Simulation of hydrodynamically interacting particles near a no-slip
boundary
James W. Swan and John F. Brady
Division of Chemistry and Chemical Engineering, California Institute of Technology,
Pasadena, California 91125, USA
Received 9 May 2007; accepted 6 September 2007; published online 14 November 2007
The dynamics of spherical particles near a single plane wall are computed using an extension of the
Stokesian dynamics method that includes long-range many-body and pairwise lubrication
interactions between the spheres and the wall in Stokes flow. Extra care is taken to ensure that the
mobility and resistance tensors are symmetric, positive, and definite—something which is
ineluctable for particles in low-Reynolds-number flows. We discuss why two previous simulation
methods for particles near a plane wall, one using multipole expansions and the other using the
Rotne-Prager tensor, fail to produce symmetric resistance and mobility tensors. Additionally, we
offer some insight on how the Stokesian dynamics paradigm might be extended to study the
dynamics of particles in any confining geometry. ©
2007 American Institute of Physics
.
DOI:
10.1063/1.2803837
I. INTRODUCTION
Computational simulations of colloidal particles typi-
cally study the properties of unbounded suspensions. How-
ever, suspensions are often bounded by walls, and a rich
assortment of phenomena including templated self-assembly
1
and shear-induced resuspension
2
arise precisely because hy-
drodynamic flows are intimately coupled to the interactions
of particles with boundaries. For colloidal particles in low-
Reynolds-number flows, a mobility tensor linearly couples
the forces and torques on particles to their velocities and
provides a complete characterization of the hydrodynamic
interactions among particles. In principle, this tensor in-
cludes the effects of a plane wall on the hydrodynamics, but
including these effects is not trivial.
It is often the case that details of the fluid motion
throughout a suspension are ignored in simulations of many
hydrodynamically interacting particles. In fact, the choice to
embed the fluid flow implicitly in mobility and resistance
tensors allows for the rapid simulation of the dynamics of
particles numbering from just a few up into the thousands.
Even though the detailed fluid velocity field is discarded in
favor of these tensors, the Stokes equations dictate specific
properties that must emerge from the hydrodynamic interac-
tions among particles. Principally, the mobility and resis-
tance tensors must be symmetric and positive-definite. While
models can be constructed which have neither of these prop-
erties, the physics governing the Brownian motion of hydro-
dynamically interacting particles require that the mobility
and resistance tensors have orthogonal eigenvectors and
strictly positive eigenvalues.
3
We have a twofold motivation for developing a new ap-
proach to this problem. First, in order to study the Brownian
motion of colloidal particles near a wall, it is essential that
the mobility tensor be symmetric.
3
Second, the relatively
simple and physically intuitive strategy first used in un-
bounded Stokesian dynamics simulations,
4
in which the mo-
bility tensor is constructed directly from Faxén formulas, is
easy to leverage and extend to study a number interesting
problems. Although others have developed simulations of
colloidal particles near a single wall, these studies lack either
the crucial symmetry of the method herein
5
,
6
or the physical
and mathematical straightforwardness of the original Stoke-
sian dynamics technique.
7
,
8
In Sec. II we discuss our ap-
proach to computing the mobility tensor from multipole ex-
pansions and Faxén formulas. Included is a discussion of
why two previous approaches failed to compute symmetric
mobility tensors and cannot be used for Brownian dynamics
simulations. Examples are given illustrating the importance
of the symmetry of the resistance and mobility tensors.
II. THEORY AND METHOD
Consider the problem of particles moving in an ambient
flow field
u
x
. The grand mobility tensor, denoted by
M
,
couples the velocity moments of the particles relative to this
imposed flow
U
−
U
,
−
,−
E
,...
to the moments of
the hydrodynamic force density on the particle surfaces
F
,
T
,
S
,...
,
U
−
U
−
−
E
]
=−
M
UF
M
UL
M
US
̄
M
F
M
L
M
S
̄
M
EF
M
EL
M
ES
̄
]]]
·
F
L
S
]
,
1
where
U
and
are the translational and rotational velocities
of the particles,
U
,
and
E
are velocity, vorticity, and
rate of strain of the imposed flow at the particles’ centers and
F
,
L
, and
S
are the total hydrodynamic force, torque, and
stresslet on the particles. It follows directly from the Stokes
equations and the reciprocal theorem that the grand mobility
tensor is always symmetric, positive, and definite.
9
PHYSICS OF FLUIDS
19
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The structure of Eq.
1
and its symmetries applies
equally well to particles in an unbounded fluid as to particles
adjacent to a wall or walls. Shown in Fig.
1
is a sample
configuration of a pair of spheres
,
above a plane wall.
The hydrodynamic flows generated by motion of these par-
ticles near the wall must produce symmetric interactions
between the pair as well as between each particle and the
wall. Two natural although not necessarily intuitive conse-
quences are that for a single particle near a wall, there
exists couplings between torque and translation and between
force and rotation, and that these couplings are symmetric
M
UL
=
M
F
T
. Some bacteria actually put these properties to
work when swimming near a plane wall, smoothing out their
almost chaotic run and tumble paths through the fluid.
