Historical.—The fundamental principles of hydrostatics were first given by Archimedes in his work Περὶ τῶν ὀχουμένων, 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 116 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.

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 ℔ 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 ℔ per sq. ft., or P/144A ℔ per sq. in. (℔/ft.2 and ℔/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 ΔP ℔ acts on a small plane area ΔA ft.2 enclosing a point B, the pressure p at B is the limit of ΔP/ΔA; and

p = lt (ΔP/ΔA) = dP/dA,

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.

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 ℔ is applied to one piston of area A ft.2, it will be balanced by a thrust W ℔ applied to the other piston of area B ft.2, where

p = P/A = W/B,

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.

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 α ft.2, the pressure at A and By by p0 and p ℔/ft.2, and by w the density of the liquid estimated in ℔/ft.3,

pα − p0α = wα·AB,

p = w·AB + p0.

Thus in water, where w = 62.4℔/ft.3, the pressure increases 62.4 ℔/ft.2, or 62.4 ÷ 144 = 0.433 ℔/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 σ rises to the height h and of density ρ 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 σh + p0 and ρk + p0, and these being equal we have

σh = ρk.

The principle is illustrated in the article Barometer, where a column of mercury of density σ and height h, rising in the tube to the Torricellian vacuum, is balanced by a column of air of density ρ, 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 = σ/ρ = 800 × 13.6 = 10,880;

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 σh at a depth h ft. in liquid of density σ 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 118 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).

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 1⁄3πy3, 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 ρ, 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,

∫ ∫ lpdS = ∫ ∫ ∫ρX dx dy dz,

where l, m, n denote the direction cosines of the normal drawn outward of the surface S.

But by Green’s transformation

∫ ∫ lp dS = ∫ ∫ ∫	dp	dx dy dz,
	dx

thus leading to the differential relation at every point

dp	= ρX,	dp	= ρY,	dp	= ρZ.
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
ρ		dx		ρ		dy		ρ		dz

are the partial differential coefficients of some function P, = ∫ dp/ρ, 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,
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	= 0,	1		dp	= 0,	1		dp	= 1,
ρ		dx		ρ		dy		ρ		dz

P = ∫ dp/ρ = z + a constant.

When the density ρ is uniform, this becomes, as before in (2) § 9

p = ρz + p0.

Suppose the density ρ varies as some nth power of the depth below O, then

dp/dz = ρ = μzn

p = μ	zn+1	=	ρz	=	ρ	(	ρ	)	1/n	,
	n + 1		n + 1		n + 1		μ

supposing p and ρ to vanish together.

These equations can be made to represent the state of convective equilibrium of the atmosphere, depending on the gas-equation

p = ρk = R ρθ,

where θ denotes the absolute temperature; and then

R	dθ	=	d	(	p	) =	1	,
	dz		dz		ρ		n + 1

so that the temperature-gradient dθ/dz is constant, as in convective equilibrium in (11).

From the gas-equation in general, in the atmosphere

1

dp

=

1

dp

−

1

dθ

=

ρ

−

1

dθ

=

1

−

1

dθ

,

ρ

dz

p

dz

θ

dz

p

θ

dz

k

θ

dz

which is positive, and the density ρ diminishes with the ascent, provided the temperature-gradient dθ/dz does not exceed θ/k.

With uniform temperature, taking k constant in the gas-equation,

dp/dz = ρ = p/k,   p = p0ez/k,

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 = loge (p2/p1) = 2.3 log10 (p2/p1).

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

ρ/ρ0 = (θ/θ0)n, p/p0 = (θ/θ0)n + 1,

dz	=	1		dp	= (n + 1)	p	= (n + 1) R, γ = 1 +	1	,
dθ		ρ		dθ		ρθ		n

where γ 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	= (n + 1)	p0		dθ	= −	a2	dr,
ρ		ρ0		θ0		r2

gravity varying inversely as the square of the distance r from the centre; so that, k = p0/ρ0, denoting the height of the homogeneous atmosphere at the surface, θ is given by

(n + 1) k (1 − θ/θ0) = a(1 − a/r),

or if c denotes the distance where θ = 0,

θ	=	a	·	c − r	.
θ0		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 (ρ − ρ0), or ρ/ρ0 = 1 + (p − p0)/λ, λ = kρ0,

so that λ is the pressure due to a head k of the liquid at density ρ0 under atmospheric pressure p0; and it is the gauge pressure required on this law to double the density. Then

dp/dz = kdρ/dz = ρ,   ρ = ρ0ez/k,   p − p0 = kρ0 (ez/k − 1);

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

kez/k − k − z = ½z2/k, when k is large.

For sea water, λ 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 λ,

dp = λdρ/ρ,   (p − p0)/λ = log (ρ/ρ0),

dp

=

λ

dρ

= ρ,   λ (

1

−

1

) = z,   1 −

ρ0

=

z

,   λ = kρ0,

zd

ρ

dz

ρ0

ρ

ρ

k

119

and the lowering of the surface is

p − p0	− z = k log	ρ	− z = −k log ( 1 −	z	) − z ≈	z2
ρ0		ρ0		k		2k

as before in (17).

Thus if the plane is normal to Oz, the resultant thrust

R = ∫ ∫ p dx dy,

and the coordinates x, y of the C.P. are given by

xR = ∫ ∫ xp dx dy,   yR = ∫ ∫ yp dx dy.

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 α + y sin α − h = 0,

R = ∫ ∫ ρ (h − x cos α − y sin α) dx dy,

xR = ∫ ∫ ρx (h − x cos α − y sin α) dx dy,

yR = ∫ ∫ ρy (h − x cos α − y sin α) dx dy.

Placing the new origin at the C.G. of the area A,

∫ ∫ xd x dy = 0, ∫ ∫ y dx dy = 0,

R = ρhA,

xhA = −cos α ∫ ∫ x2 dA − sin α ∫ ∫ xy dA,

yhA = −cos α ∫ ∫ xy dA − sin α ∫ ∫ y2 dA.

Turning the axes to make them coincide with the principal axes of the area A, thus making ∫∫ xy dA = 0,

xh = −a2 cos α, yh = −b2 sin α,

where

∫ ∫ x2dA = Aa2,   ∫ ∫ y2dA = Ab2,

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.

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 θ or a slope of 1 in m,

GM = GG1 cot θ = mc (P/W),

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

GM =	240	× 42 ×	20	2.24 ft.
	10		9000

also

cot θ = 24, θ = 2°24′.

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 ΣydA tanθ, so that there is no change of displacement if ΣydA = 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

Σwy dA tan θ·y = w tan θ Σ y2 dA = w tan θ·Ak2 ft. tons,

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 θ,

BM = BB1 cot θ = Ak2/V,

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 θ 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 θ is given by W.BM. tanθ; but if weights inside the ship are raised to bring G above B, the righting couple is diminished by W·BG.tanθ, so that the resultant righting couple is W·GM·tanθ. 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 120 the movement of a weight P through a distance c will cause the ship to heel through an angle θ 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,

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′

through an angle θ or a slope of one in m, given by

sin θ =	1	=	P	=	P	·	V	FQ sin QFF′
	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.

In a straight uniform current of fluid of density ρ, 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 θ with the velocity, is ρAq cos θ, the product of the density ρ, the area A, and q cos θ 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 θ 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, = flux across the surface into the interior = − ∫∫ ρq cos θ 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,

∫∫∫ (	dξ	+	dη	+	dζ	) dx dy dz = ∫∫ (lξ + mη + nζ) dS,
	dx		dy		dz

or individually

∫∫∫	dξ	dx dy dz = ∫∫ lξ dS, ...,
	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 ξ, η, ζ any continuous functions of x, y, z.

The integral equation of continuity (1) may now be written

∫∫∫	dρ	dx dy dz = ∫∫ (lρu + mρv + nρw) dS = 0,
	dt

which becomes by Green’s transformation

∫∫∫ (	dρ	+	d(ρu)	+	d(ρv)	+	d(ρw)	) dx dy dz = 0,
	dt		dx		dy		dz

leading to the differential equation of continuity when the integration is removed.

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

− ∫∫ ρuq cos θ dS = − ∫∫ (lρu2 + mρuv + nρuw) dS,

which by Green’s transformation is

− ∫∫∫ (	d(ρu2)	+	d(ρuv)	+	d(ρuw)	) dx dy dz.
	dx		dy		dz

The rate of generation of momentum in the interior of S by the component of force, X per unit mass, is

∫∫∫ ρX dx dy dz,

and by the pressure at the surface S is

− ∫∫ lp dS = − ∫∫∫	dp	dx dy dz,
	dx

by Green’s transformation.

The time rate of increase of momentum of the fluid inside S is

∫∫∫	d(ρu)	dx dy dz;
	dt

and (5) is the sum of (1), (2), (3), (4), so that

∫∫∫ (	dρu	+	dρu2	+	dρuv	+	dρuw	− ρX +	dp	) dx dy dz = 0,
	dt		dx		dy		dz		dx

leading to the differential equation of motion

dρu	+	dρu2	+	dρuv	+	dρuw	= ρX −	dp	,
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

dρu	+	dρu2	+	dρuv	+	dρuw
dt		dx		dy		dz

= ρ (	du	+ u	du	+ v	du	+ w	du	)
	dt		dx		dy		dz

+ u (	dρ	+	dρu	+	dρv	+	dρw	),
	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	+ u	du	+ v	du	+ w	du	= X −	1		dp	,
dt		dx		dy		dz		ρ		dx

with the two others

dv	+ u	dv	+ v	dv	+ w	dv	= Y −	1		dp	,
dt		dx		dy		dz		ρ		dy

dw	+ u	dw	+ v	dw	+ w	dw	= Z −	1		dp	.
dt		dx		dy		dz		ρ		dz

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	= lt·	F(x + uδt, y + vδt, z + wδt, t + δt) − F(x, y, z, t)
dt		δt

=	dF	+ u	dF	+ v	dF	+ w	dF	;
	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

represent the rate of change of F at the time t, at the point, x, y, z, fixed in space.

