Pressure Pulses in Liquid- Bubble Mixtures 



physical fact that during collapse of the bubbles the volume of the gas content 

 decreases, which fact is ignored in the linear case. 



Equation (4.13) shows that at y = the rate of change of R is much slower 

 than at y= \. The maximum gas pressure occurs when dR/dt = 0. From (4.13) 

 we infer that the value x^.^ at which this occurs is the same for all y. This 

 value is for p >> l approximately given by 



According to the relation (4.10) we have therefore at y = o 



- 4 



A P - X • 



- 4 '^ mi n 



- X ■ = 



cosh 



Mx,iJ»'' 



(4.14) 



(4.15) 



On account of (4.14) the maximum gas pressure is given by 



- 4 



mi n 



10-2p4. (4.16) 



Note for comparison that in the linear theory the maximum gas pressure is 

 1 + (2Ap/pq). If the value of A.(x^.^)*/Ms not small, it follows from (4.15) that 

 the maximum pressure at the wall n^^^ can increase to a significant fraction of 

 the maximum pressure in the bubbles. High average pressures can therefore 

 be expected to occur. 



5. CONCLUSION 



Although the approximations of the present analysis do not permit accurate 

 quantitative results, the linear and nonlinear theories developed here make it 

 clear that water-bubble mixtures are strongly dispersive and that due to the 

 dispersion high average pressures in the mixture can occur during the non- 

 linear collapse of individual bubbles. 



REFERENCES 



1. Minnaert, M., Phil. Mag. 16:235 (1933) 



2. Van Wijngaarden, L., XI International Congress of Applied Mechanics, 

 Munchen, 1964 



3. Lamb, Sir Horace, "Hydrodynamics," Cambridge: University Press, 1952 



4. Brillouin, L., "Wave Propagation and Group Velocity," Academic Press, 

 1960 



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