14 



Using the equation of continuity for a compressible fluid and assuming that the viscosity and 

 compressibility of the liquid are small, he writes, finally, 



P -JlE- - 





If the internal pressure is constant and the surface tension and viscosity negligible,* 

 the equation of motion can be solved analytically if only terms in dU/dR and dH/dR are re- 

 tained, thus, Gilmore finds 



3o i/2 V 



{"ii-k-^u^ 



'^^p^-Pi^' 



Neglect of the term U/SC yields the Rayleigh results. As ft -► 0, the Rayleigh theory 

 (incompressible) gives U--R~^^^, whereas Gilmore's results gives U'^R~^^^. 



Figure 3 - The Theoretical Wall- Velocity of 

 a Bubble in Water Collapsing under a 

 Constant Pressure Difference of 

 0.517 Atmospheres 







N. 



S 





\ 





>> 

























•s 



s 



*, 



\ 

 \ 































Xl 



\ 

























































































































■^ 

































































\ 































^ 



N 









Present Theory 





i 



■. 

 \ 







Incompr 



essibi 



, Compressible) 









N 



s. 





Schr 



eic 



er's N 

 1 1 



um 



ericol Co 



cula 



ion 



These curves are reproduced from Gilmore, Refer- 

 ence 21. "Present Theory" refers to Gilmore's results, 

 see text. 



0.01 

 Radius Ratio, R/Rg 



Figure 3 shows a comparison of Gilmore's theory with the results of Rayleigh, Herring, 

 and a numerical integration of the complete equation of motion carried out by Schneider. ^^ 

 It will be seen that the solutions approach each other for small ratios of the wall velocity to 

 sound velocity but diverge rapidly as the sonic velocity is approached and finally exceeded. 

 Gilmore also gives the equations of pressure and velocity throughout the fluid but did not 

 compute numerical values. 



More recently, computations have been carried out by Poritsky'^^ and Shu^^ to deter- 

 mine specifically the effects of viscosity and surface tension, but assuming the fluid to be 

 incompressible. In these papers, large effects of viscosity in retarding both the growth and 



♦ Although Gilmore only carried through the computation with surface tension and viscosity neglected, he ex- 

 amined the effects of these variables and found bounds for the ratio R/R,^ within which the effects could be 

 neglected without affecting the motion except in the very last period of coUapse (which will usually be of the 

 order of a few microseconds). 