121

The components of acceleration of a particle of fluid are consequently

Du	=	du	+ u	du	+ v	du	+ w	du	,
dt		dt		dx		dy		dz

Dv	=	dv	+ u	dv	+ v	dv	+ w	dv	,
dt		dt		dx		dy		dz

Dw	=	dw	+ u	dw	+ v	dw	+ w	dw	,
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	= 0, or	dF	+ u	dF	+ v	dF	+ w	dF	= 0,
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;

and putting

dw	−	dv	= 2ξ,	du	−	dw	= 2η,	dv	−	du	= 2ζ,
dy		dz		dz		dx		dx		dy

dξ	+	dη	+	dζ	= 0,
dx		dy		dz

the equations of motion may be written

du	− 2vζ + 2wη +	dH	= 0,
dt		dx

dv	− 2wξ + 2uζ +	dH	= 0,
dt		dy

dw	− 2uη + 2wξ +	dH	= 0,
dt		dz

where

H = ∫ dp/ρ + V + ½q2,

q2 = u2 + v2 + w2,

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)

Dξ

− ξ

du

− η

dw

− ζ

dv

+ ξ (

du

+

dv

+

dw

) = 0,

dt

dx

dx

dx

dx

dy

dz

and combining this with the equation of continuity

1		Dρ	+	du	+	dv	+	dw	= 0,
ρ		dt		dx		dy		dz

we have

D

(

ξ

) −

ξ

du

−

η

dv

−

ζ

dw

= 0,

dt

ρ

ρ

dx

ρ

dx

ρ

dx

with two similar equations.

Putting

ω2 = ξ2 + η2 + ζ2,

a vortex line is defined to be such that the tangent is in the direction of ω, the resultant of ξ, η, ζ, called the components of molecular rotation. A small sphere of the fluid, if frozen suddenly, would retain this angular velocity.

If ω 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 φ exists, called the velocity function, such that

u dx + v dy + w dz = −dφ,

and then the velocity in any direction is the space-decrease or downward gradient of φ.

25. But in the most general case it is possible to have three functions φ, ψ, m of x, y, z, such that

u dx + v dy + w dz = −dφ − m dψ,

as A. Clebsch has shown, from purely analytical considerations (Crelle, lvi.); and then

ξ = ½	d(ψ, m)	,   η = ½	d(ψ, m)	,   ζ = ½	d(ψ, m)	,
	d(y, z)		d(z, x)		d(x, y)

and

ξ

dψ

+ η

dψ

+ ζ

dψ

= 0,   ξ

dm

+ η

dm

+ ζ

dm

= 0,

dx

dy

dz

dx

dy

dz

so that, at any instant, the surfaces over which ψ and m are constant intersect in the vortex lines.

Putting

H −	dφ	− m	dψ	= K,
	dt		dt

the equations of motion (4), (5), (6) § 24 can be written

dK	− 2uζ + 2wη −	d(ψ,m)	= 0, ..., ...;
dx		d(x,t)

and therefore

ξ	dK	+ η	dK	+ ζ	dK	= 0.
	dx		dy		dz

Equation (5) becomes, by a rearrangement,

dK	−	dψ	(	dm	+ u	dm	+ v	dm	+ w	dm	)
dx		dx		dt		dx		dy		dz

+	dm	(	dψ	+ u	dψ	+ v	dψ	+ w	dψ	) = 0, ..., ...,
	dx		dt		dx		dy		dz

dK	−	dψ		Dm	+	dm		Dψ	= 0, ..., ...,
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 ψ 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	− 2vζ + 2wη = 0, ..., ...
dx

ξ

dK

+ η

dK

+ ζ

dK

= 0,   u

dK

+ v

dK

+ w

dK

= 0,

dx

dy

dz

dx

dy

dz

and

K = ∫ dp/ρ + V + ½q2 = H

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, ξ = 0; and in steady motion the equations reduce to

dH/dν = 2vζ − 2wη = 2qω sin θ,

where θ is the angle between the stream line and vortex line; and this holds for their projection on any plane to which dν is drawn perpendicular.

In plane motion (4) reduces to

dH	= 2qζ = q (	dQ	+	q	),
dν		dv		r

if r denotes the radius of curvature of the stream line, so that

1		dp	+	dV	=	dH	−	d ½q2	=	q2	,
ρ		dν		dν		dν		dν		r

the normal acceleration.

The osculating plane of a stream line in steady motion contains the resultant acceleration, the direction ratios of which are

u

du

+ v

du

+ w

du

=

d ½q2

− 2vζ + 2wη =

d ½q2

−

dH

, ...,

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/ρ + z + q2/2g = P/ρ + h,

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,

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,
dx		dy

so that we can put

u = −dψ/dy,   v = dψ/dx,

122

where ψ is a function of x, y, called the stream- or current-function; interpreted physically, ψ − ψ0, the difference of the value of ψ 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 dψ is the increase of ψ due to a displacement from P to P′, and k is the component of velocity normal to PP′, the flow across PP′ is dψ = k·PP′; and taking PP′ parallel to Ox, dψ = v dx; and similarly dψ= −u dy with PP′ parallel to Oy; and generally dψ/ds is the velocity across ds, in a direction turned through a right angle forward, against the clock.

In the equations of uniplanar motion

2ζ =	dv	−	du	=	d2ψ	+	d2ψ	= −∇2ψ, suppose,
	dx		dy		dx2		dy2

so that in steady motion

dH	+ ∇2ψ	dψ	= 0,	dH	+ ∇2ψ	dψ	= 0,	dH	+ ∇2ψ = 0,
dx		dx		dy		dy		dψ

and ∇2ψ must be a function of ψ.

If the motion ia irrotational,

u = −	dφ	= −	dψ	, v = −	dφ	=	dψ	,
	dx		dy		dy		dx

so that ψ and φ are conjugate functions of x and y,

φ + ψi = ƒ(x + yi), ∇2ψ = 0, ∇2φ = 0;

or putting

φ + ψi = w, x + yi = z, w = ƒ(z).

The curves φ = constant and ψ = constant form an orthogonal system; and the interchange of φ and ψ 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 2πm, the velocity must be m/r, and

φ = m log r, ψ = mθ, φ + ψi = m log reiθ, w = m log z.

Interchanging these values

ψ = m log r,   φ = mθ,   ψ + φi = m log reiθ

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

ψ = Σm log r = log Πrm;

the path of a vortex is obtained by equating the value of ψ at the vortex to a constant, omitting the rm 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 2θ = 2a, or x−2 + y−2 = a−2;

this is therefore the path of a single vortex in a right-angled corner; and generally, if the angle of the corner is π/n, the path is the Cotes’ spiral

r sin nθ = na.

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 2π (a2 − c2)/m.

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 θ and longitude λ can be written

dφ	=	1		dψ	,	1		dψ	= −	dψ	,
dθ		sin θ		dλ		sin θ		dλ		dθ

and then

φ + ψi = F (tan ½θ·eλi).

28. Uniplanar Motion of a Liquid due to the Passage of a Cylinder through it.—A stream-function ψ must be determined to satisfy the conditions

∇2ψ = 0, throughout the liquid;

ψ = constant, over any fixed boundary;

dψ/ds = normal velocity reversed over a solid boundary,

so that, if the solid is moving with velocity U in the direction Ox, dψ/ds = −U dy/ds, or ψ + Uy = constant over the moving cylinder; and ψ + Uy = ψ′ is the stream function of the relative motion of the liquid past the cylinder, and similarly ψ − Vx for the component velocity V along Oy; and generally

ψ′ = ψ + Uy − Vx

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,

dψ/ds = normal velocity reversed

= −Rx	dx	− Ry	dy	,
	ds		ds

ψ + ½R (x2 + y2) = ψ′,

a constant over the boundary; and ψ′ is the current-function of the relative motion past the cylinder, but now

V2ψ′ + 2R = 0,

throughout the liquid.

Inside an equilateral triangle, for instance, of height h,

ψ′ = −2Rαβγ/h,

where α, β, γ are the perpendiculars on the sides of the triangle.

In the general case ψ′ = ψ + 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

ω = U (z + a2/z),

so that

ψ = U ( r +	a2	) cos θ = U ( 1 +	a2	) x,
	r		r2

φ = U ( r +	a2	) sin θ = U ( 1 +	a2	) y.
	r		r2

Then ψ = 0 over the cylinder r = a, which may be considered a fixed post; and a stream line past it along which ψ = Uc, a constant, is the curve

( r −	a2	) sin θ = c, (x2 + y2) (y − c) − a2y = 0
	r

a cubic curve (C3).

Over a concentric cylinder, external or internal, of radius r = b,

ψ′ = ψ + U1y = [ U ( 1 −	a2	) + U1] y,
	b2

and ψ′ is zero if

U1/U = (a2 − b2)/b2;

so that the cylinder may swim for an instant in the liquid without distortion, with this velocity U1, and ω 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 = ∞; and when b = ∞, 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 ω = −Uz, we are left with

ω = Ua2/z,

φ = U (a2/r) cos θ = Ua2x/(x2 + y2),

ψ = −U (a2/r) sin θ = −Ua2y/(x2 + y2),

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 θ′ with Ox,

tan θ′ =	dφ	/	dφ	=	2xy	= tan 2θ,   θ = ½θ′,
	dy		dx		x2 − y2

and the velocity is Ua2/r2.

Along the path of a particle, defined by the C3 of (3),

sin2 ½θ′ =	y2	=	y (y − c)	,
	x2 + y2		a2

½ sin θ′	dθ′	=	2y − c		dy	,
	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 φ1 denotes the velocity function of the liquid filling the cylinder r = b, and moving bodily with it with velocity U1,

φ1 = −U1x,

and over the separating surface r = b

φ	= −	U	( 1 +	a2	) =	a2 + b2	,
φ1		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.

123

With liquid of density ρ, this gives rise to a kinetic reaction to acceleration dU/dt, given by

πρb2	a2 + b2		dU	=	a2 + b2	M′	dU	,
	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 = ∞, the extra inertia is M′.