10
Any
model of the motion of particles in the low-Reynolds-
number limit, no matter how approximate, should at a mini-
mum preserve these traits as they are fundamental to the
physics of Stokes flow. For instance, one of the reasons why
the full Rotne-Prager approximation for
M
UF
in unbounded
flows has proven so useful is that, even though it admits
unphysical situations where particles may overlap, it is al-
ways symmetric, positive, and definite.
11
We may occasion-
ally reference the grand resistance tensor as well, but this is
simply the inverse of the grand mobility tensor
R
=
M
−1
.
We proceed to demonstrate how to incorporate a plane wall
into the hydrodynamic interactions between colloidal
particles so that these tensors are symmetric, positive, and
definite.
Following the approach taken in the original Stokesian
dynamics method, we consider the disturbance velocity,
u
x
, generated by a solid sphere of radius
a
with its center
at
x
a height
h
above a plane wall with force density on its
surface denoted
f
. This disturbance field can be separated
into two parts,
u
x
=
u
i
x
+
u
w
x
,
2
where
u
i
x
is the velocity field generated by an unbounded
particle and
u
w
x
is the reflection of the unbounded field off
the plane wall such that the no-slip condition is satisfied,
u
i
x
+
u
w
x
=0,
3
when
x
is a point on the wall. We will need the Stokes flow
Green’s function, or Stokeslet, for the velocity field at
x
due
to an isolated point force at
y
,
J
x
,
y
=
1
8
I
r
+
rr
r
3
,
4
where
I
is the identity tensor,
r
=
x
−
y
,
r
2
=
r
·
r
and
is the
viscosity of the surrounding fluid. In addition, we need
Blake’s solution
12
for the image of a Stokeslet above a wall
located at
H
in the fluid with normal
3
,
J
w
x
,
y
;
H
=−
J
x
,
y
+
y
3
−
H
2
y
2
J
x
,
y
·
P
+2
y
3
−
H
P
·
y
J
x
,
y
·
3
T
,
5
where
y
=
y
−2
y
3
−
H
3
,
P
=
I
−2
3
3
, and the superscript
T
indicates transposition. With these we can write the exact
solution for the disturbance velocity caused by particle
as
u
x
=
S
G
x
,
y
;
H
·
f
y
dS
y
,
6
where the total Green’s function is
G
x
,
y
;
H
=
J
x
,
y
+
J
w
x
,
y
;
H
.
7
From this exact expression we expand in surface moments
the force density on particle
following the Stokesian dy-
namics procedure to obtain
u
x
=
1+
a
2
6
y
2
G
x
,
y
;
H
y
=
x
·
F
+
1
2
y
G
x
,
y
;
H
y
=
x
·
L
+
1+
a
2
10
y
2
K
x
,
y
;
H
y
=
x
:
S
+
̄
,
8
where
K
x
,
y
;
H
=
1
2
y
G
x
,
y
;
H
+
y
G
x
,
y
;
H
T
.
9
We choose to truncate these expressions at the stresslet level,
though there is no reason that they cannot be expanded fur-
ther to include higher order and faster decaying moments of
the Green’s function. The key step to obtaining the correct
hydrodynamic interactions is to note that care must be used
when taking the derivatives with respect to
y
of Blake’s so-
lution for the reflected velocity field,
J
w
x
,
y
;
H
depends
explicitly on both
y
and
x
−
y
, reflecting the fact that the
point force density
f
is distributed on the surface of the par-
ticle and not simply located at the particle’s center. The cor-
rect derivatives are complex, but the following chain rules:
FIG. 1. The interactions between a pair of spheres
and
near a plane
wall where
h
=
x
3
−
H
and
h
=
x
3
−
H
.
113306-2
J. W. Swan and J. F. Brady
Phys. Fluids
19
, 113306
2007
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y
f
y
3
g
R
=
3
f
y
3
g
R
−
f
y
3
P
·
R
g
R
,
10
y
2
f
y
3
g
R
=
f
y
3
g
R
+2
f
y
3
3
·
R
g
R
+
f
y
3
R
2
g
R
,
where
R
=
x
−
y
, allow for considerable simplification of this
process.
We can also use these expressions to say something
about the reflections of higher order Stokes flow singularities
from a plane wall. From Eq.
8
, we see that a spherical
particle with a constant force density on its surface denoted
F
/4
a
2
, behaves as though it generates two singularities in
the surrounding fluid at the particle’s center. These singulari-
ties are the familiar Stokeslet and the source doublet
y
2
J
x
,
y
. The same Fourier transform approach used to
compute Blake’s expression for the reflection of the Stokes-
let
J
w
x
,
y
;
H
can be used to compute the reflection of the
source doublet.
13
While this procedure is arduous, the result-
ing reflection of the source doublet could have been used to
generate the reflected field
u
w
x
directly. Knowing that the
Stokes equations are unique, we can see from Eq.
8
that the
reflected field has only two contributions: Blake’s reflection
of the Stokeslet and one other which must be the reflection of
the source doublet. This suggests a direct and facile way to
compute reflections of higher order Stokes flow singularities,
something which is quite subtle and has recently been a
source of confusion.
