Collision and Attachment Interactions of Single Air Bubbles with Flat Surfaces in Aqueous Solutions - Drops and Bubbles in Contact with Solid Surfaces - page 233

by Cox [13] considers the effect of flow inertia on the TPC motion. In this case, the
hydrodynamic zone is also controlled by the flow Reynolds number which can be
described as

g θ m + 9 Ca ln 2 L

λ Re 1 / 3

g(θ) = + Ca ln Re, (25)

where g is defined by g(x) ρUL/μ .

In the above two models, a local coordinate system with the origin located at
(and moving with) the TPC point is used. In reality, the solution has to be cor-
rected for the relative motion of the origin via the Galilean transformations which
is a drawback of this model. Moreover, the macroscopic length scale, L , is usually
not known precisely. Phan et al. [58] followed the special approach developed for
bubbles and drops by Hocking and Rivers [25] who solved the governing equations
relative to a fixed coordinate system with the origin located at the centre of the con-
tact area and is stationary during the TPC expansion. In this approach, the unknown
macroscopic length is not needed. Solving the motion equations are accompanied
by a set of integration constants. The determination of the unknown constants is
complicated since the deformed interface deviates from the spherical profile. More-
over, due to the singularity, the motion equations have been solved in three separate
domains as discussed previously. Matching the results gives

= 1 . 532 (x − sin x) and Re =

G(θ) − G(θ m )

Ca = (26)

− + − +

ln (λ) ln (r) Q 0 (θ) Q 1 (θ m )

where Q 0 (θ) = cos θ) + J(θ)/g(θ) and Q 1 (θ m ) = j(θ m )/g(θ m ) . Func-

tions G(x),g(x),J(x) and j(x) can be found from the solution of motion equations
in outer and inner regions. The details of the analysis are given in [58].
2. Molecular-Kinetic Theories
The motion of TPC line is determined by the overall statistics of the individual gas-
liquid molecular displacement on the solid substrate. This molecular displacement
is made by molecular jumping with a mean distance, λ , on the solid substrate. The
molecular-kinetic theory gives [4, 53]

1 + ln ( 1 +

2 Kλ sinh σλ 2 cos θ m ) ,

U =

2 k B T ( cos θ 0 −

(27)

where k B is the Boltzmann constant, T is the absolute temperature and θ 0 is the
equilibrium (or the so-called Young contact angle).
Phan et al. [57] examined both hydrodynamic and molecular-kinetic models ac-
curacy by comparing the experimental results of observing the motion (the radial
position) of the TPC line for a small rising bubble ruptured by a submerged hor-
izontal glass plate. It was shown that both models were not able to describe the
experimental data using the physically consistent values for the model parameters.
As shown in Fig. 13, the models fit only the first few experimental data. The differ-
ence between the model and the experimental data at long time of the TPC motion