Biomedical Engineering Reference
In-Depth Information
Figure 9. Dependence of film drainage rate on film radius as predicted by Eq. (19). The points de-
scribes the experimental data [72].
Comparison of Eq. (19) with experimental results is shown in Fig. 9.
If the spatial correlation between the film surface domains is described by a
fractal dimension, α , having a value between 0 and 2, Eqs (17) and (19) can be
generalised as [31]
R 2 P
16 γ h
2
α
d h
d t =−
2 h 3 P
3 μR 2
2
+
α
.
(20)
=
For α
2, Eq. (20) recovers the Stefan-Reynolds drainage law for planar parallel
films. If α
=
1, the film contains an axisymmetric dimple causing faster drainage. If
α
1 / 2, the film exhibits numbers of asymmetric dimples and the draining rate is
even faster [72]. For α
=
0, the film contains spatially uncorrelated domains causing
the fastest possible drainage.
In the case of bubbles in ultra-clean water, the boundary condition of tangential
immobility employed in the Stefan-Reynolds theory is no longer valid and has to
be replaced by the boundary condition of tangential mobility at the air-water film
surface. Since the air-water film surface is fully mobile, the water velocity on the
air-water surface of the film is of the order of the bubble rise velocity and the inertial
terms in the Navier-Stokes equations can become significantly large and cannot be
neglected as in the case of the lubrication approximation. This is particularly the
case of strong-slip regime at the water-solid interface, which renders a plug flow
within the film. If the slip regime is weak (i.e., the slip length is of the order of
=
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