Biomedical Engineering Reference
In-Depth Information
, the l.h.s. of Eq. (1) is evaluated on the
sphere surfaces using the stick boundary conditions. This leads to a set of boundary
integral equations, written in a compact notation as
To determine the force densities
{
f
i
}
N
Z
−
0
(i)
f
i
+
w
i
−
v
0
=
G
F
(ij)
f
j
,
for
r
∈
S
i
,
(3)
j
=
1
with the single-sphere resistance operator,
Z
0
(i)
, of an unbounded fluid, and the in-
terparticle Green propagator, which contains a part diagonal and a part off-diagonal
in the particle indices
i
and
j
,
G
F
(ij)
δ
ij
)
G
(ij),
(4)
where the Green operator for an unbounded fluid with Oseen kernel
T
0
is denoted
as
G
0
. The off-diagonal operator,
G
(ij)
δ
ij
G
0
(ii
)
=
R
F
+
(
1
−
G
0
(ij
)
R
F
,
(5)
accounts for the velocity fields generated by the force densities
f
j
of all the other
particles
j
=
G
0
(ij)
+
=
i
,incidenton
S
i
. The non-zero diagonal part of the Green propaga-
tor
G
F
(ii)
0, absent in the unbounded fluid, now does not vanish owing to the
presence of the free surface. Moreover, the off-diagonal part of
G
F
(ij)
,
i
=
j
,is
also modified with respect to the Oseen propagator for the unbounded fluid. Both
diagonal and off-diagonal extra terms, caused by the free surface, are generated by
the image spheres
i
and
j
.
The Q2D boundary integral equation (3) has the same form as in infinite space,
but with the modified Q2D single-sphere operator
=
Z
−
0
(i)
G
(ii
)
[
+
]
rather than
Z
−
0
(i)
, and the modified Q2D propagator
G
(ij)
rather than
G
0
(ij)
,
i
G
(ij)
f
j
,
for
r
Z
−
0
(i)
G
0
(ii
)
w
i
−
v
0
=[
+
]
f
i
+
∈
S
i
,
(6)
j
=
In the following, we will briefly describe the procedure of solving Eq. (6) for the
force density, and evaluating the 6
N
6
N
friction
ζ
and mobility
μ
matrices. The
focus of this paper is on the Q2D mobility matrix, which relates the hydrodynamic
forces and torques acting on the
N
spheres to the resulting translational and rota-
tional velocities [1]. To determine the leading far-field expression, we aim towards
constructing
μ
as a series of scattering sequences (reflections), which involve the
Q2D single-sphere friction operators and the Q2D propagators between pairs of
spheres.
HI between many spheres in a fluid bounded by a planar free surface have been
analyzed in Ref. [38]. In that work, the multiple expansion has been carried out
with respect to a basis set of multipole functions identical to the set used for an
unbounded fluid. However, for a genuine Q2D system where all spheres touch the
interface at
z
×
0, this basis set should be used with care. Indeed, the multipole
projection of the boundary integral equation (6), gives the degenerate matrix of the
=
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