The pressure moment of a rigid particle is defined as the trace of the first moment of the surface stress acting on the particle. We calculate the pressure moments of two unequal rigid spheres immersed in a uniform compressible linear flow, using twin multipole expansions and lubrication theory. Following the practice established in previous studies on two-body hydrodynamic interactions at low Reynolds number, the results are expressed in terms of a new (stresslet) resistance function

Brady, John F.

Khair, Aditya S.

Swaroop, Manuj

English

Caltech Authors - Main

PHYSICS OF FLUIDS 18, 043102 2006A new resistance function for two rigid spheres in a uniform
compressible low-Reynolds-number flow
Aditya S. Khair, Manuj Swaroop, and John F. Brady
Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena,
California 91125
Received 13 October 2005; accepted 10 March 2006; published online 21 April 2006
The pressure moment of a rigid particle is defined as the trace of the first moment of the surface
stress acting on the particle. We calculate the pressure moments of two unequal rigid spheres
immersed in a uniform compressible linear flow, using twin multipole expansions and lubrication
theory. Following the practice established in previous studies on two-body hydrodynamic
interactions at low Reynolds number, the results are expressed in terms of a new stresslet
resistance function. © 2006 American Institute of Physics. DOI: 10.1063/1.2194559I. INTRODUCTION
In low-Reynolds-number fluid dynamics, it is common
to characterize interactions between particles using the mo-
ments of the surface stress acting on each particle. Tradition-
ally, the first moment has been split into two parts: an anti-
symmetric part, which gives the torque acting on a particle,
and a traceless symmetric part called the stresslet. More re-
cently, Jeffrey, Morris, and Brady1 hereafter JMB calcu-
lated the trace of the first moment, which they termed the
“pressure moment,” and used it to determine the osmotic
pressure in a dilute colloidal dispersion. Specifically, they
computed the pressure moments for two unequal rigid
spheres immersed in an incompressible linear flow, i.e., a
flow with a traceless ambient rate of strain tensor. However,
just as the pressure moment is the trace of the first moment
of the surface stress, it is natural to consider the contribution
to the pressure moment from compressible linear flows for
which the trace of the rate of strain tensor is nonzero. Put
another way, if one allows the stresslet to possess a trace
i.e., the pressure moment, one must consider the contribu-
tion to it from a rate of strain tensor with nonzero trace.2
One should not think, however, that the present exercise
is of purely mathematical interest. Recently, Brady, Khair,
and Swaroop3 hereafter BKS computed the bulk viscosity
also known as the second or expansion viscosity of a col-
loidal dispersion to second order in volume fraction of the
rigid spherical suspended particles. This calculation in-
volves determination of the dispersion microstructure, which
reflects a balance between an imposed uniform expansion
flow for which the trace of the rate of strain tensor is a
nonzero constant, serving to drive the dispersion out of
equilibrium, and Brownian diffusion of the colloidal par-
ticles, which acts to restore equilibrium. There is a contribu-
tion to the bulk viscosity due to the inability of the rigid
particles to deform in the expansion flow, which is reflected
mathematically as an average of the pressure moments of the
particles weighted over the nonequilibrium microstructure.
Thus, the calculation of the pressure moments for two rigid
spheres immersed in a uniform compressible flow is indeed
of practical interest. Furthermore, we anticipate that the re-
1070-6631/2006/184/043102/5/$23.00 18, 04310
Downloaded 13 May 2006 to 131.215.240.9. Redistribution subject to sults presented in this work may be useful in future studies of
colloidal dispersions in compressible flow fields.
Finally, the function relating the pressure moment to the
trace of the rate of strain “completes” the set of two-body
hydrodynamic resistance functions for spherical particles.
We proceed by recalling the constitutive equation for the
stress tensor  in a compressible Newtonian fluid see Secs.
3.3 and 3.4 of Batchelor4
 = − pthI + 2e +  − 23 · uI ,
where pth is the thermodynamic pressure,
5 e= 12 u+ u
†
is the rate of strain tensor with u the fluid velocity,  is the
shear viscosity,  is the bulk viscosity, and I is the identity
tensor. The pressure moment S of a particle is given by
S = − x ·  · n − 3n · udA; 1
the integral is over the surface of the particle with normal n
pointing into the fluid, and x is the position vector measured
from the center of the particle. For rigid particles, the normal
component of the fluid velocity vanishes on the particle sur-
face; thus 1 reduces to
S = − x ·  · ndA . 2
In Stokes flow the velocity field can be decomposed into a
uniform rate of expansion everywhere in the fluid which, by
definition, has an isotropic rate of strain tensor and a distur-
bance flow created by any immersed particles and their
fluid-mediated interactions, which satisfies the usual incom-
pressible Stokes equations; that is,
u =
1
3
Ex + us,
with  ·u=E and  ·us=0. In turn, the fluid stress can be
decomposed into an isotropic contribution due to the uni-
form expansion flow, E=−pth−EI, and a disturbance
stress s=−psI+2es, with  ·s=0. Here, ps is the dynami-
cal pressure of the incompressible Stokes flow. Substituting
© 2006 American Institute of Physics2-1
AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
043102-2 Khair, Swaroop, and Brady Phys. Fluids 18, 043102 2006this decomposition into 2 we have for the pressure moment
S = pth − E  x · ndA − x · s · ndA , 3
where the first term on the right-hand side of 3, to be de-
noted as Sc, is the contribution to the pressure moment from
the uniform expansion flow, and the second term, to be de-
noted as Ss, is the contribution from the incompressible
Stokes disturbance flow.6
Now, suppose we have two particles in such a uniform
expansion flow. The disturbance stress on the surface of a
particle may be written as s=s,0+s,i, i.e., a sum of the
single isolated particle contribution s,0 and an “interac-
tion” contribution s,i due to the presence of the second par-
ticle. For a single spherical particle of radius a in the expan-
sion flow, the disturbance velocity is BKS
us,0 = −
1
3
Ea3x/r3 and ps,0 = 0,
where r= x. Note, for the single spherical particle Sc
=4a3pth−E. The disturbance stress is then
s,0 = −
2
3
a3E I
r3
−
3xx
r5
 .
Therefore, the disturbance flow contribution to the pressure
moment may be written as Ss=Ss,0+Ss,i, where
Ss,0 = −
16
3
a3E
is the single particle “self” contribution and
Ss,i = − x · s,i · ndA 4
is the contribution due to hydrodynamic interactions with the
second particle. Note, 4 is identical to the pressure moment
for two particles in an incompressible linear flow, as consid-
ered in JMB, for which there is no self-contribution analo-
gous to Ss,0. Henceforth, for clarity of presentation, we drop
the superscripts s and i; they are to be understood.
It is the aim of this work to calculate the disturbance
flow contribution to the pressure moments for two spherical
particles immersed in a uniform expansion flow. As per pre-
vious investigations on two-particle hydrodynamic interac-
tions Jeffrey and Onishi,7 Jeffrey,8 and JMB, twin multi-
pole expansions and lubrication theory are employed. We
follow the nomenclature of these earlier works throughout,
and, in the interest of brevity, we do not repeat their deriva-
tions and equations here.
II. RESISTANCE FUNCTIONS
Consider two rigid spherical particles of radii a1 and a2,
respectively. The pressure moment of sphere  S is related
to the translational velocities U of the spheres and their
rates of strain E via a generalized matrix equation,
Downloaded 13 May 2006 to 131.215.240.9. Redistribution subject to S1
S2
 = P11 P12 Q11 Q12
P21 P22 Q21 Q22
 ·	
U1 − Ux1
U2 − Ux2
E1 − E
E2 − E

