jj-thompson treatise on the motion of vortex rings

Kineticenergyof thesystem .Expression for the kinetic energy of a number of circular vortex ringsmovinginsideaclosed vessel Expressionfor thevelocity parallel to the axis ofxduetoan appr

A TREATISE

ON THE

IN 1882, IN THE UNIVERSITY OF CAMBRIDGE.

In this essay,in addition to the set subject, I have discussed

of matter

on the essay. Beyond these I have not made any alterations

inthefirst three parts of the essay: buttothe fourth part, which

suggestions

J J THOMSON.

TRINITY COLLEGE, CAMBRIDGE

October 1st, 1883.

Kineticenergyof thesystem .

Expression for the kinetic energy of a number of circular vortex

ringsmovinginsideaclosed vessel

Expressionfor thevelocity parallel to the axis ofxduetoan

approxi-matelycircular vortex ring

Thevelocityparallel tothe axis of

yThevelocity parallel to the axis ofz

Calculation of the coefficients in theexpansionof

368

11

13

151820

Theaction oftwovortex ringsoneachother

Theexpression for the velocity

parallel to the axis of x duetoone

vortex at a pointonthe core of the other

Thevelocityparallelto the axis ofy

Thevelocity parallel to the axis ofz

Thevelocity parallel to the axis of z

expressedasa function of the

time

Thesimilar expression for the velocity parallel to the axis ofy

Thesimilar expression for the

velocity parallel to the axis ofxTheexpression for the deflection ofone

of the vortex rings

Thechangein the radius of the vortex

ring

Thechangesin thecomponentsof themomentum

Effects of the collisiononthe sizesanddirections ofmotionof the

twovortices.

37

3940

40

414344

46

5052

51

PARAGRAPH *AGB

32. Theimpulseswhichwould producethesameeffect as the collision 56

33 ) Theeffect of the collisionupontheshapeof the vortex ring : 34.i lation of

calcu-cos nt dt . OD

2 _j_ L.2/2\i'"P""'

36. Motionofacircular vortex ring inafluidthroughoutwhichthe tribution of velocity isknown 63

dis-Oory\

PAET HI

39. Thevelocity potentialduetoandthe vibrations ofanapproximately

circular vortexcolumn 71

40 Velocity potentialduetotwovortexcolumns 74

42. Actionoftwovortexcolumns uponeachother 75

42*. Themotionoftwolinked vortices of equal strength 78

44 Calculation of themotionoftwolinked vortices of equal strength to

ahigherorder ofappproximation 88

45. Proofthat theabovesolution is theonly onefor circular vortices 92

47. Themotionof several vortex rings linked together 93

48. Theequationsgiving themotionwhenasystemofnvortexcolumns

of equal strength is slightly displacedfromits position of steady

49. Thecasewhenn=3 98

51. Thecasewhenn-5 100

52. Thecasewhenn=6 103

53. Thecasewhenn=7 105

54. Mayer's experiments withfloatingmagnets 107

slightly disturbed from its circular form

the effect of a sphereon a circular vortex ring passing near it is

circular vortex rings linked through each other; the conditions

results to the vortexatomtheoryofgases,anda sketch ofa vortex

When we have a mass of fluid under the action of no forces,

fluid particles, such asthe surface of asolid immersedin afluid or

dF dF dF dF

surface, there will be no discontinuity in the velocity, and so no

con-dition we have explicitly to consider Thus our method is very

dF dF dF dF

get differential equations sufficient to solve any of the aboveproblems

momen-tum, momentof momentum, andkinetic energy of amass of fluid

, dS

/0

or -f- (3cos6cos& cose),

& the angles their directions of motion make with the linejoining

The methodused to calculate the expressions

by Helmholtz thedistribution of velocityduetoa vortexringwhose

ni/r),

A 7n> ^n are small compared with a. The transverse section of

ed vol II. 683) Hence theaction ofa vortex ring ofthis shape

V(?-cos<9)'

case, however, when q is nearly unity is not important in that

in this case It was therefore necessary to investigate some

case; the resultof thisinvestigation isgiven in equation 25, viz.

where gm=1+i+

2m _1>

n , /3n , yn , 8

2-Tra(log 1 j (equation41),

t

- * (n>

-1} log

Thus wesee thatthe ring executesvibrations inthe period

27T

p = a + 5(ancosnty+ nsinmjr),

$ + 2(?BCOS

tti/r+Snsin711/r);

W=-5?