14
This approach has been used in simi-
lar circumstances,
15
but we state it here explicitly since it
applies to any reflection of a Stokes flow singularity.
Namely, no complicated Fourier transform or limiting pro-
cess is needed to compute these reflections. One simply
writes the higher order singularity as a derivative of the
Stokeslet or other fundamental singularity with a given con-
dition on the wall, and the reflection of that singularity is
simply the same derivative
with respect to the source point,
y
of Blake’s reflection or fundamental singularity. That is,
we can simply write the no slip condition on the wall as
y
n
J
x
,
y
+
y
n
J
w
x
,
y
;
H
=0,
11
when
x
is a point on the plane wall
x
3
=
H
. This also sug-
gests one way to simulate the hydrodynamics of particles
constrained by more than one wall or some curvilinear
geometry. If we know the reflection of the Stokeslet satisfy-
ing the boundary conditions on the wall, then we can com-
pute the disturbance velocity generated by a particle with an
arbitrary force density using the procedure just described.
The mobility tensor is constructed from the disturbance
velocity field
u
x
and Faxén formulas for a particle near a
wall, which, unfortunately are not known. However, we can
bypass the need to know the exact Faxén formulas with a
wall by noting that the flow caused by the reflection from the
wall
u
w
x
is just another disturbance flow and has no sin-
gularities within the fluid region above the wall. Therefore,
the usual, well-known
9
Faxén formulas can be applied. That
is, the Faxén formulas coupling a disturbance field in the
fluid to the translational and rotational velocities of a spheri-
cal particle
of radius
a
centered at
x
relative to the
fluid
U
−
U
,
−
and the rate of strain of the fluid
E
are
U
−
U
=
F
6
a
+
1+
a
2
6
x
2
u
x
x
,
12
−
=
L
8
a
3
+
1
2
x
u
x
x
,
13
−
E
=
S
20
3
a
3
+
1+
a
2
10
x
2
e
x
x
,
14
where the disturbance rate of strain is
e
x
=
1
2
x
u
x
+
x
u
x
T
.
15
In principle, since we have a detailed knowledge of the dis-
turbance field caused by particle
, we can compute how it
interacts hydrodynamically with particle
. For that matter, if
and
are the same particle, we can compute the hydrody-
namic interaction between just a single particle and a wall by
only considering the reflected part of the disturbance field,
u
w
x
.
In order to complete our description of the hydrody-
namic interactions, we need to compute the mobility and
resistance tensors. Each term in the grand mobility tensor is
constructed from the combination of a Faxén formula with
the singularities from a force multipole in the disturbance
velocity field. As an example, we construct the term
M
UF
explicitly using Eqs.
8
and
12
to characterize the coupling
of a force on particle
to the relative velocity of particle
,
M
UF
=
1+
a
2
6
x
2
1+
a
2
6
y
2
G
x
,
y
;
H
x
=
x
y
=
x
.
16
The other grand mobility tensor terms are constructed analo-
gously. One additional note is necessary, however. For the
coupling between particle
and itself, we discard the distur-
bance velocities generated by the Stokeslet and use the re-
flected field alone in the Faxén formulas. This is illustrated
for the term
M
UF
,
M
UF
=
I
6
a
+
1+
a
2
6
x
2
1+
a
2
6
y
2
J
w
x
,
y
;
H
x
=
x
y
=
x
.
17
All terms in the mobility tensor are given explicitly in
Appendices A–C.
The grand resistance tensor, denoted by
R
=
R
FU
R
F
R
FE
...
R
LU
R
L
R
LE
...
R
SU
R
S
R
SE
...
]]]
,
18
is simply the inverse of the grand mobility tensor. However,
since we truncate the expansion generating the grand mobil-
ity tensor after a finite number of multipoles, this tensor is
only a far-field approximation. We wish to include lubrica-
113306-3
Simulation of hydrodynamically interacting
Phys. Fluids
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, 113306
2007
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tion interactions which are only properly expressed after in-
cluding a large
infinite
number of force multipoles. After
inverting the fully truncated grand mobility tensor, we add in
the exact lubrication forms for the resistance tensor for both
the interactions between near pairs of particles
9
R
P
and a
particle near the wall
5
R
W
. In addition to this, we subtract
out the far-field contributions arising from the inversion of
the grand mobility tensor
R
P
,
+
R
W
,
to avoid over count-
ing the hydrodynamic interactions. The grand resistance ten-
sor,
R
, including these lubrication contributions is
R
=
M
−1
+
R
P
+
R
W
−
R
P
,
+
R
W
,
.
19
Elements of this resistance tensor can now be used in the
Langevin equation for particle dynamics, correctly including
the far-field, many-body hydrodynamic interactions and the
singular, pairwise lubrication forces, and is symmetric
positive-definite by construction. The details of how this is
done can be found in the thorough discussion by Phung,
Brady, and Bossis.
