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Bulk stress of a periodic suspension of solid particles of arbitrary shape
Fluid Dynamics North-Holland
Research
165
7 (1991) 165-179
Bulk stress of a periodic suspension of solid particles of arbitrary shape Yukio Kaneda Deportment Received
of Applied Physics, Nagoya
18 October
University,
Chikusa-ku,
Nagoya 464-01, Japan
1990
An analytical study is made of the bulk stress of a periodic array of solid identical and force-free particles of arbitrary shape in an incompressible Newtonian fluid. Asymptotic expressions are derived for the bulk stress of dilute periodic suspensions. Asymptotic expressions are also derived for a concentrated suspension of almost-touching ellipsoids at an instant when the suspension has orthotropic symmetry.
Abstract.
1. Introduction
This paper treats the problem of determining the bulk stress of a periodic array of particles in a Newtonian fluid. So far studies have been made of the bulk properties of a periodic suspension of spherical particles. For low volume concentration 6, Kaneda (1973) has obtained expressions of the energy dissipation up to O(Q~) of periodic suspensions of solid spheres and of spherical drops for cubic lattices. The dissipation in such suspensions depends on the orientation of the lattice with respect to the flow. Zuzovsky (1976) and Zuzovsky, Alder and Brenner (1983) have derived expressions of the effective viscosity up to O(G~“~) of periodic suspensions of solid spheres for cubic lattices. For high concentrations, Frankel and Acrivos (1967) obtained the leading-order term in the asymptotic expansions for one of the two key viscosity coefficients in a simple cubic lattice consisting of solid spheres. Zuzovsky (1976) corrected their result and also gave the leading-order term for the other coefficient. Nunan (1983) and Nunan and Keller (1984) have obtained additional terms in these expansions and a general expression for arbitrary lattice geometry. The bulk properties of periodic suspensions of solid spheres not limited to low or high concentration have been studied numerically by Nunan (1983), Nunan and Keller (1984) and Brady et al. (1988). It is impressive that their numerical results agree well with the analytical asymptotic results not only for very low or high concentrations but also for quite a wide range of $I. Since the suspended particles are not spherical in many real suspensions, it is interesting to study suspensions of non-spherical particles. Recently, Tran-Cong, Phan-Tien and Graham (1990) have numerically studied the bulk properties of a monolayered periodic suspension of solid spheroids bounded by two flat plates, where the array of the spheroids is periodic in only two directions parallel to the plates. In the present paper we study analytically the bulk stress of infinitely extended periodic (in three directions) suspensions of solid identical force-free particles of arbitrary shape. After reviewing in section 3 some characteristic features of the disturbance field due to a single particle in an unbounded linear shear flow, we consider in section 4 the bulk stress of dilute periodic suspensions, and derive asymptotic expressions of the bulk stress up to O($). The method used in section 4 is basically the same as in Kaneda (1973), in the sense that the 0169-5983/91/$4.75
0 1991 - The Japan
Society
of Fluid Mechanics
approximation of the disturbance due to the suspended particles as a sum of the derivatives of Hasimoto’s periodic Green function (Hasimoto. 1959) plays the key role in the analysis. The cases of periodic suspensions of orthotropic particles are studied in some detail in section 5. Finally, some typical cases of concentrated suspensions of almost-touching ellipsoids are analyzed in section 6.
2. Basic equations We consider a homogeneous,
a periodic isotropic
array of small identical rigid particles Newtonian fluid that obeys
of arbitrary
shape immersed
in
p he-vp=o.
W
‘c7 lv=o.
(lb)
where v, p and p are the velocity, the pressure and the viscosity of the fluid, respectively. The particles are neutrally buoyant, and their instantaneous positions are assumed to be at the lattice points r “=m,a”‘+m,a(“)+mi,(i)
(m,,
m2, m,=O,
i:l+2
,___ ),
where u(l), a(‘) and a(‘) are the basis vectors determining the unit cell of the array. The orientation of each particle is the same, and the surface of the particle at rm may therefore be expressed as g( r - P) = 0 by a certain function g which is the same for each particle. We assume that the velocity gradient is spatially periodic so that v may be written as U, = y,,r, + ui, where and II is also The
(2)
the summation convention for repeated indices is used, yI, is a constant traceless tensor, is a spatially periodic vector. Equation (1) then implies that vp is periodic, and p itself periodic because no external force is exerted on the particles. no-slip boundary condition on the surface S” of the rigid particle located at P requires u,=y,,(P),+
(2),-t
[wX
(r-P)],,
on S”,
(3)
where y,,(P),, + (v”), is the velocity of the material point of the particle coinciding instantaneously with rm, and w is the angular velocity of the particle about r”‘. Because of the periodicity, b’ and w are the same for each particle. Similarly, the force f and torque L exerted on the fluid by the particle are the same for each particle. Let “PI, be the volume average of the stress over the unit cell V of the lattice containing the particle B at the origin. It is then shown (Batchelor, 1970; Nunan and Keller. 1984) that
and the bulk deviatoric D,J = AX,
stress D, i is therefore
given by
+ Y,J + o,“,*
where S is the surface of the particle B, 7 the volume of the unit cell V, a,, the stress tensor due to (0, p), and n the outward unit vector on S pointing into the fluid. The tensor D,“/ represents the contribution to the bulk deviatoric stress due to the presence of the particles, and is called the particle deviatoric stress or simply the particle stress.
Y. Kaneda
The decomposition
of tensors
167
/ Bulk stress of a periodic suspension
y,, and D$ into the symmetric
DI”/= S,, + (1/27)e,,kLi,
Y,, = e;j-
and antisymmetric
parts gives
crjkfikr
(5)
where Si, = (1D; + D,$‘2,
L/r
ei, = (Y,, + Y,, )/2,
9, =
= E,/@$, -E,JkYlk/2-
The particles are force-free so that the force exerted on the fluid by each particle is zero. It is not difficult to verify that the specification of the torque L or the angular velocity o is then sufficient to uniquely determine the periodic field (u, p) and the particle stress 0,; for any given tensor y,,. If the torque L and the tensor yij are given, then, because of the linearity of the problem, the symmetric part of the particle stress may be written as %j = @&rYpq
+ :
c,,k
Lk
(6)
)
and traceless in their first pair of where the constant tensor CIjP4 as well as Cijk are sy~etric indices. The first term on the right-hand side of eq. (6) represents the contribution due to the field satisfying eqs. (1) and (2) for given yjj with f = L = 0, while the second term is the contribution due to the field satisfying eqs. (I) and (2) for given L with yII = 0, f= 0. If the symmetric part elj of y,, is zero, then the velocity of the former field is equivalent to a rigid rotation and the field makes no contribution to the particle stress. The term y,,, in eq. (6) may @ per unit volume is therefore be replaced by the symmetric part epy. The energy dissipation given by (cf. Batchelor, 1970)
3. Disturbances
due to a single particle immersed in an unbounded linear shear flow
We now consider a neutrally from which the velocity gradient as r=
14 -+ v,,r,
buoyant single particle immersed is uniform and given by
V
is taken at a point
P=Po+p’,
where p. is constant and u’, p’, The disturbance fields satisfy p Au’ - vp’
=
fluid,
far
Irl --f co,
where the origin of the coordinate U; = Y,j’, + u:,
in an unbounded
CT’
a,/=
inside
the particle.
Let us write
-P$,j+2Pe,j+a,l,,
are the disturbances
(8)
due to the existence
of the particle.
0,
(9a)
lu’=o,
(9b)
and U’-+O, The general
p’-+o
solution
asr-+co.
u’=v(r*+)/2-cp, where 4 is a harmonic outside the particle as
(10)
of eqs. (9) and (10) are given by (Imai, 1973; Hasimoto p’=i_1v
vector satisfying
~~=((Ai+Aipap+Aipgapaq+
and Sano, 1980)
-tp,
A+ = 0 and may be expanded -**)S{,
(111
by spherical
harmonics
(12)
Y. Kunedu / Bulk stress of a periodic suspension
168
where S; = l/r
satisfies
hs;
= -M(r).
The fundamental
solution
P Aq; - a/,
(13) (U,:, P:) of eqs. (9) and (10) satisfying = Q(r),
a,u,; = 0, is obtained
by retaining
only the first term and putting
A, = 6,,/4~~
in eq. (12) as
The field (r/;:, p,‘) represents the disturbance due to a point force (Stokeslet) in the jth direction at the origin. The coefficients A,, A,,, . . . in eq. (12) as well as the field (u’, p’) satisfying eqs. (X), (9), and (10) are in general uniquely determined for any solid particle by specifying y,,, the force f and torque L exerted on the fluid by the particle, where f, = - k,;n, It is known
dS,
L, = --~,~j$lplli
dS.
that A, and A,, are given by -47&A
=f,
so that A;, = 0. For a force-free particle, we have f = A = 0. Because of the linearity of the problem, A,, is then linear in y, j and L, and may be expressed in the form [cf. eqs. (5) and (6)]:
-4al*.A,,
= PC,‘,,&
+ ‘z(C,;J+
where the constant tensor C,‘,,, as well indices and is determined by the shape, tensor y,, in eq. (17) may be replaced In terms of the Stokeslet field given written as
p’=(A,,t$+A
,py $3,
+ e,,~%)r
(17)
as C,:, are symmetric and traceless in their first pair of size’and the orientation of the particle. The shear rate by ePy as in eq. (6). by eqs. (14) or (15), (11) with (12) and A = 0 may be
+ . . * )(4WP,‘)’
(1Xb)
sides of eqs. where in deriving eq. (18a) we have used A,, = 0. The first terms on the right-hand (18a) and (18b) (A,, terms) represent the field due to the so-called stresslet at the origin.
