ARTICLES
The pressure moments for two rigid spheres in low-Reynolds-number
flow
D. J. Jeffrey
Department
of
Applied Mathematics, The University oj. Western Ontario, London, Ontario, N6A 5B9,
Canada
J. F. Morris and J. F. Brady
Department
of
Chemical Engineering, California Institute
of
Technology, Pasadena, CaIifornia 91125
(Received 22 February 1993; accepted 2 June 1993)
The pressure moment of a rigid particle is defined to be the trace of the first moment of the
surface stress acting on the particle. A Fax&r law for the pressure moment of one spherical
particle in a general low-Reynolds-number flow is found in terms of the ambient pressure, and
the pressure moments of two rigid spheres immersed in a linear ambient flow are calculated
using multipole expansions and lubrication theory. The results are expressed in terms of
resistance functions, following the practice established in other interaction studies. The osmotic
pressure in a dilute colloidal suspension at small P&let number is then calculated, to second
order in particle volume fraction, using these resistance functions. In a second application of the
pressure moment, the suspension or particle-phase pressure, used in two-phase flow modeling, is
calculated using Stokesian dynamics and results for the suspension pressure for a sheared cubic
lattice are reported.
I. INTRODUCTION
A mechanical definition of the osmotic pressure in a
colloidal dispersion has been given by Brady’ in terms of
the hydrodynamic interactions among the suspended par-
ticles. In addition to the interactions already familiar in
low-Reynolds-number hydrodynamics, a new “pressure in-
teraction” must be defined. This new interaction is also
needed in models of particulate two-phase systems, where
particle-phase momentum balances necessitate the concept
of a solid-phase pressure2-5
in order to complete the spec-
ification of the bulk stress. The “suspension pressure” thus
introduced requires for its determination [by Stokesian
dynamics6 simulations, for example) the hydrodynamic
pressure interactions between particles. Examples of os-
motic pressure and suspension pressure calculations are
given at the end of this paper, after we have defined the
pressure interaction precisely and established how it can be
calculated.
In low-Reynolds-number hydrodynamics, interactions
between particles are frequently specified by using the mo-
ments of the surface stress acting on each particle. The first
moment of the stress has been decomposed in the past into
an antisymmetric part, which equals the couple acting on
the particle, and a traceless symmetric part called the
stresslet. These two quantities have been tabulated for two
rigid spheres in a series of papers summarized in Jeffrey
and Onishi,7 Jeffrey,’ and Kim and Karrila.’ It is the trace
of the first moment, however, that is needed for the pres-
sure interactions, and this has not been studied before. We
denote it by S and define it for a specified particle as
S=- x’*u*ndA,
f
where x’ is the position vector measured relative to the
particle center, and the integration is over the surface of
the particle. The minus sign is included because the previ-
ous studies used it in their definitions; the integral can then
be interpreted as the moment exerted
by
the sphere
on
the
fluid. Such an interpretation and sign will be used in this
paper in order to keep the equations similar to those al-
ready developed. Before we proceed with the calculations,
it is important to pause for a moment and consider termi-
nology. We should decide whether the term stresslet,
which until now has referred to a traceless quantity, is to
be expanded to include the trace of the first moment, or
whether it should be left as the traceless quantity. After
considering the equations that arise in the applications de-
scribed later in this paper, we think that it is most conve-
nient to make an analogy with the terminology used for the
stress tensor. Thus the stress tensor has a nonzero trace,
and when the traceless part of the stress tensor is referred
to separately it is called the deviatoric stress. In the same
way, a stresslet should have a nonzero trace, and if a trace-
less quantity is needed, it can be called either the traceless
part of the stresslet, or the deviatoric stresslet.
II. EXPRESSIONS FOR THE PRESSURE MOMENT OF
A SPHERE
First we derive a Fax&n law for the pressure moment
by using the reciprocal theorem. We start by recallinggl”
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that the pressure field p(x) produced by a point force F
acting at a point y is p(x) =F* P(x-y), where
P(X)+--.
Next we note that the velocity field v(x) around a point
source of fluid of strength Q located at the origin is
v=$;=QP(x).
