Encyclopaedia Britannica, 11th Edition, "Hydromechanics" to "Ichnography"
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Title: Encyclopaedia Britannica, 11th Edition, "Hydromechanics" to "Ichnography"
Author: Various
Release date: July 29, 2012 [eBook #40370]
Most recently updated: October 23, 2024
Language: English
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*** START OF THE PROJECT GUTENBERG EBOOK ENCYCLOPAEDIA BRITANNICA, 11TH EDITION, "HYDROMECHANICS" TO "ICHNOGRAPHY" ***
Transcriber's notes:
(1) Numbers following letters (without space) like C2 were originally
printed in subscript. Letter subscripts are preceded by an
underscore, like C_n.
(2) Characters following a carat (^) were printed in superscript.
(3) Side-notes were relocated to function as titles of their respective
paragraphs.
(4) Macrons and breves above letters and dots below letters were not
inserted.
(5) [root] stands for the root symbol; [alpha], [beta], etc. for greek
letters.
(6) The following typographical errors have been corrected:
ARTICLE HYDROMECHANICS: "... and [omega] the angular velocity
about it generated by an impulse couple M, and M' is the couple
required to set the surrounding medium in motion ..." 'impulse'
amended from 'impluse'.
ARTICLE HYMENOPTERA: "... see P. Cameron's British Phytophagous
Hymenoptera (4 vols., London, Roy. Soc., 1882-1893)." 'Roy'
amended from 'Ray'.
ARTICLE HYRCANUS: "During its later years his reign was much
disturbed, however, by the contentions for ascendancy which arose
between the Pharisees and Sadducees, the two rival sects or
parties which then for the first time (under those names at least)
came into prominence." 'disturbed' amended from 'distrubed'.
ARTICLE ICELAND: "Iceland is emphatically a land of proverbs,
while of folk-tales, those other keys to the people's heart, there
is plentiful store." 'people's' amended from 'poeple's'.
ENCYCLOPAEDIA BRITANNICA
A DICTIONARY OF ARTS, SCIENCES, LITERATURE
AND GENERAL INFORMATION
ELEVENTH EDITION
VOLUME XIV, SLICE II
Hydromechanics to Ichnography
ARTICLES IN THIS SLICE:
HYDROMECHANICS HYTHE
HYDROMEDUSAE I
HYDROMETER IAMBIC
HYDROPATHY IAMBLICHUS (Greek philosopher)
HYDROPHOBIA IAMBLICHUS (Greek romance writer)
HYDROSPHERE IANNINA
HYDROSTATICS IAPETUS
HYDROXYLAMINE IAPYDES
HYDROZOA IATROCHEMISTRY
HYENA IAZYGES
HYÈRES IBADAN
HYGIEIA IBAGUÉ
HYGIENE IBARRA
HYGINUS (eighth pope) IBERIANS
HYGINUS (Latin writer) IBEX
HYGINUS, GAIUS JULIUS IBIS
HYGROMETER IBLIS
HYKSOS IBN 'ABD RABBIHI
HYLAS IBN 'ARABI
HYLOZOISM IBN ATHIR
HYMEN IBN BATUTA
HYMENOPTERA IBN DURAID
HYMETTUS IBN FARADI
HYMNS IBN FARID
HYPAETHROS IBN GABIROL
HYPALLAGE IBN HAUKAL
HYPATIA IBN HAZM
HYPERBATON IBN HISHAM
HYPERBOLA IBN ISHAQ
HYPERBOLE IBN JUBAIR
HYPERBOREANS IBN KHALDUN
HYPEREIDES IBN KHALLIKAN
HYPERION IBN QUTAIBA
HYPERSTHENE IBN SA'D
HYPERTROPHY IBN TIBBON
HYPNOTISM IBN TUFAIL
HYPOCAUST IBN USAIBI'A
HYPOCHONDRIASIS IBO
HYPOCRISY IBRAHIM AL-MAUSILI
HYPOSTASIS IBRAHIM PASHA
HYPOSTYLE IBSEN, HENRIK
HYPOSULPHITE OF SODA IBYCUS
HYPOTHEC ICA
HYPOTHESIS ICE
HYPOTRACHELIUM ICEBERG
HYPSOMETER ICELAND
HYRACOIDEA ICELAND MOSS
HYRCANIA ICE-PLANT
HYRCANUS ICE-YACHTING
HYSSOP I-CH'ANG
HYSTASPES ICHNEUMON
HYSTERESIS ICHNEUMON-FLY
HYSTERIA ICHNOGRAPHY
HYSTERON-PROTERON
HYDROMECHANICS ([Greek: hydromêchanika]), the science of the mechanics
of water and fluids in general, including _hydrostatics_ or the
mathematical theory of fluids in equilibrium, and _hydromechanics_, the
theory of fluids in motion. The practical application of hydromechanics
forms the province of hydraulics (q.v.).
_Historical._--The fundamental principles of hydrostatics were first
given by Archimedes in his work [Greek: Peri tôn ochoumenôn], or _De
iis quae vehuntur in humido_, about 250 B.C., and were afterwards
applied to experiments by Marino Ghetaldi (1566-1627) in his _Promotus
Archimedes_ (1603). Archimedes maintained that each particle of a
fluid mass, when in equilibrium, is equally pressed in every
direction; and he inquired into the conditions according to which a
solid body floating in a fluid should assume and preserve a position
of equilibrium.
In the Greek school at Alexandria, which flourished under the auspices
of the Ptolemies, the first attempts were made at the construction of
hydraulic machinery, and about 120 B.C. the fountain of compression,
the siphon, and the forcing-pump were invented by Ctesibius and Hero.
The siphon is a simple instrument; but the forcing-pump is a
complicated invention, which could scarcely have been expected in the
infancy of hydraulics. It was probably suggested to Ctesibius by the
_Egyptian Wheel_ or _Noria_, which was common at that time, and which
was a kind of chain pump, consisting of a number of earthen pots
carried round by a wheel. In some of these machines the pots have a
valve in the bottom which enables them to descend without much
resistance, and diminishes greatly the load upon the wheel; and, if we
suppose that this valve was introduced so early as the time of
Ctesibius, it is not difficult to perceive how such a machine might
have led to the invention of the forcing-pump.
Notwithstanding these inventions of the Alexandrian school, its
attention does not seem to have been directed to the motion of fluids;
and the first attempt to investigate this subject was made by Sextus
Julius Frontinus, inspector of the public fountains at Rome in the
reigns of Nerva and Trajan. In his work _De aquaeductibus urbis Romae
commentarius_, he considers the methods which were at that time
employed for ascertaining the quantity of water discharged from
ajutages, and the mode of distributing the waters of an aqueduct or a
fountain. He remarked that the flow of water from an orifice depends
not only on the magnitude of the orifice itself, but also on the
height of the water in the reservoir; and that a pipe employed to
carry off a portion of water from an aqueduct should, as circumstances
required, have a position more or less inclined to the original
direction of the current. But as he was unacquainted with the law of
the velocities of running water as depending upon the depth of the
orifice, the want of precision which appears in his results is not
surprising.
Benedetto Castelli (1577-1644), and Evangelista Torricelli
(1608-1647), two of the disciples of Galileo, applied the discoveries
of their master to the science of hydrodynamics. In 1628 Castelli
published a small work, _Della misura dell' acque correnti_, in which
he satisfactorily explained several phenomena in the motion of fluids
in rivers and canals; but he committed a great paralogism in supposing
the velocity of the water proportional to the depth of the orifice
below the surface of the vessel. Torricelli, observing that in a jet
where the water rushed through a small ajutage it rose to nearly the
same height with the reservoir from which it was supplied, imagined
that it ought to move with the same velocity as if it had fallen
through that height by the force of gravity, and hence he deduced the
proposition that the velocities of liquids are as the square root of
the head, apart from the resistance of the air and the friction of the
orifice. This theorem was published in 1643, at the end of his
treatise _De motu gravium projectorum_, and it was confirmed by the
experiments of Raffaello Magiotti on the quantities of water
discharged from different ajutages under different pressures (1648).
In the hands of Blaise Pascal (1623-1662) hydrostatics assumed the
dignity of a science, and in a treatise on the equilibrium of liquids
(_Sur l'équilibre des liqueurs_), found among his manuscripts after
his death and published in 1663, the laws of the equilibrium of
liquids were demonstrated in the most simple manner, and amply
confirmed by experiments.
The theorem of Torricelli was employed by many succeeding writers, but
particularly by Edmé Mariotte (1620-1684), whose _Traité du mouvement
des eaux_, published after his death in the year 1686, is founded on a
great variety of well-conducted experiments on the motion of fluids,
performed at Versailles and Chantilly. In the discussion of some
points he committed considerable mistakes. Others he treated very
superficially, and in none of his experiments apparently did he attend
to the diminution of efflux arising from the contraction of the liquid
vein, when the orifice is merely a perforation in a thin plate; but he
appears to have been the first who attempted to ascribe the
discrepancy between theory and experiment to the retardation of the
water's velocity through friction. His contemporary Domenico
Guglielmini (1655-1710), who was inspector of the rivers and canals at
Bologna, had ascribed this diminution of velocity in rivers to
transverse motions arising from inequalities in their bottom. But as
Mariotte observed similar obstructions even in glass pipes where no
transverse currents could exist, the cause assigned by Guglielmini
seemed destitute of foundation. The French philosopher, therefore,
regarded these obstructions as the effects of friction. He supposed
that the filaments of water which graze along the sides of the pipe
lose a portion of their velocity; that the contiguous filaments,
having on this account a greater velocity, rub upon the former, and
suffer a diminution of their celerity; and that the other filaments
are affected with similar retardations proportional to their distance
from the axis of the pipe. In this way the medium velocity of the
current may be diminished, and consequently the quantity of water
discharged in a given time must, from the effects of friction, be
considerably less than that which is computed from theory.
The effects of friction and viscosity in diminishing the velocity of
running water were noticed in the _Principia_ of Sir Isaac Newton, who
threw much light upon several branches of hydromechanics. At a time
when the Cartesian system of vortices universally prevailed, he found
it necessary to investigate that hypothesis, and in the course of his
investigations he showed that the velocity of any stratum of the
vortex is an arithmetical mean between the velocities of the strata
which enclose it; and from this it evidently follows that the velocity
of a filament of water moving in a pipe is an arithmetical mean
between the velocities of the filaments which surround it. Taking
advantage of these results, Henri Pitot (1695-1771) afterwards showed
that the retardations arising from friction are inversely as the
diameters of the pipes in which the fluid moves. The attention of
Newton was also directed to the discharge of water from orifices in
the bottom of vessels. He supposed a cylindrical vessel full of water
to be perforated in its bottom with a small hole by which the water
escaped, and the vessel to be supplied with water in such a manner
that it always remained full at the same height. He then supposed this
cylindrical column of water to be divided into two parts,--the first,
which he called the "cataract," being an hyperboloid generated by the
revolution of an hyperbola of the fifth degree around the axis of the
cylinder which should pass through the orifice, and the second the
remainder of the water in the cylindrical vessel. He considered the
horizontal strata of this hyperboloid as always in motion, while the
remainder of the water was in a state of rest, and imagined that there
was a kind of cataract in the middle of the fluid. When the results of
this theory were compared with the quantity of water actually
discharged, Newton concluded that the velocity with which the water
issued from the orifice was equal to that which a falling body would
receive by descending through half the height of water in the
reservoir. This conclusion, however, is absolutely irreconcilable with
the known fact that jets of water rise nearly to the same height as
their reservoirs, and Newton seems to have been aware of this
objection. Accordingly, in the second edition of his _Principia_,
which appeared in 1713, he reconsidered his theory. He had discovered
a contraction in the vein of fluid (_vena contracta_) which issued
from the orifice, and found that, at the distance of about a diameter
of the aperture, the section of the vein was contracted in the
subduplicate ratio of two to one. He regarded, therefore, the section
of the contracted vein as the true orifice from which the discharge of
water ought to be deduced, and the velocity of the effluent water as
due to the whole height of water in the reservoir; and by this means
his theory became more conformable to the results of experience,
though still open to serious objections. Newton was also the first to
investigate the difficult subject of the motion of waves (q.v.).
In 1738 Daniel Bernoulli (1700-1782) published his _Hydrodynamica seu
de viribus et motibus fluidorum commentarii_. His theory of the motion
of fluids, the germ of which was first published in his memoir
entitled _Theoria nova de motu aquarum per canales quocunque
fluentes_, communicated to the Academy of St Petersburg as early as
1726, was founded on two suppositions, which appeared to him
conformable to experience. He supposed that the surface of the fluid,
contained in a vessel which is emptying itself by an orifice, remains
always horizontal; and, if the fluid mass is conceived to be divided
into an infinite number of horizontal strata of the same bulk, that
these strata remain contiguous to each other, and that all their
points descend vertically, with velocities inversely proportional to
their breadth, or to the horizontal sections of the reservoir. In
order to determine the motion of each stratum, he employed the
principle of the _conservatio virium vivarum_, and obtained very
elegant solutions. But in the absence of a general demonstration of
that principle, his results did not command the confidence which they
would otherwise have deserved, and it became desirable to have a
theory more certain, and depending solely on the fundamental laws of
mechanics. Colin Maclaurin (1698-1746) and John Bernoulli (1667-1748),
who were of this opinion, resolved the problem by more direct methods,
the one in his _Fluxions_, published in 1742, and the other in his
_Hydraulica nunc primum detecta, et demonstrata directe ex fundamentis
pure mechanicis_, which forms the fourth volume of his works. The
method employed by Maclaurin has been thought not sufficiently
rigorous; and that of John Bernoulli is, in the opinion of Lagrange,
defective in clearness and precision. The theory of Daniel Bernoulli
was opposed also by Jean le Rond d'Alembert. When generalizing the
theory of pendulums of Jacob Bernoulli (1654-1705) he discovered a
principle of dynamics so simple and general that it reduced the laws
of the motions of bodies to that of their equilibrium. He applied this
principle to the motion of fluids, and gave a specimen of its
application at the end of his _Dynamics_ in 1743. It was more fully
developed in his _Traité des fluides_, published in 1744, in which he
gave simple and elegant solutions of problems relating to the
equilibrium and motion of fluids. He made use of the same suppositions
as Daniel Bernoulli, though his calculus was established in a very
different manner. He considered, at every instant, the actual motion
of a stratum as composed of a motion which it had in the preceding
instant and of a motion which it had lost; and the laws of equilibrium
between the motions lost furnished him with equations representing the
motion of the fluid. It remained a desideratum to express by equations
the motion of a particle of the fluid in any assigned direction. These
equations were found by d'Alembert from two principles--that a
rectangular canal, taken in a mass of fluid in equilibrium, is itself
in equilibrium, and that a portion of the fluid, in passing from one
place to another, preserves the same volume when the fluid is
incompressible, or dilates itself according to a given law when the
fluid is elastic. His ingenious method, published in 1752, in his
_Essai sur la résistance des fluides_, was brought to perfection in
his _Opuscules mathématiques_, and was adopted by Leonhard Euler.
The resolution of the questions concerning the motion of fluids was
effected by means of Euler's partial differential coefficients. This
calculus was first applied to the motion of water by d'Alembert, and
enabled both him and Euler to represent the theory of fluids in
formulae restricted by no particular hypothesis.
One of the most successful labourers in the science of hydrodynamics
at this period was Pierre Louis Georges Dubuat (1734-1809). Following
in the steps of the Abbé Charles Bossut (_Nouvelles Experiences sur la
résistance des fluides_, 1777), he published, in 1786, a revised
edition of his _Principes d'hydraulique_, which contains a
satisfactory theory of the motion of fluids, founded solely upon
experiments. 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. Dubuat, therefore,
assumed it as a proposition of fundamental importance that, when water
flows in any channel or bed, the accelerating force which obliges it
to move is equal to the sum of all the resistances which it meets
with, whether they arise from its own viscosity or from the friction
of its bed. This principle was employed by him in the first edition of
his work, which appeared in 1779. The theory contained in that edition
was founded on the experiments of others, but he soon saw that a
theory so new, and leading to results so different from the ordinary
theory, should be founded on new experiments more direct than the
former, and he was employed in the performance of these from 1780 to
1783. The experiments of Bossut were made only on pipes of a moderate
declivity, but Dubuat used declivities of every kind, and made his
experiments upon channels of various sizes.
The theory of running water was greatly advanced by the researches of
Gaspard Riche de Prony (1755-1839). From a collection of the best
experiments by previous workers he selected eighty-two (fifty-one on
the velocity of water in conduit pipes, and thirty-one on its velocity
in open canals); and, discussing these on physical and mechanical
principles, he succeeded in drawing up general formulae, which
afforded a simple expression for the velocity of running water.
J. A. Eytelwein (1764-1848) of Berlin, who published in 1801 a
valuable compendium of hydraulics entitled _Handbuch der Mechanik und
der Hydraulik_, investigated the subject of the discharge of water by
compound pipes, the motions of jets and their impulses against plane
and oblique surfaces; and he showed theoretically that a water-wheel
will have its maximum effect when its circumference moves with half
the velocity of the stream.
J. N. P. Hachette (1769-1834) in 1816-1817 published memoirs
containing the results of experiments on the spouting of fluids and
the discharge of vessels. His object was to measure the contracted
part of a fluid vein, to examine the phenomena attendant on additional
tubes, and to investigate the form of the fluid vein and the results
obtained when different forms of orifices are employed. Extensive
experiments on the discharge of water from orifices (_Expériences
hydrauliques_, Paris, 1832) were conducted under the direction of the
French government by J. V. Poncelet (1788-1867) and J. A. Lesbros
(1790-1860). P. P. Boileau (1811-1891) discussed their results and
added experiments of his own (_Traité de la mésure des eaux
courantes_, Paris, 1854). K. R. Bornemann re-examined all these
results with great care, and gave formulae expressing the variation of
the coefficients of discharge in different conditions (_Civil
Ingénieur_, 1880). Julius Weisbach (1806-1871) also made many
experimental investigations on the discharge of fluids. The
experiments of J. B. Francis (_Lowell Hydraulic Experiments_, Boston,
Mass., 1855) led him to propose variations in the accepted formulae
for the discharge over weirs, and a generation later a very complete
investigation of this subject was carried out by H. Bazin. An
elaborate inquiry on the flow of water in pipes and channels was
conducted by H. G. P. Darcy (1803-1858) and continued by H. Bazin, at
the expense of the French government (_Recherches hydrauliques_,
Paris, 1866). German engineers have also devoted special attention to
the measurement of the flow in rivers; the _Beiträge zur Hydrographie
des Königreiches Böhmen_ (Prague, 1872-1875) of A. R. Harlacher
(1842-1890) contained valuable measurements of this kind, together
with a comparison of the experimental results with the formulae of
flow that had been proposed up to the date of its publication, and
important data were yielded by the gaugings of the Mississippi made
for the United States government by A. A. Humphreys and H. L. Abbot,
by Robert Gordon's gaugings of the Irrawaddy, and by Allen J. C.
Cunningham's experiments on the Ganges canal. The friction of water,
investigated for slow speeds by Coulomb, was measured for higher
speeds by William Froude (1810-1879), whose work is of great value in
the theory of ship resistance (_Brit. Assoc. Report._, 1869), and
stream line motion was studied by Professor Osborne Reynolds and by
Professor H. S. Hele Shaw. (X.)
HYDROSTATICS
Hydrostatics is a science which grew originally out of a number of
isolated practical problems; but it satisfies the requirement of perfect
accuracy in its application to phenomena, the largest and smallest, of
the behaviour of a fluid. At the same time, it delights the pure
theorist by the simplicity of the logic with which the fundamental
theorems may be established, and by the elegance of its mathematical
operations, insomuch that hydrostatics may be considered as the
Euclidean pure geometry of mechanical science.
1. _The Different States of a Substance or Matter._--All substance in
nature falls into one of the two classes, solid and fluid; a solid
substance, the land, for instance, as contrasted with a fluid, like
water, being a substance which does not flow of itself.
A _fluid_, as the name implies, is a substance which flows, or is
capable of flowing; water and air are the two fluids distributed most
universally over the surface of the earth.
Fluids again are divided into two classes, termed a liquid and a gas, of
which water and air are the chief examples.
A _liquid_ is a fluid which is incompressible or practically so, i.e. it
does not change in volume sensibly with change of pressure.
A _gas_ is a compressible fluid, and the change in volume is
considerable with moderate variation of pressure.
Liquids, again, can be poured from one open vessel into another, and can
be kept in an uncovered vessel, but a gas tends to diffuse itself
indefinitely and must be preserved in a closed reservoir.
The distinguishing characteristics of the three kinds of substance or
states of matter, the solid, liquid and gas, are summarized thus in O.
Lodge's _Mechanics_:--
A solid has both size and shape.
A liquid has size but not shape.
A gas has neither size nor shape.
2. _The Change of State of Matter._--By a change of temperature and
pressure combined, a substance can in general be made to pass from one
state into another; thus by gradually increasing the temperature a solid
piece of ice can be melted into the liquid state of water, and the water
again can be boiled off into the gaseous state as steam. Again, by
raising the temperature, a metal in the solid state can be melted and
liquefied, and poured into a mould to assume any form desired, which is
retained when the metal cools and solidifies again; the gaseous state of
a metal is revealed by the spectroscope. Conversely, a combination of
increased pressure and lowering of temperature will, if carried far
enough, reduce a gas to a liquid, and afterwards to the solid state; and
nearly every gaseous substance has now undergone this operation.
A certain critical temperature is observed in a gas, above which the
liquefaction is impossible; so that the gaseous state has two
subdivisions into (i.) a true gas, which cannot be liquefied, because
its temperature is above the critical temperature, (ii.) a vapour, where
the temperature is below the critical, and which can ultimately be
liquefied by further lowering of temperature or increase of pressure.
3. _Plasticity and Viscosity._--Every solid substance is found to be
plastic more or less, as exemplified by punching, shearing and cutting;
but the plastic solid is distinguished from the viscous fluid in that a
plastic solid requires a certain magnitude of stress to be exceeded to
make it flow, whereas the viscous liquid will yield to the slightest
stress, but requires a certain length of time for the effect to be
appreciable.
According to Maxwell (_Theory of Heat_) "When a continuous alteration of
form is produced only by a stress exceeding a certain value, the
substance is called a solid, however soft and plastic it may be. But
when the smallest stress, if only continued long enough, will cause a
perceptible and increasing change of form, the substance must be
regarded as a viscous fluid, however hard it may be." Maxwell
illustrates the difference between a soft solid and a hard liquid by a
jelly and a block of pitch; also by the experiment of supporting a
candle and a stick of sealing-wax; after a considerable time the
sealing-wax will be found bent and so is a fluid, but the candle remains
straight as a solid.
