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First ZDB push. Testing repo functionality.
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This is a reproduction of a library book that was digitized by Google as part of an ongoing effort to preserve the information in books and make it universally accessible.
https://books.google.com
600048675-
PRESS 4.6.
‫بر‬
SHELF...
No
30.
C 18601 e 47
A TREATISE ON THE
MOTION OF VORTEX E
F F I L C D A
CHIVERSIT
Y R A R B I L
RINGS .
20 JAN 88 MUSS
OXFOR
Cambridge : PRINTED BY C. J. CLAY, M.A. AND SON,
AT THE UNIVERSITY PRESS.
A TREATISE ON THE
MOTION OF VORTEX RINGS .
AN ESSAY TO WHICH THE ADAMS PRIZE WAS ADJUDGED IN 1882, IN THE UNIVERSITY OF CAMBRIDGE.
BY
J. J. THOMSON, M. A.
FELLOW AND ASSISTANT LECTURER. OF TRINITY COLLEGE, CAMBRIDGE .
Y R A R B I L
E
F
F
I
L
C
D
A
RSI
20 JAN 86 * MUS
OX FGR
London : MACMILLAN AND CO .
1883
[ The Right of Translation and Reproduction is reserved.]
H
1
PREFACE .
THE subject selected by the Examiners for the Adams Prize for 1882 was
"A general investigation of the action upon each other of two closed vortices in a perfect incompressible fluid."
In this essay, in addition to the set subject, I have discussed some points which are intimately connected with it, and I have endeavoured to apply some of the results to the vortex atom theory of matter.
I have made some alterations in the notation and arrangement since the essay was sent in to the Examiners, in so doing I have received great assistance from Prof. G. H. Darwin , F.R.S. one of the Examiners, who very kindly lent me the notes he had made on the essay. Beyond these I have not made any alterations in the first three parts of the essay : but to the fourth part, which treats of a vortex atom theory of chemical action , I have made some additions in the hope of making the theory more complete : paragraph 60 and parts of paragraphs 58 and 59 have been added since the essay was sent in to the Examiners.
I am very much indebted to Prof. Larmor of Queen's College, Galway, for a careful revision of the proofs and for many valuable suggestions.
J. J. THOMSON.
TRINITY COLLEGE, CAMBRIDGE. October 1st, 1883.
T.
b
CONTENTS .
PAGE
INTRODUCTION •
ix
PART I.
PARAGRAPH
868 =3
§ 4. Momentum of a system of circular vortex rings
3
§ 5. Moment of momentum of the system
§ 6. Kinetic energy of the system
·
§ 7. Expression for the kinetic energy of a number of circular vortex
rings moving inside a closed vessel .
11
§ 8. Theory of the single vortex ring
13
§ 9. Expression for the velocity parallel to the axis of x due to an approxi-
mately circular vortex ring
15
§ 10. The velocity parallel to the axis of У
18
§ 11. The velocity parallel to the axis of z
20
§ 12. Calculation of the coefficients in the expansion of 1
(1 +a² - 2a cos 0)
in the form 4, + 4₁ cos 0 + A, cos 20 + ...
22
§ 13. Calculation of the periods of vibration of the approximately circular
vortex ring
29
PART II.
§ 14. The action of two vortex rings on each other
37
§ 15. The expression for the velocity parallel to the axis of x due to one
vortex at a point on the core of the other .
39
§ 16. The velocity parallel to the axis of y
40
§ 17. The velocity parallel to the axis of z
40
§ 20. The velocity parallel to the axis of z expressed as a function of the
time .
•
• 41
§ 21. The similar expression for the velocity parallel to the axis of y • 43
§ 22. The similar expression for the velocity parallel to the axis of x
44
§ 23.)
8 24. The expression for the deflection of one of the vortex rings
46
§ 25.
§ 26. The change in the radius of the vortex ring
50
§ 27. The changes in the components of the momentum
52
§ 28.
§ 29. Effects of the collision on the sizes and directions of motion of the
§ 30.
two vortices.
54
§ 31. b2
vill
CONTENTS.
PARAGRAPH
PAGE
§ 32. The impulses which would produce the same effect as the collision . 56
§ 33. The effect of the collision upon the shape of the vortex ring : calcu-
$ 34.
lation of
+ ∞ cos nt . dt 56
−∞ (c² + k²t²)$ (2p+1)
§ 35. Summary of the effects of the collision on the vortex rings
62
355
§ 36. Motion of a circular vortex ring in a fluid throughout which the dis-
tribution of velocity is known ·
63
$§33871. Motion of a circular vortex ring past a fixed sphere
67
8
PART III.
8223
§ 39. The velocity potential due to and the vibrations of an approximately
circular vortex column ·
71
§ 40. Velocity potential due to two vortex columns
74
§ 41. Trigonometrical Lemma • § 42. Action of two vortex columns upon each other § 42 *. The motion of two linked vortices of equal strength § 43. The motion of two linked vortices of unequal strength
75 75 78 • 86
§ 44. Calculation of the motion of two linked vortices of equal strength to
a higher order of appproximation ·
88
§ 45. Proof that the above solution is the only one for circular vortices · 92
§ 46. Momentum and moment of momentum of the vortex ring
92
-§ 47. The motion of several vortex rings linked together
93
§ 48. The equations giving the motion when a system of n vortex columns
of equal strength is slightly displaced from its position of steady
motion
94
§ 49. The case when n ==3
98
§ 50. The case when n =4
99
§ 51. The case when n = 5
100
§ 52. The case when n = 6
103
§ 53. The case when n = 7
105
§ 54. Mayer's experiments with floating magnets
107
§ 55. Summary of this Part
107
PART IV.
§ 56. Pressure of a gas. Boyle's law § 57. Thermal effusion • § 58. Sketch of a chemical theory § 59. Theory of quantivalence . § 60. Valency of the various elements
109 112 114 · 118 121
INTRODUCTION .
IN this Essay the motion of a fluid in which there are circular vortex rings is discussed. It is divided into four parts, Part I. contains a discussion of the vibrations which a single vortex ring executes when it is slightly disturbed from its circular form. Part II. is an investigation of the action upon each other of two vortex rings which move so as never to approach closer than by a large multiple of the diameter of either ; at the end of this section the effect of a sphere on a circular vortex ring passing near it is found. Part III. contains an investigation of the motion of two circular vortex rings linked through each other ; the conditions necessary for the existence of such a system are discussed and the time of vibration of the system investigated. It also contains an investigation of the motion of three, four, five, or six vortices arranged in the most symmetrical way, i.e. so that any plane perpendicular to their directions cuts their axes in points forming the angular points of a regular polygon ; and it is proved that if there are more than six vortices arranged in this way the steady motion is unstable. Part IV. contains some applicatious of the preceding results to the vortex atom theory of gases, and a sketch of a vortex atom theory of chemical action.
When we have a mass of fluid under the action of no forces, the conditions that must be satisfied are, firstly, that the expressions for the components of the velocity are such as to satisfy the equation of continuity ; secondly, that there should be no discontinuity in the pressure ; and , thirdly, that if F (x, y , z, t) = 0 be the equation to any surface which always consists of the same fluid particles, such as the surface of a solid immersed in a fluid or the surface of a vortex ring, then
dF dF dF dF + u + v + w. = 0,
dt dx dy dz
where the differential coefficients are partial, and u, v, w are the velocity components of the fluid at the point x, y, z. As we use in the following work the expressions given by Helmholtz for the velocity components at any point of a mass of fluid in which there is vortex motion ; and as we have only to deal with vortex motion which is distributed throughout a volume and not spread over a surface, there will be no discontinuity in the velocity , and so no discontinuity in the pressure ; so that the third is the only con-
X
INTRODUCTION .
dition we have explicitly to consider. Thus our method is very simple. We substitute in the equation
dF dF dF dF +u + v + w =0
dt dx dy dz
the values of u, v, w given by the Helmholtz equations, and we get differential equations sufficient to solve any of the above problems.
We begin by proving some general expressions for the momentum, moment of momentum, and kinetic energy of a mass of fluid in which there is vortex motion. In equation (9) §7 we get the
following expression for the kinetic energy of a mass of fluid in which the vortex motion is distributed in circular vortex rings,
dQ dR
+9 + h
T= Σ {23V- ( ddt
dt
dt
+ ½p√√(u² + v² + w²) pdS,
where T is the kinetic energy ; 3 the momentum of a single
vortex ring ; P , Q, R the components of this momentum along the axes of x, y, z respectively ; V the velocity of the vortex ring ;
f, g, h the coordinates of its centre ; p the perpendicular from the
origin on the tangent plane to the surface containing the fluid ; and ρ the density of the fluid. When the distance between the
rings is large compared with the diameters of the rings, we prove
in § 56 that the terms
dP dD dR
Σ
+g + h
dt
dt
dt
for any two rings may be expressed in the following forms :
ds 2 πραγ dr'
12
mmπρα α'
or
(3 cose cose - cos e),
зов
where r is the distance between the centres of the rings ; m and m' the strengths of the rings, and a and a' their radii ; S the velocity due to one vortex ring perpendicular to the plane of the other ; e is the angle between their directions of motion ; and 0, ' the angles their directions of motion make with the line joining their centres.
These equations are, I believe, new, and they have an important application in the explanation of Boyle's law (see § 56).
We then go on to consider the vibrations of a single vortex ring disturbed slightly from its circular form ; this is necessary for the succeeding investigations , and it possesses much intrinsic interest. The method used is to calculate by the expressions given
INTRODUCTION.
xi
by Helmholtz the distribution of velocity due to a vortex ring whose central line of vortex core is represented by the equations
p = a + Σ (a18 cos ny +B, sin ny),
z = 3 + (y₂ cos ny + d, sin ny) ,
where p, z, and are semi-polar coordinates, the normal to the mean plane of the central line of the vortex ring through its centre being taken as the axis of z and where the quantities a Bn Yn &n are small compared with a. The transverse section of the vortex ring is small compared with its aperture. We make use of the fact that the velocity produced by any distribution of vortices is proportional to the magnetic force produced by electric currents coinciding in position with the vortex lines, and such that the strength of the current is proportional to the strength of the vortex at every point. If currents of electricity flow round an anchor ring, whose transverse section is small compared with its aperture, the magnetic effects of the currents are the same as if all the currents were collected into one flowing along the circular axis of the anchor ring (Maxwell's Electricity and Magnetism, 2nd ed. vol. II. § 683). Hence the action of a vortex ring of this shape will be the same as one of equal strength condensed at the central line of the vortex core. To calculate the values of the velocity components by Helmholtz's expressions we have to evaluate
[" cose.de , when q is very nearly unity . This integral occurs 0
in the Planetary Theory in the expansion of the Disturbing Function, and various expressions have been found for it ; the case, however, when q is nearly unity is not important in that theory, and no expressions have been given which converge quickly in this case. It was therefore necessary to investigate some
expressions for this integral which would converge quickly in this
case ; the result of this investigation is given in equation 25, viz.
2π
1
cos no.de
π 0 √ (q - cos 0)
=
-
-
16 (q + 1) - 41 + 1 +
1
·√π2 F (1 − n, 1 + n, 1 , − x ) { log 9-1
4(1 +$
2n - 1 }
XC
1 x2
-
-
+ √π2{K, (n² — })
+ K¸2 (n² − ‡) (n² — 2) (21)222
1 x³
-
+
K¸
(n²
−
1)
(n²
—
2)
(n²
—
25)
2 (31)² 23
+ ...
1 xm
-
―
+ Km (n² − 1 ) (n² − 2 ) ... (n² −1 ( 2 m − 1)³) (m !) 22m +... }.
xii
INTRODUCTION .
1
where K1 + 1 + ...
and q = 1 + x; F ( ) denotes as
2m - 1'
usual the hyper-geometrical series.
In equations 10-18 the expressions for the components of the velocity due to the disturbed vortex at any point in the fluid are given, the expressions going up to and including the squares of the small quantities a , BnY ; from these equations, and the condition that if F (x, y, z, t) = 0 be the equation to the surface of a vortex ring, then
dF dF dF dF + u + v + w ==0,
dt dx dy dz
we get
don
Myn
64a²
==
dt
1 'n παι
log 4a - 4f(n) --11 }... (equation 37),
‫وش‬
where m is the strength of the vortex, e the radius of the transverse
1
section, and f(n) = 1 + } + ... 2n - 1
d3 m
8a
dt 2Σππaα((lloogg Sa – 1 )...(equation 41),
this is the velocity of translation, and this value of it agrees very approximately with the one found by Sir William Thomson :
dy
man
64a²
dt == 11 mπαdι; (n² - 1 ) { log 64e²0 * — 45 (n ) −1} : (equation 42).
We see from this expression that the different parts of the vortex ring move forward with slightly different velocities, and that the velocity of any portion of it is Va/p, where V is the undisturbed velocity of the ring, and p the radius of curvature of the central line of vortex core at the point under consideration ; we might have anticipated this result.
These equations lead to the equation
d'a -
+ n² (n³ − 1 ) L³a„ = 0 : (equation 44) , dt²
where
L = m {log 64a²
Απα
e³
41
―
INTRODUCTION.
xiii
Thus we see that the ring executes vibrations in the period
2π
L√
{n²
(n²
-−-
1)}
; '
thus the circular vortex ring, whose transverse section is small compared with its aperture, is stable for all displacements of its central line of vortex core. Sir William Thomson has proved that
it is stable for all small alterations in the shape of its transverse section ; hence we conclude that it is stable for all small displacements. A limiting case of the circular vortex ring is the straight columnar vortex column ; we find what our expressions for the times of vibration reduce to in this limiting case, and find that they agree very approximately with those found by Sir William Thomson, who has investigated the vibrations of a straight columnar vortex. We thus get a confirmation of the accuracy of the work.
In Part II. we find the action upon each other of two vortex rings which move so as never to approach closer than by a large multiple of the diameter of either. The method used is as follows : let the equations to one of the vortices be
p = a + (a₂ cos ny + ß₁n sin ny) ,
z = 3 + Σ (y, cos ny + d„n sin ny) ;
then, if be the velocity along the radius, w the velocity perpendicular to the plane of the vortex, we have
dẞn
B=
cos ny + sin ny ),
(ddat
dt ny),
dz w= +Σ
dt
dyn
d8n
cos ny + dt sin ny) ;
and, equating coefficients of cos ny in the expression for K, we see that da /dt equals the coefficients of cos ny in that expression. Hence we expand R and w in the form
A cos + B sin + A' cos 24+ B′ sin 2¥ + .. ..
and express the coefficients A, B, A' , B' in terms of the time ; and thus get differential equations for a BBu, Yn Sn. The calculation of these coefficients is a laborious process and occupies pp. 38-46. The following is the result of the investigation : If two vortex rings (I.) and (II.) pass each other, the vortex (I.) moving with the velocity p, the vortex (II.) with the velocity q their directions of motion making an angle e with each other ; and if c is the shortest distance between the centres of the vortex rings , g the shortest distance between the paths of the vortices, m and
Xiv
INTRODUCTION.
m' the strengths of the vortices (I. ) and (II. ) respectively, a, b their radii, and k their relative velocity ; then if the equation to the plane of the vortex ring ( II . ), after the vortices have separated so far that they cease to influence each other, be
2 = + ycos + sin ¥ + ...,
where the axis of z is the normal to the undisturbed plane of vortex (II.), we have
2ma²b
=
-
y'= c*/* ' sin² e . pq (q − p cos e ) √/( c² — gº) ( 1 – 49") : (equation 69 ),
S
2ma2b ga sin² €
- 4g2
Pq ( 1 3c²
..(equation 71 ),
and the radius of the ring is increased by
mab p*q
-
4g2
c+k+ sin³ e √/ ( c² — g ″) ( 1 — 44 ") . (equation 74) ,
where √ (c² — g²) is positive or negative according as the vortex (II .) does or does not intersect the shortest distance between the paths of the centres of the vortices before the vortex (I. ).
The effects of the collision may be divided in three parts : firstly, the effect upon the radii of the vortex rings ; secondly, the deflection of their paths in a plane perpendicular to the plane containing parallels to the original directions of motion of the vortices ; and, thirdly, the deflection of their paths in the plane parallel to the original directions of motion of both the vortex rings.
Let us first consider the effect upon the radii. Let g = c cos &,
thus is the angle which the line joining the centres of the vortex rings when they are nearest together makes with the shortest distance between the paths of the centres of the vortex rings ; & is positive for the vortex ring which first intersects the shortest distance between the paths negative for the other ring.
The radius of the vortex ring (II.) is diminished by
ma²b -p²q sin³ e sin 3p.
c34
Thus the radius of the ring is diminished or increased according as sin 34 is positive or negative. Now is positive for one vortex ring negative for the other, thus sin 34 is positive for one vortex ring negative for the other, so that if the radius of one vortex ring is increased by the collision the radius of the other will be diminished . When is less than 60° the vortex ring which first passes through the shortest distance between the paths of the
INTRODUCTION .
XV
centres of the rings diminishes in radius and the other one increases. When is greater than 60° the vortex ring which first passes through the shortest distance between the paths increases in radius and the other one diminishes. When the paths of the centres of the vortex rings intersect is 90° so that the vortex ring which first passes through the shortest distance , which in this case is the point of intersection of the paths, is the one which increases in radius. When is zero or the vortex rings intersect the shortest distance simultaneously there is no change in the radius of either vortex ring, and this is also the case when is 60º.
Let us now consider the bending of the path of the centre of one of the vortex rings perpendicular to the plane which passes through the centre of the other ring and is parallel to the original paths of both the vortex rings.
We see by equation (71 ) that the path of the centre of the vortex ring (II. ) is bent towards this plane through an angle
ma2 f c³ ³ pq sin² e cos 3p,
this does not change sign with 4 and, whichever vortex first passes through the shortest distance, the deflection is given by the rule that the path of a vortex ring is bent towards or from the plane through the centre of the other vortex and parallel to the original directions of both vortices according as cos 30 is positive or negative, so that if is less than 30° the path of the vortex is bent towards, and if o be greater than 30°, from this plane. It follows from this expression that if we have a large quantity of vortex rings uniformly distributed they will on the whole repel a vortex ring passing by them.
Let us now consider the bending of the paths of the vortices in the plane parallel to the original paths of both vortex rings. Equation (69) shews that the path of the vortex ring (II .) is bent in this plane through an angle
2ma sin² e sin 34 pq (q - p cos e)
c³k
towards the direction of motion of the other vortex. Thus the direction of motion of one vortex is bent from or towards the direction of motion of the other according as sin 30 (q - p cos e) is positive or negative. Comparing this result with the result for the change in the radius, we see that if the velocity of a vortex ring (II. ) be greater than the velocity of the other vortex (I. ) resolved along the direction of motion of (II.) , then the path of each vortex will be bent towards the direction of motion of the other when its radius is increased and away from the direction of motion of the other when its radius is diminished, while if the
xvi
INTRODUCTION .
velocity of the vortex be less than the velocity of the other resolved along its direction of motion, the direction of motion will be bent from the direction of the other when its radius is increased and
vice versa. The rules for finding the alteration in the radius were given before.
Equation (75) shews that the effect of the collison is the same as if an impulse
pqJ.'J sin e sin 30,
προ
parallel to the resultant of velocities pq cos e, and q - p cos e along the paths of vortices (II . ) and ( I. ) respectively and an impulse
pqI.I' sin² e cos 30,
3προκ
parallel to the shortest distance between the original paths of the vortex rings, were given to one of the vortices and equal and opposite impulses to the other ; here I and I' are the momenta of the vortices.
We then go on to investigate the other effects of the collision. We find that the collision changes the shapes of the vortices as well as their sizes and directions of motion . If the two vortices are
equal and their paths intersect, equations (78) and (79) shew that, after collision, their central lines of vortex core are represented by the equations
mπn*a* √/√2 €¯nc/k
p = α-
sin (2 +nt + €),
4k √3 (nc/k)
4 mπn*a* √2 €-nc/k
2 = 3+ 8k5
cos (24+ nt + €),
(nc/k)
where 2π/n is the free period of elliptic vibration of the circular axis. These are the equations to twisted ellipses, whose ellipticities are continually changing ; thus the collision sets the vortex ring vibrating about its circular form.