The reciprocal theorem is used in the form’
s
vl* (02*n)dA+
s
vl- (V*a2)dV
=
s
v2- (aI-n)dA+
s
v20 (V*cr,)dV,
where n is directed outward from the particle surface into
the fluid. We take v1 to be the flow outside an expanding
sphere whose radius is a and whose rate of volume increase
is Q; the sphere center is at the origin. For v2, we take the
flow generated by a point force F at y when there is a
sphere of constant size stationary at the origin. The recip-
rocal theorem becomes
s
$$- (ai’n)dA
+
s
QP(x)
l
(-F)G(x-y)dV(x) =O.
Simplifying further, we obtain
Q
m
(-&>-QF-P~Y)=O,
and this gives an expression for the pressure moment as
S,= -47&F-P(y).
Now we observe that -F-P(y) =F*P( -y) is the pres-
sure that would exist at the origin if the point force were
acting in the absence of the sphere. This is the “ambient”
pressure as seen by the sphere, usually denoted
p” (x=0).
Thus we obtain a Fax&n law in the form
S=4n-a3p”(x=O).
(1)
The extension of this result to an arbitrary ambient flow
follows by echoing Hinch’s argument cited in Kim and
KarrilaY that any ambient flow can be modeled by a suit-
able superposition of point forces.
Next, we obtain an exact expression for the pressure
moment of one sphere in the presence of another in terms
of multipole expansions. We follow the notation of Jeffrey
and Onishi7 throughout; equations taken from their paper
will be labeled by JO. In terms of spherical coordinates
(pa ,8, ,g) centered on sphere (Y, the quantity we wish to
calculate is
S= -a,
-p + y $ ai
sin 8,
de, d$,
a
where U=U* i)a. Using (JO 2.3) and (JO 2. l), we obtain
ai
s
p
sin
8,
de, d$=itra& $ pA:-“)t,t ‘f-, .
(2)
n=O
A similar -calculation based on [JO Eqs. (2.4), (2.7), and
(2.1>] shows that the &&3p term in the integrand inte-
grates to 0. We should remember when comparing (2)
with (1) that the coefficients
pmn
have the dimensions of
velocity. Also, it is worth noting that the integration leads
to a contribution
poo
from the sphere to its own pressure
moment; however, this coefficient must always be zero,
because it implies logarithmic velocities far from the
sphere. The only contribution, then, is the pressure envi-
ronment created by the second sphere. Since only
m=O
terms appear in the expression, we can see that only axi-
symmetric motions will lead to nonzero pressure moments.
This fact can also be deduced from general vector consid-
erations.
III. RESISTANCE FUNCTIONS
As with the other interactions between spheres, the
pressure moment can be expressed as a function of the
velocities of the spheres and the ambient velocity field,
given by
U(x)=U,+&,Xx+E;x,
with constant U, , 0,) and E, , in which case we are led
to functions analogous to the resistance functions defined
in earlier papers. We write
Sl
0 i
Pll
P12
QII
Q12
s2
=p
P21
P22
Q21
Q22
(3)
The rotations of the spheres do not appear in the equation
because it can be shown that they do not contribute to the
trace. Since the only vector in the problem is d, where
d=(x2-x1)/1x2-xl]
is the dimensionless unit vector along the line of centers
directed from particle 1 to particle 2, clearly we have (in-
cluding nondimensionalizing factors)
P,=da,+a$2X $4
i4)
Qap=rr(a,+ap)3X f$(dd-$1,
(5)
andXP
n4, X af are functions only of s-
2r/( a, + ao)
, where
r=
1 x2-x1 1. The other resistance functions that are con-
tracted with the rate of strain have been made traceless, so
it seems reasonable to follow this practice here. The only
symmetries obeyed by the functions are labeling ones:
-1
x~~(s,il)=x~_,,(,-B)(s,il 1.
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Phys. Fluids A, Vol. 5, No. 10, October 1993
Jeffrey, Morris, and Brady
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IV. THE FUNCTIONS X cP
A. Method of reflections
We start by deriving the first few terms by the method
of reflections. Suppose sphere 1 is moving with velocity
U,= Ud toward sphere 2. The pressure field at the center
of sphere 2 is
p=ia+U/r”;
hence
and therefore
X,q=
4
6A3
(1+12)2 (1+A>?&
And, from the above relations,
4 6A
xp2=----
(l+A)Z (l+a)Y’
Sphere 2 responds to the ambient velocity induced near it
by sphere 1 by exerting a force on the fluid; hence
and
18A
“;=m*
B. Twin multipole expansions
These functions can be calculated using the results ob-
tained in JO Sec. III. In terms of
Pnpq
defined in [JO, Eqs.