4. _Definition of a Fluid._--A fluid is a substance which yields
continually to the slightest tangential stress in its interior; that is,
it can be divided very easily along any plane (given plenty of time if
the fluid is viscous). It follows that when the fluid has come to rest,
the tangential stress in any plane in its interior must vanish, and the
stress must be entirely normal to the plane. This mechanical axiom of
the _normality of fluid pressure_ is the foundation of the mathematical
theory of hydrostatics.
The theorems of hydrostatics are thus true for all stationary fluids,
however viscous they may be; it is only when we come to hydrodynamics,
the science of the motion of a fluid, that viscosity will make itself
felt and modify the theory; unless we begin by postulating the perfect
fluid, devoid of viscosity, so that the principle of the _normality of
fluid pressure_ is taken to hold when the fluid is in movement.
5. _The Measurement of Fluid Pressure._--The pressure at any point of
a plane in the interior of a fluid is the intensity of the normal
thrust estimated per unit area of the plane.
Thus, if a thrust of P lb. is distributed uniformly over a plane area
of A sq. ft., as on the horizontal bottom of the sea or any reservoir,
the pressure at any point of the plane is P/A lb. per sq. ft., or
P/144A lb. per sq. in. (lb./ft.² and lb./in.², in the Hospitalier
notation, to be employed in the sequel). If the distribution of the
thrust is not uniform, as, for instance, on a vertical or inclined
face or wall of a reservoir, then P/A represents the average pressure
over the area; and the actual pressure at any point is the average
pressure over a small area enclosing the point. Thus, if a thrust
[Delta]P lb. acts on a small plane area [Delta]A ft.² enclosing a
point B, the pressure p at B is the limit of [Delta]P/[Delta]A; and
p = lt([Delta]P/[Delta]A) = dP/dA, (1)
in the notation of the differential calculus.
6. _The Equality of Fluid Pressure in all Directions._--This
fundamental principle of hydrostatics follows at once from the
principle of the normality of fluid pressure implied in the definition
of a fluid in § 4. Take any two arbitrary directions in the plane of
the paper, and draw a small isosceles triangle abc, whose sides are
perpendicular to the two directions, and consider the equilibrium of a
small triangular prism of fluid, of which the triangle is the cross
section. Let P, Q denote the normal thrust across the sides bc, ca,
and R the normal thrust across the base ab. Then, since these three
forces maintain equilibrium, and R makes equal angles with P and Q,
therefore P and Q must be equal. But the faces bc, ca, over which P
and Q act, are also equal, so that the pressure on each face is equal.
A scalene triangle abc might also be employed, or a tetrahedron.
[Illustration: FIG. 1a.]
It follows that the pressure of a fluid requires to be calculated in
one direction only, chosen as the simplest direction for convenience.
7. _The Transmissibility of Fluid Pressure._--Any additional pressure
applied to the fluid will be transmitted equally to every point in the
case of a liquid; this principle of the _transmissibility of pressure_
was enunciated by Pascal, 1653, and applied by him to the invention of
the _hydraulic press_.
This machine consists essentially of two communicating cylinders (fig.
1a), filled with liquid and closed by pistons. If a thrust P lb. is
applied to one piston of area A ft.², it will be balanced by a thrust
W lb. applied to the other piston of area B ft.², where
p = P/A = W/B, (1)
the pressure p of the liquid being supposed uniform; and, by making
the ratio B/A sufficiently large, the mechanical advantage can be
increased to any desired amount, and in the simplest manner possible,
without the intervention of levers and machinery.
Fig. 1b shows also a modern form of the hydraulic press, applied to
the operation of covering an electric cable with a lead coating.
8. _Theorem._--In a fluid at rest under gravity the pressure is the
same at any two points in the same horizontal plane; in other words, a
surface of equal pressure is a horizontal plane.
This is proved by taking any two points A and B at the same level, and
considering the equilibrium of a thin prism of liquid AB, bounded by
planes at A and B perpendicular to AB. As gravity and the fluid
pressure on the sides of the prism act at right angles to AB, the
equilibrium requires the equality of thrust on the ends A and B; and
as the areas are equal, the pressure must be equal at A and B; and so
the pressure is the same at all points in the same horizontal plane.
If the fluid is a liquid, it can have a free surface without diffusing
itself, as a gas would; and this free surface, being a surface of zero
pressure, or more generally of uniform atmospheric pressure, will also
be a surface of equal pressure, and therefore a horizontal plane.
[Illustration: FIG. 1b.]
Hence the _theorem_.--The free surface of a liquid at rest under
gravity is a horizontal plane. This is the characteristic
distinguishing between a solid and a liquid; as, for instance, between
land and water. The land has hills and valleys, but the surface of
water at rest is a horizontal plane; and if disturbed the surface
moves in waves.
9. _Theorem._--In a homogeneous liquid at rest under gravity the
pressure increases uniformly with the depth.
This is proved by taking the two points A and B in the same vertical
line, and considering the equilibrium of the prism by resolving
vertically. In this case the thrust at the lower end B must exceed the
thrust at A, the upper end, by the weight of the prism of liquid; so
that, denoting the cross section of the prism by [alpha] ft.², the
pressure at A and By by p0 and p lb./ft.², and by w the density of the
liquid estimated in lb./ft.³,
p[alpha] - p0[alpha] = w[alpha]·AB, (1)
p = w·AB + p0. (2)
Thus in water, where w = 62.4lb./ft.³, the pressure increases 62.4
lb./ft.², or 62.4 ÷ 144 = 0.433 lb./in.² for every additional foot of
depth.
10. _Theorem._--If two liquids of different density are resting in
vessels in communication, the height of the free surface of such
liquid above the surface of separation is inversely as the density.
For if the liquid of density [sigma] rises to the height h and of
density [rho] to the height k, and p0 denotes the atmospheric
pressure, the pressure in the liquid at the level of the surface of
separation will be [sigma]h + p0 and [rho]k + p0, and these being
equal we have
[sigma]h = [rho]k. (1)
The principle is illustrated in the article BAROMETER, where a column
of mercury of density [sigma] and height h, rising in the tube to the
Torricellian vacuum, is balanced by a column of air of density [rho],
which may be supposed to rise as a homogeneous fluid to a height k,
called the height of the homogeneous atmosphere. Thus water being
about 800 times denser than air and mercury 13.6 times denser than
water,
k/h = [sigma]/[rho] = 800 × 13.6 = 10,880; (2)
and with an average barometer height of 30 in. this makes k 27,200
ft., about 8300 metres.
11. _The Head of Water or a Liquid._--The pressure [sigma]h at a depth
h ft. in liquid of density [sigma] is called the pressure due to a
_head_ of h ft. of the liquid. The atmospheric pressure is thus due to
an average head of 30 in. of mercury, or 30 × 13.6 ÷ 12 = 34 ft. of
water, or 27,200 ft. of air. The pressure of the air is a convenient
unit to employ in practical work, where it is called an "atmosphere";
it is made the equivalent of a pressure of one kg/cm²; and one
ton/inch², employed as the unit with high pressure as in artillery,
may be taken as 150 atmospheres.
12. _Theorem._--A body immersed in a fluid is buoyed up by a force
equal to the weight of the liquid displaced, acting vertically upward
through the centre of gravity of the displaced liquid.
For if the body is removed, and replaced by the fluid as at first,
this fluid is in equilibrium under its own weight and the thrust of
the surrounding fluid, which must be equal and opposite, and the
surrounding fluid acts in the same manner when the body replaces the
displaced fluid again; so that the resultant thrust of the fluid acts
vertically upward through the centre of gravity of the fluid
displaced, and is equal to the weight.
When the body is floating freely like a ship, the equilibrium of this
liquid thrust with the weight of the ship requires that the weight of
water displaced is equal to the weight of the ship and the two centres
of gravity are in the same vertical line. So also a balloon begins to
rise when the weight of air displaced is greater than the weight of
the balloon, and it is in equilibrium when the weights are equal. This
theorem is called generally the _principle of Archimedes_.
It is used to determine the density of a body experimentally; for if W
is the weight of a body weighed in a balance in air (strictly _in
vacuo_), and if W´ is the weight required to balance when the body is
suspended in water, then the upward thrust of the liquid or weight of
liquid displaced is W - W´, so that the _specific gravity_ (S.G.),
defined as the ratio of the weight of a body to the weight of an equal
volume of water, is W/(W - W´).
As stated first by Archimedes, the principle asserts the obvious fact
that a body displaces its own volume of water; and he utilized it in
the problem of the determination of the adulteration of the crown of
Hiero. He weighed out a lump of gold and of silver of the same weight
as the crown; and, immersing the three in succession in water, he
found they spilt over measures of water in the ratio 1/14 : 4/77 :
2/21 or 33 : 24 : 44; thence it follows that the gold : silver alloy
of the crown was as 11 : 9 by weight.
13. _Theorem._--The resultant vertical thrust on any portion of a
curved surface exposed to the pressure of a fluid at rest under
gravity is the weight of fluid cut out by vertical lines drawn round
the boundary of the curved surface.
_Theorem._--The resultant horizontal thrust in any direction is
obtained by drawing parallel horizontal lines round the boundary, and
intersecting a plane perpendicular to their direction in a plane
curve; and then investigating the thrust on this plane area, which
will be the same as on the curved surface.
The proof of these theorems proceeds as before, employing the
normality principle; they are required, for instance, in the
determination of the liquid thrust on any portion of the bottom of a
ship.
In casting a thin hollow object like a bell, it will be seen that the
resultant upward thrust on the mould may be many times greater than
the weight of metal; many a curious experiment has been devised to
illustrate this property and classed as a hydrostatic paradox (Boyle,
_Hydrostatical Paradoxes_, 1666).
[Illustration: FIG. 2.]
Consider, for instance, the operation of casting a hemispherical bell,
in fig. 2. As the molten metal is run in, the upward thrust on the
outside mould, when the level has reached PP´, is the weight of metal
in the volume generated by the revolution of APQ; and this, by a
theorem of Archimedes, has the same volume as the cone ORR´, or 1/3
[pi]y³, where y is the depth of metal, the horizontal sections being
equal so long as y is less than the radius of the outside hemisphere.
Afterwards, when the metal has risen above B, to the level KK´, the
additional thrust is the weight of the cylinder of diameter KK´ and
height BH. The upward thrust is the same, however thin the metal may
be in the interspace between the outer mould and the core inside; and
this was formerly considered paradoxical.
_Analytical Equations of Equilibrium of a Fluid at rest under any
System of Force._
14. Referred to three fixed coordinate axes, a fluid, in which the
pressure is p, the density [rho], and X, Y, Z the components of
impressed force per unit mass, requires for the equilibrium of the
part filling a fixed surface S, on resolving parallel to Ox,
_ _ _ _ _
/ / / / /
| | lp dS = | | | [rho]X dx dy dz, (1)
_/ _/ _/ _/ _/
where l, m, n denote the direction cosines of the normal drawn outward
of the surface S.
But by Green's transformation
_ _ _ _ _
/ / / / / dp
| | lp dS = | | | -- dx dy dz, (2)
_/ _/ _/ _/ _/ dx
thus leading to the differential relation at every point
dp dp dp
-- = [rho]X, -- = [rho]Y, -- = [rho]Z. (3)
dx dy dz
The three equations of equilibrium obtained by taking moments round
the axes are then found to be satisfied identically.
Hence the space variation of the pressure in any direction, or the
_pressure-gradient_, is the resolved force per unit volume in that
direction. The resultant force is therefore in the direction of the
steepest pressure-gradient, and this is normal to the surface of equal
pressure; for equilibrium to exist in a fluid the lines of force must
therefore be capable of being cut orthogonally by a system of
surfaces, which will be surfaces of equal pressure.
Ignoring temperature effect, and taking the density as a function of
the pressure, surfaces of equal pressure are also of equal density,
and the fluid is stratified by surfaces orthogonal to the lines of
force;
1 dp 1 dp 1 dp
----- --, ----- --, ----- --, or X, Y, Z (4)
[rho] dx [rho] dy [rho] dz
are the partial differential coefficients of some function P, =
[int]dp/[rho], of x, y, z; so that X, Y, Z must be the partial
differential coefficients of a potential -V, such that the force in
any direction is the downward gradient of V; and then
dP dV
-- + -- = 0, or P + V = constant, (5)
dx dx
in which P may be called the hydrostatic head and V the head of
potential.
With variation of temperature, the surfaces of equal pressure and
density need not coincide; but, taking the pressure, density and
temperature as connected by some relation, such as the gas-equation,
the surfaces of equal density and temperature must intersect in lines
lying on a surface of equal pressure.
15. As an example of the general equations, take the simplest case of
a uniform field of gravity, with Oz directed vertically downward;
employing the gravitation unit of force,
1 dp 1 dp 1 dp
----- -- = 0, ----- -- = 0, ----- -- = 1, (1)
[rho] dx [rho] dy [rho] dz
_
/
P = | dp/[rho] = z + a constant. (2)
_/
When the density [rho] is uniform, this becomes, as before in (2) § 9
p = [rho]z + p0. (3)
Suppose the density [rho] varies as some nth power of the depth below
O, then
dp/dz = [rho] = [mu]z^n (4)
z^(n+1) [rho]z [rho] /[rho]\^1/n
p = [mu]------- = ------ = ----- ( ----- ) , (5)
n + 1 n + 1 n + 1 \[mu] /
supposing p and [rho] to vanish together.
These equations can be made to represent the state of convective
equilibrium of the atmosphere, depending on the gas-equation
p = [rho]k = R[rho][theta], (6)
where [theta] denotes the absolute temperature; and then
d[theta] d / p \ 1
R-------- = -- ( ----- ) = -------, (7)
dz dz \[rho]/ (n + 1)
so that the temperature-gradient d[theta]/dz is constant, as in
convective equilibrium in (11).
From the gas-equation in general, in the atmosphere
1 dp 1 dp 1 d[theta] [rho] 1 d[theta] 1 1 d[theta]
----- -- = --- -- - ------- -------- = ----- - ------- -------- = --- - ------- --------, (8)
[rho] dz p dz [theta] dz p [theta] dz k [theta] dz
which is positive, and the density [rho] diminishes with the ascent,
provided the temperature-gradient d[theta]/dz does not exceed
[theta]/k.
With uniform temperature, taking k constant in the gas-equation,
dp/dz = [rho] = p/k, p = p0e^(z/k), (9)
so that in ascending in the atmosphere of thermal equilibrium the
pressure and density diminish at compound discount, and for pressures
p1 and p2 at heights z1 and z2
(z1 - z2)/k = log e (p2/p1) = 2.3 log10 (p2/p1). (10)
In the convective equilibrium of the atmosphere, the air is supposed
to change in density and pressure without exchange of heat by
conduction; and then
[rho]/[rho]0 = ([theta]/[theta]0)^n, p/p0 =
([theta]/[theta]0)^(n+1), (11)
dz 1 dp p 1
-------- = ----- -------- = (n + 1)------------R, [gamma] = 1 + ---,
d[theta] [rho] d[theta] [rho][theta] n
where [gamma] is the ratio of the specific heat at constant pressure
and constant volume.
In the more general case of the convective equilibrium of a spherical
atmosphere surrounding the earth, of radius a,
dp p0 d[theta] a²
----- = (n + 1)------ -------- = - --- dr, (12)
[rho] [rho]0 [theta]0 r²
gravity varying inversely as the square of the distance r from the
centre; so that, k = p0/[rho]0, denoting the height of the homogeneous
atmosphere at the surface, [theta] is given by
(n + 1) k (1 - [theta]/[theta]0) = a(1 - a/r), (13)
or if c denotes the distance where [theta] = 0,
[theta] a c - r
-------- = --- · -----. (14)
[theta]0 r c - a
When the compressibility of water is taken into account in a deep
ocean, an experimental law must be employed, such as
p - p0 = k([rho] - [rho]0), or [rho]/[rho]0 = 1
+ (p - p0)/[lambda], [lambda] = k[rho]0, (15)
so that [lambda] is the pressure due to a head k of the liquid at
density [rho]0 under atmospheric pressure p0; and it is the gauge
pressure required on this law to double the density. Then
dp/dz = kd[rho]/dz = [rho], [rho] = [rho]0e^(z/k),
p - p0 = k[rho]0(e^(z/k) - 1); (16)
and if the liquid was incompressible, the depth at pressure p would be
(p - p0)/p0, so that the lowering of the surface due to compression is
ke^(z/k) - k - z = ½z²/k, when k is large. (17)
For sea water, [lambda] is about 25,000 atmospheres, and k is then
25,000 times the height of the water barometer, about 250,000 metres,
so that in an ocean 10 kilometres deep the level is lowered about 200
metres by the compressibility of the water; and the density at the
bottom is increased 4%.
On another physical assumption of constant cubical elasticity
[lambda],
dp = [lambda]d[rho]/[rho], (p - p0)/[lambda] = log([rho]/[rho]0), (18)
dp [lambda] d[rho] / 1 1 \ [rho]0 z
-- = -------- ------ = [rho], [lambda]( ------ - ----- ) = z, 1 - ------ = ---, [lambda] = k[rho]0, (19)
zd [rho] dz \[rho]0 [rho]/ [rho] k
and the lowering of the surface is
p - p0 [rho] / z \ z²
------ - z = k log ------ - z = k log ( 1 - --- ) - z [approx] --- (20)
[rho]0 [rho]0 \ k / 2k
as before in (17).
16. _Centre of Pressure._--A plane area exposed to fluid pressure on
one side experiences a single resultant thrust, the integrated
pressure over the area, acting through a definite point called the
centre of pressure (C.P.) of the area.
Thus if the plane is normal to Oz, the resultant thrust
_ _
/ /
R = | |pdxdy, (1)
_/_/
and the coordinates [=x], [=y] of the C.P. are given by
_ _ _ _
/ / / /
[=x]R = | | xp dx dy, [=y]R = | | yp dx dy. (2)
_/_/ _/_/
The C·P. is thus the C·G. of a plane lamina bounded by the area, in
which the surface density is p.
If p is uniform, the C·P. and C·G. of the area coincide.
For a homogeneous liquid at rest under gravity, p is proportional to
the depth below the surface, i.e. to the perpendicular distance from
the line of intersection of the plane of the area with the free
surface of the liquid.
If the equation of this line, referred to new coordinate axes in the
plane area, is written
x cos [alpha] + y sin [alpha] - h = 0, (3)
_ _
/ /
R = | | [rho](h - x cos [alpha] - y sin [alpha]) dx dy, (4)
_/_/
_ _
/ /
[=x]R = | | [rho]x(h - x cos [alpha] - y sin [alpha]) dx dy, (5)
_/_/
_ _
/ /
[=y]R = | | [rho]y(h - x cos [alpha] - y sin [alpha]) dx dy.
_/_/
Placing the new origin at the C.G. of the area A,
_ _ _ _
/ / / /
| | xd x dy = 0, | | y dx dy = 0, (6)
_/_/ _/_/
R = [rho]hA, (7)
_ _ _ _
/ / / /
[=x]hA = -cos [alpha] | | x² dA - sin [alpha] | | xy dA, (8)
_/_/ _/_/
_ _ _ _
/ / / /
[=y]hA = -cos [alpha] | | xy dA - sin [alpha] | | y² dA. (9)
_/_/ _/_/
Turning the axes to make them coincide with the principal axes of the
area A, thus making [int][int] xy dA = 0,
[=x]h = -a² cos [alpha], [=y]h = -b² sin [alpha], (10)
where
_ _ _ _
/ / / /
| | x² dA = Aa², | | y² dA = Ab², (11)
_/_/ _/_/
a and b denoting the semi-axes of the momental ellipse of the area.
This shows that the C.P. is the antipole of the line of intersection
of its plane with the free surface with respect to the momental
ellipse at the C.G. of the area.
Thus the C.P. of a rectangle or parallelogram with a side in the
surface is at 2/3 of the depth of the lower side; of a triangle with a
vertex in the surface and base horizontal is ¾ of the depth of the
base; but if the base is in the surface, the C·P. is at half the depth
of the vertex; as on the faces of a tetrahedron, with one edge in the
surface.
The _core_ of an area is the name given to the limited area round its
C.G. within which the C·P. must lie when the area is immersed
completely; the boundary of the core is therefore the locus of the
antipodes with respect to the momental ellipse of water lines which
touch the boundary of the area. Thus the core of a circle or an
ellipse is a concentric circle or ellipse of one quarter the size.
The C.P. of water lines passing through a fixed point lies on a
straight line, the antipolar of the point; and thus the core of a
triangle is a similar triangle of one quarter the size, and the core
of a parallelogram is another parallelogram, the diagonals of which
are the middle third of the median lines.
In the design of a structure such as a tall reservoir dam it is
important that the line of thrust in the material should pass inside
the core of a section, so that the material should not be in a state
of tension anywhere and so liable to open and admit the water.
[Illustration: FIG. 3.]
17. _Equilibrium and Stability of a Ship or Floating Body. The
Metacentre._--The principle of Archimedes in § 12 leads immediately to
the conditions of equilibrium of a body supported freely in fluid, like
a fish in water or a balloon in the air, or like a ship (fig. 3)
floating partly immersed in water and the rest in air. The body is in
equilibrium under two forces:--(i.) its weight W acting vertically
downward through G, the C.G. of the body, and (ii.) the buoyancy of the
fluid, equal to the weight of the displaced fluid, and acting vertically
upward through B, the C.G. of the displaced fluid; for equilibrium these
two forces must be equal and opposite in the same line.
The conditions of equilibrium of a body, floating like a ship on the
surface of a liquid, are therefore:--
(i.) the weight of the body must be less than the weight of the total
volume of liquid it can displace; or else the body will sink to the
bottom of the liquid; the difference of the weights is called the
"reserve of buoyancy."
(ii.) the weight of liquid which the body displaces in the position of
equilibrium is equal to the weight W of the body; and
(iii.) the C.G., B, of the liquid displaced and G of the body, must lie
in the same vertical line GB.
18. In addition to satisfying these conditions of equilibrium, a ship
must fulfil the further condition of stability, so as to keep upright;
if displaced slightly from this position, the forces called into play
must be such as to restore the ship to the upright again. The stability
of a ship is investigated practically by inclining it; a weight is moved
across the deck and the angle is observed of the heel produced.
Suppose P tons is moved c ft. across the deck of a ship of W tons
displacement; the C.G. will move from G to G1 the reduced distance
G1G2 = c(P/W); and if B, called the centre of buoyancy, moves to B1,
along the curve of buoyancy BB1, the normal of this curve at B1 will
be the new vertical B1G1, meeting the old vertical in a point M, the
centre of curvature of BB1, called the _metacentre_.