We then go on to consider the changes in size, shape, and direction of motion , which a circular vortex ring suffers when
placed in a mass of fluid in which there is a distribution of velocity
d given by a velocity potential . We prove that if denotes
dh
differentiation along the direction of motion of the vortex ring, l, m , n the direction cosines of this direction of motion, and a the radius of the ring,
INTRODUCTION .
da
ΦΩ
Τα
dt
dh2
dl ΦΩ ΦΩ
=
dt
dh2 dxdh
dm ΦΩ =m
dt dhe
dn ΦΩ =n
dt dh"
ΦΩ dydh ΦΩ dz dh
(equation 80).
xvii
The first of these equations shews that the radius of a vortex ring placed in a mass of fluid will increase or decrease according as the velocity at the centre of the ring along the straight axis decreases or increases as we travel along a stream line through the centre. We apply these equations to the case of a circular vortex ring moving past a fixed sphere, and find the alteration in the radius and the deflection .
In Part III. we consider vortex rings which are linked through each other. We shew that if the vortex rings are of equal strengths and approximately circular they must both lie on the surface of an anchor ring whose transverse section is small compared with its aperture, the manner of linking being such that there are always portions ofthe two vortex rings at opposite extremities ofa diameter of the transverse section. The two vortex rings rotate with an angular velocity 2m/πd round the circular axis of the anchor ring, whilst this circular axis moves forward with the comparatively slow
m
64a2
velocity
log
where m is the strength and e the radius of
2πα
e²
the transverse section of the vortex ring, a is the radius of the circular axis of the anchor ring and d the diameter of its transverse section.
We begin by considering the effect which the proximity of the two vortex rings has upon the shapes of their cross sections ; since the distance between the rings is large compared with the radii of their transverse sections and the two rings are always nearly parallel, the problem is very approximately the same as that of two parallel straight columnar vortices, and as the mathematical work is more simple for this case, this is the one we consider. By means of a Lemma (§ 33) which enables us to transfer cylindrical harmonics from one origin to another, we find that the centres of the transverse sections of the vortex columns describe circles with the
centre of gravity of the two cross sections of the vortex columns as centre , and that the shapes of their transverse sections keep changing, being always approximately elliptical and oscillating about the circular shape, the ellipticity and time of vibration is given by
xviii
INTRODUCTION.
equation (89). We then go on to discuss the transverse vibrations of the central lines of vortex core of two equal vortex rings linked together. We find that for each mode of deformation there are two periods of vibration, a quick one and a slow one.
If the equations to the central line of one of the vortex rings be
p = a + acos ny +Bมsin ny,
2 = 8 + y₁₂ cos ny + dn sin ny,
and the equations to the circular axis of the other be of the same
form with a , Bn, Y , 8 , written for an' B , Y , &n,' we prove
α„ = A cos (vt + €) -− B cos (µt + €)
a = A cos (vt + e) + B cos (ut + € )
=
√
(n²
-—
Yn
n
1)
A
sin
( vt +
e)
+
B
sin
(ut
+
e')
(equation 96) ,
√ (n² - 1)
Yn'== n
A sin (ut + c) – Bsin (ut + c )
where
m
64a²
V = 2πa³√ {n² (n² — 1 )} log de
m 2 (2n² - 1)
μπ
log
π d²
4a²
Thus, if the conditions allow of the vortices being arranged in this way the motion is stable. In § 41 we discuss the condition necessary for the existence of such an arrangement of vortex rings ; the result is, that if I be the momentum, I the resultant moment of momentum, r the number of times the vortices are linked through each other, and p the density of the fluid, then I, I are constants determining the size of the system, and the conditions are that
I = 4mπρα ,
г = mπрrad².
These equations determine a and d; from these equations we get
d² 4г (4mπp)*
a²
TI
Now da² must be small, hence the condition that the rings should be approximately circular and the motion steady and stable, is that I (4mπp) /rI should be small . We then go on to consider the case of two unequal vortex rings, and in § (43) we arrive at results. similar in character to those we have been describing ; the chief difference is that the system cannot exist unless the moment of momentum has a certain value which is given in equation (105 ), and which only depends on the strengths and volumes of the
INTRODUCTION .
xix
vortices, and the number of times they are linked through each other.
In the latter half of Part III. we consider the case when n
vortices are twisted round each other in such a way that they all lie on the surface of an anchor ring and their central lines of vortex core cut the plane of any transverse section of the anchor ring at the angular point of a regular polygon inscribed in this cross section. We find the times of vibration when n equals 3, 4, 5 , or 6, and prove that the motion is unstable for seven or more vortices, so that not more than six vortices can be arranged in this way.
Part IV. contains the application of these results to the vortex atom theory of gases, and to the theory of chemical combination.
:
H
ON THE MOTION OF VORTEX RINGS .
§1. THE theory that the properties of bodies may be explained by supposing matter to be collections of vortex lines in a perfect fluid filling the universe has made the subject of vortex motion at present the most interesting and important branch of Hydrodynamics. This theory, which was first started by Sir William Thomson, as a consequence of the results obtained by Helmholtz in his epoch-making paper " Ueber Integrale der hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen " has à priori very strong recommendations in its favour. For the vortex ring obviously possesses many of the qualities essential to a molecule that has to be the basis of a dynamical theory of gases. It is indestructible and indivisible ; the strength of the vortex ring and the volume of liquid composing it remain for ever unaltered ; and if any vortex ring be knotted, or if two vortex rings be linked together in any way, they will retain for ever the same kind of be-knottedness or linking. These properties seem to furnish us with good materials for explaining the permanent properties of the molecule. Again, the vortex ring, when free from the influence of other vortices, moves rapidly forward in a straight line ; it can possess, in virtue of its motion of translation, kinetic energy ; it can also vibrate about its circular form, and in this way possess internal energy, and thus it affords us promising materials for explaining the phenomena of heat and radiation .
This theory cannot be said to explain what matter is, since
it postulates the existence of a fluid possessing inertia ; but it proposes to explain by means of the laws of Hydrodynamics all the properties of bodies as consequences of the motion of this fluid .
It is thus evidently of a very much more fundamental character
than any theory hitherto started ; it does not, for example, like the
ordinary kinetic theory of gases, assume that the atoms attract
each other with a force which varies as that power of the distance
T.
1
2
ON THE MOTION OF VORTEX RINGS.
which is most convenient, nor can it hope to explain any property of bodies by giving the same property to the atom. Since this theory is the only one that attempts to give any account of the mechanism of the intermolecular forces, it enables us to form much the clearest mental representation of what goes on when one atom influences another. Though the theory is not sufficiently developed for us to say whether or not it succeeds in explaining all the properties of bodies, yet, since it gives to the subject of vortex motion the greater part of the interest it possesses, I shall not scruple to examine the consequences according to this theory of any results I may obtain.
The present essay is divided into four parts : the first part, which is a necessary preliminary to the others, treats of some general propositions in vortex motion and considers at some length the theory of the single vortex ring ; the second part treats of the mutual action of two vortex rings which never approach closer than a large multiple of the diameter of either, it also treats of the effect of a solid body immersed in the fluid on a vortex ring passing near it ; the third part treats of knotted and linked vortices ; and the fourth part contains a sketch of a vortex theory of chemical combination, and the application of the results obtaining in the preceding parts to the vortex ring theory of gases.
It will be seen that the work is almost entirely kinematical ; we start with the fact that the vortex ring always consists of the same particles of fluid (the proof of which, however, requires dynamical considerations), and we find that the rest of the work is kinematical. This is further evidence that the vortex theory of matter is of a much more fundamental character than the ordinary solid particle theory, since the mutual action of two vortex rings can be found by kinematical principles, whilst the " clash of atoms " in the ordinary theory introduces us to forces which themselves demand a theory to explain them.
PART I.
Some General Propositions in Vortex Motion.
§ 2. WE shall, for convenience of reference , begin by quoting the formulae we shall require. We shall always denote the components of the velocity at the point (x, y, z ) of the incompressible fluid by the letters, u, v, w ; the components of the angular velocity of molecular rotation will be denoted by έ, n, S.
Velocity.
§3. The elements of velocity arising from rotations §', n', ' in the element of fluid dx'dy'dz' are given by
1
би =
{n'
(z
—
z')
— 5′
(y
— y')}
dx'dy'dz'
2πp³
1
Sv =
{5'
(x
−- x')
—
§'
(z
— 2') }
dx'dy'dz '
}
... ( 1) ,
2π³
1
δω =
{§' (y — y') — n' (x − x')} dx'dy'dz'
27.3
where r is the distance between the points (x, y, z) and (x' , y', z′) .
Momentum .
§4. The value of the momentum may be got by the following considerations : Consider a single closed ring of strength m, the velocity potential at any point in the irrotationally moving fluid
m due to it is - times the solid angle subtended by the vortex
2π
ring at that point, thus it is a many-valued function whose cyclic constant is 2m. If we close the opening of the ring by a barrier, we shall render the region acyclic. Now we know that the motion at any instant can be generated by applying an impulsive pressure
1-2
4
ON THE MOTION OF VORTEX RINGS.
to the surface of the vortex ring and an impulsive pressure over the barrier equal per unit of area to p times the cyclic constant, p being the density of the fluid. Now if the transverse dimensions of the vortex ring be small in comparison with its aperture, the impulse over it may be neglected in comparison with that over the barrier, and thus we see that the motion can be generated by a normal impulsive pressure over the barrier equal per unit of area to 2mp.
Resolving the impulse parallel to the axis of x, we get
momentum of the whole fluid system parallel to x = 2mpx (projection of area of vortex ring on plane yz),
with similar expressions for the components parallel to the axes of y and z.
Thus for a single circular vortex ring, if a be its radius and λ, μ, v the direction -cosines of the normal to its plane, the components of momentum parallel to the axes of x, y, z respectively are
2прта?),
2npma2v.
The momentum may also be investigated analytically in the following way :
Let P be the x component of the whole momentum of the fluid which moves irrotationally due to a single vortex ring of strength m.
Let be the velocity potential, then ΦΩ
P= =[[[padnx dx dy dz.
Integrating with respect to x,
P = SSp (Q - Q₂) dy dz,
where , and , are the values of 2 at two points on opposite sides of the barrier and infinitely close to it. Now the solid angle subtended by the ring increases by 47 on crossing the boundary, thus
therefore
Q₁ - L₂2 = 2m ; P = 2m ffp dy dz,
where the integration is to be taken all over the barrier closing the vortex ring ; if λ, µ, v be the direction-cosines of the normal to this barrier at any point
P = 2mp fЛxdS, where dS is an element of the barrier.
ON THE MOTION OF VORTEX RINGS.
5
Now
dz
-z ssads = }[(y ds dsy)ds,
where ds is an element of the boundary of the barrier, i.e. an element of the vortex ring, thus
dz Р = mp MУyds -
2 dy ds
ds
= PSSS(y - zn) dx dy dz,
and if we extend the integration over all places where there is vortex motion, this will be the expression for the x component of the momentum due to any distribution of vortex motion.
Thus, if P, Q, R be the components of the momentum along
x, y, z respectively,
P = p√√√(y5 — zn) dx dy dz
Q = p [ƒ(z - x ) dx dy dz
R = p √√√(xn− y§) dx dy dz
. ( 2) .
Again
dP dt = p fffddutdx dy dz.
But where a force potential V exists,
du
dx
= 2υζ – 20η - "
dt
dx
where
= Χ
[dp +
V+
(vel . )²
ρ
(Lamb's Treatise on the Motion of Fluids, p. 241 ) ; therefore
dP
-
dt
= P√√[√[((2205
-
2wn
dx) dx dy dz. - dx) dx
Since Χ is single-valued and vanishes at an infinite distance,
SfSfSfdxdx dy dz = 0.
Again,
If(v - wn) dx dy dz = 0
(Lamb's Treatise, p. 161 , equation 31 ) ; therefore
dP = 0,
dt
or P is constant. We may prove in a similar way that both Q and Rare constant ; thus the resultant momentum arising from any distribution of vortices in an unlimited mass of fluid remains constant both in magnitude and direction.
6.
ON THE MOTION OF VORTEX RINGS.
Moment of Momentum.
§ 5. Let L, M, N be the components of the moment of momentum about the axes of x, y, z respectively ; let the other notation be the same as before ; then for a single vortex ring
L = pfff(wy — vz) dx dy dz
ΦΩ ΦΩ
=p У ·SSS(3
dz
2 dya) da dy dz
=
p
SS{y
(Q,
—
Q₂)
dx
dy.—
z
(N¸1 — N¸)
dx
dz}
= =
2mp
[f(zµ
—
yv)
dS ;
this surface integral is, by Stokes' theorem, equal to the line
integral
dx + √(2² + y²) Is ds.
dx
So
L = mp √(z² + y²) ds
ds
= PSSS(2² + y²) § dx dy dz ;
and if we extend the integration over all places where there is vortex motion , this will be the expression for the a component
of the moment of momentum due to any distribution of vortices. Thus
L = p √√√(y² + z²) § dx dy dz
M= p√√√(2² + x²) ʼnη dx dy dz
.(3).
N = p√√√(x² + y²) Ç dx dy dz
Again,
dL
dw dv\
=-
Y dt 2 dx dy dz ;
dt SSSC:
dt
as before,
dv
de
= -
2w5-2u5-
dx,
dt
dy
dw
dx
= 2un dt
2v§ →
dz
thus
dL
=
-
dt · 2 ƒƒƒ{y ( un — v§) — z (w§ — u§) } dx dydz
-
dz.
+ S[S[[S((==dxy - y ddxz) dx dy da
Since Χx is a single-valued function, the last term vanishes, and
dv
dv du
–
И-
fiz (wỹ – u5) dxdydz = [[S ]z {w(dy –ddz)- dx dy)} dxdydz.
ON THE MOTION OF VORTEX RINGS.
7
Integrating this by parts, it
= [f(zw³dxdz — zwvdx dy – zuvdy dz + zu³dxdz )
dw
dw
du
du
ZV -Zv + zu -vw ) dxdydz.
- SSS(zu dy
dz
dx
dy
The surface integrals are taken over a surface at an infinite
distance R from the origin ; now we know that at an infinite 1
distance u, v, w are at most of the order while the element of R
surface is of the order R², and z is of the order R ; thus the surface 1
integral is of the order at most, and so vanishes when R is inR
definitely great.
Integrating by parts, similar considerations will shew that
dw [[[zw tdoy dxdydz = 0,
du zu dx dy dz = 0 ; SSS dy
so the integral we are considering becomes.
dw
du
20 SSSC=
dz
+
zv
dx
+
vw
)
dx dydz
;
or, since
du dv dw + + = 0,
dx dy dz
it since
-SISSfC:(zvdy - vw ) dxdydz
= fffvw dx dydz,
dv 20 dxdydz = 0. SSS= dy
Similarly 2fy (un — v§) dx dy dz = fffvwdx dydz,
dL
and thus
=
-
·2p Sƒƒ{y (un — v§) — z (wę — u ) } dx dy dz = 0 ;
dt
thus L is constant. We may prove in a similar way that M and N are also constant, and thus the resultant moment of momentum arising from any distribution of vortices in an unlimited mass of fluid remains constant both in magnitude and direction . When there are solids in the fluid at a finite distance from the vortices, then the surface integrals do not necessarily vanish, and the momentum and moment of momentum are no longer constant.
8
ON THE MOTION OF VORTEX RINGS.
Kinetic Energy.
§ 6. The kinetic energy (see Lamb's Treatise, § 136)
=
-
= 2p SSS{u (y5 — zn) + v (z§ − x§) + w (xn− y§) } dx dydz ;
this may be written, using the same notation as before,
dx
=
2p
Σ ≥ [ mf{u
(y ddsz−
z
dys )
+
v
(z ddasx
−
x
ddsz)
+
w
(x
dsy
-y
le)} ds],
where Σ means summation for all the vortices.
We shall in subsequent investigations require the expression for the kinetic energy of a system of circular vortex rings. To evaluate the integral for the case of a single vortex ring with any origin O we shall first find its value when the origin is at the centre C' ; then we shall find the additional term introduced when we move the origin to a point P on the normal to the plane of the vortex through C', and such that PO is parallel to the plane of the vortex ; and, finally, the term introduced by moving the origin from P to 0.
When the origin is at C', the integral = = 2pm J Vads,
where V is the velocity perpendicular to the plane of the vortex.
If V' be the mean value of this quantity taken round the ring, the
integral
=4πрm a²V'.
When we move the origin from C' to P, the additional term introduced
=- 2pm Sp Rds,
where R is the velocity along the radius vector measured outwards,
and p the perpendicular from O on the plane of the vortex ; thus the integral
d = - 2mp P at(πα ).
When we change the origin from P to O the additional term introduced
= = 2pm fc cos & Vds,
where c is the projection of OC' on the plane of the vortex ring, and the angle between this projection and the radius vector drawn from
the centre of the vortex ring to any point on the circumference.
Let us take as our initial line the intersection of the plane of the vortex ring with the plane through its centre containing the normal and a parallel to the axis of z.
ON THE MOTION OF VORTEX RINGS.
9
Let be the angle any radius of the vortex ring makes with
this initial line, w the angle which the projection of OC on the plane of the vortex makes with this initial line ; then
$ = 4 - w.
Let V be expanded in the form
V = V' + A cosy + B sin + C cos 24 + D sin 24 + &c.,
then
fcos & Vds = πα ( A cos w + В sin w) .
Since V is not uniform round the vortex ring, the plane of the vortex ring will not move parallel to itself, but will change its aspect. We must express A and B in terms of the rates of change of the direction-cosines of the normal to the plane of the vortex ring.
Let the perpendicular from any point on the vortex ring at the time t + dt on the plane of the ring at the time t be
83 + dx cos + Sẞ sin ;
thus the velocity perpendicular to the plane of the vortex
dz dz
dB
= + cos + sin y.
dt dt
dt
Comparing this expression with the former expression for the
velocity, we get
d V' =
dt'
da A=
dt'
dB B=
dt
Fig.1. 8
B
V
P M
đa d
We must now find
in terms of the rates of change of the
dt' dt
direction-cosines of the normals to the plane of the ring.
10
ON THE MOTION OF VORTEX RINGS.
Draw a sphere with its centre at the centre C' of the vortex ring. Let A, B, C be the extremities of axes parallel to the axes
Let I be the pole of the ring determined by e and as shewn in the figure. Let MN be the ring itself and P any point
on it defined by the angle . The displaced position of the plane of the vortex ring may be got by rotating the plane of the ring through an angle 8B/a about the radius vector M in the plane of
the ring for which = 0, and through an angle da/a about the π
radius vector N for which = . The first rotation leaves ✪ un2'
changed and diminishes e by Sẞ/a sin 0 ; the second rotation diminishes by da/a and leaves e unaltered, thus
δα 80 =
a
SB δε
a sin e
If l, m, n be the direction-cosines of I it is clear that l = sin e cos e, m = sin 0 sin e, n = cos 0, and
δα
SB
... δι =
cos e cos e + sin €,
a
a
δα
SB
Sm =
cos e sin e
COS €,
а
a
δα
Sn =
sin 0.
a
It follows at once that
da a dn =
dt sin e dt
dB
dl
dm
sin e- COS €
dt
dt
dt e),
therefore
(dn cos w dl
dm
A cos + B sin w = a
+ sin €
dt sin e dt
-ddtecos e) sin a }.
Now if X', ', ' be the direction-cosines of the projection
of OC' on the plane of the vortex ring, and f, g, h the coordinates of C',
λ=
--cos e cos e cos @ — sin e sin w,
μsin e cose cos @ + cos e sin ω, v - sin cos w.
It is also easily proved that x' =f-lp
ON THE MOTION OF VORTEX RINGS.
11
μ' =9 - mp
h - np ν=
c
p = lf+ mg + nh.
So COS W
v' -
(h― np) =__ Inf+ mng - sin³0 . h
sin
c sin e
c sin
1 sin ∞ = μ' cos e -X'sin e = sin 0 (μ'l —-X'm)
thus
1
= c
sine
(lg
-
mf)
;
A cos + B sin @ = a Sdn cos @ +
sindt
(ddtim - ddmt 1) sin o }.
This, after substituting for cos w and sin o the values given
above,
dn dm =- h + g +f
-aс (k dt dt ddtl).
Thus
2pmсπα
(A
cos
w
+
B
sin
w)
―― 2прта2
.dl dm dn
+g + h dt dt dt
Thus the kinetic energy of the vortex ring
d
dm
= = 2pm
2πα²
V'
— 2mpp
dt (πа³)
—
2πpmа³
+9 (ƒddt
dt
+h dn).