4. Dilute periodic suspension Now let us consider the periodic array discussed in section 2. One might suppose that the periodic field (u, p) satisfying eqs. (1) and (2) may be given by the form (18) with U’, P’, S;
Y. Kaneda
replaced
by the following u,,(r)
U, P, S,:
= Cl,$(r-r”), m
P(r)
= Jp’(Y-P), m
S,(r)
and with appropriate coefficients A,,, A,,,, . . . However, because the sums are divergent and not well defined. Following Hasimoto (1959) we define here the fundamental P AQ, - a,?, = Ci&,,~(rm a$;,
the
= cs;(r-P), m
form
(19)
(19)
is inconvenient,
field U, P, S, by t 204
%I,
(2Ob)
= 0,
AS,= instead
169
/ 3uNc stress of a periodic s~~en~io~
-47r
c+P)-;‘, i m
(2Oc)
i
of eqs. (13) and (14). The solutions
of eqs. (20) are given by
where ‘exp( -2aik.P)
,
k2
k-t@
,=-L
c
’ exp( - 2vik
* r)
k4
4T3T kiO
’
and k = m,b”’ + m,b’2’ + n@(3) is a vector in the reciprocal following expansions: S, = l/r
lattice,
- c + .. *(
(m,,
and
m2, m3=
b(‘) = cilx(~(‘)
0, il, X
+2,...),
atk’)/2r.
For small
Y, we have the
s* = r/2 - Cl + * . * )
and ~R/.LV, = 4a~1.q.; + K,, + K,,,r,
f K,Jmnr,,,r,, t
...,
(21)
on the lattice geometry. where c, ~2, Kilt K,,,n,, . . _ are constants depending plays an important role in the following analysis. We may assume without that K,imir is symmetric in m and n so that it is given by K i,““’ = 27~ 3, 4$4,(ri
- V:bi]
The tensor Kii,,,, loss of generality
I r=n = Ki,,nn + K&w
(22)
where K,:,,
= -
!is,j
I,=*?
aflz an[sl(r)-l/r]
K2r,mn = t a, a, a, a,[s,w
- ~21
I r=O.
Let A, = A:, A,, = Al,, . . . be the coefficients such that (u’, p’) in eq. (18) is the solution of eqs. (8), (9) and (10) for a force-free particle with given y,, = vii and L = i, i.e. the field (u, p) given by ~1,= 7, Irl + 24:)
p=pO+pr>
~~=(~;~~~+~~~~a~a~+
p’=(A;,
a, apt-A
;,,
. ..)(4~~1.!.)-~(A;,,+A;,,) a, a, a, +
satisfies eqs. (8), (9), (10) and the no-slip the particle and gives torque E.
a,a,+
. . . )(47~~;),
and force-free
. ..is.,
(234
(23b) boundary
conditions
on the surface
of
Now we assume that jut” 1 - jac2’ / - 1a(j) 1 - I is much larger than the characteristic length scale a of the particle, i.e. c:= a/! --K1, where the symbol - denotes the equivalence in order of magnitude. Then, since r/& given by eq. (20) is determined by p and the basis vectors consideration gives that ai, a2, a3> all of which are of order I, the diniensional K,,, = 0(1/Y).
K,,,, = O&P).
K:i,nn = 0(1/1”)..
..
Similarly, since c2 is the only representative length scale of the disturbance a single particle, the dimensional consideration gives that A.LV =O(A*u),
A;,<,,=O(A*u~)
0‘4) field (u’, p’) due to
,....
(25)
and also c,;,,
= tt(CI”).
C,>,* = O(I).
where A * is the characteristic from eq. (17) PA* -F*a3e*
magnitude
of A:; (e.g., it may be given
by A*’ - AI,A:,),
and
+ IL/,
(26)
in which e * is the characteristic magnitude of Cli = (Uzi + V,j)/2. The expansions (21) and (23), and the estimates (24) and (25) suggest solution of eqs. (1) and (2) for small t: may be obtained by putting
that an approximate
L’,= y,,r, + u,, U, = c; + ( AiI, $ + A.;,, a,, 3, + * . * )t4TU,_/)
- [ ( Aip, + Ai,jp) ai ap + ’ ’ . ] sl/2, j27a)
P=(A~~~~+A~~Yi?na,+
*‘.)(4Tr/&P),.I
f27b)
provided that c and r,, and z are chosen appropriately, where c is a constant vector. Once ;Ji and E are given, the coefficients A;,,, A:,,,... , are uniquely determined from them for any given solid particfe. Since eq. (27) may be written as
- [(Aip,+A:jp)a, P=~‘+(A;~ we obtain
a,+A;p,
a,+ $,a,+
***](SI-S;)/~, I. ’ )[4TPfpj-P,‘)]v
(2Xa) (2fQ7)
from eqs. (24) and (25) that for i’ - LI, 0, = y, ,‘, + #: -t (c, + II,) i Rl,Yj -k O(A*a’/P), olj = u,; -t ,u(2eti i B,, -I- B,,) + &,d+
~~~A~~,/~4),
C29a) t29b)
where
B, and d are appropriate constants, and IJ,~ and ul: are the stress tensor respectively. So far. the field (u, p) in ey, (27) or (28) may be defined for arbitrary chose c, ;i,/ and E such that c + B = 0, E = L and
due to (u, p) and (u’,
p’),
Yr,,+ 3, { = 7, JI i.e.
c, r,i and E. Now we
Y. Kunedu
171
/ Bulk stress of a perrodic suspensron
then the field (t’, p) satisfies the force-free and no-slip boundary condition on the surface S and yields torque I. within the approximation neglecting terms of 0(,4*a2/L4) in eq. (29a) and 0(~~*~/~4) in eq. (29b), because the field (D, p) given by eq. (8) with y,, replaced v,, satisfies these conditions. The substitution of eq. (29) into eq. (4) yields D:;-t~~[(~,;~,-is,,a,~~~)~*-p(l(:ni+n;,i,,]
dS+0(~A*e4/~).
Because of eqs. (16) and (17), this may be written 470:, D,; = - ~ 7
+ 0( pA*c4,+) + &(c,&
= 5 [PC/,&q Substitutillg
as
+ ‘,,&,)]
+ O(/_6A”E4/T).
(31)
eq. (30) into eq. (31) and using again eq. (17) yields
$&C&L, +ti,xL,)
+ O(yA”IE4/7-).
(32)
In the following, we consider the case when the torque z is at most O(poae*). This condition is satisfied not only when each particle is torque-free, but also when the orientation of each particle is kept fixed by external torque. We then obtain from eq. (26) that A* - a3y*, so that eq. (24) gives (K,,,,z,> + K,,prt,)&p Hence, P,, on the right-hand and we finally obtain D,~=PC,~,~~,,,+
= O(Y*~~) side of eq. (32) may be replaced
:(C,,kL,
+~-‘c~~ALA)
by ePrl within
our approximation,
(33)
+PY*O(~~)~
where
X 1 ii’4
=
(34b)
Y,A
y&C,‘,c,h ( Koca + Kud
(34c)
7
(34d)
Y,,c = & C~:a~,(Krr~<~h+ Kildc/>)C<;k.
Equations (33) with (34) are the main result of the present paper. For dilute suspensions, i.e. and T,,/T are 0(c6) = O(+‘). for e ==z1, C,;,, /T and C,‘,,/r are O(c’) = O(+), and X,,J7 The tensor Kuhcd is determined solely by the lattice geometry irrespective of the particle solely by the particle geometry irrespective of geometry, while Cl’,P, and C,:, are determined the lattice geometry.
5. Dilute periodic suspension The tensors
of orthotropic particles
C,:,,,, and C,;n, in eq. (18) in general
C;),,,,, = C,:,,, 3
satisfy
the symmetry
relation
and C,:,,,,, = 0,
c,:, = 0
We may assume without loss of generality that C,‘,,,,,,, = 0 since only traceless tensor is significant. Hinch (1972) has shown that then
the contraction
with a
C’i ,““’ = cm j . The form of C,:,,,,, may be further simplified when the surface of the particle has orthotropic symmetry, i.e. when it is symmetrical about each of three orthogonal planes. We take here a coordinate system in which the principal axes of the particle coincide with the coordinate axes. It is shown, in appendix A. that C,:,,,,, is then zero unless the indices i. j. PI, n are equal in pairs and also that C,:h is zero if any two of indices i, j, k are equal. The tensors C,‘,,,,,, and C,‘,x satisfying these relations must be of the following form: 71 C 1,‘,I>’= [ &I 6 , pm +(.--)+(...)]+[~;(P,,Pm,,l~,,~~Ql,)+(...)+(...)] (35a)
+[P;(P,Y,+p,q,)(l,,,4,,+~,IY,i)+(...)+(..’)],
(35bJ
C,‘,, = ~ilh. (47, -t Lf,r, ) + ( . . I > + ( . I. f + p, q, P are the unit
where G?, C;;:, /3,‘. y,’ (i = 1. 2, 3) are constants, principal axes, and
Q,,=w-~,,/~~
P,,=P,P,-4,/J,
vectors
parallel
to the
R,,=r,r,-&,/3.
Since Pii +
Pi, + R,, = 0,
and therefore
f’i,Qtn,, + f’n,,,Q,, = R, /R,,,,, - P, ,P,,,,, - Q, ,Qnr,,. etc., eq. (35a) may be put in the form
C’i ,r,in=
[4P,,P,,,, -I- &Q,,Q,n,, + dft,,R,,,,,]
+ [8;iu,q
+ Pi ( r, P, + r, p, ) ( r,,,P,, + 6, h, 1 + P; ( ~~4,
+ ~,~)(~,~,~, + cl,r,,)
+ p,4, I( p,d,, + ml,,, )J .
