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Physica 92A (1978) 557-570 @ North-Holland Publishrng Co.
HYDRODYNAMIC INTERACTION OF TWO PERMEABLE SPHERES II: VELOCITY FIELD AND FRICTION CONSTANTS
RB. JONES
Department of Physrcs, Queen Mary College, Mde End Road, London El 4NS, England
Received 24 January 1978
By a method of reflections described in a previous arttcle we construct the approximate
velocity field produced by the matron of two permeable spheres through an mcompressible fluid.
The permeability is described by the Debye-Bueche equatrons. We constder contributtons to the
velocity field from seven reflections. From the velocity field we evaluate the forces and torques
acting on the spheres. Expressions for the force are given through order l/1’ and for the torque
through order l/L’, where 1 is the separation of the two spheres
1. Introduction
In the preceding paper (I) we described a formal iterative procedure for
applying the method of reflections to the determination of the velocity field
which results when two permeable spherical particles move in a viscous fluid.
In the present paper we will use the method of reflections to calculate the
velocity field explicitly. Because the detailed form of the velocity field is
rather complex we will content ourselves with describing the first reflection
rather completely but then giving only sufficient summary of the higher
reflections to enable the reader to reconstruct the full calculation. At the end
of this article we use the explicit velocity field together with the general force
and torque expressions in (I) to write down the translational friction constants
to order l/1’ and the rotational friction constants to order 1/j6, where 1 is the
interparticle separation. These friction constant expressions are unfortunately
rather complex. However, in the following article (III) we invert the friction
tensors to get the mobility tensors for two freely rotating spheres and these
are very much simpler in form than are the friction tensors. In article (III) we
also give an explicit evaluation of the friction constants for the case of
uniformly permeable spheres. For an appropriate limit this enables us to
compare with previous hard sphere calculations as a check on the correctness
of the present article’s conclusions. In what follows we adhere to the notation
used in (I) and we will often refer to specific results in (I).
557
558 R.B. JONES
2. I n p u t for the f irs t r e f l e c t i o n
From (I) we see that we must repeatedly solve the equations
~lV2d~(r) - Vq~(r) - AA(r )d~(r ) = AA(r )e~- l ( r ) , (2. la)
~/V2e~(r) -- V s ' ( r ) -- AB(r)e~(r ) = AB(r)d~- l ( r ) , (2.1b)
V - d ~ = 0; V . e ~ = 0. ( 2 . 1 c )
As input for this procedure we require d~(r) and el(r). Again from (I) we
know that dl ( r ) = ViA(r), e l ( r ) = v~(r ) where, for example, V1A(r) satisfies
V2v l ( r ) - Vp ~(r) - AA(r)o ~(r) = --AA(r)u A(r),
V " VlA(r) = O,
with
uA(r) = u A + c o a x ( r -- rA).
( 2 . 2 a )
(2.2b)
(2.2c)
Here U A and ta A are translational and angular velocities of sphere A whose
centre is at rA. A similar set of equations define v~(r ) if we replace A by B in
(2.2). It is clear from (2.2) that V~A(r) is the velocity field due to sphere A
moving in the complete absence of sphere B and vice versa for v~(r) . These
in fact are the fields already computed for the case of dilute polymer
solutionsl). In the language introduced in (I) we may say that V~A(r) arises
from a source field --uA(r). The field --uA(r) is itself a sum of two terms each
of which is an irreducible divergenceless vector field. The term - U A is a
symmetric tensor source with n = 0 and the term - -¢oax( r - - rA) is an
antisymmetric tensor source with n = 1. Using the results of (I) we can
immediately solve (2.2) and hence write d I and e 1 in the asymptotic region (i.e.
the region where AA and AB vanish) as
all(r) = [ (A ' /2) (a /r ' ) - (B ' /2 ) (a / r ' )3]U A
+ [ (A ' /2) (a /r ' ) + (3B ' /2 ) (a / r ' ) 3] r ' ( u A " r') + D,(alr,)3o~A × r' (2.3a) F t2
for [r'l > a, and
el(r) = [(A"/2)(b /r") - (B"/2) (b /r")3]U B
+ [ (A"/2) (b /r" ) + (3B" /2 ) (b / r" ) 3] r " (UB" r") + D,,(b/r,,)3~oB x r" r .2
(2.3b)
for If"] > b, where r ' = r - rA and r" = r - rB. In these equations a and b are the radii (or characteristic lengths) of spheres A and B, while the constants A ' , B ' , D ' , A" , B" , D" are dimensionless quantities corresponding to the
constants C used in section 5 of (I) to describe the asymptotic velocity fields.
HYDRODYNAMIC INTERACTION OF TWO PERMEABLE SPHERES II 559
A single pr ime refers to sphere A, a double pr ime to sphere B. These constants are obtained by solving the differential equations given in (I). Thus,
A' and B ' are found by solving the coupled equations
f~ ( r ' )+ ( 4 ) f ~ , ( r ' ) - h A ( r ' ) - - r ' h A ( r ' ) - - ~ fA(r') = AA(r')~7 ' (2.4a)
(41h , ( r , ) ~ A~(r ' ) fA( r ' )_A 'A( r ' ) l h'~(r') + r ' ~ r" (2.4b)
Here r' = Ir - rA[ while the pr imes on the functions denote derivatives. In the language of (I) eqs. (2.4) are associated with the (n = 0) symmetr ic tensor source 7", = - U, A. From the results of (I) we know that for r ' > a,
fA(r') = A'(a/r ' ) + B'(a/r ' ) 3, (2.4c)
hA(r') = A ' a(1/r') 3.
To obtain D ' one solves
X'~(r') + (4/r')X'A(r') - AA(r')/~XA(r') = --XA(r')/~,
corresponding to the n = 1 ant i symmetr ic tensor source A, 1
we have
X^(r ' ) = D'(a/r ') 3. (2.5b)
These equations have been studied in some detail previously by Felderhof and Deutch~). An exact ly analogous set of equations is used to find A", B", D" which reflect the proper t ies of sphere B.
