Biomedical Engineering Reference
In-Depth Information
A Short Summary of Overdamped Hydrodynamics
The velocity field
v
(
r
) of incompressible fluid flow obeys the Navier-Stokes equation
ρ
∂
v
∂t
2
v
.
+
ρ
(
v
·∇
)
v
=
−∇p
+
η∇
(8.39)
Here,
v
=
v
(
r
) is the velocity in the fluid at position
r
,
p
=
p
(
r
) the distribution
of pressure, and
ρ
the density of the fluid with viscosity
η
. The incompressibility of
the fluid leads to
div
v
=0
.
(8.40)
The ratio between convective and viscous terms is given by the Reynolds number
ρV L
η
Re
≡
,
(8.41)
where
V
and
L
are a typical velocity and length scale associated with the flow. Here,
we are studying small objects moving with small velocities implying Re
1. In this
case, the velocity field is determined by Stokes equation
4
2
v
.
∇p
=
η∇
(8.42)
The flow field generated by a moving sphere of radius
a
at large distances
r a
can be approximated by the flow due to a point force acting within the fluid. The
corresponding velocity field is that of a stokeslet moving on a trajectory
r
t
and
obeys
2
v
−
f
δ
(
r
−
r
t
)
,
(8.43)
where
δ
(
r
) is Dirac's delta function. The Green's function (or Oseen tensor) of the
last equation is given by
∇p
=
η∇
8
πν
δ
αβ
r
3
,
1
+
r
α
r
β
G
αβ
(
r
)=
(8.44)
r
where
δ
αβ
is Kronecker's delta (i.e.,
δ
αβ
=1if
α
=
β
and
δ
αβ
=0if
α
=
β
).
The corresponding velocity field is given by
v
(
r
)=
G
S
(
r
)
f
(
r
), which reduces to
Equation (8.2).
In the presence of a no-slip wall, the velocity field must vanish on the surface of
the wall. As shown by Blake [89], mirror images can be used to fulfill this boundary
condition. This is a generalization of a method known from electrostatics where
a charge image is introduced to cancel a charge distribution's field on a surface.
However, in hydrodynamics it is not sucient to simply take into account a mirror
stokeslet. Additional contributions, denoted as doublets (D) and source doublets
(SD) are required. Thus, for a stokeslet at position
r
i
=(
x
i
,y
i
,z
i
) in the presence
of a no-slip wall at
z
= 0 the Green's function becomes
G
wall
(
r
i
,
r
)=
G
S
(
r
−
r
i
)
− G
S
(
r
−
r
i
)+
G
D
(
r
−
r
i
)
− G
SD
(
r
−
r
i
)
,
(8.45)
where
r
i
=
r
i
−
2
z
i
e
z
is the position of the image stokeslet and
G
D
and
G
SD
are
the contributions of the double and source doublet [89].
4
However, one should note that in large systems even for slow motion momentum
injection is not instantaneous everywhere, which has to be taken into account by
considering time-dependent velocity fields.