(36)
The expressions (35) and (36) are independent of the particular choice of the coordinate system. The form of the tensor K,,nl,i may be similarly simplified when the lattice has orthotropic symmetry, i.e. the configuration of the lattice is symmetrical about each of three orthogonal planes. Then, in the coordinate system with the coordinate axes coinciding with the principal axes of the lattice, K ,,,, I,, is zero unless the indices i, j, YYI,11 are equal in pairs. If the lattice has further cubic symmetry, then the tensors K’ and K’ are in general of the form kxlI ,,,iw
=
f?s,
,L
>
&2,,,,,
=
@z
/,,i,,
+
t(
4,L~
where l, g and Q are constants, and Sijmrr is unity For a cubic lattice. S, and S, may be expanded 1
S,=;-c+~;r’+
2m
+
L&
+
Lo]“’
>’
if i = j = m = n and zero otherwise. for small r (Hasimoto, 1959) as
...,
Sz=f-~,+~T?+~r4+b2~~Y~‘(x1.x2,_~~)+h~~Y~(xI, where hzo and h2i are constants
x2.x:7)+--.,
satisfying
b,, = h,,/168 (Zuzovsky,
1976)
and
Y,“‘(x,, x2, .x3) = r”P,:)‘(cos 0) cos rn+ is a solid
harmonic
in which
173
Y. Kaneda / Bulk stress of a penodic .su,spensjon
xi = r sin B cos +, x2 = r sin 8 sin +, x3 = r cos 8. It is shown sions that Kucdh + Krrd<,, = E&M + 9 (%,,&, + &$,,.)
+ 2&,&,
from eq. (22) and these expan-
(37)
>
where q=
5 = 60b,,,
-2l'l/5T-
12b,,.
The value of b,, has been calculated lattice (BCL) and face-centered cubic Zuzovsky (1976), f&l
= -0.195
(SCL).
= 0.120
@CL),
= 0.213
(FCL) >
for simple cubic lattice (SCL), body-centered cubic lattice (FCL). According to Hasimoto (1959) and
where I= la(‘) 1 = jac2)1 = 1~~~)I and 13/7 = 1, 2 and 4 for SCL, BCL and FCL, respectively. The expression (34) can be most simplified when the principal axes p, q, r of the orthotropic particles coincide with the principal axes of the cubic lattice. Then, substituting eqs. (35b), (36) and (37) into eq. (34) yields after some algebra that
A(G,P,,&,+ GzQ,,Qm,,+ W,,L,)
X ‘,,,r,, =
+ B [ p;“( q,r, + q,r,)( q,r, + x,,
= B f /3;y;p,
q,r,,)
( qir, + q,r, ) + two similar
+ two similar
(384
terms],
terms] ,
(38b)
where &+2rl_
Ii 157
12n
/ 3&o 71 ’
G,=2(a,“-+;)+I& H = +I;
G,=~(cI;~-oI;cY;)-~H,
-rl
’ G,=2(n;‘-+;)+I$,
+ a&; + a$~;.
For example, let us consider Then eq. (7) gives
Similarly,
B=11=_L_12b20 71 Sr
the case when
y,, = e,, = P,) - Qi, =pipi
- q/q,
and
L = 0.
when y, i = ei, = 4,rj + q,r, and L = 0, (39b)
Example: cubic lattice of ellipsoids. The ellipsoidal particle has the orthotropic discussed above, and the coefficients (Y:, fi,‘, y,’ may be known from Jeffery’s follows (Jeffery, 1922; Batchelor, 1970): I 45, -2GF= a=&_
3(f,Jz+JzJ~+JjJ1)’
4nabc
2 = 31, ’
62 - c2 ‘I=
-
b2+c2'
etc.,
symmetry solution as
174
Y. Kaneda
/ Bulk stress
of a periodic
suspension
Table 1 Viscosity coefficients
0.1
0.2 0.5 1.0 2.0 5.0 10.0
2.091 1.463 1.301 1.667 2.834 7.985 20.776
6.640 3.829 2.178 1.667 1.449 1.357 1.340
3.320 1.915 1.089 0.833 0.725 0.678 0.670
0.413 0.492 0.688 0.833 0.835 0.739 0.696
where a, b and c are the semi-diameters of the ellipsoid and I,, I,, I, and J,, J2, J3 are integrals like I, =
= sa
ubc(b’+
73
c’) dh
J1= / o
A(b*+X)(c’+h)’
- 0.980 - 0.923 - 0.600 0.000 0.600 0.923 0.980
in the p, q, r-directions,
respectively,
ubcX dh A(b’+h)(c’+X)
where A2=(a2+h)(b2+X)(c2+h). particle is thus equivalent to In considering the viscosity tensors C,:,, and C,:,. any orthotropic an ellipsoid, provided that the coefficients (Y:, ,8,’ and y,’ (i = 1, 2, 3) can be chosen properly. y;= -y; and y,‘=O.The For spheroid particles with b = c, it is clear that (Y;=cxi, pi=&‘. values of 6,. E:2, &. & and T? are listed in table 1 for values of u/b. For a solid spherical particle of radius u, we have LYE = ai = ai = 2EI/;,, fl,’ = ,& = & = E&, y, = y2 = yj = 0, where E = 5/2 and V, = 4au3/3. Equation (36) then reduces to ; C,‘,,, = E+(QJy and eq. (34) or (38) reduces
y,,: =
(40)
+ %,6,,, - fi&,$,), to
(41b)
0,
where (Y= 3A(2E1/,)*/~= /3 = 2B(EV,)2/7=
-2~5~$~(+
@/2w,,e,, =
+ 12b20++
-2E2#($
in agreement with Zuzovsky (1973) particles. Also, eq. (39) reduces to
- 18b,,++
where
$ = Vo/r
1 + E$ [ 1 + E+( $ - 18b2&r)],
for y,, = e,, = p, p, - q, q, and
is the volume
concentration
of the
(434
L = 0, while
@/2Pe, ,e,, = 1 + E+[l
+ E+(+ + 12b,,+)],
(43b)
for Y,, = e,, = (q,r, + q,r,) and L = 0, in agreement with Kaneda (1973) (in which only the case of simple cubic lattice is mentioned, but a generalization to include the cases of the other kind of cubic lattices is straightforward). In concluding this section, the case of a periodic suspension of spherical drops may deserve comment. It was shown in Kaneda (1973) that for a periodic suspension of spherical drops of a
175
fluid with viscosity
j& @ is given by eq. (43) but with
instead of E = l/2. From this result, the viscosity tensor be given by eqs. (33) (34), (40) (41) and (42) with (44).
6. Concentrated
periodic suspension
CiJPy in such a suspension
is seen to
of ellipsoids
We consider here a concentrated periodic suspension of almost-touching ellipsoids. The analysis is most simplified when the principal axes of the ellipsoids are parallel to the basis the fundamental unit cell. When the suspension has this vectors a(‘), at2), and af3) determining kind of symmetry, it can be shown in the same way as in deriving eqs. (35) and (36) that the tensors C, ,py, C,,, are in general of the form c ,,,nn = (aipl,jpl,t,l
+ a~Qt)Qnzn+ ~3R~~‘rnnI
+ [ &( q,r- -I-q,r,)( q,r, + qnrm) + two similar C,jk = y, pk (q,r, -k q,r,) + two similar
terms],
terms.
(45a) (45b)
Let us take the coordinate axes parallel to p, q and r, and let li = /a”’ 1, cl = 1 - (2a/i,), Ed = 1 - (2b/f,). e3 = 1 - (2c/i,), where a, b, c are, respectively, the semi-diameters of ellipsoids in the p-, q- and r-directions. We consider here only the case when f3 -=X1,
c,, E2 = O(1).
The cases when not only f3 but also ei and/or e2 are/is very small can be treated similarly. The coefficients in eq. (45) may be estimated by considering the following three cases. The outline of the analysis is sketched in appendix B. Case 1. When y,j = Pi, - R,, = p,p, - r,r, and the motions of the particles are given by eq. (3) with v” = w = 0, the particles are force- and torque-free. The energy dissipation Q, per unit volume is then given by
where 7 = f,i,f,. From the symmetry consideration or by considering the case when Y,~= Qi, R,, and v* = w = 0, we obtain a similar expression for cuz + (Ye, while if y,, = Ri, - Ptj and v0 = w = 0, then @ = O(r’), so that 0~~+ (Ye= O(<‘). From these estimates, we can get the estimates of (Ye, (Y>and LYE.It is seen that (Y,, a2 +z (Ye- l/h. Case 2. When y,, =p,r, -t-Fir, and the particles @/21*e,,e,,
are force- and torque-free,
@ is given by
= p* = O( 8).
From the symmetry consideration we obtain a similar expression the case when y,, = p,q, + p,q, under the force- and torque-free obtain @ = O( E’) so that & = 0( co).
for &_ Finally, by considering conditions of the particles, we
Case 3. When y,, =p,r, +pjr, and the motions of the particles are given by eq. (3) with v0 = w = 0, the particles are force-free. The energy dissipation per unit volume and the torque L
Y. Kane& / Bulk stress of a periodic
176
exerted
on the fluid by one particle @
are then given by
aabc = --log 7
I5L.q
+e,.,e,,
2~
suspension
c:,
L = (itrr&uabc log e)q. Hence, we obtain yz = - l/27. Similarly, by the symmetry consideration, we obtain y, = l/27. Finally, the estimate of y3 may be obtained by considering the case when y,, = 0, 0 = 0 and w = r. The particles are then force-free and the torque and the dissipation are O(E’). The energy dissipation is then at most O(cO), from which we obtain yj = O(r”).
