Page 187 - Percolation Models for Transport in Porous Media With

11.2 MOVEMENT AND GROWTH OF BUBBLES 185

Estimate the velocity of a bubble with radius a0 as it moves from a pore of
size,..., a* to a capillary ofradius r <a*. Consider the movement ofthe bubble as
though the fluid were flowing past it. The mass of the former, m 0 , is of the order
of mass of the fluid displaced by the bubble

1
When the velocity of the bubble v0 < p.a0 P/ (a0 <: 10- 4 m), the resistance
1
to the movement of the bubble is adequately described by Stokes's formula

The force that causes the movement of the bubbles for small temperature
gradients T' is determined by the following relationship [87, 88]
'T'
F. g = -x 11'ao, x' = dx/dT
2
As the bubble comes closer to a thinner capillary, both the temperature gra-
dient and the velocity go up.
The steady state velocity of the bubble is determined from the condition Fe =
Fg
vo = -x'T'aop.- /6 (11.14)
1
The transient period for the velocity of the bubble is

m 0 2 PJ 2
Ty ,..., 67rp.ao = 9 -;ao (11.15)

and is usually much less than the time r 9 needed for a bubble to reach the capillary
from the pore. This fact permits to consider the movement of the bubble to be
quasi-uniform with velocity determined by (11.14) and T' determined according
to (11.11) or the corresponding asymptotics. For example, when T' grows slowly
(when t > a* /(4Kt)), using the asymptotics of the expression (11.11), we obtain
2
the duration of the movement for the bubble
48p.Kt 1 (a*) 2
rg ~ ( -x')qp ao -:;: (11.16)

From the condition ry/r 9 < 1, the following relationship results,
216p. Kt 1 (a*) 2
q < 2 - -
P ( -x')PJ a~ r

For the period of,..., r 9 the bubbles move within a large capillary towards its
boundary. This results in the bubbles situated in the pores that surround the
"hot" capillary concentrating inside it ,..., r 9 seconds after the current is turned on.