When the cylinder r = a is moved with velocity U and r = b with velocity U1 along Ox,

φ = U	a2	(	b2	+ r ) cos θ − U1	b2	( r +	a2	) cos θ,
	b2 − a2		r		b2 − a2		r

ψ = −U	a2	(	b2	− r ) sin θ − U1	b2	( r −	a2	) sin θ,
	b2 − a2		r		b2 − a2		r

and similarly, with velocity components V and V1 along Oy

φ = V	a2	(	b2	+ r ) cos θ − V1	b2	( r +	a2	) cos θ,
	b2 − a2		r		b2 − a2		r

ψ = V	a2	(	b2	− r ) sin θ + V1	b2	( r −	a2	) sin θ,
	b2 − a2		r		b2 − a2		r

and then for the resultant motion

w = (U2 + V2)	a2		z	+	a2b2		U + Vi
	b2 − a2		U + Vi		b2 − a2		z

−(U12 + V12)	b2		z	−	a2b2		U1 + V1i	.
	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),

X = ∫ ρφ cos θ·a dθ = πρa2 ( U	b2 + a2	− U1	2b2	);
	b2 − a2		b2 − a2

and over r = b

X1 = ∫ ρφ cos θ·b dθ = πρb2 ( U	2a2	− U1	b2 + a2	),
	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

X1 = 0,	V1	=	2a2	,   X = πρa2U	b2 − a2	.
	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 σ,

X = −πρa2U,	U1	=	2ρb2	,
	U		ρ (b2 + a2) + σ (b2 − a2)

U − U1	=	(ρ − σ) (b2 − a2)	,
U1		ρ (b2 + a2) + σ (b2 − a2)

and the inside cylinder starts forward or backward with respect to the outside cylinder, according as ρ > or < σ.

30. The expression for ω in (1) § 29 may be increased by the addition of the term

im log z = −mθ + im log r,

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 ρm dA, and of the whole vortex is ρmπ (b2 − a2).

Any circular filament can be started from rest by the application of a circumferential impulse πρm dr at each end of a diameter; so that a mechanism attached to the cylinders, which can set up a uniform distributed impulse πρm across the two parts of a diameter in the liquid, will generate the vortex motion, and react on the cylinder with an impulse couple −ρmπa2 and ρmπb2, having resultant ρmπ (b2 − a2), and this couple is infinite when b = ∞, 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 θ + m/a;

and the loss of head due to this increase of velocity from U to q′ is

q′2 − U2	=	(2U sin θ + m/a)2 − U2	,
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 θ/ga, and its thrust is 2πρmU 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 2πρmV 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.) − 2πρmV, 2πρmU, due to the vortex motion;

(ii.) − πρa2 dU/dt, − πρa2 dV/dt, due to the kinetic reaction of the liquid;

(iii.) 0, −π(σ − ρ) a2g, due to gravity,

taking Oy vertically upward, and denoting the density of the cylinder by σ; so that the equations of motion are

πρa2	dU	= − πρa2	dU	− 2πρmV,
	dt		dt

πρa2	dV	= − πρa2	dV	+ 2πρmV − π (σ − ρ) a2g,
	dt		dt

or, putting m = a2ω, so that the vortex velocity is due to an angular velocity ω at a radius a,

(σ + ρ) dU/dt + 2ρωV = 0,

(σ + ρ) dV/dt − 2ρωU + (σ-ρ) g = 0.

Thus with g = 0, the cylinder will describe a circle with angular velocity 2ρω/(σ + ρ), so that the radius is (σ + ρ) v/2ρω, if the velocity is v. With σ = 0, the angular velocity of the cylinder is 2ω; 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 σ 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

ρ dy (U − Ua2r−2 cos 2θ + mr−1 sin θ) (Ua2r−2 sin 2θ + mr−1 cos θ) + ρ dy ( −U + Ua2r−2 cos 2θ + mr−1 sin θ) (Ua2r−2 sin 2θ − mr−1 cos θ) = 2ρdymUr−1 (cos θ − a2r−2 cos 3θ),

and with y = b tan θ, r = b sec θ, this is

2ρmU dθ (1 − a2b−2 cos 3θ cos θ),

and integrating between the limits θ = ±½π, the resultant, as before, is 2πρmU.

31. Example 2.—Confocal Elliptic Cylinders.—Employ the elliptic coordinates η, ξ, and ζ = η + ξi, such that

z = c ch ζ, x = c ch η cos ξ, y = c sh η sin ζ;

then the curves for which η and ξ are constant are confocal ellipses and hyperbolas, and

J =	d(x, y)	= c2 (ch2 η − cos2 ξ)
	d(η, ξ)

= (1/2)c2 (ch 2η − cos 2ξ) = r1r2 = OD2,

if OD is the semi-diameter conjugate to OP, and r1, r2 the focal distances,

r1, r2 = c (ch η ± cos ξ);

r2 = x2 + y2 = c2 (ch2 η − sin2 ξ)

= ½c2 (ch 2η + cos 2ξ).

Consider the streaming motion given by

w = m ch (ζ − γ), γ = α + βi,

φ = m ch (η − α) cos (ξ − β), ψ = m sh (η − α) sin (ξ − β).

Then ψ = 0 over the ellipse η = α, and the hyperbola ξ = β, so that these may be taken as fixed boundaries; and ψ is a constant on a C4.

Over any ellipse η, moving with components U and V of velocity,

ψ′ = ψ + Uy − Vx = [ m sh (η − α) cos β + Uc sh η ] sin ξ
- [ m sh (η − α) sin β + Vc ch η ] cos ξ;

so that ψ′ = 0, if

U = −	m		sh (η − α)	cos β, V = −	m		sh (η − α)	sin β,
	c		sh η		c		ch η

having a resultant in the direction PO, where P is the intersection of an ellipse η with the hyperbola β; and with this velocity the ellipse η can be swimming in the liquid, without distortion for an instant.

At infinity

U = −	m	e−a cos β = −	m	cos β,
	c		a − b

V = −	m	e−a sin β = −	m	sin β,
	c		a − b

a and b denoting the semi-axes of the ellipse α; so that the liquid is streaming at infinity with velocity Q = m/(a + b) in the direction of the asymptote of the hyperbola β.

An ellipse interior to η = α will move in a direction opposite to the exterior current; and when η = 0, U = ∞, but V = (m/c) sh α sin β.

Negative values of η 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.

124

The components of the liquid velocity q, in the direction of the normal of the ellipse η and hyperbola ξ, are

−mJ−1 sh (η − α) cos (ξ − β), mJ−1 ch (η − α) sin (ξ − β).

The velocity q is zero in a corner where the hyperbola β cuts the ellipse α; and round the ellipse α the velocity q reaches a maximum when the tangent has turned through a right angle, and then

q = Qea	√(ch 2α − cos 2β)	;
	sh 2α

and the condition can be inferred when cavitation begins.

With β = 0, the stream is parallel to x0, and

φ = m ch (η − α) cos ξ
= −Uc ch (η − α) sh η cos ξ/sh (η − α)

over the cylinder η, and as in (12) § 29,

φ1 = −Ux = −Uc ch η cos ξ,

for liquid filling the cylinder; and

φ	=	th η	,
φ1		th (η − α)

over the surface of η; so that parallel to Ox, the effective inertia of the cylinder η, displacing M′ liquid, is increased by M′th η/th(η- α), reducing when α = ∞ to M′ th η = M′ (b/a).

Similarly, parallel to Oy, the increase of effective inertia is M′/th η th (η − α), reducing to M′/th η = M′ (a/b), when α = ∞, and the liquid extends to infinity.

32. Next consider the motion given by

φ = m ch 2 (η − α) sin 2ξ, ψ = −m sh 2 (η − α) cos 2ξ;

in which ψ = 0 over the ellipse α, and

ψ′ = ψ + ½R (x2 + y2) = [ −m sh 2 (η − α) + ¼Rc2 ] cos 2ξ + ¼Rc2 ch 2η,

which is constant over the ellipse η if

¼ Rc2 = m sh 2 (η − α);

so that this ellipse can be rotating with this angular velocity R for an instant without distortion, the ellipse α being fixed.

For the liquid filling the interior of a rotating elliptic cylinder of cross section

x2/a2 + y2/b2 = 1,

ψ1′ = m1 (x2/a2 + y2/b2)

with

∇2ψ1′ = −2R = −2m1 (1/a2 + 1/b2),

ψ1 = m1 (x2/a2 + y2/b2) − ½R (x2 + y2)
= −½R (x2 − y2) (a2 − b2) / (a2 + b2),

φ1 = Rxy (a2 − b2) / (a2 + b2),

w1 = φ1 + ψ1i = −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).

For the liquid in the interspace between α and η,

φ	=	m ch 2 (η − α) sin 2ξ
φ1		¼ Rc2 sh 2η sin 2ξ (a2 − b2) / (a2 + b2)

= 1/th 2 (η − α) th 2η;

and the effective k2 of the liquid is reduced to

¼ c2/th 2 (η − α) sh 2η,

which becomes ¼ c2/sh 2η = 1⁄8 (a2 − b2)/ab, when α = ∞, and the liquid surrounds the ellipse η to infinity.

An angular velocity R, which gives components −Ry, Rx of velocity to a body, can be resolved into two shearing velocities, −R parallel to Ox, and R parallel to Oy; and then ψ is resolved into ψ1 + ψ2, such that ψ1 + ½Rx2 and ψ2 + ½Ry2 is constant over the boundary.

Inside a cylinder

φ1 + ψ1i = −½ iR (x + yi)2 a2 / (a2 + b2),

φ2 + ψ2i = ½ iR (x + yi)2 b2 / (a2 + b2),

and for the interspace, the ellipse α being fixed, and α1 revolving with angular velocity R

φ1 + ψ1i = −1⁄8 iRc2 sh 2 (η − α + ξi) (ch 2α + 1) / sh 2 (α1 − α),

φ2 + ψ2i = 1⁄8 iRc2 sh 2 (η − α + ξi) (ch 2α − 1) / sh 2 (α1 − α),

satisfying the condition that ψ1 and ψ2 are zero over η = α, and over η = α1

ψ1 + ½ Rx2 = 1⁄8 Rc2 (ch 2α1 + 1),

ψ2 + ½ Ry2 = 1⁄8 Rc2 (ch 2α1 − 1),

constant values.

In a similar way the more general state of motion may be analysed, given by

w = m ch 2 (ζ − γ), γ = α + βi,

as giving a homogeneous strain velocity to the confocal system; to which may be added a circulation, represented by an additional term mζ in w.