If the ship heels through an angle [theta] or a slope of 1 in m,
GM = GG1cot[theta] = mc(P/W), (1)
and GM is called the metacentric height; and the ship must be
ballasted, so that G lies below M. If G was above M, the tangent drawn
from G to the evolute of B, and normal to the curve of buoyancy, would
give the vertical in a new position of equilibrium. Thus in H.M.S.
"Achilles" of 9000 tons displacement it was found that moving 20 tons
across the deck, a distance of 42 ft., caused the bob of a pendulum 20
ft. long to move through 10 in., so that
240 20
GM = --- × 42 × ---- = 2.24 ft.; (2)
10 9000
also
cot [theta] = 24, [theta] = 2°24´. (3)
In a diagram it is conducive to clearness to draw the ship in one
position, and to incline the water-line; and the page can be turned if
it is desired to bring the new water-line horizontal.
Suppose the ship turns about an axis through F in the water-line area,
perpendicular to the plane of the paper; denoting by y the distance of
an element dA if the water-line area from the axis of rotation, the
change of displacement is [sum]ydA tan[theta], so that there is no
change of displacement if [sum]ydA = 0, that is, if the axis passes
through the C.G. of the water-line area, which we denote by F and call
the centre of flotation.
The righting couple of the wedges of immersion and emersion will be
[Sigma]wy dA tan [theta]·y = w tan [theta] [Sigma] y² dA
= w tan [theta]·Ak² ft. tons, (4)
w denoting the density of water in tons/ft.³, and W = wV, for a
displacement of V ft.³
This couple, combined with the original buoyancy W through B, is
equivalent to the new buoyancy through B, so that
W.BB1 = wAk² tan [theta], (5)
BM = BB1 cot [theta] = Ak²/V, (6)
giving the radius of curvature BM of the curve of buoyancy B, in terms
of the displacement V, and Ak² the moment of inertia of the water-line
area about an axis through F, perpendicular to the plane of
displacement.
An inclining couple due to moving a weight about in a ship will heel
the ship about an axis perpendicular to the plane of the couple, only
when this axis is a principal axis at F of the momental ellipse of the
water-line area A. For if the ship turns through a small angle [theta]
about the line FF´, then b1, b2, the C·G. of the wedge of immersion
and emersion, will be the C·P. with respect to FF´ of the two parts of
the water-line area, so that b1b2 will be conjugate to FF´ with
respect to the momental ellipse at F.
The naval architect distinguishes between the _stability of form_,
represented by the righting couple W.BM, and the _stability of
ballasting_, represented by W.BG. Ballasted with G at B, the righting
couple when the ship is heeled through [theta] is given by W.BM.
tan[theta]; but if weights inside the ship are raised to bring G above
B, the righting couple is diminished by W.BG.tan[theta], so that the
resultant righting couple is W·GM·tan[theta]. Provided the ship is
designed to float upright at the smallest draft with no load on board,
the stability at any other draft of water can be arranged by the
stowage of the weight, high or low.
19. Proceeding as in § 16 for the determination of the C.P. of an
area, the same argument will show that an inclining couple due to the
movement of a weight P through a distance c will cause the ship to
heel through an angle [theta] about an axis FF´ through F, which is
conjugate to the direction of the movement of P with respect to an
ellipse, not the momental ellipse of the water-line area A, but a
confocal to it, of squared semi-axes
a² - hV/A, b² - hV/A, (1)
h denoting the vertical height BG between C.G. and centre of buoyancy.
The varying direction of the inclining couple Pc may be realized by
swinging the weight P from a crane on the ship, in a circle of radius
c. But if the weight P was lowered on the ship from a crane on shore,
the vessel would sink bodily a distance P/wA if P was deposited over
F; but deposited anywhere else, say over Q on the water-line area, the
ship would turn about a line the antipolar of Q with respect to the
confocal ellipse, parallel to FF´, at a distance FK from F
FK = (k² - hV/A)/FQ sin QFF´ (2)
through an angle [theta] or a slope of one in m, given by
1 P P V
sin [theta] = --- = ----- = --- · -------- FQ sin QFF´, (3)
m wA·FK W Ak² - hV
where k denotes the radius of gyration about FF´ of the water-line
area. Burning the coal on a voyage has the reverse effect on a
steamer.
HYDRODYNAMICS
20. In considering the motion of a fluid we shall suppose it
non-viscous, so that whatever the state of motion the stress across any
section is normal, and the principle of the normality and thence of the
equality of fluid pressure can be employed, as in hydrostatics. The
practical problems of fluid motion, which are amenable to mathematical
analysis when viscosity is taken into account, are excluded from
treatment here, as constituting a separate branch called "hydraulics"
(q.v.). Two methods are employed in hydrodynamics, called the Eulerian
and Lagrangian, although both are due originally to Leonhard Euler. In
the Eulerian method the attention is fixed on a particular point of
space, and the change is observed there of pressure, density and
velocity, which takes place during the motion; but in the Lagrangian
method we follow up a particle of fluid and observe how it changes. The
first may be called the statistical method, and the second the
historical, according to J. C. Maxwell. The Lagrangian method being
employed rarely, we shall confine ourselves to the Eulerian treatment.
_The Eulerian Form of the Equations of Motion._
21. The first equation to be established is the _equation of
continuity_, which expresses the fact that the increase of matter within
a fixed surface is due to the flow of fluid across the surface into its
interior.
In a straight uniform current of fluid of density [rho], flowing with
velocity q, the flow in units of mass per second across a plane area
A, placed in the current with the normal of the plane making an angle
[theta] with the velocity, is [rho]Aq cos [theta], the product of the
density [rho], the area A, and q cos [theta] the component velocity
normal to the plane.
Generally if S denotes any closed surface, fixed in the fluid, M the
mass of the fluid inside it at any time t, and [theta] the angle which
the outward-drawn normal makes with the velocity q at that point,
dM/dt = rate of increase of fluid inside the surface, (1)
= flux across the surface into the interior
_ _
/ /
= - | | [rho]q cos [theta] dS,
_/_/
the integral equation of continuity.
In the Eulerian notation u, v, w denote the components of the velocity
q parallel to the coordinate axes at any point (x, y, z) at the time
t; u, v, w are functions of x, y, z, t, the independent variables; and
d is used here to denote partial differentiation with respect to any
one of these four independent variables, all capable of varying one at
a time.
To transfer the integral equation into the differential equation of
continuity, Green's transformation is required again, namely,
_ _ _ _ _
/ / / /d[xi] d[eta] d[zeta] \ / /
| | | ( ----- + ------ + ------- )dx dy dz = | | (l[xi] + m[eta] + n[zeta]) dS, (2)
_/_/_/ \ dx dy dz / _/_/
or individually
_ _ _ _ _
/ / / d[xi] / /
| | | ----- dx dy dz = | | l[xi] dS,..., (3)
_/_/_/ dx _/_/
where the integrations extend throughout the volume and over the
surface of a closed space S; l, m, n denoting the direction cosines of
the outward-drawn normal at the surface element dS, and [xi], [eta],
[zeta] any continuous functions of x, y, z.
The integral equation of continuity (1) may now be written
_ _ _ _ _
/ / / d[rho] / /
| | | ----- dx dy dz = | | (l[rho]u + m[rho]v + n[rho]w) dS = 0, (4)
_/_/_/ dt _/_/
which becomes by Green's transformation
_ _ _
/ / / /d[rho] d([rho]u) d([rho]v) d([rho]w)\
| | | ( ------ + --------- + --------- + -------- ) dx dy dz = 0, (5)
_/_/_/ \ dt dx dy dz /
leading to the differential equation of continuity when the
integration is removed.
22. The equations of motion can be established in a similar way by
considering the rate of increase of momentum in a fixed direction of the
fluid inside the surface, and equating it to the momentum generated by
the force acting throughout the space S, and by the pressure acting over
the surface S.
Taking the fixed direction parallel to the axis of x, the time-rate of
increase of momentum, due to the fluid which crosses the surface, is
_ _ _ _
/ / / /
- | | [rho]uq cos [theta] dS = - | | (l[rho]u² + m[rho]uv + n[rho]uw) dS, (1)
_/_/ _/_/
which by Green's transformation is
_ _ _
/ / / /d([rho]u²) d([rho]uv) d([rho]uw)\
- | | | (---------- + ---------- + ---------- ) dx dy dz. (2)
_/_/_/ \ dx dy dz /
The rate of generation of momentum in the interior of S by the
component of force, X per unit mass, is
_ _ _
/ / /
| | | [rho]X dx dy dz, (3)
_/_/_/
and by the pressure at the surface S is
_ _ _ _ _
/ / / / / dp
- | | lp dS = - | | | -- dx dy dz, (4)
_/_/ _/_/_/ dx
by Green's transformation.
The time rate of increase of momentum of the fluid inside S is
_ _ _
/ / / d([rho]u)
| | | --------- dx dy dz; (5)
_/_/_/ dt
and (5) is the sum of (1), (2), (3), (4), so that
_ _ _
/ / / /d[rho]u d[rho]u² d[rho]uv d[rho]uw dp \
| | | ( ------- + -------- + -------- + -------- - [rho]X + -- ) dx dy dz = 0, (6)
_/_/_/ \ dt dx dy dz dx /
leading to the differential equation of motion
d[rho]u d[rho]u² d[rho]uv, d[rho]uw dp
------- + -------- + -------- + -------- = [rho]X - --, (7)
dt dx dy dz dx
with two similar equations.
The absolute unit of force is employed here, and not the gravitation
unit of hydrostatics; in a numerical application it is assumed that
C.G.S. units are intended.
These equations may be simplified slightly, using the equation of
continuity (5) § 21; for
d[rho]u d[rho]u² d[rho]uv d[rho]uw
------- + -------- + -------- + --------
dt dx dy dz
/du du du du \
= [rho] ( -- + u-- + v-- + w-- )
\dt dx dy dz /
/d[rho] d[rho]u d[rho]v d[rho]w \
+ u ( ------ + ------- + ------- + ------- ), (8)
\ dt dx dy dz /
reducing to the first line, the second line vanishing in consequence
of the equation of continuity; and so the equation of motion may be
written in the more usual form
du du du du 1 dp
-- + u-- + v-- + w-- = X - ----- --, (9)
dt dx dy dz [rho] dx
with the two others
dv dv dv dv 1 dp
-- + u-- + v-- + w-- = Y - ----- --, (10)
dt dx dy dz [rho] dy
dw dw dw dw 1 dp
-- + u-- + v-- + w-- = Z - ----- --. (11)
dt dx dy dz [rho] dz
23. As a rule these equations are established immediately by determining
the component acceleration of the fluid particle which is passing
through (x, y, z) at the instant t of time considered, and saying that
the reversed acceleration or kinetic reaction, combined with the
impressed force per unit of mass and pressure-gradient, will according
to d'Alembert's principle form a system in equilibrium.
To determine the component acceleration of a particle, suppose F to
denote any function of x, y, z, t, and investigate the time rate of F
for a moving particle; denoting the change by DF/dt,
DF F(x + u[delta]t, y + v[delta]t, z + w[delta]t, t + [delta]t) - F(x, y, z, t)
-- = lt·----------------------------------------------------------------------------
dt [delta]t
dF dF dF dF
= -- + u-- + v-- + w--; (1)
dt dx dy dz
and D/dt is called particle differentiation, because it follows the
rate of change of a particle as it leaves the point x, y, z; but
dF/dt, dF/dx, dF/dy, dF/dz (2)
represent the rate of change of F at the time t, at the point, x, y,
z, fixed in space.
The components of acceleration of a particle of fluid are consequently
Du du du du du
-- = -- + u-- + v-- + w--, (3)
dt dt dx dy dz
Dv dv dv dv dv
-- = -- + u-- + v-- + w--, (4)
dt dt dx dy dz
Dw dw dw dw dw
-- = -- + u-- + v-- + w--, (5)
dt dt dx dy dz
leading to the equations of motion above.
If F (x, y, z, t) = 0 represents the equation of a surface containing
always the same particles of fluid,
DF dF dF dF dF
-- = 0, or -- + u-- + v-- + w-- = 0, (6)
dt dt dx dy dz
which is called the differential equation of the _bounding surface_. A
bounding surface is such that there is no flow of fluid across it, as
expressed by equation (6). The surface always contains the same fluid
inside it, and condition (6) is satisfied over the complete surface,
as well as any part of it.
But turbulence in the motion will vitiate the principle that a
bounding surface will always consist of the same fluid particles, as
we see on the surface of turbulent water.
24. To integrate the equations of motion, suppose the impressed force
is due to a potential V, such that the force in any direction is the
rate of diminution of V, or its downward gradient; and then
X = -dV/dx, Y = -dV/dy, Z = -dV/dz; (1)
and putting
dw dv du dw dv du
-- - -- = 2[xi], -- - -- = 2[eta], -- - -- = 2[zeta], (2)
dy dz dz dx dx dy
d[xi] d[eta] d[zeta]
----- + ------ + ------- = 0, (3)
dx dy dz
the equations of motion may be written
du dH
-- - 2v[zeta] + 2w[eta] + -- = 0, (4)
dt dx
dv dH
-- - 2w[xi] + 2u[zeta] + -- = 0, (5)
dt dy
dw dH
-- - 2u[eta] + 2w[xi] + -- = 0, (6)
dt dz
where
_
/
H = | dp/[rho] + V + ½q², (7)
_/
q² = u² + v² + w², (8)
and the three terms in H may be called the pressure head, potential
head, and head of velocity, when the gravitation unit is employed and
½q² is replaced by ½q²/g.
Eliminating H between (5) and (6)
D[xi] du dv dw / du dv dw \
----- - [xi]-- - [eta]-- - [zeta]-- + [xi]( -- + -- + -- ) = 0, (9)
dt dx dx dx \ dx dy dz /
and combining this with the equation of continuity
1 D[rho] du dv dw
----- ------ + -- + -- + -- = 0, (10)
[rho] dt dx dy dz
we have
D /[xi] \ [xi] du [eta] dv [zeta] dw
-- ( ----- ) - ----- -- - ----- -- - ------ -- = 0, (11)
dt \[rho]/ [rho] dx [rho] dx [rho] dx
with two similar equations.
Putting
[omega]² = [xi]² + [eta]² + [zeta]², (12)
a _vortex line_ is defined to be such that the tangent is in the
direction of [omega], the resultant of [xi], [eta], [zeta], called the
components of molecular rotation. A small sphere of the fluid, if
frozen suddenly, would retain this angular velocity.
If [omega] vanishes throughout the fluid at any instant, equation (11)
shows that it will always be zero, and the fluid motion is then called
_irrotational_; and a function [phi] exists, called the _velocity
function_, such that
udx + vdy + wdz = -d[phi], (13)
and then the velocity in any direction is the space-decrease or
downward gradient of [phi].
25. But in the most general case it is possible to have three
functions [phi], [psi], m of x, y, z, such that
udx + vdy + wdz = -d[phi] - md[psi], (1)
as A. Clebsch has shown, from purely analytical considerations
(_Crelle_, lvi.); and then
d([psi], m) d([psi], m) d([psi], m)
[xi] = ½ -----------, [eta] = ½ -----------, [zeta] = ½ -----------, (2)
d(y, z) d(z, x) d(x, y)
and
d[psi] d[psi] d[psi] dm dm dm
[xi]------ + [eta]------ + [zeta]------ = 0, [xi]-- + [eta]-- + [zeta]-- = 0, (3)
dx dy dz dx dy dz
so that, at any instant, the surfaces over which [psi] and m are
constant intersect in the vortex lines.
Putting
d[phi] d[psi]
H - ------ - m ------ = K, (4)
dt dt
the equations of motion (4), (5), (6) § 24 can be written
dK d([psi],m)
-- - 2u[zeta] + 2w[eta] - ---------- = 0, ..., ...; (5)
dx d(x,t)
and therefore
dK dK dK
[xi]-- + [eta]-- + [zeta]-- = 0. (6)
dx dy dz
Equation (5) becomes, by a rearrangement,
dK d[psi] /dm dm dm dm \
-- - ------ ( -- + u-- + v-- + w-- )
dx dx \dt dx dy dz /
dm / d[psi] d[psi] d[psi] d[psi] \
+ -- ( ------ + u------ + v------ + w------ ) = 0, ..., ..., (7)
dx \ dt dx dy dz /
dK d[psi] Dm dm D[psi]
-- - ------ -- + -- ------ = 0, ..., ..., (8)
dx dx dt dx dt
and as we prove subsequently (§ 37) that the vortex lines are composed
of the same fluid particles throughout the motion, the surface m and
[psi] satisfies the condition of (6) § 23; so that K is uniform
throughout the fluid at any instant, and changes with the time only,
and so may be replaced by F(t).
26. When the motion is _steady_, that is, when the velocity at any
point of space does not change with the time,
dK
-- - 2v[zeta] + 2w[eta] = 0, ..., ... (1)
dx
dK dK dK dK dK dK
[xi]-- + [eta]-- + [zeta]-- = 0, u-- + v-- + w-- = 0, (2)
dx dy dz dx dy dz
and
_
/
K = | dp/[rho] + V + ½q² = H (3)
_/
is constant along a vortex line, and a _stream line_, the path of a
fluid particle, so that the fluid is traversed by a series of H
surfaces, each covered by a network of stream lines and vortex lines;
and if the motion is irrotational H is a constant throughout the
fluid.
Taking the axis of x for an instant in the normal through a point on
the surface H = constant, this makes u = 0, [xi] = 0; and in steady
motion the equations reduce to
dH/d[nu] = 2v[zeta] - 2w[eta] = 2q[omega] sin [theta], (4)
where [theta] is the angle between the stream line and vortex line;
and this holds for their projection on any plane to which d[nu] is
drawn perpendicular.
In plane motion (4) reduces to
dH / dQ q \
----- = 2q[zeta] = q ( -- + --- ), (5)
d[nu] \ dv r /
if r denotes the radius of curvature of the stream line, so that
1 dp dV dH d½q² q²
----- ----- + ----- = ----- - ----- = ---, (6)
[rho] d[nu] d[nu] d[nu] d[nu] r
the normal acceleration.
The osculating plane of a stream line in steady motion contains the
resultant acceleration, the direction ratios of which are
du du du d½q² d½q² dH
u-- + v-- + w-- = ---- - 2v[zeta] + 2w[eta] = ---- - --, ..., (7)
dx dy dz dx dx dx
and when q is stationary, the acceleration is normal to the surface H
= constant, and the stream line is a geodesic.
Calling the sum of the pressure and potential head the statical head,
surfaces of constant statical and dynamical head intersect in lines on
H, and the three surfaces touch where the velocity is stationary.
Equation (3) is called Bernoulli's equation, and may be interpreted as
the balance-sheet of the energy which enters and leaves a given tube
of flow.
If homogeneous liquid is drawn off from a vessel so large that the
motion at the free surface at a distance may be neglected, then
Bernoulli's equation may be written
H = p/[rho] + z + q²/2g = P/[rho] + h, (8)
where P denotes the atmospheric pressure and h the height of the free
surface, a fundamental equation in hydraulics; a return has been made
here to the gravitation unit of hydrostatics, and Oz is taken
vertically upward.
In particular, for a jet issuing into the atmosphere, where p = P,
q²/2g = h - z, (9)
or the velocity of the jet is due to the head k - z of the still free
surface above the orifice; this is Torricelli's theorem (1643), the
foundation of the science of hydrodynamics.
27. _Uniplanar Motion._--In the uniplanar motion of a homogeneous
liquid the equation of continuity reduces to
du dv
-- + -- = 0, (1)
dx dy
so that we can put
u = -d[psi]/dy, v = d[psi]/dx, (2)
where [psi] is a function of x, y, called the stream- or
current-function; interpreted physically, [psi] - [psi]0, the
difference of the value of [psi] at a fixed point A and a variable
point P is the flow, in ft.³/second, across any curved line AP from A
to P, this being the same for all lines in accordance with the
continuity.
Thus if d[psi] is the increase of [psi] due to a displacement from P
to P´, and k is the component of velocity normal to PP´, the flow
across PP´ is d[psi] = k·PP´; and taking PP´ parallel to Ox, d[psi] =
vdx; and similarly d[psi]= -udy with PP´ parallel to Oy; and generally
d[psi]/ds is the velocity across ds, in a direction turned through a
right angle forward, against the clock.
In the equations of uniplanar motion
dv du d²[psi] d²[psi]
2[zeta] = -- - -- = ------ + ------ = -[Nabla]²[psi], suppose, (3)
dx dy dx² dy²
so that in steady motion
dH d[psi] dH d[psi] dH
-- + [Nabla]²[psi]------ = 0, -- + [Nabla]²[psi]------ = 0, ------ + [Nabla]²[psi] = 0, (4)
dx dx dy dy d[psi]
and [Nabla]²[psi] must be a function of [psi].
If the motion ia irrotational,
d[phi] d[psi] d[phi] d[psi]
u = - ------ = - ------, v = - ----- = ------, (5)
dx dy dy dx´
so that [psi] and [phi] are conjugate functions of x and y,
[phi] + [psi]i = [f](x + yi), [Nabla]²[psi] = 0, [Nabla]²[phi] = 0; (6)
or putting
[phi] + [psi]i = w, x + yi = z, w = [f](z).
The curves [phi] = constant and [psi] = constant form an orthogonal
system; and the interchange of [phi] and [psi] will give a new state
of uniplanar motion, in which the velocity at every point is turned
through a right angle without alteration of magnitude.
For instance, in a uniplanar flow, radially inward towards O, the flow
across any circle of radius r being the same and denoted by 2[pi]m,
the velocity must be m/r, and
[phi] = m log r, [psi] = m[theta],
[phi] + [psi]i = m log re^(i[theta]), w = m log z. (7)
Interchanging these values
[psi] = m log r, [phi] = m[theta],
[psi] + [phi]i = m log re^(i[theta]) (8)
gives a state of vortex motion, circulating round Oz, called a
straight or columnar vortex.
A single vortex will remain at rest, and cause a velocity at any point
inversely as the distance from the axis and perpendicular to its
direction; analogous to the magnetic field of a straight electric
current.
If other vortices are present, any one may be supposed to move with
the velocity due to the others, the resultant stream-function being
[psi] = [Sigma]m log r = log [Pi]r^m; (9)
the path of a vortex is obtained by equating the value of [psi] at the
vortex to a constant, omitting the r^m of the vortex itself.
When the liquid is bounded by a cylindrical surface, the motion of a
vortex inside may be determined as due to a series of vortex-images,
so arranged as to make the flow zero across the boundary.
