Biomedical Engineering Reference
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where μ r 0
ηa 2 ) 1 .
Finally, let us now discuss the far-field approximation of the many-sphere mobil-
ity. In general, the mobility matrix
=
( 4
π
¯
μ ( 1
···
N) for N spheres is not pairwise-additive,
¯
= ¯
¯
= ¯
i.e.,
μ 12 ( 12 ) . However, the far-field ap-
proximation of the N -sphere mobility contains only a single propagator G 0 ,and
therefore it is the superposition of the two-sphere mobilities specified in Eqs (24)-
(28),
μ 11 ( 1
···
N)
μ 11 ( 1 ) and
μ 12 ( 1
···
N)
μ 11 ( 1
¯
···
N)
= ¯
μ 11 ( 1 )
μ F ,
(29)
¯
= ¯
μ 12 ( 1
···
N)
μ 12 ( 12 ).
(30)
E. Point-Particle Mobility
In discussing the far-field approximation of the Q2D mobility for a system of many
spheres, it is interesting to investigate also the point-particle model, where the
spheres touching a flat interface are approximated by points located at their centers.
The image points are now constructed by reflection with respect to the free surface,
and, according to the method of images, [1], the original system is thus replaced
by a doubled number of points and their images placed in an unbounded fluid. The
subtlety of the Q2D point-particle model is that, in addition to the horizontal radial
and transversal external forces
F ix = F i x and
F iy = F i y , additional constraint
forces perpendicular to the interface
F iz are required to keep the point particles i
at the fixed distance z
=
a from the interface (and
F i z =− F iz to keep the images
i
F iz can be
eliminated, leading to the Q2D N-point mobilities [42]. Unlike in case of spheres,
for points the mobility matrix contains only the translation-translational tensors. In-
deed, for point particles of zero extension, it does not make sense to specify angular
velocities and torques. Therefore, for two points the mobility relation has the form,
at z
=−
a ). Using the condition U iz =
U i z =
0, the constraint forces
2
tt
U i =
1 M
ij · F j ,i
=
1 , 2 ,
(31)
j
=
and the corresponding mobility tensors in the dyadic notation are expressed as
1 i R R
1 i R R ,i
t 1 i
M
= M
+ M
=
1 , 2 ,
(32)
with the radial and transversal Q2D mobility coefficients for two points given by,
11
8
1 ) 3 / 2 μ t 0 ,
3 RA
8 (R 2
11
R
T
8 μ t 0 ,
M
11 =
M
11 =
(33)
+
2
R +
μ t 0 ,
2 R 2
3
8
+
1
+
RAB
12
M
=
(R 2
+
1 ) 3 / 2
1
R +
μ t 0 ,
(34)
3
8
1
R 2
12
=
M
+
1
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