
81Hydrodynamics of Bubble Swarms
4
23
=
µρ−ρ
ρσ
M
g
f
(5.12)
which, in conjunction with a denition of a Eötvös number, allows the
terminal velocity to be very approximately estimated via Re
∞
. However, for
a more reliable method of estimating the terminal velocity, across a range of
parameters, Wallis (1974) correlated a dimensionless bubble radius:
*
2
2
1/3
=
ρ
µ
rr
g
b
f
(5.13)
with a dimensionless terminal velocity:
VV
g
f
*
1/3
=
µ
ρ
∞
−
(5.14)
A dimensionless group, as follows, is used to include a dependency on the
surface tension:
P
g
f
3
4
=
σρ
µ
(5.15)
Wallis suggested a number of dimensionless expressions based on P, r
*
, V
*
and the group:
2
=
ρ
B
gr