The small variations of L and k with temperature are neglected. It will be assumed that 

 the temperature of the liquid is initially uniform 



T(r,0) = To. (14) 



The bubble growth is supposed to begin at / = 0; at a distance from the bubble the 

 temperature will be uniform, undisturbed by the cooling produced by the bubble growth. 

 We have, then, 



D 



T x = TXoo,*) = T +-q(t). (15) 



k 



We shall admit, for the present, heat sources which correspond to relatively slow 

 temperature changes. For example, a temperature rise of l°C/sec would correspond 

 to an insignificant change in T x in the time of a few milliseconds over which we follow 

 the bubble growth. The initial superheat temperature T will be essentially the liquid 

 superheat throughout the bubble growth. It is convenient to introduce a radius R 

 defined by the relation 



2(7 



— = P ,(r ) - p . (16) 



Ro 



Physically, R is the effective initial radius of the bubble; it represents an extrapolation 

 of the free spherical bubble down to the equilibrium radius for the given initial condi- 

 tions. It should be noted that a bubble at rest with radius R is in unstable equilibrium. 

 The actual nucleus from which the bubble grows is not necessarily spherical and its 

 surface energy may be less than 4tt(tR 2 (it may be zero); however, such a nucleus and 

 the free spherical bubble of radius R will have the same growth behavior when their 

 radii are too or three times R . 



In terms of the parameter R , the equation of motion (3) may be written 



p v (T) - p v (T ) + (2a/R )(l - Ro/R) 

 RR + %R 2 = . ; (17) 



or 



2R 2 R dt 



d . p v (T) - p v (T ) + (2(r/JBo)(l - Ro/R) 

 (R*R 2 ) = ■ . (18) 



If the cooling effect of evaporation is disregarded so that p v {T) zr p v (T ), 

 Eq. (18) may be integrated to give 



R\ 4cr / R S \ 2<r/ #o 2 \ 



R* = — R 2 + 1 1 ) . (19) 



# 3 3pR \ R 3 I pR\ R~ I 



This solution will be called the Rayleigh solution. For R^>R , Eq. (19) gives 



4(r 2 



£2 ^ = _ [pv{To) _ p o] _ (2Q) 



SpRo 3p 



The actual motion may be expected to deviate markedly from the Rayleigh solution be- 

 cause of the cooling effect. 



Let us denote the temperature at which the vapor pressure equals the external 

 pressure P by T h , which is thus the "boiling temperature," 



Vv{T h ) = P . (21) 



300 