Hence we expandHi and w in theform

Acos^ + Bsin^ + A'cos2^ +B'sin2>|r+ .

moving with the velocityp, the vortex (II.) with the velocity q,

m the strengths of the vortices (I.) and (II.) respectively, a, b

$+ y COST/r+ &sin

sm8

eV(c3

-g

2 )

The effects ofthe collision may be divided in three parts:

firstly, the effect upon the radii of the vortex rings; secondly,

parallel to the original directions of motion of both the vortex

rings

Letus firstconsider the effectupon the radii. Letg=ccos

</>,

thus

When <t> is greater than 60 the vortex ring which first

passes

firstpasses through theshortest distance, whichinthis case is the

radius When<j> is zero orthe vortex rings intersect the shortest

them

.

,

^ pq ^ ~ pcos6^

ring (II.) be greater than the velocity of the other vortex (I.)

velocity ofthe vortexbe less thanthevelocity ofthe otherresolved

vice versa The rulesforfinding thealteration inthe radius were

impulse

parallel to the shortest distance betweentheoriginal paths of the

thevortices

We find that the collision

after collision, their central linesof vortex core are representedby

axis These are the equationsto twistedellipses,whoseellipticities

-The first of these equations shews that the radius of a

eachother We shewthat ifthe vortex rings are ofequal strengths

2

circular axis of the anchor ring and d the diameter of its

parallel, theproblem isvery approximately thesame as thatoftwo

parallel straight columnar vortices,and as the mathematical work

of a Lemma ( 33) whichenables us to transfer cylindrical

thecircular the and time of vibration is

equation (89) We then go onto discuss the transverse vibrations

ofmomentum, r thenumberoftimes thevorticesarelinkedthrough

rmrprad

2

Now c^/a2

isthatF(4<m7rp)h

vortices,and the numberof times they are linked through each

other

this cross section Wefind the times of vibration when n equals

3, 4, 5, or6, and prove that the motion is unstable for seven or

more vortices,so that not more thansix vorticescan be arranged

ON THE MOTION OF VORTEX KINGS.

filling the universe has made the subject of vortex

ent-^en

essential to a molecule that has to be the basis of a dynamical

vortices, moves rapidly forward

of trld?8 Vll

1e; lk an P ssess' in virtue * otion

radEn matenals for explaiuing the phenomena ofheatand

>sesto explain by meansofthe laws of

2 ON THE MOTION VORTEX

mechanism ofthe intermolecularforces,itenables us toform much

the clearestmentalrepresentation of whatgoes on when one atom

gives^to the subject of

effect of a solid body immersed in the fluid on a vortex^ ring

vortices

gases

we startwith the fact that the vortex ring always consists of the

same particles of fluid (the proof of which, however, requires

"

PAKT I.

2. WE shall, for convenience of reference,

fluid by the letters,u,v,w; thecomponents ofthe angularvelocity

Velocity

weshall render theregion acyclic. Now we know that the motion

ON THE MOTION OF VORTEX RINGS.

barrier, and thuswe see that the motion can be generated by a

to 2m/?.

are

P.

sides of the barrier and infinitely close to it. Nowthe solidangle

dS

Now

* /("$

component of

x, y,zrespectively,

(2).

y& dx dy dz

ON THE MOTION OF VORTEX RINGS.

Moment of Momentum.

ofthemoment ofmomentum due to any distribution ofvortices

///{y(uij vf) z(w% u%)}dx dydz

tft * + *\ ^ 7 7 fff f (dw dv\ dv du

Xf/ K

-ON THE MOTI-ON OF VORTEX RINGS.

fluid remains constant both in magnitude and direction. When

ON THE MOTION OF VORTEX RINGS.

Kinetic Energy

=

f dy c

We shall in subsequent investigations require the expression

(7; then we shall find the additional term introduced when we

Pto 0

integral

When we move the origin from Cr

introduced

= - 2pm

fp9lds,

Let ustake asour initial linethe intersection of the plane of

ON THE MOTION OF VORTEX RINGS 9Let

this initial line, o> the angle which the projection of 0(7 on the

<f> Vds =ira(Acoso>+ Bsino>).

We must nowfind

10 ON THE MOTION OF VOKTEX RINGS.

ring Let A, B, C bethe extremities of axes parallel to the axes

shewn inthe figure. Let MN be the ring itself and P any point

-. The firstrotation leaves z

a

s

'

Z=sin6cose, m =sin6sine, n =cos0, and

V = cosecos6cos&> sin6sin ,

&>,

i/= sin cosco.