Acknowledgement I wish to express my cordial thanks to Professor H. Kimura for stimulating liquid crystals, which have revived my interest in the present problem.
discussions
on
Appendix A. Symmetry relations of C&,,, and C;6 The symmetry relations of C,‘,,,,, and C,lk can be obtained in a way similar to that used by Brenner (1964) Zuzovsky (1976) and Nunan and Keller (1984) as follows. Let T = { T, } be any orthogonal linear transformation, and i, d, fi, j?~be defined by i=Tr,
;(;)
=Tv(r),
;(;)
= TM(~),
j(F)
‘P’(Y),
then G,(i)
= ~~~~~~~~j+
G,(F),
(A-1)
where (G, $) satisfies p 8;-Ojj=o,
(A.2a)
e lD=O,
(A.2b)
and ir -4 0,
B-0
as ?=
Ii1 + z~,
(A.3)
in whiche,=3j=El/a?,, Asa, 3,. Since eqs. (A.I), (A.2) and (A.3) are of the same form as eqs. (8), (9) and (10) but with v,, replaced by qpl;uUpu> we may write, in the same way as in eq. (17),
where Cijnln and C,,k are appropriate (r,,(;)-$,G,(i)+^a,ti,(i)]
constant
tensors,
-S,,,b(f),
and ri=Tn,
i, = -41.r~E,,kajk. On the other hand, since A,, and tor, we have in general i$, = 7;,7;,A,,,
L, are respectively
i, = IT j 7&,,
a second-order
tensor
and a pseudo-vec-
(A-5)
Y. Kaneda / Bulk stress
where
of a periodic surpension
11-i
IT 1 = det T. Hence, eq. (A.4) gives * ,. - +b(~,mITITmk~k+~,,,ITITrnx~k). -4V%, = C!/rn?lTmpTnqYpq
Now suppose
T maps
that the transformation
c’,,mn = &?In,
of the particle
onto itself. Then,
c,‘,, = C,,, 3
so that from eqs. (16) (A.4) -4nM,,
the surface
(A.6)
(AS)
= C,;,,U,,
and (A.6)
+ :C,&
•t te,,,&
= TP,T4,( C;~mnTmp~qYpq + :Cd,r
IT I LA)
+ &Ldw
Since this holds for any u,, and L,, we have C,;,,
>
= Tpr7’&rnLCP’~o~
and C,:,, = T$T&T,,
IT I Cp’qr.
If the surface of the particle is symmetrical about the plane orthogonal i.e. invariant under the transformation 7;, = (- l)sL,a,i, then c,:,,
= ( - 1) “C&
c,:rn = w>“c,~m~ Hence, if the surface of in the coordinate system C r,mn is zero unless the i, j, m are all different
Appendix B. Asymptotic
a = a,, + S,, + S,, + S,, ,
3
b = 1 + S,, + S,, + S,, .
the particle is symmetrical about each of three orthogonal planes, then, whose coordinate axes coincide with the principal axes of the particle, indices i, j, m, n are equal in pairs and Crrn is zero unless the indices from each other.
analysis of flow in a narrow gap between two ellipsoids
We consider the flow in a narrow and S- are given by -2
&+P+ a2
b2
to the k th direction,
(TfD)2 c2
gap between
two ellipsoids
B+ and B-, whose surfaces
Sf
=l,
where ~=l-cC/D
i?=u’+w’x(F-r*),
be analyzed by a slight spheres (Zuzovsky, 1976;
on S’,
(B.1)
where F = (X, j, Z), fi = (U, U, W) with U, U and W being, respectively, the velocity components in the X, j and F directions, v’ and w* are the translational and angular velocities of the particles B ‘. and r ’ = (0, 0, f 0). It is convenient to introduce the following stretched variables x=c
y=c P’/2DP1X, u = D-‘U “CD-$j ,
where p is the pressure.
-1/2D-1-
We consider
w=c
Y?
z zz E-lD--lF
-1/2D-1, 9
here the following
p
=
p-
1~3/2~
three typical
cases separately.
Y.Kaneda / Bulk stress of a yrriodic suspension
178
Case I. When
o*= (0, 0, f D) and w*= 0, eq. (B.l) yields the boundary
u=u=o,
w = _t~-‘/~,
Hence we assume
the following
these expansions
ape a2uo
-=-
ax
az
Ql, w(), PO) + E”2(U,,
+, Wl, Pl )+
...
into eqs. (3) and (B.l) yields
aPo
a%, aPo
aY
az2,
-=-
2'
on S’.
form of expansions:
(u, U, w, p) = ~-“+(), Substituting
condition:
o
-=
az
1
!!!5+!!3+$+),
ay
and UC,= ua = 0,
wo= t-1,
on z=
kH(x,
y),
where
y)=l+i
H(x, These conditions
g +g
i a
are satisfied
S) = (a/D, b/D).
(a,
) i
by the following uo=
1 ape uo= -p(z’-H2),
wo=f
1
solution:
ape
z7&-(z2-H2), Apo(H2-~2)H+~,
2 ay
for which the energy
Case 2. When
u= Hence
dissipation
6 in the gap is given by
v*= (+ D, 0, 0) and w*= 0, eq. (B.l) yields the boundary +1,
we assume
u=w=O,
0nS’.
the following
form of expansions:
(u, u, w, P> = (UO> ql, w,, PO) + C(%> 013 WI? PI) + . . . Substituting
these into eqs. (3) and (B.l) yields
aPo azuo ax
ape a2uo ape -
-=-
-=_
az2 7
ay
az2 ’
0
BL -
2
and 24()= L-1, These conditions u.
=
u() = w, = 0,
are satisfied
I
H’
by the following
ug ‘PO = 0,
for which the energy dissipation 6 = - 4q.wbc
on z = &H(x,
log C,
1aH WO=Tax
y).
solution: --z2 i
H2
1 1
6 in the gap is given by
>
condition:
179
and the torque
2 exerted
on one particle
by the fluid in the gap is given by
f. = - (2n/Aahc log f)il. Case 3. When v*= (i: D, 0, 0) and toi= (0, I, O), eq. (B.1) yields G=w*x);) This boundary
on Sk,
condition
CZ@‘Xi;
3
for which the dissipation
and eqs. (3), (4) are satisfied
by
p = 0, @ in the gap is zero.
Finally, we consider the periodic array of ellipsoids considered in section 6 for which we may take 20 = 13. Let w’ be the angular velocity of the particles for which the particles are torque-free under given yiI. Because of the linearity of the problem, ec is linear in yii and L may be written in the form
where Tlk is a constant tensor. By considering the difference field between the fields of case 2 and case 3 [for which y,j = 0, uc = 0, w = (0, 1, O)], it is seen that TJk = O(log E). Since L = O(f’) in case 3, we have (0, 1,O) = wc + O((log z)- ‘) for y,, =pir, + p,~~. Moreover, we have @ = C&r”) if y, i = p,r, -I-p,r, and if the particles are torque-free, because cfi = Oft*) in case 3 and the difference between the angular velocities w” and w = (0, 1, 0) in case 3 is @(log E:$- I) as shown above.
Batch&x. G.K. (1970) The stress system in a suspension of force-free particles, J. Fluid Mech. 41, 545-570. Brady, J.F., R.J. Phillips, J.C. Lester and G. Bossis (1988) Dynamic simulation of hydrodynamically interacting suspensions, J. Fluid Mech. 195, 257-280. Brenner. H. (1964) The Stokes resistance of an arbitrary particle-III. Shear fields, Chem. Eng. SC;. 19. 631-651. Frank& N.A. and A. Acrivos (1967) On the viscosity of a concentrated suspension of solid spheres, Chem. Eng. Sci. 22, 847-853. Hasimoto, H. (1959) On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres, S. F&id Mech. 5, 317-328. Hasimoto, H. and 0. Sano (1980) Stokeslets and eddies in cr&%ping flow, Ann. Reu. FItlid Me&. 12, 335-363. Hinch, E.J. (1472) Note on symmetries of certain material tensors for a particle in Stokes flow, J. Fluid Me&. 34, 423-425. Imai, I. (1973) Ryuutai Rikiguku (Fluid Mechanics) (Shokabo, Tokyo) fin Japanese& Jeffery, G.B. (X922) The motion of ellipsoidal particles immersed in a viscous fluid, Prof. Roy. SK A f&T. 161-179. Kaneda, Y. (1973) Viscous flow through porous media, M.S. Thesis, Tokyo University [in Japanese]. Nunan, KC. (1983) Effective properties of composite media containing periodic arrays of spheres, P/XL). Thesis, Stanford University. Nunan, KC. and J.B. Keller (1984) Effective viscosity of a periodic suspension, J. Fluid Mech. 142, 269-287. Tran-Cong, T., N. Phan-Tien and A.L. Graham (1990) Stokes problems of multiparticle systems: Periodic arrays, Phys. Fluids A 2, 666-673. Zuzovsky, M. (1976) Transport processes in spatialIy periodic suspensions, Ph.D. Thesis, Carnegie-Mellon University. Zuzovsky, M., P.M. Adler and H. Brenner {1983) Spatially periodic suspensions of convex particles in linear shear flows. III. Diiute arrays of spheres suspended in Newtonian fluids, Phys. &ids 26, 1714-1723.