 , 5
where the ambient velocity field at a point x is given by
Ux = U +   x + E · x ,
with constant U, , and E. Note that the pressure mo-
ments in 5 are for the incompressible Stokes disturbance
flow only; the contribution from the uniform expansion, Sc,
is not included. The vectors P relate the pressure moment
to the translational velocities of the particles and are given
by
P = a + a2X
P d ,
where d= x2−x1 / x2−x1 is the dimensionless unit vector
directed from particle 1 to particle 2. The resistance func-
tions X
P have been calculated by JMB. The second-rank
tensors Q relate the pressure moment to the rate of strain,
and JMB write Q=a+a3X
Q dd− 13I, so that, as with
the other resistance tensors contracted with the rate of strain,
the Q are traceless. This is, however, only valid for an
incompressible linear flow for which the rate of strain is
traceless; for a spatially uniform compressible linear flow
there is an additional contribution to the Q arising from the
trace of the rate of strain tensor, with a new resistance func-
tion that we designate as T
Q . The complete Q function is
then
Q = a + a3XQ dd − 13I + TQ 13I;
hence, by construction, Qii
=a+a3T
Q . The T
Q are
functions of s=2r / a1+a2 with r= x2−x1 and =a2 /a1
only. The symmetries obeyed by the functions are labeling
ones,
T
Q s, = T3−,3−
Q s,−1 .
III. THE FUNCTIONS T
Q
A. Method of reflections
We start by deriving the first couple of terms via the
method of reflections. Let sphere 1 deform uniformly at a
rate E1=
1
3E1I. There is no pressure field created by this de-
formation; however, there is a velocity field u
=1/3E1a1a1 /r2d. Sphere 2 responds to this ambient veloc-
ity field by exerting a force on the fluid, which leads to an
induced S1 of
S1 =
8
3
a1
3E1T11
Q − 2 = − a1 + a22X12
P 1
3
a1
3/r2E1,
implying
T11
Q = 2 +
12
4 4 + ¯ .1 +  s
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043102-3 A new resistance function for two rigid spheres Phys. Fluids 18, 043102 2006B. Twin multipole expansions
Consider sphere 1 deforming at a rate E1=
1
3E1I and
sphere 2 deforming at E2=
1
3E2I. We follow Jeffrey and
Onishi7 hereafter JO in equating the surface velocities of
the spheres, i.e., we set a1E1=a2E2. The functions 	, 
, and
 defined in JO equation 2.6 take the values
	mn
 = −
1
2

mn
 =
1
3
aE0m0n, mn
 = 0.
As in JO, defining the problem this way ensures that the
coefficients are the same for each sphere, allowing one to
drop the label . We expand p0n and v0n as a double series in
t=a /r, viz.,
p0n =
2
3
aE
p=0


