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OF THE ASTRONOMICAL SOCIETYFACiFJC

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.

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:

MACMILLAN AND

HYDRODYNAMICS

BY

HORACE LAMB,

M.A.,

F.R.S.

PBOFESSOB OF MATHEMATICS IN THE OWENS COLLEGE,VICTORIA UNIVERSITY, MANCHESTER FORMERLY FELLOW OF TRINITY COLLEGE, CAMBRIDGE.;

CAMBRIDGE1895

:

AT THE UNIVERSITY

PRESS.

[All Rights reserved.]

ASffiONOMY LIBRARY

Cambridge:

PRINTED BY

J.

&

C.

F.

CLAY,

AT THE UNIVERSITY PRESS.

ASTRONOMY

PREFACE.rilHIS book may be regarded-- ontheas a second edition of a"

Treatise

Theory of the Motion of Fluids," in 1879, but the additions and alterations are so ex published tensive that it has been thought proper to make a change in theMathematicaltitle.

have attempted to frame a connected account of the principal theorems and methods of the science, and of such of the moreI

important applications as admit of being presented within a moderate compass. It is hoped that all investigations of funda

mental importance will be found to have been given with sufficient detail, but in matters of secondary or illustrative interest I haveoften condensed the argument, or merely stated results, leaving the full working out to the reader.

In making a selection of the subjects to be treated I have been guided by considerations of physical interest. Long analyticalinvestigations,

have asis

far

leading to results which cannot be interpreted, as possible been avoided. Considerable but, it

hoped, not excessive space has been devoted to the theory of waves of various kinds, and to the subject of viscosity. On theother hand, some readers

may be disappointed to find that the theory of isolated vortices is still given much in the form in which it was. left by the earlier researches of von Helrnholtz and LordKelvin,

and that

little

reference

is

made

to

the

subsequent

investigations of J. J.field.

Thomson, W. M. Hicks, and others, in this The omission has been made with reluctance, and can beon the ground that the investigations in questionb

justified onlyL.

M6772O1

VI

PREFACE.

derive most of their interest from their bearing on kinetic theories of matter, which seem to lie outside the province of a treatise like

the present.I

have ventured, in one important particular, to make a serious

innovation in the established notation of the subject, by reversing the sign of the velocity -potential. This step has been taken not

without hesitation, and was only finally decided upon when I found that it had the countenance of friends whose judgment I couldtrust

but the physical interpretation of the function, and the far-reaching analogy with the magnetic potential, are both so much;

improved by the change thator later, inevitable.I

its

adoption appeared to be, sooner

have endeavoured, throughout the book, to attribute to their

proper authors the more important steps in the development of the subject. That this is not always an easy matter is shewn by the fact that it has occasionally been found necessary to modifyreferences given in the former treatise, and generally accepted ascorrect.

I trust, therefore, thatwill

any

errors of ascriptionIt

whichwell,

remain

be

viewed

with

indulgence.

may

be

moreover, to

warn the reader, once for all, that I have allowed myself a free hand in dealing with the materials at my disposal, and that the reference in the footnote must not always be taken to imply that the method of the original author has beenclosely followed

in

the text.

I

will

confess, indeed,

that

my

ambition has been not merely to produce a text-book giving a faithful record of the present state of the science, with its

achievements and

its imperfections, but, if possible, to carry it a further here and there, and at all events by the due coordina step tion of results already obtained to lighten in some degree the

labours of future investigators. I shall be glad if I have at least succeeded in conveying to my readers some of the fascination which the subject has exerted on so long a line of distinguishedwriters.

In the present subject, perhaps more than in any other depart ment of mathematical physics, there is room for Poinsot s warning

PREFACE."

Vll

reduite a des formules

Gardens nous de croire qu une science soit faite quand on 1 a I have endeavoured to make analytiques."whichit is

the analytical results as intelligible as possible, by numericalillustrations,

hoped

will

be found correct, and by theof stream-lines

insertion of acurves,

numberto scale,

of diagrams

and other

drawn

and reduced by photography.

Some

of

these cases have, of course, been figured by previous writers, but many are new, and in every instance the curves have beencalculated and

drawn independently

for the

purposes of this work.

I am much indebted to various friends who have kindly taken an interest in the book, and have helped in various ways, but who would not care to be specially named. I cannot refrain, however,

from expressing my obligations to those who have shared in the tedious labour of reading the proof sheets. Mr H. M. Taylor hasincreased the debt I was under in respect of the former treatise by giving me the benefit, so long as he was able, of his vigilantcriticism.

On

his enforced retirement his place

was kindly taken

by Mr R. F. Gwyther, whose care has enabled me to correct many Mr J. Larmor has read the book throughout, and has errors.freely

placed;

histo

great

knowledge

of

the

subject

at

my

disposal

I

owe

him many valuable

suggestions.