(2.5a)
A For r ' > a - - ~lp] 0.1 p .
3 . T h e f i r s t r e f l e c t i o n
The two fields d~(r), el(r) are input sources for the calculation of fields d2(r) and e2(r). Thus, we now try to solve
r/V2d2(r) - Vq2(r) - AA(r)d2(r) = AA(r)el(r). (3.1)
Although in section 2 we could solve in an exact sense for dl(r) , el(r) it is not
possible to solve exact ly for d2(r). Rather, we obse rve that e~(r) is a source field centred at rB but we need its values in (3.1) over the region occupied by sphere A centred at rA. From (2.3b) we see that el(r) is of order 1/l over the region occupied by sphere A (! = ]rB-rA[). Therefore , if we expand el(r) in
series about rA, each term in such a series is smaller than the previous one by 1/l. As was shown in (I) such an expansion procedure enables us to decom- pose the divergenceless source field e~(r) into a sum of irreducible diver- genceless parts. For each such irreducible part of the source we can solve (3.1) exactly. By keeping only a finite number of such parts we calculate the veloci ty field d2(r) to some finite order in 1/l.
560 R B. JONES
The Tay lo r ' s series for e l ( r ) about rA becomes
e~(r) = h~ + h~,,r] I + h~,,,2r;~r] 2 + h~,~,2,3r',,r]2r',3 +" • ", (3.2a)
with
_ 1 a"e~(r ) h,',, ,. n] 0-~,~_?_~, . . . . a" (3.2b)
The first four te rms of the series (3.2a) suffice to enable us to calculate the
fricti6n constants to the desired order in III. Because the expansion in (3.2a) is about rA (which lies in the asymptot ic region as regards sphere B) we use (2.3b) to calculate the derivatives. The first term of (3.2a) is
h 1 = [ ( a " 1 2 ) ( b l l ) - ( B " 1 2 ) ( b l l ) 3 l U " + [ ( a " 1 2 ) ( b l l ) + ( 3 B " 1 2 ) ( b l l ) 3 ] l ( U " . l ) l l z - D " ( b l l ) 3 o j B x l, (3.3)
where ! = r B - rg. The next term gives
h~qr;, = T~,,r;1 + A~qr'q, (3.4a)
where T~,, and A~,, are symmetr ic and ant i symmetr ic tensors as explained in
(I). Explicit ly we have
T,II = - (3B"12)(b3115)[UB, l I + U~l ,]
- [ (A"12 ) (b l l ) + ( 3 B " / 2 ) ( b l l ) 3 ] S o ( U B . l ) l l z
+ [ ( 3 A " 1 2 ) ( b l l ) + (15B"12)(b l l )3] l , l s (U B . l ) l l 4
- (3D"12)(b3]lS)[l ,(o~ B × !)1 + ll(oa a x / ) , ] , (3.4b)
A ~j = e,p,l~ ~, (3.4c)
with
I I 1 = ( A " / 2 ) ( b l l 3 ) ( l × U ~) - (D"12)(b l l )3oa B + (3D"/2)(b3115)l(oa B . l). (3.4d)
The second order te rm in (3.2a) becomes
hi , t _ _ 1 t ~ 4 1 t t t 2 1 t 1 t ,,~,2r,,r,2 - T,,~,~r,lr, 2 + (~)A,,t.,2r,,r,2 + (2r 15)h,qq - ( r , 1 5 ) ( h m r j ) . (3.5a)
I t is now stra ightforward but tedious to check that
T,~,k = [ ( A " 1 2 0 ) ( b l i ) + (3B"14) (b l l )3] ( l l12){8 , ,U~ + 8jkU B, + 8~,U, B}
- [ ( 1 3 A " 1 2 0 ) ( b l l ) + ( 1 5 B " 1 4 ) ( b l l ) 3 ] ( U " . lll4){&,lk + 8,d, + 8k, l,}
- - [ (A"14) (b l l ) + (15B"/4) (bl l)3](l l l4){l , l~U~ + l, l kU ~, + lkl, U ~ }
+ [ ( 1 5 A " / 4 ) ( b l l ) + ( 1 0 5 B ' 1 4 ) ( b l i ) 3 ] ( U " . lll6)l,l~lk
- - ( 5 D " 1 2 ) ( b 3 1 1 7 ) { l , lj( ¢o B x I)k + Ijlk( to " X l), + lkl,( oJ " × !),}
+ (D"12)(b3/15){So(oa B × i)k + 8~k(tO B × !), + 8k,(~o B × l)j}, (3.5b)
HYDRODYNAMIC INTERACTION OF TWO PERMEABLE SPHERES II 561
A,~j.k = ( 3 A " [ 8 ) ( b/13){ 8 , k U ~ - 8,kUB, } -- (3 A " / 4 ) ( b l lS){l , lkUaj - ljlkUB, }
- ( 3 A " / 8 ) ( b / : ) ( U ~" l){8,kl~ -- 8~gl,}
+ (15D"14) (b3[17){ l , lk(OJ B × 1) s - l j l k ( ~ ~ × !),}
- - ( 3 D " / 4 ) ( b 3 / 1 5 ) { 8 , ~ ( t o B x l)~ - 81k(tO B X l ) , }
- - ( 3 D " / 4 )( b 3/15) { l,%pktO ~ - l~,pktO ~ -- 21ke,p,tO ~},
h , ~ = ( A " 1 2 ) ( b l l 3) U B _ ( 3 A " 1 2 ) ( b l : ) ( U B . I)l,.