q=0

Pnpqt
pt3−
q ,
v0n =
1
3
aE
p=0


q=0

1
2n + 1
Vnpqt
pt3−
q .
The recurrence relations for this problem are
V0pq = − 0p0q, P0pq = 0,
Vnpq = Pnpq −
2n
2n + 3n + 1s=1
q n + s
n
Psq−sp−n−1,
Pnpq = 
s=0
q
1
n + 1
n + s
n

n2n + 12ns + 2 − n − s
22s − 1n + s
Psq−sp−n+1
−
n2n + 12n − 1
22s + 1
Vsq−2−sp−n+1
−
n2n − 1
2n + 1
Psq−sp−n−1 .
The pressure moment of sphere 1 is
S1 =
8
3
a1
3E1T11Q + 1 + 38 T12Q  ,
and from this we find
T11
Q +
1 + 3
8
T12
Q = 
n=1


p=0


q=0

Pnpqt2
n+pt1
q+1.
We define a complimentary problem with a1E1=−a2E2,
which gives another expression for the pressure moment of
sphere 1,
T11
Q −
1 + 3
8
T12
Q = 
n=1


p=0


q=0

− 1n+p+q+2Pnpqt2
n+pt1
q+1.Thus, we conclude
Downloaded 13 May 2006 to 131.215.240.9. Redistribution subject to T11
Q = 
m=0
m even


q=1
m−1

n=1
q+1/2
Pnq−nm−q−1
2m
1 + msm
q
= 
m=0
m even

fm
1 + msm
6
and
T12
Q =
8
1 + 3 m=1
m odd

fm
1 + msm
. 7
The first few terms of the fm are
f0 = 2, f1 = f2 = f3 = 0, f4 = 12 ,
f5 = 36
4, f6 = 108
2 + 3203,
f7 = 432
4 + 3245 + 4326,
f8 = 432
2 + 9723 + 25924 + 33605.
Strictly speaking, through the twin-multipole expansion we
calculate the interaction contribution Ss,i to the disturbance
pressure moment; however, the self-contribution Ss,0 is in-
cluded in the definition of T11
Q by requiring f0=2. For the
complete pressure moment see 3 one must, of course, add
the contribution due to the uniform expansion flow Sc.
C. Lubrication theory
Let sphere 1 deform with its center fixed at a rate E1
= 1  3E1I in close proximity to sphere 2 which is at rest.
Following Jeffrey,9 we write the pressure moment of sphere
1 as
S1 = − x ·  · ndA
= a1F1 · d − a1 n − d ·  · ndA .
Substituting the known result for F1, we obtain
T11
Q =
1
2
X11
P −
3
8a1
2E1
 n − d ·  · ndA .
Performing the integration above see the Appendix yields
T11
Q = g1
−1 + g2ln 
−1 + Q11
T + g3 ln 
−1. 8
In a similar vein, consideration of the pressure moment for
sphere 2 gives
1
8
1 + 3T12
Q = g4
−1 + g5ln 
−1 +
1
8
1 + 3Q12
T + g6 ln 
−1,
where
g1 =
32
3 , g2 =
3 − 132 − 93
3 ,21 +  201 + 
AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
043102-4 Khair, Swaroop, and Brady Phys. Fluids 18, 043102 2006g3 =
5 − 274 − 2012 − 1903 − 12692 
4 + 5
2801 + 3
,
9
g4 =
33
21 + 3
, g5 =
3− 42 − 133 − 44
201 + 3
,
g6 =
1 − 43 − 922 + 1232 
3 − 924 − 5695
2801 + 3
.
The nondimensional gap width is =s−2, and the Q
T  are
functions that we tabulate here. Note, in the special case of
equal spheres, =1, the singular terms in T11
Q are the same as
those in T12
Q , but we see that in general this is not true.
IV. ARBITRARY SEPARATIONS
The singularities cause slow convergence of the series
6 and 7 when s is near 2; following Jeffrey8, we remove
them by writing
T11
Q = g1
4
s2 − 4
+ g2 + g314s2 − 1ln s2s2 − 4 − g3 + 2
+ 
m=2
m even
  f̃m
1 + m
− g1 −
2g2
m
+
4g3
mm + 2
2
s
m,
10
where f̃m=2−mfm, and the gi take on the values from
9. Mathematically, 10 is equivalent to 6; numerically,
the rate of convergence has improved as the coefficients of
s−m now decay faster by a factor of m−2. Similarly, for T12
Q we
have
1 + 3
8
T12
Q = g4
2s
s2 − 4
+ g5 + g614s2 − 1ln s + 2s − 2 − g6s,
+ 
m=1
m odd
  f̃m
1 + m
− g1 −
2g2
m
+
4g3
mm + 2

