Journal of Applied Mathematics and Physics (ZAMP) 0044-2275/84/005634-08 $ 3.10/0 Vol. 35, September 1984 �9 Birkh/iuser Verlag Basel, 1984

The forces and couples acting on two nearly touching spheres in low-Reynolds-number flow

By D. J. Jeffrey, Pulp and Paper Research Institute of Canada and Dept. of Mechanical Engineering, McGill University, Montreal, P, Q., Canada and Y. Onishi, Dept. of Mechanical Engineering, University of Osaka Prefecture, Sakai 591, Japan

1. Introduction

Two spheres, whose radii are a and b, are separated by a gap of width e. Asymptotic expressions for the forces and couples acting on the spheres have been calculated by O'Neill & Majumdar [1] and by Majumdar [2] correct to order in e for all ratios k = b/a. In addition, O'Neill & Stewartson [3] and Cooley & O'Neill [4] have calculated results correct to order e ln~ for the special case of a sphere near a plane (k = or). The object of this paper is to extend O'Neill & Majumdar's work to order e In e for all k. It is important to do this because terms 0(1) appear in the asymptotic expressions between the In e and e In e terms, and these 0(1) terms are the ones needed to estimate the mobility of unequal spheres acted on by specified forces [5]. During the calculations, some new results of physical interest were noticed and an error in [4] corrected. Otherwise the only new principle in the calculation is the handling of long algebraic expressions, which was accomplished by using the computer algebra systems Carnal and Reduce [6, 7].

We follow the notation of [1] and consider a moving sphere S A of radius a and a stationary sphere Sn of radius - a/2. (Unlike [1] we shall consider only externai spheres, so 2 is always negative.) Three problems will be studied (see Fig. 1). As in [1], problems I and II are those in which sphere S A rotates about or translates along an axis perpendicular to the line of centres, while problem III is that studied in [2] and requires SA to rotated about the line of centres.

2. Problem I

The singular terms in the expressions for the forces and couples are deter- mined by examining the flow in the gap between the spheres. We take cylindrical coordinates (a r, 0, a z) with the z-axis along the line of centres and the origin at

Vol. 35, 1984 The forces and couples acting on two nearly touching spheres 635

Figure 1 The three types of motion studied

SA ~ Uu

\

the point of SB that is nearest to SA. Denoting the gap width by a a, so that e ~ 1, we stretch the coordinates according to

R = g - 1 / 2 r and Z = e - l z . (2.1)

The surfaces of the spheres are then given by

R 2 1 R 4 Z A = I + � 8 9 + e g +0(e2) , (2.2)

12R2 1 3R4 Z~ = g + e g 2 + 0 (a2). (2.3)

If sphere S A rotates with angular velocity Qt, we can write the pressure field in the form p =/~ O z cos 0 P (r, z) and the cylindrical velocity components in the form f2Xa [U cos 0, V sin 0, W cos 0]. These quantities are now expanded in powers of ~:

P (r, z) = s -3 /ZP 0 (R, Z) + e-1 /2P 1 (R, Z) + 0 (el/2),

U (r, z) = Uo (R, Z) + e U~ (R, Z) + 0 (e2),

V (r, z) = Vo (R, Z) + c V 1 (R, Z) + 0 (e2),

W (r, z) = e 1/2 W 0 (R, Z) + g3/2 Wl (R, Z) + 0 (e5/2).

(2.4)

(2.5)

(2.6)

(2.7)

The solutions for Po, Uo, Vo and W o were given in [1]; here we wish to find P1, Ua, V~ and W 1 . The equations governing the next order are as follows. Define an operator

F-- ~2/~R2 + R - 1 ~/~R, (2.8)

636

then

D. J. Jeffrey and Y. Onishi ZAMP

8P /Sz = 8 2 W o / S Z 2 ,

_ _ 2 8P~sR - 828z 2U~ + YUo _ ~ (U ~

Pa 82V~ + Y V o _ 2 R - 8 Z 2 ~ (U~

8W~/SZ = - (Ua + VO/R - 8Ua/SR.