16
III. RESULTS AND DISCUSSION
A. Symmetric mobility and resistance tensors
The above approach follows the well-established Stoke-
sian dynamics procedure for constructing a symmetric,
positive-definite mobility tensor, and so it is somewhat sur-
prising that two prior studies of the problem of spherical
particles moving in Stokes flow near a plane wall failed to
compute symmetric mobility and resistance tensors. The first
approach by Bossis, Meunier, and Sherwood
5
used a tech-
nique similar to the one discussed above in which they ex-
pand the reflection of the Stokeslet in force multipoles to
derive the wall contribution to the mobility tensor. However,
they did not appreciate that the reflection of the Stokeslet is
a function of how far the point force is from the wall as well
as the separation vector
x
−
y
. We take care to write the
reflection as
J
w
x
,
y
;
H
which is an explicit function of a
destination, a source, and the location of the wall. Their mul-
tipole expansion presumes a dependence on
x
−
y
only and
as a result, the mobility tensors stemming from this approach
are not symmetric. In a quantitative sense, this error may not
be large; however, their approach is unusable in the context
of Brownian motion which requires a symmetric mobility
tensor. Similarly, in their study of confined chains of Brown-
ian particles, Jendrejack
et al.
6
numerically compute the re-
flection off a wall of the Rotne-Prager tensor. Recall that the
R-P tensor is simply
J
R
r
=
1+
a
2
/3
r
2
J
r
. They call this
reflection the contribution to the mobility tensor due to the
wall, but do not recognize that the Rotne-Prager tensor lacks
a direct connection with the fluid velocity field surrounding
the particle. It cannot be used to generate the reflection off
the wall since it depends on a linear combination of the dis-
turbance velocity generated by an isolated particle and its
source doublet and not just the disturbance velocity itself. As
shown earlier, the reflected velocity field is still subject to the
Faxén formulas which produces extra quadrupolar and octu-
polar contributions to the mobility that are essential to main-
taining symmetry. Again this method is unsuitable for simu-
lating Brownian particles. Jendrejack
et al.
recognized that
their mobility tensor was not symmetric and simply symme-
trized it by adding the transpose and dividing by two to
conduct Brownian simulations. There is, unfortunately, no
way to assess to accuracy of this manipulation.
B. Co-rotation of a doublet of particles
Illustrated in Fig.
2
is a pair of spherical particles of the
same radius with equal and opposite torques applied along
their line of centers. One can imagine accomplishing this by
connecting a pair of spherical particles with a slender torsion
wire and twisting those particles to load a torque into the
wire. For a pair of particles in an unbounded fluid, the torque
along the line of centers causes the particles to rotate about
their line of centers in opposite directions while remaining
otherwise still. However, when the doublet is brought near a
wall, the particles both rotate and translate because the wall
induces an additional coupling between torque and transla-
tion. Since these particles might be connected by a wire, the
separation between the particles remains the same and the
doublet spins about its center of mass. We can measure the
tensile force on the wire as the doublet rotates, but one can
show that because the grand mobility tensor is symmetric,
this tensile force is exactly zero regardless of separation and
height above the wall. In a model lacking symmetry, some
nonzero force along the line of centers is necessary to main-
tain the separation between the particles. Interestingly, as in-
dicated in Fig.
3
, the rate of rotation of the doublet about its
center of mass,
, normalized by the torque on the particles,
T
/8
a
3
, is a nonmonotonic function of both the separation
between the particles,
r
, and the height of the doublet above
the wall,
h
. When the particles are far apart, the rate of ro-
tation of the doublet is decreasing as
r
increases because the
translational speed of a particle in the doublet
r
/2
is set
only by the coupling of a single particle to the wall. When
the particles are far from the wall, the rate of rotation of the
doublet is also decreasing as
h
increases. However this is
caused by weakening interactions with the wall. Conversely,
when the separation between the particles becomes quite
small or the doublet is close to the wall, the rotation rate of
FIG. 2. A side-on and top-down view of a pair of particles above a plane
wall with equal and opposite torques
T
applied along their line of centers.
The particles have the same radius
a
, are separated by a distance
r
, and are
a height
h
above the wall.
113306-4
J. W. Swan and J. F. Brady
Phys. Fluids
19
, 113306
2007
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the doublet also decreases because in this limit the resistance
to motion of the particles becomes singular. Since the nor-
malized rotation rate is decreasing in the limits that the par-
ticles are both near and far apart and the doublet is both near
and far from the wall, it must reach a global maximum where
the doublet rotates most quickly.
Since the torques on the particles are equal and opposite,
the doublet is a force and torque free object. It can be
thought of as a model for a number of different interesting
systems. A recent study of bacteria swimming near a wall
found that the head and flagellum of the organism co-rotate
just like this doublet and for the exact same reason.
10
A
swimming bacterium is a force and torque free object and in
order to propel itself by torquing its flagellum it must also
have an equal and opposite torque on its head. The study
found that the same coupling to the wall that cause the dou-
blet to co-rotate also cause a bacterium to sweep out arcs as
it swims near a wall rather than swim straight. The only
difference between the doublet and the bacterium is that the
bacterium is also propelled along its line of centers.