2
s
m.
Numerical tabulations of T
Q are not given because the ex-
pressions and data given above are accurate to at least two
significant digits for all s. However, we do tabulate the Q
T
as they provide a good test of the convergence of the series
and may be useful in studies of nearly touching spheres.
Expanding 10 about s=2 and comparing to 8, we find
Q11
T = −
1
4
g1 − g3 + 2 + 
m=2
m even
  f̃m
1 + m
− g1 −
2g2
m
+
4g3
mm + 2
 , 11and similarly
Downloaded 13 May 2006 to 131.215.240.9. Redistribution subject to 1 + 3
8
Q12
T =
1
4
g4 + g5 ln 4 − 2g6 + 
m=1
m odd
  f̃m
1 + m
− g1
−
2g2
m
+
4g3
mm + 2
 . 12
V. RESULTS FOR T
Q
To illustrate the behavior of the pressure moment, in Fig.
1 we plot T11
Q and T12
Q as a function of s for identical spheres,
=1. Note, the singular behavior of T11
Q and T12
Q as s→2 is
proportional to the corresponding force to rate of strain cou-
pling i.e., to the functions X11
P and X12
P , respectively. In
Table I we give results of summing the series 11 and 12
to obtain the Q
T . For =1 the series were summed to
300 terms, while for 1 they were summed to 125 terms—
the data in Table I are accurate to a minimum of two decimal
places and more where shown. Of course, by summing fur-
ther terms in the series one can obtain increasingly accurate
values of the Q
T .
FIG. 1. The functions T11
Q solid line and T12
Q dashed line for equal-sized
spheres plotted versus the dimensionless separation distance s.
TABLE I. Values of the function Q
T , with =a2 /a1 the size ratio of the
two spheres, appearing in the asymptotic form of T
Q for small separation.
 Q11
T Q12
T Q21
T Q22
T
1 2.355 0.210 0.210 2.355
2 2.80 0.21 0.202 2.170
3 3.20 0.19 0.179 2.120
4 3.56 0.17 0.154 2.096
5 3.87 0.15 0.131 2.080
10 4.96 0.08 0.064 2.043AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
043102-5 A new resistance function for two rigid spheres Phys. Fluids 18, 043102 2006APPENDIX: LUBRICATION THEORY CALCULATIONS
Here, we comment briefly on the solution of the Stokes
equations in the lubrication approximation used in Sec. III C
to calculate the pressure moments of two nearly touching
spheres. The flow problem is almost identical to that consid-
ered in Sec. 3 of Jeffrey and Corless10—the only difference
being that the surface velocity of the deforming sphere in
their notation is now
u = −
aE
3
1 +  − z/ak +
rE
3
r ,
corresponding to an isotropic rate of strain uniform
expansion/contraction E= 13EI throughout the sphere. Con-
sequently, in stretched coordinates choosing the velocity
scale V= 1  3aE the boundary conditions on the deform-
ing sphere are
W = − 1 − 1 − Z and U = R .
The solution of the lubrication equations proceeds as per
Jeffrey and Corless;10 full details of the calculations are
available upon request.
1D. J. Jeffrey, J. F. Morris, and J. F. Brady, “The pressure moments for two
rigid spheres in low-Reynolds number flow,” Phys. Fluids A 5, 2317
1993.
2Note that the contribution to the force/torque from the trace of the rate of
strain can be found from the pressure moments computed by JMB via the
symmetry of the grand resistance matrix; the P functions are sufficient.Downloaded 13 May 2006 to 131.215.240.9. Redistribution subject to 3J. F. Brady, A. S. Khair, and M. Swaroop, “On the bulk viscosity of
suspensions,” J. Fluid Mech. 554, 109 2006.
4G. K. Batchelor, An Introduction to Fluid Dynamics Cambridge Univer-
sity Press, Cambridge, 1973.
5For a material element in the fluid, pth is the equilibrium or thermody-
namic pressure, as computed from the fluid’s equation of state, corre-
sponding to the instantaneous density and internal energy of the element.
The mechanical pressure, pmech, is defined as the mean normal stress,
pmech−
1
3 :I= pth− ·u. For an incompressible fluid,  ·u=0, the me-
chanical and thermodynamic pressures are equivalent and the former is
known to within an arbitrary constant; only the dynamical pressure gradi-
ent, p, enters the momentum balance.
6Note, we assume that the uniform expansion/compression of the fluid does
not lead to macroscopic i.e., on length scales much larger than the particle
size spatial gradients in the fluid density. Thus, from the continuity equa-
tion, the density decreases increases exponentially fast in time for a
uniform expansion compression, but it is not a function of spatial posi-
tion. From the fluid’s equation of state, this implies that pth is, likewise, a
function of time only. Hence, one is permitted to take pth outside the
integral for Sc. Finally, note that the expressions for the pressure moments
and formulation of the resistance matrix see 5 are valid at any instant in
time and are not, therefore, affected by temporal variation of the fluid
density and thermodynamic pressure.
7D. J. Jeffrey and Y. Onishi, “Calculation of the resistance and mobility
functions for two unequal rigid spheres in low-Reynolds-number-flow,” J.
Fluid Mech. 139, 261 1984.
8D. J. Jeffrey, “The calculation of the low Reynolds number resistance
functions for two unequal spheres,” Phys. Fluids A 4, 16 1992.
9D. J. Jeffrey, “Higher-order corrections to the axisymmetric interactions of
nearly-touching spheres,” Phys. Fluids A 1, 1740 1989.
10D. J. Jeffrey and R. M. Corless, “Forces and stresslets for the axisymmet-
ric motion of nearly touching unequal spheres,” PCH, PhysicoChem. Hy-
drodyn. 10, 461 1988.AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp


A new resistance function for two rigid spheres in a uniform compressible low-Reynolds-number flow

PHYSICS OF FLUIDS 18, 043102 2006A new resistance function for two rigid spheres in a uniform
compressible low-Reynolds-number flow
Aditya S. Khair, Manuj Swaroop, and John F. Brady
Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena,
California 91125
Received 13 October 2005; accepted 10 March 2006; published online 21 April 2006
The pressure moment of a rigid particle is defined as the trace of the first moment of the surface
stress acting on the particle. We calculate the pressure moments of two unequal rigid spheres
immersed in a uniform compressible linear flow, using twin multipole expansions and lubrication
theory. Following the practice established in previous studies on two-body hydrodynamic
interactions at low Reynolds number, the results are expressed in terms of a new stresslet
resistance function. © 2006 American Institute of Physics. DOI: 10.1063/1.2194559I. INTRODUCTION
In low-Reynolds-number fluid dynamics, it is common
to characterize interactions between particles using the mo-
ments of the surface stress acting on each particle. Tradition-
ally, the first moment has been split into two parts: an anti-
symmetric part, which gives the torque acting on a particle,
and a traceless symmetric part called the stresslet. More re-
cently, Jeffrey, Morris, and Brady1 hereafter JMB calcu-
lated the trace of the first moment, which they termed the
“pressure moment,” and used it to determine the osmotic
pressure in a dilute colloidal dispersion. Specifically, they
computed the pressure moments for two unequal rigid
spheres immersed in an incompressible linear flow, i.e., a
flow with a traceless ambient rate of strain tensor. However,
just as the pressure moment is the trace of the first moment
of the surface stress, it is natural to consider the contribution
to the pressure moment from compressible linear flows for
which the trace of the rate of strain tensor is nonzero. Put
another way, if one allows the stresslet to possess a trace
i.e., the pressure moment, one must consider the contribu-
tion to it from a rate of strain tensor with nonzero trace.2
One should not think, however, that the present exercise
is of purely mathematical interest. Recently, Brady, Khair,
and Swaroop3 hereafter BKS computed the bulk viscosity
also known as the second or expansion viscosity of a col-
loidal dispersion to second order in volume fraction of the
rigid spherical suspended particles. This calculation in-
volves determination of the dispersion microstructure, which
reflects a balance between an imposed uniform expansion
flow for which the trace of the rate of strain tensor is a
nonzero constant, serving to drive the dispersion out of
equilibrium, and Brownian diffusion of the colloidal par-
ticles, which acts to restore equilibrium. There is a contribu-
tion to the bulk viscosity due to the inability of the rigid
particles to deform in the expansion flow, which is reflected
mathematically as an average of the pressure moments of the
particles weighted over the nonequilibrium microstructure.
Thus, the calculation of the pressure moments for two rigid
spheres immersed in a uniform compressible flow is indeed
of practical interest. Furthermore, we anticipate that the re-
1070-6631/2006/184/043102/5/$23.00 18, 04310
Downloaded 13 May 2006 to 131.215.240.9. Redistribution subject to sults presented in this work may be useful in future studies of
colloidal dispersions in compressible flow fields.
Finally, the function relating the pressure moment to the
trace of the rate of strain “completes” the set of two-body
hydrodynamic resistance functions for spherical particles.
We proceed by recalling the constitutive equation for the
stress tensor  in a compressible Newtonian fluid see Secs.
3.3 and 3.4 of Batchelor4
 = − pthI + 2e +  − 23 · uI ,
where pth is the thermodynamic pressure,5 e=
1
2 u+ u
†
is the rate of strain tensor with u the fluid velocity,  is the
shear viscosity,  is the bulk viscosity, and I is the identity
tensor. The pressure moment S of a particle is given by
S = − x ·  · n − 3n · udA; 1
the integral is over the surface of the particle with normal n
pointing into the fluid, and x is the position vector measured
from the center of the particle. For rigid particles, the normal
component of the fluid velocity vanishes on the particle sur-
face; thus 1 reduces to
S = − x ·  · ndA . 2
In Stokes flow the velocity field can be decomposed into a
uniform rate of expansion everywhere in the fluid which, by
definition, has an isotropic rate of strain tensor and a distur-
bance flow created by any immersed particles and their
fluid-mediated interactions, which satisfies the usual incom-
pressible Stokes equations; that is,
u =
1
3
Ex + us,
with  ·u=E and  ·us=0. In turn, the fluid stress can be
decomposed into an isotropic contribution due to the uni-
form expansion flow, E=−pth−EI, and a disturbance
stress s=−psI+2es, with  ·s=0. Here, ps is the dynami-
cal pressure of the incompressible Stokes flow. Substituting
© 2006 American Institute of Physics2-1
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043102-2 Khair, Swaroop, and Brady Phys. Fluids 18, 043102 2006this decomposition into 2 we have for the pressure moment
S = pth − E  x · ndA − x · s · ndA , 3
where the first term on the right-hand side of 3, to be de-
noted as Sc, is the contribution to the pressure moment from
the uniform expansion flow, and the second term, to be de-
noted as Ss, is the contribution from the incompressible
Stokes disturbance flow.6
Now, suppose we have two particles in such a uniform
expansion flow. The disturbance stress on the surface of a
particle may be written as s=s,0+s,i, i.e., a sum of the
single isolated particle contribution s,0 and an “interac-
tion” contribution s,i due to the presence of the second par-
ticle. For a single spherical particle of radius a in the expan-
sion flow, the disturbance velocity is BKS
us,0 = −
1
3
Ea3x/r3 and ps,0 = 0,
where r= x. Note, for the single spherical particle Sc
=4a3pth−E. The disturbance stress is then
s,0 = −
2
3
a3E I
r3
−
3xx
r5
 .
Therefore, the disturbance flow contribution to the pressure
moment may be written as Ss=Ss,0+Ss,i, where
Ss,0 = −
16
3
a3E
is the single particle “self” contribution and
Ss,i = − x · s,i · ndA 4
is the contribution due to hydrodynamic interactions with the
second particle. Note, 4 is identical to the pressure moment
for two particles in an incompressible linear flow, as consid-
ered in JMB, for which there is no self-contribution analo-
gous to Ss,0. Henceforth, for clarity of presentation, we drop
the superscripts s and i; they are to be understood.
It is the aim of this work to calculate the disturbance
flow contribution to the pressure moments for two spherical
particles immersed in a uniform expansion flow. As per pre-
vious investigations on two-particle hydrodynamic interac-
tions Jeffrey and Onishi,7 Jeffrey,8 and JMB, twin multi-
pole expansions and lubrication theory are employed. We
follow the nomenclature of these earlier works throughout,
and, in the interest of brevity, we do not repeat their deriva-
tions and equations here.
II. RESISTANCE FUNCTIONS
Consider two rigid spherical particles of radii a1 and a2,
respectively. The pressure moment of sphere  S is related
to the translational velocities U of the spheres and their
rates of strain E via a generalized matrix equation,
Downloaded 13 May 2006 to 131.215.240.9. Redistribution subject to S1S2  = P11 P12 Q11 Q12P21 P22 Q21 Q22 ·	
U1 − Ux1
U2 − Ux2
E1 − E
E2 − E