(3.4)~( 3.9)], we have
xg-~ (l+A>2xf2= jTil $ q$o ;P,$$+v+‘*
For the complementary problem defined in [JO, Eq.
(3.11)], we have
1
xp +- (l+a)2XP
I1 4
12
= 2 2 f$ (-1)rt+F+4+2~Pnppt~+nff+l.
n=l p=o
q=o
From these equations, we see that the pattern observed
with the earlier functions continues to hold, namely that
the even and odd powers of s divide between the functions.
Thus
m odd
Y
i
f,(a)
X(l+ay,+"= m=l 0'"s"
(64
TABLE I. Values of the function P$(/2), with il the size
ratio
of the two
spheres, appearing in the asymptotic form of X iB for small separation.
/z
pi”l pi”2
pfl
p&
1
-0.0118 -0.1435 0.1435 0.0118
2 0.0930 -0.1279 0.258 1 -0.0634
3
0.3236 -0.1283 0.3199 -0.0963
4
0.5662 -0.1186 0.3337 -0.1024
5 0.7925 -0.1059 0.3249 -0.0988
10
1.6347
-0.0576 0.2286
-0.0653
20
2.5543 -0.0237 0.1193 -0.0322
100
4.0499
-0.0016 0.0114 -0.0029
-4
x.~~(s,/z)=----
(l+W
(6b)
where
fpf1=0, f*=6il, f3=18a,
f4=54a2, fs=--24a+i62a2+2i6a3,
f,=432a2+498a3+2592a4+1440a5,
fs=864a2+3888a3+7446a4+7128a5+3456a6.
C. Lubrication theory
The flow between nearly touching spheres has been
studied in Jeffrey and Corless” and Jeffrey.12 Solutions
based on an expansion in the small parameter E were given
there. In the latter paper, it was shown that, when higher
orders are included, not all quantities can be approximated
successfully by considering only the flow in the gap. We
follow the method given there to circumvent that difficulty
by writing the pressure moment as
s,=-
s
x’*u*n
dA
=arF.d-al
s
(n+d) *a*n
dA.
Using the known result for F and integrating the previ-
ously obtained solution (which had been found using the
algebra system
Maple,
and which was therefore easy to
reprogram), we obtain
and
-g3g In ,$-‘+0(6),
0)
m odd
and
where
2319 Phys. Fluids A, Vol. 5, No. 10, October 1993
Jeffrey, Morris, and Brady
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gr=3A2/(1+A)3,
g2=&l-4a2)/( 1 +a)2,
5 -97A + 64L2- 44A3 -&I4
(8)
g3=
140(1+/2)2 ’
the nondimensional gap width is g=s-2 and the
P$
are
functions that we shall tabulate here (cf. Table I). We can
notice that, as with the XA functions, the singular terms
cancel if the two spheres have the same velocity.
D. Arbitrary separations
The singularities cause slow convergence of the series
(6a) and (6b) when s is near 2. We remove them from the
series by giving the gi appearing in (7a)-(7b) the values
defined by (8) and then adding the left-hand side of (7a)
to (6a), while at the same time subtracting the right-hand
side of (7a) from (6a). If f(/Z) =2-mf(A), we write
20'
:
:
:
:
:
:
IO- \,
:
z
;
5
. . . . .
'...
xp, ,
--._
X'
*-..
- **
$ o-
-.-----*-.------..........~.~...........~...~.~~~~..~
___..--.._---_.
-10-
x t =g*
~+[g2+g3(~s’-1)]ln~-g3s
+ T,
m odd
B-4
Mathematically this is equivalent to (6a), but numerically
the rate of convergence has improved, because the coeffi-
cients of Sern
now decay faster by a factor
me’,
owing to
cancellation. Similarly,
(l+A12
------x p2
4
-
it2 (
s!m
a2
4g3
2
m
(l+A)”
-g1-L+m(m+2)
I( 1
S ’
m even
(9b)
Numerical tabulations of X & are not given because the
expressions and data given above are accurate to at least
two significant digits for all s. We do tabulate the
PC&A),
however, because they provide a good test of the conver-
gence of the series, as well as being useful in studies of
nearly touching spheres. Expanding the leading terms in
(9a) and comparing with (7a), we obtain
Pt=agl+g2 ln4-2g3
z?m
2g2 4g3
(1+/2)m-g1-~+m(m+2)
’
(104
m odd
-20 -I
I I
I 1
I
T
2.1
2.2
2.3
2.4 2.5
2.6
3
FIG. 1. The functions X f1 and X p2 relating velocities to the trace of the
first moment of the surface force distribution for equal-sized spheres are
plotted against the separation distance scaled to the particle radius.