(3.5c)
(3.5d)
The last term of (3.2a) becomes
h i . . . . l . . . . t3~--i ' r ' ' + ( 5 r ' Z / 7 ) S ~ p r ' p - ( 2 r , / 7 ) S q p r q r p U l l 2 t 3 r l l r l 2 r | 3 ~ l l l l / 2 1 3 r t l r l 2 r / 3 y 1 2 j l t ~ l l l , t 2 t 3 r t l t 2 r t3 t 1 t t
+ ( 3 r ' 2 / 5 ) a ~pr'p. (3.6a)
For tunate ly , only the tensor S,1~ contributes to the desired order. Explicitly it is given by
- (2)[h,q~ + h,oq,]
= (A"12)(bllS){UB, l, + U~,l, + ( U R. t)8,, - 5t, h(U B. Ill2)}. (3.6b)
Given the above linear combinat ion of sources we can write an approximate solution for d2(r). If we go to the asymptot ic region for sphere A, Ir'l > a it will be
d 2, = - [ ( A ' / 2 ) ( a / r ' ) - ( B ' / 2 ) ( a / r ' ) 3 ] h ] - [ ( A ' 1 2 ) ( a / r ' ) + (3B' /2)(a/r ' ) 3]
X r ' ( h 1 r ' ) / r ' 2 - D ' ( a / r ' ) 3 ( ~ l x r ' ) , ' , 5 t , • - ( 2 K / 3 ) ( a / r ) T, , r , 1 + [ ( j , / 2 ) ( a [ r , ) 3 + , , 5 , l , , ,z ( 5 K / 3 ) ( a / r ) ] r , ( T p q r p r q ) / r
[ ( N , / l O ) ( a / r , ) 5 + , , 7 1 , , - ( 3 P / 4 ) ( a / r ) ] T , , ~ , : , l r , 2
+ [ ( N , / 2 ) ( a / r , ) 5 + , , 7 , 1 . . . . 2 ( 7 P / 4 ) ( a i r ) ] r , ( T p q t r p r q r t ) l r t 1 5 1 t t + S ( a / r ) A, , , . ,2r, , r, 2 + [(V'/2)(a/r ' ) - ( W ' 1 2 ) ( a / r ' ) 3 ] a 2h ~,lq
+ [(V'/2)(a/r ' ) 3 + (3 W'/2 ) (a / r ' )5]r ] (h]q~r '~ )
- ( 2 a ' / 3 ) ( a / r ' ) S a2S~pr'p + [ ( Z ' 1 2 ) ( a / r ' ) 5 + ( 5 a ' / 3 ) ( a / r ' ) 7]
t ~ l t ! x r ,~pqrpr ,~ + • • .. (3.7)
The terms omit ted are of order I / l 4 or smaller and they fall off at large r ' at
least as rapidly as l / r '4. Many dimensionless constants have appeared in (3.7) whose values depend upon the permeabil i ty of sphere A. In fact , to evaluate all friction constants through order 1/! 7 would require the constants A' , B ' , D ' ,
J ' , K ' , N ' , S ' , V', W', Z ' .
To clarify how these constants enter the force and torque express ions we can use the relations given in (I) to find the force on sphere A due to all(r) and dE(r). It is
- 4 r r * l a [ A ' U A - A ' h ~ + V ' a2h~qq],
562 R.B JONES
while the torque would be
- 87rrla3[DIto g - D ' I ~ ] .
The other constants in (3.7) will not contribute to the force and torque except
through their contribution in higher reflections. Indeed, the statement just after (3.7) implies that P ' and or' can only contribute to the force in order III 8
so that these two constants can henceforth be ignored. Since no additional
constants arise in later reflections we can at this stage tabulate the differential
equations which define J ' , K ' , N ' , S ' , V ' , W ' and Z'. This is done in the
Appendix. The constants A' , B' , D ' were defined in the previous section.
4. Higher order reflections
Given the expression (3.7) for d2(r) we can now obtain ¢2(r) by utilizing the
symmetry of the problem. To find eE(r) w e simply take the expression (3.7)
then change all primes to double primes, interchange the lengths a and b
(a~-~-b), and replace ! by - l wherever it occurs in the various tensors given
above. The parts of the field eE(r) which are of interest can then be written as
e2(r ) = - [ ( A " / 2 ) ( b / r " ) - ( B " / 2 ) ( b / r " ) 3 ] i ~
- [ (A" /2 ) (b / r " ) + (3B"12)(b[r")3]r' , '( l i ~. r " ) l ( r "2)
- D " ( b l r " ) 3 ( ~ l x r"), - . . . . 5 A l ,, (2K / 3 ) ( b / r ) T , , r , 1
+ [(j, ,]2)(b[r,,)3 + . . . . 5 ,, ^ 1 . . . . . . 2 (5K [3 ) (b l r ) ] r , ( T p q r p r q ) / ( r )
it it 5 ^ 1 I t t t - ( N / l O ) ( b / r ) T,,~,2r,~r, 2 + ( N " 1 2 ) ( b l r " ) 5
.-,~1 . it i p . . I t 2 - - ~ t f f h I it)5~l t i t It X r , [ l p q t r p r q r t ) l r t ~ , ~ , r , .lul.,2-qrt2
+ [(V"/2)(blr") _ ( WI'/2)(blrI')3]b21~qq
+ [( V'I/2)(b[ r") 3 + (3 . . . . 5 ,, "1 W [2 ) (b / r ) ] r , ( h m r ~ )
. . . . 5 , 1 .... (4.1) + ( Z 12) (b /r ) r, S p q r p r q + . . . .