For a plane boundary the image is the optical reflection of the
vortex. For example, a pair of equal opposite vortices, moving on a
line parallel to a plane boundary, will have a corresponding pair of
images, forming a rectangle of vortices, and the path of a vortex will
be the Cotes' spiral
r sin 2[theta] = 2a, or x^(-2) + y^(-2) = a^(-2); (10)
this is therefore the path of a single vortex in a right-angled
corner; and generally, if the angle of the corner is [pi]/n, the path
is the Cotes' spiral
r sin n[theta] = na. (11)
A single vortex in a circular cylinder of radius a at a distance c
from the centre will move with the velocity due to an equal opposite
image at a distance a²/c, and so describe a circle with velocity
mc/(a² - c²) in the periodic time 2[pi](a² - c²)/m. (12)
Conjugate functions can be employed also for the motion of liquid in a
thin sheet between two concentric spherical surfaces; the components
of velocity along the meridian and parallel in colatitude [theta] and
longitude [lambda] can be written
d[phi] 1 d[psi] 1 d[psi] d[psi]
-------- = ----------- ---------, ----------- --------- = - --------, (13)
d[theta] sin [theta] d[lambda] sin [theta] d[lambda] d[theta]
and then
[phi] + [psi]i = F(tan ½[theta]·e^([lambda]i)). (14)
28. _Uniplanar Motion of a Liquid due to the Passage of a Cylinder
through it._--A stream-function [psi] must be determined to satisfy
the conditions
[Nabla]²[psi] = 0, throughout the liquid; (1)
[psi] = constant, over any fixed boundary; (2)
d[psi]/ds = normal velocity reversed over a solid boundary, (3)
so that, if the solid is moving with velocity U in the direction Ox,
d[psi]/ds = -Udy/ds, or [psi] + Uy = constant over the moving
cylinder; and [psi] + Uy = [psi]´ is the stream function of the
relative motion of the liquid past the cylinder, and similarly [psi] -
Vx for the component velocity V along Oy; and generally
[psi]´ = [psi] + Uy - Vx (4)
is the relative stream-function, constant over a solid boundary moving
with components U and V of velocity.
If the liquid is stirred up by the rotation R of a cylindrical body,
d[psi]/ds = normal velocity reversed
dx dy
= -Rx-- - Ry--, (5)
ds ds
[psi] + ½R(x² + y²) = [psi]´, (6)
a constant over the boundary; and [psi]´ is the current-function of
the relative motion past the cylinder, but now
V²[psi]´ + 2R = 0, (7)
throughout the liquid.
Inside an equilateral triangle, for instance, of height h,
[psi]´ = -2R[alpha][beta][gamma]/h, (8)
where [alpha], [beta], [gamma] are the perpendiculars on the sides of
the triangle.
In the general case [psi]´ = [psi] + Uy - Vx + ½R(x² + y²) is the
relative stream function for velocity components, U, V, R.
29. _Example 1._--Liquid motion past a circular cylinder.
Consider the motion given by
[omega] = U(z + a²/z), (1)
so that
/ a²\ / a² \
[phi] = U ( r + -- ) cos [theta] = U ( 1 + -- )x, (2)
\ r / \ r² /
/ a²\ / a² \
[psi] = U ( r - -- ) sin [theta] = U ( 1 - -- )y.
\ r / \ r² /
Then [psi] = 0 over the cylinder r = a, which may be considered a
fixed post; and a stream line past it along which [psi] = Uc, a
constant, is the curve
/ a²\
( r - -- ) sin [theta] = c, (x² + y²)(y - c) - a²y = 0 (3)
\ r /
a cubic curve (C3).
Over a concentric cylinder, external or internal, of radius r = b,
/ a²\
[psi]´ = [psi] + U1y = [U ( 1 - --- ) + U1] y, (4)
\ b²/
and [psi]´ is zero if
U1/U = (a² - b²)/b²; (5)
so that the cylinder may swim for an instant in the liquid without
distortion, with this velocity U1, and [omega] in (1) will give the
liquid motion in the interspace between the fixed cylinder r = a and
the concentric cylinder r = b, moving with velocity U1.
When b = 0, U1 = [oo]; and when b = [oo], U1 = -U, so that at infinity
the liquid is streaming in the direction xO with velocity U.
If the liquid is reduced to rest at infinity by the superposition of
an opposite stream given by [omega] = -Uz, we are left with
[omega] = Ua²/z, (6)
[phi] = U(a²/r) cos [theta] = Ua²x/(x² + y²), (7)
[psi] = -U(a²/r) sin [theta] = -Ua²y/(x² + y²), (8)
giving the motion due to the passage of the Cylinder r = a with
velocity U through the origin O in the direction Ox.
If the direction of motion makes an angle [theta]´ with Ox,
d[phi] / d[phi] 2xy
tan[theta]´ = ----- / ----- = ------ = tan 2[theta], [theta] = ½[theta]´, (9)
dy / dx x² - y²
and the velocity is Ua²/r².
Along the path of a particle, defined by the C3 of (3),
y² y(y - c)
sin² ½[theta]´ = ------- = -------, (10)
x² + y² a²
d[theta]´ 2y - c dy
½ sin [theta]´ --------- = ------ --, (11)
ds a² ds
on the radius of curvature is ¼a²/(y - ½c), which shows that the curve
is an Elastica or Lintearia. (J. C. Maxwell, _Collected Works_, ii.
208.)
If [phi]1 denotes the velocity function of the liquid filling the
cylinder r = b, and moving bodily with it with velocity U1,
[phi]1 = -U1x, (12)
and over the separating surface r = b
[phi] U / a²\ a² + b²
--------- = - -- ( 1 + -- ) = -------, (13)
[phi]1 U1 \ b²/ a² - b²
and this, by § 36, is also the ratio of the kinetic energy in the
annular interspace between the two cylinders to the kinetic energy of
the liquid moving bodily inside r = b.
Consequently the inertia to overcome in moving the cylinder r = b,
solid or liquid, is its own inertia, increased by the inertia of
liquid (a² + b²)/(a² - b²) times the volume of the cylinder r = b;
this total inertia is called the effective inertia of the cylinder r =
b, at the instant the two cylinders are concentric.
With liquid of density [rho], this gives rise to a kinetic reaction to
acceleration dU/dt, given by
a² + b² dU a² + b² dU
[pi][rho]b² ------- -- = ------- M´--, (14)
a² - b² dt a² - b² dt
if M´ denotes the mass of liquid displaced by unit length of the
cylinder r = b. In particular, when a = [oo], the extra inertia is M´.
When the cylinder r = a is moved with velocity U and r = b with
velocity U1 along Ox,
a² / b² \ b² / a²\
[phi] = U ------- ( --- + r ) cos [theta] - U1------- ( r + --- ) cos [theta], (15)
b² - a² \ r / b² - a² \ r /
a² / b² \ b² / a²\
[psi] = -U ------- ( --- - r ) sin [theta] - U1------- ( r - --- ) sin [theta]; (16)
b² - a² \ r / b² - a² \ r /
and similarly, with velocity components V and V1 along Oy
a² / b² \ b² / a²\
[phi] = V ------- ( --- + r ) sin [theta] - V1------- ( r + --- ) sin [theta], (17)
b² - a² \ r / b² - a² \ r /
a² / b² \ b² / a²\
[psi] = V ------- ( --- - r ) cos [theta] + V1------- ( r - --- ) cos [theta], (18)
b² - a² \ r / b² - a² \ r /
and then for the resultant motion
a² z a²b² U + Vi
w = (U² + V²) ------- ------ + ------- ------
b² - a² U + Vi b² - a² z
a² z a²b² U1 + V1i
-(U1² + V1²) ------- -------- + ------- --------. (19)
b² - a² U1 + V1i b² - a² z
The resultant impulse of the liquid on the cylinder is given by the
component, over r = a (§ 36),
_
/ / b² + a² 2b² \
X = | [rho][phi] cos [theta]·ad[theta] = [pi][rho]a² ( U ------- - U1 ------- ); (20)
_/ \ b² - a² b² - a²/
and over r = b
_
/ / 2a² b² + a² \
X1 = | [rho][phi] cos [theta]·bd[theta] = [pi][rho]b² ( U ------ - U1------- ), (21)
_/ \ b² - a² b² - a² /
and the difference X - X1 is the component momentum of the liquid in
the interspace; with similar expressions for Y and Y1.
Then, if the outside cylinder is free to move
V1 2a² b² - a²
X1 = 0, -- = -------, X = [pi][rho]a²U -------. (22)
U b² + a² b² + a²
But if the outside cylinder is moved with velocity U1, and the inside
cylinder is solid or filled with liquid of density [sigma],
U1 2[rho]b²
X = -[pi][rho]a²U, -- = --------------------------------,
U [rho](b² + a²) + [sigma](b² - a²)
U - U1 ([rho] - [sigma])(b² - a²)
------ = ---------------------------------, (23)
U1 [rho](b² + a²) + [sigma](b² - a²)
and the inside cylinder starts forward or backward with respect to the
outside cylinder, according as [rho] > or < [sigma].
30. The expression for [omega] in (1) § 29 may be increased by the
addition of the term
im log z = -m[theta] + im log r, (1)
representing vortex motion circulating round the annulus of liquid.
Considered by itself, with the cylinders held fixed, the vortex sets
up a circumferential velocity m/r on a radius r, so that the angular
momentum of a circular filament of annular cross section dA is
[rho]mdA, and of the whole vortex is [rho]m[pi](b² - a²).
Any circular filament can be started from rest by the application of a
circumferential impulse [pi][rho]mdr at each end of a diameter; so
that a mechanism attached to the cylinders, which can set up a uniform
distributed impulse [pi][rho]m across the two parts of a diameter in
the liquid, will generate the vortex motion, and react on the cylinder
with an impulse couple -[rho]m[pi]a² and [rho]m[pi]b², having
resultant [rho]m[pi](b² - a²), and this couple is infinite when b =
[oo], as the angular momentum of the vortex is infinite. Round the
cylinder r = a held fixed in the U current the liquid streams past
with velocity
q´ = 2U sin [theta] + m/a; (2)
and the loss of head due to this increase of velocity from U to q´ is
q´² - U² (2U sin [theta] + m/a)² - U²
-------- = ----------------------------, (3)
2g 2g
so that cavitation will take place, unless the head at a great
distance exceeds this loss.
The resultant hydrostatic thrust across any diametral plane of the
cylinder will be modified, but the only term in the loss of head which
exerts a resultant thrust on the whole cylinder is 2mU sin[theta]/ga,
and its thrust is 2[pi][rho]mU absolute units in the direction Cy, to
be counteracted by a support at the centre C; the liquid is streaming
past r = a with velocity U reversed, and the cylinder is surrounded by
a vortex. Similarly, the streaming velocity V reversed will give rise
to a thrust 2[pi][rho]mV in the direction xC.
Now if the cylinder is released, and the components U and V are
reversed so as to become the velocity of the cylinder with respect to
space filled with liquid, and at rest at infinity, the cylinder will
experience components of force per unit length
(i.) - 2[pi][rho]mV, 2[pi][rho]mU, due to the vortex motion;
(ii.) - [pi][rho]a² dU/dt, -[pi][rho]a² dV/dt, due to the kinetic
reaction of the liquid;
(iii.) 0, -[pi]([sigma] - [rho])a²g, due to gravity,
taking Oy vertically upward, and denoting the density of the cylinder
by [sigma]; so that the equations of motion are
dU dU
[pi][rho]a²-- = - [pi][rho]a²-- - 2[pi][rho]mV, (4)
dt dt
dV dV
[pi][rho]a²-- = - [pi][rho]a²-- + 2[pi][rho]mV - [pi]([sigma] - [rho])a²g, (5)
dt dt
or, putting m = a²[omega], so that the vortex velocity is due to an
angular velocity [omega] at a radius a,
([sigma] + [rho])dU/dt + 2[rho][omega]V = 0, (6)
([sigma] + [rho])dV/dt - 2[rho][omega]U + ([sigma] - [rho])g = 0. (7)
Thus with g = 0, the cylinder will describe a circle with angular
velocity 2[rho][omega]/([sigma] + [rho]), so that the radius is
([sigma] + [rho])v/2[rho][omega], if the velocity is v. With [sigma] =
0, the angular velocity of the cylinder is 2[omega]; in this way the
velocity may be calculated of the propagation of ripples and waves on
the surface of a vertical whirlpool in a sink.
Restoring [sigma] will make the path of the cylinder a trochoid; and
so the swerve can be explained of the ball in tennis, cricket,
baseball, or golf.
Another explanation may be given of the sidelong force, arising from
the velocity of liquid past a cylinder, which is encircled by a
vortex. Taking two planes x = ± b, and considering the increase of
momentum in the liquid between them, due to the entry and exit of
liquid momentum, the increase across dy in the direction Oy, due to
elements at P and P´ at opposite ends of the diameter PP´, is
[rho]dy (U - Ua²r^(-2) cos 2[theta] + mr^(-1) sin [theta])(Ua²r^(-2) sin 2[theta] + mr^(-1) cos [theta])
+ [rho]dy (- U + Ua²r^(-2) cos 2[theta] + mr^(-1) sin [theta])(Ua²r^(-2) sin 2[theta] - mr^(-1) cos [theta])
= 2[rho]dymUr^(-1)(cos [theta] - a^2r^(-2)cos 3[theta]), (8)
and with y = b tan [theta], r = b sec [theta], this is
2[rho]mUd[theta] (1 - a²b^(-2) cos 3[theta] cos [theta]), (9)
and integrating between the limits [theta] = ±½[pi], the resultant, as
before, is 2[pi][rho]mU.
31. _Example 2.--Confocal Elliptic Cylinders._--Employ the elliptic
coordinates [eta], [xi], and [zeta] = [eta] + [xi]i, such that
z = c ch[zeta], x = c ch [eta] cos [xi], y = c sh [eta] sin [zeta]; (1)
then the curves for which [eta] and [xi] are constant are confocal
ellipses and hyperbolas, and
d(x, y)
J = -------, [xi]) = c²(ch²[eta] - cos² [xi])
d([eta]
= ½c²(ch 2[eta] - cos 2[xi]) = r1r2 = OD², (2)
if OD is the semi-diameter conjugate to OP, and r1, r2 the focal
distances,
r1, r2 = c(ch[eta] ± cos [xi]); (3)
r² = x² + y² = c²(ch²[eta] - sin² [xi])
= ½c²(ch 2[eta] + cos 2[xi]). (4)
Consider the streaming motion given by
w = m ch([zeta] - [gamma]), [gamma] = [alpha] + [beta]i, (5)
[phi] = m ch([eta] - [alpha]) cos ([xi] - [beta]),
[psi] = m sh([eta] - [alpha]) sin ([xi] - [beta]). (6)
Then [psi] = 0 over the ellipse [eta] = [alpha], and the hyperbola
[xi] = [beta], so that these may be taken as fixed boundaries; and
[psi] is a constant on a C4.
Over any ellipse [eta], moving with components U and V of velocity,
[psi]´ = [psi] + Uy - Vx = [m sh([eta] - [alpha]) cos [beta] + Uc sh[eta]] sin [xi]
-[m sh ([eta] - [alpha]) sin [beta] + Vc ch [eta] cos [xi]; (7)
so that [psi]´ = 0, if
m sh([eta] - [alpha]) m sh([eta] - [alpha])
U = - --- ------------------- cos [beta], V = - --- - ------------------- sin [beta], (8)
c sh[eta] c ch[eta]
having a resultant in the direction PO, where P is the intersection of
an ellipse [eta] with the hyperbola [beta]; and with this velocity the
ellipse [eta] can be swimming in the liquid, without distortion for an
instant.
At infinity
m m
U = - --- e^(-a) cos [beta] = - ----- cos [beta],
c a - b
m m
V = - --- e^(-a) sin [beta] = - ----- sin [beta], (9)
c a + b
a and b denoting the semi-axes of the ellipse [alpha]; so that the
liquid is streaming at infinity with velocity Q = m/(a + b) in the
direction of the asymptote of the hyperbola [beta].
An ellipse interior to [eta] = [alpha] will move in a direction
opposite to the exterior current; and when [eta] = 0, U = [oo], but V
= (m/c) sh [alpha] sin [beta].
Negative values of [eta] must be interpreted by a streaming motion on
a parallel plane at a level slightly different, as on a double Riemann
sheet, the stream passing from one sheet to the other across a cut SS´
joining the foci S, S´. A diagram has been drawn by Col. R. L.
Hippisley.
The components of the liquid velocity q, in the direction of the
normal of the ellipse [eta] and hyperbola [xi], are
-mJ^(-1)sh([eta] - [alpha]) cos([xi] - [beta]),
mJ^(-1)ch([eta] - [alpha]) sin ([xi] - [beta]). (10)
The velocity q is zero in a corner where the hyperbola [beta] cuts the
ellipse [alpha]; and round the ellipse [alpha] the velocity q reaches
a maximum when the tangent has turned through a right angle, and then
[root](ch 2[alpha] - cos 2[beta])
q = Qe^a ---------------------------------; (11)
sh 2[alpha]
and the condition can be inferred when cavitation begins.
With [beta] = 0, the stream is parallel to x0, and
[phi] = m ch([eta] - [alpha])cos [xi]
= -Uc ch([eta] - [alpha])sh [eta] cos [xi]/sh([eta] - [alpha]) (12)
over the cylinder [eta], and as in (12) § 29,
[phi]1 = -Ux = -Uc ch [eta] cos [xi], (13)
for liquid filling the cylinder; and
[phi] th [eta]
------ = --------------------, (14)
[phi]1 th ([eta] - [alpha])
over the surface of [eta]; so that parallel to Ox, the effective
inertia of the cylinder [eta], displacing M´ liquid, is increased by
M´th [eta]/th([eta]- [alpha]), reducing when [alpha] = [oo] to M´th
[eta] = M´(b/a).
Similarly, parallel to Oy, the increase of effective inertia is M´/th
[eta] th([eta] - [alpha]), reducing to M´/th [eta] = M´(a/b), when
[alpha] = [oo], and the liquid extends to infinity.
32. Next consider the motion given by
[phi] = m ch 2([eta] - [alpha]) sin 2[xi],
[psi] = -m sh 2([eta] - [alpha]) cos 2[xi]; (1)
in which [psi] = 0 over the ellipse [alpha], and
[psi]´ = [psi] + ½R(x² + y²)
= [-m sh 2([eta] - [alpha]) + ¼Rc²] cos 2[xi] + ¼Rc² ch 2[eta], (2)
which is constant over the ellipse [eta] if
¼Rc² = m sh 2([eta] - [alpha]); (3)
so that this ellipse can be rotating with this angular velocity R for
an instant without distortion, the ellipse [alpha] being fixed.
For the liquid filling the interior of a rotating elliptic cylinder of
cross section
x²/a² + y²/b² = 1, (4)
[psi]1´ = m1(x²/a² + y²/b²) (5)
with
[nabla]²[psi]1´ = -2R = -2m1(1/a² + 1/b²),
[psi]1 = m1(x²/a² + y²/b²) - ½R(x² + y²)
= -½R(x² - y²)(a² - b²)/(a² + b²), (6)
[phi]1 = Rxy(a² - b²)/(a² + b²),
w1 = [phi]1 + [psi]1i = -½iR(x + yi)²(a² - b²)/(a² + b²).
The velocity of a liquid particle is thus (a² - b²)/(a² + b²) of what
it would be if the liquid was frozen and rotating bodily with the
ellipse; and so the effective angular inertia of the liquid is (a² -
b²)²/(a² + b²)² of the solid; and the effective radius of gyration,
solid and liquid, is given by
k² = ¼(a² + b²), and ¼(a² - b²)²/(a² + b²). (7)
For the liquid in the interspace between [alpha] and [eta],
[phi] m ch 2([eta] - [alpha]) sin 2[xi]
------ = -------------------------------------------
[phi]1 ¼Rc² sh 2[eta] sin 2[xi](a² - b²)/(a² + b²)
= 1/th 2([eta] - [alpha])th 2[eta]; (8)
and the effective k² of the liquid is reduced to
¼c²/th 2([eta] - [alpha]) sh 2[eta], (9)
which becomes ¼c²/sh 2[eta] = 1/8 (a² - b²)/ab, when [alpha] = [oo],
and the liquid surrounds the ellipse [eta] 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 [psi] is resolved into [psi]1 + [psi]2,
such that [psi]1 + ½Rx² and [psi]2 + ½Ry² is constant over the
boundary.
Inside a cylinder
[phi]1 + [psi]1i = -½iR(x + yi)²a²/(a² + b²), (10)
[phi]2 + [psi]2i = ½iR(x + yi)²b²/(a² + b²), (11)
and for the interspace, the ellipse [alpha] being fixed, and [alpha]1
revolving with angular velocity R
[phi]1 + [psi]1i = -1/8 iRc²sh 2([eta] - [alpha]
+ [xi]i)(ch 2[alpha] + 1)/sh 2([alpha]1 - [alpha]), (12)
[phi]2 + [psi]2i = 1/8 iRc²sh 2([eta] - [alpha]
+ [xi]i)(ch 2[alpha] - 1)/sh 2([alpha]1 - [alpha]), (13)
satisfying the condition that [psi]1 and [psi]2 are zero over [eta] =
[alpha], and over [eta] = [alpha]1
[psi]1 + ½Rx² = 1/8 Rc²(ch 2[alpha]1 + 1), (14)
[psi]2 + ½Ry² = 1/8 Rc²(ch 2[alpha]1 - 1), (15)
constant values.
In a similar way the more general state of motion may be analysed,
given by
w = m ch 2([zeta] - [gamma]), [gamma] = [alpha] + [beta]i, (16)
as giving a homogeneous strain velocity to the confocal system; to
which may be added a circulation, represented by an additional term
m[zeta] in w.
Similarly, with
x + yi = c[root][sin ([xi] + [eta]i)] (17)
the function
[psi] = Qc sh ½([eta] - [alpha]) sin ½([xi] - [beta]) (18)
will give motion streaming past the fixed cylinder [eta] = [alpha],
and dividing along [xi] = [beta]; and then
x² - y² = c² sin [xi] ch [eta], 2xy = c² cos [xi] sh [eta]. (19)
In particular, with sh [alpha] = 1, the cross-section of [eta] =
[alpha] is
x^4 + 6x²y² + y^4 = 2c^4, or x^4 + y^4 = c^4 (20)
when the axes are turned through 45°.
33. _Example 3._--Analysing in this way the rotation of a rectangle
filled with liquid into the two components of shear, the stream
function [psi]1 is to be made to satisfy the conditions
(i.) [nabla]²[psi]1 = 0,
(ii.) [psi]1 + ½Rx² = ½Ra², or [psi]1 = 0 when x = ±a,
(iii.) [psi]1 + ½Rx² = ½Ra², [psi]1 = ½R(a² - x²), when y = ± b.
