BULLETIN OF MATHEMATICAL BIOPHYSICS

VOLUME 2, 19.40

A GENERALIZATION OF CUNNINGHAM'S EXTENSION OF STOKE'S LAW FOR THE FORCE ON A SPHERE

GALE Y O U N G

THE UNIVERSITY OF CHICAGO

C u n n i n g h a m ' s f o r m u l a for the force on a sphere m o v i n g w i t h i n a l a r g e r concen t r i c spher i ca l b o u n d a r y is ex tended to cover a gen- e ra l s t a t e of moti.on of t he fluid be tween them.

Stoke's formula for the force on a sphere moving slowly through a viscous fluid of infinite extent has played an important role in many physical investigations, and has been the subject of much experimen- tal and theoretical study. Various modifications of it have been worked out by different authors; for some account of these see Smo- luchowski (1913), Lamb (1924, p. 565-583, 608), and Bulletin of the National Research Council (1931, Chapter 7).

Cunnigham (1910) worked out the force on a sphere moving through a fluid enclosed within a larger concentric sphere, and from this tried to estimate the effect of the presence of many other moving spheres. His results have been used (Heilbrunn, 1928, Chapter 5) in determining the viscosity of a fluid from observation of the rate of motion through it of a large number of particles under the action of applied forces or thermal agitation. For a discussion of Cunning- ham's work and the many sphere problem see Smoluchowski (1913).

Cunningham found the force on the inner sphere under the as- sumption that the fluid is at rest* over the surface of the outer sphere, and then considered various sizes for the larger sphere to somewhat correspond to the presence of other particles. Since there is in such a situation no surrounding sphere where the fluid is at rest, and since it is quite easy to remove the assumption of an exactly speci- fied flow at either sphere, it may be worth while to note the general result. This will also be of use in other connections.

I The complete solution of the equations of slow, potential force

motion of an incompressible fluid is given by Lamb (p. 562-564) in a form adapted to a region with concentric spherical boundaries. In

* Wi l l i ams (1915} t r e a t e d t he s ame concen t r i c sphe re p rob l em by a d i f fe ren t method. We shal l he r e more n e a r l y fol low C u n n i n g h a m . Wi l l i ams gives some d i a g r a m s showing flow p a t t e r n s , bo th theore t i ca l a n d observed.

105

106 MATHEMATICAL BIOPHYSICS

such a motion the mean pressure p of the fluid is a harmonic funct ion and m a y thus be wr i t ten as

p = 2 : p , (1)

where p . is a solid harmonic of degree n . The summation is over n and over t h e yarious independent solid harmonics ( there a re 2n ~ 1 of these Of positive degree n or of negative degree - - n - - 1) of each degree. The fluid motion cannot give rise to a pressure te rm of de- gree - - 1 , so tha t p-1 is always zero. Denoting by u , v , w the compo- nent velocities of the fluid along the (x , y , z)-axes respectively we have

{ nr2n+~ ~ p , } 12: r 2 ~P"-} - (n 1) (2n-}-1) (2n-}-3) ~x r 2"+1 U =

2(2n-~- 1) ax -~- (2)

+ 2 : + ,

with the corresponding expressions for v and w obtained by cyclical in terchange of x , y , and z . Here ~ is the coefficient of viscosity of the fluid, while q~, and Z- are a rb i t r a ry solid harmonics of degree n .

The stress exerted in the x-direction across a spherical surface of radius r by the fluid outside the sphere is p'~, where

/ n---1 r2~p~ 2n2-t-4n-[ -3 r~,§ ~ P'~ } rp~ = 2:[2n---1 ~ - ( n + l ) ( 2 n + l ) (2n+3) ~ x r 2"+---V

(3)

_[_ ~ 2: ( n _ _ 1) { 2 ~ x , z ~ g , , ~ z - ]

The corresponding expressions for rp,,j and rp,~ are obtained by cyc- lical in terchange of x , y and z . The total force exerted on the sphere in the x direction is given by the integral over the sphere surface

F = f f p~ ds . (4)

I t is evident f rom the orthogonali ty propert ies of surface harmonics tha t only those in (3) of zero order survive the integration, and fur- ther consideration shows tha t all of these drop out except the t e rm E x / r ~ in p_~. The final result is

F = - - 4 h E , (5)

so tha t in any such f luid motion the total x-force on a sphere depends only on the coefficient of x / r 3 in the expansion of the pressure field in solid harmonics about the center of the sphere as origin. Similarly, the total y-force on the sphere depends only upon the coefficient of y / r 3 , and the z-force only upon tha t of the z / r 3 term.