If I be the momentum of the vortex ring, viz. 2πpma², and
P , Q, R the components of I along the axes of x, y, z respectively,
this may be written, since plf+ mg + nh,
.dv da dR
2IV'
+9 + h
-( 9dt
dt
dt
and thus the kinetic energy of any system of circular vortex rings
dP da dR
-
+g +h
= {21V ( ddt
dt
dt
(8) .
This expression for the kinetic energy is closely analogous to Clausius expression for the virial in the ordinary molecular theory of gases.
§ 7. We have in the preceding investigation supposed that the bounding surfaces were infinitely distant from the vortices, so that the surface integrals might be neglected ; we shall, however, ' require the expression for the kinetic energy when this is not the case.
12
ON THE MOTION OF VORTEX RINGS.
The expression -
2p Лff{u (y5− zn) + v (z§− x5) + w (xn− y§) } dx dydz
becomes, on integrating by parts and retaining the surface integrals, supposing, however, that the boundaries are fixed so that
lu + mv + nw = 0,
if l, m, n are the direction-cosines of the normal to the boundary surface,
½p SSS(u² + v² + w²) dx dydz − } p SS (u² + v² + w³) (x dy dz + y dxdz + z dxdy) ,
or if dS be an element of the surface and p the perpendicular
from the origin on the tangent plane
=
-
{P SSS(u² + v² + w³) dx dydz − p ƒƒ (u² + v² + w²) pdS.
But by the preceding investigation it also equals
dv dQ
-
+9 + h
Σ {2IV® – (. dt
dt ddot).
Thus T, the kinetic energy, is given by the equation
da
T=
-
+ g + hd}') } +{pƒƒ(u² + v² + w")pd‚…§ … ..(9).
" = {2IV" — ( ddt) + dt dt
MOTION OF A SINGLE VORTEX.
§ 8. HAVING investigated these general theorems we shall go on to consider the motion of a single approximately circular vortex ring. We shall suppose that the transverse section of the vortex core is small compared with the aperture of the ring. We know that the velocity produced by any distribution of vortices is proportional to the magnetic force produced by electric currents coinciding in position with the vortex lines, and such that the strength of the current is proportional to the strength of the vortex at every point. Now if currents of electricity flow round an anchor ring, whose transverse section is small compared with its aperture, the magnetic effects of the currents are the same as if all the currents were collected into one flowing along the central line of the anchor ring ( Maxwell's Treatise on Electricity and Magnetism, 2nd edition, vol. II. , § 683 ). Hence the action of a vortex ring of this shape will be the same as one of equal strength condensed at the central line of the vortex core.
Let the equation to this central line be p = a + acos ny + ẞ,13 sin ny, z = 3 + Y₂n cos ny + d₁ sin ny,
when ' , p, are the cylindrical coordinates of a point on the central line of the vortex core, the normal of the vortex ring being taken as the axis of z, the axis of x being the initial line from which the angle is measured. a is the mean radius of the central line of the vortex core, the perpendicular from the origin on the mean plane of the vortex, and a B Yn & quantities which are very small compared with a. Let m be the strength of the vortex ring, e the radius of the transverse section of the core. Now, by equations (1 ) , the velocity components due to a vortex of this
14
ON THE MOTION OF VORTEX RINGS.
strength, situated at the central line of the vortex core, are
given by
m1 u=
2π
- dy' z')
dz't
- (y - y')
ds' ,
ds'
ds')
m1
dz'
dx'
v= 2π
(x − x')
³
ds'
(z - z') dsfds,'
m W=
2π
dx' -
{(y - y) ds' (x − x') dsy' } ds',
where r is the distance between the points (x, y , z) and ( x', y' , z') , and the integrals are taken all round the vortex ring.
Now
x = p cos y = a cos y + α, cos ny cos y + ẞn sin ny cos y,
y' = p sin = a sin + a, cos ny sin y + ßn, sin ny sin y, therefore
da
a
sin
c
−
sin
g
(an
cos
nh
t
ơn
cos
ny)
dy
— n cos (an sin ny – ß„ cos ny),
dy' = a cos + cos (a, cos ny + B, sin ny),
dy
-n sin
dz
= ―n (y, sin ny – d₁n cos ny). dy
(a sin ny - B₁ cos ny ),
In calculating the values of u, v, w we shall retain small quantities up to and including those of the order of the squares of an Bn, Y Sn" Although, for our present purpose, which is to find the time of oscillation of the vortex about its circular form, we only require to go to the first powers of a,, &c., yet we go to the higher order of approximation because, when we come to consider thẹ question of knotted vortices, we require the terms containing the squares of these quantities.
If R, 4, z be the cylindrical coordinates of the point x, y, z,
-73 = {p² + R² − 2p R cos ( − ) + (≈ — 2') ²} ",
1 now when we substitute for p its value it is evident that 28 can be
expanded in the form
Σ (s) (4¸ + B¸ cos n↓ + C¸ sin ny + D¸cos 2n↓ + E, sin 2n↓)
× cos s ( − ),
where A,8 contains terms independent of a,..., B, and C, are of the first, and D, and E, of the second order in these quantities.
MOTION OF A SINGLE VORTEX.
15
The part of A, which is independent of a ... is evidently
1 (2π
cos sᎾ ᏧᎾ
π= (a² + R² + z² −- 2aR cos 0)¹ ›
but we shall investigate the values of all these coefficients later.
Velocity parallel to the axis of x.
§9. In the equation
m (2π 1 u=
2π. 0 зов
dy
-
— (y − y) d ) dự, dy
the expression to be integrated becomes, when the values for Y', z, dy dz
are substituted and the terms arranged in order of ds' ' ds'
magnitude, being written for z 8,
1 (Ça cos y + ny (y, sin ny — §,n cos ny)
+ } { (n + 1 ) 5x„n + (n −- 1 ) ay } cos (n + 1 ) † ↓
- } {(n · · 1 ) 5ª, + ( n + 1 ) ay } cos (n − 1 )
+ } { (n + 1 ) ¿B₂ + (n − 1 ) ad„ } sin (n + 1 ) y
— - { (n − 1) 5Bn + (n + 1 ) ad } sin (n - 1)
+ n (1,8d,n - Y₂n B ) sin - ½ (2, Yn + B₁n d ) cos y ↓
-
-
− 1 (ªn Yn − B₂n Ɛn ) { cos ( 2n + 1 ) ¥ + cos (2n − 1 ) ¥}
−
1
(an dn +
Yn Bn)
{sin
( 2n +
1)↓
+ sin
(2n
−
1)
↓} )
.
m 2 La cos
Let us consider the term
dy.
2π
для
1 Expanding this equals
2π m[* dy ga cos y Σ. (s) { (4¸8 + B¸ cos nỵ + C¸ sin nỵ 2π
+ D, cos 2n + E, sin 2n ) cos s ( − $)} .
Remembering that 2π cos my cos ny dy = 0 if m does not equal n,
this equals
16
ON THE MOTION OF VORTEX RINGS.
+
2π
m[*dy sa cosy x 2π
+ [{4,1 cos ( - )]
+ [Bn++1 cos ( -− ( n + 1 ) p } + BB₂, -1 cos { + ( n - 1 ) } ]
1 + [Cn--1 sin { + ( n - 1 ) 4p } - CC₂ +1 sin {
( n + 1 ) ø} ]
+ [Dan+1 CcOoSs { -— (2n + 1 ) $} + Dan-1 cos (y + (2n - 1 ) p} ]
+ [[EE2nn-s1in {{y + (2n - 1 ) p} - E2 +1 sin { (2n + 1 ) $} ])
= = { ma } [A¸ cos &
+ ½ {Bati cos ( n + 1 ) + B -1 cos ( n - 1 ) + CN-,1 sin ( n - 1 ) p
+ Cnt sin (n + 1 ) ø}
+ } {D₂n+1 cos (2n + 1 ) + Dan-1 cos (2n - 1) + E2n--1 sin (2n - 1 ) p
+ Egnti sin (2n + 1) ø}].
Similarly, we may prove that
m 2T 1
-
2π[ * дов ny (v sin ny — §, cos ny) dự
= mny {A, (y, sin no- 8, cos no) + Coy₂- BB¸odnμ
++ (B₂n Yn - Ca2nn Sn) sin 2n - (B₂n En + Can Yn) cos 2n p},
and that
m 2″ 1
4π
[2" 0
2018
{(n
+
1)
5²n
+
(n
−
1)
ay„ }
cos
(n
+
1) ydy
== ‡m { (n + 1 ) 5αn + (n − 1 ) aɣn}
x {Annt+i1cos (n + 1) + (B, cos & -C, sin & + Ba2nnt+1 cos (2n + 1 )
and that
+ Can +1 sin (2n + 1) $)},
2″ -
4mπ(* 17 {(n − 1 ) 5x + (n + 1 ) ay } cos (n −− 1) ↓dự
= ‡m { (n - − 1 ) 5α„ + (n + 1) ayn}
x {{A4 -1 cos (n - 1) 4 + ½ (B, cos & + C, sin & + B₂n-1 CcOoSs (2n - 1 )
and that
+ Can-1 sin (2n - 1) 4)} ,
2T
m(* 13 { (n + 1) 58, + (n − 1 ) aò̟„} sin (n + 1) ↓dự Ꮞ .0
== ‡ m { (n + 1 ) [B„ + (n − 1) ad„}
× {An+1 sin (n +1 ) 4 + ½ (B¸ sin 4 + C₁cos + B₂n+1 sin (2n + 1 ) ø -- Can+1 Cos (2n +1 ) 6)} ,
MOTION OF A SINGLE VORTEX.
17
and
m 2″ 1
4π[0** 2/03
{ (n
−
1)
¿³,n„
+
(n
+
1)
ad „ }
sin
(n − 1 )
↓dự
=
-
1m { (n − 1 ) SB₂ + (n + 1 ) ad„ }
× {4,-, sin (n − 1) ☀ + † ( − B¸ sin + C, cos & + B,2n--11 sin (2n - 1 )
---- Can-1 cos (2n - 1) 6 ) }
The integral of the terms involving the products a,, Bn ,...
= = {m [ n A¸1 (a„§„ — B„ „) sin & – A₂ (ªn¥n + ß„dn) cos &
− — (ann - B₂dn) { A2n+1 cos (2n + 1 )
+ A₂n-1 cos ( 2n − 1 ) 4}
- ‡ (and + BY ) {A2n +1 sin (2n + 1 ) † + 4A-,2n1-1 sin (2n − 1 ) 4} ] .
Thus u terms not containing a +terms containing a ... to the first power + terms containing a ... to the second power.
The term not containing a
= = 1mga A ,1 cos p………...
.. (10).
The terms containing a ... to the first power
= = m [ 2ny A₂n (Y₂n, sin no - 8n, cos no)
+ {Ça B₂ +1 + [ (n + 1 ) 5x + (n − 1 ) ay, ] 4(n+1)} cos (n + 1 ) $
+ { a B -1 - [(n - 1 ) an + (n + 1 ) ay ] A-n-1} cos (n - 1) φ
+ {5a COn+1 + [(n + 1 ) B + (n - 1 ) ad ] 4n++1) sin (n + 1 ) -
+ { % a C n-1 −· [ (n − 1 ) $ 3 , + ( n + 1 ) a8d„,] A ,n--1} sin ( n − 1 ) ¢] ( 11 ).
The terms containing a ... to the second power = = { m [ny {Y₂ ( 2C, + B₂2,n sin 2nd - C₂2n cos 2np) - „ (2B。
+ B₂2nn cos2no + C₁2₂n sin 2nd) }
+ { − (ªn¥n + Bndµ) A¸1 + (Çαn − aɣn) B₂1 + ( Çß₂ − ad₁ ) C₁} cos &
-
-
+ n {2 (andn − Bn¥n) A₁1 + ( SBn„ + ad„) B¸ − (Ǫ„ + ay ) C₁ } sin &
+ { - } (~₂Y₂- B₂n8 ) A 2n +1 + } [ ( n + 1 ) a + ( n - 1 ) ay ] B₂n +1
- [(n + 1 ) SBn + (n − 1 ) ad ] Can+1 ++ aα¿DD2ann+1}} cos (2n + 1) p
+ { - } (α- Bß₂„S§„) A2n-1 - [ (n − 1 ) 5¹„ + (n + 1 ) ay ] B₂2n-1 -
+ } [(n − 1) SB,n + ( n + 1 ) ad ] C₂n-1 + açDan- 1} cos (2n − 1 ) p
+ { - } (0,n8d₂n + BnYn) A2n +1 + }½ [ ( n + 1 ) (BB₁n + (n -− 1 ) ad„ ] B₂n +1
+ ½ [ (n + 1 ) Sa₂ + ( n − 1 ) ayn] C₂n+1 + a 【E2n +1} sin ( 2n + 1 ) ø
+ { − & (and₂ + B₂Yn) A2n-1 -—- } [ ( n - − 1 ) ¿B₁n + (n + 1 ) ad„ ] B₂n-1
−}
[ (n −
1)
5¼„
+
(n
+
1)
ayn ]
C₂n - 1
+
a 【E2n-1 }
sin
(2n -−
1)
ø]
( 12)
T. 2
14
ON THE MOTION OF VORTEX RINGS.
strength, situated at the central line of the vortex core, are
given by
m u=
2π
dzt
(z
—
2 ) ddsy'
-— (y
— y)
des }
ds' ,
m1
บ= 2π
--
³
(x − x ') dz' — (z — 2 ') dx } ds ',
- Is
ds'j
m1
dx'
w= 2π
гов
(y — y')
ds'
dy (x - x')ds'}ds,
where r is the distance between the points (x, y, z) and (x', y' , z ') , and the integrals are taken all round the vortex ring.
Now
x
= p cos y = a cosy + a„n cos ny cos
+ B₁n sin ny cos y,
y' = p sin = a sin + a, cos ny sin + Bከ sin ny sin y, therefore
dx' dy
dy' = dy
dz =
dy
a sin - sin (a, cos ny + B₂n cos ny)
-n cos (a sin ny -B cos ny),
a cos y + cos ( a„ cos n¥ + ßn„ sin ny),
-
-
— n sin f (a sin ny – ẞn„ cos ny),
n (y, sin ny — §n cos ny).
In calculating the values of u, v, w we shall retain small
quantities up to and including those of the order of the squares of
ann, B Y
Although, for our present purpose , which is to find
the time of oscillation of the vortex about its circular form, we only
require to go to the first powers of a,, &c., yet we go to the higher order of approximation because, when we come to consider the question of knotted vortices, we require the terms containing the squares of these quantities.
If R, 4, ≈ be the cylindrical coordinates of the point x, y, z,
203 =
-
{p³ + R² – 2p R cos († − y) + (≈ − 2 ')²} ",
1 can be
now when we substitute for p its value it is evident that
expanded in the form
Σ (s) (A¸8 + B¸ cos ny + C¸ sin ny + D¸ cos 2ny + E, sin 2ny) -
× cos s ( − ), where A, contains terms independent of a,..., B, and C, are of the first, and D, and E, of the second order in these quantities.
MOTION OF A SINGLE VORTEX.
15
The part of A, which is independent of a ... is evidently
1 2π
cos sᎾ de
-
√
-
π 。 (a² + R² + z² — 2aR cos 0)
but we shall investigate the values of all these coefficients later.
Velocity parallel to the axis of x.
§9. In the equation
m (2″ 1 u=
2π 0
dy' 2')
-- (y —
y')
dz'
dy
ddyź) dy,
the expression to be integrated becomes, when the values for dy dz
y', ' , ds are substituted and the terms arranged in order of
magnitude , being written for z -8,
1
203· (Ça
cos
y
+
ny
(y,
sin
ny
– -
8n,
cos
ny)
+ } { (n + 1 ) 5ª„ + (n − 1 ) ay, } cos (n + 1 ) ↓
-
-
− 1 { (n − 1 ) 5ª„ + (n + 1 ) ay„ } cos (n − 1) ↓
+ } { (n + 1 ) [B₂ + ( n − 1 ) ad„ } sin (n + 1 ) ↳
—
-
-
− 1 { (n − 1 ) SB„ + (n + 1 ) ad } sin (n − 1) ↳
-+ n (¼,น Ɛn − Yn ẞ„) sin ↓ – ½ ( „น V₂n + Bn Sn) cos y
-
-
-
− 1 ( ªn Yn − Bn Ɛn ) { cos (2n + 1 ) ¥ + cos (2n − 1 ) y}
− 1 (αmn d₂ + Yn B₂) { sin ( 2n + 1 ) ♣ + sin (2n − 1) †} ).
m 2π COS зв
Let us consider the term
dy.
2π 0 зов
1 Expanding this equals
зод
m 2π [* dy ça cos ↳ Σ (s) { (4¸ + B¸ cos n¥ + C¸8 sin n↓
2π
+ D, cos 2n↓ + E, sin 2ny) cos s (¥ − p) }.
Remembering that 2π cos my cos ny dy = 0 if m does not equal n,
this equals
16
ON THE MOTION OF VORTEX RINGS.
2π 4x
2mπfh0 dy ga cos y
+ [{4 , cos ( − )]
+
[B + cos {
− ( n + 1 ) p } + B -1 cos {
+ ( n - 1)
}]
+ [CC²n-s sin {{ ++ (( nn − 1 ) p } - Cant+1 sin { ( n + 1 ) p} ]
+ [D₂n+1 COS { - −· (2n + 1 ) p} + D₂n -1 cos ( + (2n - 1 ) p} ]
+ [En2n--1 sin { + (2n - 1 ) p} - E2q21un+1 sin { (2n + 1 ) 6} ] )
= = ma [ A1, cos &
+ {Bunt+1 cos ( n + 1 ) + B -1 cos ( n - 1 ) + C , sin ( n - 1) p
+ Cn+1 sin (n + 1 ) ø}
+ } {D₂n+1 cos (2n + 1 ) + Dan-1 cos (2n - 1) + E2-n-,1 sin (2n - 1 )
+ Egn+1 sin (2n + 1 ) $ } ] .
Similarly, we may prove that
m 2″ 1 -
- ny (y₁ sin ny — 8, cos ny) dy 2πS0
== ½
mny
{A„
(Y₂n sin
no
-
d,n
cos
np)
+
Coyn― B¸dn
+ (B₂2mn Y Ca2n Sn) sin 2n - ½ ( B₂n Ɛn + C₂n Yn) cos 2n p} ,
and that
m 2″ 1
4п:Јо
дов {(n + 1 ) 5x
+ (n − 1 ) ay } cos (n + 1) Yd¥ −
=·‡m
{(n
+
1)
5x₂n
+
(n
−
1)
aɣ„ }
x {Annt+i1cos (n + 1 ) + (B, cos -C, sin + Ba2nnt+1 cos (2n +1 )
and that
+ Ca2nnt+i1 sin (2n + 1 ) ø) } ,
m 2π 1 -
4π [** = { (n − 1 ) 51, + (n + 1) ay.) cos (n − 1) ódó
= ‡m {(n − 1 ) 5amn + (n + 1 ) ayn} x {{AA₂-1 cos (n - 1) + (B, cos
and that
+ C, sino + B2n-11 Cos (2n - 1)
+ C₂n-1 sin (2n − 1) 4)},
m 2π 1 {(n + 1 ) ¿Bn + (n − 1 ) ad„ } sin ( n + 1) ydy
° 1
== {m { ( n + 1 ) {B₂ + (n − 1 ) ad„}
x {Antisin (n +1) + (B, sin + C, cos + Ban+1 sin ( 2n + 1 ) p - C₂n +1 cos (2n +1 ) ø) } ,
MOTION OF A SINGLE VORTEX .
17
and
m 2″ 1 {(n -1)
4π зав
+ (n + 1 ) ad, } sin (n - 1 ) ydy
= ầm { (n − 1 ) $ Bn + (n + 1 ) a8, }
× { 4„-, sin (n − 1 ) 4 + ½ ( − B, sin
+ C, cos + B2n-1 sin ( 2n − 1 ) ø
- C₂n-1 cos (2n - 1) 6) }
The integral of the terms involving the products a,, B. ,...
=
-
-
= {m [nA¸ (αn„§n„ — By ) sin – A¸ (αn¥n + ßndn) cos &
- (α -– BB„8dn) ( 42n +1 cos (2n + 1 ) + A2n-1 cos (2n- 1 ) }
- 1 (andn + BB₂„¥Y„)) {A2n +1 sin (2n + 1 ) + A,2n-1 sin (2n - 1) 4} ].