+ Vo),

+ Vo),

(2.9)

(2.10)

(2.11)

(2.12)

The boundary conditions imposed on the surface Z = 1 + (1/2)R 2 a r e

1 4 ~Uo/SZ (2.13) U I = Z - I - ~ R

1 R 4 8Vo/SZ ' (2.14)

Wx = - g 4 8Wo/SZ, (2.15)

while on Z = (1/2) 2 R 2, the boundary conditions are

U~ = - ~ 2 s R 4 8Uo/eZ, (2.16)

V~ = - ~ 2 3 R 4 eVo/SZ, (2.17)

Wa = - ~ 2 3 R" 8Wo/SZ. (2.18)

The key step in solving these equations is the derivation of a differential equation for the pressure (the Reynolds equation). From (2.9), we find

t'1 = 8Wo/8Z + Q (R), (2.19)

where Q is to be determined. The integration of the remaining equations can be easily performed by a

computer, because integration is always with respect to Z and this variable appears in the equations only as Z", for some n. On the other hand the fact that the equations contain terms divided by the factor

H(R)= 1 +(1/2)(1--2)R 2

tended to produce ungainly expressions. It was found that the computer could be helped in its simplification of the expressions if 2 was replaced by q = (1 - 2)/2, thus reducing H to the sum of two terms. In terms of q, the equation for Q was found to be

R 2 Q" + [R + 6 q R 3 H -~] Q' -- Q

72 qS R 9 _ (40 q4 224- ~3 102 R 7 5 = [-- 25 -- ~ - q + -g- q2) + ...] H - (2.20)

Vol. 35, 1984 The forces and couples acting on two nearly touching spheres 637

The lower powers of R will not be presented. A particular integral of this equat ion is

Q 12 ~4 R 7 (596 q3 _ ~52 q2 + 51 R 5 .. H - 4 = [ ~ q + ,125 ~ q ) + .] . (2.21)

As in [3] and [4], the complementary functions need not be considered. We now wish to obtain the singular contributions to the forces and couples

following the method of [3], which is based on extracting the R - 1 term in the integrands for the forces and couples. The integrands for sphere Sa are given in [3], and the integrands for sphere SB are as follows. The force is - 6 7c # a 2 (2Ifl 2, and for this we consider

- ~ R l + ~ e 2 2 R 2 R(Po+eP,)+ 1 - ~ e 2 2 ] e z ( U - V )

- ~ 2 2 R OU/~R + e ~Wo/SR + e 2 R 2 ~ ( R - 1 V)/~R - ~ 2 U + ~ R - i Wol ,

while for the couple - 8 zcga 3 ~'~I912 w e consider

_ e 22 R2 ~ U ( 2 oW~ ~ U 0 (R - 1 V) b + 2R\ e2 2 +R eR

(2.22)

+ ~ - + e

The conclusions are that

2 (1 - - 4 2 ) 86 -- 1662 -- 6622 -- 6423 f*l - 15 (1 -- 2) 2 In e + ' A t , (2) + 375 (1 -- 2) 3 e In ~ + 0(e)

(2.24)

2 2 2 ( 1 - - 42) In g + A12 (2) 15 (1 - 2) 2 A2-

22 (86 - 166 2 - 66 22 - - 64 23) - 375 (1 -- 2) 3 e In e + 0 (~) (2.25)

-- 2 66 -- 12 2 + 16 2 2 g,1 - 5 (1 - 2) In e + B1, (2) - 125 (1 -- 2) 2 e In e + 0(~), (2.26)

22 22 (43 + 24 2 + 43 22) 912 -- 10 (1 -- 2) In e + B , 2 (2) --~ 250 (1 - - 2) 2 g In e + 0 (e). (2.27)

The functions A and B must be found from the numerical da ta in [8]. Compar ing these results in the special case 2 = 0 with those in [4], we see that there is a sign difference in the e In e terms. However, not only do the present signs improve the agreement with numerical data, they also agree with the results of problem II as the reciprocal theorem requires.

638

3. Problem II

D. J. Jeffrey and Y. Onishi ZAMP

The governing equations for the previous section remain unchanged, pro- vided we replace O r with UU/a. The boundary conditions (2.13)-(2.15) must be replaced by

1 R 48Uo/~Z , UI = - ~

1 R 4 ~Vo/~Z ' VI = - ~

R 4 ~Wo/~Z. W i = - g

(3.1)

(3.2)

(3.3)