GALE YOUNO 107

II Such a fluid motion as considered above is uniquely determined

throughout a region by its velocity components over the boundary (Lamb, p. 584). We proceed now to determine the value of E in terms of the boundary velocities on two concentric spherical surfaces of radii t and T.

To this end we consider the quantity 1 defined by

1 = x u + yv -+ z w , (6)

which is simply r times the outward radial velocity of the fluid. In terms of the above quantities its value (Lamb, p. 564) is

l = l x nr' 2 (2n+3) Pn -}- ~ n ~n. (7)

In this expression E appears only in connection with the first order surface harmonic x / r . The following quantities are thus involved with E in the boundary conditions on 1 ;

p~ : A x

P-2 : E x / r a

q,1 : H x (8)

q)-2 : G x / r 3

Upon equating coefficients of the x / r surface harmonic we obtain

A t~ + E 1 2G 1 It 10'7 - - ~ - u _ _ =

(9) A E 1 1

T~ + - - ~ - ~ , + H - - 2G = lr ,

where 3

l~ -- 4nr~ f f x lds (10)

is 1 / r times the coefficient of x / r in the expansion of I in surface har-

monics on a sphere of radius r . It is the average over the sphere sur- face of 3 times the x component of the outward radial velocity of the fluid.

Similarly, considering the boundary values of the velocity com- ponent u and picking out the terms involving zero order surface har-

108 MATHEMATICAL BIOPHYSICS

monics gives

where

A ~ 2 E 1 ~ t q - ~ T § = u,

A T +2E 1 + H -T

(11)

1 u~ -- 4 n r 2 f f uds (12)

is the coefficient of the zero order surface harmonic in the expansion of u over a sphere of radius r . I t is the average over the sphere sur- face of the x-component of the fluid velocity.

Solving f rom (9) and (11)

where

E ---- 3 ~ t 2 T ( T ~ - - t s ) A u - - 5 T ( T 2 - - t 2) ( A : u - - A 1 ) (13) - 5 T t ( T 2 - - t ~ ) 2 - - 4 ( T - - t ) ( T ~ - - t ~)

z~U ~- UT"--Ut

A:u -~ T3uT - - t3ut (14)

A1 = T31~ ~ t ~ l ~ .

Thus the x force on a sphere due to any such f l u i d m o t i o n is ex- p r e s s e d in t e r m s of the quan t i t i e s (10) and (12) w h i c h are cer ta in ave rages of the f l u id ve loci t ies over the sur faces of t w o concentr ic spheres . I f v ---- w = 0 on these surfaces, while u assumes constant values, then A~u = A1 and the resul t as expressed by (5) and (13) reduces to tha t of Cunningham; if then T is made much larger than t it goes over into Stoke's formula.

This work was aided in pa r t by a g ran t f rom the Dr. Wallace C. and Clara A. Abbot t Memorial Fund of the Univers i ty of Chicago.

LITERATURE Cunningham, E. 1910. "On the velocity of steady fall of spherical particles

through a fluid medium." Proc. Roy. Soc. A., 83, 357-365. Heilbrunn, L. V. 1928. The CaUoid Chemistry of P~otopla~m. Berlin: Lamb, H. 1924. Hyd~'odynamics, 5th Edition. Cambridge. National Research Council, 1931, Report of Committee on Hydrodynamics, Bulle-

t in 84. Washington. Smoluchowski, M. S. 1913. "On the practical applicability of Stoke's law of

resisiance, and the modification of it required in certain cases." P~oc. 5th Int. Congress of Math., 2, 192-291. Cambridge:

W~lliams, W. 1915. "On the motion of a sphere in a viscous fluid." Ph. Mag., 29, 526-552.