Biomedical Engineering Reference
In-Depth Information
Figure 1.
The system of spheres embedded in a semi-infinite fluid (
z>
0). The spheres are in
single-point contact with a planar fluid-gas interface at
z
=
0.
B. Theoretical Method
We consider
N
identical spherical particles immersed in a low-Reynolds number
fluid of shear viscosity
η
, which occupies the region
z>
0. The particles touch the
planar free interface located at
z
=
0 , so that during their motion, the sphere centers
remain within the plane
z
a
(cf. Fig. 1). To satisfy the free surface boundary
conditions of zero normal fluid velocity and zero tangential-normal components of
the fluid stress tensor, we apply the method of images [38]. Within this method, the
fluid is mentally extended to fill the whole space, by reflecting the
N
spheres and
the incident flow field
v
0
at
z
=
0 into the mirror region
z<
0. For creeping flow as
described by the Stokes equations [1], the fluid velocity field,
v
(
r
)
, can be written
as [44],
=
N
f
j
(
r
)d
3
−
=
T
F
(
r
,
r
)
·
r
,
v
(
r
)
v
0
(
r
)
(1)
j
=
1
with the integral kernel,
T
F
(
r
,
r
)
, equal to [45]
¯
r
)
T
F
(
r
,
¯
r
)
=
T
0
(
r
− ¯
r
)
+
T
0
(
r
− ¯
·
R
F
.
(2)
Here,
T
0
(
r
)
=
(
I
+
rr
)/(
8
π
ηr)
is the Oseen tensor, and
R
F
=
1
−
2
zz
is the reflec-
tion operator with respect to
z
=
0. The reflection of a position vector
¯
r
=
(
x,
¯
y,
¯
¯
z)
r
=
R
F
¯
is given by
z)
. The subscript
F
is used to indicate a system
with a planar free interface. The induced force density,
f
i
(
r
)
, exerted by the sphere
i
on the fluid and located on the surface
S
i
of the sphere
i
, is determined by the
stick boundary conditions
v
(
r
)
¯
r
=
(
x,
¯
y,
¯
−¯
=
w
i
(
r
)
≡
U
i
+
i
×
(
r
−
r
i
)
,for
r
∈
S
i
,where
U
i
and
i
are the translational and angular velocities of sphere
i
.
Search WWH ::
Custom Search