proc. london math. soc. 1915 jeffery 327 38

8/13/2019 Proc. London Math. Soc. 1915 Jeffery 327 38

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S fEADY ROTATION OF A SOLID OF REVOLU l ION IN A VISCOUS FLUID. 32 7

ON THE STEADY ROTATION OF A SOLID OF REVOLUTION INA VISCOUS FLUID

y G B JEFFERY.

[Received February 10th , 1915. Read February 11th ,1915 ]

r rHE only possible nlotion of a viscous fluid whicll is syulm E: t r ica labout an axis , and for which the stream lines are circles having theircentres on th e axis and their planes perpendicular to the axis , is themotion generated by the rotation of two infinite concentric circularcylinders about their common axis * If , however , the motion is slo\y sothat th e syuares and products of the velocity components lnay be neglected ,th er e a re other possible solutions Let z be cylindrical coordinates ,and t V w the corresponding components of velocity , then neglecting thesquares and products of v lV the e q u a o n sof steady nlotion in thesymmetrical case are

~

\ 7l 0 ,

?? 9 . 0 ,

: r

L V

O= 0 , OZwhere is the kinematic viscosity , and X V p p where V is thepotential of the externa .l forces , p the lnean pressure , and th e density.These equat ions ar e satisfied by X const , w 0 , while 7. is afunct ion of l U , Z only and

\Vriting

this equ a.tion becomes

y 2V _

l s n

y 2 =

1)

* The Equations of Motion of a Viscous Fluid , Phi l Apri11915 , p 445


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,,.l lL

mthef

1\fR G. B. JEFFERY

Hence if we have an y solution of Laplace s equationj T z sin a solution of t he p re sent problem is given by

32 8

v f U.Tz . f th e motion is generated by the rotation of a solid of revolution

ahout the axis of z with angular velocity we have to make l vanish forlarge values of U.Twhile , on the hypothesis of no slipping at th e solid-fluidsurface , we have 1 over the surface of the solid.

f a , y r e a system of orthogonal curvi linear coord inates , andl v W the corresponding components of velocity , and if elements of arcmeas J1red along th e no rmals to the surfaces a , y const. , are o hI

h2 oy j h 3 respectively , we have with the usual notation for stress components

h 3 a n n I hI 3 17 1ay fJ ~ 1: 3 ; lu + ~ h i l U J ) f- j t Uy h3 Q 9 Jwhere is the coefficient of viscosity.

Let a , be conj ugate functions of z while y then

h o. = = -- I h 2 2+

f th e solid is defined by one of the surfaces a const , we have for thetangential stress on it s surface in the direction perpendicular to the axis

hU T ) ,

or th e to tal couple exerted by th e fluid on the solid is

G = = 2 ) 2)

the value of th e integrand being taken on the surface of the solid an d theintegration extending round the contour of the solid.

f n . a re measu red along the norma] and the arc of the contourrespecti vely

3

\Ve have a solution of Laplace s equation in spherical polar coordinates

sIn n 1 n 1 - } ~ t cos 8) ,

G = = 2 JU.T3 ) ds

where P ~ cos 8) is the associated Legendre function of the first kind , ofdegree n and of th e fir s t order. Hence we have a solution of the present

Lame , Ol donnees r i l i g n e s p. 284.


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1915.J STEADY ROTATION OF A SOLID OF REVOLUTION IN A VISCOUS FLUID 329

problemv ~ { . A n r n B . - - } P ~ c o s8 .

As a particular case t a k e n 1 We have

v = = l A r B /sin 8

which is the well known solution for the motion of a fluid cont tine be-tween two concentric spheres which ar e constrained to rotate about thesame dianleter. In the case of a single sphere of radius rotating withangular velocity in an infinite fiuid we have

3 v== s f G H 7 i w a 3 4

The general case of the rotation of an ellipsoid of any shape about aprincipal axis has been solved in terms of ell ipsoidal harmonics by NI Edwardes Q r t e r l y J o u r n l of J lL t h e m t c s Vo l XX p. 70 . Thesimpler case of an ellipsoid of revolution can however be very readilysolved by t he m et ho d of this paper.

R o t t i o nof v r y llipsoid

Take coordinates defined by

z i c cosh f

so that z c cosh cos T J c sinh ~ sin .

Th e surfaces const. ar e a series of confocal ovary ellipsoids whilethe surfaces const. are a se t of confocal hyperboloids of revolution th e foci in each case being z C Ve I l known s oluti onsof Laplace s equat ion a re

p;;t cos P : ~ t cosh ~ sin In

cos Q ; ~ cosh ~ si n n

alId hence we have solutions of th e problem in hand

v P ~ t cos P; cosh tand 1 P;t cos Q ~ c o s hfthe former is f in ite when cosh ~ 1 i e on th e line joining the foci butit becomes infinite w i t h ~ .The second solution on the other hand be-comes infinite on the l ine joining th e foci but vanishes at infinity. t is


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3 ;J0 ~ I H .G B. J ~ F F E R Y [Feb. ,

therefore applicable to th e case of an ellipsoid in an infinite extent of fluid.We will , however , take the more general case of the I l l otion of a fluid be-tween two confocal O Y r y ellipsoids rotating with different angular velo-cities. For this purpose both solutions are required. Put n 1 in theabove solutions and assume

Sin { A P ~ cosh f + B Q ~ c o s hf }.

