ”

“ No one has yet examined what happens when a rough sphere enters a liquid with a very high velocity. That the motion set up must differ from that at a low velocity is apparent to any one who has thrown stones from a low bridge into deep water below. ”

“ When several gases are mixed, as oxygen and nitrogen are in our atmosphere, the molecules of each gas continue to move with their own characteristic velocities. So far as we can estimate the temperature at the boundary of the earth's atmosphere, we may assume that the average of the velocities of the oxygen molecules there found is about a quarter of a mile per second. The velocities for nitrogen are much the same, while the average speed of a molecule of hydrogen is about one mile per second, being, in fact, by far the greatest molecular velocity possessed by any gas. ”

“ It therefore follows that a body falling at the sun's surface receives in each second an increment of velocity to the extent of 864 feet per second. But the visible parts of the sun are composed of gaseous or vaporous materials. From the molecular spectrum of gases we have been taught to believe that the molecules of which the sun is composed are in incessant motion. Gravitation constantly tends to impart to each molecule an increase of velocity downwards. It is quite true that we cannot expect each molecule should actually acquire an additional velocity of 864 feet each second. ”

“ If the quantity of motion is so great that the attraction of the molecules can no longer be in equilibrio with their angular velocities, the body will remain in the gaseous state till it has transmitted to other bodies, the excess of velocity which it possesses. ”

“ The speed with which the surface waves move outwards in all directions, varies not only with the force of the wave, but also with the kind of material through which they pass. This velocity may be in the neighborhood of twenty miles per second, while in others the velocity is as great as 140 miles per second. ”

“ But the liquid after it leaves the vessel is subject to no forces except gravity, the pressure of the air, and its own surface-tension. Of these gravity has no effect on the form of the stream except in drawing asunder its parts in a vertical direction, because the lower parts are moving faster than the upper parts. The resistance of the air produces little disturbance until the velocity becomes very great. ”

“ In short, the high normal velocity of the particles of the medium necessarily renders it in all respects completely impalpable, or its presence impossible to detect by the senses. The high velocity of the particles is only naturally adapted to the minute size of the particles. ”

“ We know by experiment what the effect of increased velocity has upon any whirling body; it tends to enlarge the body at those parts where the velocity is the greatest, the consequence being that the bulging out of the atmosphere would be greatest at the equator. ”