Research
165
7 (1991) 165-179
Bulk stress of a periodic suspension of solid particles of arbitrary shape Yukio Kaneda Deportment Received
of Applied Physics, Nagoya
18 October
University,
Chikusa-ku,
Nagoya 464-01, Japan
1990
An analytical study is made of the bulk stress of a periodic array of solid identical and force-free particles of arbitrary shape in an incompressible Newtonian fluid. Asymptotic expressions are derived for the bulk stress of dilute periodic suspensions. Asymptotic expressions are also derived for a concentrated suspension of almost-touching ellipsoids at an instant when the suspension has orthotropic symmetry.
Abstract.
1. Introduction
This paper treats the problem of determining the bulk stress of a periodic array of particles in a Newtonian fluid. So far studies have been made of the bulk properties of a periodic suspension of spherical particles. For low volume concentration 6, Kaneda (1973) has obtained expressions of the energy dissipation up to O(Q~) of periodic suspensions of solid spheres and of spherical drops for cubic lattices. The dissipation in such suspensions depends on the orientation of the lattice with respect to the flow. Zuzovsky (1976) and Zuzovsky, Alder and Brenner (1983) have derived expressions of the effective viscosity up to O(G~“~) of periodic suspensions of solid spheres for cubic lattices. For high concentrations, Frankel and Acrivos (1967) obtained the leading-order term in the asymptotic expansions for one of the two key viscosity coefficients in a simple cubic lattice consisting of solid spheres. Zuzovsky (1976) corrected their result and also gave the leading-order term for the other coefficient. Nunan (1983) and Nunan and Keller (1984) have obtained additional terms in these expansions and a general expression for arbitrary lattice geometry. The bulk properties of periodic suspensions of solid spheres not limited to low or high concentration have been studied numerically by Nunan (1983), Nunan and Keller (1984) and Brady et al. (1988). It is impressive that their numerical results agree well with the analytical asymptotic results not only for very low or high concentrations but also for quite a wide range of $I. Since the suspended particles are not spherical in many real suspensions, it is interesting to study suspensions of non-spherical particles. Recently, Tran-Cong, Phan-Tien and Graham (1990) have numerically studied the bulk properties of a monolayered periodic suspension of solid spheroids bounded by two flat plates, where the array of the spheroids is periodic in only two directions parallel to the plates. In the present paper we study analytically the bulk stress of infinitely extended periodic (in three directions) suspensions of solid identical force-free particles of arbitrary shape. After reviewing in section 3 some characteristic features of the disturbance field due to a single particle in an unbounded linear shear flow, we consider in section 4 the bulk stress of dilute periodic suspensions, and derive asymptotic expressions of the bulk stress up to O($). The method used in section 4 is basically the same as in Kaneda (1973), in the sense that the 0169-5983/91/$4.75
0 1991 - The Japan
Society
of Fluid Mechanics
approximation of the disturbance due to the suspended particles as a sum of the derivatives of Hasimoto’s periodic Green function (Hasimoto. 1959) plays the key role in the analysis. The cases of periodic suspensions of orthotropic particles are studied in some detail in section 5. Finally, some typical cases of concentrated suspensions of almost-touching ellipsoids are analyzed in section 6.
2. Basic equations We consider a homogeneous,
a periodic isotropic
array of small identical rigid particles Newtonian fluid that obeys
of arbitrary
shape immersed
in
p he-vp=o.
W
‘c7 lv=o.
(lb)
where v, p and p are the velocity, the pressure and the viscosity of the fluid, respectively. The particles are neutrally buoyant, and their instantaneous positions are assumed to be at the lattice points r “=m,a”‘+m,a(“)+mi,(i)
(m,,
m2, m,=O,
i:l+2
,___ ),
where u(l), a(‘) and a(‘) are the basis vectors determining the unit cell of the array. The orientation of each particle is the same, and the surface of the particle at rm may therefore be expressed as g( r - P) = 0 by a certain function g which is the same for each particle. We assume that the velocity gradient is spatially periodic so that v may be written as U, = y,,r, + ui, where and II is also The
(2)
the summation convention for repeated indices is used, yI, is a constant traceless tensor, is a spatially periodic vector. Equation (1) then implies that vp is periodic, and p itself periodic because no external force is exerted on the particles. no-slip boundary condition on the surface S” of the rigid particle located at P requires u,=y,,(P),+
(2),-t
[wX
(r-P)],,
on S”,
(3)
where y,,(P),, + (v”), is the velocity of the material point of the particle coinciding instantaneously with rm, and w is the angular velocity of the particle about r”‘. Because of the periodicity, b’ and w are the same for each particle. Similarly, the force f and torque L exerted on the fluid by the particle are the same for each particle. Let “PI, be the volume average of the stress over the unit cell V of the lattice containing the particle B at the origin. It is then shown (Batchelor, 1970; Nunan and Keller. 1984) that
and the bulk deviatoric D,J = AX,
stress D, i is therefore
given by
+ Y,J + o,“,*
where S is the surface of the particle B, 7 the volume of the unit cell V, a,, the stress tensor due to (0, p), and n the outward unit vector on S pointing into the fluid. The tensor D,“/ represents the contribution to the bulk deviatoric stress due to the presence of the particles, and is called the particle deviatoric stress or simply the particle stress.
Y. Kaneda
The decomposition
of tensors
167
/ Bulk stress of a periodic suspension
y,, and D$ into the symmetric
DI”/= S,, + (1/27)e,,kLi,
Y,, = e;j-
and antisymmetric
parts gives
crjkfikr
(5)
where Si, = (1D; + D,$‘2,
L/r
ei, = (Y,, + Y,, )/2,
9, =
= E,/@$, -E,JkYlk/2-
The particles are force-free so that the force exerted on the fluid by each particle is zero. It is not difficult to verify that the specification of the torque L or the angular velocity o is then sufficient to uniquely determine the periodic field (u, p) and the particle stress 0,; for any given tensor y,,. If the torque L and the tensor yij are given, then, because of the linearity of the problem, the symmetric part of the particle stress may be written as %j = @&rYpq
+ :
c,,k
Lk
(6)
)
and traceless in their first pair of where the constant tensor CIjP4 as well as Cijk are sy~etric indices. The first term on the right-hand side of eq. (6) represents the contribution due to the field satisfying eqs. (1) and (2) for given yjj with f = L = 0, while the second term is the contribution due to the field satisfying eqs. (I) and (2) for given L with yII = 0, f= 0. If the symmetric part elj of y,, is zero, then the velocity of the former field is equivalent to a rigid rotation and the field makes no contribution to the particle stress. The term y,,, in eq. (6) may @ per unit volume is therefore be replaced by the symmetric part epy. The energy dissipation given by (cf. Batchelor, 1970)
3. Disturbances
due to a single particle immersed in an unbounded linear shear flow
We now consider a neutrally from which the velocity gradient as r=
14 -+ v,,r,
buoyant single particle immersed is uniform and given by
V
is taken at a point
P=Po+p’,
where p. is constant and u’, p’, The disturbance fields satisfy p Au’ - vp’
=
fluid,
far
Irl --f co,
where the origin of the coordinate U; = Y,j’, + u:,
in an unbounded
CT’
a,/=
inside
the particle.
Let us write
-P$,j+2Pe,j+a,l,,
are the disturbances
(8)
due to the existence
of the particle.
0,
(9a)
lu’=o,
(9b)
and U’-+O, The general
p’-+o
solution
asr-+co.
u’=v(r*+)/2-cp, where 4 is a harmonic outside the particle as
(10)
of eqs. (9) and (10) are given by (Imai, 1973; Hasimoto p’=i_1v
vector satisfying
~~=((Ai+Aipap+Aipgapaq+
and Sano, 1980)
-tp,
A+ = 0 and may be expanded -**)S{,
(111
by spherical
harmonics
(12)
Y. Kunedu / Bulk stress of a periodic suspension
168
where S; = l/r
satisfies
hs;
= -M(r).
The fundamental
solution
P Aq; - a/,
(13) (U,:, P:) of eqs. (9) and (10) satisfying = Q(r),
a,u,; = 0, is obtained
by retaining
only the first term and putting
A, = 6,,/4~~
in eq. (12) as
The field (r/;:, p,‘) represents the disturbance due to a point force (Stokeslet) in the jth direction at the origin. The coefficients A,, A,,, . . . in eq. (12) as well as the field (u’, p’) satisfying eqs. (X), (9), and (10) are in general uniquely determined for any solid particle by specifying y,,, the force f and torque L exerted on the fluid by the particle, where f, = - k,;n, It is known
dS,
L, = --~,~j$lplli
dS.
that A, and A,, are given by -47&A
=f,
so that A;, = 0. For a force-free particle, we have f = A = 0. Because of the linearity of the problem, A,, is then linear in y, j and L, and may be expressed in the form [cf. eqs. (5) and (6)]:
-4al*.A,,
= PC,‘,,&
+ ‘z(C,;J+
where the constant tensor C,‘,,, as well indices and is determined by the shape, tensor y,, in eq. (17) may be replaced In terms of the Stokeslet field given written as
p’=(A,,t$+A
,py $3,
+ e,,~%)r
(17)
as C,:, are symmetric and traceless in their first pair of size’and the orientation of the particle. The shear rate by ePy as in eq. (6). by eqs. (14) or (15), (11) with (12) and A = 0 may be
+ . . * )(4WP,‘)’
(1Xb)
sides of eqs. where in deriving eq. (18a) we have used A,, = 0. The first terms on the right-hand (18a) and (18b) (A,, terms) represent the field due to the so-called stresslet at the origin.
