Biomedical Engineering Reference
In-Depth Information
2. Augmented Young-Laplace Equation
After rebound and oscillation, the bubble can rest at the solid surface with or with-
out an intervening liquid film. In addition to the capillary pressure and hydrostatic
pressure, the bubble surface can subjected to one more pressure, called the disjoin-
ing pressure,
, which arises from the intermolecular interactions within the liquid
film [16, 28]. The components of the disjoining pressure are briefly described in
Section D. Shown below is a modification of the classical Young-Laplace equation
for the bubble deformation in the presence of the disjoining pressure.
When the bubble rests at the solid surface, all the transient terms in Eq. (5) be-
come zero and only the buoyancy and bubble-surface interaction force on the right
hand side remain. These two terms can be further refined to give the Young-Laplace
equation for a sessile bubble. Specifically, the equation can be directly established
by minimizing the surface energy, the gravitational potential energy, and the surface
free energy due to intermolecular interactions between the bubble and the surfaces
at close proximity. The minimization [1, 6] yields the augmented Young-Laplace
equation described as
d
r
d
r
r(
d
h/
d
r)
1
1
(
d
h/
d
r)
2
∂σ
∂h
,
(6)
where
r
is the radial coordinate measured from the axis of symmetry,
h
describes
the bubble surface coordinate measured from the solid surface as a function of the
radial coordinate
r
,
p
is the pressure at the lower bubble apex and
γ
is the surface
tension.
σ(r,h)
is the surface interaction free energy per unit area between the
bubble and solid surfaces separated by the distance,
s
, directed from the bubble
centroid.
By definition, the surface interaction energy is related to the disjoining pressure
by
(s)
+
(
d
h/
d
r)
2
[
γ
+
σ(r,h)
]
=
p
+
gρh
+
+
∂σ/∂s
, where the partial derivative can be linked with t
he bubb
le
coordinate on the basis of the distance balance. It gives
=−
(
√
s
2
2
h
2
[
R
b
+
s(r)
]
=
−
+
r)
2
2
.
Assuming that air is incompressible under the normal condition and therefore the
bubble volume remains constant during the interaction, giving
V
b
=
+[
+
R
b
]
h(
0
)
π
rh
d
r
=
const
. This volume constancy has to be considered when integrating Eq. (6). The
initial conditions for integrating Eq. (6) include:
h
2
0
(at the upper bubble apex). The disjoining pressure as a function of separation is
described in the next section.
Differential Eq. (6) with implicitly unknown pressure
p
presents an initial-
boundary problem which can successfully be solved by the shooting technique. The
integration starts at
r
=
h(
0
)
and d
h/
d
r
=
0at
r
=
0 with a guess for
p
and ends at the lower bubble apex.
The guess for
p
is updated until bubble volume constancy is met. The fourth-
order Runge-Kutta method can be used to carry out the integration. In particular,
following the traditional numerical integration of the Young-Laplace equation [23],
the integration of Eq. (6) is improved by integrating along the arc length of the
bubble surface. Figure 7 show the numerical results.
=
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