Biomedical Engineering Reference
In-Depth Information
4.3.3 Dynamic Contact Angle
The contact angle formed between a flowing liquid front (advancing or receding)
and a solid surface is not constant but reflects the balance between capillary forces
and viscous forces. The relative importance of these forces is often expressed by the
nondimensional capillary number
Ca
defined by
µ
γ
U
Ca
=
(4.51)
where
m
is the dynamic viscosity of the moving fluid (unit kg/m/s),
U
its velocity
(m/s), and
g
its surface tension (N/m). The capillary number is a scale of the ratio
between the drag force of the flow on a plug and the capillary forces. In a cylindri-
cal tube of radius
R
, the friction pressure drop for a plug of length
L
is given by the
Washburn law
8
UL
µ
(4.52)
D =
P
2
R
We deduce an order of magnitude of the drag force (force necessary to push the
plug in the tube)
2
F
P R
UL
(4.53)
» D
π
»
µ
drag
On the other hand, the capillary/wetting force is given by
(4.54)
F
»
γ
R
cap
From (4.53) and (4.54) we deduce
F
µ
γ
U L
L
drag
»
»
Ca
(4.55)
F
R
R
cap
Hoffman first proposed an expression for the dynamic contact angle as a func-
tion of the capillary number
Ca
based on experimental observations [32]. However,
this correlation is rather complicated and Voinov and Tanner have established the
more workable correlation
3
3
(4.56)
θ θ
-
=
ACa
d
s
where
q
d
and
q
s
are the dynamic and static contact angles. The value of the coef-
ficient
A
is
A
~ 94 when
q
is expressed in radians. Tanner's law is plotted in Figure
4.39.
For microflows, using the approximate values
m
~10
-3
kg/m/s,
U
~ 10
m
m/s to
1 cm/s, and
g
~50 10
-3
N/m, the typical values of the capillary number are in the
range 2.10
-7
to 2.10
-4
. The capillary number is then small and corresponds to the
linear part of the Tanner law. Linearization of (4.56) yields [33]
1
æ
ö
1
ACa
3
3
(4.57)
θ θ
=
(
+
ACa
)
»
θ
1
+
d
s
s
ç
÷
3
3
è
ø
θ
s