4. Dilute periodic suspension Now let us consider the periodic array discussed in section 2. One might suppose that the periodic field (u, p) satisfying eqs. (1) and (2) may be given by the form (18) with U’, P’, S;
Y. Kaneda
replaced
by the following u,,(r)
U, P, S,:
= Cl,$(r-r”), m
P(r)
= Jp’(Y-P), m
S,(r)
and with appropriate coefficients A,,, A,,,, . . . However, because the sums are divergent and not well defined. Following Hasimoto (1959) we define here the fundamental P AQ, - a,?, = Ci&,,~(rm a$;,
the
= cs;(r-P), m
form
(19)
(19)
is inconvenient,
field U, P, S, by t 204
%I,
(2Ob)
= 0,
AS,= instead
169
/ 3uNc stress of a periodic s~~en~io~
-47r
c+P)-;‘, i m
(2Oc)
i
of eqs. (13) and (14). The solutions
of eqs. (20) are given by
where ‘exp( -2aik.P)
,
k2
k-t@
,=-L
c
’ exp( - 2vik
* r)
k4
4T3T kiO
’
and k = m,b”’ + m,b’2’ + n@(3) is a vector in the reciprocal following expansions: S, = l/r
lattice,
- c + .. *(
(m,,
and
m2, m3=
b(‘) = cilx(~(‘)
0, il, X
+2,...),
atk’)/2r.
For small
Y, we have the
s* = r/2 - Cl + * . * )
and ~R/.LV, = 4a~1.q.; + K,, + K,,,r,
f K,Jmnr,,,r,, t
...,
(21)
on the lattice geometry. where c, ~2, Kilt K,,,n,, . . _ are constants depending plays an important role in the following analysis. We may assume without that K,imir is symmetric in m and n so that it is given by K i,““’ = 27~ 3, 4$4,(ri
- V:bi]
The tensor Kii,,,, loss of generality
I r=n = Ki,,nn + K&w
(22)
where K,:,,
= -
!is,j
I,=*?
aflz an[sl(r)-l/r]
K2r,mn = t a, a, a, a,[s,w
- ~21
I r=O.
Let A, = A:, A,, = Al,, . . . be the coefficients such that (u’, p’) in eq. (18) is the solution of eqs. (8), (9) and (10) for a force-free particle with given y,, = vii and L = i, i.e. the field (u, p) given by ~1,= 7, Irl + 24:)
p=pO+pr>
~~=(~;~~~+~~~~a~a~+
p’=(A;,
a, apt-A
;,,
. ..)(4~~1.!.)-~(A;,,+A;,,) a, a, a, +
satisfies eqs. (8), (9), (10) and the no-slip the particle and gives torque E.
a,a,+
. . . )(47~~;),
and force-free
. ..is.,
(234
(23b) boundary
conditions
on the surface
of
Now we assume that jut” 1 - jac2’ / - 1a(j) 1 - I is much larger than the characteristic length scale a of the particle, i.e. c:= a/! --K1, where the symbol - denotes the equivalence in order of magnitude. Then, since r/& given by eq. (20) is determined by p and the basis vectors consideration gives that ai, a2, a3> all of which are of order I, the diniensional K,,, = 0(1/Y).
K,,,, = O&P).
K:i,nn = 0(1/1”)..
..
Similarly, since c2 is the only representative length scale of the disturbance a single particle, the dimensional consideration gives that A.LV =O(A*u),
A;,<,,=O(A*u~)
0‘4) field (u’, p’) due to
,....
(25)
and also c,;,,
= tt(CI”).
C,>,* = O(I).
where A * is the characteristic from eq. (17) PA* -F*a3e*
magnitude
of A:; (e.g., it may be given
by A*’ - AI,A:,),
and
+ IL/,
(26)
in which e * is the characteristic magnitude of Cli = (Uzi + V,j)/2. The expansions (21) and (23), and the estimates (24) and (25) suggest solution of eqs. (1) and (2) for small t: may be obtained by putting
that an approximate
L’,= y,,r, + u,, U, = c; + ( AiI, $ + A.;,, a,, 3, + * . * )t4TU,_/)
- [ ( Aip, + Ai,jp) ai ap + ’ ’ . ] sl/2, j27a)
P=(A~~~~+A~~Yi?na,+
*‘.)(4Tr/&P),.I
f27b)
provided that c and r,, and z are chosen appropriately, where c is a constant vector. Once ;Ji and E are given, the coefficients A;,,, A:,,,... , are uniquely determined from them for any given solid particfe. Since eq. (27) may be written as
- [(Aip,+A:jp)a, P=~‘+(A;~ we obtain
a,+A;p,
a,+ $,a,+
***](SI-S;)/~, I. ’ )[4TPfpj-P,‘)]v
(2Xa) (2fQ7)
from eqs. (24) and (25) that for i’ - LI, 0, = y, ,‘, + #: -t (c, + II,) i Rl,Yj -k O(A*a’/P), olj = u,; -t ,u(2eti i B,, -I- B,,) + &,d+
~~~A~~,/~4),
C29a) t29b)
where
B, and d are appropriate constants, and IJ,~ and ul: are the stress tensor respectively. So far. the field (u, p) in ey, (27) or (28) may be defined for arbitrary chose c, ;i,/ and E such that c + B = 0, E = L and
due to (u, p) and (u’,
p’),
Yr,,+ 3, { = 7, JI i.e.
c, r,i and E. Now we
Y. Kunedu
171
/ Bulk stress of a perrodic suspensron
then the field (t’, p) satisfies the force-free and no-slip boundary condition on the surface S and yields torque I. within the approximation neglecting terms of 0(,4*a2/L4) in eq. (29a) and 0(~~*~/~4) in eq. (29b), because the field (D, p) given by eq. (8) with y,, replaced v,, satisfies these conditions. The substitution of eq. (29) into eq. (4) yields D:;-t~~[(~,;~,-is,,a,~~~)~*-p(l(:ni+n;,i,,]
dS+0(~A*e4/~).
Because of eqs. (16) and (17), this may be written 470:, D,; = - ~ 7
+ 0( pA*c4,+) + &(c,&
= 5 [PC/,&q Substitutillg
as
+ ‘,,&,)]
+ O(/_6A”E4/T).
(31)
eq. (30) into eq. (31) and using again eq. (17) yields
$&C&L, +ti,xL,)
+ O(yA”IE4/7-).
(32)
In the following, we consider the case when the torque z is at most O(poae*). This condition is satisfied not only when each particle is torque-free, but also when the orientation of each particle is kept fixed by external torque. We then obtain from eq. (26) that A* - a3y*, so that eq. (24) gives (K,,,,z,> + K,,prt,)&p Hence, P,, on the right-hand and we finally obtain D,~=PC,~,~~,,,+
= O(Y*~~) side of eq. (32) may be replaced
:(C,,kL,
+~-‘c~~ALA)
by ePrl within
our approximation,
(33)
+PY*O(~~)~
where
X 1 ii’4
=
(34b)
Y,A
y&C,‘,c,h ( Koca + Kud
(34c)
7
(34d)
Y,,c = & C~:a~,(Krr~<~h+ Kildc/>)C<;k.
Equations (33) with (34) are the main result of the present paper. For dilute suspensions, i.e. and T,,/T are 0(c6) = O(+‘). for e ==z1, C,;,, /T and C,‘,,/r are O(c’) = O(+), and X,,J7 The tensor Kuhcd is determined solely by the lattice geometry irrespective of the particle solely by the particle geometry irrespective of geometry, while Cl’,P, and C,:, are determined the lattice geometry.
5. Dilute periodic suspension The tensors
of orthotropic particles
C,:,,,, and C,;n, in eq. (18) in general
C;),,,,, = C,:,,, 3
satisfy
the symmetry
relation
and C,:,,,,, = 0,
c,:, = 0
We may assume without loss of generality that C,‘,,,,,,, = 0 since only traceless tensor is significant. Hinch (1972) has shown that then
the contraction
with a
C’i ,““’ = cm j . The form of C,:,,,,, may be further simplified when the surface of the particle has orthotropic symmetry, i.e. when it is symmetrical about each of three orthogonal planes. We take here a coordinate system in which the principal axes of the particle coincide with the coordinate axes. It is shown, in appendix A. that C,:,,,,, is then zero unless the indices i. j. PI, n are equal in pairs and also that C,:h is zero if any two of indices i, j, k are equal. The tensors C,‘,,,,,, and C,‘,x satisfying these relations must be of the following form: 71 C 1,‘,I>’= [ &I 6 , pm +(.--)+(...)]+[~;(P,,Pm,,l~,,~~Ql,)+(...)+(...)] (35a)
+[P;(P,Y,+p,q,)(l,,,4,,+~,IY,i)+(...)+(..’)],
(35bJ
C,‘,, = ~ilh. (47, -t Lf,r, ) + ( . . I > + ( . I. f + p, q, P are the unit
where G?, C;;:, /3,‘. y,’ (i = 1. 2, 3) are constants, principal axes, and
Q,,=w-~,,/~~
P,,=P,P,-4,/J,
vectors
parallel
to the
R,,=r,r,-&,/3.
Since Pii +
Pi, + R,, = 0,
and therefore
f’i,Qtn,, + f’n,,,Q,, = R, /R,,,,, - P, ,P,,,,, - Q, ,Qnr,,. etc., eq. (35a) may be put in the form
C’i ,r,in=
[4P,,P,,,, -I- &Q,,Q,n,, + dft,,R,,,,,]
+ [8;iu,q
+ Pi ( r, P, + r, p, ) ( r,,,P,, + 6, h, 1 + P; ( ~~4,
+ ~,~)(~,~,~, + cl,r,,)
+ p,4, I( p,d,, + ml,,, )J .