Thus u terms not containing a + terms containing a ... to the first power + terms containing a ... to the second power.
The term not containing a
= = ½ma A 1, cos ……...
.. (10).
The terms containing a ... to the first power = = 1m [ 2ny A₂n (Y₂ sin np - 8, cos no)
+ {Ça B₂+1 + [ (n + 1 ) n + (n − 1 ) ay, ] Annt+1} cos (n + 1)
+ {ca B -1 - [ (n - 1 ) (an + ( n + 1 ) ay ] A-n-1} cos (n - 1 )
+ {5a Cnnt+₁1 + [(n + 1 ) SB + (n - 1 ) ad ] 4+n+1)) sin (n + 1 ) $
+ { a C₁ -- [(n - 1 ) B₂n + (n + 1 ) ad ] 4.n-1} sin (n - 1 ) ] (11).
The terms containing a ...to the second power
= =
{m
[ny
{ Y₂
( 2C
+
B₂2,n
sin
2nd
–
Ca2nn
cos
2nd) -—
8n,
(2B。
+ B₂2nn cos2n + C₂2nn sin 2nd) }
+
{
−
(αn¥n
+
Bndµ)
A¸1
+
( Ǫ ‚
−
aɣn)
B₁1
+
(§ß„ -− ad„)
C₁}
cos
&
+n
{2 (ªndn − BnYn)
A₁1
+
( 5ßn
+
ad„ )
B¸ -−-
( Çα„
+
ay₁)
C₁ }
sin &
+ { − 1 ((~a₂nYn - 3B8₂,8„ ) A 2n +1 + } [ ( n + 1 )
an + ( n − 1 ) ay ] B2n +1
― − } [ ( n + 1 ) 5B₂n + ( n − 1 ) ad„] C'2n+1 + a¿D2n+1} cos ( 2n + 1 ) p
+
{-
} (α-
BB₂„d„)
A2n
-1
—−
1§
[ (n
-− 1 )
5¹„
+ (n
+
1 ) ay
] B₂n-1
+ } [(n − 1) SB₂ + (n + 1 ) ad ] C₂n-1 + a¿D₂n- 1} cos (2n -− 1 ) p
+
{-
1 (0n8n
+
BnYn)
Agn +1
+
½
[ (n
+ 1)
SBn,
+
(n -− 1 )
a8
]B 2n+1
+ } [ (n
+ 1 ) Ša„ +
(n
−
1)
ayn]
C2n+1 + açE2n+1} sin (2n + 1 )
ø
+ { - } (anndn + B₂Yn) A2n-1 -
[ ( n − 1 ) B₂n + ( n + 1 ) ad ] B₂2n-1
− } [ (n − 1) 5ºn„ + (n + 1 ) ayn ] C₂n-1 + açE2n-1 } sin (2n -− 1) ø ] (12)
T.
2
18
ON THE MOTION OF VORTEX RINGS.
§ 10.
m [2″ 1.
dz
v= 2π 0 2.3
(x
-- x')
dy
(2
— 2)
day} dy .
The expression to be integrated becomes on substitution
[ça sin ynx (y, sin ny — 8n, cos n¥)
− } { (n + 1 ) SB„ + (n − 1 ) ad„ } cos (n + 1) ↓
-− } { (n − 1 ) 5B,„ + (n + 1 ) ad, } cos (n − 1) ↓
+ } { (n + 1) 5α, + (n − 1 ) ay,} sin (n + 1) ↓
+ } { (n − 1 ) Sa„ + (n + 1 ) ay„} sin (n − 1 ) ♣
-
---
+ n (BnYn − α„dn) cos y − 1 (ª‚Yn + Bß„§„) sin ¥
―
-
-
· ‡ (αnyn — B₂dn) { sin (2n + 1 ) − sin ( 2n − 1) ↓}
− 1 (andn + Bn¥n) {cos (2n − 1 ) ↓ — cos (2n + 1) y } ] .
The term
m 2T 1 2π 0 2.3 La sin y dy
= = ½ maš [ aA¸1 sin & + ½ {B₂+1 sin (n + 1 ) † − B„-1 sin (n − 1) ø
-
-
Cnnt+i1 cos (n + 1 ) + C₁-1 cos (n − 1 ) p}
+ } {D₂n+1 sin (2n + 1 ) - D₂n-1 sin (2n - 1 ) p - En +1 cos (2n + 1 ) $
The term
+ E2n-1 cos (2n − 1 ) }] .
m 2πnx -
(y sin ny — §, cos ny) d¥ 2π 0
= - mnx {4,n (yn, sin np − d, cos np) + CoYn − BƐn
+ (B₂nyn - C₂2nnd ) sin 2nd
The term
- (B₂8 + C₂nyn) cos 2np} .
m
1
-
4π
{ (n
+
1)
58,
+
(n
−
1)
ad„} [**1./7cos 0
(n
+
1)
v
.
dy
== 1 m {(n + 1 ) [B₂n + (n − 1) ad„}
x {4n+ cos (n + 1 ) 4 + ½ (B, cos - C, sin + BBa,n+1 cos (2n + 1) $
The term
+ Ca2nnt+i1 sin (2n + 1) 4)} .
m
4π {(n
−
1)
5B₁
+
(n
+
1)
ad „}
[* 0
2+3
cos (n
-
1) y dy
= 1m {(n - 1 ) SB + (n + 1 ) ad }
x× {A4n,-1 cos (n - 1) ☀ + ½ (B, cos & + C, sin & + B2n-1 cos (2n - 1 ) + C₂2n-1 sin (2n - 1 ) 6) }
MOTION OF A SINGLE VORTEX.
19
The term
m
(2π 1
4π { (n
+ 1 ) $5ª,
++
(n
− 11 )) aayn }
[]0
nдля
sin
(n
+
1)
dy nhưng
= } m { (n + 1 ) 51„ + (n − 1 ) aɣ„}
× { An+1 sin (n + 1 ) $ + ½ ( B, sin 4 + C₁1 cos The term
+ B₂n+1 sin (2n + 1 ) $ ― -C2n+1 cos (2n + 1 ) )}
m
(2π 1
4π
{ (n
−−
1)
g5axn, ++
(n
+
1
)
aayy₂}
[* 0
↓ d↓ asin (n − 1 ) ch
= = 1 m { ( n − 1 ) 5x„ + (n + 1 ) ayn}
× [4,n-,1 sin (n − 1 ) 4 + 1 { − B, sin & + C, cos + B2n-11 sin (2n − 1 )
-Can-1 cos (2n - 1 ) } ]
The integral of the terms involving the products a ,, P ...
=
-
-
= { m [ n (ẞ„Y„ — αn„ §„ ) 41, cos & — § ( a„ß„ + ß„§„) A¸ sin &
-− } ( αnYn − Budn) { A2n +1 sin ( 2n + 1 ) † - An-1 sin (2n − 1 ) 6}
− 1 (and₂ + B„vn) {A2n-1 cos ( 2n − 1 ) † - A2n +1 cos (2n + 1 ) p} ] .
Thus terms not containing a ... + terms containing a ...to the first power + terms containing a ... to the second power.
The term not containing a ... ma 41, sin o
. (13) .
The terms containing a ... to the first power
=
-
= { m [— 2nxA₂ (y, sin no — 8n, cos no)
-—− { [ (n
+
1)
5ßn„ +
(n
−
1)
ad„ ]
An++1 +
a¿C₁₂ +1 }
cos
(n
+
1)
p
-
-
− { [ (n − 1 ) 5ßn„ + (n + 1 ) ad„] A „-1 − a¿Сn„-1.} cos (n − 1 ) ø
+ { [(n + 1) Sa„ + (n − 1 ) ayn] Antı + a¿B₁n+₁1 } sin ( n + 1 ) ø
+ {[(n − 1) Sa„ + (n + 1 ) ayn ] A n„-ı1 — a¿Bn-1.} sin ( n − 1 ) ] ... ( 14) .
The terms containing a ...to the second power
== = 1m
[ — nx {yn (2C¸ + B½
sin 2np – C₂2n cos 2np)
- 8 (2B + B₂2,n cos 2nd + C₂2nn sin 2nd)}
+ n {2 (B₂Yn − α„n§„) A¸1 − ( §ß„ + ad„) B₁1 + ( §ª„n + ay ) C₁ } cos
-
{− (α₂Yn + B₂dn) A¸ + ( Çî„ − aɣ„) B₁1 + ( ¿ß₂ − aồn )
C₁ } sin &
+ {1 (a'ndµ + Bn¥n) A2n+1 − † [ ( n + 1 ) Çß„ + ( n − 1 ) a§„ ] B₂n+1
- [(n + 1 ) 5x + (n - 1 ) adμ ] Canti - agE2n+1} cos ( 2n + 1 ) p
+ {- } (αdn + BnYn) A2n-1 - [ ( n -− 1 ) SBn + (n + 1 ) ad ] B₂2n-1 − 1 [(n −- 1 ) 5x + ( n + 1 ) ayn ] C₂n-1 + acE2n--1} cos (2n - 1 ) ø
2--2
20
ON THE MOTION OF VORTEX RINGS.
-
·
+ {— § (ª‚„Y„ − B₂dn) A2n + 1 + ½ [ (n + 1 ) ²n + ( n − 1 ) ay„ ] B2n+1
-
-
− } [ (n + 1 ) 5ß, + (n − 1 ) adn] Câ₂n+1 + açƊ2n+1} sin (2n + 1) $
+ {}† (ªnYn − B₂dn) A2n−1 + ½ [ ( n − 1 ) 52n + (n + 1 ) ay₂ ] BB₂, -1
− § [(n − 1 ) SB„ + (n + 1 ) aồn] C₂n-1 — a¿D₂n-1} sin (2n + 1) p... (15)
m 12π 1
dx'
§ 11 .
w= 2π 0
(y - y) d↓ --・ (x −- x') dy') dy. dys
The expression to be integrated becomes after substitution
1 -
дов [a² - a (y sin + cos y) + 2a (2, cos n¥ + ß„ sin n¥)
+ } (n + 1 ) (yß₂ − xx„) cos̟ (n + 1 ) †
+ 1 (n − 1 ) (xa„ + yẞ„) cos (n − 1 ) †
-} (n + 1 ) (31,+8, ) sin (n + 1 )
-
-
- * ( n −1) (y2n = 8, ) sin (n − 1 )
+ (a²„ cos³n¥ + 21,ßn, cos ny sin ny + ß²„ sin³ n¥)].
m 2π
The term 2π
a² зов
dy
= {ma³ (2A¸ + B„ cos no + C₂ sin no + D₂2n cos 2nd + E½2n sin 2nd).
ทาง 2π
The term 2π
зов {y sin ↓ + x cos ) dự
putting a = R cos p, y = Rsin & becomes
- maR (*2π 1 2π 0 cos (p − y) dy
=- †maR [2A¸ + (Bn+1 + Bn-1) cos no + ( C'n+1 + C₂-1) sin nø
+ (D2n+1 + D2n-1) cos 2nd + (E2n +1 + En-1) sin 2nd]
(2π a
The
term
mπJcohnзов
(a,
cos
n nh
+
B,
sin
nh)
ảnh
= ma [4, (a,n cos no + ß„ sin np) + Bo₂ + Coßn + ((Ba₂n'ºn - Ca2un ẞn) cos 2np + ( C₂2nnn + B₂2nnßn) sin 2nd]
m
1
The
term
(n 4π
+
1)
( yẞ,
−
xa
)
[2
, cos
(n
+
1)
dy
m
==
147· (n
+
1)
( yẞ₂
−
xx„)
× {Annt+i1 cos (n + 1) + (B, cos - C₁ sin
+ B2n+1 Cos (2n + 1 ) + C2n +1 sin (2n + 1) ø)} .
MOTION OF A SINGLE VORTEX .
21
m
2″ 1
The term
-
4.πT (n − 1) (x² + yẞ, ) [** — cos (n − 1 ) ydy
m =
4 (n − 1 ) (xamn + yẞ„)
× {4 „-1 cos (n − 1 ) 4 + 1 (B₁ cos & + C₁ sin &
+ B₂n-1 cos (2n − 1 ) + C₂2n-1 sin ( 2n − 1 ) $) }
m
•2″ 1
The term - (n + 1 ) ( y² + xẞn)
sin (n + 1 ) ydy
4πT
[0
m -
4 (n + 1 ) ( y²„ + xß„)
× {An+1 sin (n + 1 ) ☀ + ½ [ B¸ sin 4 + C₁ cos &
+ Ba2nnt+i1 sin (2n + 1) - Ca2nnt+i1 cos (2n + 1) $]} .
m
1
The term 477 (n − 1) ( ya,„ — æxß߸„)) [*” , sin (n − 1 ) ♣dy
.m
-
-
= − ™4 (n − 1 ) (y¹„ — xß„)
× {A „-, sin (n − 1 ) 4 + ½ ( − B¸1 sin + C₁ cos &
+ B₂n-1 sin (2n − 1 ) - C₂n-1 cos (2n − 1 ) 4)} .
The term containing the second powers of a ... --
= } m {(x²„ + ß³„) A。 + † ( a³n − ß³„) Aa2nn cos 2n + aẞn422n sin 2np}.
Thus w terms not involving a, +terms containing a ... to the first power + terms containing a ... to the second power.
The terms not involving a
= = ½ m (2a³Ã¸ — aRA¸) ………………………….
...... (16).
The terms involving a ...to the first power become after substi-
tuting for x and y, R cos & and R sin & respectively
½m [(a²В₂ — 1aR (B₂+1 + B₂-1) + 2uï„An
+ †Rî„ {(n − 1) A₂-1 − (n + 1 ) An+1}) cos no
+ (a² C₂ − aR ( Cn+1 + Cn-1) + 2aß„ªn
+ } Rß„ {(n − 1 ) A „-1 − ( n + 1 ) A „-1}) sin nø]………….…(17) .
The term involving a ... to the second power
= m [aª‚„B¸0 + aẞ„ С. — § α„ B₁1 — {ẞn C₁1 + ½ (a³n + B²„) A。
+
}į
{ R²,
[ (n
−
1)
-B₂n- 1 − (n
+
1)
B₂n+1]
--
-
- RB₁n [(n − 1) CC₂,n-1 − ( n
+
1)
C2n +1]
+
44αa
a (B₂2nnªn
´--
C₂2unẞn)
+ 4a³D₂2nn - 2aR (D2n+1 + D2n-1) + 2 ( a²n - ẞß³„ ) A₂2n} cos 2nd
22
ON THE MOTION OF VORTEX RINGS.
-
-
+ į {Ra, [(n − 1 ) C₂n-1 − ( n + 1 ) C₂n+1]
-
RB₂
[ (n
+
1)
B₂n+1
-−·
(n
−
1)
B₂n-1]
+
44aα
( C₂n²n
+
BB₂n8 ß₂))
+ 4a³E₂ - 2aR (E2n +1 + E2n-1) + 47'n ẞn422n} sin 2nd] ......
...... (18 ) .
§ 12. We must now proceed to determine the values of the quantities which we have denoted by the symbols A , B , C , &c. We have, in fact, to determine the coefficients in the expansion of
1
{p² + R² + 5² − 2Rp cos ( 0 − 4) )
or, as it is generally written for symmetry, of
1
in the form
-
,
{1 + a² — 2a cos (0 – $) }
AΟ + A₁cos (0 −4) +….. A„ cos n (0 − $) +…...
This problem also occurs in the Planetary Theory in the expansion of the disturbing function, and consequently these coefficients have received a good deal of attention ; they have been considered by, amongst others, Laplace, in the Mécanique Céleste, t. I. § 49 ; Pontecoulant, Du Système du Monde, vol . III. chap. II.
These mathematicians obtain series for these coefficients proceeding by ascending powers of a. The case we are most concerned with is when the point whose coordinates are R, z, & is close to the vortex ring, and then R is very nearly equal to p and is very small, so that a is very nearly equal to unity, and thus the series given by these mathematicians converge very slowly, and are almost useless for our present purpose. We must investigate some expression which will converge quickly when a is nearly unity.
Our problem in its simplest form may be stated as follows , if
1
= co
+
c,
cos
0
+...c,
cos
no
+... ,
(q - cos 0)
we have to determine c, in a form which will converge rapidly if q be nearly unity.
1
Let
= b + b₁ cos 0 +...b₁ cos ne +....
(q - cos 0) ³
Then by Fourier's theorem,
1 2π cos no
= Cn
de,
π
0
(q — cos 0) *
1 2π cos no
=
de,
π 0 (q - cos 0) ¹
1 2π de Co =
2πS0 (q·- cos 0) ·
1 2π de
b。 = 2π 0 (q - cos 0)2
MOTION OF A SINGLE VORTEX.
23
Now
d sin no
n cos no
=
de (q - cos 0) (q - cos 0)
-
{cos (n - 1 ) 0. cos (n + 1 ) 0}
-}
...(19) .
(q - cos 0) *
Integrating both sides with respect to 0 between the limits O
and 27, we have
-
-
0 = nb₁ — 4 (C1„2--1 — Cn+1),
or
4nbn = Cn-1 4 Cot
... (20).
Reducing the right-hand side of equation ( 19) to a common
denominator, we have
d sin no 4
do (q - cos 0)*
-
-
4ng cos no — { ( 2n + 1 ) cos ( n − 1 ) 0 + (2n − 1 ) cos (n + 1 ) 0}
(g- cos 0)
Integrating both sides with respect to 0 between the limits 0 and 27 , we get
0 = 4nqc, — { (2n + 1) Cn-1 + ( 2n − 1 ) C₂+1} ... .. (21 ) .
By means of this and equation (20), we easily get
2n + 1
= Cn
n n+1
(q² - 1)
. (22) ;
and thus, if we know the values of the b's, we can easily get those
of the c's, and as the b's are easier to calculate we shall determine
them and deduce the values of the c's.
1
Let V :=
= b₁₂ + b₁1 cos 0 +... bn, cos no +……..
(q - cos 0)3
By differentiation we have
d²V
dV
d2V
-
=
(1 − q³)
2q - IV:
;
dq²
dq
d02
hence, substituting for V the value just written and equating the
coefficients of cos no we have
d2bn - dbn
-
(1 − q²) dqª 29 dq + b₂ (n² − 4) = 0 .
Let
9-1 b₁ = $(2) log 16 (q + 1) + Y (2),
where (9) and (q) are rational and integral algebraic functions of q.
Substituting in the differential equation , we find
- do- 29 ddoq
-
(1 − q²)
+ (n² − 1) $ = 0,
dq²
аф
ď² - 29 ddyq
-
-4 dq + (1 − g³) dq²
+ (n² − 1) ↓ = 0 .
24
ON THE MOTION OF VORTEX RINGS.
Let us change the variable from q to x, where x = q - 1, the equations then become
d'o
do
x (2 + x)
-- +2 (1 + x) dx − (n² — 1) p = 0,
dx²
dk
do
d
dy -
4 + x (2 + x) +2 (1 + x)
(n² − 1) y = 0.
dx
dx²
dx
Let
$ = а + a₁x +... amxm + ....
Substituting in the differential equation for 4, we find
am+1 = n² - 1 - m.m + 1 ami 2 (m + 1)²
therefore
$ (x) = a { 1 + (n² − 1) %/
-
2 n² - 41. n² - 2 . n² - 25
+
(n²
—
¹) (n² 22
—
3)
(
)*
+
(3 !)²
(~)*.+..}.. (23),
or, with the ordinary notation for the hypergeometrical series,
$ (x) = a¸F (± −n , § + n, 1 , − { x). Let
† (x) = α + α¸x + x,x² + ... αmxm + ....
Substituting in the differential equation for y (x), we find
n² − 1 − m . m +1 am+1 =
2 (m + 1 )²
2
am+1 m +1
So
↓ (x) = a¸ F (} — n, § + n, 1 , − 1x)
1 x²
-
-
-
-
− a {2 (n − 1) 2 + 3 (n ² − 4) (n − 1) 2 22722
1 23
-
-
+¹¹ (n² − 1) (n² − 2) (n² — 25) (3 !)2 29
1
+ 25 (n² — 4) (n² − 2) (n² — 25) (n³ — 49).