ON THE MOTION OF VORTEX RINGS 11

2

12 ON THE MOTION VORTEX RINGS.

in-tegrals,supposing, however, thatthe boundariesare fixed so that

=iP///(? ^ +v*+ ^2

)dxdydz - ^//(w

2

+v2

-0)

MOTION OF A SINGLE VORTEX.

8. HAVING investigated these general theorems we shall go

ring We shall suppose that the transverse section of the vortex

ring, whosetransverse section issmall comparedwith its aperture,

2nd edition, vol II., 683) Hence the action of a vortex ring of

n , yn , B

ring, e the radius of the transverse section of the core Now, by

ON THE MOTION OF VORTEX RINGS.

t" a( *-*>-(8-* '

thetimeofoscillation ofthe vortex aboutitscircularform,we only

IfR, <f>, zbe thecylindrical coordinates ofthe pointx, y, z,

now when we substitute forp itsvalue it is evident that -3 can be

2(s)(At+ Bscosnty+ C

ssinn-^r cos sn

MOTION SINGLE VORTEX. 15

but we shallinvestigatethe valuesofallthese coefficients later

8(facosty+ ny(yusinn^ S

16 ON THE MOTION OF VORTEX KINGS.

SINGLE VORTEX. 17and

x (yl(,_lsia(>i-1) +

thefirstpower + termscontaining a

= %m&Al cos<f> (10)

=Jra[2nyAn(ynsin

n to the second power

(2<?+ B sin27i>- C cos

ON THE MOTION OF VORTEX RINGS.

C 1A

-3[fasinty nx(<ynsinnty Sncosn

the first power4-terms containingan to the second power.

n to the firstpower

ON THE MOTION OF VORTEX RINGS.

!-4(,.%,

-su

+( 1)08J M+,+ ar

-1)89.+( +1) o8J C^ - arZ^Jsin(2n+1) <...(15)

ncos2?i>|r+ z cos^>rsn?i\r+ sn

MOTION OF A SINGLE VOBTEX. 21

The term T (n 1)(x*n -fy/3 ) I

first power +terms containingan . tothesecond power

n

22 ON THE MOTION OF VOKTEX KINGS.

MOTION OP A SINGLE VORTEX 23

Now

By meansofthisand equation (20), we easilyget

coefficientsofcosnd we have

24 ON THE MOTION OF VORTEX RINGS.

and . Weshall do- this by determining the valueof bnwhen q is

(20), that

or 4>n + = - 6^ +

A SINGLE VORTEX 25

.)b +(A'

lt

26 ON THE MOTION VORTEX RINGS.

A SINGLE VORTEX 27

numbers up toand includingn, then

28 ON THE MOTION VORTEX RINGS.

_ 2Rp

MOTION OF SINGLE VORTEX 29

dS

we shall neglect all terms containing the squares of thosequantities

Fig 2.

3C

e

produced

NowifF(x,y,z, t)= be an equation to a surface which as

(*> V>4

ctt (it

ON.THE MOTION OF VORTEX RINGS.

ring,

con-tainsan andj3ntothe firstpower; and a^P will be of the second

mfy e sin^.X = & (31).

But ucos^r+vsin*fy=i&.

we have R = a +ancosnty+finsinnty-t-ecos

%,

8 ,

X = ft>,

6

32 ON THE MOTION OF VORTEX RINGS.

Cill tit Cit

But we know by equations (16) and (17) that

cos

+ 2aAn@n ]sinn

x, the coefficient of cos^ and the coefficients of cos nty

The first equation gives the velocity of translation of the

t wefind

8a

1 (42)>

34 ON THE

=c^ a

dt p"

this isthe same coefficientaswe had inthe equation giving dzjdt

MOTION OF A SINGLE VORTEX.

-1)}

it is stable forallsmall displacements. The time of vibration

Thusfor ellipticdeformation the time ofvibration is"289 times

circumference

32

ON THE MOTION OF VORTEX RINGS.

of vibration

27T

2o>7rV(log

PART II.

ofeither.

ring, which we investigated in the previous part, will enable

will not differ much from circles; hence in finding the velocity

p =b+ 2(a'ncosnty'+f?nsinnty'),

= '

38 ON THE MOTION OF VORTEX RINGS.

We shall have to express or n , y M , yw , 8B, a'n, /3'M , y'n> S' n as

Y drawn upwards fromthe plane ofthepaper.

Let Z, X (fig.4) be the pointswhere the axesofZ and X cut

Flg.4.