I f th e two s u lar e 0 l t >1 respectively , th e boundary conditions give

APf cosh o) B Q ~ c o s h fo) C 0 sinh fo

A P ~ c o s hf l + B Q ~ c o s hfl = C1 sinh 1

We may conveniently write n A O }

ref log coth~ J - tS V U U 2 sinh2:;

Then P ~ cosh f sinh and Q1 cosh = sinh gI g .

Solving for B we have1-. ~ : ~ f f - f 1 l f fo I

= C sInn sin l oI o f 3 ; - l f f o - / J

rro calculate th e couple on either ellipsoid necessary to maintain therotations we may use th e formula 2 , and we have at once

G 1 i r M C3

- - - C~ I fo} _ j 1 .

In the case of a single ellipsoid = fo) in an inf inite exten t of fluid , wehave , putting 1 = = , 1 =0 ,

j c sinh g sin I f o

and c 3 /j f o

I f b ar e the polar and equator ia l radi i respectively ,

a = cosh fa b = c sinh foand we have

U

p

+=a

/I

ll\

l2

T-3

5

where c 2 b 2


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1 1 5 . JS n ~ A D YROTATION OF A SOLID 0 1 REVOLUTION IN A VISCOUS , LUID. 331

Vhell b c 0 , andG == 8 7 r 3

which is the value a l r e a ] ygiven for a sphere.

Rotat ion o f P l n e t l Ellipsoid.

In the case of an oblate or planetary ellipsoid we take coordinatesdefined by

z+iro == c s i n h )

so that z

c sinh [ cos ccosh sin 1].The surface8 const. and const. are ellipsoids and h y p e r bo i d sofrevolution having a COUlmon focal circle , Z 0 , c. We have assolut ions of Laplace s equation*

P : ~ ~(cos 1] p sinh [ sin P CO ) q sinh [ sin 1 n

1 nwhere P : ~ ~ )== ~ 2 +l ) ~ ~ n; ~ m P n ~ ) ~ 2 +l ) ~ ~ nA :in q n ~ ) - \ ~ I . . I ~ mj / n \ ~ :l n \ ~ I -

the functions p qn being connected with the ord inary Legendre funct ionsby th e relations

P n ~ ) i n P n

~ ) q n ~ )= i n 1 Qn )

Hence we have solutions

== P ~ C08 1] p ~ 8 i n h)

P ~ C 0 8 ) s i n h f .

As inthe previous cas e the former

becomesinfinite ,

while: le

lat tervanishes at infinity.

Let the two ellipsoids [0 have angular velocities l Assunle

v == -l A p ~ sinh f + B q ~ s i n hf } sin .

The boundary conditions give

Now

A p ~ s i n h[ o ) + B s i n h = c 0 cosh fo 4 .p ~ (sinh 1) B (sinh f1 c 1 cosh fl

(sinh f == cosh

Lamb , Hydrodynamics , p. 136.


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and wri ti ng

we have

MR. G. B. JEFFERY

F f) ~ o t 1 sinh f ash:(sinh f) cosh F f) .

[Feb. 11

Solving for A B v c cosh f sin F f l ) - F F ) - F f o )l

l 0 F l ) - F f o ) 1F l ) - F f o ) ).

Using equation (2) to evaluate th e couple which must be applied toeither ellipsoid to maintain the rotation we have without difficulty

G ]. ; c 1 - - . 7 r F f l ) -F fo)

the fluid is not bounded externally so that we have a single ellipsoid inan infinite extent of fluid we ma y put 1 0 an d {I . I n this case

?) c cosh sIn F ) / F { o )

and G - c 3 o F 0

a b are th e polar and equatorial radii respectively

c sinh{o b = c coshan d we have

G = = ] l

where b 2 2 c

We may note that this is the form taken by the result (5) for an ovaryellipsoid when b is greater than for

i b 2 . . . ; . -a 2 -

log b 2

- jcot-1

The R o t t i o nof Circular D isc in n I n n e F l i d .

The solution of this problem may be obtained by putting = =0 inthe solution for a planetary ellipsoid. Hence

l - ~ c cosh SIn F (6)7 r

and the couple necessary to maintain the rotation is

G== c S

where c is the radius of the disc.


