Biomedical Engineering Reference
In-Depth Information
3.6.5 Capillary Pumping
If the microchannel is placed horizontally instead of vertically, the weight of the
liquid cannot balance the capillary force and a continuous flow is set up that lasts
as long as there is liquid available in the entry port (Figure 3.34). Let us assume
that the reservoir is large so that the curvature of the nearly flat horizontal interface
can be neglected. The pressure at
x
= 0 is then
P
0
, the atmospheric pressure. Let us
assume also that the channel is rectangular (width
w
, depth
d
, with
d
<
w
).
Following Bruus [19], we assume that the continuous flow in the horizontal
channel is from the Hagen-Poiseuille type and the flow rate can be symbolically
written (Chapter 2)
η
L
D =
P R V
=
V
(3.47)
V
f w d
(
,
)
where
f
is a function depending on the aspect ratio. The velocity of the flow
V
is
given by
V dL d t
=
and the driving pressure D
P
is
D =
P
2 cos
γ θ
d
After substitution, we find a differential equation for
L
(
)
2 cos
γ θ
η
f w d
,
L dL
=
dt
(3.48)
d
which can easily be integrated, yielding
(
)
γ θ
η
cos
f w d
,
L
=
2
t
(3.49)
d
The flow velocity
V
is then
(
)
γ θ
η
cos
f w d
,
1
V
=
(3.50)
d
t
Figure 3.34
Principle of capillary pumping.
