Hydrodynamics

“ All that we have done has been merely to follow the rapid changes of form that take place in the bounding surface of the liquid. The interior particles of the liquid itself have remained invisible to us. But it is precisely the motion of these particles that the student of hydrodynamics desires to be able to trace. His study is so difficult that even in the apparently simple case of the gently-undulating surface of deep water, the reasoning necessary to discover the real path of any particle can at present only be followed by the highly-trained mathematician. ”

“ Motion of Fluids.—The only branch of mathematical mechanics which remains to be considered, is that which is, we may venture to say, hitherto incomparably the most incomplete of all,—Hydrodynamics. It may easily be imagined that the mere hypothesis of absolute relative mobility in the parts, combined with the laws of motion and nothing more, are conditions too vague and general to lead to definite conclusions. ”

“ It treats of motion in a resisting medium and of various problems connected with waves. At the end of it, it is shown that the Cartesian theory of vortices is inconsistent with the laws of motion, and necessarily leads to incorrect results. This book opened another world to the application of mathematics and, in effect, created the science of hydrodynamics. ”

“ If a wave is made and travels away over water, it carries with it energy, and hence it can only be created if we have a store of energy to draw upon. If we suppose that skin friction is absent, and that the ship floats upon a perfect fluid, it would nevertheless be true that, if the moving object creates waves, it will thereby reduce its own movement and require the application of force to it to keep it going. We may say therefore that if any floating object creates waves on a liquid over which it moves, these waves rob the floating body of some of its energy of motion. ”

“ But, further, in fluids such as water, air, steam, &c., to which the present division of the article relates, the tangential stresses that are called into action between contiguous portions during distortion or change of figure are always small compared with the weight, inertia, pressure, &c., which produce the visible motions it is the object of hydraulics to estimate. On the other hand, while a fluid passes easily from one form to another, it opposes considerable resistance to change of volume. ”

“ The fluid is here supposed to have no inertia, and its motion is opposed by the action of a force which we may conceive to be due to the resistance of a medium through which the fluid is supposed to flow. This resistance depends on the nature of the medium, and will in general depend on the direction in which the fluid moves, as well as on its velocity. For the present we may restrict ourselves to the case of a uniform medium, whose resistance is the same in all directions. ”

“ Then the whole velocity at any point will be inversely proportional to the radius of the point, and the fluid will describe stream lines having a constant inclination to the radius drawn to the axis of the current. That is, the stream lines will be logarithmic spirals. When water is delivered from the circumference of a centrifugal pump or turbine into a chamber, it forms a free vortex of this kind. The water flows spirally outwards, its velocity diminishing and its pressure increasing according to the law stated above, and the head along each spiral stream line is constant. ”

“ What he received was water with a certain pressure; in other words, a certain form of energy, which he transforms into rotation by the appropriate means; but by substituting other means he can make the same water pressure maintain a vibratory motion, as with the hydraulic ram valve, or let it waste itself by open flow, in which case it becomes ultimately molecular vibration that is heat. The analogy holds strictly. The trouble all comes from neglecting to distinguish between different forms of energy—energy in matter and energy in the ether. ”

“ Each particle of water, through its friction upon neighboring particles, is able to transmit its motion both along the surface and downward into the water below. The force which starts the water in motion and develops the wave, is the friction of wind blowing over the water surface, and the size of the orbit of the water particle at any point is proportional to the wind’s force and to the stretch of water over which it has blown. The wind’s effect is, therefore, cumulative—the wave is proportional to the wind’s effect upon all water particles in its rear, added to the local wind friction. ”

“ Relation of Pressure and Velocity in a Stream in Steady Motion when the Changes of Section of the Stream are Abrupt.—When a stream changes section abruptly, rotating eddies are formed which dissipate energy. The energy absorbed in producing rotation is at once abstracted from that effective in causing the flow, and sooner or later it is wasted by frictional resistances due to the rapid relative motion of the eddying parts of the fluid. In such cases the work thus expended internally in the fluid is too important to be neglected, and the energy thus lost is commonly termed energy lost in shock. ”

“ Finally, in confirmation of our explanation, we have the fact that with a liquid of small density and surface-tension, such as paraffin oil, a much smaller velocity of impact with a highly polished sphere suffices to give a "rough" splash than with water, a liquid of greater density and surface-tension, the reason being without doubt that the tangential velocity given by the impact is greater with the lighter liquid, as, indeed, is proved to be the case by the greater height to which the surrounding sheath is thrown up. ”

“ When dealing with real liquids not entirely free from viscosity, such resistance as does exist is due, as already mentioned, to skin friction and eddy formation. In the next place, leaving the consideration of the movement of wholly submerged bodies through liquids whether perfect or imperfect, we shall proceed to discuss the important question of the resistance offered by water to the motion through it of a floating object, such as a ship or swan. We have in this case to take into consideration the wave-making properties of the floating solid. ”

“ Let AB (fig. 25) be any one elementary stream, in a steadily moving fluid mass. Then, from the steadiness of the motion, AB is a fixed path in space through which a stream of fluid is constantly flowing. Let OO be the free surface and XX any horizontal datum line. Let ω be the area of a normal cross section, v the velocity, p the intensity of pressure, and z the elevation above XX, of the elementary stream AB at A, and ω1, p1, v1, z1 the same quantities at B. Suppose that in a short time t the mass of fluid initially occupying AB comes to A′B′. ”