Similarly, with

x + yi = c√[ sin (ξ + ηi) ]

the function

ψ = Qc sh ½ (η − α) sin ½ (ξ − β)

will give motion streaming past the fixed cylinder η = α, and dividing along ξ = β; and then

x2 − y2 = c2 sin ξ ch η, 2xy = c2 cos ξ sh η.

In particular, with sh α = 1, the cross-section of η = α is

x4 + 6x2y2 + y4 = 2c4, or x4 + y4 = c4

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 ψ1 is to be made to satisfy the conditions

(i.) ∇2ψ1 = 0, (ii.) ψ1 + ½Rx2 = ½Ra2, or ψ1 = 0 when x = ±a, (iii.) ψ1 + ½Rx2 = ½Ra2, ψ1 = ½R (a2 − x2), when y = ± b.

Expanded in a Fourier series,

a2 − x2 =	32	a2 Σ	cos (2n + 1) ½ πx/a	,
	π3		(2n + 1)3

so that

ψ1 = R	16	a2 Σ	cos (2n + 1) ½πx/a · ch (2n + 1) ½πy/a)	,
	π3		(2n + 1)3 · ch (2n + 1) ½πb/a

w1 = φ1 + ψ1i = iR	16	a2 Σ	cos (2n + 1) ½πz/a	,
	π3		(2n + 1)3 ch (2n + 1) ½πb/a

an elliptic-function Fourier series; with a similar expression for ψ2 with x and y, a and b interchanged; and thence ψ = ψ1 + ψ2.

Example 4.—Parabolic cylinder, axial advance, and liquid streaming past.

The polar equation of the cross-section being

r1/2 cos ½θ = a1/2, or r + x = 2a,

the conditions are satisfied by

ψ′ = Ur sin θ − 2Ua1/2r1/2 sin ½θ = 2Ur1/2 sin ½θ (r1/2 cos ½θ − a1/2),

ψ = 2Ua1/2r1/2 sin ½θ = −U √ [ 2a (r − x) ],

w = −2Ua1/2z1/2,

and the resistance of the liquid is 2πρaV2/2g.

A relative stream line, along which ψ′ = Uc, is the quartic curve

y − c = √ [ 2a (r − x) ],   x =	(4a2y2 − (y − c)4	,   r =	4a2y2 + (y − c)4	,
	4a (y − c)2		4a (y − c)2

and in the absolute space curve given by ψ,

dy	= −	(y − c)2	, x =	2ac	− 2a log (y − c).
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 ψ 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 2π (ψ − ψ0); and, as before, if dψ is the increase in ψ 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 2πdψ = 2πyk·PP′; and taking PP′ parallel to Oy and Ox,

u = −dψ/ydy,   v = dψ/ydx,

and ψ 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 dψ/yds is the component velocity across ds in a direction turned through a right angle forward.

In this symmetrical motion

ξ = 0, η = 0, 2ζ =

d

(

1

dψ

) +

d

(

1

dψ

)

dx

y

dx

dy

y

dy

=

1

(

d2ψ

+

d2ψ

−

1

dψ

) = −

1

∇2ψ,

y

dx2

dy2

y

dy

y

suppose; and in steady motion,

dH	+	1		dψ	∇2ψ = 0,	dH	+	1		dψ	∇2ψ = 0,
dx		y2		dx		dy		y2		dy

so that

2ζ/y = −y−2∇2ψ = dH/dψ

is a function of ψ, say ƒ′(ψ), and constant along a stream line;

dH/dv = 2qζ,   H − ƒ(ψ) = constant,

throughout the liquid.

When the motion is irrotational,

ζ = 0,  u = −

dφ

= −

1

dψ

,  v = −

dφ

=

1

dψ

,

dx

y

dy

dy

y

dx

∇2ψ = 0, or	d2ψ	+	d2ψ	−	1		dψ	= 0.
	dx2		dy2		y		dy

125

Changing to polar coordinates, x = r cos θ, y = r sin θ, the equation (2) becomes, with cos θ = μ,

r2	d2ψ	+ (1 − μ2)	d2ψ	= 2 ζr3 sin θ,
	dr2		dμ

of which a solution, when ζ = 0, is

ψ = ( Arn+1 +	B	) (1 − μ2)	dPn	= ( Arn − 1 +	B	) y2	dPn	,
	rn		dμ		rn+2		dμ

φ = { (n + 1) Arn − nBr−n−1 } Pn,

where Pn denotes the zonal harmonic of the nth order; also, in the exceptional case of

ψ = A0 cos θ, φ = A0/r; ψ = B0r, φ = −B0 log tan ½θ = −½B0 sh−1 x/y.

Thus cos θ is the Stokes’ function of a point source at O, and PA − PB of a line source AB.

The stream function ψ of the liquid motion set up by the passage of a solid of revolution, moving with axial velocity U, is such that

1		dψ	= −U	dy	, ψ + ½Uy2 = constant,
y		ds		ds

over the surface of the solid; and ψ must be replaced by ψ′ = ψ + 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

ψ′ = ψ + ½Uy2 = ½U (r2 − a3/r) sin2 θ, ψ = −½Ua3 sin2 θ/r;

and then

φ′ = Ux (1 + ½a3/r2), φ = ½Ua3 cos θ/r2,

−	dφ	= U	a3	cos θ,   −	dφ	= ½U	a3	sin θ,
	dr		r3		r dθ		r3

so that, if the direction of motion makes an angle ψ with Ox,

tan (ψ − θ) = ½ tan θ, tan ψ = 3 tan θ/(2 − tan2 θ),

Along the path of a liquid particle ψ′ is constant, and putting it equal to ½Uc2,

(r2 − a3/r) sin2 θ = c2, sin2 θ = c2r / (r3 − a3),

the polar equation; or

y2 = c2r3 / (r3 − a3), r3 = a3y2 / (y2 − c2),

a curve of the 10th degree (C10).

In the absolute path in space

cos ψ = (2 − 3 sin2 θ) / √ (4 − sin2 θ), and sin3 θ = (y3 − c2y) / a3,

which leads to no simple relation.

The velocity past the surface of the sphere is

1		dψ′	= ½U ( 2r +	a3	)	sin2 θ	= 3⁄2 U sin θ, when r = a;
r sin θ		dr		r2		r sin θ

so that the loss of head is

(9⁄4 sin2 θ − 1) U2/2g, having a maximum 5⁄4 U2/2g,

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

ψ = −½ Uy2a4 μ/r4,   ψ′ = ½ Uy2 (1 − a4 μ/r4),

φ′ = Ux + φ,   φ = −1⁄3 U (a4 / r3) P2,   P2 = 3⁄2 μ2 − ½,

representing a stream past the surface r4 = a4μ.

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 φ is given by

T = ½ρ ∫ ∫ ∫ [ (

dφ

)

2

+ (

dφ

)

2

+ (

dφ

)

2

] dx dy dz,

dx

dy

dz

= ½ρ ∫ ∫ φ	dφ	dS
	dν

by Green’s transformation, dν denoting an elementary step along the normal to the exterior of the surface; so that dφ/dν = 0 over the surface makes T = 0, and then

(

dφ

)

2

+ (

dφ

)

2

+ (

dφ

)

2

= 0,

dφ

= 0,

dφ

= 0,

dφ

= 0.

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 ῶ satisfies the equations

1		dῶ	= −u,	1		dῶ	= −v,	1		dῶ	= ῶ,
ρ		dx		ρ		dy		ρ		dz

ῶ = ρφ + a constant,

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 φ, the liquid can circulate in the circuits independently of any motion of the surface; thus, for example,

φ = mθ = m tan−1 y/x

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 φ in any circuit, or the value by which φ has increased in completing the circuit, the values of φ on the two sides of the membrane are taken as differing by k, so that the integral over the membrane

∫ ∫ φ	dφ	dS = k ∫ ∫	dφ	dS,
	dν		dν

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 = 2πm.

In plane motion the kinetic energy per unit length parallel to Oz

T = ½ρ ∫ ∫ [ (

dφ

)

2

+ (

dφ

)

2

] dx dy = ½ρ ∫ ∫ [ (

dψ

)

2

+ (

dψ

)

2

] dx dy

dx

dy

dx

dy

= ½ρ ∫ φ	dφ	ds = ½ρ ∫ ψ	dψ	ds.
	dν		dν

For example, in the equilateral triangle of (8) § 28, referred to coordinate axes made by the base and height,

ψ′ = −2Rαβγ/h = −½ Ry [ (h − y)2 − 3x2 ] /h

ψ = ψ′ − ½R [ ( 1⁄3 h − y)2 + x2 ] = −½R [ ½h3 + 1⁄3 h2y + h) (x2 − y2) − 3x2y + y3 ] /h

and over the base y = 0,

dx/dν = −dx/dy = + ½R ( 1⁄3 h2 − 3x2) / h, ψ = −½R ( 1⁄9 h2 + x2).

Integrating over the base, to obtain one-third of the kinetic energy T,

1⁄3T = ½ρ ∫	h / √3	¼R2 (3x4 − 1⁄27 h4) dx/h = ρR2 h4 / 135 √3
	−h / √3

so that the effective k2 of the liquid filling the triangle is given by

k2 = T / ½ρR2A = 2h2 / 45 = 2⁄5 (radius of the inscribed circle)2,

or two-fifths of the k2 for the solid triangle.

Again, since

dφ/dν = dψ/ds,   dφ/ds = −dψ/dν,

T = ½ρ ∫ φ dψ = −½ρ ∫ ψ dφ.

With the Stokes’ function ψ for motion symmetrical about an axis.

T = ½ρ ∫ φ	dψ	2πy ds = πρ ∫ φ dψ.
	y ds

37. Flow, Circulation, and Vortex Motion.—The line integral of the tangential velocity along a curve from one point to another, defined by

∫ ( u	dx	+ v	dy	+ w	dz	) ds = ∫ (u dx + v dy + z dz),
	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 φ, the flow

−∫ dφ = φ1 − φ2,

126

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 φ 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 ζ, the circulation round a curve in the plane xy is

∫ ζ ( x	dy	− y	dx	) ds = ζ times twice the area.
	ds		ds

In a fluid, the circulation round an elementary area dxdy is equal to

u dx + ( v +	dv	dx ) dy − ( u +	du	dy ) dx − vdy = (	dv	−	du	) dx dy,
	dx		dy		dx		dy

so that the component spin is

½ (	dv	−	du	) = ζ,
	dx		dy

in the previous notation of § 24; so also for the other two components ξ and η.

