Collision and Attachment Interactions of Single Air Bubbles with Flat Surfaces in Aqueous Solutions - Drops and Bubbles in Contact with Solid Surfaces - page 228

Figure 10. Comparison between the experimental (points) and the inertial drainage model described
by Eq. (22) for the dimensionless drainage rate ( d h/ d t)/U (dash lines) and film thickness h/d eq (solid
lines) versus time tU/d eq for varying M = 5 , 10 and 50 [82].

the film thickness or smaller) and the inertial terms can be ignored, the lubrication
approximation with zero stress at the air-water surface gives
∂h
∂t =

( 4 h 3 ,

1
3 rμ

∂
∂r

12 βh 2 )r ∂P
∂r

+ (21)

where β is the slip length of the water-solid surface. In the case of strong-slip
regime, the inertial terms should be retained. The viscous terms can be ignored in
the modelling of the inertial drainage. The water velocity at the air-water film sur-
face is of the order of the bubble rise velocity. The inertial drainage can be described
as follows [82]:

U

3 / 2

d h
d t =

2 M + 1

, (22)

2 M + d eq /h

where h is the film thickness, i.e., the distance between the bubble and solid sur-
faces, at the upper bubble apex, the initial thickness at t

0 is equal to the bubble
equivalent (geometric mean) diameter. The other model parameter is described as
M

=

32 C AM / 3, which is O (10). Figure 10 shows a comparison between Eq. (22)
and experimental data. The inertial drainage mechanism fits the experimental more
closely at the beginning and the deviation between the model and the theory indi-
cates that viscous effects can be dominated for very thin films.
It should be noticed that as in reality there are usually contaminants or surfactants
in the liquid medium, the surfaces are between these two extreme Stefan-Reynolds
and Scheludko models for fully immobile and mobile surfaces respectively. In this

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