+......... ( 24),
(4 !)² 2+
where the general term inside the bracket
1 xm =
2 (1 + 4 +... —1 ) (n² − 1 ) (n ² — £). …… ..(n² — 1 (2m − 1 )º) (m ! )² 2m ·
To complete the solution we have to determine the values of аo
and
We shall do this by determining the value of b, when q is
very nearly unity, or when x is small.
We may prove, in exactly the same way as we proved equation (20), that
4ngb₁ = (2n - 1) bn-1 + (2n + 1 ) bn+1
or
=3 4n (1 + x) b₁n = (2n − 1 ) bn-1 + (2n + 1) b₂n+1°
MOTION OF A SINGLE VORTEX.
25
By the help of this sequence equation we can express b, in terms of b, and b, in the form
b₁ = (A + Bx + Cx² + Dx³ +... ) b + ( A ' + B'x + C'x² + ... ) b₁.
We only want now to determine a, and a , i.e. the parts of and & independent of x, thus we only want the coefficients A and A' in the equation just written ; now evidently A and A' will be the same as if we put x = 0 in the sequence equation and then determine bn in terms of b, and b₁,
The sequence equation becomes, when x = 0,
4nbN
= (2n
-
1)
b, -1
+
( 2n
+
1)
bu+1'
the solution of this is
1
b₁n₂ =
C
+
C'
(1
+
}
+ .· .· .· 2n 2n ·
where C and C' are arbitrary constants.
Determining the arbitrary constants in terms of b, and b, we find
b₁ = 2b¸ + (b¸ − 2b,¸) (1 + +} +... 2n - 1).
for in the sequence equation involving b。, 2b, must be written instead of b
Now
1 2π do
1 2″ de
26. =
=-
π 。 √(q - cos 0) π 。 √(q + cos 0)
2π 1 =
πSo √(a + 1 -- 2 sin² 30)
1
2π
=
T√ ( +1 ) √ √ (1 -− l² sin'0) •
where
2 k² =
q + 1'
4
π √ (q + 1 ) √ √ (1 − sin³¿)
Now, when k is very nearly unity, we know that
аф
4 =
log approximately,
√ √(1 -1 sin' )
k₁
where
k₁
=
√
(1
−
k²) ,
in
our
case
=
+
26
ON THE MOTION OF VORTEX RINGS.
Therefore, when q is very nearly unity
= 26, -2π2 log ( + √ ( 1) } approximately.
1 2π cose de
b₁1 = =√ π
√ (q-= cos 0)
1 2π -
2π q
do
-- π 0 √ (q − cos 0) do + π45%。 √ (q - cose)
4 π √ (q + 1 ) '[ ** √ ( 1 − k² sin²p) dp + 2b,q.
When k is very nearly unity
[** √(1 − kº sin³p) do = 1 approximately ;
therefore
4/2
b₁
+ 2b。.
π
Therefore, when q is very nearly unity,
b₂ = √2 log 16 (q + 1 ) -_ 2 / 2 ( 1 + + +...2n1-
ba
1
;
П
9-1
π
comparing this with our former solution for b , we find
=- √2 a。
π
and Thus
4/2
1
ao
...
π (1 +3 + ··· 2n − 1
1 2π cos no
ba =
de
π
。 √ (q — cos 0)
= √2 ( į - π F () -n, 1 + n, 1, -ja) { log 16 (2+
) 4 (1 + 1 .+.. 22n 1 - ) }
√2
-
1 x²
+
K₁ (n²
П
1) ½2 • + K¸2 (n² − 1) (n² — ³) (2 !)2 22
1 x8
+ K¸ (n³ − 1) (n² — 2) (n³ — 25) (3 !) 2 23
+ Km (n² − 1) (n³ — 2) ... (n³ —— — ( 2m —−11))²))
xm ) 2m+
( 25 ).
where
=2 K = 2 (1 ++... ) so that
K₁ = 2, K₁2 = 3, K¸3 = ¹¹ ‚ K₁ = 25, K¸ = 137 &c.
MOTION OF A SINGLE VORTEX.
27
If n denote the sum of the reciprocals of the natural numbers up to and including n, then
1
1
+
1
+
1
+ .. ..
2n
-
= 1
2n
—
1S₁n
= ƒ (n)
say.
Now
11 S₁n₂=.577215 + log n + 2n - 12n2 + ' ,
see Boole's Finite Differences, 2nd edition, p. 93.
1
Thus
f(n)
=
.288607
+
log
2n
−†
log
n
+ 48n2 + ....
We only want the value of b, when x is very small, and thus we have approximately
b,
= 1/П2² (1
− 1 (↓ —− n°)x}
{
log _1166
((22 ++ XC
æx))
—
4ƒ
(n )}
- √2 -π·x (‡ − n²) ….. (26) .
By equation ( 22)
2n + 1
Cn =
(qbn - bn+1).
(q³- 1)
If we substitute for b, and bn,+1 their values, as given above, we find that approximately
Cn
=
√252 ~π~ [2-
((nn² -
1)
{ log
((22
++
2)
116
—
4ƒ (n ) } − (n² + 3 ) ]...( 27).
The integrals we have to evaluate are of the form
1 2π
cos ne.de
> = π√₂。 (R² + p² + y² −- 2Rp cos 0)
which may be written 1
2π cos no.de
where therefore and
Thus
π (2 Rp) o (q - cose) ,
R² + p² + Dz
q=
;
2Rp
{(R − p)² + 5"} x=q - 1 =
2Rp
(R + p)² + y² 2 +x=1 + q=
2Rp
2 + x = (R + p)² + y²
OC
(R − p)² + y²
28
ON THE MOTION OF VORTEX RINGS.
1
and the integral we wish to find =
cn.,if the value
(2Rp)*
{(R − p)² + y²} x=
2Rp be put for x in equation (27) .
Let
us
1 denote(2R )
Cn,' when
x has
this
value
by
S
Then 1
-
-
{R² + p² + 5² − 2Rp cos (↓ − p) } *
= S'₂ + S' , cos (y- 4) +…..S', cos n († − p) +……..
Now in S' , p and are functions of y,
and
p = a + a₁cosny + B₁n sin ny,
Y = z — 2' = (≈ — 3) — (Y₂ cos ny + d„ sin ny).
Now let S, be the value of S' when p = a and = (z — 3) .
By Taylor's theorem ,
dsn S₂ = S₂ + (ªa‚„, cos n¥ + B, sin ny) da
dSn (y,n cos ny + d, sin ny) dz
d'Sn + ½ (a, cos n↓ + ß„ sin n¥)² -da2
d'Sn − (a₂n cos ny + ẞ₂n sin ny) (Y½ cos n¥ + d„ sin n¥) da dz
d's + ½ (Y₂ cos ny + d₁ sin ny)²
dz + terms involving the cubes and higher powers of a &c.
d&Sn
d'Sn
S'₂ = S₂ + 1 (an³ + ß„²) da - ½ (αnYn + B₂dn) dadz
2
d'S"
+ 1 (yn² + 8,²) dz
dSpn
ds
dsn dS
+ cos nyan da --Yn'
{
dz
+ sin ny ( Bn da
Enn dz
-
Sn
d'Sn
+ ‡ cos 2n↓ { (a„² —— ßB¸„”²)) dd'aS" - 2 ((aan¥nn- ẞndn) dadz
+ (Yn² - En²) d²S₂) dz2
d'S
d'S
d'S
-
On + √ndn
+ } sin 2n¥ {a‚‚ß„
− (ª‚„d„ + B₂Yn) dadz
dz2
+ terms of higher orders.
MOTION OF A SINGLE VORTEX .
29
Hence, comparing these equations with § 8, we see that
de S18 -
d'Sท
d2 Sn
A₁n = S₂n + 1 (a ,n³ + ß„²) da
- ½ (αmYn + B₂dn) dadz + 1 (Y, ² + 8,3) dz
B₁28 =
ddsan
- Yn
dSn dz
C₂
=
ds n
Bn
- Enddsz12
da
D=
-
d² Sn
d'S²²
d'S
da² · 2 (αnn - B₂Sn) dadz + (1₂² - 8₂²) dz2
d'S
d'Sn
B₁n = } {13Bnddsa².
- (andn + BnYn)
+ YnEn
dadz
dz2
( 28) .
§ 13. We can now go on to find the motion of a vortex ring disturbed slightly from its circular form. It will here be only necessary to retain the first power of the quantities a ,..., so that we shall neglect all terms containing the squares of those quantities.
Ө
Fig.2. 0
Let fig. 2 represent a section of the vortex ring by the plane
of the paper. Let O be the origin of coordinates, and let C be the centre of the transverse section of the vortex core ; let the
radius CP of this
section
= e;
let
CP make an angle x with OC
produced .
Then the equations to the surface of the vortex ring are p = a + a₁ cos n¥ + ẞ„n sin ny + e cos x......... (29 ), z = 8 + y, cos ny + d, sin ny + e sin x ......... (30).
Now if F (x, y, z, t) = 0 be an equation to a surface which as it moves always consists of the same particles of the fluid , then we know that
dF dF dF dF + u + v + w = 0,
dt dx dy dz
where the differential coefficients are partial, and where u, v, w are the x, y, z components of the velocity of the fluid at the point (x, y, z).
The surface of a vortex ring is evidently a surface of this kind ; we may therefore apply this result to its equation.
If we apply this theorem to equation ( 29) , we find
dan
dB
dt
cosny +
" dt
sin
ny
—
n
( ” „ sinny
–ẞ„ cosny) V - esinx.X - K = 0,
30
ON THE MOTION OF VORTEX RINGS.
where is the velocity of the fluid along the radius vector, Y the
angular velocity of the fluid round the normals to the vortex ring, X the angular velocity round a tangent to the central line of the vortex core.
Now if the vortex be truly circular, I vanishes ; thus contains an and B, to the first power ; and a will be ofthe second order in a,, and may for our present purpose be neglected . Neglecting such terms, the equation becomes
dan
dß
cos ny + sin nye sin
dt
dt
Χ
X = R ......... (31).
But
u cos y + v sin ¥ = K.
Since R and are now the coordinates of a point on the surface of the vortex ring,
we have
R = a + a, cos ny + B, sin ny + e cos x,
y= Y cos ny + sin nye sin X,
and writing instead of in equations (11 ) and (14), we find, neglecting terms of the order a,,
u cos
+ v sin ✈ = †ma (yn, cos n¥ + d, sin ny + e sin x) A¸
-· + ‡m { (n − 1 ) A, +1 − (n + 1 ) A -1} ayn cos ny
+ ≥m {(n − 1) An+t1i n + 1 ) A₂n--1} ad, sin ny
= = ½me A , sin x + 1 ma { 2A¸ + (n − 1 ) Anti
But
− (n + 1 ) An+1} (Y, cos ny + 8, sin ny).
dan
R = cos ny +ddeßn siinn nn y – e sin X.X ;
dt
dt
therefore, equating coefficients of sin x, cos ny, sin ny, we get
X = -1mA
. (32),
dan = = ½may,
[ A¸1
+
½
{ (n
−
1)
Antı
- − ( n
+
1)
A „-1} ] ….… (33) ,
dt
dßnn = = ½mad„
dt
[ 4 1, +
½
{ (n
−
1)
An+1
− (n
+ 1)
An-1}] ….… (34).
Now as we neglect the squares of a ..., we may put A„ = S„
e²
and R = a + e cos x,
= e sin X;
that is, x = in the 2a2
quantity
denoted by S
Making these substitutions in equation (27) , we get
1 4a²
64a2
Sn=
--
-
S. - 22-παa³ [ +3 — (n ° — 4) {log 4e² – 45ƒ(n)} −- (nn²° ++ 3) ]... (35 ) ;
MOTION OF A SINGLE VORTEX.
31
thus therefore
1 (4a² S=
2πα e
64a³ (log e²
m
3m
64a2
X == - + πρ 16πα (log e²
;
or, if be the angular velocity of molecular rotation, so that m = ωπε ,
64a²
X = -w + 38 w
-
& (log e² $)
. (36),
64a²
and since is small, log
will be small ; thus we have
a
a²
e²
approximately
X = - w,
which agrees with the result given by Sir William Thomson in a note to Professor Tait's translation of Helmholtz's paper, Phil. Mag. 1867.
Substituting in equation (33) the values of A,, An+19 An-1, i.e. in this case S, S -1 Sn1+1 given in equation (35) , we find
dan
Myn
64a²
dt
=-1 παι
n²
{loogg
4f ( ) - 1 }......( 37 ),
where we have neglected terms of the form Af (n) + C, where A
and C are numerical coefficients, since when n is small ƒ (n) is
64a²
small compared with n² log
> and when ʼn is large it is small
e³
compared with n²ƒ (n) .
64a²
Now unless n be very large, log
is very large compared
e*
with ƒ (n), and the equation becomes
ddzxn ==
myn n² log 64a
dt
παλ
e²
.. (38) .
64a2 But iff (n) be so large that ƒ (n) is comparable with log e² then, since approximately
ƒ(n) = · 288607 + log 2n − 1 log n (Boole's Finite Differences, p. 93)
equation (37) becomes
Myn 2
4a²
4............ (39) ..
ddat, = -1mπα² n ' (log n²e² - 2 ·1544)
32
ON THE MOTION OF VORTEX RINGS.
This formula must be used when n is so large that ne is comparable with a.
We have exactly the same relation between dß /dt and 8, as between da /dt and YYn.
If we make the second of the equations to the surface of the vortex ring satisfy the condition necessary for it to be the equation to a surface which always consists of the same particles, we get, using the same notation as before,
dz
+ dt
dyn dt'cosny
d&n + dt
sinny
–
n
(ỵ,n
sin
ny
–
S„
cos
ny)
¥
+
e
cos
x
.
X
-— w = 0 ;
or, neglecting we find
(y, sin ny - 8 cos ny) I as before
dz , dyn
d8
+ dt
dt
cos ny +
" sin nye cos x . X = w .. ........ (40) . dt
But we know by equations (16) and ( 17) that w = m (2a³A - aRA ,)
+ }m [a²B„ − }aR (B₂+1 + Bn+1) + ½ { (n − 1 ) A „-ı− (n + 1 ) An+1} Rïn + 2a4n ] cos n¥
+ {m [a³C„ −1aR ( Cn+1 + Cn-1) + ½ { (n − 1 ) An-ı− (n + 1 ) An+1 } Rßn
+ 2aAnẞ ] sin ny,
where
R = a + a, cos ny + B, sin ny + e cos x, &c.
Substituting this value for R and the values of A, B , &c. given in equation ( 28) , we find
w = 1m (2a³S - a²S₁) - mae cos x . S₁
d
+
ma [ da
{Sn − 1
(Sn+1
+
Sn-1) }
+
}
{ (n
− 1)
S n--1- · (n
+
1)
Sn+1}
+
2S,
- S₁ + a dR (25, - S )] a (2, cos ny + B₂ sin ny).
Where in S , R after differentiation is put equal to a + e cos x,
e² and x =
2a2,
Equating in the two expressions for w, the term independent of and x, the coefficient of cos x and the coefficients of cos ny and sin ny, we get
MOTION OF A SINGLE VORTEX.
33
22
=
-
dt †m (2a³S — a³S¸),
X = -mae S₁,
ddyt. =
¡ maz.
[
a
d da
(8.n
-
†
(S5,n+1
+
B -1)}
+ } { (n − 1 ) Sn„-,1 − (n + 1 ) S₁₂+1 } + 2S₂- S₁
d
+ a1d7R (25, -5)] .
with a similar equation between dƐ„/dt and ß„.
Sn before differentiation
1
2
2=
(Ra)+
Ꮳ (n² – [ = − ( a* − 1)
-4 {log (2 + # 16 ) − + ƒ(m)}
− (x² + 1)] ,
where
(R − a)² + 5² X=
2Ra
When S has not to be differentiated, it equals
1 [4a²
−
Σπα
*ρ** —
(n²
—
1) { log 64ea²²
—
4ƒ,(n)}
−
(n² + 2)] .
The first equation gives the velocity of translation of the
vortex ring, substituting the values for S, and S, we find
dz m
64a2
=
dt Απα (log e² 2)
= 2πα(log e -11..1)............ ....... (41 ).
In a note to Professor Tait's translation of Helmholtz's paper
on Vortex Motion , Phil. Mag., 1867, Sir William Thomson states
that the velocity of translation of a circular vortex ring is
m
8a
2πα (log Sea - 4).
This agrees very approximately with the result we have just obtained, and Mr T. C. Lewis, in the Quarterly Journal of Mathematics, vol. XVI. obtains the same expression as we have for the velocity of translation .
The second expression gives the same value for the angular velocity X as we had before.
The third equation gives on substitution and differentiation
dy.
==
man 22,
dt Απα
(w²
–
1)
{ log
64a² e³
-
4
(n ) −1} .
T.
3
34
ON THE MOTION OF VORTEX RINGS.
neglecting as before terms of the form Af(n) + C, where A and C are numerical coefficients.
We have a similar equation between dƐ /dt and ẞ„•
dyn d8n
Substituting these values for and in equation (40) , we
dt
dt
find that the velocity of translation W at any point on the ring
is given by
m
64a2
W= ddt + 4πa² (n - 1){log e² - 4ƒ(n ) -1} (a, cos ny + 8 sinny) ;
or, neglecting 4f(n),
dz
-
W= dd³t {1
+
n²
α–
1
(xn, cos ny
+ B₁n
sin nų)} .
If p' be the radius of curvature at any point of the central line of vortex core, we can easily prove that
1 1 n² - 1
=+
·(am cos ny + ẞ₂n sin ny),
Р α a²
L
so that the velocity of translation of any point of the vortex ring
= dz
a ;
dt ρ
thus those portions of the axis which at any time have the greatest curvature will have the greatest velocity.
dyn Returning to the equation for , we have as before
dt
dyn =
dt · (n² — 1) La₂ ·
.. (43),
where
m
64a2
L==
log
Απα
e³
except when n is so large that ne is at all comparable with a, then
m
4a2
L=AΑmπaα² (log
nn²ee² - 2: 1544),
approximately ; the accurate value of L is
64a2
log
- 4f(n) -1 ;
this is the same coefficient as we had in the equation giving da /dt
so that our equations are
dxn - n²LY
dt
dyn
=
-
(n² − 1) La„.
dt
MOTION OF A SINGLE VORTEX.
35
dyn Differentiating the first of these, and substituting for from
dt
the second, we find
d'a '+ n² (n² — 1 ) L³a„ = 0,
dt2
the solution of which is -
an = A cos [L √/ {n³ (n² − 1 ) } t + B] and therefore
Yn =A
sin [L √ {n² (n² – 1 ) } t + B]
√ (n²n=1),
.. (44),
where A and B are arbitrary constants.
We can shew by work of an exactly similar kind, that
B₁n = A' cos [ L√ {n² (n² − 1 ) } t + B'] S₂n = A' √✓ (n² n-³ ¹ ) sin [ L √ { n² (n² − 1 ) } t + B]
(45) .
These equations shew that the circular vortex ring is stable for all small displacements of its central line of vortex core. Sir
William Thomson has proved, that it is stable for all small alterations in the shape of its transverse section, hence we conclude that it is stable for all small displacements. The time of vibration
2π =
L √ {n² (n² − 1 ) } '
2π =
´√{n² (n³ — 1 ) }
Απα
64a2 -
m (log te²a* —- 4f (n) – 1)
. (46),
where
1
f (n )= 1 + 3 +
+
... ' 2n
-
1'
Thus, unless n be very large, the time of vibration
2π =
2πα
√ { n³ (n² − 1 ) }
8a
m log e
or, if V be the velocity of translation of the vortex ring
2π
α
=
√ {n² (n² − 1 ) } V ·
Thus for elliptic deformation the time of vibration is 289 times the time taken by the vortex ring to pass over a length equal to its circumference.
3-2
36
ON THE MOTION OF VORTEX RINGS.
When ne is at all comparable with a, the time of vibration is
approximately
2π
2πα
;
√ {n² (n² — 1 )} m
2a -1.0772
(log 2n0e
or, since we may write, as n is large, n² instead of n² — 1 , it equals, Σπα
if I be the wave length n
2π
2ωπ ρ ―- 1.0772
12 (log Te
Now this case agrees infinitely nearly with the transverse vibrations of a straight columnar vortex which have been investigated by Sir William Thomson.