The hat occurring on the tensors above means that they are obtained from the
corresponding tensors in section 3 by interchanging labels A and B, inter-
changing primes and double primes, and replacing l by - l . The field eE(r) n o w is used as a source for field d3(r):
r/V2d3(r) - Vq3(r ) - Ag(r )d3(r ) = Ak(r )eZ(r ) . (4.2)
Once again we must develop eE(r) in series about rk and then solve (4.2) by superposition, as in the first reflection. It is evident that such a procedure is much more laborious than at the first step. One simplification results from the
observation that the first three terms in (4.1) have exactly the same form as the field e l ( r ) if, in (2.3b), we replace U a by - / ~ and to B by -1~ I. Thus, the first part of e2(r) has a series expansion identical to the already calculated
HYDRODYNAMIC INTERACTION OF TWO PERMEABLE SPHERES II 563
expansion of e~(r), apart f rom this substitution. Unfor tunate ly , the remainder
of (4.1) contr ibutes many new terms not seen in the first reflection. By
observing the order of magnitude of the tensors occurring in (4.1) we see that e2(r ) is of order 1/12 in the region occupied by sphere A whereas e ' ( r ) was of
order l / l . This fact , that e ~ ( r ) is of order 1/l ~ in the region of sphere A, eventual ly makes the calculations simple again since we are only working to a fixed order in l[l . Howeve r , the intermediate calculations for d3(r ) and d4(r) are sufficiently involved that we will no longer a t tempt a detailed description.
Rather, we will quote the result obtained for the fields d ~ ( r ) for c~ = 3, 4 . . . . . 8. The interested reader, by following the pat tern of the calculation given for dE(r), may reconst ruc t and check these fields for himself.
For d3(r) one obtains the following result:
d3,(r) = - [ ( A ' / 2 ) ( a ! r ' ) - ( B ' / 2 ) ( a / r ' ) a ] ( h 2, + Q~)
- [ ( A ' / 2 ) ( a / r ' ) + (3B ' /2 ) (a[r ' )a]r ' , ( (h 2 + Q I ) . r ,) /r ,2
- D ' ( a / r ' ) 3 ( ( ~ 2 +dp 1) x r ') , - ( 2 K ' / 3 ) ( a / r ' ) 5 ( T ~ + cr,,~)r',~
+ [ ( J ' / 2 ) ( a / r ' ) 3 + (5K ' /3 ) (a / r ' )5]r ' , ( (TZq + trp~)r'pr'~)/r '2 I i 5 2 t I - ( N / l O ) ( a / r ) T,,~,2r,lr, 2 + (N ' /2 ) (a / r ' )S r ' ,T2q , r'p r q r d r . . . . 2
+ S'(a/r ')SA21.,2r; ~r; 2
+ [(V'/2)(a/r ' ) - (W' /2 ) (a / r ' )3]a: (hEqq + P~)
+ [(V' /2)(a/r ' ) 3 + (3 W'/2 ) (a / r ' )5] r ' , ( (h~q + P,)r,)l ,
, ,5 , 2 ' ' (4.3a) + ( Z / 2 ) ( a / r ) r , Spqrprq + . • ..
In this express ion tensors with superscr ipt 2 are defined by taking the
corresponding tensor f rom section 3 and making the substitution U,B-~/~, to, B ~ - l ) ~ . There are additional tensors whose definitions we now give:
Q~ = (2K"/3)(b / l )S f '~ , , l , t - [ ( j , , / 2 ) (b / l ) 3 + (5 K " / 3 ) ( b / l ) 5]
× l,(T~,~lplq)/l 2 - (N"/lO)(b/l)Sf'~,~,21,~l,2
+(N, , /2)(b/ l )51,~lpq, l f l f l j l2 + ,, 5 "1 S (b / l ) A,,,.,21,~l, 2
+ [(V"/2)(b/l) 3 - ( W"/2)(b/l)5]121~oq + [(V"/2)(b[l) 3 + (3 W " / 2 ) ( b / l ) 5]
× l,(l~qqlj) - (Z" /2) (b[ l )S l , S~qlflq, (4.3b)
~b~ i i (4.3c) = - ( g e , p q a pq,
ot,~ = (J" /2)(b/ l )3{ l t f '~qlq - Ijf'~qlq}/l 2 - (V"/2)(b/ l )3{ l , l~qq - l,f~qq}, (4.3d)
tr,, = (J"/2)(b/l)3{8,,?'~polpl~ - 5l, l , ( f '~ol f lq) / l 2
^~ I j '~ f l~} / l 2 - (V" /2) (b l l )3{8 , , - 31,1j/12}(i~,,lq), (4.3e) + l, Tjqiq +
and
P ~ = - J" ( b/ l )3{ ~ ~qlfl l 2 - (~)l,( T ~flflq)/ l 4}
+ ( V"/2)(b/ l )3{t~pp - 31,(fz}pplj)/12}. (4.3f)
564 R.B JONES
We have omitted from d3( r ) terms which would contribute to the torque on sphere A in order l [ l 7. However , force terms in d3(r) are correct ly given through order 1[17 and torque terms may be computed to order I I I 6.