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, ξ, η, ζ, 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 ωα is constant where α is the cross-section of the filament and ω 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	= 0,	Dv	+	dQ	= 0,	Dw	+	dQ	= 0,
dt		dx		dt		dy		dt		dz

Q = ∫ dp/ρ + V,

and taking dx, dy, dz in the direction of u, v, w, and

dx : dy : dz = u : v : w,

D	( u dx + v dy + w dz ) =	Du	dx + u	D dx	+ ... = −dQ + ½ dq2,
dt		dt		dt

and integrating round a closed curve

D	∫ (u dx + v dy + w dz) = 0,
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 αω 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 α of a filament by dS and its mass by dm, the quantity ωdS/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 ω cosεdS/dm, if dS is the oblique section of which the normal makes an angle ε with the filament, while the aggregate vorticity of a mass M inside a surface S is

M−1 ∫ ω cos ε dS.

Employing the equation of continuity when the liquid is homogeneous,

2 (	dζ	−	dη	) = ∇2u, ... , ∇2 = −	d2	−	d2	−	d2	,
	dy		dz		dx2		dy2		dz2

which is expressed by

∇2 (u, v, w) = 2 curl (ξ, η, ζ), (ξ, η, ζ) = ½ curl (u, v, w).

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, 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;

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	u +	dm	v +	dn	w
dt		dt		dt		dt

+ l (	du	+ u′	du	+ v′	du	+ w′	du	)
	dt		dx		dy		dz

+ m (	dv	+ u′	dv	+ v′	dv	+ w′	dv	)
	dt		dx		dy		dz

+ n (	dw	+ u′	dw	+ v′	dw	+ w′	dw	).
	dt		dx		dy		dz

But as l, m, n are the direction cosines of a line fixed in space,

dl	= mR − nQ,	dm	= nP − lR,	dn	= lQ − mP;
dt		dt		dt

so that

Dk	= l (	du	− vR + wQ + u′	du	+ v′	du	+ w′	du	) + m (...) + n (...)
dt		dt		dx		dy		dz

= l ( X −

1

dp

) + m ( Y −

1

dp

) + n ( Z −

1

dp

),

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

u = −

dφ

− m

dψ

, v = −

dφ

− m

dψ

, w = −

dφ

− m

dψ

,

dx

dx

dy

dy

dz

dz

as in § 25 (1), a first integral of the equations in (5) may be written

∫	dp	+ V + ½q2 −	dφ	− m	dψ	+ (u − u′) (	dφ	+ m	dψ	)
	ρ		dt		dt		dx		dx

+ (v − v′) (	dφ	+ m	dψ	) + (w − w′) (	dφ	+ m	dψ	) = F(t),
	dy		dy		dz		dz

in which

dφ	− (u − u′)	dφ	− (v − v′)	dφ	− (w − w′)	dφ
dt		dx		dy		dz

=	dφ	− (U − yR + zQ)	dφ	− (V − zP + xR)	dφ	− (W − xQ + yP)	dφ
	dt		dx		dy		dz

is the time-rate of change of φ 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/ρ + V + ½q′2 − ƒ (ψ′) = constant

becomes transformed into

p	+ V + ½q2 −	U		dψ	+ ½U2 − ƒ (ψ + ½Uy2) = constant,
ρ		y		dy

ψ′ = ψ + ¼U y2,

subject to the condition, from (4) § 34,

y−2 ∇2ψ′ = −ƒ′(ψ′),   y−2 ∇2ψ = −ƒ′ (ψ + ½Uy2).

Thus, for example, with

ψ′ = ¾U y2 (r2a−2 − 1), r2 = x2 + y2,

for the space inside the sphere r = a, compared with the value of ψ′ in § 34 (13) for the space outside, there is no discontinuity of the velocity in crossing the surface.

Inside the sphere

2ζ =

d

(

1

dψ′

) +

d

(

1

dψ′

) =

15

U

y

,

dx

y

dx

dy

y

dy

2

a2

so that § 34 (4) is satisfied, with

ƒ′ (ψ′) =	15	Ua−2, ƒ (ψ′) =	15	Uψ′ a−2;
	2		2

and (10) reduces to

p	+ V −	9	U { (	x2	− 1 )	2	− (	y2	− ½ )	2	} = constant;
ρ		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;
a2		b2		c2

and first suppose the liquid to be frozen, and the ellipsoid to be 127 rotating about the centre with components of angular velocity ξ, η, ζ; then

u = − yζ + zη, v = − zξ + xζ, w = − xη + yξ.

Now suppose the liquid to be melted, and additional components of angular velocity Ω1, Ω2, Ω3 communicated to the ellipsoidal case; the additional velocity communicated to the liquid will be due to a velocity-function

φ = − Ω1	b2 − c2	yz − Ω2	c2 − a2	zx − Ω3	a2 − b2	xy,
	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,

u′ = u + yR − zQ =	2a2	Ω3y −	2a2	Ω2z,
	a2 + b2		c2 + a2

v′ = v + zP − xR =	2b2	Ω1z −	2b2	Ω3x,
	b2 + c2		a2 + b2

w′ = w + xQ − yP =	2c2	Ω2x −	2c2	Ω1y,
	c2 + a2		b2 + c2

P = Ω1 + ξ, Q = Ω2 + η, R = Ω3 + ζ.

Thus

u′	x	+ v′	y	+ w′	z	= 0,
	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	+ 4πρAx +	du	− vR + wQ + u′	du	+ v′	du	+ w′	du	= 0, ... , ... ,
ρ		dx		dt		dx		dy		dz

where

A, B, C = ∫∞0	abcdλ
	(a2 + λ, b2 + λ, c2 + λ) P

P2 = 4 (a2 + λ) (b2 + λ) (c2 + λ).

With the values above of u, v, w, u′, v′, w′, the equations become of the form

1		dp	+ 4πρ Ax + αx + hy + gz = 0,
ρ		dx

1		dp	+ 4πρBy + hx + βy + fz = 0,
ρ		dy

1		dp	+ 4πρCz + gx + fy + γz = 0,
ρ		dz

and integrating

pρ−1 + 2πρ (Ax2 + By2 + Cz2)
+ ½ (αx2 + βy2 + γz2 + 2fyz + 2gzx + 2hxy) = const.,

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

α =	4c2(c2 − a2)	Ω22 − (	c2 − a2	Ω2 − η )	2
	(c2 + a2)2		c2 + a2

−	4b2(a2 − b2)	Ω32 − (	a2 − b2	Ω3 − ζ )	2	,
	(a2 + b2)2		a2 + b2

with similar equations for β and γ.

If we can make

(4πρA + α) x2 = (4πρB + β) b2 = (4πρC + γ) c2,

the surfaces of equal pressure are similar to the external case, which can then be removed without affecting the motion, provided α, β, γ remain constant.

This is so when the axis of revolution is a principal axis, say Oz; when

Ω1 = 0, Ω2 = 0, ξ = 0, η = 0.

If Ω3 = 0 or θ3 = ζ 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, Ω3 may be replaced by zero, and the equations, hydrodynamical and dynamical, reduce to

dξ	= −	2c2	Ω2 ζ,	dη	=	2a2	Ω1 ζ,	dζ	=	2c2	(Ω2 ξ − Ω2 η)
dt		a2 + c2		dt		a2 + c2		dt		a2 + c2

dΩ1	= Ω2 ζ +	a2 + c2	ηζ,	dΩ2	= −Ω1 ζ −	a2 + c2	ξζ;
dt		a2 − c2		dt		a2 − c2

of which three integrals are

ξ2 + η2 = L −	a2	ζ2,
	c2

Ω12 + Ω22 = M +	(a2 + c2)2	ζ2,
	2c2 (a2 − c2)

Ω1 ξ + Ω2 ηN = +	a2 + c2	ζ2;
	4c2

and then

(	dζ	)	2	=	4c4	(Ω2ξ − Ω12η)2
	dt				(a2 + c2)

=	4c4	[ (ξ2 + η2) (Ω12 + Ω22) − (Ω1ξ + Ω2η)2 ]
	(a2 + c2)2

=	4c4	[ LM − N2 + {	(a2 + c2)2	− M	a2	− N	a2 + c2	} ζ2
	(a2 + c2)2		2c2 (a2 + c2)		c2		2c2

−	(a2 + c2) (9a2 − c2)	ζ4 ] = Z,
	16c4 (a2 − c2)

where Z is a quadratic in ζ2, so that ζ is an elliptic function of t, except when c = a, or 3a.

Put Ω1 = Ω cos φ, Ω2 = −Ω sin φ,

Ω2	dφ	=	dΩ1	Ω2 − Ω1	dΩ2	= Ω2ζ −	(a2 + c2)	(Ω1ξ + Ω2η) ζ,
	dt		dt		dt		(a2 − c2)

dφ	= ζ −	(a2 + c2)	·	N + a2 + c2 4c2	,
dt		(a2 − c2)		M + (a2 + c2)2 ζ2 2c2 (a2 − c2)

φ = ∫	ζ dζ	−	a2 + c2	∫	N + a2 + c2 4c2	·	ζ dζ	,
	√Z		a2 − c2		M + (a2 + c2)2 ζ2 2c2 (a2 − c2)		√Z

which, as Z is a quadratic function of ζ2, are non-elliptic integrals; so also for ψ, where ξ = ω cos ψ, η = −ω sin ψ.

In a state of steady motion

dζ	= 0,	Ω1	=	Ω2	,
dt		ξ		η

φ = ψ = nt, suppose,

Ω1ξ + Ω2η = Ωω,

dφ	= ζ −	a2 + c2		ω	ζ,
dt		a2 − c2		Ω

dψ	= −	2a2		Ω	ζ,
dt		a2 + c2		ω

1 −	a2 + c2		ω	= −	2a2		Ω	,
	a2 − c2		Ω		a2 + c2		ω

(	ω	− ½	a2 + c2	)	2	=	(a2 − c2) (9a2 − c2)	,
	Ω		a2 − c2				4 (a2 + c2)

and a state of steady motion is impossible when 3a > c > a.