Expanded in a Fourier series,
32 __ cos (2n + 1) ½[pi]x/a
a² - x² = ----- a² \ ---------------------, (1)
[pi]³ /__ (2n + 1)³
so that
16 __ cos (2n + 1) ½[pi]x/a · ch(2n + 1) ½[pi]y/a)
[psi]1 = ----- a² \ ---------------------------------------------,
[pi]³ /__ (2n + 1)^3 · ch(2n + 1) ½[pi]b/a
16 __ cos (2n + 1) ½[pi]z/a
w1 = [phi]1 + [psi]1i = iR ----- \ ------------------------------, (2)
[pi]³ /__ (2n + 1)^3 ch(2n + 1) ½[pi]b/a
an elliptic-function Fourier series; with a similar expression for
[psi]2 with x and y, a and b interchanged; and thence [psi] = [psi]1 +
[psi]2.
_Example 4._--Parabolic cylinder, axial advance, and liquid streaming
past.
The polar equation of the cross-section being
r^½ cos ½[theta] = a^½, or r + x = 2a, (3)
the conditions are satisfied by
[psi]´ = Ur sin [theta] - 2Ua^½ r^½ sin ½[theta]
= 2Ur^½ sin ½[theta](r^½ cos ½[theta] - a^½), (4)
[psi] = 2Ua^½ r^½ sin ½[theta] = -U[root][2a(r-x)], (5)
w = -2Ua^½ z^½, (6)
and the resistance of the liquid is 2[pi][rho]aV²/2g.
A relative stream line, along which [psi]´ = Uc, is the quartic curve
(4a²y² - (y - c)^4 4a²y² + (y-c)^4
y - c = [root][2a(r - x)], x = -------------------, r = ---------------, (7)
(4a(y - c)² 4a(y - c)²
and in the absolute space curve given by [psi],
dy (y - c)² 2ac
-- = - --------, x = - ----- 2a log (y - c). (8)
dx 2ay y - c
34. _Motion symmetrical about an Axis._--When the motion of a liquid
is the same for any plane passing through Ox, and lies in the plane, a
function [psi] can be found analogous to that employed in plane
motion, such that the flux across the surface generated by the
revolution of any curve AP from A to P is the same, and represented by
2[pi]([psi] - [psi]0); and, as before, if d[psi] is the increase in
[psi] due to a displacement of P to P´, then k the component of
velocity normal to the surface swept out by PP´ is such that
2[pi]d[psi] = 2[pi]yk.PP´; and taking PP´ parallel to Oy and Ox,
u = -d[psi]/ydy, v = d[psi]/ydx, (1)
and [psi] is called after the inventor, "Stokes's stream or current
function," as it is constant along a stream line (_Trans. Camb. Phil.
Soc._, 1842; "Stokes's Current Function," R. A. Sampson, _Phil.
Trans._, 1892); and d[psi]/yds is the component velocity across ds in
a direction turned through a right angle forward.
In this symmetrical motion
d / 1 d[psi] \ d / 1 d[psi]\
[xi] = 0, [eta] = 0, 2[zeta] = -- ( --- ------ ) + -- ( --- ------ )
dx \ y dx / dy \ y dy /
1 /d²[psi] d²[psi] 1 d[psi]\ 1
= --- ( ------- + ------- - --- ------ ) = - ---[nabla]²[psi], (2)
y \ dx² dy² y dy / y
suppose; and in steady motion,
dH 1 d[psi] dH 1 d[psi]
-- + --- ----- [nabla]²[psi] = 0, -- + --- ------ [nabla]²[psi] = 0, (3)
dx y² dx dy y² dy
so that
2[zeta]/y = -y^(-2)[nabla]²[psi] = dH/d[psi] (4)
is a function of [psi], say [f]´([psi]), and constant along a stream line;
dH/dv = 2q[zeta], H - [f]([psi]) = constant, (5)
throughout the liquid.
When the motion is irrotational,
d[phi] 1 d[psi] d[phi] 1 d[psi]
[zeta] = 0, u = - ------ = - --- ------, v = - ------ = --- ------, (6)
dx y dy dy y dx
d²[psi] d²[psi] 1 d[psi]
[nabla]²[psi] = 0, or ------- + ------- - --- ------ = 0. (7)
dx² dy² y dy
Changing to polar coordinates, x = r cos[theta], y = r sin[theta], the
equation (2) becomes, with cos[theta] = [mu],
d²[psi] d²[psi]
r²------- + (1 - [mu]²) ------- = 2[zeta]r³ sin [theta], (8)
dr² d[mu]²
of which a solution, when [zeta] = 0, is
/ B \ dPn / B \ dPn
[psi] = ( Ar^(n+1) + --- ) (1 - [mu]²) ----- = ( Ar^(n-1) + ------- ) y²-----, (9)
\ r^n / d[mu] \ r^(n+2) / d[mu]
[phi] = {(n + 1)Ar^n - nBr^(-n-1)} Pn, (10)
where Pn denotes the zonal harmonic of the nth order; also, in the
exceptional case of
[psi] = A0 cos[theta], [phi] = A0/r;
[psi] = B0r, [phi] = -B0 log tan ½[theta]
= -½B0 sh(-1) x/y. (11)
Thus cos[theta] is the Stokes' function of a point source at O, and PA
- PB of a line source AB.
The stream function [psi] of the liquid motion set up by the passage
of a solid of revolution, moving with axial velocity U, is such that
1 d[psi] dy
--- ------ = -U --, [psi] + ½Uy² = constant, (12)
y ds ds
over the surface of the solid; and [psi] must be replaced by [psi]´ =
[psi] + ½Uy² in the general equations of steady motion above to obtain
the steady relative motion of the liquid past the solid.
For instance, with n = 1 in equation (9), the relative stream function
is obtained for a sphere of radius a, by making it
[psi]´ = [psi] + ½Uy² = ½U(r² - a³/r) sin² [theta],
[psi] = -½Ua³ sin² [theta]/r; (13)
and then
[phi]´ = Ux(1 + ½a³/r²), [phi] = ½Ua³ cos [theta]/r², (14)
d[phi] a³ d[phi] a³
- ------ = U -- cos [theta], - --------- = ½U -- sin [theta], (15)
dr r³ rd[theta] r³
so that, if the direction of motion makes an angle [psi] with Ox,
tan ([psi] - [theta]) = ½ tan [theta],
tan [psi] = 3 tan [theta]/(2 - tan² [theta]), (16)
Along the path of a liquid particle [psi]´ is constant, and putting it
equal to ½Uc²,
(r² - a³/r) sin² [theta] = c², sin² [theta] = c²r/(r³ - a³), (17)
the polar equation; or
y² = c²r³/(r³ - a³), r³ = a³y²/(y² - c²), (18)
a curve of the 10th degree (C10).
In the absolute path in space
cos [psi] = (2 - 3 sin² [theta])/[root](4 - sin² [theta]),
and sin³ [theta] = (y³ - c²y)/a³, (19)
which leads to no simple relation.
The velocity past the surface of the sphere is
1 d[psi]´ / a³ \ sin² [theta]
------------ ------- = ½U ( 2r + -- ) ------------- = 3/2 U sin [theta], when r = a; (20)
r sin[theta] dr \ r² / r sin [theta]
so that the loss of head is
(9/4 sin² [theta] - 1) U²/2g, having a maximum 5/4 U²/2g, (21)
which must be less than the head at infinite distance to avoid
cavitation at the surface of the sphere.
With n = 2, a state of motion is given by
[psi] = -½Uy²a^4[mu]/r^4, [psi]´ = ½Uy²(1 - a^4[mu]/r^4), (22)
[phi]´ = Ux + [phi], [phi] = -1/3 U(a^4/r³)P2, P2 = 3/2 [mu]² - ½, (23)
representing a stream past the surface r^4 = a^4[mu].
35. A circular vortex, such as a smoke ring, will set up motion
symmetrical about an axis, and provide an illustration; a half vortex
ring can be generated in water by drawing a semicircular blade a short
distance forward, the tip of a spoon for instance. The vortex advances
with a certain velocity; and if an equal circular vortex is generated
coaxially with the first, the mutual influence can be observed. The
first vortex dilates and moves slower, while the second contracts and
shoots through the first; after which the motion is reversed
periodically, as if in a game of leap-frog. Projected perpendicularly
against a plane boundary, the motion is determined by an equal opposite
vortex ring, the optical image; the vortex ring spreads out and moves
more slowly as it approaches the wall; at the same time the molecular
rotation, inversely as the cross-section of the vortex, is seen to
increase. The analytical treatment of such vortex rings is the same as
for the electro-magnetic effect of a current circulating in each ring.
36. _Irrotational Motion in General._--Liquid originally at rest in a
singly-connected space cannot be set in motion by a field of force due
to a single-valued potential function; any motion set up in the liquid
must be due to a movement of the boundary, and the motion will be
irrotational; for any small spherical element of the liquid may be
considered a smooth solid sphere for a moment, and the normal pressure
of the surrounding liquid cannot impart to it any rotation.
The kinetic energy of the liquid inside a surface S due to the
velocity function [phi] is given by
_ _ _ _ _
/ / / | /d[phi]\² /d[phi]\² /d[phi]\² |
T = ½[rho] | | | | ( ------ ) + ( ------ ) + ( ------ ) | dx dy dz,
_/_/_/ |_ \ dx / \ dy / \ dz / _|
_ _
/ / d[phi]
= ½[rho] | | [phi] ------ dS (1)
_/_/ d[nu]
by Green's transformation, d[nu] denoting an elementary step along the
normal to the exterior of the surface; so that d[phi]/d[nu] = 0 over
the surface makes T = 0, and then
/d[phi]\² /d[phi]\² /d[phi]\² d[phi] d[phi] d[phi]
( ------ ) + ( ------ ) + ( ------ ) = 0, ------ = 0, ------ = 0, ------ = 0 (2)
\ dx / \ dy / \ dz / dx dy dz
If the actual motion at any instant is supposed to be generated
instantaneously from rest by the application of pressure impulse over
the surface, or suddenly reduced to rest again, then, since no natural
forces can act impulsively throughout the liquid, the pressure impulse
[~[omega]] satisfies the equations
1 d[~omega] 1 d[~omega] 1 d[~omega]
----- --------- = -u, ----- --------- = -v, ----- --------- = -[~omega], (3)
[rho] dx [rho] dy [rho] dz
[~omega] = [rho][phi] + a constant, (4)
and the constant may be ignored; and Green's transformation of the
energy T amounts to the theorem that the work done by an impulse is
the product of the impulse and average velocity, or half the velocity
from rest.
In a multiply connected space, like a ring, with a multiply valued
velocity function [phi], the liquid can circulate in the circuits
independently of any motion of the surface; thus, for example,
[phi] = m[theta] = m tan^(-1) y/x (5)
will give motion to the liquid, circulating in any ring-shaped figure
of revolution round Oz.
To find the kinetic energy of such motion in a multiply connected
space, the channels must be supposed barred, and the space made
acyclic by a membrane, moving with the velocity of the liquid; and
then if k denotes the cyclic constant of [phi] in any circuit, or the
value by which [phi] has increased in completing the circuit, the
values of [phi] on the two sides of the membrane are taken as
differing by k, so that the integral over the membrane
_ _ _ _
/ / d[phi] / / d[phi]
| | [phi] ------ dS = k | | ------ dS, (6)
_/_/ d[nu] _/_/ d[nu]
and this term is to be added to the terms in (1) to obtain the
additional part in the kinetic energy; the continuity shows that the
integral is independent of the shape of the barrier membrane, and its
position. Thus, in (5), the cyclic constant k = 2[pi]m.
In plane motion the kinetic energy per unit length parallel to Oz
_ _ _ _ _ _ _ _
/ / | /d[phi]\² /d[phi]\² | / / | /d[psi]\² /d[psi]\² |
T = ½[rho] | | | ( ------ ) + ( ------ ) | dx dy = ½[rho] | | | ( ------ ) + ( ------ ) | dx dy
_/_/ |_ \ dx / \ dy / _| _/_/ |_ \ dx / \ dy / _|
_ _
/ d[phi] / d[phi]
= ½[rho] | [phi] ------ ds = ½[rho] | [psi] ------ ds. (7)
_/ d[nu] _/ d[nu]
For example, in the equilateral triangle of (8) § 28, referred to
coordinate axes made by the base and height,
[psi]´ = -2R[alpha][beta][gamma]/h = -½Ry[(h - y)² - 3x²]/h (8)
[psi] = [psi]´ - ½R [(1/3h - y)² + x²]
= -½R [½h³ + 1/3 h²y + h) (x² - y²) - 3x²y + y³] /h (9)
and over the base y = 0,
dx/d[nu] = -dx/dy = + ½R(1/3 h² - 3x²)/h, [psi] = -½R(1/9 h² + x²). (10)
Integrating over the base, to obtain one-third of the kinetic energy
T,
_
/ h/[root]3
1/3 T = ½[rho] | ¼R²(3x^4 - 1/27 h^4) dx/h
_/ -h/[root]3
= [rho]R²h^4/135[root]3 (11)
so that the effective k² of the liquid filling the triangle is given
by
k² = T/½[rho]R²A = 2h²/45
= 2/5 (radius of the inscribed circle)², (12)
or two-fifths of the k² for the solid triangle.
Again, since
d[phi]/d[nu] = d[psi]/ds, d[phi]/ds = -d[psi]/d[nu], (13)
_ _
/ /
T = ½[rho] | [phi] d[psi] = -½[rho] | [psi] d[phi]. (14)
_/ _/
With the Stokes' function [psi] for motion symmetrical about an
axis.
_ _
/ d[psi] /
T = ½[rho] | [phi] ------ 2[pi]y ds = [pi][rho] | [phi] d[psi]. (15)
_/ yds _/
37. _Flow, Circulation, and Vortex Motion._--The line integral of the
tangential velocity along a curve from one point to another, defined
by
_ _
/ / dx dy dz \ /
| ( u-- + v-- + w-- ) ds = | (u dx + v dy + z dz), (1)
_/ \ ds ds ds / _/
is called the "flux" along the curve from the first to the second
point; and if the curve closes in on itself the line integral round
the curve is called the "circulation" in the curve.
With a velocity function [phi], the flow
_
/
- | d[phi] = [phi]1 - [phi]2, (2)
_/
so that the flow is independent of the curve for all curves mutually
reconcilable; and the circulation round a closed curve is zero, if the
curve can be reduced to a point without leaving a region for which
[phi] is single valued.
If through every point of a small closed curve the vortex lines are
drawn, a tube is obtained, and the fluid contained is called a _vortex
filament_.
By analogy with the spin of a rigid body, the component spin of the
fluid in any plane at a point is defined as the circulation round a
small area in the plane enclosing the point, divided by twice the
area. For in a rigid body, rotating about Oz with angular velocity
[zeta], the circulation round a curve in the plane xy is
_
/ / dy dx \
| [zeta] ( x -- - y -- ) ds = [zeta] times twice the area. (3)
_/ \ ds ds /
In a fluid, the circulation round an elementary area dxdy is equal to
/ dv \ / du \ / dv du \
udx + ( v + --dx )dy - ( u + --dy )dx - vdy = ( -- - -- )dx dy, (4)
\ dx / \ dy / \ dx dy /
so that the component spin is
/ dv du \
½ ( -- - -- ) = [zeta], (5)
\ dx dy /
in the previous notation of § 24; so also for the other two components
[xi] and [eta].
Since the circulation round any triangular area of given aspect is the
sum of the circulation round the projections of the area on the
coordinate planes, the composition of the components of spin, [xi],
[eta], [zeta], is according to the vector law. Hence in any
infinitesimal part of the fluid the circulation is zero round every
small plane curve passing through the vortex line; and consequently
the circulation round any curve drawn on the surface of a vortex
filament is zero.
If at any two points of a vortex line the cross-section ABC, A´B´C´ is
drawn of the vortex filament, joined by the vortex line AA´, then,
since the flow in AA´ is taken in opposite directions in the complete
circuit ABC AA´B´C´ A´A, the resultant flow in AA´ cancels, and the
circulation in ABC, A´B´C´ is the same; this is expressed by saying
that at all points of a vortex filament [omega][alpha] is constant
where [alpha] is the cross-section of the filament and [omega] the
resultant spin (W. K. Clifford, _Kinematic_, book iii.).
So far these theorems on vortex motion are kinematical; but
introducing the equations of motion of § 22,
Du dQ Dv dQ Dw dQ
-- + -- = 0, -- + -- = 0, -- + -- = 0, (6)
dt dx dt dy dt dz
_
/
Q = | dp/[rho] + V, (7)
_/
and taking dx, dy, dz in the direction of u, v, w, and
dx : dy : dz = u : v : w,
D / \ Du D dx
-- (u dx + v dy + w dz ) = -- dx + u ---- + ... = -dQ + ½dq², (8)
dt \ / dt dt
and integrating round a closed curve
_
D /
-- | (u dx + v dy + w dz) = 0, (9)
dt _/
and the circulation in any circuit composed of the same fluid
particles is constant; and if the motion is differential irrotational
and due to a velocity function, the circulation is zero round all
reconcilable paths. Interpreted dynamically the normal pressure of the
surrounding fluid on a tube cannot create any circulation in the tube.
The circulation being always zero round a small plane curve passing
through the axis of spin in vortical motion, it follows conversely
that a vortex filament is composed always of the same fluid particles;
and since the circulation round a cross-section of a vortex filament
is constant, not changing with the time, it follows from the previous
kinematical theorem that [alpha][omega] is constant for all time, and
the same for every cross-section of the vortex filament.
A vortex filament must close on itself, or end on a bounding surface,
as seen when the tip of a spoon is drawn through the surface of water.
Denoting the cross-section [alpha] of a filament by dS and its mass by
dm, the quantity [omega]dS/dm is called the _vorticity_; this is the
same at all points of a filament, and it does not change during the
motion; and the vorticity is given by [omega] cos[epsilon]dS/dm, if dS
is the oblique section of which the normal makes an angle [epsilon]
with the filament, while the aggregate vorticity of a mass M inside a
surface S is
_
/
M^(-1) | [omega] cos [epsilon] dS.
_/
Employing the equation of continuity when the liquid is homogeneous,
/ d[zeta] d[eta]\ d² d² d²
2( ------ - ------ ) = [nabla]²u, ... , [nabla]² = - --- - --- - ---, (10)
\ dy dz / dx² dy² dz²
which is expressed by
[nabla]²(u,v,w) = 2 curl ([xi], [eta], [zeta]),
([xi], [eta], [zeta]) = ½ curl (u, v, w). (11)
38. _Moving Axes in Hydrodynamics._--In many problems, such as the
motion of a solid in liquid, it is convenient to take coordinate axes
fixed to the solid and moving with it as the movable trihedron frame
of reference. The components of velocity of the moving origin are
denoted by U, V, W, and the components of angular velocity of the
frame of reference by P, Q, R; and then if u, v, w denote the
components of fluid velocity in space, and u´, v´, w´ the components
relative to the axes at a point (x, y, z) fixed to the frame of
reference, we have
u = U + u´ - yR + zQ, (1)
v = V + v´- zP + xR,
w = W + w´ - xQ + yP.
Now if k denotes the component of absolute velocity in a direction
fixed in space whose direction cosines are l, m, n,
k = lu + mv + nw; (2)
and in the infinitesimal element of time dt, the coordinates of the
fluid particle at (x, y, z) will have changed by (u´, v´, w´)dt; so
that
Dk dl dm dn
-- = --u + --v + --w
dt dt dt dt
/ du du du du \
+ l( -- + u´-- + v´-- + w´-- )
\ dt dx dy dz /
/ dv dv dv dv \
+ m( -- + u´-- + v´-- + w´-- )
\ dt dx dy dz /
/ dw dw dw dw \
+ n( -- + u´-- + v´-- + w´-- ). (3)
\ dt dx dy dz /
But as l, m, n are the direction cosines of a line fixed in space,
dl dm dn
-- = mR - nQ, -- = nP - lR, -- = lQ - mP; (4)
dt dt dt
so that
Dk / du du du du \
-- = l( -- - vR + wQ + u´-- + v´-- + w´-- ) + m(...) + n(...)
dt \ dt dx dy dz /
/ 1 dp \ / 1 dp \ / 1 dp \
= l( X- --- -- ) + m( Y - --- -- ) + n( Z - --- -- ), (5)
\ p dx / \ p dy / \ p dz /
for all values of l, m, n, leading to the equations of motion with
moving axes.
When the motion is such that
d[phi] d[psi] d[phi] d[psi] d[phi] d[psi]
u = - ------ - m------, v = - ------ - m------, w = - ------ - m------, (6)
dx dx dy dy dz dz
as in §25 (1), a first integral of the equations in (5) may be written
_
/ dp d[phi] d[psi] / d[phi] d[psi] \
| ----- + V + ½q² - ------ - m------ + (u - u´) ( ------ + m------ )
_/ [rho] dt dt \ dx dx /
/ d[phi] d[psi] \ / d[phi] d[psi] \
+ (v - v´)( ------ + m------ ) + (w - w´)( ------ + m------ ) = F(t), (7)
\ dy dy / \ dz dz /
in which
d[phi] d[phi] d[phi] d[phi]
------ - (u - u´)------ - (v -v´)------ - (w - w´)------
dt dx dy dz
d[phi] d[phi] d[phi] d[phi]
= ------ - (U - yR + zQ)------ - (V - zP + xR)------ - (W - xQ + yP)------ (8)
dt dx dy dz
is the time-rate of change of [phi] at a point fixed in space, which
is left behind with velocity components u - u´, v - v´, w - w´.
In the case of a steady motion of homogeneous liquid symmetrical about
Ox, where O is advancing with velocity U, the equation (5) of § 34
p/[rho] + V + ½q´² - [f]([psi]´) = constant (9)
becomes transformed into
p U d[psi]
----- + V + ½q² - --- ------ + ½U² - [f]([psi] + ½Uy²) = constant, (10)
[rho] y dy
[psi]´ = [psi] + ¼U², (11)
subject to the condition, from (4) §34,
y^(-2)[nabla]²[psi]´ = -[f]´([psi]´),
y^(-2)[nabla]²[psi] = -[f]´([psi] + ½Uy²). (12)
Thus, for example, with
[psi]´ = ¾Uy²(r²a^(-2) - 1), r² = x² + y², (13)
for the space inside the sphere r = a, compared with the value of
[psi]´ in §34 (13) for the space outside, there is no discontinuity of
the velocity in crossing the surface.
