Biomedical Engineering Reference
In-Depth Information
In Ref. [41], we have obtained these numerical values from a truncated multi-
pole expansion of order
L
=
80. Our numerical values agree with earlier calcu-
lations of the reduced inverse drag and turn coefficients of two touching spheres
[47], which are given by 1
.
3801 and 4
/(
3
ζ(
3
))
1
.
1092098301, respectively. The
single-sphere mobilities in presence of the interface are larger than in the bulk since
the particle experiences less solvent friction, in particular regarding its rotational
motion. Note that the method developed in this work allows to evaluate with high
numerical precision additional hydrodynamic coefficients which characterize the
response of a single sphere to an incident fluid flow.
≈
D. Many-Sphere Mobility Tensor for Large Interparticle Distances
The Q2D mobility tensor for two spheres can be expressed in dyadic notation by
employing the unit vectors
R
R
⊥
= ˆ
×
R
depicted in Fig. 1.
z
,
ˆ
=
(
r
2
−
r
1
)/r
and
z
The basic mobility relation (10) acquires here the form,
2
μ
tt
μ
tr
U
i
=
ij
·
F
j
+
ij
·
T
j
,
(22)
j
=
1
2
μ
rt
μ
rr
i
=
ij
·
F
j
+
ij
·
T
j
.
(23)
j
=
1
The corresponding translation-translational, translation-rotational, rotation-transla-
tional and rotation-rotational mobilities can be interpreted likewise as 2
×
×
2, 2
1,
1
μ
, which describes
only the relevent degrees of freedom, or, alternatively, as the standard 3
×
2and1
×
1 tensors, the components of the reduced matrix
¯
3tensors
μ
with a number of zeros as elements, which eliminate the unphysical motions.
These tensors are specified by 4 scalar functions of the dimensionless interparticle
distance
R
×
.
The long-distance approximation of the Q2D mobility to
=
r/(
2
a)
, with
r
=|
r
2
−
r
1
|
(
1
/R
3
)
, is con-
structed from Eq. (13) by considering only terms in the scattering expansions
with no more than a single propagator
G
0
(for details, see [41]). This yields the
translation-translational long-distance part of the Q2D mobility matrix as
μ
t
11
/μ
t
0
O
[
R R
+
R
⊥
R
⊥
]
=
1
.
3799554
,
(24)
3
2
R
1
1
R
3
[
μ
t
12
/μ
t
0
R R
2
R
⊥
R
⊥
1
.
159862
R R
0
.
111686
R
⊥
R
⊥
]
=
+
−
+
.
(25)
Likewise, the long-distance asymptotic translation-rotational, rotation-transla-
tional and rotation-rotational mobilities are obtained to
(
1
/R
3
)
as
O
μ
t
11
/μ
r
0
μ
t
12
/μ
r
0
=−
R
⊥
z
/(
4
R
2
),
=
0
,
(26)
μ
r
11
/μ
r
0
μ
r
12
/μ
r
0
=
z R
⊥
/(
4
R
2
),
=
0
,
(27)
μ
r
11
/μ
r
0
μ
r
12
/μ
r
0
z
/(
8
R
3
),
=
1
.
10920983
z
ˆ
z
,
ˆ
=−ˆ
z
ˆ
(28)
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