Uniplanar motion alone is so far amenable to analysis; the velocity function φ and stream function ψ are given as conjugate functions of the coordinates x, y by

w = ƒ(z) where z = x + yi, w = φ + ψi,

and then

dw	=	dφ	+ i	dψ	= −u + vi;
dz		dx		dx

so that, with u = q cos θ, v = q sin θ, the function

ζ = −Q	dz	=	Q	=	Q	(u + vi) =	Q	(cos θ + i sin θ),
	dw		(u − vi)		q2		q

gives ζ 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 ζ must be Constructed so as to have a constant 128 direction θ 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

Ω = log ζ = log (Q/q) + θi

Fig. 4.

so that the polygon representing Ω conformally has a boundary given by straight lines parallel to the coordinate axes; and then to determine Ω 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 Ω and w polygon is made to coincide with the real axis of u.

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 β = ½π/n.

The theory of conformal representation shows that the motion is given by

ζ = [	√ (b − a′·u − a) + √(b − a·u − a′)	]	1/n	, u = ae−πw/m;
	√ (a − a′·u − b)

where u = a, a′ at the edge A, A′; u = b at a corner B; u = 0 across xx′ where φ = ∞; and u = ∞, φ = ∞ 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 ψ = 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 ζ is the product of corresponding factors, such as in (5).

Restricting the attention to a single corner B,

ζn = (	Q	)	n	(cos nθ + i sin nθ) =	√ (b − a′·u − a) + √ (b − a·u − a′)	,
	q				√ (a − a′·u − b)

ch nω = ch log (	Q	)	n	cos nθ + i sh log (	Q	)	n	sin nθ
	q				q

= ½(ζn + ζ−n) = √	b − a′	√	u − a	,
	a − a′		u − b

sh nΩ = sh log (	Q	) cos nθ + i ch log (	Q	)	n	sin nθ
	q		q

= ½(ζn + ζ−n) = √	b − a	√	u − a′	,
	a − a′		u − b

∞ > a > b > 0 > a′ > −∞

and then

dΩ	= −	1		√ (b − a′·b − a′)	,	dw	= −	m	,
du		2n		(u − b) √ (a − a·u − a′)		du		πu

the formulas by which the conformal representation is obtained.

For the Ω polygon has a right angle at u = a, a′, and a zero angle at u = b, where θ changes from 0 to ½π/n and Ω increases by ½iπ/n; so that

dΩ	=	A	, where A =	√ (b − a·b − a′)	.
du		(u − b) √ (u − a·u − a′)		2n

And the w polygon has a zero angle at u = 0, ∞, where ψ changes from 0 to m and back again, so that w changes by im, and

dw	=	B	, where B = −	m	.
du		u		π

Along the stream line xBAPJ,

ψ = 0,   u = ae−πφ/m;

and over the jet surface JPA, where the skin velocity is Q,

dφ	= −q = −Q,   u = aeπsQ/m = aeπs/c,
ds

denoting the arc AP by s, starting at u = a;

ch nΩ = cos nθ = √	b − a′	√	u − a	,
	a − a′		u − b

sh nΩ = i sin nθ = i √	a − b	√	u − a′	,
	a − a′		u − b

∞ > u = aeπs/c > a,

and this gives the intrinsic equation of the jet, and then the radius of curvature

ρ = −

ds

=

1

dφ

=

i

dw

=

i

dw

/

dΩ

dθ

Q

dθ

Q

dΩ

Q

du

du

=	c	·	u − b		√ (u − a·u − a′)	,
	π		u		√ (a − b·b − a′)

not requiring the integration of (11) and (12)

If θ = α across the end JJ′ of the jet, where u = ∞, q = Q,

ch nΩ = cos nα = √	b − a′	, sh nΩ = i sin nα= i √	a − b	,
	a − a′		a − a′

Then

cos 2nα − cos 2nθ = 2	a − b·b − a′	= ½sin2 2nα	a − a′
	a − a′·u − b		u − b

sin 2nθ = 2	√ (a − b.b − a′) √ (u − a·u − b′)
	a − a′·u − b

= sin 2nα	√ (a − a·b − a′)	;
	u − b

2n		c	( 1 +	b	)	√ (a − b·b − a′)
φ		ρ		u − b		√ (u − a·u − a′)

=	a − a′ + (a + a′) cos 2nα − [ a + a′ + (a − a′) cos 2nα ] cos 2nθ	×	cos 2nα − cos 2nθ	.
	(a − a′) sin2 2nα		sin 2nθ

Along the wall AB, cos nθ = 0, sin nθ = 1,

a > u > b,

ch nΩ = i sh log (	Q	)	n	= i √	b − a′	√	a − u	,
	q				a − a′		u − b

sh nΩ = i ch log (	Q	)	n	= i √	a − b	√	u − a′	,
	q				a − a′		u − b

ds	=	ds		dφ	=	m	=	c		Q
du		dφ		dt		πqu		π		qu

π

AB

= ∫ab

Q

du

∫ [

√ (a − b) √ (u − a′) + √ (b − a′) √ (a − u)

]

1/n

du

.

c

q

u

√ (a − a′) √ (u − b′)

u

Along the wall Bx, cos nθ = 1, sin nθ = 0,

b > u > 0

ch nΩ = ch log (	Q	)	n	= √	b − a′	√	a − u	,
	q				a − a′		b − u

sh nΩ = sh log (	Q	)	n	= √	a − b	√	u − a′	.
	q				a − a′		b − u

At x where φ = ∞, u = 0, and q = q0,

(

Q

)

n

= √

b − a′

√

a

+ √

a − b

√

−a′

.

q0

a − a′

b

a − a′

q

In crossing to the line of flow x′A′P′J′, ψ 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′

JJ′

=

q0

[ √

b − a′

√

a

− √

a − b

√

−a′

]

1/n

,

xx′

Q

a − a′

b

a − a′

b

giving the contraction of the jet compared with the initial breadth of the stream.

Along the line of flow x′A′P′J′, ψ = m, u = a′e−πφ/m, and from x′ to A′, cos nθ = 1, sin nθ = 0,

ch nΩ = ch log (	Q	)	n	= √	b − a′	√	a − u	,
	q				a − a′		b − u

sh nΩ = sh log (	Q	)	n	= √	a − b	√	u − a′	.
	q				a − a′		b − u

0 > u > a′.

Along the jet surface A′J′, q = Q,

ch nΩ = cos nθ = √	b − a′	√	a − u	,
	a − a′		b − u

sh nΩ = i sin nθ = i √	a − b	√	u − a′	.
	a − a′		b − u

a′ > u = a′eπ/sc > −∞,

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′ = −∞, as in fig. 5,

ch Ω = √	u − a	, sh Ω = √	− a	;
	u		u

and along the jet APJ, ∞ > u = aeπs/c > a,

sh Ω = i sin θ − i √	a	= ie−1/2 πs/c,
	u

PM = ∫∞s sinθ ds = ∫ e−½πs/c ds =	c	e−1/2 πs/c =	c	sin θ,
	½π		½π

129

so that PT = c/½π, and the curve AP is the tractrix; and the coefficient of contraction, or

breadth of the jet	=	π	.
breadth of the orifice		π + 2

A change of Ω and θ into nΩ and nθ will give the solution for two walls converging symmetrically to the orifice AA1 at an angle π/n. With n = ½, the reentrant walls are given of Borda’s mouthpiece, and the coefficient of contraction becomes ½. Generally, by making a′ = −∞, 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.

Fig. 5.	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 π/n as in Bobyleff’s problem; and, by making a = ∞, the wedge extends to infinity; then

ch nΩ = √	b	, sh nΩ = √	n	.
	b − u		b − u

Over the jet surface ψ = m, q = Q,

u = − eπφ/m = − beπ2/c,

ch Ω = cos nθ = √	1	, sh Ω = i sin nθ = i √	eπ2/c	,
	eπ2/c + 1		eπ2/c + 1

e½π2/c = tan nθ,	½π		ds	=	2n	.
	c		dθ		sin 2nθ

For a jet impinging normally on an infinite plane, as in fig. 6, n = 1,

e½π2/c = tan θ, ch (½πs/c) sin 2θ = 1,

sh ½πx/c = cot θ, sh ½πy/c = tan θ,

sh ½πx/c sh ½πy/c = 1, e½π(x + y)/c = e1/2 πx/c + e1/2 πy/c + 1.

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,

ch nΩ = √	b	√	u − a	, sh nΩ = √	b − a	√	u	,
	a		u − b		a		u − b

cos nα = √	b	, sin nα = √	a − b	,
	a		a

and along the free surface APJ, q = Q, ψ = 0, u = e−πφ/m = aeπs/c,

cos nθ = cos nα √	eπ2/c − 1	,
	eπ2/c − cos2 nα

eπ2/c =	cos2 nα sin2 nθ	,
	sin2 nθ − sin2 nα

the intrinsic equation, the other free surface A′P′J′ being given by

eπ2/c =	cos2 nα sin2 nθ	,
	sin2 nα − sin2 nθ

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, α = 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 ≈ a − (a − b)w′,

ch nΩ = √	w′ + 1	, sh nΩ = √	1	,
	w′		√ w′

in which we may write

w′ = φ + ψi.

Along the stream line xABPJ, ψ = 0; and along the jet surface APJ, −1 > φ > −∞; and putting φ = −πs/c − 1, the intrinsic equation is

πs/c = cot2 nθ,

which for n = 1 is the evolute of a catenary.

