Biomedical Engineering Reference
In-Depth Information
Figure 2.
Experimental (symbols) [24, 55] and model (lines) results for terminal rise velocity of N
2
bubbles as a function of bubble size in ultra-clean water and salt solutions.
liquid interface. The shape that the bubble assumes is in turn a complex function
of the hydrodynamic, viscous and interfacial forces. The studies have shown the
importance of the local water flow passing air bubbles in determining the bubble
rise velocity. In particular, the mobility of the air bubble surface appears to be crit-
ical. The rise velocity of bubbles with an immobile surface can be determined by
the method for solid spheres. The rise velocity of bubbles with a mobile surface
in clean water was investigated by a number of researchers [53]. The Hadamard-
Rybczynski-Boussinesq equation for the terminal bubble rise velocity,
U
,isde-
scribed as follows [53]:
2
R
b
δg
9
μ
3
+
2
Bou
U
=
2
Bou
,
(1)
2
+
where
R
b
is the bubble radius,
δ
and
μ
are the water density and viscosity, re-
spectively, and
g
is the acceleration due to gravity. The Boussinesq number,
Bou
=
μ
s
/(μR
b
)
,where
μ
s
is the surface shear viscosity, is determined by ratio of the sur-
face to bulk shear stresses, based on the hypothesis that a thin liquid layer of higher
viscosity exists near the water-air interface. If the bubble surface is rigid, Eq. (1) re-
duces to the Stokes equation for the bubble terminal velocity:
U
Stokes
=
2
R
b
δg/
9
μ
.
If the bubble surface is fully mobile, Eq. (1) reduces to the Hadamard-Rybczynski
equation:
U
HR
=
R
b
δg/
3
μ
. Experiments have confirmed that the rise velocity of
small bubbles in clean water follows the Hadamard-Rybczynski prediction (Fig. 2)
and the bubble surface in clean (surfactant-free) aqueous solutions should be mo-
bile.
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