C. Grand mobility tensors for any confining
boundaries
The procedure described in Sec. II is hardly limited to
the single plane wall geometry. For any confining geometry,
we can write the disturbance velocity field at
x
generated by
a point force at
y
as
u
x
=
G
x
,
y
;
H
1
,
H
2
, ...
·
f
y
=
J
x
,
y
·
f
y
+
J
w
x
,
y
;
H
1
,
H
2
, ...
·
f
y
,
20
where
J
w
x
,
y
;
H
1
,
H
2
,...
is the reflection of the Stokeslet
off the confining surfaces and
H
1
,
H
2
,... are
just geometrical
parameters. It is clear that the terms of the mobility tensor
are constructed in exactly the same way as for a single plane
wall. No changes in procedure are necessary since the pro-
cess only relies on properties of the Stokes equations and not
on the geometrical constraints.
As a specific example, consider particles in a fluid
bounded by a flat and nondeformable free surface; the fluid
is constrained only by a no-penetration condition at the sur-
face. The Green’s function in this case is similar to the plane
wall Green’s function. It is the sum of a Stokeslet above the
interface and a stokeslet with a reflected force density below
the interface, viz.,
G
x
,
y
;
H
=
J
x
,
y
+
J
x
,
y
·
P
.
21
Nothing about this Green’s function precludes using Eqs.
16
and
17
and their analogs to build up the grand mobility
tensor. In fact, the same caveats for taking the derivatives
with respect to the source,
y
, apply to this Green’s function
too. A clever implementation of a traditional Stokesian dy-
namics simulation can model this interface as well, by rec-
ognizing that a particle near a free surface acts as though
there were an image particle with a reflected force density on
its surface below the interface, one can simulate
N
particles
near a free surface using those
N
particles and
N
image par-
ticles. The extra particles increase the computational time for
conventional Stokesian dynamics by a factor of 8 over that
for
N
unbounded particles. Using the method described
herein with the Green’s function in Eq.
21
, that multiplica-
tive factor is near unity.
It is worth noting further that the grand mobility tensor
for particles near interfaces with even more complicated
boundary conditions may be constructed by using this
method directly or by using the linear combination of mobil-
ity tensors with simpler confinements. Consider particles
moving near a flat and nondeformable liquid-liquid interface
where the ratio of the viscosity of the confining liquid to the
viscosity of the embedding liquid is
. Constructing a simu-
lation using image particles as in the free surface problem is
now quite difficult. However, the Green’s function in this
case is a linear combination of the Green’s functions for a
free surface
G
F
x
,
y
;
H
, Eq.
21
and a solid plane wall
G
W
x
,
y
;
H
, Eq.
7
G
x
,
y
;
H
=
1
1+
G
F
x
,
y
;
H
+
1+
G
W
x
,
y
;
H
,
22
so that changing
from zero to infinity transitions from the
free surface boundary to the plane wall boundary.
17
Taking
this same linear combination of the grand mobility tensors
for particles near a free surface and particles near a plane
wall generates the grand mobility tensor for particles near a
liquid-liquid interface. As before, a resistance tensor may be
constructed from the mobility tensor, where now one needs
the resistance interactions for a single particle adjacent to a
free surface or a liquid-liquid interface.
IV. CONCLUSIONS
We have constructed a symmetric, positive-definite mo-
bility tensor for particles near a plane wall. Additionally, we
have shown how to compute a symmetric, positive-definite
mobility tensor for particles in any confining geometry given
FIG. 3. The rotation rate of the doublet about its center of mass normalized
by the torque on the particles about their line of centers
8
a
3
/
T
.The
normalized rate of rotation is maximum near
r
=2.1
a
,
h
=1.01
a
.
113306-5
Simulation of hydrodynamically interacting
Phys. Fluids
19
, 113306
2007
Downloaded 16 Nov 2007 to 131.215.225.181. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
the reflection of the Stokeslet off the confining walls. Inter-
estingly, the scaling of the Stokesian dynamics algorithm
with respect to the number of particles is independent of any
confining geometry and only a few more computations per
particle pair are necessary to build up the resistance and mo-
bility tensors. While the expressions for the elements of the
grand mobility terms in Appendices A–C are algebraically
complicated, they may be tabulated and used with efficient
table lookup procedures for dynamic simulation. Following
this approach, one can now use Stokesian dynamics to study
the behavior of suspensions of spherical particles in arbitrary
confining geometries. In an even simpler approach, one may
approximate certain confinements such as the parallel plate
geometry by superimposing the mobility tensors due to a pair
of plane walls with opposing normals. There are some indi-
cations that this approximation can be quite accurate,
18
though it does trade the convenience of superposition for the
fidelity of the exact two wall reflection.
Symmetry and positive-definiteness of the mobility ten-
sor are fundamental to the hydrodynamic interactions be-
tween particles in Stokes flow and are essential to simulating
Brownian motion. We have chosen to study the static prop-
erties of particle pairs here, illustrating the importance of
symmetry in the physically interesting problem of a force
and torque free doublet rotating above a plane wall. The
Brownian motion of particles may be generated directly from
the tensors derived herein, and can be used to study various
dynamic processes of Brownian particles near a plane wall
including self-assembly and colloidal crystallization using
these tensors in Stokesian dynamics simulations.
