Page 193 - Percolation Models for Transport in Porous Media With

11.2 MOVEMENT AND GROWTH OF BUBBLES 191

Derivatives with respect to 0. will be denoted by a prime.
In this case the equations (11.30), (11.31) and the initial conditions can be
represented in the following form, which is convenient for the further investigation,

1
1
3
G:/G. + 3xow~jw. = DH A - G; w; ; (11.36)
w.(w~Y)' + 3/2(w~- Y) =(G.- 1- Zjw.}M
2
-(w~jw. - Y)N; (11.37)
G,.(O} = 1 + Z, w.(O) = 1 (11.38)
where M = p00j(pta~X ), N = 4p,f(pta~X), Y = {12/(ap,X), Lo =
2
p 00(3Lppf)- 1 (1- r; 1 ), D = 3Aw~(Y +LoG:).
Note that as the liquid reaches its boiling point T.(Poo) = n, the temperature
inside the bubble T.(pp) can be substantially higher if the size of the latter is
sufficiently small (Z > 1}. Moreover, if the conditions (11.35) are met, then it
is possible to assume that the temperature outside the bubble undergoes little
change. Therefore (forT< 0.8n) the parameters of the liquid can be considered
practically constant, whereas the parameters of the vapor can change significantly.
In the general case, a solution to a system of nonlinear equations (11.36) -
(11.37) with conditions (11.38) can be obtained only numerically.
To obtain a qualitative dependence w.(O.), transform (11.36) to

(11.39)

Since Y "' X, M "' x- 2 , N "' x- 1 , it follows that as q0 "' X goes down, the
term "' M becomes the principal term in the equation (11.36). (This follows from
the obvious fact that w~ and w~ decrease as X goes down.) In this case we have

G.,~ 1 +Z/w. (11.40}
After substituting (11.40) in (11.39), integrating, and taking into account the
initial value (11.38), we obtain

e.
{(0.} = {(0) + (Axo)- j D dO., {(0} = 1 + Z(3xo - 1)(2xo)- 1 (11.41)
1
0
The following facts are taken into account in (11.41). Z depends on the tem-
perature of the fluid Tt and changes little in time. H "' T(pp) and depends on
G.; however, this dependence is weak -In G •. The second term in (11.41) can be
significant only for very small bubbles (Z ::» 1) at the initial stage ofthe movement
w. > 1, when it can be approximately taken that