The conditions (2.16)-(2.18) remain unchanged. The equat ion for Q (R) is now

R2Q" + [R + 6qR3H - 1 ] Q ' - Q

=[72 q 4 ( q - 1 ) R 9 +(40q 4 2 5 _~_q464-3 + ~ _ 3 9 6 q 2 _ ~ _ q ) R + . . . ] H - 5

with the particular integral

Q = [ _ t2 q3 R 7 (596 q3 2g (q -- 1) -- ,a-gg - - - 1256125 q2 + ~ q _ 2~) Rs + ' " .l H - 4

(3.4)

Following the same steps as before, we obtain the forces and couples for this problem. They are

f 2 1 = - 4 ( 2 - 2 + 2 2 2 ) l n e + A 2 1 ( 2 ) 15 (1 - 2) 3

4 (16 + 45 2 + 58 22 + 45 23 -t- 16 24) -- 375 (1 -- 2) 4 e In e + 0 (5). (3.5)

f22 = -- 4 2 (2 - 2 + 2 22) 15 (1 - 2) 3 In g -t'- A22 (2)

42(16 + 452 + 5 8 2 2 4- 4523 + 162'*) - e In ~ + 0 (~). (3.6)

375 (1 - 2)'*

1 - 4 2 921 -- 10 (1 - 2) 2 In e + B21 (2) -t-

2 (4 - 2) 922 -- 10 (1 - 2) 2

43 - 83 2 -- 33 22 - - 32 23 In e + 0 (e),

250 (1 - 2) 3 (3.7)

2(32 + 332 + 8322 - 4323) e lne + 0(e). In e + B22 (2) -{- 250 (1 - 2) 3

(3.8)

with the special cases. The fact that 3 f l l = 4921 and because of the reciprocal theorem has been pointed

These results agree 3f12 = 422922 (2-x) out in [81.

Vol. 35, 1984 The forces and couples acting on two nearly touching spheres

4. Physical aspects of the results

639

The results (2.24) (2.27) and (3.5)-(3.8) imply results of physical signifi- cance, or, if one prefers it, there are physical requirements that prevent the results from being independent of each other. Singular forces and couples arise because of the lubricating effects of viscous fluid between close surfaces that are in relative motion. Thus if the spheres were to have velocities and rotations that resulted in no relative motion of their surfaces in the neighbourhood of the gap, then the forces and couples acting should be non-singular.

To discuss this further, we change to the notation of O'Neill & Majumdar [8] in which k = - 2-~. We combine problems I and II by giving S~ angular velocity f21, and velocity U1, and S B angular velocity ~2 and velocity U 2. The force and couple acting on sphere S A are given by [8] as

F A = - - 67c#a [f21 a f l 1 (k, e) - - ~ ~ 2 af12( k - l , ek -1 ) + U1 f21 (k, ~)

+ U2f22 ( k-~, ~ k- l ) ] , (4.1)

a A = - - 8 g # a 2 [~"~1 a gll (k, e) - 02 a 912 (k- 1, e k-J) + U1 g21 (k, e)

-~ U2 g22 (k- 1, 6 k - 1)]. (4.2)

There are two motions that lead to no relative movement of the surfaces in the gap region. The first motion is that for which/21 = U 2 and O 1 = f22 = 0, i.e. side by side translation. The second motion is rotation as a rigid dumbbell, for which f2, = ~22 and U 2 = U1 - a (1 + k + 6) f2~. From (4.1) and (4.2) we therefore conclude that the following combinations should be independent of terms in In 6 and e In 6:

f21 (k, 6) + f22(k -1, gk-t),

g21(k, 6) -/- g22(k -1, e k - 1 ) ,

f l l (k, 6) - f12(k -1, 6k -1) -- (1 + k + 6)f22(k -1, ~k-1),

gl l (k ,e) - g~2(k -1, 6k -~) - (1 + k + 6)g22 (k -~, ~ k - l ) ,

as indeed they are. A motion which seems to offer the possibility of non-singular forces is that of rolling, in which U 1 = U 2 = 0 and O 1 = - kf2 a. Such is not the case, however, because the couple remains singular.

5. Numerical results

The asymptotic results obtained above can be combined with the numerical results given in [8] to obtain values for the functions A u and Bq appearing in (2.24)-(2.27) and (3.5)-(3.8). These functions have been used extensively in cal- culations of the mobilities of spheres [5] and it is important to provide an

640 D. J. Jeffrey and Y. Onishi Z A M P

independent check of the accuracy of the tabulations. This was done by follow- ing the methods of [6]. F rom the numerical tabulations of [8], we subtract the singular terms and then a straight line is fitted to the non-singular data thus obtained and the intercept with ~ = 0 taken as the value of the required function. The values thus obtained for the case 2 = - 1 (equal spheres) are A ~ = 0.1594, A12 = - 0.0011, A21 = 0.9983, A22 --- - 0.2737 and Bl l = 0.7029, B12 = 0.0274. These agree to within 0.0001 with the corresponding results used in [5].

