Biomedical Engineering Reference
In-Depth Information
Stefan-Reynolds theory by a factor, which is expressed as an effective viscosity and
replaces the dynamic viscosity in the Stefan-Reynolds Eq. (17), as follows:
+
+
+
+
μ
Ma
Na
Ap
MaNa
ApNa/
2
μ
e
=
,
(23)
12
+
4
Ma
+
4
Na
+
MaNa
ahE
g
μ(aD
s
+
hβ
μ
where
Ma
=
Dh)
is the Marangoni number,
Na
=
is the Navier number,
ah
2
μ(aD
s
+
∂
∂
Ap
∂/∂c
is the adsorp-
tion length (
is the adsorption density and
c
is the surfactant concentration),
E
G
=−
=−
is the adsorption number. Here
a
=
Dh)
∂γ/∂
ln
is the surface Gibbs elasticity,
D
s
and
D
are the surface and bulk
diffusion coefficients of the surfactants. The Marangoni number and Navier number
account for the mobility of the air-water film surface and water-solid film surface,
respectively. In the limit as
Ma
1 which corresponds to the tan-
gentially immobile film surfaces, Eq. (23) gives
μ
e
=
μ
and the Stefan-Reynolds
theory is recovered. The adsorption-pressure number,
Ap
, takes into account the de-
pendence of the disjoining pressure on the adsorption on the film surfaces. Unlike
the other dimensionless numbers in Eq. (23), the adsorption number can be positive
and negative.
The above models do not account for the dynamic effects produced by the elec-
trical double layer, which can arise with the ionic surfactants present in the film
solution. Therefore, an electrical term can be added into the bulk pressure stress
tensor and the electrostatic potential into the film to make the theory valid for ionic
surfactants [73, 76]. Depending on the adsorption layer at the film interface, the
interfacial charge density can vary with the film thickness, the so-called charge reg-
ulation. Two particular models are well known in the literature, namely constant
interfacial charge density and constant interfacial potential. According to the nu-
merical analysis of the governing equations, ionic surfactants can influence the film
drainage in two ways: at high surfactant or salt concentration, the interfaces become
tangentially immobile and then dynamic changes in the concentration, adsorption,
electric charge and middle plane potential affect the film thinning due to the change
in the non-equilibrium part of the electrostatic disjoining pressure. At small sur-
factant concentration, it can influence the film drainage by reducing the surface
mobility.
1and
Na
F. Spreading and Relaxation of Contact Lines
The classical hydrodynamics of the spreading and relaxation of the three-phase con-
tact (TPC) lines leads to a singularity due to a non-integratable stress balance and an
infinite curvature of the interface when the contact line is approached. A slip length
of nanometre scales is introduced to describe the unique hydrodynamic mecha-
nisms operating in a very small neighbourhood of the contact line which is called
the inner region. To ease the modelling the TPC motion, the governing equations
are solved in three different domains (Fig. 12): (1) the outer region, which is far
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