ACKNOWLEDGMENTS
This work was supported in part by NSF Grant No.
CBET 0506701.
APPENDIX A: REFLECTED STOKESLET
CONTRIBUTIONS TO THE GRAND MOBILITY TENSOR
As in Eq.
16
, we can write explicit expressions for all
of the terms in the mobility tensor. For the purposes of this
Appendix we only concern ourselves with the contributions
due to reflection of the Stokeslet. The interactions due to the
Stokeslet itself yield the well-known far-field grand mobility
tensor for unbounded pairs of particles and can be found in
Ref.
4
. We write the contribution to the grand mobility ma-
trix due to the reflected field,
J
w
x
,
y
;
H
,as
M
ˆ
, and the
contributions to the subtensors are denoted
M
ˆ
UF
,
M
ˆ
UL
,....
When this is added to the grand mobility tensor for an un-
bounded set of particles, we recover the complete grand mo-
bility tensor,
M
. Without loss of generality, these expres-
sions can be used to generate the single particle-wall and
particle pair-wall interactions as described in Sec. II. Note
that we define the operator
x
T
such that
x
T
u
x
=
x
u
x
T
,
M
ˆ
UF
=
1+
a
2
6
x
2
1+
a
2
6
y
2
J
w
x
,
y
;
H
x
=
x
y
=
x
,
A1
M
ˆ
F
=
1
2
x
1+
a
2
6
y
2
J
w
x
,
y
;
H
x
=
x
y
=
x
,
A2
M
ˆ
EF
=
1
2
x
+
x
T
1+
a
2
10
x
2
1+
a
2
6
y
2
J
w
x
,
y
;
H
x
=
x
y
=
x
,
A3
M
ˆ
UL
=
1+
a
2
6
x
2
1
2
y
J
w
x
,
y
;
H
x
=
x
y
=
x
,
A4
M
ˆ
L
=
1
2
x
1
2
y
J
w
x
,
y
;
H
x
=
x
y
=
x
,
A5
M
ˆ
EL
=
1
2
x
+
x
T
1+
a
2
10
x
2
1
2
y
J
w
x
,
y
;
H
x
=
x
y
=
x
,
A6
M
ˆ
US
=
1+
a
2
6
x
2
1+
a
2
10
y
2
K
w
x
,
y
;
H
x
=
x
y
=
x
,
A7
M
ˆ
S
=
1
2
x
1+
a
2
10
y
2
K
w
x
,
y
;
H
x
=
x
y
=
x
,
A8
M
ˆ
ES
=
1
2
x
+
x
T
1+
a
2
10
x
2
1+
a
2
10
y
2
K
w
x
,
y
;
H
x
=
x
y
=
x
.
A9
APPENDIX B: PARTICLE-WALL “SELF” MOBILITY
TENSOR
„
...
Presented below is the contribution to the grand mobility
tensor due to a single particle
interacting with a plane
wall. All of the following terms are normalized by 6
a
n
,
where
n
is chosen to keep things dimensionally consistent.
Additionally,
h
is the normalized height of the particle above
the wall such that
h
=
h
/
a
. Note that since the mobility
tensor is symmetric by construction, we include only six of
nine subtensors. The other three can be computed directly
taking a transposition,
113306-6
J. W. Swan and J. F. Brady
Phys. Fluids
19
, 113306
2007
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M
ˆ
UF
,
ij
=−
1
16
9
h
−1
−2
h
−3
+
h
−5
ij
−
i
3
j
3
−
1
8
9
h
−1
−4
h
−3
+
h
−5
i
3
j
3
,
B1
M
ˆ
F
,
ij
=
3
32
h
−4
3
ij
,
B2
M
ˆ
L
,
ij
=−
15
64
h
−3
ij
−
i
3
j
3
−
3
32
h
−3
i
3
j
3
,
B3
M
ˆ
EF
,
ijk
=−
3
160
15
h
−2
−12
h
−4
+5
h
−6
ik
−
i
3
k
3
j
3
+
jk
−
j
3
k
3
i
3
+
3
32
3
h
−2
−3
h
−4
+
h
−6
ij
−
i
3
j
3
k
3
−
3
16
3
h
−2
−3
h
−4
+
h
−6
i
3
j
3
k
3
,
B4
M
ˆ
EL
,
ijk
=−
9
320
5
h
−3
−4
h
−5
j
3
3
ik
+
i
3
3
jk
,
B5
M
ˆ
ES
,
ijkl
=−
3
640
10
h
−3
−24
h
−5
+9
h
−7
ij
−
i
3
j
3
kl
−
k
3
l
3
−
9
640
10
h
−3
−8
h
−5
+3
h
−7
ik
−
i
3
k
3
jl
−
j
3
l
3
+
il
−
i
3
l
3
jk
−
j
3
k
3
+
3
160
20
h
−3
−24
h
−5
+9
h
−7
ij
−
i
3
j
3
k
3
l
3
+
kl
−
k
3
l
3
i
3
j
3
−
9
320
15
h
−3
−16
h
−5
+6
h
−7
ik
−
i
3
k
3
j
3
l
3
+
il
−
i
3
l
3
j
3
k
3
+
jk
−
j
3
k
3
i
3
l
3
+
jl
−
j
3
l
3
i
3
k
3
−
3
80
20
h
−3
−24
h
−5
+9
h
−7
i
3
j
3
k
3
l
3
.