(1 +A12
-p:2=;g1+g3- mz2
4
(
(~;;;m-gl-~
(lob)
E. Results for X $
To illustrate the behavior of the pressure moment, in
Fig., 1 we plot X r1
and X fz as a function of s for the case
of identical spheres A= 1. Note that the singular behavior
of X fi (and X f2) is proportional to the corresponding
resistance functions X;li, etc., relating forces to transla-
tional velocities. In Table I we give the results of summing
the series (lOa)-( lob) for
P
zfi to 300 terms. We can es-
timate the rate of convergence by making a comparison
with sums to 200 terms, and that shows that the results are
generally accurate to four significant figures.
V. THE FUNCTIONS X $
A. Method of reflections
If sphere 1 deforms at a rate Et =
E,
(kk-+I), the pres-
sure at the center of sphere 2 is
p=qaipE1/?.
Hence,
and therefore
x,q =
8
2oi13
(I+/%)3 (l+#s3’
and similarly
2320
Phys. Fluids A, Vol. 5, No. 10, October 1993
Jeffrey, Morris, and Brady
2320
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There
will
be an ambient flow ( $alEl (ai/?)d created by
sphere 1, which will lead to an induced S, of
S,=8n&#($> E1=-~(ai+a2>2X~2(~)(a:/3)E1,
imp1 ying
Xfl =
30/I
Tm-FF
B. Twin multipole expansions
The calculation in Jeffrey8 used the problem
E,=E,(kk-$I),
together with the condition
a,Er=azE2.
This means that
S1=8q&
@EI+qdal+a2)3X
f2$T2,
6
5
4
-3
0
x
3
2
1
0
2.0 2.1 2.2 2.3 2.4 2.5 2.6
s
and therefore
FIG. 2. The functions X ?, and X & relating the rate of strain to the trace
of the first moment of the surface force distribution for equal-sized
spheres are plotted against the separation distance scaled to the particle
radius.
Prom this we find
x?l+
~xf2= i 2 i ;p,,$+“rf+?
n=l p=o q=o
The complementary problem adds a factor ( - l)n+p+q+3
to the summation, so we conclude
x ?lw> = z, (l-$p
m even
and
(114
X~2=g4S-*+g51n~-‘+Q~+gsSIn~-‘,
(12b)
where
g,=$12/(l+a)3,
g2=&(a+2a2-9a3)/(i+a)3,
s-a-2oia2+25ia3-i84a4
g3=
28G(l+a)3 ’
g4=i2a3/(i+a)6,
g5=~(-2a2+a3~2a4)/(i+a)6,
-65a+34a2-4411a3+76a4-44a5
a=
wl+U
m odd
(lib)
AS a check on our working we have the identity
4x~~(1+a)(1+~~)-8x~~-(l+a)3x~2=o(1).
where
D. Arbitrary separations
fo=fl=f2=o,
f3=20a3,
f4=30a,
fs=90a4,
f6=-72a+270a2+680a3,
f,=720A4-t-810A5+864A6,
As before, we have the following expressions for Qfi
and Q F2 defined in Eqs. (12a>-( 12b):
1
Q;‘=
-p--83
f8=864/22+3888i13+7446/24+7128i15+3456A6.
C. Lubrication theory
We expect
+ f2
(13a)
m even
and
Xna& X fl $ E,=a$?*d-al
(q+d) *a-n &I,
utW x
8
Q
12=igl+g2 In 4-k
==4m3X GE -a
lllu 1
s
(n+d) .a*n
dA,
and we obtain
(~~;;m-&-~+m(~;2)
m odd
x ffl=glg-l +g2 In
g-‘+Q
$+g& ln C-t,
(1W
(1%)
2321 Phys. Fluids A, Vol. 5, No. 10, October 1993 Jeffrey, Morris, and Brady
2321
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