Biomedical Engineering Reference
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Z
−
0
(i)
G
0
(ii
)
multipole elements of the single-particle operator
. Inverting
it would lead to spurious divergences of most of the elements. To solve the problem,
Eq. (6) has been first projected onto those multipole functions, which are symmetric
with respect to reflections in the plane
z
[
+
R
F
]
=
0. The corresponding projection operator
has the form,
1
2
[
Pv
(
r
)
=
v
(
r
)
+
R
F
v
(
R
F
r
)
]
.
(7)
In this way the boundary conditions at the free surface are automatically satisfied.
The multipole matrix elements of
P
(i)
are determined from displacement theo-
rems [46]. We refer to [41, 42] for a derivation of the explicit formulas for these
elements.
The projection operator
P
is used to define the Q2D one-sphere resistance oper-
ator
Z
F
(i)
, as the inversion in the projected subspace of the symmetric multipole
functions,
P
(i)
}
−
1
,
(8)
where the supscript
T
denotes the operation of transposition. The multipole ele-
ments of the re-defined one-sphere resistance operator,
Z
F
(i)
, are finite even when
sphere
i
is in contact with the interface.
For the Q2D off-diagonal propagator, the following relation has been proved [43],
G
(ij)
P
T
(i)
[
Z
−
0
(i)
+
G
0
(ii
)
R
F
]
Z
F
(i)
={
2
P
T
(i)
G
0
(ij)
P
(j).
(9)
The physical implication of the spurious divergences noted above is that, once
sphere
i
touches the interface, lubrication effects forbid its translational mo-
tion perpendicular to the interface, i.e.,
U
iz
=
=
0, and its rotation along an axis
0.
1
For more information, we refer
to [41]. For zero incident flow, the translational and rotational velocities,
parallel to the interface, i.e.,
ix
=
iy
=
U
≡
(U
1
x
,U
1
y
,
1
z
,...,U
Nx
,U
Ny
,
Nz
)
depend linearly on the horizontal compo-
nents of the external forces and the vertical components of the external torques,
F
≡
(
F
1
x
,
F
1
y
,
T
1
z
,...,
F
Nx
,
F
Ny
,
T
Nz
)
,
F
=
ζ
·
U
,
(10)
with the
N
-particle friction tensor,
ζ
Z
−
1
2
G
0
]
−
1
=
P
F
[
+
P
F
≡
P
F
[
Z
F
−
Z
F
2
G
0
Z
F
+···]
P
F
,
(11)
F
where the operator
P
F
selects those spherical multipoles, which correspond to the
components of forces and torques (and to translational and angular particle veloc-
ities), symmetric under reflection at the interface, and the inverse, limited to the
1
In reality, the surfaces of spheres are not perfectly smooth, the free surface may be curved rather than
flat in vicinity of a touching particle and, moreover, a liquid-gas interface always shows thermally induced
undulations. Therefore, it would be of interest to check experimentally to what extent the constrained-
motion predictions of lubrication theory are met in real quasi-two-dimensional systems and how accurately
the model analyzed in this paper approximates the real geometry of the system.
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