B6
APPENDIX C: PARTICLE-WALL “PAIR” MOBILITY
TENSOR
„
...
Here we present the contributions to the grand mobility
tensor due to interactions between an identically sized par-
ticle pair
and a plane wall. One can generate these
tensors for particles of different sizes just as easily using the
expressions in Appendix A. However, the expressions for
those tensors are significantly longer. As above, the mobility
terms are normalized by 6
a
n
, where
a
=
a
=
a
and
n
is
selected to provide the correct dimensionality. We define the
following as well:
R
=
1
a
x
−
x
+2
h
3
C1
and
e
=
R
/
R
, where
R
=
R
·
R
. Additionally, we define a res-
caled height above the wall as
h
ˆ
=
h
/
aR
3
. As in the previ-
ous section we include six of the nine subtensors. The others
may be generated through transposition,
M
ˆ
UF
,
ij
=−
1
4
3
1+2
h
ˆ
1−
h
ˆ
e
3
2
R
−1
+2
1−3
e
3
2
R
−3
−2
1−5
e
3
2
R
−5
ij
−
1
4
3
1−6
h
ˆ
1−
h
ˆ
e
3
2
R
−1
−6
1−5
e
3
2
R
−3
+10
1−7
e
3
2
R
−5
e
i
e
j
+
1
2
e
3
3
h
ˆ
1−6
1−
h
ˆ
e
3
2
R
−1
−6
1−5
e
3
2
R
−3
+10
2−7
e
3
2
R
−5
e
i
j
3
+
1
2
e
3
3
h
ˆ
R
−1
−10
R
−5
i
3
e
j
−
3
h
ˆ
2
e
3
2
R
−1
+3
e
3
2
R
−3
+
2−15
e
3
2
R
−5
i
3
j
3
,
C2
M
ˆ
F
,
ij
=
3
4
R
−2
ijk
e
k
+
3
2
6
h
ˆ
e
3
2
R
−2
+
1−10
e
3
2
R
−4
3
ki
e
k
j
3
−
3
2
e
3
3
h
ˆ
R
−2
−5
R
−4
3
ki
e
k
e
j
−
3
2
e
3
h
ˆ
R
−2
−
R
−4
3
ij
,
C3
M
ˆ
L
,
ij
=
3
8
1−6
e
3
2
R
−3
ij
−
9
8
R
−3
e
i
e
j
+
9
4
e
3
R
−3
i
3
e
j
+
9
4
R
−3
3
ki
3
lj
e
k
e
l
,
C4
M
ˆ
EF
,
ijk
=
6
5
e
3
5−2
h
ˆ
R
−4
−15
R
−6
i
3
j
3
k
3
+3
R
−6
i
3
j
3
e
k
−
3
10
e
3
5
h
ˆ
1−6
1−
h
ˆ
e
3
2
R
−2
−2
5−
h
ˆ
−5
5−2
h
ˆ
e
3
2
R
−4
+10
2−7
e
3
2
R
−6
ij
k
3
−
3
10
5−2
h
ˆ
R
−4
−10
R
−6
e
3
ik
j
3
+
i
3
jk
+
3
2
e
3
5−2
h
ˆ
R
−4
−14
R
−6
j
3
e
i
e
k
+
i
3
e
j
e
k
+
3
2
6
h
ˆ
1−
h
ˆ
e
3
2
R
−2
+
1−4
5−2
h
ˆ
e
3
2
R
−4
−2
2−21
e
3
2
R
−6
e
i
j
3
k
3
+
e
j
i
3
k
3
−
3
10
15
h
ˆ
1−
h
ˆ
e
3
2
R
−2
+
4−5
5−2
h
ˆ
e
3
2
R
−4
−5
1−7
e
3
2
R
−6
e
i
jk
+
e
j
ik
+
3
2
e
3
3
h
ˆ
1−10
1−
h
ˆ
e
3
2
R
−2
−2
5−
h
ˆ
−7
5−2
h
ˆ
e
3
2
R
−4
+14
2−9
e
3
2
R
−6
e
i
e
j
k
3
+
3
20
5
1−6
h
ˆ
1−
h
ˆ
e
3
2
R
−2
−2
4−5
5−2
h
ˆ
e
3
2
R
−4
+10
1−7
e
3
2
R
−6
ij
e
k
−
3
4
3
1−10
1−
h
ˆ
e
3
2
R
−2
−2
4−7
5−2
h
ˆ
e
3
2
R
−4
−14
1−9
e
3
2
R
−6
e
i
e
j
e
k
,
C5
113306-7