(36)
The expressions (35) and (36) are independent of the particular choice of the coordinate system. The form of the tensor K,,nl,i may be similarly simplified when the lattice has orthotropic symmetry, i.e. the configuration of the lattice is symmetrical about each of three orthogonal planes. Then, in the coordinate system with the coordinate axes coinciding with the principal axes of the lattice, K ,,,, I,, is zero unless the indices i, j, YYI,11 are equal in pairs. If the lattice has further cubic symmetry, then the tensors K’ and K’ are in general of the form kxlI ,,,iw
=
f?s,
,L
>
&2,,,,,
=
@z
/,,i,,
+
t(
4,L~
where l, g and Q are constants, and Sijmrr is unity For a cubic lattice. S, and S, may be expanded 1
S,=;-c+~;r’+
2m
+
L&
+
Lo]“’
>’
if i = j = m = n and zero otherwise. for small r (Hasimoto, 1959) as
...,
Sz=f-~,+~T?+~r4+b2~~Y~‘(x1.x2,_~~)+h~~Y~(xI, where hzo and h2i are constants
x2.x:7)+--.,
satisfying
b,, = h,,/168 (Zuzovsky,
1976)
and
Y,“‘(x,, x2, .x3) = r”P,:)‘(cos 0) cos rn+ is a solid
harmonic
in which
173
Y. Kaneda / Bulk stress of a penodic .su,spensjon
xi = r sin B cos +, x2 = r sin 8 sin +, x3 = r cos 8. It is shown sions that Kucdh + Krrd<,, = E&M + 9 (%,,&, + &$,,.)
+ 2&,&,
from eq. (22) and these expan-
(37)
>
where q=
5 = 60b,,,
-2l'l/5T-
12b,,.
The value of b,, has been calculated lattice (BCL) and face-centered cubic Zuzovsky (1976), f&l
= -0.195
(SCL).
= 0.120
@CL),
= 0.213
(FCL) >
for simple cubic lattice (SCL), body-centered cubic lattice (FCL). According to Hasimoto (1959) and
where I= la(‘) 1 = jac2)1 = 1~~~)I and 13/7 = 1, 2 and 4 for SCL, BCL and FCL, respectively. The expression (34) can be most simplified when the principal axes p, q, r of the orthotropic particles coincide with the principal axes of the cubic lattice. Then, substituting eqs. (35b), (36) and (37) into eq. (34) yields after some algebra that
A(G,P,,&,+ GzQ,,Qm,,+ W,,L,)
X ‘,,,r,, =
+ B [ p;“( q,r, + q,r,)( q,r, + x,,
= B f /3;y;p,
q,r,,)
( qir, + q,r, ) + two similar
+ two similar
(384
terms],
terms] ,
(38b)
where &+2rl_
Ii 157
12n
/ 3&o 71 ’
G,=2(a,“-+;)+I& H = +I;
G,=~(cI;~-oI;cY;)-~H,
-rl
’ G,=2(n;‘-+;)+I$,
+ a&; + a$~;.
For example, let us consider Then eq. (7) gives
Similarly,
B=11=_L_12b20 71 Sr
the case when
y,, = e,, = P,) - Qi, =pipi
- q/q,
and
L = 0.
when y, i = ei, = 4,rj + q,r, and L = 0, (39b)
Example: cubic lattice of ellipsoids. The ellipsoidal particle has the orthotropic discussed above, and the coefficients (Y:, fi,‘, y,’ may be known from Jeffery’s follows (Jeffery, 1922; Batchelor, 1970): I 45, -2GF= a=&_
3(f,Jz+JzJ~+JjJ1)’
4nabc
2 = 31, ’
62 - c2 ‘I=
-
b2+c2'
etc.,
symmetry solution as
174
Y. Kaneda
/ Bulk stress
of a periodic
suspension
Table 1 Viscosity coefficients
0.1
0.2 0.5 1.0 2.0 5.0 10.0
2.091 1.463 1.301 1.667 2.834 7.985 20.776
6.640 3.829 2.178 1.667 1.449 1.357 1.340
3.320 1.915 1.089 0.833 0.725 0.678 0.670
0.413 0.492 0.688 0.833 0.835 0.739 0.696
where a, b and c are the semi-diameters of the ellipsoid and I,, I,, I, and J,, J2, J3 are integrals like I, =
= sa
ubc(b’+
73
c’) dh
J1= / o
A(b*+X)(c’+h)’
- 0.980 - 0.923 - 0.600 0.000 0.600 0.923 0.980
in the p, q, r-directions,
respectively,
ubcX dh A(b’+h)(c’+X)
where A2=(a2+h)(b2+X)(c2+h). particle is thus equivalent to In considering the viscosity tensors C,:,, and C,:,. any orthotropic an ellipsoid, provided that the coefficients (Y:, ,8,’ and y,’ (i = 1, 2, 3) can be chosen properly. y;= -y; and y,‘=O.The For spheroid particles with b = c, it is clear that (Y;=cxi, pi=&‘. values of 6,. E:2, &. & and T? are listed in table 1 for values of u/b. For a solid spherical particle of radius u, we have LYE = ai = ai = 2EI/;,, fl,’ = ,& = & = E&, y, = y2 = yj = 0, where E = 5/2 and V, = 4au3/3. Equation (36) then reduces to ; C,‘,,, = E+(QJy and eq. (34) or (38) reduces
y,,: =
(40)
+ %,6,,, - fi&,$,), to
(41b)
0,
where (Y= 3A(2E1/,)*/~= /3 = 2B(EV,)2/7=
-2~5~$~(+
@/2w,,e,, =
+ 12b20++
-2E2#($
in agreement with Zuzovsky (1973) particles. Also, eq. (39) reduces to
- 18b,,++
where
$ = Vo/r
1 + E$ [ 1 + E+( $ - 18b2&r)],
for y,, = e,, = p, p, - q, q, and
is the volume
concentration
of the
(434
L = 0, while
@/2Pe, ,e,, = 1 + E+[l
+ E+(+ + 12b,,+)],
(43b)
for Y,, = e,, = (q,r, + q,r,) and L = 0, in agreement with Kaneda (1973) (in which only the case of simple cubic lattice is mentioned, but a generalization to include the cases of the other kind of cubic lattices is straightforward). In concluding this section, the case of a periodic suspension of spherical drops may deserve comment. It was shown in Kaneda (1973) that for a periodic suspension of spherical drops of a
175
fluid with viscosity
j& @ is given by eq. (43) but with
instead of E = l/2. From this result, the viscosity tensor be given by eqs. (33) (34), (40) (41) and (42) with (44).
6. Concentrated
periodic suspension
CiJPy in such a suspension
is seen to
of ellipsoids
We consider here a concentrated periodic suspension of almost-touching ellipsoids. The analysis is most simplified when the principal axes of the ellipsoids are parallel to the basis the fundamental unit cell. When the suspension has this vectors a(‘), at2), and af3) determining kind of symmetry, it can be shown in the same way as in deriving eqs. (35) and (36) that the tensors C, ,py, C,,, are in general of the form c ,,,nn = (aipl,jpl,t,l
+ a~Qt)Qnzn+ ~3R~~‘rnnI
+ [ &( q,r- -I-q,r,)( q,r, + qnrm) + two similar C,jk = y, pk (q,r, -k q,r,) + two similar
terms],
terms.
(45a) (45b)
Let us take the coordinate axes parallel to p, q and r, and let li = /a”’ 1, cl = 1 - (2a/i,), Ed = 1 - (2b/f,). e3 = 1 - (2c/i,), where a, b, c are, respectively, the semi-diameters of ellipsoids in the p-, q- and r-directions. We consider here only the case when f3 -=X1,
c,, E2 = O(1).
The cases when not only f3 but also ei and/or e2 are/is very small can be treated similarly. The coefficients in eq. (45) may be estimated by considering the following three cases. The outline of the analysis is sketched in appendix B. Case 1. When y,j = Pi, - R,, = p,p, - r,r, and the motions of the particles are given by eq. (3) with v” = w = 0, the particles are force- and torque-free. The energy dissipation Q, per unit volume is then given by
where 7 = f,i,f,. From the symmetry consideration or by considering the case when Y,~= Qi, R,, and v* = w = 0, we obtain a similar expression for cuz + (Ye, while if y,, = Ri, - Ptj and v0 = w = 0, then @ = O(r’), so that 0~~+ (Ye= O(<‘). From these estimates, we can get the estimates of (Ye, (Y>and LYE.It is seen that (Y,, a2 +z (Ye- l/h. Case 2. When y,, =p,r, -t-Fir, and the particles @/21*e,,e,,
are force- and torque-free,
@ is given by
= p* = O( 8).
From the symmetry consideration we obtain a similar expression the case when y,, = p,q, + p,q, under the force- and torque-free obtain @ = O( E’) so that & = 0( co).
for &_ Finally, by considering conditions of the particles, we
Case 3. When y,, =p,r, +pjr, and the motions of the particles are given by eq. (3) with v0 = w = 0, the particles are force-free. The energy dissipation per unit volume and the torque L
Y. Kane& / Bulk stress of a periodic
176
exerted
on the fluid by one particle @
are then given by
aabc = --log 7
I5L.q
+e,.,e,,
2~
suspension
c:,
L = (itrr&uabc log e)q. Hence, we obtain yz = - l/27. Similarly, by the symmetry consideration, we obtain y, = l/27. Finally, the estimate of y3 may be obtained by considering the case when y,, = 0, 0 = 0 and w = r. The particles are then force-free and the torque and the dissipation are O(E’). The energy dissipation is then at most O(cO), from which we obtain yj = O(r”).