The field e3(r) is obtained from d3( r ) by the same symmetry operation as was used to get e2(r ) . Using e3(r) as a source field gives for d4( r ) the following:
d4,(r) = - [ ( A ' / 2 ) ( a / r ' ) - ( B ' / 2 ) ( a / r ' ) 3 ] ( h 3 + Q2 + R , )
- [ ( A ' / 2 ) ( a / r ' ) + ( 3 B ' / 2 ) ( a / r ' ) 3 ] r ; ( ( h 3 + Q2 + R ) . r ' ) / r '2
_ D , ( a / r , ) 3 ( ( l ~ a + d p 2 ) × r , ) , + , , 3 , 3 , , ,2 ( J / 2 ) ( a i r ) r , T p q r p r q / r
+ ( V , / 2 ) ( a / r , ) a 2 ( h ~ q q + p 2 ) + ( V , / 2 ) ( a / r , ) 3 r',((h~qq + p 2 ) r ; ) + . • .. (4.4a)
Again we have written explicitly only sufficient terms to compute the force to order l [ l 7 and the torque to order 1]16. The superscript 3 on the tensors in (4.4a) means that they are computed from the tensors with superscript 1 of section 3 after first replacing U a by _(/~2+ (~) and to a by - ( 1 ~ 2 + ~ ) . The vectors Q2, p 2 , and ~2 are obtained from (4.3b), (4.3c, d) and (4.3f) by simply changing all superscripts there from 1 to 2. The one new tensor in (4.4a) is R which may be written as
R, = - (J" /2)(b3/15) l , (d-pJf lq) + ( V " / 2 ) ( b / l ) b E f f ~ + ( V " / 2 ) ( b / l ) 3 1 , ( P ~ l j ) . (4.4b)
For d~(r ) we have
d ~ ( r ) = - [ ( A ' / 2 ) ( a / r ' ) - ( B ' / 2 ) ( a / r ' ) 3 ] ( h 4, + Q~)
- [ ( A ' / 2 ) ( a / r ' ) + ( 3 B ' / 2 ) ( a / r ' ) 3 ] r ; ( ( h 4 + Q 3 ) . r , ) / r ,2
- D ' ( a / r ' ) 3 ( 1 2 4 × r ' ) , + ' , 3 , 4 , , ,2 ( J / 2 ) ( a / r ) r , ( T p q r p r q ) / r
+ ( V ' / 2 ) ( a / r ' ) a 2 h ) ~ q + (V ' /2) (a /r ' )ar ' , (h~qqr; ) + . • .. (4.5)
Tensors with superscript 4 arise f rom the tensors with superscript 1 in section 3 after replacing U B by - ( / 1 3 + 0 2 ) and to a by -(1~3+ ~2). The vector Q3 arises f rom (4.3b) by changing all superscripts 1 to 3. In fact, most of Q3 is higher order in l / l than we require and only the terms containing J " or V" need be kept.
The remaining fields rapidly simplify. Thus, for d6(r) we have
d 6 ( r ) = - [ ( A ' / 2 ) ( a / r ' ) - ( B ' / 2 ) ( a / r ' ) a ] ( h ~ + Q4)
- [ ( A ' / 2 ) ( a / r ' ) + (3B ' / 2 ) (a / r ' ) 3 ] r ' , ( ( h 5 + Q 4 ) . r , ) / r ,2
- D ' ( a / r ' ) 3 ( l l 5 × r'), + (V'/2)(a/r')a2h~qq
+ (V ' /2) (a /r ' )3r ' , (h~qqr; ) + . •. , (4.6)
where h~, f~ , h~q arise f rom the corresponding tensors of section 3 after replacing U a by _(/~4+ 03) and to a by _124. Again, Q4 is given by (4.3b) if
superscripts are changed from 1 to 4.
where l = r B - r g , I a = (l|]12), and 1 is the unit dyadic. The various friction
HYDRODYNAMIC INTERACTION OF TWO P E R M E A B L E S P H E R E S II 565
For d7(r) and dS(r) we have
d 7 ( r ) = - [ ( A ' / 2 ) ( a / r ' ) - ( B ' / 2 ) ( a / r ' ) 3 ] h 6
- [ ( A ' 1 2 ) ( a l r ' ) + ( 3 B ' 1 2 ) ( a l r ' ) 3 ] r ; ( h 6 . r ' ) l r ' 2 + • • . , (4.7)
dS, ( r ) = - [ ( A ' / 2 ) ( a / r ' ) - ( B ' / 2 ) ( a / r ' ) 3 ] h 7
- [ ( A ' / 2 ) ( a / r ' ) + ( 3 B ' / 2 ) ( a / r ' ) 3 ] r ; ( h 7 . r ' ) l r '2 + . • . , (4.8)
where /16 is ob ta ined f rom h I by the subst i tut ion U a ~ - / ~ 6 , t o8~_1~5 , and h 7
is ob ta ined f rom h 1 by the subst i tu t ion U B ~ - / ~ 6, toB--*-l~ 6.
5. Forces and torques
F r o m a knowledge of d~(r) , a = 1 . . . . . 8, we can calculate the fo rce and
to rque on sphere A as expla ined in (I). In fact , the express ion given above enables us to calcula te the fo rce th rough order 1/ l 7 and the to rque th rough
order l / l 6. I f we deno te by - F A and - T A the fo rce and torque exer ted on
sphere A by the fluid, then f rom the results of (I) and the explicit express ions
above we m a y wri te
F A 4 ~ l a { a , [ u A l 3 4 5 6 7 = - h , - h 2 - h , - h , - h , - h , - h , - Q ~ - Q 2 _ Q 3 _ Q 4 _ R , ]
+ V , a 2 [ h ~ q q + 2 3 4 s + p ~ + p 2 ] . (5.1) h ,qq "Jr- h tqq - I- h ,qq "t- h ~qq
For the to rque we obtain
3 4 5 T, A = 8~r~a3D'[~o, A - 1)~ - f~2 _ I~, - ~ , - fL - d~ - 4~2]. (5.2)
As men t ioned before the torque express ion is co r rec t only th rough order l / l 6.
The var ious tensors that o c c u r above have all been defined in sect ions 3 and
4. T h e y mus t now be eva lua ted explici t ly to show the linear d e p e n d e n c e o f fo rce and torque on the driving terms U A, U B, to A, oJ B. The calcula t ion o f
these tensors involves the i terative subst i tu t ion o f one a l ready calcula ted tensor into ano the r a l ready k n o w n tensor . This calcula t ion is tedious and is
only par t ly simplified at the later s tages by the f r e edom to discard any
cont r ibu t ion to the fo rce of order greater than l / ! 7 and any cont r ibut ion to the
to rque of o rder grea ter than l / l 6. We will now write the final result one obtains for fo rce and torque. Us ing the nota t ion of (I) we have
F A = 4~rr/[(~AA1 + ZAAP) " U A + ~bAA/ × tO A + (~AB1 + ZAaP) " U a + ~bAal × tOB],
(5.3)
T ^ = 87rr/[(~AA1 + EAAP) " toA+ I~AA(! X u A ) / I 2
-t-(~AB1 -t- EABP) " ~ B "1- ~bAB(/ × u s ) l / 2 ] , (5.4)
566 R B. JONES
cons t an t s in (5.3) and (5.4) have the fo l lowing values .