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

dΩ	= −	√ (b − a·b − a′)
du		(u − b) √ (u − a·u − a′)

dw	= −	m		1	−	m′		1	,
du		π		u − j		π		u − j

= −	m + m′	·	u − b	,
	π		u − j·u − j′

b =	mj′ + m′j	,
	m + m′

taking u = ∞ at the source where φ = ∞, u = b at the branch point B, u = j, j′ at the end of the two diverging streams where φ = −∞; while ψ = 0 along the stream line which divides at B and passes through A, A′; and ψ = 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

ch ½Ω = √	b − a′	√	u − b	, sh ½Ω = √	b − a	√	u − a′	,
	a − a′		u − b		a − a′		u − b

ch Ω =	2b − a − a′	−	N	,
	a − a′		u − b

sh Ω = √ N	√ (2·a − u·u − a′)	,
	u − b

N = 2	a − b·b − a′	.
	a − a′

Along a jet surface, q = Q, and

chΩ = cos θ = cos α − ½sin2 α(a − a′) / (u − b),

if θ = α at the source x of the jet xB, where u = ∞; and supposing θ = β, β′ at the end of the streams where u = j, j′,

u − b	=	½sin2 α	,	u − j	½sin2 α	cosθ − cosβ	,
a − a′		cos α − cos θ		a − a′		(cos α − cos β) (cos α − cos θ)

u − j′	= ½sin2 α	cos θ − cos β′	;
a − a′		(cosα − cos β′) (cos α − cosθ)

and ψ being constant along a stream line

dφ	=	dw	, Q	ds	=	dφ	=	dw		du	,
du		du		dθ		dθ		du		dθ

πQ		ds	=	π		ds	=	(cos α − cos β) (cos α − cos β′) sin θ	,
m + m′		dθ		c		dθ		(cos α − cos θ) (cos θ − cos β) (cos θ − cos α′)

=	sin θ	+	cos α − cos β′	·	sin θ
	cos α − cos θ		cos β − cos β′		cos θ − cos β

cos α − cos β	·	sin θ	,
cos β − cos β′		cos θ − cos β′

giving the intrinsic, equation of the surface of a jet, with proper attention to the sign.

From A to B, a > u > b, θ = 0,

ch Ω = ch log	Q	= cos α − ½ sin2 α	a − a′
	q		a − b

sh Ω = sh log	Q	=	√ (a − u·u − a′)	sinα
	q		u − b

Q	=	(u − b) cos α − ½ (a − a′) sin2 α + √ (a − u·u − a′) sin α
q		u − b

Q	ds	= Q	ds		dφ	= −	Q		dw
	du		dφ		du		q		du

=	m + m′	·	(u − b) cos α − ½ (a − a′) sin2 α + √ (a − u·u − a′) sin α
	π		j − u·u − j′

π	AB	= ∫ab	(2b − a − a′) (u − b) − 2(a − b) (b − a′) + 2√ (a − b·b − a′·a − u·u − a′)	du,
	c		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, α = β = β′, 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′ = −∞, 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.

130

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 nΩ = sn w, sh nΩ = i cn w

so that, round the boundary of the polygon, ψ = K′, sin nθ = 0; and on the surface of the vortex ψ = 0, q = Q, and

cos nθ = sn φ, nθ = ½π − am s/c,

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

φ = Uφ1 + Vφ2 + Wφ3 + Pχ1 + Qχ2 + Rχ3,

where the φ’s and χ’s are functions of x, y, z, depending on the shape of the body; interpreted dynamically, C − ρφ represents the impulsive pressure required to stop the motion, or C + ρφ to start it again from rest.

The terms of φ may be determined one at a time, and this problem is purely kinematical; thus to determine φ1, 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 φ1 must be determined to satisfy the conditions

To determine χ1 the angular velocity P alone is introduced, and the conditions to be satisfied are

For a cavity filled with liquid in the interior of the body, since the liquid inside moves bodily for a motion of translation only,

φ1 = −x, φ2 = −y, φ3 = −z;

but a rotation will stir up the liquid in the cavity, so that the χ’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
a2 + λ		b2 + λ		c2 + λ

and a velocity function of the form

φ = xψ,

where ψ is a function of λ only, so that ψ is constant over an ellipsoid; and we seek to determine the motion set up, and the form of ψ which will satisfy the equation of continuity.

Over the ellipsoid, p denoting the length of the perpendicular from the centre on a tangent plane,

l =	px	,   m =	py	,   n =	pz
	a2 + λ		b2 + λ		c2 + λ

1 =	p2x2	+	p2y2	+	p2z2	,
	(a2 + λ)2		(b2 + λ)2		(c2 + λ)2

p2 = (a2 + λ) l2 + (b2 + λ) m2 + (c2 + λ) n2, = a2l2 + b2m2 + c2n2 + λ,

2p	dp	=	dλ	;
	ds		ds

Thence

dφ	=	dx	ψ + x	dψ
ds		ds		ds

=	dx	ψ + 2 (a2 + λ)	dψ	l	dp	,
	ds		dλ		ds

so that the velocity of the liquid may be resolved into a component -ψ parallel to Ox, and −2(a2 + λ)l dψ/dλ 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

dφ	{ ψ + 2(a2 + λ)	dψ	} l,
ds		dλ

so that over the surface of an ellipsoid where λ and ψ are constant, the normal velocity is the same as that of the ellipsoid itself, moving as a solid with velocity parallel to Ox

U = −ψ − 2 (a2 + λ)	dψ	,
	dλ

and so the boundary condition is satisfied; moreover, any ellipsoidal surface λ 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 λ and λ1, 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 ψ across the annular area α1 − α of the two ellipsoids made by x = 0; or if

αU − α1U1 = ∫λ1λ ψ	dα	dλ,
	dλ

α = π√ (b2 + λ.c2 + λ).

Expressed as a differential relation, with the value of U from (11),

d	[ αψ + 2 (a2 + λ) α	dψ	] − ψ	dα	= 0,
dλ		dλ		dλ

3α	dψ	+ 2 (a2 + λ)	d	( α	dψ	) = 0,
	dλ		dλ		dλ

and integrating

(a2 + λ)3/2 α	dψ	= a constant,
	dλ

so that we may put

ψ = ∫	M dλ	,
	(a2 + λ) P

P2 = 4 (a2 + λ) (b2 + λ) (c2 + λ),

where M denotes a constant; so that ψ is an elliptic integral of the second kind.

The quiescent ellipsoidal surface, over which the motion is entirely tangential, is the one for which

2 (a2 + λ)	dψ	+ ψ = 0,
	dλ

and this is the infinite boundary ellipsoid if we make the upper limit λ1 = ∞.

The velocity of the ellipsoid defined by λ = 0 is then

U = −2a2	dψ0	− ψ0
	dλ

=	M	− ∫∞0	M dλ
	abc		(a2 + λ)P

=	M	(1 − A0),
	abc

with the notation

A or Aλ = ∫∞λ	abc dλ
	(a2 + λ) P

= −2abc	d	∫∞λ	dλ	,
	da2		P

so that in (4)

φ =	M	xA =	UxA	,   φ1 =	xAλ	,
	abc		1 − A0		1 − A0

in (1) for an ellipsoid.

The impulse required to set up the motion in liquid of density ρ is the resultant of an impulsive pressure ρφ over the surface S of the ellipsoid, and is therefore

∫ ∫ ρφl dS = ρψ0 ∫ ∫ xl dS = ρψ0 (volume of the ellipsoid) = ψ0W′,

where W′ denotes the weight of liquid displaced.

Denoting the effective inertia of the liquid parallel to Ox by αW′. the momentum

αW′U = ψ0W′

α =	ψ0	=	A0	;
	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

βW′ =	B0	W′,   γW′ =	C0	W′,
	1 − B0		1 − C0

Bλ, Cλ = ∫∞λ	abc dλ	;
	(b2 + λ, c2 + λ) P

and

A + B + C = abc / ½P,   A0 + B0 + C0 = 1.

For a sphere

a = b = c,   A0 = B0 = C0 = 1⁄3,   α = β = γ = ½,

131

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.
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 λ = λ1, a slight extension will give the velocity function φ of the liquid in the interspace as the ellipsoid λ = 0 is passing with velocity U through the confocal position; φ must now take the form x(ψ + N), and will satisfy the conditions in the shape

φ = Ux	A + B1 + C1	= Ux	abc + ∫λ1λ abcdλ a1b1c1 (a2 + λ) P	,
	B0 + C0 − B1 − C1		1 − abc − ∫λ10 abcdλ a1b1c1 (a2 + λ) P

and any confocal ellipsoid defined by λ, internal or external to λ = λ1, may be supposed to swim with the liquid for an instant, without distortion or rotation, with velocity along Ox

U	Bλ + Cλ − B1 − C1	.
	B0 + C0 − B1 − C1

Since − Ux is the velocity function for the liquid W′ filling the ellipsoid λ = 0, and moving bodily with it, the effective inertia of the liquid in the interspace is

A0 + B1 + C1	W′.
B0 + C0 − B1 − C1

If the ellipsoid is of revolution, with b = c,

φ = ½Ux	A + 2B1	,
	B0 − B1

and the Stokes’ current function ψ can be written down

ψ = − ½ Uy2	B − B1	;
	B0 − B1

reducing, when the liquid extends to infinity and B1 = 0, to

φ = ½ Ux	A	,   ψ = − ½ Uy2	B	;
	B0		B0

so that in the relative motion past the body, as when fixed in the current U parallel to xO,

φ′ = ½Ux ( 1 +	A	),   ψ′ = ½Uy2 ( 1 −	B	).
	B0		B0

Changing the origin from the centre to the focus of a prolate spheroid, then putting b2 = pa, λ = λ′a, and proceeding to the limit where a = ∞, we find for a paraboloid of revolution

B = ½	p	,	B	=	p	,
	p + λ′		B0		p + λ′

y2	= p + λ′ − 2x,
p + λ′

with λ′ = 0 over the surface of the paraboloid; and then

ψ′ = ½ U [ y2 − p √ (x2 + y2) + px ];

ψ = −½ Up [ √ (x2 + y2) − x ];

φ = −½ Up log [ √ (x2 + y2) + x ].

The relative path of a liquid particle is along a stream line

ψ′ = ½ Uc2, a constant,

x =	p2y2 − (y2 − c2)2	,   √ (x2 + y2) =	p2y2 − (y2 − c2)2
	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	,
dx		y		2py

y2 − c2 = a2 e−x/p.

