Biomedical Engineering Reference
In-Depth Information
Figure 4.42
Schematic view of a liquid plug in a capillary tube with the advancing and receding
contact angles.
The pressure drop due to friction on the solid walls is given by the Washburn
law [37]
8
U
D
P
=
(
µ
L
+
µ
L
)
(4.61)
drag
1 1
2
2
2
R
where indices 1 and 2 address to liquid 1 (liquid plug) and liquid 2 (surrounding
carrier fluid).
R
is the radius of the capillary,
U
the average liquid velocity and
L
1
,
L
2
the total length of contact of liquid 1, 2 with the solid wall (
L
1
+
L
2
=
L
, total
length of the tube). Each interface—advancing and receding—contributes (posi-
tively or negatively in function of the contact angles) to the capillary pressure drop.
The capillary pressure drop derives directly from the Laplace law, which relates the
pressure difference at a spherical interface of curvature radius
a
by
D =
2
γ
P
(4.62)
a
The meniscus has a spherical shape (if the capillary is small enough); as shown
in Figure 4.44. The contact angle is related to the tube radius
R
and the curvature
radius
a
by
R
a
cos
θ
= -
Substitution of this equation into (4.62) yields
= -
2
γ
D
P
cos
θ
(4.63)
a
a
R
Similarly, the receding front contribution is given by
Figure 4.43
Decomposition of a two-phase flow in a lumped element. Between points A and B, C
and D, and E and F, the pressure drop is due to friction; between B and C, and D and E the pressure
drop results from capillary forces.