 , 5
where the ambient velocity field at a point x is given by
Ux = U +  x + E · x ,
with constant U, , and E. Note that the pressure mo-
ments in 5 are for the incompressible Stokes disturbance
flow only; the contribution from the uniform expansion, Sc,
is not included. The vectors P relate the pressure moment
to the translational velocities of the particles and are given
by
P = a + a2X
P d ,
where d= x2−x1 / x2−x1 is the dimensionless unit vector
directed from particle 1 to particle 2. The resistance func-
tions X
P have been calculated by JMB. The second-rank
tensors Q relate the pressure moment to the rate of strain,
and JMB write Q=a+a3XQ dd− 13I, so that, as with
the other resistance tensors contracted with the rate of strain,
the Q are traceless. This is, however, only valid for an
incompressible linear flow for which the rate of strain is
traceless; for a spatially uniform compressible linear flow
there is an additional contribution to the Q arising from the
trace of the rate of strain tensor, with a new resistance func-
tion that we designate as T
Q
. The complete Q function is
then
Q = a + a3XQ dd − 13I + TQ 13I;
hence, by construction, Qii=a+a3TQ . The TQ are
functions of s=2r / a1+a2 with r= x2−x1 and =a2 /a1
only. The symmetries obeyed by the functions are labeling
ones,
T
Q s, = T3−,3−
Q s,−1 .
III. THE FUNCTIONS TQ
A. Method of reflections
We start by deriving the first couple of terms via the
method of reflections. Let sphere 1 deform uniformly at a
rate E1=
1
3E1I. There is no pressure field created by this de-
formation; however, there is a velocity field u
=1/3E1a1a1 /r2d. Sphere 2 responds to this ambient veloc-
ity field by exerting a force on the fluid, which leads to an
induced S1 of
S1 =
8
3
a1
3E1T11
Q
− 2 = − a1 + a22X12
P 1
3
a1
3/r2E1,
implying
T11
Q
= 2 +
12
4 4 + ¯ .1 +  s
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043102-3 A new resistance function for two rigid spheres Phys. Fluids 18, 043102 2006B. Twin multipole expansions
Consider sphere 1 deforming at a rate E1=
1
3E1I and
sphere 2 deforming at E2=
1
3E2I. We follow Jeffrey and
Onishi7 hereafter JO in equating the surface velocities of
the spheres, i.e., we set a1E1=a2E2. The functions 	, 
, and
 defined in JO equation 2.6 take the values
	mn

