Then, for superheats not too far above the boiling temperature, the vapor pressure may 

 be approximated by a linear function of the temperature so that we may write 



p v (T) - P p v (T) - p v (T b ) 



= = A (T _ Tb ). (22) 



p p 



While a more accurate linear approximation to p v (T) has been used over the interval 

 T -T b , A is very nearly the slope of the vapor pressure, p v (T), at T divided by the 

 liquid density. The equation of motion now becomes 



1 d 2<7 



■ - (R S R 2 ) = A(T - T h ) . (23) 



2R 2 R dt R 



When T is given as a function of R, the solution of the problem is of course determined. 

 To find this relation, it is necessary to solve the heat transport problem at the 

 moving boundary of the bubble wall. Because of the boundary condition at the moving 

 bubble wall it is convenient to transform the heat flow equation to Lagrangian coordi- 

 nates. It does not appear possible to solve this heat equation in closed form; however, 

 physical considerations suggest an approximation which gives an analytic solution. The 

 thermal diffusivity of the liquid is so small that the characteristic diffusion length \/D t 

 is small compared with the bubble radius R(t). In other words, the liquid layer over 

 which the temperature changes from its value at infinity to its value at the bubble wall 

 may be taken to be small. This "thermal boundary layer" assumption leads to an analytic 

 solution in successive approximations [3]. The first and second approximations have 

 been found; the second approximation has been used to demonstrate that the first 

 approximation has acceptable accuracy. This first approximation to the temperature 

 field in the liquid gives for the temperature at the bubble wall 



t d 



f -(R s p')dr 



L I dr 



T(R) = T„ / . (24) 



3fc J ° [.f<fl 4 (e)de]* 



The behavior of the solution for bubble growth 



The bubble growth problem is now determined, and we may consider the be- 

 havior of the solution for vapor bubbles growing in moderately superheated water. The 

 initial conditions considered are specifically that the bubble is spherical of radius R 

 and at rest for t = 0. Such a bubble is unstable. The growth is initiated by the rising 

 temperature of the bulk liquid for which we have 



D 



T„=T a +-q{t) . (25) 



k 



The heat source is supposed to give slow changes in this external temperature. As we 

 have already remarked, for ordinary heating rates the bulk temperature of the liquid 

 will remain essentially unchanged during the few milliseconds of bubble growth which 

 will be followed here. The external temperature rise of the liquid is really only a device 

 for initiating the growth. It may be emphasized that the important range of bubble 

 growth is not appreciably affected by the initial conditions. Thus, one might suppose 

 that the growth was initiated by expansion of a permanent gas in the bubble. The 

 permanent gas content of the nucleus will soon become unimportant and the subsequent 

 behavior will be essentially that of the pure vapor bubble described here. Another pos- 

 sibility is that the initial surface tension condition is not that of the free bubble, but 



301 