Finally, I

have had the advantage, in the revision of the last chapter, of Mr A. E. H. Love s special acquaintance with the problems theretreated.

Notwithstanding so much friendly help I cannot hope to have escaped numerous errors, in addition to the few which have beenesteem myself fortunate if those which remain should prove to relate merely to points of detail and not of principle. In any case I shall be glad to have my attention called to them.detected.I shall

HORACE LAMB.May, 1895.

CONTENTS.CHAPTERART.I, 2.

I.

THE EQUATIONS OF MOTION.PAGE

3-8.

Fundamental property of a fluid Eulerian form of the equations of motion...

.

.

.

.

1

Dynamical.

equations, equation of continuity9.

.

.

.

.

37

10.

Physical equations Surface-conditions

.

.

.

.

.

.

.

.

.

.

.

8

II.12.

13, 14.

Equation of energy . . Impulsive generation of motion Lagrangian forms of the dynamical equations, and of the...

.10.

.

.

.

12

equation of continuity15. 16, 17.

14.

Weber s Transformation.

.

15

Extension of the Lagrangian notation. two forms

Comparison

of the

16

CHAPTER

II.

INTEGRATION OF THE EQUATIONS IN SPECIAL CASES.18.

Velocity-potential.

Lagrange s theorem

m,

or,

for

an incompressible

d

(x, y, z)

d(a;Qt

d(a,b,c)If

y0) z ) d(a,b,c)

we compare the two forms of the fundamental equations to which 17. we have been led, we notice that the Eulerian equations of motion are linearand of the first order, whilst the Lagrangian equations are of the second order, and also contain products of differential coefficients. In Weber s transfor mation the latter are replaced by a system of equations of the first order, andof the second degree.

The Eulerian equation

of continuity

is

also

much

simpler than the Lagrangian, especially in the case of liquids. In these respects, therefore, the Eulerian forms of the equations possess great ad vantages. Again, the form in which the solution of the Eulerian equations appears corresponds, in many cases, more nearly to what we wish to knowas to the motion of a fluid, our object being, in general, to gain a knowledge of the state of motion of the fluid mass at any instant, rather than to trace

the career of individual particles.

On the other hand, whenever the fluid is bounded by a moving surface, the Lagrangian method possesses certain theoretical advantages. In the Eulerian method the functions u, v, w have no existence beyond this surface,and hence the range of values of #, y, z for which these functions exist varies In in consequence of the motion which is itself the subject of investigation. the other method, on the contrary, the range of the independent variables,

6, c is

given once fordifficulty,

all

by the

initial conditions

*.

The

however, of integrating the Lagrangian equations has

hitherto prevented

their application except in certain very special cases. Accordingly in this treatise we deal almost exclusively with the Eulerian forms. The simplification and integration of these in certain cases form the

subject of the following chapter.*

H. Weber,

I.e.

CHAPTER

II.

INTEGRATION OF THE EQUATIONS IN SPECIAL CASES.18.

velocities u,,

IN a large and important class of cases the component v, w can be expressed in terms of a single function:

as follows

u

=is

d(f>

f dv

t

v

=

dd>

-7 -,

1

w=

dd>

dy

dz

f

,, x

(1).

Such a functionAttractions,

called a

with the potentialis

function

velocity-potential/ from its analogy which occurs in the theories of&c.

Electrostatics,

The

general

theory

of

thegive

reserved for the next chapter; but velocity-potential at once a proof of the following important theorem:

we

If

finite portion of

a velocity-potential exist, at any one instant, for any a perfect fluid in motion under the action of

forcesfluid

which have a potential, then, provided the density of the be either constant or a function of the pressure only, a

velocity-potential exists for the instants before or after*.

same portion of the

fluid at all

In the equations of Art. 15,"

let

the instant at which the

* Me moire sur la Th6orie du Mouvement des Fluides," Nouv. Lagrange, mim. de VAcad. de Berlin, 1781 Oeuvres, t. iv. p. 714. The argument is repro duced in the Mecanique Anatytique. Lagrange s statement and proof were alike imperfect the first rigorous demon Memoire sur la Th6orie des Ondes," Mem. de VAcad. stration is due to Cauchy,; ;"

Oeuvres Completes, Paris, 1882..., l re Se"rie, t. i. p. 38 roy. des Sciences, t. i. (1827) Another proof is given by Stokes, Camb. Trans, t. the date of the memoir is 1815.; ;

viii.

(1845) (see also Math,t.ii.

and

p. 36), together

and Pliys. Papers, Cambridge, 1880..., t. i. pp. 106, 158, with an excellent historical and critical account of the

whole matter,

18-19]velocity-potential

VELOCITY-POTENTIAL.