46. Between two concentric spheres, with

a2 + λ = r2, a2 + λ1 = a12,

A = B = C = a3 / 3r3,

φ = ½ Ux	a3/r3 + 2 a3/a13	,   ψ = ½ Uy2	a3/r3 − a3/a13	;
	1 − a4/a12		1 − a3/a13

and the effective inertia of the liquid in the interspace is

A0 + 2A1	W′ = ½	a13 + 2a3	W′.
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 μ at S outside a sphere of radius a is a source of strength μa/ƒ at H, where OS = ƒ, OH = a2/ƒ, and a line sink reaching from the image H to the centre O of line strength −μ/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 μ cos PSx, and of the source H and line sink OH is μ(a/ƒ) cos PHx and −(μ/a)(PO − PH); so that

ψ = μ ( cos PSx +	a	cos PHx −	PO − PH	),
	ƒ		a

and ψ = −μ, 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

m	d	cos PSx = −m	y2	;
	dƒ		PS3

and for its image at H the Stokes’ function is

m	d	cos PHx = −m	a3		y2	;
	dƒ		ƒ3		PH3

so that for the combination

ψ = my2 (

a3

1

−

1

) = m

y2

(

a3

−

ƒ3

),

ƒ3

PH3

PS3

ƒ3

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 ψ′ to be inferred from a distribution of doublets, finite in number when the surface is composed of two spheres intersecting at an angle π/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

ψ′ = ½ Uy2 ( 1 −	a13	−	a23	+	a3	),
	r13		r23		r3

where a1, a2, a = a1a2/√ (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

ψ′ = Uy ( 1 −	a12	−	a22	+	a2	).
	r12		r22		r2

With a2 = ∞, these reduce to

ψ′ = ½Uy2 ( 1 −	a5	)	x	, or Uy ( 1 −	a4	)	x	,
	r5		a		r4		a

for a sphere or cylinder, and a diametral plane.

Two equal spheres, intersecting at 120°, will require

ψ′ = ½Uy2 [	x	−	a3	+	a4 (a − 2x)	+	a3	−	a4 (a + 2x)	],
	a		2r13		2r15		2r23		2r25

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

φ = xyχ

for a rotation R about Oz; and a similar procedure shows that an ellipsoidal surface λ may be in rotation about Oz without disturbing the motion if

R = −	[ 1/ (a2 + λ) + 1/ (b2 + λ) ] χ + 2 dx/dλ	,
	1 / (b2 + λ) − 1 / (a2 + λ)

and that the continuity of the liquid is secured if

(a2 + λ)3/2 (b2 + λ)3/2 (c2 + λ) ½	dχ	= constant,
	dλ

χ = ∫∞λ	N dλ	=	N	·	Bλ − Aλ	;
	(a2 + λ) (b2 + λ) P		abc		a2 − b2

and at the surface λ = 0,

R = −	[ (1/a2 + 1/b2) · N/abc · (B0 − A0)/(a2 − b2) ] − N/abc · 1/a2b2	,
	1/b2 − 1/a2

N	= R	1/b2 − 1/a2	,
abc		1/a2b2 − [ (1/a2 + 1/b2) · (B0 − A0) / (a2 − b2) ]

= R	(a2 − b2)2 / (a2 + b2)
	(a2 − b2) / (a2 + b2) − (B0 − A0)

132

The velocity function of the liquid inside the ellipsoid λ = 0 due to the same angular velocity will be

φ1 = Rxy (a2 − b2) / (a2 + b2),

and on the surface outside

φ0 = xyχ0 = xy	N		B0 − A0	,
	abc		a2 − b2

so that the ratio of the exterior and interior value of φ at the surface is

φ0	=	B0 − A0	,
φ1		(a2 − b2) / (a2 + b2) − (B0 − A0)

and this is the ratio of the effective angular inertia of the liquid, outside and inside the ellipsoid λ = 0.

The extension to the case where the liquid is bounded externally by a fixed ellipsoid λ = λ1 is made in a similar manner, by putting

φ = xy (χ + M),

and the ratio of the effective angular inertia in (9) is changed to

(B0 − A0) − (B1 − A1) + a12 − b12   abc a12 + b12 a1b1c1	.
a2 − b2 − a12 − b12   abc − (B0 − A0) + (B1 − A1) a2 + b2 a12 + b12 a1b1c1

Make c = ∞ for confocal elliptic cylinders; and then

Aλ = ∫∞λ	ab	=	ab	( 1 − √	b2 + λ	),
	(a2 + λ) √ (4a2 + λb2 + λ)		a2 − b2		a2 + λ

Bλ =	ab	( √	a2 + λ	− 1 ),   Cλ = 0;
	a2 − b2		b2 + λ

and then as above in § 31, with

a = c ch α, b = c sh α, a1 = √ (a2 + λ) = c ch α1, b1 = c sh α1

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 φ = xyzΩ, where Ω can be expressed by the elliptic integrals Aλ, Bλ, Cλ, in a similar manner, since

Ω = L ∫∞λ dλ / P3.

48. The determination of the φ’s and χ’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

x1 =	dT	, x2 =	dT	, x3 =	dT	,
	dU		dV		dW

y1 =	dT	, y2 =	dT	, y3 =	dT	;
	dP		dQ		dR

but when it is expressed as a quadratic function of x1, x2, x3, y1, y2, y3,

U =	dT	, V =	dT	, W =	dT	,
	dx1		dx2		dx3

P =	dT	, Q =	dT	, R =	dT	.
	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

X =	dx1	− x2	dT	+ x3	dT	, Y = ..., Z = ...,
	dt		dy3		dy2

L =	dy1	− y2	dT	+ y3	dT	− x2	dT	+ x3	dT	, M = ..., N = ...,
	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	= mR − nQ,	dm	= nP − lR,	dn	= lQ − mP.
dt		dt		dt

If P denotes the resultant linear impulse or momentum in this direction

P = lx1 + mx2 + nx3,

dP	=	dl	x1 +	dm	x2 +	dn	x3
dt		dt		dt		dt

+ l	dx1	+ m	dx2	+ n	dx3	,
	dt		dt		dt

= l (	dx1	− x2R + x3Q )
	dt

+ m (	dx2	− x3P + x1R )
	dt

+ n (	dx3	− x1Q + x2P )
	dt

= lX + mY + nZ,

for all values of l, m, n.

Next, taking a fixed origin Ω 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 Ω in the direction (l, m, n)

G = l (y1 − x2z + x3y) + m (y2 − x3x + x1z) + n (y3 − x1y + x2x).

Differentiating with respect to t, and afterwards moving the fixed origin up to the moving origin O, so that

x = y = z = 0, but	dx	= U,	dy	= V,	dz	= W,
	dt		dt		dt

dG	= l (	dy1	− y2R + y3Q − x2W + x3V )
dt		dt

+ m (	dy2	− y3P + y1R − x3U + x1W )
	dt

+ n (	dy3	− y1Q + y2P − x1V + x2U )
	dt

= lL + mM + nN,

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

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,

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

so that a fourth integral is given by

dy3 / dt = 0, y3 = constant;

dx3	= x1 (qx2 + ry2) − x2 (qx1 + ry1) = r (x1y2 − x2y1),
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,

in which

x12 + x22 = F2 − x32, x1y1 + x2y2 = FG − x3y3,

r (y12 + y22) = 2T − p(x12 + x22) − p′x32 − 2q (x1y1 + x2y2) − 2q′x3y3 − r′y32 = (p − p′) x32 + 2 (q − q′) x3y3 + m1,

m1 − 2T − pF2 − 2qFG − r1y32

so that

1	(	dx3	)	2	= X3
r2		dt

where X3 is a quartic function of x3, and thus t is given by an elliptic 133 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 √ X3,

y1 + y2i	=	FG − x3y3 + i √ X3	,
x1 + x2i		x12 + x22

d	(x1 + x2i) = −i [ (q′ − q) x3 + r′y3 ] + irx3 (y1 + y2i),
dt

d	log (x1 + x2i) = −(q′ − q) x3 − r′y3 + rx3	FG − x3y3 + i √ X3	,
dti		F2 − x32

d	log √	x1 + x2i	= −(q′ − q) x3 − (r′ − r) y3 − Fr	Fy3 − Gx3	,
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 θ, φ, ψ,

x1 = F sin θ sin φ,   x2 = F sin θ cos φ, x1 + x2i = iF sin θε−ψi,   x3 = F cos θ;

sin θ	dψ	= P sin φ + Q cos φ,
	dt

F sin2 θ	dψ	=	dT	x1 +	dT	x2
	dt		dy1		dy2

= (qx1 + ry1) x1 + (qx2 + ry2) x2 = q (x12 + x22) + r (x1y1 + x2y2) = gF2 sin2 θ + r (FG − x3y3),

ψ − qFt = ∫	FG − x3y3		Fr dx3	,
	F2 − x32		√ 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,

FY = −y1 cos xX + y2 cos yX + y3 cos zX,

G = y1 cos xZ + y2 cos yZ + y3 cos zZ,

F2(X2 + Y2) = y12 + y22 + y32 − G2,

F(X + Yi) =	Fy3 − Gx3 + i √ X3	εψi.
	√ (F2 − x32)

Suppose x3 − F is a repeated factor of X3, then y3 = G, and

X3 = (x3 − F)2 [	p′ − p	(x3 + F)2 + 2	q′ − q	G (x3 + F) − G2 ],
	r		r

and putting x3 − F = y,

(

dy

)

2

= r2y2 [ 4

p′ − p

F2 + 4

q′ − q

FG − G2 + 2 ( 2

p′ − p

F +

q′ − q

G ) y +

p′ − p

y2 ],

dt

r

r

r

r

r

so that the stability of this axial movement is secured if

A = 4	p′ − p	F2 + 4	q′ − q	FG − G2
	r		r

is negative, and then the axis makes r√(-A)/π nutations per second. Otherwise, if A is positive

rt = ∫	dy
	y √ (A + 2By + Cy2)

=

1

sh−1

√ A √ (A + 2By + Cy2)

=

1

ch−1

A + By

,

√ A

ch−1

y√ (B2 ~ AC)

√A

sh−1

y √ (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 ½θ = x,

(Nx eψi)3/2 = (x + 1) √ X1 + i (x − 1) √ X2,

X1	= ± ax4 + 2ax3 ± 3 (a + b) x2 + 2bx ± b,
X2

N3 = −8 (a + b),

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).