Simulation of hydrodynamically interacting
Phys. Fluids
19
, 113306
2007
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M
ˆ
EL
,
ijk
=−
9
8
R
−3
e
j
ikl
e
l
+
e
i
jkl
e
l
−
9
10
R
−5
j
3
3
ik
+
i
3
3
jk
+
9
4
R
−3
−2
R
−5
e
i
j
3
3
kl
e
l
+
i
3
e
j
3
kl
e
l
−
9
4
e
3
1−2
h
ˆ
R
−3
−2
R
−5
e
j
3
ik
+
e
i
3
jk
+
9
2
e
3
1−
h
ˆ
R
−3
−
R
−5
ij
3
kl
e
l
−
9
2
e
3
5
1−
h
ˆ
R
−3
−7
R
−5
e
i
e
j
,
C6
M
ˆ
ES
,
ijkl
=
9
5
R
−7
i
3
j
3
kl
−
18
5
e
3
5
R
−5
−21
R
−7
i
3
j
3
e
k
l
3
+
i
3
j
3
k
3
e
l
+
9
10
e
3
5
R
−5
−14
R
−7
e
i
j
3
kl
+
i
3
e
j
kl
+
il
j
3
e
k
+
i
3
jl
e
k
+
i
3
jk
e
l
+
ik
j
3
e
l
+
9
5
5
h
ˆ
1−
h
ˆ
e
3
2
R
−3
+
1−10
e
3
2
R
−5
−
2−21
e
3
2
R
−7
ik
j
3
l
3
+
il
j
3
k
3
+
i
3
jk
l
3
+
i
3
jl
k
3
−
9
10
5
h
ˆ
1−
h
ˆ
e
3
2
R
−3
+
1−5
e
3
2
R
−5
−
1−7
e
3
2
R
−7
il
jk
+
ik
jl
+
3
20
5
1−6
h
ˆ
1−
h
ˆ
e
3
2
R
−3
−6
1−5
e
3
2
R
−5
+6
1−7
e
3
2
R
−7
ij
kl
−
36
5
5
h
ˆ
1−
h
ˆ
e
3
2
R
−3
+
1−15
e
3
2
R
−5
−3
1−14
e
3
2
R
−7
i
3
j
3
k
3
l
3
−
9
5
5
1−10
h
ˆ
1−
h
ˆ
e
3
2
e
3
2
R
−3
−10
2−7
e
3
2
R
−5
−
1−42
e
3
2
+ 126
e
3
4
R
−7
ij
k
3
l
3
−
9
5
e
3
5
1−20
h
ˆ
1−
h
ˆ
e
3
2
R
−3
−30
1−7
e
3
2
R
−5
+84
1−6
e
3
2
R
−7
e
i
j
3
k
3
l
3
+
i
3
e
j
k
3
l
3
+
9
20
e
3
5
1−20
h
ˆ
1−
h
ˆ
e
3
2
R
−3
−10
3−14
e
3
2
R
−5
+14
2−9
e
3
2
R
−7
e
i
jk
l
3
+
e
i
jl
k
3
+
ik
e
j
l
3
+
il
e
j
k
3
−
9
20
5
1−10
h
ˆ
1−
h
ˆ
e
3
2
R
−3
−10
1−7
e
3
2
R
−5
+14
1−9
e
3
2
R
−7
e
i
e
j
kl
+
ij
e
k
e
l
+
9
5
25
1−14
h
ˆ
1−
h
ˆ
e
3
2
e
3
2
R
−3
−70
2−9
e
3
2
R
−5
−7
1−54
e
3
2
+ 198
e
3
4
R
−7
e
i
e
j
k
3
l
3
+
9
10
e
3
5
1−10
h
ˆ
1−
h
ˆ
e
3
2
R
−3
−5
3−14
e
3
2
R
−5
+14
2−9
e
3
2
R
−7
ij
e
k
l
3
+
ij
k
3
e
l
−
9
40
5
1−20
h
ˆ
1−
h
ˆ
e
3
2
R
−3
−20
1−14
e
3
2
R
−5
+28
1−9
e
3
2
R
−7
e
i
jl
e
k
+
il
e
j
e
k
+
e
i
jk
e
l
+
ik
e
j
e
l
+
9
20
5
1−20
h
ˆ
1−
h
ˆ
e
3
2
R
−3
−20
1−14
e
3
2
R
−5
+28
2−27
e
3
2
R
−7
e
i
j
3
e
k
l
3
+
i
3
e
j
e
k
l
3
+
e
i
j
3
k
3
e
l
+
i
3
e
j
k
3
e
l
−
9
10
e
3
25
1−14
h
ˆ
1−
h
ˆ
e
3
2
R
−3
− 105
1−6
e
3
2
R
−5
+ 126
2−11
e
3
3
R
−7
e
i
e
j
e
k
l
3
+
e
i
e
j
k
3
e
l
−
63
5
R
−7
i
3
j
3
e
k
e
l
−
63
10
5
R
−5
−9
R
−7
i
3
e
j
e
k
e
l
+
e
i
j
3
e
k
e
l
+
9
20
25
1−14
h
ˆ
1−
h
ˆ
e
3
2
R
−3
−70
1−9
e
3
2
R
−5
+ 126
1−11
e
3
2
R
−7
e
i
e
j
e
k
e
l
.
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