Inside the sphere
d / 1 d[psi]´\ d / 1 d[psi]´\ 15 y
2[zeta] = --- ( --- ------- ) + --- ( --- ------- ) = ---U ---, (14)
dx \ y dx / dy \ y dy / 2 a²
so that §34 (4) is satisfied, with
15 15
[f]´([psi]´)= ---Ua^(-2), [f]([psi]´) = ---U[psi]´a^(-2); (15)
2 2
and (10) reduces to
_ _
p 9 | / x² \² / y² \² |
----- + V - ---U | ( --- -1 ) - ( --- - ½ ) | = constant; (16)
[rho] 8 |_ \ a² / \ a² / _|
this gives the state of motion in M. J. M. Hill's spherical vortex,
advancing through the surrounding liquid with uniform velocity.
39. As an application of moving axes, consider the motion of liquid
filling the ellipsoidal case
x² y² z²
--- + --- + --- = 1; (1)
a² b² c²
and first suppose the liquid to be frozen, and the ellipsoid to be
rotating about the centre with components of angular velocity [xi],
[eta], [zeta]; then
u = - y[zeta] + z[eta], v = - z[xi] + x[zeta],
w = - x[eta] + y[xi]. (2)
Now suppose the liquid to be melted, and additional components of
angular velocity [Omega]1, [Omega]2, [Omega]3 communicated to the
ellipsoidal case; the additional velocity communicated to the liquid
will be due to a velocity-function
b² - c² c² - a² a² - b²
[phi] = - [Omega]1 ------- yz - [Omega]2 -------zx - [Omega]3 -------xy, (3)
b² + c² c² + a² a² + b²
as may be verified by considering one term at a time.
If u´, v´, w´ denote the components of the velocity of the liquid
relative to the axes,
2a² 2a²
u´ = u + yR - zQ = ------- [Omega]3 y - ------- [Omega]2 z, (4)
a² + b² c² + a²
2b² 2b²
v´ = v + zP - xR = ------- [Omega]1 z - ------- [Omega]3 x, (5)
b² + c² a² + b²
2c² 2c²
w´ = w + xQ - yP = ------- [Omega]2 x - ------- [Omega]1 y, (6)
c² + a² b² + c²
P = [Omega]1 + [xi], Q = [Omega]2 + [eta], R = [Omega]3 + [zeta]. (7)
Thus
x y z
u´ --- + v´ --- + w´ --- = 0, (8)
a2 b2 c2
so that a liquid particle remains always on a similar ellipsoid.
The hydrodynamical equations with moving axes, taking into account the
mutual gravitation of the liquid, become
1 dp du du du du
----- -- + 4[pi][rho]Ax + -- - vR + wQ + u´-- + v´-- + w´-- = 0, ... , ... , (9)
[rho] dx dt dx dy dz
where
_
/ [oo] abcd[lambda]
A, B, C = | ----------------------------------------------
_/ 0 (a² + [lambda], b² + [lambda], c² + [lambda])P
P² = 4(a² + [lambda]) (b² + [lambda]) (c² + [lambda]). (10)
With the values above of u, v, w, u´, v´, w´, the equations become of
the form
1 dp
----- -- + 4[pi][rho]Ax + [alpha]x + hy + gz = 0, (11)
[rho] dx
1 dp
----- -- + 4[pi][rho]By + hx + [beta]y + fz = 0, (12)
[rho] dy
1 dp
----- -- + 4[pi][rho]Cz + gx + fy + [gamma]z = 0, (13)
[rho] dz
and integrating
p[rho]^(-1) + 2[pi][rho](Ax² + By² + Cz²)
+ ½([alpha]x² + [beta]y² + [gamma]z² + 2fyz + 2gzx + 2hxy) = const., (14)
so that the surfaces of equal pressure are similar quadric surfaces,
which, symmetry and dynamical considerations show, must be coaxial
surfaces; and f, g, h vanish, as follows also by algebraical
reduction; and
4c²(c² - a²) / c² - a² \²
[alpha] = ------------[Omega]2² - ( -------[Omega]2 - [eta] )
(c² + a²)² \ c² + a² /
4b²(a² - b²) / a² - b² \²
- ------------[Omega]3² - ( -------[Omega]3 - [zeta] ), (15)
(a² + b²)² \ a² + b² /
with similar equations for [beta] and [gamma].
If we can make
(4[pi][rho]A + [alpha])x² = (4[pi][rho]B + [beta])b²
= (4[pi][rho]C + [gamma])c², (16)
the surfaces of equal pressure are similar to the external case, which
can then be removed without affecting the motion, provided [alpha],
[beta], [gamma] remain constant.
This is so when the axis of revolution is a principal axis, say Oz;
when
[Omega]1 = 0, [Omega]2 = 0, [xi] = 0, [eta] = 0. (17)
If [Omega]3 = 0 or [theta]3 = [zeta] in addition, we obtain the
solution of Jacobi's ellipsoid of liquid of three unequal axes,
rotating bodily about the least axis; and putting a = b, Maclaurin's
solution is obtained of the rotating spheroid.
In the general motion again of the liquid filling a case, when a = b,
[Omega]3 may be replaced by zero, and the equations, hydrodynamical
and dynamical, reduce to
d[xi] 2c² d[eta] 2a²
----- = - -------[Omega]2 [zeta], ------ = -------[Omega]1 [zeta],
dt a² + c² dt a² + c²
d[zeta] 2c²
------- = -------([Omega]2 [xi] - [Omega]2 [eta]) (18)
dt a² + c²
d[Omega]1 a² + c²
--------- = [Omega]2 [zeta] + -------[eta][zeta],
dt a² - c²
d[Omega]2 a² + c²
--------- = [Omega]1 [zeta] + -------[xi][zeta]; (19)
dt a² - c²
of which three integrals are
a²
[xi]² + [eta]² = L - --[zeta]², (20)
c²
(a² + c²)²
[Omega]1² + [Omega]2² = M + ------------ [zeta]², (21)
2c²(a² - c²)
a² + c²
[Omega]1 [xi] + [Omega]2 [eta]N = + ------- [zeta]²; (22)
4c²
and then
/ d[zeta]\² 4c^4
( ------- ) = --------- ([Omega]2[xi] - [Omega]1²[eta])²
\ dt / (a² + c²)
4c^4
= ---------- [([xi]² + [eta]²)([Omega)1² + [Omega]2²) -([Omega]1[xi] + [Omega]2[eta])²]
(a² + c²)²
_
4c^4 | / (a² + c²)² a² (a² + c²)\
= ---------- | LM - N² + ( L------------ - M --- - N --------- ) [zeta]²
(a² + c²)² |_ \ 2c²(a² + c²) c² 2c² /
_
(a² + c²)(9a² - c²) |
- ------------------- [zeta]^4 | = Z, (23)
16c^4(a² - c²) _|
where Z is a quadratic in [zeta]², so that [zeta] is an elliptic
function of t, except when c = a, or 3a.
Put [Omega]1 = [Omega] cos [phi], [Omega]2 = -[Omega] sin [phi],
d[phi] d[Omega]1 d[Omega]2 (a² + c²)
[Omega]2 ------ = -----------[Omega]2 - [Omega]1 --------- = [Omega]²[zeta] - ---------([Omega]1 [xi] + [Omega]2 [eta])[zeta], (24)
dt dt dt (a² - c²)
a² + c²
N + -------
d[phi] (a² + c²) 4c²
------ = [zeta] - --------- · -------------------------, (25)
dt (a² - c²) (a² + c²)²
M + ------------[zeta]²
2c²(a² - c²)
a² + c²
_ _ N + -------[zeta]²
/ [zeta]d a² + c² / 4c² [zeta] d[zeta]
[phi] = | ------- - ------- | ------------------------ · --------------, (26)
_/ [root]Z a² - c² _/ (a² + c²)² [root]Z
M + ------------[zeta]²
2c²(a² - c²)
which, as Z is a quadratic function of [zeta]², are non-elliptic
integrals; so also for [psi], where [xi] = [omega] cos [psi], [eta] =
-[omega] sin [psi].
In a state of steady motion
d[zeta] [Omega]1 [Omega]2
------- = 0, -------- = --------, (27)
dt [xi] [eta]
[phi] = [psi] = nt, suppose, (28)
[Omega]1[xi] + [Omega]2 [eta] = [Omega][omega], (29)
d[phi] a² + c² [omega]
------ = [zeta]- ------- -------[zeta], (30)
dt a² - c² [Omega]
d[psi] 2a² [Omega]
------ = - ------- -------[zeta], (31)
dt a² + c² [omega]
a² + c² [omega] 2a² [Omega]
1 - ------- ------- = - ------- -------, (32)
a² - c² [Omega] a² + c² [omega]
/ [omega] a² + c² \² (a² - c²)(9a² - c²)
( ------- - ½ ------- ) = -------------------, (33)
\ [Omega] a² - c² / 4(a² + c²)
and a state of steady motion is impossible when 3a > c > a.
An experiment was devised by Lord Kelvin for demonstrating this, in
which the difference of steadiness was shown of a copper shell filled
with liquid and spun gyroscopically, according as the shell was slightly
oblate or prolate. According to the theory above the stability is
regained when the length is more than three diameters, so that a modern
projectile with a cavity more than three diameters long should fly
steadily when filled with water; while the old-fashioned type, not so
elongated, would be highly unsteady; and for the same reason the gas
bags of a dirigible balloon should be over rather than under three
diameters long.
40. _A Liquid Jet._--By the use of the complex variable and its
conjugate functions, an attempt can be made to give a mathematical
interpretation of problems such as the efflux of water in a jet or of
smoke from a chimney, the discharge through a weir, the flow of water
through the piers of a bridge, or past the side of a ship, the wind
blowing on a sail or aeroplane, or against a wall, or impinging jets of
gas or water; cases where a surface of discontinuity is observable, more
or less distinct, which separates the running stream from the dead water
or air.
Uniplanar motion alone is so far amenable to analysis; the velocity
function [phi] and stream function [psi] are given as conjugate
functions of the coordinates x, y by
w = [f](z) where z = x + yi, w = [phi] + [psi]i, (1)
and then
dw d[phi] d[psi]
-- = ------ + i------ = -u + vi; (2)
dz dx dx
so that, with u = q cos [theta], v = q sin [theta], the function
dz Q Q Q
[zeta] = -Q -- = -------- = ---(u + vi) = --- (cos [theta] + i sin [theta]), (3)
dw (u - vi) q² q
gives [zeta] as a vector representing the reciprocal of the velocity q
in direction and magnitude, in terms of some standard velocity Q.
To determine the motion of a jet which issues from a vessel with plane
walls, the vector [zeta] must be Constructed so as to have a constant
direction [theta] along a plane boundary, and to give a constant skin
velocity over the surface of a jet, where the pressure is constant.
It is convenient to introduce the function
[Omega] = log [zeta] = log(Q/q) + [theta]i (4)
so that the polygon representing [Omega] conformally has a boundary
given by straight lines parallel to the coordinate axes; and then to
determine [Omega] and w as functions of a variable u (not to be
confused with the velocity component of q), such that in the conformal
representation the boundary of the [Omega] and w polygon is made to
coincide with the real axis of u.
[Illustration: FIG. 4.]
It will be sufficient to give a few illustrations.
Consider the motion where the liquid is coming from an infinite
distance between two parallel walls at a distance xx´ (fig. 4), and
issues in a jet between two edges A and A´; the wall xA being bent at
a corner B, with the external angle [beta] = ½[pi]/n.
The theory of conformal representation shows that the motion is given
by
_ _
| [root](b - a´·u - a) + [root](b - a·u - a´) |^1/n
[zeta] = | ------------------------------------------- | , u = ae^(-[pi]w/m); (5)
|_ [root](a - a´·u - b) _|
where u = a, a´ at the edge A, A¹; u = b at a corner B; u = 0 across
xx´ where [phi] = [oo]; and u = [oo], [phi] = [oo] across the end JJ´
of the jet, bounded by the curved lines APJ, A´P´J´, over which the
skin velocity is Q. The stream lines xBAJ, xA´J´ are given by [psi] =
0, m; so that if c denotes the ultimate breadth JJ´ of the jet, where
the velocity may be supposed uniform and equal to the skin velocity Q,
m = Qc, c = m/Q.
If there are more B corners than one, either on xA or x´A´, the
expression for [zeta] is the product of corresponding factors, such as
in (5).
Restricting the attention to a single corner B,
/ Q \^n [root](b - a´·u - a) + [root](b - a·u - a´)
[zeta]^n = ( --- ) (cos n[theta] + i sin n[theta] = -------------------------------------------, (6)
\ q / [root](a - a´.u - b)
/ Q \^n / Q \^n
ch n[omega] = ch log( --- ) cos n[theta] + i sh log ( --- ) sin n[theta]
\ q / \ q /
/b - a´ /u - a´
= ½([zeta]^n + [zeta]^(-n)) = / ------ / ------ (7)
\/ a - a´ \/ u - b´
/ Q \ / Q \^n
sh n[Omega] = sh log ( --- ) cos n[theta] + i ch log ( --- ) sin n[theta]
\ q / \ q /
/b - a´ /u - a´
= ½([zeta]^n - [zeta]^(-n)) = / ------ / ------ (8)
\/ a - a´ \/ u - b´
[oo] > a > b > 0 > a´ > -[oo] (9)
and then
d[Omega] 1 [root](b - a´·b - a´) dw m
-------- = - -- ---------------------------, -- = - ------ (10)
du 2n (u - b)[root](a - a·u - a´) du [pi]u´
the formulas by which the conformal representation is obtained.
For the [Omega] polygon has a right angle at u = a, a´, and a zero
angle at u = b, where [theta] changes from 0 to ½[pi]/n and [Omega]
increases by ½i[pi]/n; so that
d[Omega] A [root](b - a·b - a´)
-------- = ---------------------------, where A = --------------------. (11)
du (u - b)[root](u - a·u - a´) 2n
And the w polygon has a zero angle at u = 0, [oo], where [psi] changes
from 0 to m and back again, so that w changes by im, and
dw B m
-- = ---, where B = - ----. (12)
du u [pi]
Along the stream line xBAPJ,
[psi] = 0, u = ae^(-[pi][phi]/m); (13)
and over the jet surface JPA, where the skin velocity is Q,
d[phi]
------ = -q = -Q, u = ae^([pi]sQ/m) = ae^([pi]s/c), (14)
ds
denoting the arc AP by s, starting at u = a;
/b - a´ /u - a´
ch n[Omega] = cos n[theta] = / ----- / ------ (15)
\/ a - a´ \/ u - b´
/a - b /u - a´
sh n[Omega] = i sin n[theta] = i / ------ / ------ (16)
\/ a - a´ \/ u - b´
[oo] > u = ae^([pi]s/c) > a, (17)
and this gives the intrinsic equation of the jet, and then the radius
of curvature
ds 1 d[phi] i dw i dw / d[Omega]
[rho] = - -------- = --- -------- = --- ------- = --- -- / -------
d[theta] Q d[theta] Q d[Omega] Q du / du
c u - b [root](u - a·u - a´)
= ----·2n----- --------------------, (18)
[pi] u [root](a - b·b - a´)
not requiring the integration of (11) and (12)
If [theta] = [alpha] across the end JJ´ of the jet, where u = [oo], q
= Q,
/b - a´ /a - b
ch n[Omega] = cos n[alpha] = /-------, sh n[Omega] = i sin n[alpha]= / ------, (19)
\/ a - a´ \/ a - a´
Then
a - b·b - a´ a - a´
cos 2n[alpha] - cos 2n[theta] = 2------------ = ½ sin² 2n[alpha]------
a - a´·u - b u - b
[root](a - b.b - a´)[root](u - a·u - b´)
sin 2n[theta] = 2---------------------------------------- (20)
a - a´·u - b
[root](a - a·b - a´)
= sin 2n[alpha]--------------------;
u - b
2n c / b \ [root](a - b·b - a´)
----- ----- = ( 1 + ----- ) -------------------- (21)
[phi] [rho] \ u - b/ [root](u - a·u - a´)
a - a´ + (a + a´) cos 2n[alpha] - [a + a´ + (a - a´) cos 2n[alpha] cos 2n[theta]
= --------------------------------------------------------------------------------
(a - a´) sin² 2n[alpha]
cos 2n[alpha] - cos 2n[theta]
× -----------------------------
sin 2n[theta]
Along the wall AB, cos n[theta] = 0, sin n[theta] = 1,
a > u > b, (22)
/ Q \^n /b - a´ /a - u
ch n[Omega] = i sh log ( --- ) = i / ------ / ------ (23)
\ q / \/ a - a´ \/ u - b´
/ Q \^n /a - b /u - a´
sh n[Omega] = i ch log ( --- ) = i / ------ / ------ (24)
\ q / \/ a - a´ \/ u - b´
ds ds d[phi] m c Q
-- = ------ ------ = ------ = --- -- (25)
du d[phi] dt [pi]qu [pi] qu
_
AB / a Q du
[pi]-- = | --- --
c _/ b q u
_ _ _
/ | [root](a - b)[root](u - a´) + [root](b - a´)[root](a - u) |^1/n du
= | | --------------------------------------------------------- | --. (26)
_/ |_ [root](a - a´)[root](u - b´) _| u
Along the wall Bx, cos n[theta] = 1, sin n[theta] = 0,
b > u > 0 (27)
/ Q \^n /b - a´ /a - u
ch n[Omega] = ch log ( --- ) = / ------ / ------ (28)
\ q / \/ a - a´ \/ b - u´
/ Q \^n /a - b /u - a´
sh n[Omega] = sh log ( --- ) = / ------ / ------. (29)
\ q / \/ a - a´ \/ b - u
At x where [phi] = [oo], u = 0, and q = q0,
/ Q \^n /b - a´ / a /a - b / -a´
( --- ) = / ------ / --- + / ------ / ---. (30)
\ q0 / \/ a - a´ \/ b \/ a - a´ \/ q
In crossing to the line of flow x´A´P´J´, [psi] changes from 0 to m,
so that with q = Q across JJ´, while across xx´ the velocity is q0, so
that
m = q0.xx´ = Q.JJ´ (31)
_ _
JJ´ q0 | /b - a´ / a /a - b / -a´ |^1/n
-- = --- = | / ------ / --- - / ------ / --- | , (32)
xx´ Q |_ \/ a - a´ \/ b \/ a - a´ \/ q _|
giving the contraction of the jet compared with the initial breadth of
the stream.
Along the line of flow x´A´P´J´, [psi] = m, u = a´e^(-[pi][phi]/m),
and from x´ to A´, cos n[theta] = 1, sin n[theta] = 0,
/ Q \^n /b - a´ /a - u
ch n[Omega] = ch log ( --- ) = / ------ / ------, (33)
\ q / \/ a - a´ \/ b - u´
/ Q \^n /a - b /u - a´
sh n[Omega] = sh log ( --- ) = / ------ / ------. (34)
\ q / \/ a - a´ \/ b - u´
0 > u > a´. (35)
Along the jet surface A´J´, q = Q,
/b - a´ /a - u
ch n[Omega] = cos n[theta] = / ------ / ------, (36)
\/ a - a´ \/ b - u´
/a - b /a´ - u
sh n[Omega] = i sin n[theta] = i / ------ / ------, (37)
\/ a - a´ \/ b - u´
a´ > u = a´e^([pi]/sc) > -[oo], (38)
giving the intrinsic equation.
41. The first problem of this kind, worked out by H. v. Helmholtz, of
the efflux of a jet between two edges A and A1 in an infinite wall, is
obtained by the symmetrical duplication of the above, with n = 1, b =
0, a´ = -[oo], as in fig. 5,
/u - a / -a
ch [Omega] = / -----, sh [Omega] = / ---; (1)
\/ u \/ u
and along the jet APJ, [oo] > u = ae^([pi]s/c) > a,
sh [Omega] = i sin [theta] - i[root](a/u) = ie^(-½[pi]s/c), (2)
_ _
/ [oo] / c c
PM = | sin [theta] ds = | e^(-½[pi]s/c) ds = ----- e^(-½[pi]s/c) = ----- sin [theta], (3)
_/ s _/ ½[pi] ½[pi]
so that PT = c/½[pi], and the curve AP is the tractrix; and the
coefficient of contraction, or
breadth of the jet [pi]
---------------------- = --------. (4)
breadth of the orifice [pi] + 2
A change of [Omega] and [theta] into n[Omega] and n[theta] will give
the solution for two walls converging symmetrically to the orifice AA1
at an angle [pi]/n. With n = ½, the reentrant walls are given of
Borda's mouthpiece, and the coefficient of contraction becomes ½.
Generally, by making a´ = - [oo], the line x´A´ may be taken as a
straight stream line of infinite length, forming an axis of symmetry;
and then by duplication the result can be obtained, with assigned n,
a, and b, of the efflux from a symmetrical converging mouthpiece, or
of the flow of water through the arches of a bridge, with wedge-shaped
piers to divide the stream.
[Illustration: FIG. 5.]
[Illustration: FIG. 6.]
42. Other arrangements of the constants n, a, b, a´ will give the
results of special problems considered by J. M. Michell, _Phil.
Trans._ 1890.
Thus with a´ = 0, a stream is split symmetrically by a wedge of angle
[pi]/n as in Bobyleff's problem; and, by making a = [oo], the wedge
extends to infinity; then
/ b / n
ch n[Omega] = / -----, sh n[Omega] = / -----. (1)
\/ b - u \/ b - u
Over the jet surface [psi] = m, q = Q,
u = - e^([pi][phi]/m) = - be^([pi]²/c),
/ 1 / e^([pi]²/c)
ch [Omega] = cos n[theta] = / ---------------, sh [Omega] = i sin n[theta] = i / ---------------, (2)
\/ e^([pi]²/c) + 1 \/ e^([pi]²/c) + 1
½[pi] ds 2n
e^(½[pi]²/c) = tan n[theta], ----- -------- = -------------. (3)
c d[theta] sin 2n[theta]
For a jet impinging normally on an infinite plane, as in fig. 6, n = 1,
e^(½[pi]²/c) = tan [theta], ch (½[pi]s/c) sin 2[theta] = 1, (4)
sh ½[pi]x/c = cot [theta], sh ½[pi]y/c = tan [theta],
sh ½[pi]x/c sh ½[pi]y/c = 1, e^(½[pi](x + y)/c) = e^(½[pi]x/c) + e^(½[pi]y/c) + 1. (5)
With n = ½, the jet is reversed in direction, and the profile is the
catenary of equal strength.
