Biomedical Engineering Reference
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with the following functions
A
and
B
of the interparticle distance
R
,
1
R
−
.
R
2
3
R
3
2
+
2
A
=
B
2
)
,
B
=−
(35)
2
(R
2
+
1
)
3
/
2
(
1
−
(R
2
+
1
)
3
/
2
As the consequence of the constraints, the two-point Q2D mobilities, restricted
to the Q2D degrees of freedom, are different from the Oseen tensors, and the N-
point Q2D mobilities are not pairwise additive, unlike in the 3D unbounded fluid
with 6N degrees of freedom. These features, paradoxically, make the concept and
evaluation of the Q2D many-point mobility significantly more complicated than the
concept and evaluation of the far-field Q2D many-sphere mobility, s since the latter
is constructed by superposition of the pairwise mobilities (24)-(28).
F. Discussion
1. Comparison with the Three-Dimensional Unbounded Fluid
The Q2D mobility, given in Eq. (25), will be now compared with the corresponding
3D Rotne-Prager tensor [48],
−
R R
)
R R
−
R R
)
,
3
4
R
1
2
(
1
1
8
R
3
1
2
(
1
R R
μ
t
12
/μ
t
0
=
+
−
−
(36)
which is the asymptotic form of the three-dimensional trans lational-translational
mobility of two spheres moving without constraints in an unbounded fluid. The
leading order terms in the inverse distance expansion of the rotational-rotational
(
1
/R
3
), rotational-translational (
1
/R
2
) and translational-translational (
1
/R
)
parts of the Q2D mobility
μ
12
are twice as large as their three-dimensional coun-
terparts (cf. [1] and Eq. (25)). This observation is easily understood qualitatively
within the method of images. To the lowest order in the 1
/R
expansion, the posi-
tion of sphere 2 as “seen” from the distant sphere 1, coincides with the position of
its image 2
. Therefore the presence of the image effectively doubles the forces and
torques acting on sphere 2, and also the resulting translational and angular velocities
of sphere 1. The next order correction (
∼
∼
∼
1
/R
3
) to the translational-translational
∼
quasi-two-dimensional mobility
μ
t
12
in Eq. (25) cannot be interpreted in a simple
way.
2. Accuracy of Far-Field Approximations
The accuracy of the far-field approximations described above was extensively dis-
cussed in Ref. [42], where the two-particle translational-translational mobilities
μ
t
11
and
μ
t
12
were evaluated very precisely by the multipole method with a very high or-
der of truncation
L
=
20, see Ref. [42] for the definition of
L
, resulting in two
μ
t
1
i
·
radial coefficients:
R
· ¯
R
,self(
i
=
1) and distinct (
i
=
2), and two transversal
μ
t
1
i
·
ones,
R
⊥
· ¯
2). These values were used
as a reference for testing accuracy of the far field approximation for spheres, see
R
⊥
,alsoself(
i
=
1) and distinct (
i
=
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