In the sub-case in which 7/e is large, he finds that the period of vibration
2π =
2.2 2wπ
log + 1159 + 1 12 (10% 2πе
(Phil. Mag., Sept. 1880, p. 167 eq. 61) ; or, since log, 2 = '62314, this equals
2π
2ωπ ρ 3272
12 (log we
and thus agrees very approximately with the value we have just found.
Since the amplitudes of a and Bn, when n is large are approximately the same as those of yn, and 8 , we can represent a displacement of this kind by conceiving the central line of the vortex core to be wound round an anchor ring of small transverse section so as to make n turns round the central line of the vortex ring, and
this form to travel along the anchor ring with velocity , where ▾
is the time of vibration just found and 7 the wave length.
( 37 )
PART II .
To find the action of two vortices upon each other which move so as never to approach closer than a large multiple of the diameter of either.
§ 14. The expressions for the velocity due to a circular vortex ring, which we investigated in the previous part, will enable us to solve this problem. If we call the two vortices AB and CD, then in order to find the effect of the vortex AB on CD we must find the velocity at CD due to AB. Now, since the vortices never approach very closely to each other, they will not differ much from circles ; hence in finding the velocity due to one of them at a point remote from its core , say at the surface of the other, we may, without appreciable error, suppose the vortex ring to be circular.
Let the shortest distance between the directions of motion of the vortices be perpendicular to the plane of the paper ; thus the plane of the paper will be parallel to the directions of motion of both vortices .
Let the semi-polar equations to the central line of the vortex AB of strength m (fig. 3) be
Fig.3. B
C D
p = a + Σ (2₁ cos nó + ß„ sin np) , z = 3 + Σ (y₂ cos no + d„ sin np) , when z is measured perpendicularly to the plane of the vortex AB and is measured from the intersection of the plane of the vortex AB with the plane of the paper ; a B Yn n are all very small compared with a. Let m be the strength of the vortex AB. Let the equations to the central line of the vortex CD of strength m' be p' = b + Σ (a', cos ny + B'n, sin ny ), z' = ' + Σ (y'n cos ny' + d'„ sin ny'),
38
ON THE MOTION OF VORTEX RINGS.
where is measured perpendicularly to the plane of the vortex CD, and from the intersection of the plane of this vortex with the plane of the paper ; a'n B'nነ 'n d'n are all very small in comparison with b.
We shall have to express a Bun Yn En a'n B'n ' n ' n as functions of the time ; we shall then have found the action of the two vortices on each other.
To find the action of AB on CD let us take as our axis of Z the perpendicular to the plane of the vortex AB through its centre, the plane of XZ parallel to the plane of the paper and the axis of Y drawn upwards from the plane of the paper.
Let e be the angle between the direction of motion of the two vortices ; l, m, n the direction-cosines of a radius vector of the vortex CD drawn from the centre of that vortex.
Let Z, X (fig. 4) be the points where the axes of Z and X cut Z
K
Fig.4.
a sphere whose centre is at the origin of coordinates, K the point where a parallel to the direction of motion of the vortex CD cuts this sphere, and P the point where a parallel to the radius vector of the vortex CD cuts the sphere : KP will be a quadrant of a circle.
Then we easily see, by Spherical Trigonometry, that l = cos e cos y, m = sin y,
n = -sin e cosy.
Now by equations (10, 13, 16) the velocities u, v, w parallel to the axes of X, Y, Z due to the vortex AB supposed circular are given by the equations
MOTION OF TWO VORTEX RINGS.
39
1
u = maZXA₁, 2R
1 v = maZYA,,
2R
w=
m (2a³A¸ — aRA₁),
where
R = √X² + Y².
Since
1 (a² + R² + Z² − 2aR cos 0)
1
(a² - 2aR cos 0)
noo
(R² + Z²)*
(R² + Z²)
4a2R2 cos20
+1.5
+.
(R² + Z³)
where, since R + Z ' is very great compared with a, the terms diminish rapidly,
1 A. ==
colo
a² op
a²R2 +
(R² + Z²)
(R² + Z2)
(R² + Z²)
3aR and A₁ =
(R² + Z *)$ *
Nowif f, g, h be the coordinates of the centre of the vortex
CD, and X, Y, Z the coordinates of a point on the central line of that vortex,
X = ƒ + bl = ƒ + b cos e cosy,
therefore
Y = g + bm = g + b sin , Z = h + bn = h ― b sin e cos ;
R² + Z² = X² + Y² + Z²
=ƒ² + g² + h² + 2b (ƒ cos e cos y + g sin -h sin e cos ¥) + b².
$ 15.
1
XZ
u = maXZA₁ = 3 ma²·
2R
(X² + Y² + Z²)
Substituting the values given above for X, Y, Z and writing ď²
for f² + g² + h² + b² we find that approximately u = 3ma²
[ƒh
5fh
bcos
+ d's
hcos e -fsin e
d2 (fcos e --h sinnee))) b cdoss
5fgh sin d
-)+
5 + sin e cos e- (h cos e -fsin e) (fccooss ee-h sin e)
d
35
b2
+
2d,Jfh
{ (ƒcos
e
−
h
sin
e)²
—
g"}) 2d
cos
24
+ (fsin e − h cose + fg (ƒ cos e -h sin e))
sin 24 …+.+..... … .. ( 47).
40
ON THE MOTION OF VORTEX RINGS.
When in these expressions we have a coefficient consisting of several terms of different orders of small quantities we only retain the largest term .
§ 16.
1
YZ
v = ma YZA, =• § ma²
5
2R
(X² + Y² + Z²)
Substituting as before we find v = {ma³
gh - g sin e + 5kg (fcos e -- h sine)) cos y
d's
d
d5
5g2 b + (1 - 5d0") , sin
gh + } (cos eg (ƒ sin e− h cose) + ď ( (cose - hsine )" - 9" )
5g2 -
5h sin e-
-
+ ((507 — 1) sin dz (fcos e − h sin e)
cos 24
b2
+
35hg" d+
(fcos
e−
h
sin
e))
2
sin 24
. (48).
§ 17.
w = m (2a A - aRA )
2a2 = &m.
3a* (a² + R²) + 15
a¹ R²
(X² + Y² + Z²)* (X² + Y² + Z³)
(X² + Y² + Z²)
ma²
X d³
-
-
[ 2 –−d¿32ª (ƒª + g²) + 3 (2 (h sin e − 2ƒfcos e)
b
+
-
52 (ƒ²+g³) (ƒcos e − h sin e))
{cos ↳¥ + 3g (ƒ² + g³ — 4h³)
sin
5 + } ( sin²e + d (ƒcos e − h sin e) (3ƒ cos e — h sin e) — 9º}
35
b2
-
— 2¿¹ (ƒª + 9") { ( ƒcos e −— h sin e) " — gº ) d2a cos 24
+15 (3ƒcos e -h sin e
7
big
- 2d2 ( +9 ) (ƒcos e-k sine) d+ sin 24 .+..] ... ((449).
MOTION OF TWO VORTEX RINGS.
41
§18. In using these expressions to find the effect of the vortex
AB on CD, we have to find the velocity perpendicular to the
plane of CD and along the radius vector. Then, as in the case of
da'n
the single vortex, we have equations of the type
= coefficient
dt
of cos ny in the expression for the velocity along the radius vector of CD.
To solve these differential equations, we must have the quantities on the right-hand side expressed in terms of the time. Hence we must express the value for u, v, w which we have just obtained in terms of the time.
§ 19. In the small terms which express the velocity at the vortex CD due to the vortex AB, we may, for a first approximation, calculate the quantities on the supposition that the motion is undisturbed.
Let us reckon the time from the instant when the centres of the vortices are nearest together.
Let p and q be the velocities of the vortices AB and CD respectively ; k the relative velocity, viz. √ (p² + q² − 2pq cos €) ; c the shortest distance between their centres.
Then, since f, g, h are the coordinates of the centre of CD at the time t,
f= f + qsin e . t,
g = g, -
h = h + (q cos e − p) t,
where f, g, h are the values of f, g, h when t = 0 ; since the distance between the centres of the vortices, viz. √ (ƒ² + g² + h³) is
a minimum when t = 0 ,
therefore
fq sin e +h (q cos e − p) = 0 ;
-f q cos € -p
=
q
h sin
e
=
√
(c² — kc
a²)
;
therefore if ħ be positive, we have
h = q sin e√ (c² — g²)
kc
. (50),
f = _ (q cos e −- − p) √ (c² – g²) kc
• (51 ) ,
and
ƒ² + g² + h² = c² + k²t².
§ 20. If we substitute for f, g, h in the expression for w their values in terms of the time, we find that as far as the term independent of goes,
42
ON THE MOTION OF VORTEX RINGS.
W=
ma2
3
- q² sin² e g²)
k2
(c² + k²t²) ½
2-
6√c² g
+ 6 √ok =
(q cos e -− p) qsine.t + [{22 ( q cose — p) '—q′sin'e} ¿"]… ..( 52 ).
The coefficient of cos y
-
= -8
ma³
{2
g2 (√c² =gr
k
{q
(sin²
e
+
2
cos²
e)
—
2p
cos
e}
1
-sin e (q cos e + p) t cos e + p) t )
+ (L + Mt + Nť² + Pť³) ·
(c² + k²t²)
(c² + k²t²)
......... (53),
where
L=
-
-
ka
(p cos e 9) ( c² (p − q cos e)² + g²q² sin² ²e)
sin €
M=
-
k2 (c² (p − q cos e) (p² + pq cos e − 2q*) + g°q (pq (3 + cos² e)
N= 'c² — g² sin² e .. q {2p² —- qp cos e — qº}
k
P = sin³ e qp.
- 2 (p² + q²) cos
The coefficient of sin
3ma bg
(c² - g²)
{c² - 5
q2 sin' e
k2
2 (c² + k²t²)
10√c² - g²
+
k
q sine (p - q cos e) t + (5q² sin² e -—- 4k²)ť²}……….…. (54) .
The coefficient of cos may be written
sin² €
c²- g3
3 ma²b²
+ k² (pcose - q) (3 cose (p -qcose) -qsin❜e) -— g³
·(c² + k³t²)
+
2 √c² k
g²
sin e
{ 2p (p cos e -−
q)
+q
(p
−-
q
cos
e) } t
1 + sin e p (q sin 2e + p sin e) ť² )
(c² + k²ť²) {
-
1
— 35 ( L' + M′t + N'ť² + P't³ + Q't¹)
(c² + k²ť²)
.. (55),
MOTION OF TWO VORTEX RINGS.
43
where
1 -
I '′ = }, ( c ° (q cos e − p )* + g′qª sin² e) (c² (p cos e − q)ª
-g³ {(p cos e - q)² + k²}),
-
M = 2√c² — ª² sin e {c² (p² — q²) (p cos e − q) (p − q cos €) k³
+ g³q ((p cos e - ·q) ( 2pq — cos e (p² + q²) )
+ (q cos e − p) (p² + q² − 2pq cos e)) } ,
sin2 €
N=
{cp (q cos e - p)
-
g'q²
(p
cos
e
· − q)²
k²
-
G
-
— 4 pq (c² — g²) (p − q cos e) (p cos e − q)} ,
2 sin³ € P' = k √c² — g³ pq (p² — q²),
Q' = sin' e p²q².
The coefficient of sin 24 may be written.
-
√c²
—
ª² { 3p cos € {3p cos e -
q
(3
cos²
e
+
sin²
e)}
¥ maʼg {{√o² k
1
+ (q sin 2e + p sin e) t (c² + k²ť²) ½
1 − 1 (L + Mt + Nť² + Pť³)
(c² + kit²) 3
(56) ,
where L, M, N, P have the same values as in equation (53 ) .
§ 21. The velocity parallel to the axis of y.
The term independent of
ma²g (√c²- q²
= 33
k q sin e + (q cooss ee -− p) t.….…..(57) .
(c² + k²t²) ²
The coefficient of cos
==• ma'bg
sin €
+
(c² - g²) 5
:((62) 12
sin
e .q (p
cos
― e − q)
·(c² + k²t²)
1
+ √c² = g* {2pq — cose (p² + qq³')} t + sine.p (qcossee −—pp)) ťť))
k
(c²+ k²t²)=
. (58).
The coefficient of sin
qsine
1
5g²
k = { ma°¿g(√/c²—gº2³in²
+ (qcose —p)
t)
`((¿c² +kk²vt²)) ²½ ¯
(c² + k²ťA²))$².) ..(59) .
44
ON THE MOTION OF VORTEX RINGS.
The coefficient of cos 24
-
1
➡= 4
mal°g
{cos
«e√e² — hc
g²
(psin e -qsin 2e)
(c² + k²t²)
where
1
+ } (L” + M″ t + N'ť² + P''ť³)
... (60)
(c² + k²t²)
-
L' = c* — g³
-
{(c² — g³) (p sin e − q) ³ — g² k³},
1
M"
= - 11
{{ (( c²
—
g³)
(p
cos
e −— q)
[ (2
+
sin³
e)
pq − cos
e (p²
+ q³)]
k²
-
-
— g³ k² (q cos e − p)} ,
N"
=√c² k
g² -p
sin
e
{pq
(3
+
cos²
e)
—
2
(p²
+
q³)
cos
e},
P" = sin² e . p² (q cos e − p).
The coefficient of sin 24
= —-— -§& mar³
5g²
1
{sin e (1 − ((c¿² +4Pk²Tt²))) (& + ke
+5 ( 1 +5 (1-(c²7
'c² — g² sin e . q (p cos e — Q) ) ( k sine .
g2
1
+ k {2pq— cos e (p² + gº)} t + sin e .p (qcos e —−pp)) ťťº)) (♬ +1ºť)i,
(c² k³t³)
(61).
§ 22. The velocity parallel to the axis of x.
The term independent of
-
´(c² — g³)
==3 ma²
sin e. q (p - qcos €)
2
√c²- g²
1
k {qª cos 2e −2pq cos e +p”} t + sin e.q (q cose—p) ťª} (c²+ 1²²)
...... (62) .
The coefficient of cos
1
=}
ma′b
{{√o* =gº k
(q
sin2e −p
sine) +
(gcos
2e
−p
cos
e) t)
(c² + k²t²) §
1 − 5 ( L'" + M''t + N'''ť² + P'''t³)
(c² + k³t²)
MOTION OF TWO VORTEX RINGS.
45
where
-
(c³ — g³) ³
L' =
sin e .q (q cos e - p) (q − p cos e),
k
M" = c² =ª² (q³cos 2e -pq'cos e (cos² e + 2) + p³q (2 + cos²e) — p³ cos e) ,
k?
N" = sin € .
g² (— q³ cos e + q²p ( 1 + sin² e) + qp² cos e — p³) ,
k
P"" = sin² e . pq (q cos e − p).
The coefficient of sin
(c² - α²) = - 15 mmaa³bbga {(c°k sin e . q (p -q cose) –
-² (q² cos 2e k
1
-2pq cose + p²) t + sin e.q (q cos e − p) (c + k²t²):
.. ( 64 ) .
The coefficient of cos 24
sin ecose = ma²b²
((c² + k²t²) *
- - (c³ — g²) sin e {2q2 cos e - pq ( 1 + 2 cos² €) +p² cos e}
k²
1
(q² cos 2e - 2pq cose +p³)t + sine.p (q cos 2e -pcose)ť²)
k
(c²+ k²t²)
1
+ 35 (L₁ + M¸t + N₁ť² + P¸ť³ + Q₁t*)
. (65) ,
(c² + k²t²) +
where
(c² - g²)
=L₁
sin e . q (q cose - p) {c² (p cos e −q)² k*
— g² [(p cos e − q) ² + k²] } ,
/c³ — g² { (c² - — g³) ( p cos e -− q) (q³ cos 2e -−pq² cos e ( cos 2e + cos²e) M.
+p²q (1 + 2 cos 2e) -— p³ cos e) + g²² (q² cos 2e - − 2pq cos e + p²)} ,
(c² - g²)
―
sin e {q (q cos e - p) (p² cos 2e − 2pq cos e + q²)
N₁ = k
+ 2p (p cos e −- q) (q²' cos 2e - 2pq cos e + p") },
P₁
=
√cc²³ — k
ª²
sin²
e
.
p
(3pq²
— p³
—
2q³
cos
e) ,
Q₁ = sin³ e . p³q (q cos e − p) .
46
ON THE MOTION OF VORTEX RINGS.
The coefficient of sin 2
== 15 ma²b²
/c² - g² (q sin 2e -p sin e)
k
+ (q cos 2e − p cos e ) t)
1
(c²— g²)
+7
(pcose - q) (p -q cos e)
k²
(c² + k²tu)
-2
1
+ kc sin e (p - q ) t +pq sin³ e€ t² (c + kt²)
. (66).
§ 23. To find the effect of the vortex AB on CD we require the expressions for the velocity perpendicular to the plane of the vortex ring CD and along its radius vector.
The velocity perpendicular to the plane of CD = w cos e + u sin e.
Now in this expression , the most important terms are the coefficients of cos and sin y, because these terms, as we shall see, determine the deflection of the vortex. We shall therefore proceed to find these terms first.
The coefficient of cos in the expression for the velocity per-
pendicular to the plane of CD may be written as ma b
3(c³ + k²t³)}·( A + Bt + Ctª + Dť ) ,
where
A
√√ =
(c² —
g³)
{c²
(pcos
2e
- qcos
e)
-—
5
sin³e
(c²-
g³)
pq
(p
cos
ee −-
q) } ,
k
k²
-B = c² (p.sin 2e q sin e)
5 (c² - g²)
+ k2
{q³ - q² p cos e - qp³ (1 + sin³ e) + p³ cos e},
C == /√c³ — 9ª
k
{4k²
(q cos e − p )
+ sin²
e . p (8pº
— 7q′
— pq cos e) } ,
D = sin e {5p³q sin³ e - k³ (q + 3p cos e) } .
The coefficient of sin
ma ba
=
(A' + B't + Ct).
(c² + k²ť²)z
where
... (67),
5 (c³ - g²)
A' = c² cos e-
pqsin² e,
k2
5√ (c² - g²)
B' =
{2pq sin e cos e - 2q² sin e + k²},
k
C'
= k²
cos
€
·
5
(q
cos
e
−
p)
(q
−p
cos
e) .
MOTION OF TWO VORTEX RINGS.
47
Now, since the equation to the vortex CD is
z' = ' + Σ (y'n, cos ny + d'n sin ny).
The velocity perpendicular to the plane of the vortex
dz + Σ (dy" cosny + do, sin ny).
= dt
dt
dt
since as 'n, ' and I are all small quantities we may neglect -
n (S'n₂ cos ny - y' ,n sin ny) Y.
dy,
Thus
= coefficient of cos in the expression for the velo-
dt
city perpendicular to the plane of the vortex CD.
A reference to equation (43) will shew that the vortex CD con-
tributes nothing to this term, so that
dy',
ma2b
(A + Bt + Ct² + Dt³). dt (c² + k²ť²) ½
Integrating we find
Dc²/k* - B/k2
D/k
A - Cc /k² t
Y'₁ = } mab {t
13/08
+ 1/1/0
c²
(c² + k²t²)*
(c² + k²ť²) *
(c² + k²²)
{4A/c* + C/c²k²} t
8A 20
t
+ 1/8 (c² +k²t²) *
+15 Cε + ck
+ ·(c² + k²t²) ³ }k)}
where the arbitrary constant arising from the integration has been determined so as to make y'1, = 0 when t = − ∞ .
If we substitute for A, B, C, D the values given above, we shall
get the value for y, at each instant of the collision ; but at present we shall only consider the change in y' , when it has got so far away
from the vortex AB that its motion is again undisturbed. We can
find this change in y' , by putting to in the above formula, on doing this we find
4A C \ 2
Y'₁ = { ma²b
+
св c4k² k
or substituting for A and C their values ,
2ma❜bpq
- 4g
-
r₁= c*k*
(1 − p cos e) (1 — ¹ª³) √ (c² — g³) sin³ .……….(69).
§ 24. We have similarly ds'1, = coefficient of sin in the expression for the velocity perpendt
dicular to the plane of the vortex
ma ba (A' + B't + C'ť²).
(c² + k²t²) =
=
48
ON THE MOTION OF VORTEX RINGS.