In Bobyleff's problem of the wedge of finite breadth,
/ b /u - a /b - a / u
ch n[Omega] = / --- / -----, sh n[Omega] = / ----- / -----, (6)
\/ a \/ u - b \/ a \/ u - b
/ b /a - b
cos n[alpha] = / ---, sin n[alpha] = / -----, (7)
\/ a \/ a
and along the free surface APJ, q = Q, [psi] = 0, u = e^(-[pi][phi]/m) = ae^([pi]s/c),
/ e^([pi]²/c) - 1
cos n[theta] = cos n[alpha] / ---------------------------,
\/ e^([pi]²/c) - cos² n[alpha]
cos² n[alpha] sin² n[theta]
e^([pi]²/c) = -----------------------------, (8)
sin² n[theta] - sin² n[alpha]
the intrinsic equation, the other free surface A´P´J´ being given by
cos² n[alpha] sin² n[theta]
e^([pi]²/c) = -----------------------------. (9)
sin² n[alpha] - sin² n[theta]
Putting n = 1 gives the case of a stream of finite breadth disturbed
by a transverse plane, a particular case of Fig. 7.
When a = b, [alpha] = 0, and the stream is very broad compared with
the wedge or lamina; so, putting w = w´(a - b)/a in the penultimate
case, and
u = ae^(-w) [asympt] a - (a - b)w´, (10)
/w´ + 1 / 1
ch n[Omega] = / ------, sh n[Omega] = / --------, (11)
\/ w´ \/ [root]w´
in which we may write
w´ = [phi] + [psi]i. (12)
Along the stream line xABPJ, [psi] = 0; and along the jet surface APJ,
-1 > [phi] > -[oo]; and putting [phi] = -[pi]s/c - 1, the intrinsic
equation is
[pi]s/c = cot² n[theta], (13)
which for n = 1 is the evolute of a catenary.
[Illustration: FIG. 7.]
43. When the barrier AA´ is held oblique to the current, the stream
line xB is curved to the branch point B on AA´ (fig. 7), and so must
be excluded from the boundary of u; the conformal representation is
made now with
d[Omega] [root](b - a·b - a´)
-------- = - ---------------------------- (1)
du (u - b) [root](u - a·u - a´)
dw m 1 m´ 1 m + m´ u - b
-- = - ---- ----- - ---- -----, = - ------ · -------------,
du [pi] u - j [pi] u - j [pi] u - j·u - j´
mj´ + m´j
b = ---------, (2)
m + m´
taking u = [oo] at the source where [phi] = [oo], u = b at the branch
point B, u = j, j´ at the end of the two diverging streams where [phi]
= -[oo]; while [psi] = 0 along the stream line which divides at B and
passes through A, A´; and [psi] = m, -m´ along the outside boundaries,
so that m/Q, m´/Q is the final breadth of the jets, and (m + m´)/Q is
the initial breadth, c1 of the impinging stream. Then
/b - a´ /u - a /b - a /u - a´
ch ½[Omega] = / ------ / -----, sh ½[Omega] = / ------ / ------, (3)
\/ a - a´ \/ u - b \/ a - a´ \/ u - b
2b - a - a´ N
ch [Omega] = ----------- - -----,
a - a´ u - b
/ [root](2·a - u·u - a´)
sh [Omega] = / N----------------------,
\/ u - b
a - b·b - a´
N = 2------------. (4)
a - a´
Along a jet surface, q = Q, and
ch[Omega] = cos [theta] = cos [alpha] - ½sin² [alpha](a - a´)/(u - b), (5)
if [theta] = [alpha] at the source x of the jet xB, where u = [oo];
and supposing [theta] = [beta], [beta]´ at the end of the streams
where u = j, j´,
u - b ½ sin² [alpha] u - j cos[theta] - cos[beta]
----- = -------------------------, ------ = ½ sin² [alpha]-----------------------------------------------------,
a - a´ cos [alpha] - cos [theta] a - a´ (cos [alpha] - cos [beta])(cos [alpha] - cos [theta])
u - j´ cos [theta] - cos [beta]´
----- = ½ sin² [alpha]------------------------------------------------------; (6)
a - a´ (cos [alpha] - cos [beta]´)(cos [alpha] - cos [theta])
and [psi] being constant along a stream line
d[phi] dw ds d[phi] dw du
------ = --, Q -------- = -------- = -- --------,
du du d[theta] d[theta] du d[theta]
[pi]Q ds [pi] ds (cos [alpha] - cos [beta])(cos [alpha] - cos [beta]´)sin[theta]
------ -------- = ---- -------- = ---------------------------------------------------------------------------------,
m + m´ d[theta] c d[theta] (cos [alpha] - cos [theta])(cos [theta] - cos [beta])(cos [theta] - cos [alpha]´)
sin [theta] cos [alpha] - cos [beta]´ sin [theta]
= ------------------------- + ------------------------- · ------------------------
cos [alpha] - cos [theta] cos [beta] - cos [beta]´ cos [theta] - cos [beta]
cos [alpha] - cos [beta] sin [theta]
------------------------ · -------------------------, (7)
cos [beta] - cos [beta]´ cos [theta] - cos [beta]´
giving the intrinsic, equation of the surface of a jet, with proper
attention to the sign.
From A to B, a > u > b, [theta] = 0,
Q a - a´
ch [Omega] = ch log --- = cos [alpha] - ½ sin² [alpha] ------
q a - b
Q [root](a - u·u - a´)
sh [Omega] = sh log --- = -------------------- sin [alpha]
q u - b
Q (u - b) cos [alpha] - ½(a - a´) sin² [alpha] + [root](a - u·u - a´)sin[alpha]
--- = ----------------------------------------------------------------------------- (8)
q u - b
ds ds d[phi] Q dw
Q -- = Q ------ ------ = - --- --
du d[phi] du q du
m + m´ (u - b) cos [alpha] - ½(a - a´) sin² [alpha] + [root](a - u·u - a´) sin [alpha]
= ------ · ------------------------------------------------------------------------------- (9)
[pi] j - u·u - j´
_
AB / a (2b - a - a´)(u - b) - 2(a - b)(b - a´) + 2[root](a - b·b -a´·a - u·u -a´)
[pi]-- = | -------------------------------------------------------------------------- du, (10)
c _/ b a - a´·j - u·u - j´
with a similar expression for BA´.
The motion of a jet impinging on an infinite barrier is obtained by
putting j = a, j´ = a´; duplicated on the other side of the barrier,
the motion reversed will represent the direct collision of two jets of
unequal breadth and equal velocity. When the barrier is small compared
with the jet, [alpha] = [beta] = [beta]´, and G. Kirchhoff's solution
is obtained of a barrier placed obliquely in an infinite stream.
Two corners B1 and B2 in the wall xA, with a´ = -[oo], and n = 1, will
give the solution, by duplication, of a jet issuing by a reentrant
mouthpiece placed symmetrically in the end wall of the channel; or
else of the channel blocked partially by a diaphragm across the
middle, with edges turned back symmetrically, problems discussed by J.
H. Michell, A. E. H. Love and M. Réthy.
When the polygon is closed by the walls joining, instead of reaching
back to infinity at xx´, the liquid motion must be due to a source,
and this modification has been worked out by B. Hopkinson in the
_Proc. Lond. Math. Soc._, 1898.
Michell has discussed also the hollow vortex stationary inside a
polygon (_Phil. Trans._, 1890); the solution is given by
ch n[Omega] = sn w, sh n[Omega] = i cn w (11)
so that, round the boundary of the polygon, [psi] = K´, sin n[theta] =
0; and on the surface of the vortex [psi] = 0, q = Q, and
cos n[theta] = sn [phi], n[theta] = ½[pi] - am s/c, (12)
the intrinsic equation of the curve.
This is a closed Sumner line for n = 1, when the boundary consists of
two parallel walls; and n = ½ gives an Elastica.
44. _The Motion of a Solid through a Liquid._--An important problem in
the motion of a liquid is the determination of the state of velocity
set up by the passage of a solid through it; and thence of the
pressure and reaction of the liquid on the surface of the solid, by
which its motion is influenced when it is free.
Beginning with a single body in liquid extending to infinity, and
denoting by U, V, W, P, Q, R the components of linear and angular
velocity with respect to axes fixed in the body, the velocity function
takes the form
[phi] = U_[phi]1 + V_[phi]2 + W_[phi]3 + P_[chi]1 + Q_[chi]2 + R_[chi]3, (1)
where the [phi]'s and [chi]'s are functions of x, y, z, depending on
the shape of the body; interpreted dynamically, C - [rho][phi]
represents the impulsive pressure required to stop the motion, or C +
[rho][phi] to start it again from rest.
The terms of [phi] may be determined one at a time, and this problem
is purely kinematical; thus to determine [phi]1, the component U alone
is taken to exist, and then l, m, n, denoting the direction cosines of
the normal of the surface drawn into the exterior liquid, the function
[phi]1 must be determined to satisfy the conditions
(i.) [nabla]²[phi]1 = 0. throughout the liquid;
(ii.) d[ph]1/d[upsilon] = -l, the gradient of [phi] down the normal at
the surface of the moving solid;
(iii.) d[ph]1/d[upsilon] = 0, over a fixed boundary, or at infinity;
similarly for [phi]2 and [phi]3.
To determine [chi]1 the angular velocity P alone is introduced, and
the conditions to be satisfied are
(i.) [nabla]²[chi]1 = 0, throughout the liquid;
(ii.) d[chi]1/d[upsilon] = mz - ny, at the surface of the moving body,
but zero over a fixed surface, and at infinity; the same for [chi]2
and [chi]3.
For a cavity filled with liquid in the interior of the body, since the
liquid inside moves bodily for a motion of translation only,
[phi]1 = -x, [phi]2 = -y, [phi]3 = -z; (2)
but a rotation will stir up the liquid in the cavity, so that the
[chi]'s depend on the shape of the surface.
The ellipsoid was the shape first worked out, by George Green, in his
_Research on the Vibration of a Pendulum in a Fluid Medium_ (1833);
the extension to any other surface will form an important step in this
subject.
A system of confocal ellipsoids is taken
x² y² z²
------------- + ------------- + ------------- = 1 (3)
a² + [lambda] b² + [lambda] c² + [lambda]
and a velocity function of the form
[phi] = x[psi], (4)
where [psi] is a function of [lambda] only, so that [psi] is constant
over an ellipsoid; and we seek to determine the motion set up, and the
form of [psi] which will satisfy the equation of continuity.
Over the ellipsoid, p denoting the length of the perpendicular from
the centre on a tangent plane,
px py pz
l = -------------, m = -------------, n = ------------- (5)
a² + [lambda] b² + [lambda] c² + [lambda]
p²x² p²y² p²z²
1 = ---------------- + ---------------- + ----------------, (6)
(a² + [lambda])² (b² + [lambda])² (c² + [lambda])²
p² = (a² + [lambda])l² + (b² + [lambda])m² + (c² + [lambda])n², (7)
= a²l² + b²m² + c²n² + [lambda],
dp d[lambda]
2p-- = ---------; (8)
ds ds
Thence
d[phi] dx d[psi]
------ = --[psi] + x------
ds ds ds
dx d[psi] dp
= --[psi] + 2(a² + [lambda])--------- l--, (9)
ds d[lambda] ds
so that the velocity of the liquid may be resolved into a component
-[psi] parallel to Ox, and -2(a² + [lambda])l d[psi]/d[lambda] along
the normal of the ellipsoid; and the liquid flows over an ellipsoid
along a line of slope with respect to Ox, treated as the vertical.
Along the normal itself
d[phi] / d[psi] \
----- = ( [psi] + 2(a² + [lambda])-------- )l, (10)
ds \ d[lambda] /
so that over the surface of an ellipsoid where [lambda] and [psi] are
constant, the normal velocity is the same as that of the ellipsoid
itself, moving as a solid with velocity parallel to Ox
d[psi]
U = -[psi] - 2(a² + [lambda])---------, (11)
d[lambda]
and so the boundary condition is satisfied; moreover, any ellipsoidal
surface [lambda] may be supposed moving as if rigid with the velocity
in (11), without disturbing the liquid motion for the moment.
The continuity is secured if the liquid between two ellipsoids
[lambda] and [lambda]1, moving with the velocity U and U1 of equation
(11), is squeezed out or sucked in across the plane x = 0 at a rate
equal to the integral flow of the velocity [psi] across the annular
area [alpha]1 - [alpha] of the two ellipsoids made by x = 0; or if
_
/ [lambda]1 d[alpha]
[alpha]U - [alpha]1U1 = | [psi]-------- d[lambda], (12)
_/ [lambda] d[lambda]
[alpha] = [pi][root](b² + [lambda]·c² + [lambda]). (13)
Expressed as a differential relation, with the value of U from (11),
_ _
d | d[psi] | d[alpha]
--------- | [alpha][psi] + 2(a² + [lambda])[alpha]--------- | - [psi]--------- = 0, (14)
d[lambda] |_ d[lambda] _| d[lambda]
d[psi] d / d[psi] \
3[alpha]--------- + 2(a² + [lambda])-------- ( [alpha]-------- ) = 0, (15)
d[lambda] d[lambda] \ d[lambda]/
and integrating
d[psi]
(a² + [lambda])^3/2 [alpha]-------- = a constant, (16)
d[lambda]
so that we may put
_
/ Md[lambda]
[psi] = | ----------------, (17)
_/ (a² + [lambda])P
P² = 4(a² + [lambda])(b² + [lambda])(c² + [lambda]), (18)
where M denotes a constant; so that [psi] is an elliptic integral of
the second kind.
The quiescent ellipsoidal surface, over which the motion is entirely
tangential, is the one for which
d[psi]
2(a² + [lambda])--------- + [psi] = 0, (19)
d[lambda]
and this is the infinite boundary ellipsoid if we make the upper limit
[lambda]1 = [oo].
The velocity of the ellipsoid defined by [lambda] = 0 is then
d[psi]0
U = -2a²--------- - [psi]0
d[lambda]
_
M / [oo] Md[lambda]
= --- - | ----------------
abc _/ 0 (a² + [lambda])P
M
= --- (1 - A0), (20)
abc
with the notation
_
/ [oo] abc d[lambda]
A or A_[lambda] = | ----------------
_/[lambda] (a² + [lambda])P
_
d / [oo] d[lambda]
= -2abc--- | ---------, (21)
da²_/ [lambda] P
so that in (4)
M UxA xA_[lambda]
[phi] = ---xA = ------, [phi]1 = -----------, (22)
abc 1 - A0 1 - A0
in (1) for an ellipsoid.
The impulse required to set up the motion in liquid of density [rho]
is the resultant of an impulsive pressure [rho][phi] over the surface
S of the ellipsoid, and is therefore
_ _ _ _
/ / / /
| | [rho][phi]l dS = [rho][psi]0 | | xl dS
_/_/ _/_/
= [rho][psi]0 (volume of the ellipsoid) = [psi]0 W´, (23)
where W´ denotes the weight of liquid displaced.
Denoting the effective inertia of the liquid parallel to Ox by
[alpha]W´. the momentum
[alpha]W´U = [psi]0W´ (24)
[psi]0 A0
[alpha] = ------ = ------; (25)
U 1 - A0
in this way the air drag was calculated by Green for an ellipsoidal
pendulum.
Similarly, the inertia parallel to Oy and Oz is
B0 C0
[beta]W´ = ------ W´, [gamma]W´ = ------ W´, (26)
1 - B0 1 - C0
_
/ [oo] abc d[lambda]
B_[lambda], C_[lambda] = | -------------------------------; (27)
_/[lambda] (b² + [lambda], c² + [lambda])P
and
A + B + C = abc/½P, A0 + B0 + C0 = 1. (28)
For a sphere
a = b = c, A0 = B0 = C0 = 1/3, [alpha] = [beta] = [gamma] = ½, (29)
so that the effective inertia of a sphere is increased by half the
weight of liquid displaced; and in frictionless air or liquid the
sphere, of weight W, will describe a parabola with vertical
acceleration
W - W´
------- g. (30)
W + ½W´
Thus a spherical air bubble, in which W/W´ is insensible, will begin
to rise in water with acceleration 2g.
45. When the liquid is bounded externally by the fixed ellipsoid
[lambda] = [lambda]1, a slight extension will give the velocity
function [phi] of the liquid in the interspace as the ellipsoid
[lambda] = 0 is passing with velocity U through the confocal position;
[phi] must now take the form x([psi] + N), and will satisfy the
conditions in the shape
_
abc /[lambda]1 abcd[lambda]
------ + | ----------------
A + B1 + C1 a1b1c1 _/ [lambda] (a² + [lambda])P
[phi] = Ux ----------------- = Ux ------------------------------------------, (1)
B0 + C0 - B1 - C1 abc /[lambda]1 abcd[lambda]
1 - ------ - | ----------------
a1b1c1 _/0 (a² + [lambda])P
and any confocal ellipsoid defined by [lambda], internal or external
to [lambda] = [lambda]1, may be supposed to swim with the liquid for
an instant, without distortion or rotation, with velocity along Ox
B_[lambda] + C_[lambda] - B1 - C1
U ----------------------------------
B0 + C0 - B1 - C1
Since - Ux is the velocity function for the liquid W´ filling the
ellipsoid [lambda] = 0, and moving bodily with it, the effective
inertia of the liquid in the interspace is
A0 + B1 + C1
----------------- W´. (2)
B0 + C0 - B1 - C1
If the ellipsoid is of revolution, with b = c,
A + 2B1
[phi] = ½Ux -------, (3)
B0 - B1
and the Stokes' current function [psi] can be written down
B - B1
[psi] = - ½Uy² -------; (4)
B0 - B1
reducing, when the liquid extends to infinity and B1 = 0, to
A B
[phi] = ½Ux --, [psi] = - ½Uy² --; (5)
B0 B0
so that in the relative motion past the body, as when fixed in the
current U parallel to xO,
/ A \ / B \
[phi]´ = ½Ux ( 1 + -- ), [psi]´ = ½Uy² ( 1 - -- ). (6)
\ B0 / \ B0 /
Changing the origin from the centre to the focus of a prolate
spheroid, then putting b² = pa, [lambda] = [lambda]´a, and proceeding
to the limit where a = [oo], we find for a paraboloid of revolution
p B p
B = ½ -------------, -- = -------------, (7)
p + [lambda]´ B0 p + [lambda]´
y²
------------- = p + [lambda]´ - 2x, (8)
p + [lambda]´
with [lambda]´ = 0 over the surface of the paraboloid; and then
[psi]´ = ½U [y² - p[root](x² + y²) + px]; (9)
[psi] = -½Up [[root](x² + y²) - x]; (10)
[phi] = -½Up log [ [root](x² + y²) + x]. (11)
The relative path of a liquid particle is along a stream line
[psi]´ = ½Uc², a constant, (12)
p²y² - (y² - c²)² p²y² - (y² - c²)²
x = -----------------, [root](x² + y²) = ----------------- (13)
2p(y² - c²) 2p(y² - c²)
a C4; while the absolute path of a particle in space will be given by
dy r - x y² - c²
-- = - ----- = -------, (14)
dx y 2py
y² - c² = a²e^(-x/p). (15)
46. Between two concentric spheres, with
a² + [lambda] = r², a² + [lambda]1 = a1², (1)
A = B = C = a³/3r³,
a³ a³ a³ a³
-- + 2 --- -- + 2 ---
r³ a1³ r³ a1³
[phi] = ½Ux -----------, [psi] = ½Uy² ----------; (2)
1 - a^4/a1² 1 - a³/a1³
and the effective inertia of the liquid in the interspace is
A0 + 2A1 a1³ + 2a³
--------- W´ = ½ --------- W´. (3)
2A0 - 2A1 a1³ - a³
When the spheres are not concentric, an expression for the effective
inertia can be found by the method of images (W. M. Hicks, _Phil.
Trans._, 1880).
The image of a source of strength [mu] at S outside a sphere of radius
a is a source of strength [mu]a/[f] at H, where OS = [f], OH = a²/f,
and a line sink reaching from the image H to the centre O of line
strength - [mu]/a; this combination will be found to produce no flow
across the surface of the sphere.
Taking Ox along OS, the Stokes' function at P for the source S is [mu]
cos PSx, and of the source H and line sink OH is [mu](a/[f]) cos PHx
and -([mu]/a)(PO - PH); so that
/ a PO - PH \
[psi] = [mu] (cos PSx + --- cos PHx - ------- ), (4)
\ [f] a /
and [psi] = -[mu], a constant, over the surface of the sphere, so that
there is no flow across.
When the source S is inside the sphere and H outside, the line sink
must extend from H to infinity in the image system; to realize
physically the condition of zero flow across the sphere, an equal sink
must be introduced at some other internal point S´.
When S and S´ lie on the same radius, taken along Ox, the Stokes'
function can be written down; and when S and S´ coalesce a doublet is
produced, with a doublet image at H.
For a doublet at S, of moment m, the Stokes' function is
d y²
m-- cos PSx = -m---; (5)
df PS³
and for its image at H the Stokes' function is
d a³ y²
m-- cos PHx = -m-- ---; (6)
df f³ PH³
so that for the combination
/a³ 1 1 \ y² / a³ f³\
[psi] = my² ( -- --- - --- ) = m-- ( --- - --- ), (7)
\f³ PH³ PS³/ f³ \PH³ PS³/
and this vanishes over the surface of the sphere.
There is ao Stokes' function when the axis of the doublet at S does
not pass through O; the image system will consist of an inclined
doublet at H, making an equal angle with OS as the doublet S, and of a
parallel negative line doublet, extending from H to O, of moment
varying as the distance from O.
A distribution of sources and doublets over a moving surface will
enable an expression to be obtained for the velocity function of a
body moving in the presence of a fixed sphere, or inside it.
The method of electrical images will enable the stream function [psi]´
to be inferred from a distribution of doublets, finite in number when
the surface is composed of two spheres intersecting at an angle
[pi]/m, where m is an integer (R. A. Herman, _Quart. Jour. of Math._
xxii.).
Thus for m = 2, the spheres are orthogonal, and it can be verified
that
/ a1³ a2³ a³ \
[psi]´ = ½Uy² ( 1 - --- - --- + -- ), (8)
\ r1³ r2³ r³ /
where a1, a2, a = a1a2/[root](a1² + a2²) is the radius of the spheres
and their circle of intersection, and r1, r2, r the distances of a
point from their centres.
The corresponding expression for two orthogonal cylinders will be
/ a1² a2² a² \
[psi]´ = Uy ( 1 - --- - --- + -- ). (9)
\ r1² r2² r² /
With a2 = [oo], these reduce to
/ a^5 \ x / a^4 \ x
[psi]´ = ½Uy² ( 1 - --- ) ---, or Uy ( 1 - --- ) ---, (10)
\ r^5 / a \ r^4 / a
for a sphere or cylinder, and a diametral plane.
Two equal spheres, intersecting at 120°, will require
_ _
| x a³ a^4(a - 2x) a³ a^4(a + 2x) |
[psi]´ = ½Uy² | --- - ---- + ----------- + ---- - ----------- |, (11)
|_ a 2r1³ 2r1^5 2r2³ 2r2^5 _|
with a similar expression for cylinders; so that the plane x = 0 may
be introduced as a boundary, cutting the surface at 60°. The motion of
these cylinders across the line of centres is the equivalent of a line
doublet along each axis.