~AA = a A ' { 1 + ( A ' A " [ 4 ) ( a b [ l 2)
+ [ A ' ( D " / 2 - B " / 4 - V " / 4 ) a b 3 + (A '2A"2[16)a2b 2
- ( A " / 4 ) ( B ' + V ' ) a 3 b ] ( 1 / I 4) + [ A ' ( - N " / 5 0 - 3 S " / 8 + W " / 4 ) a b 5
+ ( A ' 2 A ' / 8 ) ( 2 D " - B " - V " ) a 2 b 4 + ( B ' B " / 4 + V ' V ' [ 4 + A ' 3 A ' 3 [ 6 4 ) a 3 b 3
+ ( A ' A " 2 [ 8 ) ( D ' - B ' - V ' ) a 4 b 2 + ( A " / A ' ) ( V 'B ' [4 )aSb] (1 /16)} , (5.5)
ZAA = a A ' { ( 3 A ' A " / 4 ) ( a b / l 2) + [ ( A ' / 4 ) ( 5 B " - 2 D " + 5 V " - 2 J " ) a b 3
+ (15A 'ZA"Z[16)aZb z + ( 5 A " / 4 ) ( B ' + V')a3b](1]l 4)
+ [ A ' ( - K " - 23N"/50 + 3S"18 + 3 W " I 4 + Z"12)ab ~
+ ( A ' Z A " 1 8 ) ( 1 7 B " - 2 D " + 17 V " - 8 J " ) a Z b 4
+ ( 3 B ' B " I 4 + 3 V ' V " I 4 - 3 B ' J " / 2 - 3 V ' J " / 2 + 63A 'aA"3164)a3b 3
+ ( A ' A " Z [ 8 ) ( 1 7 B ' - D ' + 17 V ' - 4 J ' ) a 4 b z + ( A " / A ' ) ( 3 V 'B ' / 4 )aSb] (1 /16 ) } ,
(bAg = a a ' D ' { - ( a " / 2 ) ( a 3 b / l 4) (5.6)
+ [ ( B " - D " ) ( a3b3 / 2 ) - ( A ' a " E / 8 ) a 4 b 2 + ( A " / a ' ) ( V ' / 2 ) a S b ] ( 1 / 1 6 ) } ,
~AB
ZAB ~-
(5.7)
a A ' { - ( A " / 2 ) ( b l l ) + [ (B"12)b 3 - ( A ' A " Z l 8 ) a b 2
+ ( A " / A ' ) ( V ' / 2 ) a Z b ] ( l / l 3) + [ ( A ' A " / 8 ) ( 2 B " - 2 D " + V " ) a b 4
- (A '2A"3132)aZb3 + (A"218)(B ' - 2 D ' + 2 V ' ) a 3 b 2 ] ( l l l 5)
+ [ ( ( A ' A " ) ( - W " I 8 + 3S"/16 + N " / I O 0 ) + ( A ' B " / 8 ) ( 2 D " - B " - V " ) ) a b 6
+ (A '2A"Z132) (3B" + 2 V " - 4 D " ) a Z b 5
+ ( A " I 4 ) ( - V ' V " - B ' B " - D ' D " + V ' D " - V ' B " + D ' B " - (A '3A"3132))a364
+ ( A ' A " 3 1 3 2 ) ( 2 B ' - 4 D ' + 3 V ' ) a 4 b 3 + ((A")Z( - W ' I 8 + 3 S ' 1 1 6 + N ' 1 1 0 0 )
+ ( l / A ' ) ( A " 2 / 8 ) ( 2 V ' D ' - V '2 - B ' V ' ) )aSbZ]( l / /7 ) , (5.8)
a a ' { - (A"12 ) ( b l l ) + [ - (3B"12)b 3 - ( 7 A ' A " 2 l S ) a b z
- ( A " / A ' ) ( 3 V ' / 2 ) a Z b ] ( l l l 3)
+ [ ( A ' A " 1 8 ) ( - 1 8 B " + 2 D " + 4 J " - 9 V " ) a b 4 - (31A'ZA"3132)aEb3
+ ( A " E 1 8 ) ( - 9 B ' + 2 D ' + 4 J ' - 18 V ' ) a 3 b Z ] ( l l l 5)
+ [ ( ( A ' A " ) ( 4 7 N " / l O 0 - 3S"116 - 7 W " / 8 + K " - Z"/2)
+ ( A ' I S ) ( - 7 B "2 - 7 V " B " - 2 B " D " + 4 J " B " ) ) a b 6
+ (A 'ZA"2132) (4D" - 66 V " - 9 9 B " + 3 2 J " ) a Z b 5
+ ( A " 1 4 ) ( - 7 V ' V " - 7 B ' B " - 7 V ' B " - D ' B " - V ' D " + D ' D "
- 2 J ' J " + 8 V ' J " + 6 J ' V " + 8 J ' B " + 6 B ' J " - 127A'3A"3132)a3b4
+ ( A ' A " 3 1 3 2 ) ( 4 D ' - 99 V ' - 66B ' + 3 2 J ' ) a 4 b 3
+ ( ( A " Z ) ( 4 7 N ' / l O 0 - 3S ' /16 - 7 W ' / 8 + K ' - Z'/2)
+ ( 1 / A ' ) ( A ' a / 8 ) ( - 7 V 'z - 7 B ' V ' - 2 V ' D ' + 4 J ' V ' ) )aSb2]( l /17)} , (5.9)
HYDRODYNAMIC INTERACTION OF TWO PERMEABLE SPHERES II 567
d?AB = a A ' D " { - (b / l ) 3 - ( A ' A " / 4 ) ( a b 4/l 5)
+ [ ( - A ' D " [ 2 + A ' V" /4 )ab 6 - (A'2A"2/8)a2b5
+ ( A " / 4 ) ( B ' - D ' + V')a3b4](1]lT)},
~AA = a3D'{1 + (D 'A"[2) (a3b / l 4)
+ [ ( D ' / 4 ) ( D " - 3J")a3b 3 + (A'D'A"2/8)a4b2](1/16)},
--=AA = a 3 D ' { - ( D ' A " / 2 ) ( a 3bl 14)
+ [(3D'[4)(D" + J")a3b 3 - (A'D'A"2[8)a4b2](l[16)},
~AB = a3D'{(D"[2)(b[ l ) 3 + (A'A"D"14)(ab4[15)}
~Aa = a3D'{ - (3O" /2) (b l l ) 3 - ( A ' A " D " / 4 ) ( a b 4/IS)}
I~AA
I~AB
(5.10)
(5.11)
(5.12)
(5.13)
(5.14)
a 3D'{(A'A"/4)(abll 2)
+ [ ( A ' / 4 ) ( D " - V" )ab 3 + (A'2A"2/16)a2b ~ - (B 'A"/4)a3b]( l /14)} , (5.15)
a3D'A"{ - (½)(b[l) - (A 'A"18) (ab2/ l 3)
- [ (A ' /8 ) (D" - B" - V " )a b 4 + (A'2A"2/32)a2b 3
+ (A" /8 ) (2D ' - B ' - V')a3b2](1/15)}. (5.16)