Then we easily see,by Spherical Trigonometry,that

I= COS COS

-vjr,

m = sin

-fy, 7i= sinecos^Jr.

the equations

MOTION OF TWO VORTEX RINGS 39

+f sinecose -~(hcose-/sine)(/cose-^sine)

ON THE MOTION OF VORTEX

MOTION OF TWO VORTEX RINGS.

of cosnijr in the expression for the velocity along the radius

Hence we must express the value for u, v,w which we havejust

Let p and q be the velocities of the vortices AB and CD

2

+2* 2p<?cose);

inde-of

ON THE MOTION OF VORTEX RINGS.

1 ma* [f 2 fsin2e 2\

2 ^i (3(<r g):?

p c )

-(qcose p)qsine.t+(2(grcose pf<fsin2e} 2 (52).

The coefficientof cos

J /^*_R_^_* ' xv* BraTk*^

r 10 i)-T- (/ bill C

MOTION OF TWO VORTEX RINGS 43 where

The coefficientof cos^

ON THE MOTION VORTEX RINGS.

The coefficientof cos 2\/r

MOTION OF TWO VORTEX RINGS.

46 ON THE MOTION OF VORTEX RINGS.

The coefficientof sin 2ty

=J 7/za262

1-( ^~

g2

(gsin2e-psine)+(qcos2e

/?cose)t

) -.+7

'-* ^ *

23 To find the effectof the vortex AB on CD we require

= wcose+ usine.

The coefficient of costyinthe expression forthe velocity

MOTION OF TWO VORTEX RINGS.

j'+ 2 (y'ncosn-\fr+S' nsinn-ty}.

Thus -7^ =coefficientof cos-Jr in the expression forthedt

&

we shall onlyconsiderthe change in7^ when ithas gotso far away

CLO

~\=coefficientof sin1^ in theexpression forthe velocity

ON THE MOTION OF VORTEX RINGS.

t

(70)'

i fy

) l- sin3e.pq(q-pcose).

Thusif A,B, C(fig. 5) be the pointswhere the axes ofx, y, z

a parallel through this centre to the direction of motion of the

MOTION OF TWO VORTEX RINGS 40

Fig.G.

q^ pcosebe positive. We may

the impact

2

for f and j; this will change the sign of <y\ but will leave S\

50 ON THE MOTION OF VORTEX RINGS.

move so as to come as close together as possible, then c=

g,

when theydo not

~ =coefficient of the term independent of -^in the expression

cLu

MOTION OF TWO VORTEX RIN

X = COS COS>/r,

^,

=coefficientofthe termindependentof^r in

5*

42

52 ON THE MOTION VORTEX RINGS.

*

a'

by the collision Thus we see for our present purpose we may

in velocity.

27 Havingfound the change in the radius andthe change

axes

MOTION OF TWO VORTEX

*',,

b

similarly, 8&V =2-~<S'+ '

I, ^P, (01,HI denote the same quantitiesforthe vortexAB as the

same letters accented do for the vortex CD, then it is easy to

are -an?cose an -air cose

\vith symmetrical expressionsfor Bmf

54 ON THE MOTION OF VORTEX EINGS.

- (75),

28 We can now sum upthe effectsofthecollisionupon the

29 Let us first consider the effect of the collision on the

radiiofthe vortex

MOTION OF TWO VORTEX RINGS 55

firstpasses through theshortest distance between the paths of the

firstpasses through the shortest distance,which inthis case is the

radius When is zero orthe vortex rings intersecttheshortest

both the vortex

rings

rings Equation (69) shewsthat the path ofthe vortexring CDis

.o .

,

esm ^ pq(q"~

pcose)

ON THE MOTION OF VORTEX RINGS.

viceversa Therules for findingthealteration in the radiuswere

'

2 ,

sm esin36,

parallel to the resultant of velocities p qcose and q pcose

impulse

parallelto theshortest distance between the original paths of the

thevortices

33 We have so far been engagedwith the changes in the

thecollision These changeswill be expressed bythe quantities

CDitselfcontributes tothis coefficient theterm

MOTION OF TWO VORTEX RINGS 57

JfT = 3^sin2e 5pqsin2e(q pcose),

Now -~ =thecoefficient of cos

2>|r in the expression for the

. m' 86 ,

f^log^-.a,

and (65),the term

-58 ON THE MOTION OF VOETEX KINGS.

MOTION OF TWO VORTEX RINGS 59

the effect of the vortex AB on CD Now the vortex AB will

vibra-tion, and after the collision is over the period of the vibration is