19;

$

exists

be taken as the origin of time,

we

have then

u da

+ v db + w dc =

c

throughout the portion of the mass in question. Multiplying the equations (2) of Art. 15 in order by da, db, dc, and adding,

we

get

X ~*~ffi^or,

dt

^ ^~dt^Z ~notation,

(u

^ a + Vodb + wodc) = - dx,+ %) =mayc&,say.

in the

Eulerianudoc -f

vdy + wdzt

=

d((/>

Since the upper limit ofthis proves the theorem.

in Art. 15 (1)

be positive or negative,

of a

It is to be particularly noticed that this continued existence velocity-potential is predicated, not of regions of space,

A portion of matter for which a moves about and carries this property with it, but the part of space which it originally occupied may, in the course of time, come to be occupied by matter which did riot originally possess the property, and which thereforebut of portions of matter.velocity-potential exists

cannot have acquired

it.

cludes

of cases in which a velocity-potential exists in those where the motion has originated from rest under the action of forces of the kind here supposed for then we have,classall;

The

initially,

u da + v dbor

+ w dc =

0,

= const.

under which the above theorem has been proved must be carefully remembered. It is assumed not only that the external forces X, F, Z, estimated at per unit mass, have a potential, but that the density The latter condition is violated p is either uniform or a function of p only.restrictionsfor example, in the case of the convection currents generated

The

application of heat to a fluid

;

and again,

in the

by the unequal wave-motion of a hetero*

geneous but incompressible fluid arranged originally in horizontal layers of equal density. Another important case of exception is that of electro-magneticrotations.

comparison of the formulae (1) with the equations (2) of Art. 12 leads to a simple physical interpretation of 0.19.

A

Any actual state of motion of a liquid, for which a (single-valued)velocity-potential exists, could be produced instantaneously from rest

22

20

INTEGRATION OF THE EQUATIONS IN SPECIAL CASES.

[CHAP.

II

by the application of a properly chosen system of impulsive pressures. This is evident from the equations cited, which shew, moreover,gives the requisite sys gives the system of impulsive which would completely stop the motion. The occur pressures rence of an arbitrary constant in these expressions shews, what is

that

= tr/p + const.

;

so that

w=

4-

C

p

tem.

In the same way

^=

p(f>

+C

otherwise evident, that a pressure uniform throughout a liquid mass produces no effect on its motion.

In the case of a gas, may be interpreted as the potential of the external impulsive forces by which the actual motion at any instant could be produced instantaneously from rest.

A

state of motion for

which a velocity-potential does not exist

cannot be generated or destroyed by the action of impulsive pressures, or of extraneous impulsive forces having a potential.

The existence of a velocity-potential indicates, besides, 20. certain kinematical properties of the motion.

A line of motion or drawn from point to point,of the motion

stream-line *

is

defined to be a line

so that its direction is everywhere that of the fluid. The differential equations of the

system of such lines are

dx U

= dy == dz V W

"

"

\

/

The

lines of

relations (1) shew that when a velocity-potential exists the motion are everywhere perpendicular to a system of sur

faces, viz.

theif

equipotential

surfaces

dz

,

.

,

_

d(/>

dz dsis

ds

decrease of

velocity in any direction in that direction.

therefore equal to the rate of

Taking 8s in the direction of the normal

to the surface

= const,to

we*

see that if a series of such surfaces be

drawn correspondingstream-line

Some

writers prefer to restrict the use of the term

to the case of

steady motion, as defined in Art. 22.

19-21]

VELOCITY-POTENTIAL.

21

the common difference being infinitely equidistant values of the velocity at any point will be inversely proportional to the small, distance between two consecutive surfaces in the neighbourhood,

of the point.Hence,if

the intersection.

any equipotential surface intersect itself, the velocity is zero at The intersection of two distinct equipotential surfaces would

imply an21.

infinite velocity.

Under the circumstances

stated in Art. 18, the equations

of motion are at once integrable throughout that portion of the For in virtue fluid mass for which a velocity-potential exists.

of the relations

dv dz

_dw ~dy(1),

dw __ dudxdz

du _ dv dy dx

which are implied in2

the equations of Art. 6

mays

be writtens &c.

dl 1 dp dw d6 du dv ---T-^ t + u^- + Vj- + w-j-=- dx j dx dx dx dxdt dxp

f

fee.,

These have the integral

where q denotes the resultant velocity (u2 + v 2 + w 2 )*, and F (t) is an arbitrary function of t. It is often convenient to suppose thisthis arbitrary function to be incorporated in the value of w are not thereby is permissible since, by (1), the values of u, v,d