Integrating we find
-
B'/k2
(A' — C' c²/k³)
t
8',1 = { ma²bg { − +
+}
c²
(c² + k²t²)*
(c² + k²t²) #
(4A '/c*+ C'/c²k³)t
18A' 20"
t
+18
++
(c² + k³t²) #
+
+
20)(
·(c² + k²t²)·
where the arbitrary constant arising from the integration has been determined so as to make 8,0 when t∞ . The change in
S',1 when the vortex CD is so far away from AB that its motion is undisturbed is given by
18A' 20'' 2
5₁1 = {} maba is (34
+ ck
k
Substituting we find
δ', = 2ma²bª sin² e . pq 1 -4g2\
C¹³
3c²
...........(71) .
§ 25. We have in paragraph (6) investigated the changes in
the direction cosines of the direction of motion of the vortex ring
due to changes in the
coefficients
y' , and 1
' 1°
From that investi-
gation we find that the direction cosines of the direction of motion
of the vortex CD after the impact are
sin € -γι COS €, b
S1 b
COS € + sin €, b
or substituting for y' , and '1, the values just found, the direction cosines become
2ma2
sin € c*k*
√ (c² — ggº²)) (( 11 - 40²)
sin'e cos e. pq (q − p cos e) ,
2ma² sin²e .
4g
ck sin'e . pq ( 1 - 319c7) "
2ma²
COS E +
-
4g2
-
c*k* √ (c² — g³) ( 1 − 1.9 ") sin²e. pq (q − p cos e).
Thus if A, B, C (fig. 5) be the points where the axes of x, y, z cut a sphere with the origin for centre and P the point where a parallel through this centre to the direction of motion of the vortex CD before the collision cuts the sphere.
Then if the vortex CD be the first to intersect the shortest
MOTION OF TWO VORTEX RINGS.
49
distance between the directions of motion of the vortices, P' will be the point where a parallel to the direction of motion after impact
Fig.5 .
B
cuts the sphere, supposing g to be positive and < c and the velocity of CD greater than the velocity of AB resolved along the direction of motion of CD, i.e. if q - p cos e be positive. We may describe this by saying that the direction of motion of the vortex ring is altered in the same way as it would be if the vortex ring received an impulse parallel to the shortest distance between the directions of motion of the vortices and another impulse perpendicular both to its own direction of motion and the shortest
distance ; the first impulse being from and the second towards the vortex AB. In this case the angle between the direction of motion of CD and the original direction of motion of AB is diminished by the impact.
If the vortex AB be the first to intersect the shortest distance
then we must change the sign of √ (c² - g ) in the expressions
for f and h ; this will change the sign of y' , but will leave S',
unaltered, and consequently P" the point where the direction of
motion of CD after the impact intersects the sphere of reference
will be situated as in the figure ; in this case the angle between
the direction of motion of CD and the original direction of
motion of AB is increased by the impact. The angle through
which the direction of motion of CD is deflected
2
=
8, +
b2
2ma²pq = 2maq sin'e
2
(9 - pense)
4g2 (1-19′)
(~(c"² — g)
+ g°
(1
4a0²22 −- +0 ) " }
k
3c2
... (72 ) .
If the paths of the vortices intersect so that g = 0, this becomes
2ma' sin²e -
-pq (q − p cos e) c³k+
. (73),
T.
50
ON THE MOTION OF VORTEX RINGS.
or the deflection is cæteris paribus inversely proportional to the cube of the shortest distance between the vortices.
If the paths of the vortices do not intersect, but the vortices
move so as to come as close together as possible, then c = g,
and the deflection
2ma sin❜e.pq =
3c³k³
..... (74).
This is again inversely proportional to the cube of the distance.
If in the two cases above, c be the same, then the deflection when the paths of the vortices intersect will be greater, equal to or less than when they do not, according as 8 (q - p cos e) is greater, equal to, or less than p' sin e ; thus, unless the relative velocity of the vortices perpendicular to the direction of motion of CD is great compared with that along CD, the deflection will be greater when the directions of motion of the vortices intersect than when they do not.
The expression for the deflection simplifies when the line.
joining the vortices at the instant when they are nearest
together is inclined at an angle of 30° to the shortest distance between their directions of motion, in this case g = c cos 30° =c√3,
thus &', == 0 as 1 ―
4g² vanishes, and the deflection
3c²
2ma² sin²e . pq (q — p cos e) ck
which, if c be the same, is the same as when the vortices intersect .
§ 26. We have next to consider how the vortex CD is altered in size by the collision.
We know that if a 0 be the alteration in the radius of the vortex CD that da'
= dt coefficient of the term independent of in the expression for the velocity along the radius vector of CD.
Now a reference to equation (38) will shew that the vortex CD contributes nothing to this term itself, so that da'.
= coefficient of the term independent of in the expression dt for the velocity along the radius vector of CD due to the vortex AB.
Since λ, u, v, the direction-cosines of a radius vector, are by § 6 given by the equations
MOTION OF TWO VORTEX RINGS.
51
λ = cos e cos y,
μπ sin ,
v = sin e cosy,
da0。 = coefficient of the term independent of in
dt
u cos e cos + v sin - w sin e cos y.
Hence by equations (53), (59), (63) ,
dz
mab
=
(F+ Gt + H + Kt),
dt (c² + k²t²)=
where
-
F= √ (c²— gº)
-
k³ sin e [c² {p³q (4 — cos³e) — 2p³cos e — q³ } — 5gª sin³e . p²q] ,
(p² c² COS
G= & { (qcos e - p) (2-5 (q - poose) ") + sin'e .p ( 3-5 (pk" -9" ) } ,
H = √ (c² — g³) sin e (8p³ cos e - p²q cos² e - 11p³q + 4q³),
k
K = k² {2 (q cos e − p) + 3p sin³ e} - 5pq sin'e (q − p cos e) .
Integrating, we find
a' = 2 ma²b
Kc²/k - G/k² - K/k
F- Hek
t
-}
+ 1/3 c²
(c² + k²t²)#
(c²+ k³t²)}
(c²+ k²t") =
(4F/c* + H/c²k²) t
18F 2H
t
+ 1/8
+ 1/15
+
+
C6 ck
(c² + k²ť²)*
(c² + k³t²)
where the arbitrary constant arising from the integration has been determined so as to make a = 0 when t = -∞ . If we
substitute for F, G, H, K the values just written we shall get
the change in the radius at any instant, but at present we shall only consider the change in the radius of CD when it has got so far away from the vortex AB that its motion is again undisturbed. We can find this change in the radius by putting t = ∞ in the above formula ; doing this we find
ma2b /4F H +
5k C6 ck
Substituting for F and H their values, we find
ma2b sin³ € . p²q
4g
=
1-
2
--√ (c² — g²) ........ ( 74 *) .
k³c
Thus we see that the radius of the vortex which first passes through the shortest distance between their directions of motion is increased, provided c > 2g. If AB had first intersected the shortest
4-2
52
ON THE MOTION OF VORTEX RINGS.
distance we should have had to change the sign of (c - g³) , then a would be negative, and the radius of CD would be diminished.
If the directions of motion of the vortices intersect, so that g = 0, then
ma2b sin³ e. p³q άπ =
k³c³
or the increase in radius is cæteris paribus inversely proportional to the cube of the shortest distance between the vortices.
If the directions of motion of the vortices do not intersect, but the vortices move so as to come as close together as possible, then c = α, and a = 0, and the radius of the vortex in this case is not altered by the collision.
If c = 2g, or if the line joining the vortices when they are nearest together be inclined at an angle of 60° to the shortest distance between the directions of motion of the vortices, then a = 0, or in this case again the radius of the vortex is not altered by the collision. Thus we see for our present purpose we may divide collisions into two classes. In the first class the line joining the centres of the vortices when they are nearest together is inclined at an angle greater than 60° to the shortest distance between the directions of motion of the vortices. In this case the vortex
which first passes through the shortest distance increases in radius, and consequently decreases in velocity and increases in energy, while the other vortex decreases in radius and energy and increases in velocity.
In the second class of collisions the line joining the centres of the vortices when they are nearest together is inclined at an angle less than 60° to the shortest distance between the directions of motion of the vortices. In this case the vortex which first passes
through the shortest distance decreases in radius, and consequently increases in velocity and decreases in energy, while the other vortex increases in radius and energy and decreases in velocity.
§ 27. Having found the change in the radius and the change in the direction of motion of the vortex, we can find the changes
in the components of the momentum of the vortex referred to any axes .
Let I ' be the momentum of the vortex CD ; P', Q', R′ its com-
ponents parallel to the axes of x, y, z respectively, l', m', n' the
direction-cosines of the normal to the plane of the vortex.
Thus
I' = m'πb², P' = m'πb²l' ,
SO
SP ' = 2πm'bdbl' + m'πb²Sl′
= 2 %/% P' + 'Sʊ',
MOTION OF TWO VORTEX RINGS.
53
similarly,
¿Q² = 2 ºº Q' + I'dm ',
SR' = 2 7°0 R' +F'dn'. b
It remains to find dl', dm', Ɛn' in terms of y', and S. Now if E, P, Q, R denote the same quantities for the vortex AB as the same letters accented do for the vortex CD, then it is easy to
prove that the direction-cosines of the old axes referred to the new are as follows.
The direction -cosines of the old axis of x are
P'E - PE' cos e Q'I - QI' cos € RI - RI' cos e
I.I' sin e
I.I' sin e
I.I'sin e
The direction-cosines of the old axis of y are
OR - RO RP - PR' PO' - OP '
I.I'sin e
>
I.I' sin e
I.I'sin e
The direction-cosines of the old axis of z are
P છดે R
E
" I
I
Thus if λ, μ, v be the direction-cosines of the normal to the
plane of the vortex CD referred to the old axes, then
-
δι'
= dλ
(P'X
–
PF'
cos
e)
+
dµ
(QR' -
RQ )
, +
Sv.P
I. ' sin e
I. I' sin e
I
with symmetrical expressions for dm' and Ɛn'. Now by § 6 δλ == COS E,
δ'
δμ = " b
δν = Y¹ sin e. b
Substituting for y' 1, and ' 1, their values, we find
δι'
=2ma pq sin e c¹k³ . I'
(q
− p cos €
k
√ (c² —— g³)
(1
4g2 − 4c9² ")
(pr′
– EEPp'′
cos
e)
+ g (1-2 3c2) (OR = RQ)} with symmetrical expressions for Sm ' and Sn'.
54
ON THE MOTION OF VORTEX RINGS.
Thus
2ma pq sin e (√(c— g³)
843' = c'k³1 . I'
k (1 - 42")
× F' { PF '(q — poos e) – PE (q cos e - p ) ) +g ( 1-4 ) F((@ RR´-_RRO Q '))}}
=_pq sin e √√√(c²- g²) k (1-4 )
wo
-
- 4g
× {PF' (2 −pcos e) — P´E (q cos e −p)} +g ( 1 3c
(QR′ – RQ )}
with symmetrical expressions for dQ' and R'.
.. (75) ,
If be the angle which the line joining the centres of the vortices when they are nearest together makes with the shortest
distance between the paths of the centres of the vortex rings, then
g = c coso,
SO
-
- =--c sin 34,
(1.– 4 º ) √c — g² = c sin 4 (4 sin² + — 3) =
and
4g2
4 cos²
g (1 - 138c0²)7. = c cos + (1
14)=-3 cos 34. 3
Thus
4pq sin €
-
sap' = πολ sin 34 { X' (q — p cos e) - P'I (q cos e − p) }
-cos 34 R(QR' –-RO') RQ)] ,
with symmetrical expressions for SQ' and R'.
Since similarly
+ ' is constant throughout the motion = - sap',
SQ = - SQ ', SR =-- SR'.
§ 28. We can now sum up the effects of the collision upon the vortex rings AB and CD. We shall find it convenient to express
them in terms of the angle & used in the last paragraph : is the angle which the line joining the centres of the vortex rings when they are nearest together makes with the shortest distance between
the paths of the centres of the vortex rings, & is positive for the vortex ring which first intersects the shortest distance between the
paths, negative for the other ring, so that with a given a, may be regarded as giving the delay of one vortex behind the other.
§ 29. Let us first consider the effect of the collision on the radii of the vortex rings.
MOTION OF TWO VORTEX RINGS.
55
The radius of the vortex ring CD is diminished by ma²b ck p³q sin³ e sin 3p.
Thus the radius of the ring is diminished or increased according as sin 34 is positive or negative. Now & is positive for one vortex ring negative for the other, thus sin 34 is positive for one vortex ring negative for the other, so that if the radius of one vortex ring is increased by the collision the radius of the other will be diminished . When is less than 60° the vortex ring which first passes through the shortest distance between the paths of the centres of the rings diminishes in radius and the other one increases. When is greater than 60° the vortex ring which first passes through the shortest distance between the paths increases in radius and the other one diminishes. When the paths of the centres of the vortex rings intersect & is 90°, so that the vortex ring which first passes through the shortest distance, which in this case is the point of intersection of the paths, is the one which increases in radius. When is zero or the vortex rings intersect the shortest
distance simultaneously there is no change in the radius of either vortex ring, and this is also the case when is 60º.
§30. Let us now consider the bending of the path of the centre of one of the vortex rings perpendicular to the plane through the centre of the other ring and parallel to the original paths of both the vortex rings.
We see by equation (71) that the path of the centre of the vortex ring CD is bent towards this plane through an angle
ma2 pq sin² e cos 34 ;
c³k³
this does not change sign with 4, and whichever vortex first passes through the shortest distance the deflection is given by the rule that the path of a vortex ring is bent towards or from the plane through the centre of the other vortex and parallel to the original directions of both vortices according as cos 36is positive or negative, so that if is less than 30° the path of the vortex is bent towards, and if o be greater than 30° from this plane. It follows from this expression for the deflection that if we have a large quantity of vortex rings uniformly distributed they will on the whole repel a vortex ring passing by them .
§31 . Let us now consider the bending of the paths of the vortices in the plane parallel to the original paths of both vortex rings. Equation (69) shews that the path of the vortex ring CD is bent in this plane through an angle
2ma² sin² e sin 34 pq (q − p cos €)
56
ON THE MOTION OF VORTEX RINGS.
towards the direction of motion of the other vortex. Thus the direction of motion of one vortex is bent from or towards the direction of motion of the other according as sin 34 (q - p cos e) is positive or negative. Comparing this result with the result for the change in the radius, we see that if the velocity of a vortex ring CD be greater than the velocity of the other vortex AB resolved along the direction of motion of CD, then the path of each vortex will be bent towards the direction of motion of the other when its radius is increased and away from the direction of motion of the other when its radius is diminished, while if the velocity of the vortex be less than the velocity of the other resolved along its direction of motion, the direction of motion will be bent from the direction of the other when its radius is increased and vice versa. The rules for finding the alteration in the radius were given before.
§32. Equation (75) shews that the effect of the collision is the same as if an impulse
pqł .I' sin² e sin 30,
pc k
parallel to the resultant of velocities p - q cos e and q ― p cos e along the paths of vortices ( CD) and (AB) respectively, and an impulse
pqt .F sin² e cos 30,
3προκ
parallel to the shortest distance between the original paths of the vortex rings, were given to one of the vortices and equal and opposite impulses to the other ; here I and I' are the momenta of the vortices.
§ 33. We have so far been engaged with the changes in the magnitude and position of the vortex ring CD, and have not considered the changes in shape which the vortex ring suffers from the collision. These changes will be expressed by the quantities ag B₂, ag, B₂3,9 &c. We must now investigate the values of these quantities.
Now we know da',
= coefficient of cos 24 in the expression for the velocity along dt
the radius vector.
A reference to equation (38) will shew that the vortex ring CD itself contributes to this coefficient the term
2m' 8b log
πb2
MOTION OF TWO VORTEX RINGS.
57
The vortex ring AB contributes, as we see from equations (53),
(59), and (63), a term ma²b {F" + G′t + Hť² + K'ť³),
(c² + k²t²) where
C³ -
F = sin e { p³q (2 — cos³e) + 4pqª cos e —·3q³ - 2p³ cos e} , k2
G' = c²
-
-
-
(q -p cos e)2) -
{p sin² e ( 3 — 5 (p² - 9 " ) — 5 (q cose −p)
c sin e
H' =
-
k {8p³ cos e − p³q (2 + cos² € ) + pq² (4 cos e − 11 ) + 2q³} ,
K = 3k'p sin'e - 5pq sin'e (q - p cos e), where, in order to make the work as simple as possible, we have put g = 0 ; so that the undisturbed paths of the vortices intersect.
Thus
da'.
2m' 8b
=
log
dt
πb2 é
ma2b
·(F″ + G′t + H'ť² + K't³), } (c² + k²t²)
say
dx, =
dt
2m' 8b log · Ÿ'₂ +ƒ(t).
πb2 é
dý 2 Now dy = the coefficient of cos 24 in the expression for the
dt
velocity perpendicular to the plane of the vortex CD.
The vortex CD itself contributes to this coefficient the term
m' 8b The log · 2*
é
The vortex AB contributes, as we see from equations (55)
and (65), the term ma²b (F"' + G″ t + H''t² + K't³ + L″ t ).
(c² + k²t²) 3
Say for brevity F(t), where if, as before, we put g = 0,
F" = (p cos e q) {35 (p cos e − q) (3pq sin² e — k³ cos e)
— 5k² (½p sin² e + p − q cos €)} ,
5c³ sin e
G"
k³ [(1² — p³) {21 (p cos e − q)² + žk²}
-
-
+ (p cos e − q) p ( 41 pq sin² e — 5k²) ] ,
58
ON THE MOTION OF VORTEX RINGS.
pc² sin' e
H" =
-
k² [{p³q sin³ e + 2 (p cos e − q) (p² — q³)
- (p cos e - q) } (q cos e − p)
— ½ 5k² {7 cos e . k² + ( p² + q²) cos e − 2pq} ],
pc sin e
K" =
k {21p sin² e (q² — p³) — k² (5p cos e + 6q) },
L" = p sin² e {21p (q —p cos e) (q cos e -− p ) — k² ( 5p cos e + 6g) }.
Thus
m'
=
ddy'ts
-
1 m, logo. d', По e'
+
F ( 6)
;
dx' differentiating this equation, and substituting for from the
dt
other equation, we find
d'y'2
'm'
m' 26
ddrt²2 +3 ( 6- ; log 2b) , = F' ( t) + 4 mb²2 log 25 f (t)
= x (t) say ;
or writing n² for 3 (wπbόa log 20) ;
P✅2 ' ď''½ + n³y'₂2 = x (t). dt2
The solution of this differential equation is
y'₂ = A cos nt + B sin nt cos nt t
+ n I'x (t') sin nť dť
sin nt ft
x (t) cos nt' dt', n
or choosing the arbitary constants so that y' , and dy2 both dt
vanish when t := -∞ , we find
cos nt "t
sin nt rt
=
X (t') sin nt' dt'
X (t') cos nt' dt'.
น n[ x
- n fx
The integral
x (t') cos nt' dt'
involves integrals of the form
rt cos nť dť and
-∞ (c² + k²t²) $(2p+1)
t' cos nt' dť -∞ (c² + k²t²) $ (2p +1 ) °
I have not been able to evaluate these integrals except when t∞ .
MOTION OF TWO VORTEX RINGS.
59
9 20
In the expression for y',2 the terms under the integral express the effect of the vortex AB on CD. Now the vortex AB will
only exert an appreciable effect on CD during the time the vortices are in the neighbourhood of the place where they are
nearest together ; and thus, after the collision , we may, without appreciable error, write the equation for y', as
P cos nt =
Q sin nt
ช่ 2
n
n
∞ +.
where
P=
sin nt . x (t) dt,
81
•+ ∞
= Q
cos nt . x (t) dt.
Thus the vortex rings are thrown by the collision into vibration, and after the collision is over the period of the vibration is 2π
the same as the period of the corresponding free vibration of n the vortex CD.
To find P and Q we have to find •+00 cos nt.dt , (c² +k²²)& (2p+1)
or if we write q for k',
+00 cos nt.dt
88
(q²
+
(2p+1) t²)$ (2)
Now q is the time taken by the vortices to separate by a
2π distance c, while is (§ 13) of the same order as the time taken
n
by the vortex CD to pass over a length equal to its diameter ; but, since c is large compared with the diameter of the vortex,
2 or ng is large. 2π
∞ cos nt.dt
Let
} } (2p+ 1) = N² V₂ •
(q² + t²)*
By differentiation we find
-2
Vp+1 = -
dvp -P
(2p + 1) \dn nv).