The sign error in [4] was corrected and the new values agree more closely with the numerical values. The terms 3.0 ~ and 7.2 e estimated by [4] now become 0.1 ~ and 0.3 ~.

6. Problem III

No previous asymptotic analysis exists for this problem, the work of Ma jumdar [2] being confined to the case of touching spheres. The velocity field when S a rotates about the line of centres with angular velocity f2 m is a f2 m [o, V, o] and the pressure field is zero. In unstretched coordinates, V obeys

8 2 V/~Z 2 "[- 8 2 V/~? "2 --]- r -1 ~V/Sr - - r - 2 V = O. (6.1)

We look for a solution in the form

V = e~/2 Vo (R, Z) , (6.2)

which is stretched coordinates obeys

82 Vo/OZ 2 = 0, (6.3)

subject to the boundary conditions

1 V o -- R on Z = 1 + 2 R 2, (6.4)

and

Vo = 0 on Z = �89 2 R 2 . (6.5)

The solution is

1 j~ R 3) H - 1 (6.6) Vo = ( R z +

The singular contr ibution to the couple is obtained from the R-1 term in the integrand 2 n p a s s m e R z OVo/SZ. In this case the 0(1) term has been found in closed form by Majumdar [2], so we can write the couple - 8 ~c # a 3 0 m h l~ acting on SA as

(3, (1 - 2)- 1) 1 h l l = (1 - 2) 3 2(1 - 2) 2 e l n e + 0(e). (6:7)

Vol. 35, 1984 The forces and couples acting on two nearly touching spheres 641

On SB, the couple - 8 ~z# ff~III b 3 hi 2 is

- 2 3 ~ (3, 1) 2 3 h 1 2 - ( 1 - 2 ) 3 2 ( 1 - 2 ) 2 e l n e + 0 ( e ) " (6.8)

The function ~ (z, a) is defined as

(z, a) = ~ (k + a)-Z. (6.9) k=O

During this work, D. J. Jeffrey was supported by the Natural Environment Research Council of Great Britain.

References

[1] M. E. O'Neill and S. R. Majumdar, Z. angew. Math. Phys. 21, 180 (1970). [2] S. R. Majumdar, Mathematica 14, 43 (1967). [3] M. E. O'Neill and K. Stewartson, J. Fluid Mech 27, 705 (1967). [4] M. D. A. Cooley and M. E. O'Neill, J. Inst. Maths. Applics 4, 163 (1967). [5] D. J. Jeffrey and Y. Onishi, J. Fluid Mech. 139, 261 (1984). [6] D. J. Jeffrey, Mathematica 29, 58 (1982). [7] D. Barton and J. P. Fitch, Computer J. 15, 362 (1972). [8] M. E. O'Neill and S. R. Majumdar, Z. angew. Math. Phys. 21, 164 (1970).

Abstract

When two unequal spheres are very close, the low-Reynolds-number flow in the narrow gap between them can be analysed using lubrication approximations, and asymptotic formulae for the forces and couples acting on the spheres deduced. The expressions for the forces and couples have previously been regarded as independent, but it is shown here that they are linked by simple physical considerations. The new formulae can be used to improve the accuracy of companion calculations which apply to cases in which the spheres are not close.

Zusammenfassung

Es wird die Str6mung bei kleinen Reynoldszahlen untersucht in einem Spalt zwischen zwei ungleichen Kugetn, die sich nahezu beriihren. Asymptotische Formeln fiir die Krfifte und Momente an den Kugeln werdeu hergeleitet. Es wird gezeigt dab die Ausdrficke fiir die Kr/ifte und Momente, die bis jetzt als unabh/ingig betrachtet wurden, tatsfichlich durch einfache physikalische Betrach- tungen verknfipft werden k6nnen. Die neuen Formeln k6nnen zur Verbesserung der Genauigkeit von Rechnungen fiir gr613ere Kugelabst~inde bentitzt werden.

(Received: October 6, 1983; revised: May 25, 1984)