Biomedical Engineering Reference
In-Depth Information
When approaching and interacting with the solid surface, the bubble surface is
further deformed by (1) the liquid pressures produced by the flows confined be-
tween the bubble and solid surfaces, and/or (2) the intermolecular forces within
the intervening liquid film. The dynamic deformation and static deformation of a
bubble interacting with the solid surface are described below.
1. Deceleration and Oscillation of Bubbles Interacting with the Surface
When a bubble rises towards a horizontal surface, the velocity of the bubble cen-
troid decelerates from the terminal rise velocity to zero. The bubble can rebound
and oscillates at the solid surface. For slowly rising bubbles, no significant rebound
and oscillation are expected. For fast rising bubbles with sufficiently high values of
the Reynolds and Weber numbers, the bubble rebound and oscillation are signifi-
cant. Quantitatively, air bubbles rising in water can undergo shape oscillation upon
collision without a clearly visible rebound if their Weber numbers based on their
terminal rise velocity are less than about 0.3 [71].
Evidently, the bubble oscillation and rebound result from the interchange be-
tween the bubble kinetic energy, surface energy and potential energy. The bubble
dynamic behaviour at the solid surface can be analyzed using the equation of motion
written for the bubble centroid which gives [34, 59, 82]:
L
2
)
ρ
L
U
2
2
ρ
G
V
d
U
C
AM
ρ
L
V
d
U
d
t
d
t
=
Vρ
L
g
−
C
d
(
π
−
μR
b
t
−∞
r
max
(5)
Re) U(s)
d
s
−
6
π
K(t,s
;
−
2
π
rp
d
r.
0
Equation (5) accounts for various forces acting on a bubble rising towards a solid
surface in a still aqueous solution. The left hand side of Eq. (5) describes the bub-
ble inertia. The terms on the right hand side describe the bubble buoyancy, liquid
resistance, added-mass force, history force and bubble-surface interaction force,
respectively. The hydrodynamic equation for the bubble-surface interaction force
can be inferred and modelled from the pressure field obtained from the 2D Navier-
Stokes equations simplified for flows within the liquid films. The bubble weight is
neglected in the motion equation due to the big difference between the air and water
densities. Equation (5) is featured by various factors, such as the drag coefficient
C
d
, the added mass coefficient
C
AM
and the kernel
K(t,s
Re)
of the history force,
associated with the surface deformation of bubbles interacting with the solid sur-
faces. These factors are known for motions of bubbles and particles far away from
the solid surface [10, 53] and are less known for bubbles interacting with the solid
surface. The factors for the bubble motion are generally complicated as it can be
appreciated by the dependence of the drag coefficient on bubble Reynolds number
shown in Fig. 4. Nonetheless, progress has been made and good agreement be-
tween the theory and experiments has been established. Figure 5 shows an example
available in the literature, where the model prediction uses the lubrication theory in
determining the bubble-surface interaction force in Eq. (5).
;
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