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1915.J S ADY ROTATION OF A SOLID OF REVOLUTION IN A VISCOUS FLUID. 333

From 6 t he norma l gradient of the velocity over the surface of thedisc can be calculated ,

)~ = O= - o ~=o4

- 0v ). 7)

8)

9

I t appears that , while the velocity is everywhere finite , th e velocitygradient , and therefore th e shearing stress , become infinite at the edgeof th e disc. I t is probable that in th e neighbourhood of the sharp edgethe condition of nO relat ive motion between th e solid and the tiuid breaksdown. I t would be interesting to see whe ther thi s infinity disappeared ina more accurate solution which did no t neglect the terms involving thesquares of the velocity components.

Another solution of this problenl lllay be obtained in terms of Besselfunctions , for we have a solution of Laplace s equation

e- k z J 1 k sin

and hence a solution of equation 1

v = J ~f k e -kzJ 1 d kIn the case of the disc we have

1 = = ot O for z 0 , < c for z = 0 , > c

and hence f k is determined by

J ~ f k J 1 dk 0 to < c ) J 1 kUT dk = > c

I t ha s not been found possible to solve these integral equations directly ,but if we make use of 7 , together with 9) , and t he theorem* that if

- = F ) .for z 0 , J Z

then 1 J ~e -k ) C k ) Ad dk .. Gray and Mathew8 , Bessel Functions p. 80. The inf in it y of E = c does not in

validate this theorem.


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,3 34 ; ln. G. B . . J F F RY [Feb. 11 ,

we obtain without d i f c u l t y

[ dk1 = 0 e-kZ(sin c k c k cos ck J r - Jo

and this will he found to satisfy both conditions 8) and 9).

The R o t t i o nof Two Non-Concentric Spheres.

This method may be applied to the problem of the m.otion of a viscousfluid generated by the rotation of any two sp he re s w ith different angularvelocities abou t the ir common diameter. f we take coordinates defined by

u r + i z + t + i = 1 0 e : r i z lQ

in an y 111eridian plane the curves t = const. , = const. are the systemsof coaxial circles b ou t the points z = and through these points respectively. The surfaces f = const. will be a family of coaxial sphereshaving the C l n m o n radical plane z f = 0 , an d we can choose the axes ofreference and the constant a so that any two given spheres are membersof this family. \Ve have a solution of Laplace s equation in these C001-

,

d ina tes of which the following is a particular case* c o s h c o s ~ : A n cosh ~ n B s inh n+i ) pj cos sin .

t ollo\vs that a solution of the present problem is given by

n I c o s h c o s )~ {An cosh n+i) f B sinh n ) p ~ (cos .

Let the two spheres be defined by f = ~ l and le t them have angularvelocities l 2 . \Vithout loss of generality we lnay take 1 positive , andf2 positive or negative according as the two spheres do or do no t lie one

within the other. \Ve have to determine A B n so that v is equal to l when f = 1 and equal to 2 when = . From 10) ,

nl

and we obtain

An s inh n+ ) 1 )= a [ 1 e - n + ~ )1 s i n h n + 2 e n + s inb n+ ) 1J

n s i n h n +) (

= 2 2 [ 1e - + ~ 1 c 0 8 h n + ~ ) 2 e + ) ~ 2c o s h n + l J

Heine, K : g e lf u n c t i o n en I I , p. 268; see also a paper by the author On a Form of theSolution of Laplaces Equation , Proc Roy Soc A , 87, 1912, p. 109


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1915 J STEADY nOTATION OF A ROLID OF REVOLUTION IN .\ V I C O U RFLUID 335

where the upper Or lower sign is taken according as is negative OrpO itive

We will investigate the resultant couple on either sphere

Iv _ sinh f ~F \ r J - 2 sin U

cosh t - t U ~ {A l St = l

where ~ == cos .

The couple on the sphere f1 is given by

G == 7 r I 3 a f == f1

; C l ( 1 - ~ 2 )= 7 r a s 1sinh f1 , 7 r 2Z 1 ( + { A s i n h n + ~f1 + B o s h n I } Ill

where I .. = +1 1 - d P ~ t 5 ~dt. J-1 cosh f t d ~ ~

Now I+1 n f F4 d ~ = c o s e c hl

and it remains to evaluate In Now

, mcosh = 2 e - l It +~ El Pm ~ .

Differentiating with respect to ~

J _ \ /\ 1 n ~ J f .... 4 1 t . dP m J = ~ e - ( m + ~ ) cosh U S/ : 1 ~ d

and so

l/

p

P vgZ

m

OA r

U-U

,

9

n n + l= = 41 2 H ~ HI ~ I e2 n + l


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33G 1\I R. G. B. JEFF RY [Feb. 11

Hence

G 4 3 1cosech 3 ti+ 4 / 2 7 T 2 L n n + 1 ) {An s i n h n +) l + B cosh n + ) } e - ( n + )

or inserting the values of the constan ts G 4 3 1cosech 3 1

+ l f i7 T 3~ n n+1) .1 1e - ( 2 1) coth n + f 1 f 2

2 e - H l ) cosech n + }~

T/he two series lnult iplying 1 an d 2 are convergent whatever ma y be provided that f1 is posit ive; they may t h e t f o r e : b esunllued separately.