47. The extension of Green's solution to a rotation of the ellipsoid
was made by A. Clebsch, by taking a velocity function
[phi] = xy[chi] (1)
for a rotation R about Oz; and a similar procedure shows that an
ellipsoidal surface [lambda] may be in rotation about Oz without
disturbing the motion if
/ 1 1 \ dx
( ------------ + ------------ ) [chi] + 2---------
\ a² + [lambda] b² + [lambda] / d[lambda]
R = - -----------------------------------------------------, (2)
1/(b² + [lambda] - 1/(a² = [lambda])
and that the continuity of the liquid is secured if
d[chi]
(a² + [lambda])^3/2 (b² + [lambda])^3/2 (c² + [lambda]) ½--------- = constant, (3)
d[lambda]
_
/ [oo] Nd[lambda] N B_[lambda] - A_[lambda]
[chi] = | ------------------------------- = --- . -----------------------; (4)
_/[lambda] (a² + [lambda])(b² + [lambda])P abc a² - b²
and at the surface [lambda] = 0,
/ 1 1\ N B0 - A0 N 1
( -- + -- ) --- ------- - --- ----
\a² b²/ abc a² - b² abc a²b²
R = - ----------------------------------, (5)
1/b² - 1/a²
N 1/b² - 1/a²
--- = R --------------------------, (6)
abc 1 / 1 1\ B0 - A0
---- - ( -- + -- ) -------
a²b² \a² b²/ a² - b²
(a² - b²)²/(a² + b²)
= R -------------------------------.
(a² - b²)/(a² + b²) - (B0 - A0)
The velocity function of the liquid inside the ellipsoid [lambda] = 0
due to the same angular velocity will be
[phi]1 = Rxy(a² - b²)/(a² + b²), (7)
and on the surface outside
N B0 - A0
[phi]0 = xy[chi]0 = xy--- -------, (8)
abc a² - b²
so that the ratio of the exterior and interior value of [phi] at the
surface is
[phi]0 B0 - A0
------ = -------------------------------, (9)
[phi]1 (a² - b²)/(a² + b²) - (B0 - A0)
and this is the ratio of the effective angular inertia of the liquid,
outside and inside the ellipsoid [lambda] = 0.
The extension to the case where the liquid is bounded externally by a
fixed ellipsoid [lambda] = [lambda]1 is made in a similar manner, by
putting
[phi] = xy([chi] + M), (10)
and the ratio of the effective angular inertia in (9) is changed to
a1² - b1² abc
(B0 - A0) - (B1 - A1) + --------- ------
a1² + b1² a1b1c1
--------------------------------------------------. (11)
a² - b² a1² - b1² abc
------- - --------- ------ - (B0 - A0) + (B1 - A1)
a² + b² a1² + b1² a1b1c1
Make c = [oo] for confocal elliptic cylinders; and then
_
/[oo] ab ab / /b² + [lambda] \
A[lambda] = | ----------------------------------------------------- = ------- ( 1 - / ------------- ), (12)
_/[lambda] (a² + [lambda])[root]([4·a² + [lambda]·b² + [lambda]) a² - b² \ \/ a² + [lambda] /
ab / /a² + [lambda] \
B[lambda] = ------- ( / ------------- - 1 ), C[lambda]= 0;
a² - b² \ \/ b² + [lambda] /
and then as above in § 31, with
a = c ch [alpha], b = c sh [alpha],
a1 = [root](a² + [lambda]) = c ch [alpha]1, b1 = c sh [alpha]1 (13)
the ratio in (11) agrees with § 31 (6).
As before in § 31, the rotation may be resolved into a shear-pair, in
planes perpendicular to Ox and Oy.
A torsion of the ellipsoidal surface will give rise to a velocity
function of the form [phi] = xyz[Omega], where [Omega] can be
expressed by the elliptic integrals A_[lambda], B_[lambda],
C_[lambda], in a similar manner, since
_
/ [oo]
[Omega] = L | d[lambda]/P³
_/ [lambda]
48. The determination of the [phi]'s and [chi]'s is a kinematical
problem, solved as yet only for a few cases, such as those discussed
above.
But supposing them determined for the motion of a body through a
liquid, the kinetic energy T of the system, liquid and body, is
expressible as a quadratic function of the components U, V, W, P, Q,
R. The partial differential coefficient of T with respect to a
component of velocity, linear or angular, will be the component of
momentum, linear or angular, which corresponds.
Conversely, if the kinetic energy T is expressed as a quadratic
function of x1, x2, x3, y1, y2, y3, the components of momentum, the
partial differential coefficient with respect to a momentum component
will give the component of velocity to correspond.
These theorems, which hold for the motion of a single rigid body, are
true generally for a flexible system, such as considered here for a
liquid, with one or more rigid bodies swimming in it; and they express
the statement that the work done by an impulse is the product of the
impulse and the arithmetic mean of the initial and final velocity; so
that the kinetic energy is the work done by the impulse in starting
the motion from rest.
Thus if T is expressed as a quadratic function of U, V, W, P, Q, R,
the components of momentum corresponding are
dT dT dT
x1 = --, x2 = --, x3 = --, (1)
dU dV dW
dT dT dT
y1 = --, y2 = --, y3 = --;
dP dQ dR
but when it is expressed as a quadratic function of x1, x2, x3, y1,
y2, y3,
dT dT dT
U = ---, V = ---, W = ---, (2)
dx1 dx2 dx3
dT dT dT
P = ---, Q = ---, R = ---.
dy1 dy2 dy3
The second system of expression was chosen by Clebsch and adopted by
Halphen in his _Fonctions elliptiques_; and thence the dynamical
equations follow
dx1 dT dT
X = --- - x2--- + x3---, Y = ..., Z = ..., (3)
dt dy3 dy2
dy1 dT dT dT dT
L = --- - y2--- + y3--- - x2--- + x2---, M = ..., N = ..., (4)
dt dy3 dy2 dx3 dx2
where X, Y, Z, L, M, N denote components of external applied force on
the body.
These equations are proved by taking a line fixed in space, whose
direction cosines are l, m, n, then
dl dm dn
-- = mR - nQ, -- = nP - lR, -- = lQ - mP. (5)
dt dt dt
If P denotes the resultant linear impulse or momentum in this
direction
P = lx1 + mx2 + nx3, (6)
dP dl dm dn
-- = --x1 + --x2 + --x3
dt dt dt dt
dx1 dx2 dx3
+ l--- + m--- + n---,
dt dt dt
/ dx1 \
= l ( --- - x2R + x3Q )
\ dt /
/ dx2 \
+ m ( --- - x3P + x1R )
\ dt /
/ dx3 \
+ n ( --- - x1Q + x2P )
\ dt /
= lX + mY + nZ, (7)
for all values of l, m, n.
Next, taking a fixed origin [Omega] and axes parallel to Ox, Oy, Oz
through O, and denoting by x, y, z the coordinates of O, and by G the
component angular momentum about [Omega] in the direction (l, m, n)
G = l(y1 - x2z + x3y)
+ m(y2 - x3x + x1z)
+ n(y3 - x1y + x2x). (8)
Differentiating with respect to t, and afterwards moving the fixed
origin up to the moving origin O, so that
dx dy dz
x = y = z = 0, but -- = U, -- = V, -- = W,
dt dt dt
dG / dy1 \
-- = l ( --- - y2R + y3Q - x2W + x3V )
dt \ dt /
/ dy2 \
+ m ( --- - y3P + y1R - x3U + x1W )
\ dt /
/ dy3 \
+ n ( --- - y1Q + y2P - x1V + x2U )
\ dt /
= lL + mM + nN, (9)
for all values of l, m, n.
When no external force acts, the case which we shall consider, there
are three integrals of the equations of motion
(i.) T = constant,
(ii.) x1² + x2² + x3² = F², a constant,
(iii.) x1y1 + x2y2 + x3y3 = n = GF, a constant;
and the dynamical equations in (3) express the fact that x1, x2, x3
are the components of a constant vector having a fixed direction;
while (4) shows that the vector resultant of y1, y2, y3 moves as if
subject to a couple of components
x2W - x3V, x3U - x1W, x1V - x2U, (10)
and the resultant couple is therefore perpendicular to F, the
resultant of x1, x2, x3, so that the component along OF is constant,
as expressed by (iii).
If a fourth integral is obtainable, the solution is reducible to a
quadrature, but this is not possible except in a limited series of
cases, investigated by H. Weber, F. Kötter, R. Liouville, Caspary,
Jukovsky, Liapounoff, Kolosoff and others, chiefly Russian
mathematicians; and the general solution requires the double-theta
hyperelliptic function.
49. In the motion which can be solved by the elliptic function, the
most general expression of the kinetic energy was shown by A. Clebsch
to take the form
T = ½p(x1² + x2²) + ½p´x3²
+ q(x1y1 + x2y2) + q´x3y3
+ ½r(y1² + y2²) + ½r´y3² (1)
so that a fourth integral is given by
dy3/dt = 0, y3 = constant; (2)
dx3
--- = x1(qx2 + ry2) - x2(qx1 + ry1) = r(x1y2 - x2y1), (3)
dt
1 / dx3 \²
-- ( --- ) = (x1² + x2²)(y1² + y2²) - (x1y1 + x2y2)²
r² \ dt /
= (x1² + x2²)(y1² + y2²) - (FG - x3y3)²
= (x1² + x2²)(y1² + y2² + y3² - G²) - (Gx3 - Fy3)², (4)
in which
x1² + x2² = F² - x3², x1y1 + x2y2 = FG - x3y3, (5)
r(y1² + y2²) = 2T - p(x1² + x2²) - p´x3²
- 2q(x1y1 + x2y2) - 2q´x3y3 - r´y3²
= (p - p´)x3² + 2(q - q´)x3y3 + m1, (6)
m1 - 2T - pF² - 2qFG - r1y3² (7)
so that
1 / dx3 \²
-- ( --- ) = X3 (8)
r² \ dt /
where X3 is a quartic function of x3, and thus t is given by an
elliptic integral of the first kind; and by inversion x3 is in
elliptic function of the time t. Now
(x1 - x2i)(y1 + y2i) = x1y1 + x2y2 + i(x1y2 - x2y1)
= FG - xy3y3 + i[V-]X3, (9)
y1 + y2i FG - x3y3 + i[root]X3
-------- = --------------------- , (10)
x1 + x2i x1² + x2²
d
-- (x1 + x2i) = -i[(q´ - q)x3 + r´y3] + irx3(y1 + y2i), (11)
dt
d FG - x3y3 + i[root]X3
--- log (x1 + x2i) = dti -(q´ - q)x - r´y + rx ---------------------, (12)
dti F² - x3²
d /x1 + x2i Fy3 - Gx3
--- log \ / -------- = -(q´ - q)x3 - (r´ - r)y3 - Fr---------, (13)
dti \/ x1 - x2i F² - x3²
requiring the elliptic integral of the third kind; thence the
expression of x1 + x2i and y1 + y2i.
Introducing Euler's angles [theta], [phi], [psi],
x1 = F sin [theta] sin [phi], x2 = F sin [theta] cos [phi],
x1 + x2i = iF sin [theta][epsilon]^(-[psi]i), x3 = F cos [theta]; (14)
d[psi]
sin [theta] ------ = P sin [phi] + Q cos[phi], (15)
dt
d[psi] dT dT
F sin²[theta] ------ = --- x1 + --- x2
dt dy1 dy2
= (qx1 + ry1)x1 + (qx2 + ry2)x2
= qx1² + x2²) + r (x1y1 + x2y2)
= gF² sin² [theta] + r(FG - x3y3), (16)
_
/ FG - x3y3 Fr dx3
[psi] - qFt = | --------- --------, (17)
_/ F² - x3² [root]X3
elliptic integrals of the third kind.
Employing G. Kirchhoff's expressions for X, Y, Z, the coordinates of
the centre of the body,
__ __ __
FX = y1 cos xY + y2 cos yY + y3 cos zY, (18)
__ __ __
FY = -y1 cos xX + y2 cos yX + y3 cos zX, (19)
__ __ __
G = y1 cos xZ + y2 cos yZ + y3 cos zZ, (20)
F²(X² + Y²) = y1² + y2² + y3² - G², (21)
Fy3 - Gx3 + i[root]X3
F(X + Yi) = --------------------- [epsilon]^[psi]_i. (22)
[root](F² - x3²)
Suppose x3 - F is a repeated factor of X3, then y3 = G, and
_ _
| p´ - p q´ - q |
X3 = (x3 - F)² | ------(x3 + F)² + 2------G(x3 + F) - G² |, (23)
|_ r r _|
and putting x3 - F = y,
_
/ dy \² | p´ - p q´ - q
( -- ) = r²y² | 4 ------ F² + 4 ------ FG - G²
\ dt / |_ r r
_
/ p´ - p q´ - q \ p´ - p |
+ 2 ( 2 ------ F + ------ G ) y + ------ y² |, (24)
\ r r / r _|
so that the stability of this axial movement is secured if
p´ - p q´ - q
A = 4 ------F² + 4 ------FG - G² (25)
r r
is negative, and then the axis makes r[V-](-A)/[pi] nutations per
second. Otherwise, if A is positive
_
/ dy
rt = | ----------------------
_/ y[root](A + 2By + Cy²)
1 sh^(-1) [root]A[root](A + 2By + Cy²) 1 ch^(-1) A + By
= ------- ---------------------------- = ------- ---------------, (26)
[root]A ch^(-1) y[root](B² ~ AC) [root]A sh^(-1) y[root](B² ~ AC)
and the axis falls away ultimately from its original direction.
A number of cases are worked out in the American Journal of
Mathematics (1907), in which the motion is made algebraical by the use
of the pseudo-elliptic integral. To give a simple instance, changing
to the stereographic projection by putting tan ½[theta] = x,
(Nxe[psi]i)^3/2 = (x + 1)[root]X1 + i(x - 1)[root]X2, (27)
X1
-- = ±ax^4 + 2ax³ ± 3(a + b)x² + 2bx ± b, (28)
X2
N³ = -8(a + b), (29)
will give a possible state of motion of the axis of the body; and the
motion of the centre may then be inferred from (22).
50. The theory preceding is of practical application in the
investigation of the stability of the axial motion of a submarine boat,
of the elongated gas bag of an airship, or of a spinning rifled
projectile. In the steady motion under no force of such a body in a
medium, the centre of gravity describes a helix, while the axis
describes a cone round the direction of motion of the centre of gravity,
and the couple causing precession is due to the displacement of the
medium.
In the absence of a medium the inertia of the body to translation is the
same in all directions, and is measured by the weight W, and under no
force the C.G. proceeds in a straight line, and the axis of rotation
through the C.G. preserves its original direction, if a principal axis
of the body; otherwise the axis describes a cone, right circular if the
body has uniaxial symmetry, and a Poinsot cone in the general case.
But the presence of the medium makes the effective inertia depend on the
direction of motion with respect to the external shape of the body, and
on W´ the weight of fluid medium displaced.
Consider, for example, a submarine boat under water; the inertia is
different for axial and broadside motion, and may be represented by
c1 = W + W´[alpha], c2 = W + W´[beta], (1)
where [alpha], [beta] are numerical factors depending on the external
shape; and if the C.G. is moving with velocity V at an angle [phi]
with the axis, so that the axial and broadside component of velocity
is u = V cos [phi], v = V sin [phi], the total momentum F of the
medium, represented by the vector OF at an angle [theta] with the
axis, will have components, expressed in sec. lb.,
u V v V
F cos [theta] = c1 --- = (W + W´[alpha]) --- cos [phi], F sin [theta] = c2 --- = (W + W´[beta]) --- sin [phi]. (2)
g g g g
Suppose the body is kept from turning as it advances; after t seconds
the C.G. will have moved from O to O´, where OO´ = Vt; and at O´ the
momentum is the same in magnitude as before, but its vector is
displaced from OF to O´F´.
For the body alone the resultant of the components of momentum
V V V
W --- cos [phi] and W --- sin [phi] is W --- sec. lb., (3)
g g g
acting along OO´, and so is unaltered.
But the change of the resultant momentum F of the medium as well as of
the body from the vector OF to O´F´ requires an impulse couple,
tending to increase the angle FOO´, of magnitude, in sec. foot-pounds
F·OO´·sin FOO´ = FVt sin ([theta] - [phi]), (4)
equivalent to an incessant couple
N = FV sin ([theta] - [phi])
= (F sin [theta] cos [phi] - F cos [theta] sin [phi])V
= (c2 - c1)(V²/g) sin [phi] cos [phi]
= W´([beta] - [alpha]uv/g). (5)
This N is the couple in foot-pounds changing the momentum of the
medium, the momentum of the body alone remaining the same; the medium
reacts on the body with the same couple N in the opposite direction,
tending when c2-c1 is positive to set the body broadside to the
advance.
An oblate flattened body, like a disk or plate, has c2 - c1 negative,
so that the medium steers the body axially; this may be verified by a
plate dropped in water, and a leaf or disk or rocket-stick or piece of
paper falling in air. A card will show the influence of the couple N
if projected with a spin in its plane, when it will be found to change
its aspect in the air.
An elongated body like a ship has c2-c1 positive, and the couple N
tends to disturb the axial movement and makes it unstable, so that a
steamer requires to be steered by constant attention at the helm.
Consider a submarine boat or airship moving freely with the direction
of the resultant momentum horizontal, and the axis at a slight
inclination [theta]. With no reserve of buoyancy W = W´, and the
couple N, tending to increase [theta], has the effect of diminishing
the metacentric height by h ft. vertical, where
c1 u²
Wh tan[theta] = N = (c2 - c1) -- --- tan [theta], (6)
c2 g
c2 - c1 c1 u² 1 + [alpha] u²
h = ------- -- --- = ([beta] - [alpha] ----------- ---. (7)
W c2 g 1 + [beta] g
51. An elongated shot is made to preserve its axial flight through the
air by giving it the spin sufficient for stability, without which it
would turn broadside to its advance; a top in the same way is made to
stand upright on the point in the position of equilibrium, unstable
statically but dynamically stable if the spin is sufficient; and the
investigation proceeds in the same way for the two problems (see
GYROSCOPE).
The effective angular inertia of the body in the medium is now
required; denote it by C1 about the axis of the figure, and by C2
about a diameter of the mean section. A rotation about the axis of a
figure of revolution does not set the medium in motion, so that C1 is
the moment of inertia of the body about the axis, denoted by Wk1². But
if Wk2² is the moment of inertia of the body about a mean diameter,
and [omega] the angular velocity about it generated by an impulse
couple M, and M´ is the couple required to set the surrounding medium
in motion, supposed of effective radius of gyration k´,
Wk2²[omega] = M - M´, W´k´²[omega] = M´, (1)
Wk2² + W´k´²[omega] = M, (2)
C2 = Wk2² + W´k´² = (W + W´[epsilon])k2², (3)
in which we have put k´² = [epsilon]k², where [epsilon] is a numerical
factor depending on the shape.
If the shot is spinning about its axis with angular velocity p, and is
preceding steadily at a rate [mu] about a line parallel to the
resultant momentum F at an angle [theta], the velocity of the vector
of angular momentum, as in the case of a top, is
C1p[mu] sin [theta] - C2[mu]² sin [theta] cos [theta]; (4)
and equating this to the impressed couple (multiplied by g), that is,
to
c1
gN = (c1 - c2)-- u² tan [theta], (5)
c2
and dividing out sin[theta], which equated to zero would imply perfect
centring, we obtain
c1
C2[mu]² cos [theta] - C1p[mu] + (c2 - c1)-- u² sec [theta] = 0. (6)
c2
The least admissible value of p is that which makes the roots equal of
this quadratic in [mu], and then
C1
[mu] = ½ --p sec [theta], (7)
C2
the roots would be imaginary for a value of p smaller than given by
c1
C1²p² - 4(c2 - c1)-- C2u² = 0, (8)
c2
p² c1 C2
-- = 4 (c2 - c1) -- ---. (9)
u² c2 C1²
_Table of Rifling for Stability of an Elongated Projectile, x Calibres
long, giving [delta] the Angle of Rifling, and n the Pitch of Rifling
in Calibres._
+-----------------------+-----------------+-----------------+-----------------+-----------------+
| | Cast-iron Common| Palliser Shell | Solid Steel | Solid Lead |
| | Shell [f] = 2/3,| [f] = ½, S.G. 8.| Bullet | Bullet [f] = 0, |
| | S.G. 7.2. | | [f] = 0, S.G. 8.| S.G. 10.9. |
+--------+--------------+--------+--------+--------+--------+--------+--------+--------+--------+
| x |[beta]-[alpha]|[delta] | n |[delta] | n |[delta] | n |[delta] | n |
+--------+--------------+--------+--------+--------+--------+--------+--------+--------+--------+
| 1.0 | 0.0000 | 0° 0´|Infinity| 0° 0´|Infinity| 0° 0´|Infinity| 0° 0´|Infinity|
| 2.0 | 0.4942 | 2 49 | 63.87 | 2 32 | 71.08 | 2 29 | 72.21 | 2 08 | 84.29 |
| 2.5 | 0.6056 | 3 46 | 47.91 | 3 23 | 53.32 | 3 19 | 54.17 | 2 51 | 63.24 |
| 3.0 | 0.6819 | 4 41 | 38.45 | 4 13 | 42.79 | 4 09 | 43.47 | 3 38 | 50.74 |
| 3.5 | 0.7370 | 5 35 | 32.13 | 5 02 | 35.75 | 4 58 | 36.33 | 4 15 | 42.40 |
| 4.0 | 0.7782 | 6 30 | 27.60 | 5 51 | 30.72 | 5 45 | 31.21 | 4 56 | 36.43 |
| 4.5 | 0.8100 | 7 24 | 24.20 | 6 40 | 26.93 | 6 32 | 27.36 | 5 37 | 31.94 |
| 5.0 | 0.8351 | 8 16 | 21.56 | 7 28 | 23.98 | 7 21 | 24.36 | 6 18 | 28.44 |
| 6.0 | 0.8721 | 10 05 | 17.67 | 9 04 | 19.67 | 8 56 | 19.98 | 7 40 | 23.33 |
| 10.0 | 0.9395 | 16 57 | 10.31 | 15 19 | 11.47 | 15 05 | 11.65 | 13 00 | 13.60 |
|Infinity| 1.0000 | 90 00 | 0.00 | 90 00 | 0.00 | 90 00 | 0.00 | 90 00 | 0.00 |
+--------+--------------+--------+--------+--------+--------+--------+--------+--------+--------+
If the shot is movin
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