Acknowledgement I wish to express my cordial thanks to Professor H. Kimura for stimulating liquid crystals, which have revived my interest in the present problem.
discussions
on
Appendix A. Symmetry relations of C&,,, and C;6 The symmetry relations of C,‘,,,,, and C,lk can be obtained in a way similar to that used by Brenner (1964) Zuzovsky (1976) and Nunan and Keller (1984) as follows. Let T = { T, } be any orthogonal linear transformation, and i, d, fi, j?~be defined by i=Tr,
;(;)
=Tv(r),
;(;)
= TM(~),
j(F)
‘P’(Y),
then G,(i)
= ~~~~~~~~j+
G,(F),
(A-1)
where (G, $) satisfies p 8;-Ojj=o,
(A.2a)
e lD=O,
(A.2b)
and ir -4 0,
B-0
as ?=
Ii1 + z~,
(A.3)
in whiche,=3j=El/a?,, Asa, 3,. Since eqs. (A.I), (A.2) and (A.3) are of the same form as eqs. (8), (9) and (10) but with v,, replaced by qpl;uUpu> we may write, in the same way as in eq. (17),
where Cijnln and C,,k are appropriate (r,,(;)-$,G,(i)+^a,ti,(i)]
constant
tensors,
-S,,,b(f),
and ri=Tn,
i, = -41.r~E,,kajk. On the other hand, since A,, and tor, we have in general i$, = 7;,7;,A,,,
L, are respectively
i, = IT j 7&,,
a second-order
tensor
and a pseudo-vec-
(A-5)
Y. Kaneda / Bulk stress
where
of a periodic surpension
11-i
IT 1 = det T. Hence, eq. (A.4) gives * ,. - +b(~,mITITmk~k+~,,,ITITrnx~k). -4V%, = C!/rn?lTmpTnqYpq
Now suppose
T maps
that the transformation
c’,,mn = &?In,
of the particle
onto itself. Then,
c,‘,, = C,,, 3
so that from eqs. (16) (A.4) -4nM,,
the surface
(A.6)
(AS)
= C,;,,U,,
and (A.6)
+ :C,&
•t te,,,&
= TP,T4,( C;~mnTmp~qYpq + :Cd,r
IT I LA)
+ &Ldw
Since this holds for any u,, and L,, we have C,;,,
>
= Tpr7’&rnLCP’~o~
and C,:,, = T$T&T,,
IT I Cp’qr.
If the surface of the particle is symmetrical about the plane orthogonal i.e. invariant under the transformation 7;, = (- l)sL,a,i, then c,:,,
= ( - 1) “C&
c,:rn = w>“c,~m~ Hence, if the surface of in the coordinate system C r,mn is zero unless the i, j, m are all different
Appendix B. Asymptotic
a = a,, + S,, + S,, + S,, ,
3
b = 1 + S,, + S,, + S,, .
the particle is symmetrical about each of three orthogonal planes, then, whose coordinate axes coincide with the principal axes of the particle, indices i, j, m, n are equal in pairs and Crrn is zero unless the indices from each other.
analysis of flow in a narrow gap between two ellipsoids
We consider the flow in a narrow and S- are given by -2
&+P+ a2
b2
to the k th direction,
(TfD)2 c2
gap between
two ellipsoids
B+ and B-, whose surfaces
Sf
=l,
where ~=l-cC/D
i?=u’+w’x(F-r*),
be analyzed by a slight spheres (Zuzovsky, 1976;
on S’,
(B.1)
where F = (X, j, Z), fi = (U, U, W) with U, U and W being, respectively, the velocity components in the X, j and F directions, v’ and w* are the translational and angular velocities of the particles B ‘. and r ’ = (0, 0, f 0). It is convenient to introduce the following stretched variables x=c
y=c P’/2DP1X, u = D-‘U “CD-$j ,
where p is the pressure.
-1/2D-1-
We consider
w=c
Y?
z zz E-lD--lF
-1/2D-1, 9
here the following
p
=
p-
1~3/2~
three typical
cases separately.
Y.Kaneda / Bulk stress of a yrriodic suspension
178
Case I. When
o*= (0, 0, f D) and w*= 0, eq. (B.l) yields the boundary
u=u=o,
w = _t~-‘/~,
Hence we assume
the following
these expansions
ape a2uo
-=-
ax
az
Ql, w(), PO) + E”2(U,,
+, Wl, Pl )+
...
into eqs. (3) and (B.l) yields
aPo
a%, aPo
aY
az2,
-=-
2'
on S’.
form of expansions:
(u, U, w, p) = ~-“+(), Substituting
condition:
o
-=
az
1
!!!5+!!3+$+),
ay
and UC,= ua = 0,
wo= t-1,
on z=
kH(x,
y),
where
y)=l+i
H(x, These conditions
g +g
i a
are satisfied
S) = (a/D, b/D).
(a,
) i
by the following uo=
1 ape uo= -p(z’-H2),
wo=f
1
solution:
ape
z7&-(z2-H2), Apo(H2-~2)H+~,
2 ay
for which the energy
Case 2. When
u= Hence
dissipation
6 in the gap is given by
v*= (+ D, 0, 0) and w*= 0, eq. (B.l) yields the boundary +1,
we assume
u=w=O,
0nS’.
the following
form of expansions:
(u, u, w, P> = (UO> ql, w,, PO) + C(%> 013 WI? PI) + . . . Substituting
these into eqs. (3) and (B.l) yields
aPo azuo ax
ape a2uo ape -
-=-
-=_
az2 7
ay
az2 ’
0
BL -
2
and 24()= L-1, These conditions u.
=
u() = w, = 0,
are satisfied
I
H’
by the following
ug ‘PO = 0,
for which the energy dissipation 6 = - 4q.wbc
on z = &H(x,
log C,
1aH WO=Tax
y).
solution: --z2 i
H2
1 1
6 in the gap is given by
>
condition:
179
and the torque
2 exerted
on one particle
by the fluid in the gap is given by
f. = - (2n/Aahc log f)il. Case 3. When v*= (i: D, 0, 0) and toi= (0, I, O), eq. (B.1) yields G=w*x);) This boundary
on Sk,
condition
CZ@‘Xi;
3
for which the dissipation
and eqs. (3), (4) are satisfied
by
p = 0, @ in the gap is zero.
Finally, we consider the periodic array of ellipsoids considered in section 6 for which we may take 20 = 13. Let w’ be the angular velocity of the particles for which the particles are torque-free under given yiI. Because of the linearity of the problem, ec is linear in yii and L may be written in the form
where Tlk is a constant tensor. By considering the difference field between the fields of case 2 and case 3 [for which y,j = 0, uc = 0, w = (0, 1, O)], it is seen that TJk = O(log E). Since L = O(f’) in case 3, we have (0, 1,O) = wc + O((log z)- ‘) for y,, =pir, + p,~~. Moreover, we have @ = C&r”) if y, i = p,r, -I-p,r, and if the particles are torque-free, because cfi = Oft*) in case 3 and the difference between the angular velocities w” and w = (0, 1, 0) in case 3 is @(log E:$- I) as shown above.
Batch&x. G.K. (1970) The stress system in a suspension of force-free particles, J. Fluid Mech. 41, 545-570. Brady, J.F., R.J. Phillips, J.C. Lester and G. Bossis (1988) Dynamic simulation of hydrodynamically interacting suspensions, J. Fluid Mech. 195, 257-280. Brenner. H. (1964) The Stokes resistance of an arbitrary particle-III. Shear fields, Chem. Eng. SC;. 19. 631-651. Frank& N.A. and A. Acrivos (1967) On the viscosity of a concentrated suspension of solid spheres, Chem. Eng. Sci. 22, 847-853. Hasimoto, H. (1959) On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres, S. F&id Mech. 5, 317-328. Hasimoto, H. and 0. Sano (1980) Stokeslets and eddies in cr&%ping flow, Ann. Reu. FItlid Me&. 12, 335-363. Hinch, E.J. (1472) Note on symmetries of certain material tensors for a particle in Stokes flow, J. Fluid Me&. 34, 423-425. Imai, I. (1973) Ryuutai Rikiguku (Fluid Mechanics) (Shokabo, Tokyo) fin Japanese& Jeffery, G.B. (X922) The motion of ellipsoidal particles immersed in a viscous fluid, Prof. Roy. SK A f&T. 161-179. Kaneda, Y. (1973) Viscous flow through porous media, M.S. Thesis, Tokyo University [in Japanese]. Nunan, KC. (1983) Effective properties of composite media containing periodic arrays of spheres, P/XL). Thesis, Stanford University. Nunan, KC. and J.B. Keller (1984) Effective viscosity of a periodic suspension, J. Fluid Mech. 142, 269-287. Tran-Cong, T., N. Phan-Tien and A.L. Graham (1990) Stokes problems of multiparticle systems: Periodic arrays, Phys. Fluids A 2, 666-673. Zuzovsky, M. (1976) Transport processes in spatialIy periodic suspensions, Ph.D. Thesis, Carnegie-Mellon University. Zuzovsky, M., P.M. Adler and H. Brenner {1983) Spatially periodic suspensions of convex particles in linear shear flows. III. Diiute arrays of spheres suspended in Newtonian fluids, Phys. &ids 26, 1714-1723.