To obtain the forces and torques associated with sphere B one takes the expressions (5.3)-(5.16) and makes the following substitutions: interchange labels A and B, interchange primes and double primes, interchange a and b, replace ! by -1. The above expressions are somewhat daunting in their complexity. However , we will show in the next paper in this series (III) that, on imposing the condition of free rotation and then inverting the relation (5.3) (and its associated relation for sphere B) we obtain mobility tensors which are very markedly simpler than the friction tensors given above.
One may also ask at this point how reliable are the above expressions. We have repeatedly checked the calculation leading to the expressions (5.3)- (5.16). Moreover , as we show in (III), one can regain the corresponding expressions for hard spheres with stick boundary conditions by taking a limit as ;tA, An--~ 0o in the above calculation. In this hard sphere limit the dimension- less coefficients used above have the following values:
A' = A " =~; B' = B " = - ½ ,
D ' = D " = 1 ; J ' = J " = - 5 ,
K ' = K" = 3; N ' = N" 35 (5.17)
s ' = s " = _ 4 v ' = w - - -½,
W ' = W"=~0; Z' = Z " = - 3 .
On inserting these values in (5.5)-(5.16) we obtain results which agree with the general results quoted in Happel and Brenner 2) for the hard sphere problem treated to order l / l 5 by reflections. In addition, for the special
568 R.B JONES
geometry when the two hard spheres are equal-sized and move along their line of centres, our result for the force agrees through order l / l 7 with the series obtained by Dahl3). Our quantities g,l and ,=,j agree in the hard sphere limit with results given by Montgomery and Berne 4) through order 1]16. A further check can be applied by comparing the mobility tensors (given in III) arising from the above friction tensors with the recent hard sphere calculation through order I I I 7 by FelderhofS). All of these checks suggest that the results given above are indeed correct.
In the following paper we will investigate the mobility and diffusion tensors for the two spheres in free rotation assuming either a uniform or shell-like distribution of the inverse permeability ;t(r). However, from the equations given in the Appendix and in section 2 it is possible to see the additional information which the hydrodynamic interaction of two permeable spheres contains compared with the simpler hydrodynamic force that acts on one isolated permeable sphere. The dimensionless constants A, D contain the same information about ;t(r) that we could obtain from one isolated perme- able sphere. The extra constants B, J, K, N, S, V, W, Z, give information about higher order moments of A(r). More specifically, A depends upon ;t(r), B, D, J, V depend upon r2;t(r), while K, N, S, W, Z depend upon r4X(r). In short, the hydrodynamic interaction of two spheres gives us an additional probe of the matter distribution within each sphere which may be particularly sensitive to deviations from uniformity near the edge of each sphere.
Appendix
Here we gather together the differential equations whose asymptotic solu- tions determine the dimensionless constants J' , K', N', P ' , S', V', W', Z' and a ' which appeared in section 3. We first consider J', K'. In the language of (I) we may say that J ' , K' arise from solving the Debye-Bueche equation with symmetric tensor source term (n = 1) of the form
S , ( r ' ) = T~,~r;~. (A.1)
The velocity and pressure fields have the form
v , ( r ' ) L f ( r ' ) + , , , 1 . . . . . i , , = ( r f ( r ) 1 3 ] T , l r , , - f ( r ) l ( 3 r ) r , T p q r p r q ,
= * l h ( r ) T p q r p r ~ , p ( r ' ) , l , , (A.2)
where f ' ( r ' ) denotes the derivative of f ( r ' ) . The functions f ( r ' ) , h ( r ' ) satisfy the differential equations
f ' ( r ' ) + ( 6 1 r ' ) f ' ( r ' ) - 2h(r') - r h ' ( r ' ) - ( ~ A ( r ' ) / n ) f ( r ' ) = XA(r')/~, (A.3)
h'(r') + ( 6/ r ' ) h '( r ' ) + (A'A(r')/~l)(f(r')/r') = (-- X 'A( r ' ) l *l )( l l r ' ) .