= −
1
2

mn

=
1
3
aE0m0n, mn

= 0.
As in JO, defining the problem this way ensures that the
coefficients are the same for each sphere, allowing one to
drop the label . We expand p0n and v0n as a double series in
t=a /r, viz.,
p0n =
2
3
aE
p=0


q=0

Pnpqt
pt3−
q
,
v0n =
1
3
aE
p=0


q=0
 1
2n + 1
Vnpqt
pt3−
q
.
The recurrence relations for this problem are
V0pq = − 0p0q, P0pq = 0,
Vnpq = Pnpq −
2n
2n + 3n + 1s=1
q n + s
n
Psq−sp−n−1,
Pnpq = 
s=0
q 1
n + 1n + sn 
n2n + 12ns + 2 − n − s22s − 1n + s Psq−sp−n+1
−
n2n + 12n − 1
22s + 1
Vsq−2−sp−n+1
−
n2n − 1
2n + 1
Psq−sp−n−1 .
The pressure moment of sphere 1 is
S1 =
8
3
a1
3E1T11Q + 1 + 38 T12Q  ,
and from this we find
T11
Q +
1 + 3
8
T12
Q
= 
n=1


p=0


q=0

Pnpqt2
n+pt1
q+1
.
We define a complimentary problem with a1E1=−a2E2,
which gives another expression for the pressure moment of
sphere 1,
T11
Q
−
1 + 3
8
T12
Q
= 
n=1


p=0


q=0

− 1n+p+q+2Pnpqt2
n+pt1
q+1
.Thus, we conclude
Downloaded 13 May 2006 to 131.215.240.9. Redistribution subject to T11
Q
= 
m=0
m even


q=1
m−1

n=1
q+1/2
Pnq−nm−q−1
2m
1 + msm
q
= 
m=0
m even
 fm
1 + msm
6
and
T12
Q
=
8
1 + 3 m=1
m odd
 fm
1 + msm
. 7
The first few terms of the fm are
f0 = 2, f1 = f2 = f3 = 0, f4 = 12 ,
f5 = 364, f6 = 1082 + 3203,
f7 = 4324 + 3245 + 4326,
f8 = 4322 + 9723 + 25924 + 33605.
Strictly speaking, through the twin-multipole expansion we
calculate the interaction contribution Ss,i to the disturbance
pressure moment; however, the self-contribution Ss,0 is in-
cluded in the definition of T11Q by requiring f0=2. For the
complete pressure moment see 3 one must, of course, add
the contribution due to the uniform expansion flow Sc.
C. Lubrication theory
Let sphere 1 deform with its center fixed at a rate E1
= 1  3E1I in close proximity to sphere 2 which is at rest.
Following Jeffrey,9 we write the pressure moment of sphere
1 as
S1 = − x ·  · ndA
= a1F1 · d − a1 n − d ·  · ndA .
Substituting the known result for F1, we obtain
T11
Q
=
1
2
X11
P
−
3
8a1
2E1
 n − d ·  · ndA .
Performing the integration above see the Appendix yields
T11
Q
= g1−1 + g2ln −1 + Q11T + g3 ln −1. 8
In a similar vein, consideration of the pressure moment for
sphere 2 gives
1
8
1 + 3T12
Q
= g4−1 + g5ln −1 +
1
8
1 + 3Q12T + g6 ln −1,
where
g1 =
32
3 , g2 =
3 − 132 − 93
3 ,21 +  201 + 
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043102-4 Khair, Swaroop, and Brady Phys. Fluids 18, 043102 2006g3 =
5 − 274 − 2012 − 1903 − 12692 4 + 5
2801 + 3
,
9
g4 =
33
21 + 3
, g5 =
3− 42 − 133 − 44
201 + 3
,
g6 =
1 − 43 − 922 + 1232 
3
− 924 − 5695
2801 + 3
.
The nondimensional gap width is =s−2, and the QT  are
functions that we tabulate here. Note, in the special case of
equal spheres, =1, the singular terms in T11
Q are the same as
those in T12
Q
, but we see that in general this is not true.
IV. ARBITRARY SEPARATIONS
The singularities cause slow convergence of the series
6 and 7 when s is near 2; following Jeffrey8, we remove
them by writing
T11
Q
= g1
4
s2 − 4
+ g2 + g314s2 − 1ln s2s2 − 4 − g3 + 2
+ 
m=2
m even
  f˜m
1 + m
− g1 −
2g2
m
+
4g3
mm + 2
2
s
m,
10
where f˜m=2−mfm, and the gi take on the values from
9. Mathematically, 10 is equivalent to 6; numerically,
the rate of convergence has improved as the coefficients of
s−m now decay faster by a factor of m−2. Similarly, for T12Q we
have
1 + 3
8
T12
Q
= g4
2s
s2 − 4
+ g5 + g614s2 − 1ln s + 2s − 2 − g6s,
+ 
m=1
m odd
  f˜m
1 + m
− g1 −
2g2
m
+
4g3
mm + 2

