where R(t) is the liquid interface or bubble wall. Further, the liquid equation of 

 motion gives the Bernoulli integral 



dcp p{r,t) 



- + K(W) 2 = - - - + C(t). (2) 



dt P 



Evaluation of this equation at r — R gives the equation of motion of the bubble wall 



p(R) - Po 



RR + %R 2 = (3) 



p 



where p is the liquid density, p(R) is the pressure in the liquid at the bubble boundary, 

 and P is the external pressure in the liquid or the pressure at infinity. The energy 

 equation for the liquid may be written as 

 IbT _ \ 



pc v — + v ■ VT = kAT + q + p.Av 2 (4) 



\dt ) 



where /a is the viscosity, k the thermal conductivity of the liquid, and q is the heat 



source per unit volume per unit time. With the velocity field v zz R 2 R/r-, the viscous 

 heat generated is maximum at the bubble wall and has the value 



R 2 

 (nAv 2 ) r =R = 12 M — . (5) 



R 2 



In a typical bubble growth situation (103°C, R max — 32 cm/sec, R — 3 X 10 -3 cm) 

 this term is of the order of 0.1 cal/(cm 3 sec). The rate of heat loss by conduction at the 

 dT(R) 



pc v , is of the order of 10 4 calf (cm 3 sec) . Clearly the viscous 



d t 

 heat generation is negligible. 



In considering the conditions within the bubble, one has the simplification that 

 the vapor density is so small compared with the liquid density that inertia effects in the 

 vapor may be neglected. The vapor density is about 10 -3 of the liquid density. Actually 

 the pressure gradients in the vapor are an additional order of magnitude smaller than 

 10 -3 of the liquid pressure gradients. The reason for the additional reduction is that the 

 velocity and variation in velocity in the vapor are at least an order of magnitude smaller 

 than the liquid velocity. Evaporation at the liquid interface fills the bubble as it grows 

 and there is negligible motion of the vapor. We may therefore take the pressure in the 

 bubble as uniform and, since the bubble wall moves at a velocity small compared with 

 the sound velocity in the vapor, we may say further that the pressure within the vapor 

 follows practically instantaneously its value at the bubble wall. We now argue that the 

 uniform pressure within the bubble is the equilibrium vapor pressure of the liquid. 



When the liquid interface moves with a velocity R, then the pressure in the vapor, p vap , 

 is related to the equilibrium vapor pressure p eq (the vapor pressure for a stationary inter- 

 face) by [2] 



Pvap Pvap C 



= — = : (6) 



Peg Peq C* + R 



where c* is connected with the sound velocity c by 



ac 



c* = -7= • (7) 



V2tt7 

 298 



bubble wall 