Hence we find
= (−1)" q² Up 1.3.5... (2(p - 1 ) (d
−
-
p
) (d = 2)...d &
n
dn
60
ON THE MOTION OF VORTEX RINGS.
This may be written
= (-2n)'q-²
d
Vp
1.3.5
...
(2p- - 1
)
\d
(n²).
.. (76).
We can easily verify that , satisfies the differential equation 2
dv, + 1 dv, - (P² + q²) v₂ = 0. dn² n dn -C-
Let us assume
-nq
A,
= Vp
x
((-4₂ + 4n
+4
+ ...) .
If we substitute this expression for v, in the differential
equation, and equate to zero the various powers of n, we get the
equations
A¸ (2qx q) = 0 ,
2q (x + 1 ) A₁1 + x (x + 1) A, -qA, -
xA。0 -p³A = 0,
2q (x + 2) A₂ + (x + 1 ) (x + 2 ) A¸ −qA¸− ( x + 1 ) A¸ −µ³Ã¸ =0,
2q(x+m) Д + (x+m− 1) (x+ m) Am -1 − qAm− (x+ m − 1 ) Am-1 - p³Am-1 = 0 ;
these give
x = 1,
2q4, +
(1 − p²) A¸ = 0,
44qq4A2, +
(2 - p²) AA₁, = 0,
therefore
2mqAm + [ † (2m — 1 ) ³ — p³] Am-1 = 0 ;
e-ng
v= p
4.1+
η
1 -
22
+ 2nq
2
2- 321
11) p 22
(2ng) 2.2
32
52、
22
22
22
+ C-D(C 2n- g)³D .3 !C-D ....
and Aalone remains to be determined ; if we can determine A, for any value of p, we can find it for any other by means of equation (76 ) . Now when p = 0,
cos nt . dt
and
• (9² + t²)$ ∞
cos nt.dt == K (i.ng)
° (q² + ť²)
MOTION OF TWO VORTEX RINGS.
61
(Heine, Kugelfunctionen, vol . II . § 50) , where K is the second kind
of Bessel's function of zero order and i = √ − 1 .
When ny is large, eng
K (i.ng) = √√ (31π) (ng)
(Heine, vol. I. § 61 ) ; hence
e-ng
1
1.32
v₂ = √ (1π)
-
+
(1 (ng)2
23. 2nq 25. (2nq)²
- ...) ;
and, by equation (76), we find on comparing the coefficient of eng
that
(ng)2
q
e-ng
A₁ = √ ( π)
1.3.5 ... (2p - 1) (ng)
therefore e-nq
v, = √ (3π ) 1.3.5 ... (2p - 1 ) (ng)
p
x1 +
+
2nq
22
32 -
22
(2ng)* . 2
cos nt . dt
no
1
e-ng
=
[+8010
(2p+1) (q² + t²) $ ( 2p
√ (2π) q1.3.5 ... (2p− 1 ) ( q) *
22
+
x {1
2ng
-
2 - 32
p 22
+ ...
(2ng) .2
. (77) ,
and this series converges rapidly when nq is large.
The other integrals in Q are of the form t cos nt.dt (2p+1)
S (q² + t²)
∞-
and these evidently vanish.
The integrals in P are of the forms
∞
sin nt . dt
and
-2
(2p+1 )
(q² + t²)
t sin nt . dt (2p+1) *
-∞ • (q² + t²)²
The first of these evidently vanishes, and the second
00
d
cos nt . dt
==
dn 81 (
(2p+1)
(q² + ť²)'
and we have just found the value of the integral.
62
ON THE MOTION OF VORTEX RINGS .
§ 34. We can now find the values of y' , and a „
By § 28,
Pcos nt Qsin nt
.
n
n
where
Р=
sin nt . x (t) dt,
-∞
= cos nt . x (t) dt.
If we substitute for x (t) its value, and evaluate the integrals
by means of formula (77), and retain only the largest terms, we shall find
m√ (2π) a²b²
P:=
8k⁹
{4p cos e (p cos e − q)²
e-nc/k
-
-
-
— 4p (q² — p²) (g — p cos e) + cos e (q² — p²)²} .n³
(nc/k) *
m√ (2π) a²b²
e-nc/k
= Q
--
8k5
sin e {4p (p cos e − q) ² — (q² — p²)²} . nº
(nc/k)
If the vortices move with equal velocities these expressions
simplify very much and become
so that
m √ (2π) a²b³n"
e-nc/k
P=
COS €
8k5 (nc/k)
5 m √ (2π) a²b²n°
e-nc/k
= Q
sin e
8k
(nc/k) +'
m √ (2π) a²b³n¹ e-nc/k
8/5
cos (nt + e) ........ (78) ;
(nc/k)
therefore
m√ (2π) a²b³n e-nc/k
√3.4k
sin (nt + €) ......... (79). (nc/k)*
These equations represent twisted ellipses whose greatest ellipticity is
m√ (2π) a²bnt e-nc/k √3.2k (nc/k)**
The time of vibration is the corresponding free period.
§ 35. We can now sum up the effects of the collision of two vortices AB and CD.
The collisions must be divided into two classes, (1 ) those in which the shortest distance between the vortices is greater than twice the shortest distance between the directions of motion of
the vortices ; (2) those in which it is less.
MOTION OF TWO VORTEX RINGS.
63
Class I.
If the vortex CD be the first to intersect the shortest distance between the directions of motion of the vortices its radius is
increased, and if its velocity is greater than the velocity of AB, resolved along the direction of motion of CD, it is bent towards the direction of motion of AB, and away from the plane containing the path of AB, and a parallel to that of CD. If its velocity is less than the value stated above it is bent from the direction of motion of AB and away from the plane containing the path of the centre of AB and a parallel to that of CD. This is the direction in which the path of CD is deflected if AB first intersects the shortest distance between the directions of motion of the vortices , but in this case the radius of CD is diminished.
Class II.
If the vortex CD be the first to intersect the shortest distance between the directions of motion of the vortices its
radius is diminished by the collision. It is bent from or towards the direction of motion of AB according as its velocity is greater or less than the velocity of AB resolved along the direction of motion of CD, and away from or towards the plane containing the path of AB and a parallel to that of CD, according as the shortest distance between the vortices is greater or less than 2
times the shortest distance between their directions of motion. √3 The deflection of AB with reference to this plane is the same whether AB or CD first intersect the shortest distance. If AB be the first to intersect the shortest distance, the radius of CD is increased, and the deflection of the path of CD relative to the direction of motion of AB is the opposite of that when CD was the first to intersect the shortest distance.
When the directions of motion of the vortices intersect, these results admit of much simpler statement, and, though included in Class I., it may be worth while to restate them. In this case the result is that the vortex which first passes through the point of intersection of the directions of motion of the vortices is deflected towards the direction of motion of the other ; it increases in radius and energy, and its velocity is decreased ; the other vortex is deflected in the same direction, it decreases in radius and energy, and its velocity is increased.
§ 36. Very closely allied to the problem of finding the action of two vortices on each other is the problem of finding the motion of one vortex when placed in a mass of fluid throughout which
64
ON THE MOTION OF VORTEX RINGS.
the distribution of velocity is known. We proceed to consider this problem, using the notation of § 14. Let be the velocity potential of that part of the motion which is not due to the vortex ring itself. Let the equations to the central line of the vortex core be
p = a + Σ (a, cos n¥ +ß„n sin ny) , z = 3 + Σ (Y₂ cos ny + ♪22 sin ny).
Let Twe² be the strength of the vortex ; let l, m, n be the direction-cosines of the normal to its plane, λ, μ, v the directioncosines of a radius vector of the vortex ; then (§ 6)
l = sin e cos e,
m = sin 0 sin e,
n = cos 0,
λ=
cos e cos cos
— sin e sin y,
με sin e cose cos + cos e sin y, y = – sin cosy.
Let x, y, z be the coordinates of the centre of the vortex ;
if u, v, w be the velocities parallel to the axes of x, y, z at a point
on the vortex ring, then, by Taylor's theorem,
ΦΩ
d
d
ΦΩ
u = + αλ + μ + v
dx
dx dy ddz) dasx
d
+ 1 a² (λ dx
d + v d d) z ,adxn +
with symmetrical expressions for v and w.
The velocity along the radius vector = λu + µv + vw
d
dd
2
d
dd
λ + μ + v Ω + αλ + μ + v Ω
=(x dx dy dz
dx dy dz
d
d
d \³
+ $& a²(λ —d² /x ++μμ dy + v dz Ω + .
da
-term in the expression for the velocity along the radius vector, dt
which is independent of y.
As λ, u, v all involve , the first powers of these quantities
furnish nothing to this term .
-
-
a² = { ( 1 − 1²) + cos 24 (cos³0 cos² e — sin²e) — sin 2 sinecos e cose,
µ² = 1 ( 1 − m²) + cos 24 ( cos²0 sin²e — cos³e) + sin 2 sinecos e cose, ນ := (1 - n²) + cos 2 sin² 0,
λµ= -1 lm + cos 24 (1 +cos²0) sin ecos e + sin 24 cos 0 cos 2e,
\v = -1 In +1 cos 24 ( − sin 0 cos 0 cos e) + sin 24 sin 0 sin e,
μυ
-==− 1
mn
+
½
cos
24
(−
sin
✪
cos
0
sin
e)
–
sin 24 sin 0 cos e.
MOTION OF TWO VORTEX RINGS.
65
The vortex itself contributes no term independent of to the
expression for the velocity along the radius vector ; thus if the
radius of the ring be small, we have approximately
da
ΦΩ
ΦΩ
ΦΩ
= } a {(1 - . dt a--
dx2
+ (1 − m²³) dy³
+ (1 − n³)
dz2
ΦΩ
ΦΩ
ΦΩ
―-2lm
- 2ln
2mn
dx dy
dx dz
dy dz
or, since
ΦΩ ΦΩΤΩ
+
+
= 0,
dx² + dy dz2
2
da
d
d
d
+m +n
Ω;
dt § a (1 dx dy
d or, if denote differentiation along the normal to the plane of
dh
the vortex ring,
da
ΦΩ
=-
dt τα dh2 .
From this equation we see that the radius of a vortex ring placed in a mass of fluid will increase or decrease according as the velocity along the normal to the plane of the vortex ring at the centre of the ring decreases or increases as we travel along a stream line through the centre. A simple application of this result is to the case when we have a fixed ring placed near a fixed barrier parallel to the plane of the ring. The effect of the barrier is to superpose on the distribution of velocity due to the vortex
ring a velocity from the barrier which decreases as we recede from the barrier ; it is this superposed velocity which affects the size of the ring, and, since the velocity decreases as we go along a stream line (which flows from the barrier), the preceding rule shews that the vortex will increase in size, which agrees with the well-known result for this case.
Let us now find how the vortex ring is deflected.
The velocity perpendicular to the plane of the vortex
ΦΩ
d
d
d do
= dh + alλ dx + μ dy+ v dz dh
d
d d \' dn
+ ½ a² (λ + μ + v
+
(2ddax dy dz/ dh
The coefficient of cos y
d
d
d\ dQ
= α COS E COS
+ sin e
sin
+ terms in a³.
(
dx
dy
d dz dh
The coefficient of sin
d
d\ dn
= aa (( -– :sin e dx + sin e cos
dy
+ terms in a³. dh
T.
5
66
ON THE MOTION OF VORTEX RINGS.
dy, = coefficient of cos in the expression for the velocity perpen-
dt
dicular to the plane of the vortex.
The vortex itself contributes nothing to the coefficients of either cos or sin in the expression for the velocity perpendicular to the plane of the vortex (see equation 43) .
Thus
dy₁
d
d
ΦΩ
= acos e cos dt
+ sin €
dx
dy
sin
0. ddz)
adhn
approximately,
d\ dn
ddôt = a(-sin e d
+ sin e cos 0 dy/ dh
Now by § 6,
dl =
dt
1
d8,
sin
e
1 dy, --
COS e
cose,
a dt
a dt
dm
1 d8,
1 dy,
=
COS €
cos e sin e,
dt
a dt
a dt
dn
1 dy₁ sin 0.
=
dt
a dt
Substituting the
values just
found
for
ds1, ,
dt
dy, dt
in
these
expressions, we find
dl
ΦΩ dan
=
dt
dh2 dh dx
dm
ΦΩ ΦΩ
=m
dt
dha dh dy
.(80) .
dn ΦΩ ΦΩ
-n
dt
dh2 dh dz
These equations enable us to find the orientation of the plane of the vortex at any time.
To find the change in the shape of the vortex, we have
dx2= coefficient
of
cos 24
in the
expression
for
the
velocity
along
dt
the radius vector.
Now the vortex itself contributes to this coefficient the term
2we² 8a a² log e • Y₂ (see equation 38) .
MOTION OF TWO VORTEX RINGS.
67
And if we pick out the coefficient of cos 24 arising from the velocity potential , we shall find that it reduces to
ΦΩ ΦΩ
-1a
dh²
+
2dk
d
where
denotes differentiation along an axis coinciding in
dk
direction with the radius of the vortex ring for which ↓ = ½π.
da, Thus 2
dt
2we 8a
ΦΩ
a²
log
e
•
Y2
- ta
(ddahn2
+2
dk2
Again, dy2
= coefficient of cos 24 in the expression for the velocity dt
perpendicular to the plane of the vortex.
Now the vortex itself contributes to this coefficient the term
we² 8a
3334 log · a, (see equation 43).
a²
e
And if we pick out the coefficient of cos 24 arising from the velocity potential N , we shall find that it reduces to
d² dQ
+2 - ta²(ddh dle dh '
Thus
dy2 we2 8a
d2
d² dn
dt
= 94 a²
log
-'a₂ — e
a²(d1h² +2 dk
; dh
d72 and
and this, with the preceding equation connecting dt
Y2 enables
us to find da and Y2'
We have two exactly analogous equations connecting dß/dt and
d
d
8, the only difference being that we substitute for where
dk dk'
d
denotes differentiation with respect to an axis passing through dk
the centre and coinciding in direction with the radius of the vortex
ring for which ¥ = 0.
§ 37. We can apply these equations to find the motion of a vortex ring which passes by a fixed obstacle. We shall suppose that the distance of the vortex from the obstacle is large compared with the diameter of the vortex, and that the obstacle is a sphere.
Let the plane containing the centre of the fixed sphere B, the centre of the vortex A, and a parallel to the direction of motion of the vortex be taken as the plane of xy. Let the axis of x be parallel to the direction of motion of the vortex. Let m' be the strength of the vortex, and a its radius.
5-2
68
ON THE MOTION OF VORTEX RINGS.
The velocity potential due to the vortex at a point P
Now
==
•
Im'a²dxd (1 ) approximately.
1
1 BP
BP2
=+ AP AB
AB²
Q,1 +
AB³
Q₂
+
....
(fig.
6),
Fig.6.
8
if BP < AB, and Q , Q,...are spherical harmonics with AB for axis.
At the surface of the sphere the velocity parallel to x
d2 1
3 cos 0-1
= { m'a²
|= }m'a²²
+ smaller terms,
dx² (AP)=
AB3
where is the angle AB makes with the axis of x.
The velocity parallel to the axis of y
d2
3 cos e sin
== m'a²
dxdy (A1P1) = {m'a² AB³
+ smaller terms .
Now at the surface of the sphere the velocity must be entirely
tangential, hence we must superpose a distribution of velocity,
giving a radial velocity over the sphere equal and opposite to the
radial velocity due to the vortex ring, i. e. equal to
m' a²
ABS
У (3 cos³0-1 ) +3 cos✪ sin
if xᏆ and y be the coordinates of a point on the sphere, b the
radius of the sphere. Let AB = R. Q, the velocity potential
which will give this radial velocity, is given by the equation
m'b³a²
d1
d 1)
Ω1
(3 cos20-1 )
+3 cos 0 sin
R³
dx r
dy r
where r = BP.
2 is approximately the value of the velocity potential which produces the disturbance of the motion of the vortex.
MOTION OF TWO VORTEX RINGS.
69
The equation becomes in this case
da
ΦΩ
--
-La
dt
dh
da m'a³ ³
d³ 1
=} dt
R³
-
+3 cos e sin
{(3 cos³0 – 1) dx³r
Now
d³ das (
= - 3 (5 cos 0-3 cos 0) R$
d³
3 sin (1-5 cos² )
=
da³dy (1)
Ꭱ
da 1) da'dy r
We must express the quantities on the right-hand side of the equation in terms of the time.
Let us measure the time from the instant when the line joining the centre of the sphere to the centre of the vortex is perpendicular to the direction of motion of the vortex. Let u be the velocity of the vortex ; then we have, accurately if the motion were undisturbed, and very approximately as the motion of the vortex is only slightly disturbed,
R² = c² + u²t², ut
cos == (c² + u³t³) + '
с sin =
(c² + u²²) + '
where c is the shortest distance between the centre of the vortex
and the centre of the sphere.
Substituting we find
da
u³t³ m'a³b³
=-
dt
132 (c² + u²t*,³ ³
thus the vortex expands until it gets to its shortest distance from the centre of the sphere, after passing its shortest distance it ceases to expand and begins to contract.
Integrating the differential equation, we get 1 m'a²2b³
a= α + {1 16u R (1 + 3 cos³0) } ,
where ао is the value of a before the vortex got near the sphere.
Thus we see that the radius is the same after the vortex
has passed quite away from the sphere as it was before it got near to it, since in both cases R = ∞ ; in intermediate positions it is always greater.
70
ON THE MOTION OF VORTEX RINGS.
The greatest value of the radius is b
a. (1 + 116um'a,' ');
the greatest increase in the radius is thus proportional to the
volume of the sphere, and inversely proportional to the sixth power of the shortest distance between the vortex and the
sphere.
§38. To find the way in which the direction of motion of the vortex is altered we have, if l, m are the x and У direction
cosines of the normal to its plane,
dm
ΦΩ ΦΩ
=m
dt
dx dxdy
Now in the undisturbed motion m = 0, so we may write this
equation
dm
ΦΩ
dt dady
dm m'b³aª
=4
cos 0-1)
+3 cos 0 sin
P
dt
R3 {(3.
dx dy (1)
Now
d³ 1 de'dy (+ )
=
330y(1-5a²)
ds dy'de (-)}·
d³ dy'de (- ) = 3.x (r² - 5y")
pr
Substituting these values, we find
dm
2 m'b³a⁹
=-
sin (1 +4 cos* 0) ;
dt
Ꭱ.
dm thus is always negative , or the vortex moves as if attracted
dt
by the sphere ; expressing the right-hand side in terms of the
time, we get
dm
5
4c2
dt = - {m′ba²c { (c² + ut*)** (c² + u³t²)5})
Integrating both sides from t = − ∞ to t = + ∞ , we find that
m, the whole angle turned through by the vortex, is given by the
equation
πm'b³a²
m
=
-
45 128
9
and this effect varies inversely as the sixth power of the shortest distance between the vortex ring and the sphere, and directly as the volume of the sphere. Sir William Thomson shewed by general reasoning that a vortex passing near a fixed solid will appear to be attracted by it ("Vortex Motion ," Edinburgh Transactions, vol. xxv. p. 229) ; and this result agrees with the results we have obtained for the sphere.
( 71 )
PART III.
Linked Vortices.
§ 39. WE must now pass on to discuss the case of Linked Vortices. We shall suppose that we have two vortex rings linked one through the other in such a way that the shortest distance between the vortex rings at any point is small compared with the radius of the aperture of either vortex ring, but large compared with the radius of the cross section of either of them. Thus, the circumstances in this case are the opposite to those in the case we have just been considering, when the shortest distance between the vortices was large compared with the diameter of either.
In the present case it is important to examine the changes in the shape of the cross section of the vortices, in order to see that they remain approximately circular. We shall, therefore, discuss this problem first.
Since the distance between the vortices is very small compared with the radii of the apertures of the vortices, the changes in their cross sections will be very approximately the same as the changes in the cross sections of two infinitely long straight cylindrical vortex columns placed in the same mass of fluid in such a manner that the distance between them is great compared with the radius of either of their cross sections.
We shall prove that if the cross sections of two such vortex columns are at any moment approximately circular they will always remain so.
We must first find the velocity potential due to such a vortex column.
Let the equation to the cross section be p = a + a₁ cos no + B sin no,
where a and Bn are small compared with a, the mean radius of the section. Let w be the angular velocity of molecular rotation.