2
s
m.
Numerical tabulations of T
Q are not given because the ex-
pressions and data given above are accurate to at least two
significant digits for all s. However, we do tabulate the QT
as they provide a good test of the convergence of the series
and may be useful in studies of nearly touching spheres.
Expanding 10 about s=2 and comparing to 8, we find
Q11T = −
1
4
g1 − g3 + 2 + 
m=2
m even
  f˜m
1 + m
− g1 −
2g2
m
+
4g3
mm + 2
 , 11and similarly
Downloaded 13 May 2006 to 131.215.240.9. Redistribution subject to 1 + 3
8
Q12T =
1
4
g4 + g5 ln 4 − 2g6 + 
m=1
m odd
  f˜m
1 + m
− g1
−
2g2
m
+
4g3
mm + 2
 . 12
V. RESULTS FOR TQ
To illustrate the behavior of the pressure moment, in Fig.
1 we plot T11
Q and T12Q as a function of s for identical spheres,
=1. Note, the singular behavior of T11Q and T12Q as s→2 is
proportional to the corresponding force to rate of strain cou-
pling i.e., to the functions X11
P and X12P , respectively. In
Table I we give results of summing the series 11 and 12
to obtain the QT . For =1 the series were summed to
300 terms, while for 1 they were summed to 125 terms—
the data in Table I are accurate to a minimum of two decimal
places and more where shown. Of course, by summing fur-
ther terms in the series one can obtain increasingly accurate
values of the QT .
FIG. 1. The functions T11Q solid line and T12Q dashed line for equal-sized
spheres plotted versus the dimensionless separation distance s.
TABLE I. Values of the function QT , with =a2 /a1 the size ratio of the
two spheres, appearing in the asymptotic form of T
Q for small separation.
 Q11T Q12T Q21T Q22T
1 2.355 0.210 0.210 2.355
2 2.80 0.21 0.202 2.170
3 3.20 0.19 0.179 2.120
4 3.56 0.17 0.154 2.096
5 3.87 0.15 0.131 2.080
10 4.96 0.08 0.064 2.043AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
043102-5 A new resistance function for two rigid spheres Phys. Fluids 18, 043102 2006APPENDIX: LUBRICATION THEORY CALCULATIONS
Here, we comment briefly on the solution of the Stokes
equations in the lubrication approximation used in Sec. III C
to calculate the pressure moments of two nearly touching
spheres. The flow problem is almost identical to that consid-
ered in Sec. 3 of Jeffrey and Corless10—the only difference
being that the surface velocity of the deforming sphere in
their notation is now
u = −
aE
3
1 +  − z/ak +
rE
3
r ,
corresponding to an isotropic rate of strain uniform
expansion/contraction E= 13EI throughout the sphere. Con-
sequently, in stretched coordinates choosing the velocity
scale V= 1  3aE the boundary conditions on the deform-
ing sphere are
W = − 1 − 1 − Z and U = R .
The solution of the lubrication equations proceeds as per
Jeffrey and Corless;10 full details of the calculations are
available upon request.
1D. J. Jeffrey, J. F. Morris, and J. F. Brady, “The pressure moments for two
rigid spheres in low-Reynolds number flow,” Phys. Fluids A 5, 2317
1993.
2Note that the contribution to the force/torque from the trace of the rate of
strain can be found from the pressure moments computed by JMB via the
symmetry of the grand resistance matrix; the P functions are sufficient.Downloaded 13 May 2006 to 131.215.240.9. Redistribution subject to 3J. F. Brady, A. S. Khair, and M. Swaroop, “On the bulk viscosity of
suspensions,” J. Fluid Mech. 554, 109 2006.
4G. K. Batchelor, An Introduction to Fluid Dynamics Cambridge Univer-
sity Press, Cambridge, 1973.
5For a material element in the fluid, pth is the equilibrium or thermody-
namic pressure, as computed from the fluid’s equation of state, corre-
sponding to the instantaneous density and internal energy of the element.
The mechanical pressure, pmech, is defined as the mean normal stress,
pmech−
1
3 :I= pth− ·u. For an incompressible fluid,  ·u=0, the me-
chanical and thermodynamic pressures are equivalent and the former is
known to within an arbitrary constant; only the dynamical pressure gradi-
ent, p, enters the momentum balance.
6Note, we assume that the uniform expansion/compression of the fluid does
not lead to macroscopic i.e., on length scales much larger than the particle
size spatial gradients in the fluid density. Thus, from the continuity equa-
tion, the density decreases increases exponentially fast in time for a
uniform expansion compression, but it is not a function of spatial posi-
tion. From the fluid’s equation of state, this implies that pth is, likewise, a
function of time only. Hence, one is permitted to take pth outside the
integral for Sc. Finally, note that the expressions for the pressure moments
and formulation of the resistance matrix see 5 are valid at any instant in
time and are not, therefore, affected by temporal variation of the fluid
density and thermodynamic pressure.
7D. J. Jeffrey and Y. Onishi, “Calculation of the resistance and mobility
functions for two unequal rigid spheres in low-Reynolds-number-flow,” J.
Fluid Mech. 139, 261 1984.
8D. J. Jeffrey, “The calculation of the low Reynolds number resistance
functions for two unequal spheres,” Phys. Fluids A 4, 16 1992.
9D. J. Jeffrey, “Higher-order corrections to the axisymmetric interactions of
nearly-touching spheres,” Phys. Fluids A 1, 1740 1989.
10D. J. Jeffrey and R. M. Corless, “Forces and stresslets for the axisymmet-
ric motion of nearly touching unequal spheres,” PCH, PhysicoChem. Hy-
drodyn. 10, 461 1988.AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp


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A new resistance function for two rigid spheres in a uniform compressible low-Reynolds-number flow

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