HYDRODYNAMIC INTERACTION OF TWO PERMEABLE SPHERES II 569
F r o m (I) w e see tha t the so lu t i on to t hese e q u a t i o n s in the r eg ion r ' > a
[where ; tA( r ' )= 0] is
f ( r ' ) = ( J ' / 2 ) ( a / r ' ) 3 + K ' ( a / r ' ) 5,
h ( r ' ) = J ' (a3 l r '~ ) . (A.4)
F i t t i ng this so lu t i on to the in t e r io r so lu t ion fo r r ' < a g ives exp l i c i t va lues for
J ' , K ' w h i c h d e p e n d on the i nve r se p e r m e a b i l i t y )tA(r'). T h e c o n s t a n t s N ' , P ' a r i se f r o m the n -- 2 s y m m e t r i c t e n s o r s o u r c e (3.5a)
S , ( r ' ) ~ l , t = l,,1,2r,~r, 2. (A.5)
V e l o c i t y and p r e s s u r e f ields have the fo rm
v , ( r ' ) = I f ( r ' ) , , , 1 . . . . . . , , , , + ( r I" (r) /4)]T, ,1 ,Er, lr ,2 - f ( r ) / ( 4 r ) r , T p q t r p r q r t ,
p ( r ' ) = , 1 , , , ~ h ( r ) T p q t r p r q r t , (A.6)
w h e r e / ( r ' ) , h ( r ' ) sa t i s fy
/ " ( r ' ) + ( 8 / r ' ) f ' ( r ' ) - 3 h ( r ' ) - r ' h ' ( r ' ) - ( A g ( r ' ) / ~ ) f ( r ' ) = XA(r')/7/, (A.7)
h" ( r ' ) + ( 8 / r ' ) h ' ( r ' ) + ( A ' A ( r ' ) / r l ) ( f ( r ' ) / r ' ) = ( - - A ' A ( r ' ) / r l ) l / r ' .
F o r r ' > a we have
f ( r ' ) = ( 2 N ' / 5 ) ( a / r ' ) 5 + p ' ( a / r ' ) 7,
h ( r ' ) = N ' ( a S / r ' 7 ) . (A.8)
The c o n s t a n t S ' a r i ses f r o m the n -- 2 a n t i s y m m e t r i c sou rce (3.5a)
S , ( r ' ) 4 , , , = (~)A,,~.,2r,,r, 2. (A.9)
F o r such a s o u r c e t he re is a van i sh ing p r e s s u r e field whi le the v e l o c i t y has the
fo rm
v , ( r ' ) = X ( r')A~,,.,2r;~r; 2, (A.10)
w h e r e
X " ( r ' ) + ( 6 / r ' ) X ' ( r ' ) - ( A ^ ( r ' ) / n ) X ( r ' ) = 4A^(r ' ) / (3rD. (A. 11)
F o r r ' > a we have
X ( r ' ) = S ' ( a / r ' ) 5. (A.12)
The c o n s t a n t s V', W ' a r i se f rom the n = 0 s y m m e t r i c s o u r c e (3.5a)
S , ( r ' ) (2r'2/5)h~qq ' x , = - ( r , / 5 ) ( h m r j ) . (A.13)
V e l o c i t y and p r e s s u r e a re
v , ( r ' ) = [ / ( r ' ) + r ' f ' ( r ' ) /2]h~qq . . . . . l , f ( r ) / ( 2 r ) r , ( h m r , ) ,
p ( r ' ) = r lh(r ' )h~qqr; , (A.14)
570 R.B. JONES
w h e r e f ( r ' ) a n d h ( r ' ) s a t i s f y
f " ( r ' ) + ( 4 / r ' ) f ' ( r ' ) - h ( r ' ) - r ' h ' ( r ' ) - (AA( r ' ) / r l ) f ( r ' ) = (AA(r ' ) / r t ) r ' 2 /5 , ( A . 1 5 )
h " ( r ' ) + ( 4 1 r ' ) h ' ( r ' ) + ( A ' A ( r ' ) / n ) f ( r ' ) / r ' = (--X 'A ( r ' ) l n ) r ' / 5 .
F o r r ' > a o n e h a s
f ( r ' ) = V ' ( a 3 / r ') + W ' ( a S / r ' 3 ) ,
h ( r ' ) = V ' ( a / r ' ) 3. ( A . 1 6 )
F i n a l l y w e o b t a i n Z ' , a ' f r o m t h e n = 1 s y m m e t r i c s o u r c e (3 .6a)
S , ( r ' ) = (5r '2 /7)S~pr ' , , 1 , , - ( 2 r , / 7 ) S p q r p r q . ( A . 1 7 )
V e l o c i t y a n d p r e s s u r e a r e
v , ( r ' ) [ f ( r ' ) + ' ' ' i . . . . . , , , = - [ ( r ) / ( 3 r ) r , S p q r p r q , r f ( r ) / 3 ] S , p r p p ( r ' ) , 1 , , ( A . 1 8 )
= r t h ( r ) S p q r p r q ,
w h e r e
f " ( r ' ) + ( 6 / r ' ) f ' ( r ' ) - 2 h ( r ' ) - r ' h ' ( r ' ) - ( A A ( r ' ) / r t ) f ( r ' )
= (AA(r ' ) /~q)(3r '2 /7) , (A . 19)
h " ( r ' ) + ( 6 1 r ' ) h ' ( r ' ) + (X ' A ( r ' ) / n ) Y ( r ' ) l r ' = -- (X k ( r ' ) l n ) 3 r ' 1 7 .
T h e s o l u t i o n f o r r ' > a is
f ( r ' ) = ( Z ' / 2 ) ( a S l r '3) + ot ' (aT/r '5) ,
h ( r ' ) = Z ' ( a l r ' ) 5. ( A . 2 0 )
References
1) B.U. Felderhof and J.M. Deutch, J Chem. Phys. 62 (1975) 2391 Also, B U. Felderhof, Physica 80A (1975) 63
2) J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, 2nd edn. (Noordhoff, Leyden, 1973) ch 6.
3) Dahl's result Is quoted in ref 2 4) J.A. Montgomery, Jr. and B.J. Berne, J. Chem. Phys 67 (1977) 4589 5) B U. Felderhof, Physica 89A (1977) 373
