Biomedical Engineering Reference
In-Depth Information
Figure 12.
Magnified view of the molecular, inner, intermediate and outer regions of a dynamic menis-
cus with the outer (macroscopic),
θ
and inner (microscopic),
θ
m
contact angles.
from the contact line, (2) the inner region, which is close to the contact line and (3)
the intermediate region which is between the inner and outer regions.
The molecular-kinetic theory and the hydrodynamic theory can be used to es-
tablish the relationship between the velocity of the three phase contact line and the
dynamic contact angle. These two models differ in the mode of energy dissipa-
tion. The molecular-kinetic approach emphasizes dissipation due to displacements
of molecules in the immediate vicinity of the contact line while the hydrodynamic
approach focuses on dissipation due to viscous flow within the wedge of liquid near
the moving contact line.
1. Hydrodynamic Theories
The hydrodynamic theory links the molecular region (described by the dynamic
contact angle,
θ
m
and the molecular characteristic length scale,
λ)
to the static-like
(outer) region described by the macroscopic contact angle,
θ
and the macroscopic
(capillary) length,
L
.
Voinov [78] and Cox [11, 12] were among the first who developed useful hy-
drodynamic models for the dependence of the moving line velocity,
U
,onthe
macroscopic dynamic contact angle,
θ
. The Voinov model was derived from the
continuity and Stokes equations near the contact line and can be described as fol-
lows:
θ
m
+
1
/
3
,
θ
={
9
Ca
[
ln
(L/λ)
]}
(24)
where
Ca
μU/σ
is the capillary number and
U
is the TPC line velocity.
When the inertia is significant, the convective term in the Navier-Stokes equation
is expected to affect the fluid flow and gas-liquid interface shape. The model given
=
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