Now when f > 0 we havecosech 3f 8 e -3 / 1 _ e -2 1 ; 3 4 ~ lt+1 e- 2u+1 S

an d so the tenns containing 1 in G1 ca n be written

8 3 1cosech 3 1 1 6 3 01 L n n + 1 ) e - 21 { c o t h n +) - f 2 ) - I }~

Also coth n + ) 1 2 L e - ( 2 n + 1 ) m ( - E2)

and so ~ n n+1)e- 2n+l) l;t { c o t h n + - l t l

~ ~ 2 n n~ 1 e - ( 2 nn +1 { 1 1 = 1m l = 1 1

T he t er ms in this double series are all positive and the series is convergentwhen summed in the present way; \ e may therefore interchange theorder of sumluation an d we have

~ L cosech3

{ + 1 l m .

We now see th a.t the terms containing 1 in G1 ca n be brought in to thesingle sunl

8 7 1 WI L cosech 3 { 1n+ 1 ) f 1 - m} ~

The terms containing can be transformed similarly using the identi ties

L n n + 1 e - ( nH ) cosech n + i f l - f2

= L~ 2 n l e - 2 + 1 )

[ 7n+1I ;t - rn + )

1 rn O

= L cosech 3 [ ( 1 1 ~ +1 1 - m + ) f2J


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1915.J STEADY ROTATION OF A SOLID OF REVOLUTION IN A VISCOUS FLUID. 337

On combining the results we readily obtain

G 87 r a [ cosech [ m ]

cosech 3[

one sphere encloses the other f2 is positive and the lower sign mustbe taken and

1 = 8 7 r 3 1 -

f on the other hand 2 is negative the spheres ar e s epa ra te and theupper sign must be taken. In this case

1 = 87 r [ cosechh [ n m cosecchh n

Suppose that the sphere is constrained to rotate with a given spin 2 There will be a certain value of 1 for which G vanishes. This isthe steady angular velocity with \vhich the sphere f l would rotate ifallowed to move freely. Fr01ll 11) we see that in the case Th e n onesphere encloses th e o th er this gives 2 . In this case then if onesphere be allowed to move freely it will acquire the same angular velocityas the other sphere and the fluid will move as a rigid body. In the caseof two non-enclosing spheres if is constrained to rotate with angular

velocity while

is left free to lllove under th e fluid stresses i t willrotate with angular v e l o c i t :

~ cosech 3 m+ 1 ) f 2

1 = 2 ~ cosech 3 [ m 1 n z ]

In the case of two equal spheres = - f l and from 1 we obtain

diameter of either sphere = sech distance apart of centresThe character of this influence of one sphere upon the other is exhibited

S E R . 2. VOL. 14. NO . 1242.


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338 STEADY ROTATION OF A SOLID OF REVOLUTION IN A VISCOUS FLUID

in the following table

Two q l Spheres ne e

~Diameter I

Centre Dis tance

2 9803 1278 4 9250 1023 6 8435 0759 8 7477 0524

1 0 6481 03401 2 5523 02111 4 4649 01261 6 3880 00731 8 3218 00422 0 2658 00232 5 1631 00053 0 0993 0001

t appears that even when th e distance between the spheres is as smallas one fiftieth part of the distance between their centres one sphere CO l11-municates only one eighth of its spin to the o ther sphere

we put = an d = 0 we obtain the solut ion for the rotationof a sphere in a viscous fluid in the presence of a fixed infinite plane per-pendicl lla r to the axis of rotation The plane will cause the sphere to

experience an increased res is tance to it s rotation and we ma y comparethe resisting couple with it s value in the absence of the plane

Th e Increase to the e s i s t n c eto the Rotation of Sphere owing to thePresence of n I n i e l n e

~

2

4 6 8

1 01 21 41 61 82 02 53 0

Radius of Sphere I Ratio of Increase ofDistance of Centre from Plane I Couple du e to Plane

9803 1 171

9250 1 126 84 5 l OR7 7477 1 057 6481 1 036 5523 1 022 4649 1 013 3880 1 007 3218 1 004 2658 1 002 1631 1 0005 0993 1 0001

Here again the effect is surprisingly small. the fixed plane isbrought so close to t he rot at ing sphere that their distance apart is but one-fiftieth of th e radius of the latter the couple required to maintain therotation is only increased by 17 per cent These resu lt s it will be noted are independent of the degree of viscosity of the fluid

proc. london math. soc. 1915 jeffery 327 38

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