“ The processes concerned would seem to be of a rather intricate character, and largely dependent upon fluid viscosity. It may be noted, indeed, that most of the still obscure phenomena of hydro-dynamics require for their elucidation a better comprehension of the laws of viscous motion. The subject is one which offers peculiar difficulties. In some problems in which I have lately been interested, a circulating motion presents itself of the kind which the mathematician excludes from the first when he is treating of fluids destitute altogether of viscosity. ”

“ The air in the tube is an elastic substance, and it resists compression. At constant temperature the volume into which the air is squeezed is inversely as the pressure applied.
The air, therefore, possesses elasticity of bulk, as it is called, and it resists being made to occupy a smaller volume. Again, the air possesses inertia, and when it is set in motion it continues to move like any other heavy body, after the moving force is withdrawn. We have, therefore, present in it the two essential qualities for the production of a wave-motion, as explained in the first lecture. ”

“ Perhaps I ought rather to say transformed, for of course it is now well known that consumption of energy in the sense of absolute loss is impossible. Thus, when energy is expended in moving a body in opposition to the force of [41] friction, or in agitating a liquid, the energy which disappears in its mechanical form reappears in the form of heat. The agitation of water by paddles moving through it warms the water, and the accession of heat thus acquired measures the energy which has been expended in making the paddles rotate. ”

“ Dubuat considered that if water were a perfect fluid, and the channels in which it flowed infinitely smooth, its motion would be continually accelerated, like that of bodies descending in an inclined plane. But as the motion of rivers is not continually accelerated, and soon arrives at a state of uniformity, it is evident that the viscosity of the water, and the friction of the channel in which it descends, must equal the accelerating force. ”

“ As it thickens, as the globe contracts, and as the gravitative force augments, the pressure will increase; and the evolution and radiation of heat will go on more rapidly. Eventually, however, when the liquid shell becomes very thick, and the internal cavity relatively small, the obstacle put to the escape of heat by this thick liquid shell, with its slowly-circulating currents, will turn the scale: the temperature of the outer surface will begin to diminish, and a solid crust will form while the internal cavity is yet unobliterated. ”

“ A medium of this kind, filled with molecular vortices having their axes parallel, differs from an ordinary fluid in having different pressures in different directions. If not prevented by properly arranged pressures, it would tend to expand laterally. In so doing, it would allow the diameter of each vortex to expand and its velocity to diminish in the same proportion. In order that a medium having these inequalities of pressure in different directions should be in equilibrium, certain conditions must be fulfilled, which we must investigate. ”

“ If the motion is rectilinear and uniform, the variation of pressure is the same as in a fluid at rest. In a stream flowing in an open channel, for instance, when the effect of eddies produced by the roughness of the sides is neglected, the pressure at each point is simply the hydrostatic pressure due to the depth below the free surface.
(b) If the velocity of the fluid is very small, the distribution of pressure is approximately the same as in a fluid at rest. ”

“ If, then, in any case of motion, we can draw the pressure surfaces, and the tubes of flow, we can determine the motion of the fluid completely. Now, the same mathematical expressions which appear in the hydro-dynamical theory occur also in the theory of electricity, the meaning only of the symbols is changed. For velocity of fluid we have to write electrical force. For difference of fluid pressure we substitute work done, or difference of electrical potential or pressure. ”

“ In this case it is not the gravity but the elasticity of the water that is the urging force. Every liquid particle pushed against its neighbor delivers up its motion with extreme rapidity, and the pulse is propagated as a thrill. The incompressibility of water, as illustrated by the famous Florentine experiment, is a measure of its elasticity, and to the possession of this property in so high a degree the rapid transmission of a sound-pulse through water is to be ascribed. ”

“ For without resistance the stream would have fallen upon the plane at the distance of 40 inches, the latus rectum of the parabolic stream being 80 inches.
Case 4. If the effluent water tend upwards, it will still issue forth with the same velocity. For the small stream of water springing upwards, ascends with a perpendicular motion to GH or GI the height of the stagnant water in the vessel; excepting in so far as its ascent is hindered a little by the resistance of the air; and therefore it springs out with the same velocity that it would acquire in falling from that height. ”

“ Fluid Pressure on a Body in Motion. — The pressure system about a body in motion in a fluid may be regarded as composed of two distinct component systems, i.e., the accelerative system, being that developed the instant a force is applied to a body at rest, which is essentially identical with the field of velocity potential, and the steady motion system, in which we have seen the fluid experiences a tension everywhere proportional to the energy density (compare "Dynamical Equations," § 59) . ”

“ The water was banked up by its own mass so that the surface fall was less than at the point of previous measurements. "In obedience to known hydraulic law, this decreased surface-inclination was indicated by decreased velocity of surface-current. But the power of the whole moving mass was greater in proportion to its depth. And thus, while to the superficial gaze the velocity was less, the mean velocity was greater; and the river swept with more resistless energy over its bed. ”

“ Again, the energy of a clear stream is entirely consumed in the friction of flow; and the friction bears a direct relation to its velocity. But if detritus be added to the water, then a portion of its energy is diverted to the transportation of the load; and this is done at the expense of the friction of flow, and hence at the expense of velocity. As the energy expended in transportation increases, the velocity diminishes. If the detritus be composed of uniform particles, then we may also say that as the load increases the velocity diminishes. ”