Collision and Attachment Interactions of Single Air Bubbles with Flat Surfaces in Aqueous Solutions - Drops and Bubbles in Contact with Solid Surfaces - page 226

Figure 8. Numerical results (solid lines) of Eq. (18) and experimental results (dash lines) for wet-
ting film thickness profiles between a bubble and a silica surface in an aqueous solution at different
times [7].

Due to the radial pressure gradient by the out-flow (the hydrodynamic pressure)
in draining films, a variation in film thickness is expected and when the approach
speed is sufficiently high and the separation is small, the hydrodynamic pressure
can be large enough to invert the curvature of the fluid interface to form a dimple
at the film centre (Fig. 8). Recently, a more complex interface shape was observed
by experiments [74] which can be referred to as a wimple (with two local minima
and one local maximum on the film profile) and a pimple (the film profile has two
inflection points but just one local minimum at the centre).
Films with diameter smaller than 100 µm are known to follow the Stefan-
Reynolds the ory [44]. T he smallness of the film radius is quantitatively described
by R rim

12 γ h/P ,where R rim is the film radius at the rim, h is the profile-
average film thickness and P is the pressure on the meniscus outside the film
minus the disjoining pressure calculated with the average thickness. For films with
larger radius, a dimple is often formed during the film drainage. If the film thick-
ness fluctuation is small, averaging Eq. (18) gives the R 4 / 5 -prediction for the fast
drainage of dimpled thin films [72]:
d h
d t =−

h 12 P 8
4 γ 3 R 4 .

1
6 μ

5

(19)