Equilibrium Solutions and Their Stability

If we refer back to the Rayleigh-Plesset equation, (4-208), it is evident that a bubble in equilibrium must have a radius RE that satisfies the condition 

is a constant, ambient pressure and pv is the vapor pressure at the ambient temperature. 

if we solve this cubic equation for Re, we find that there are three real roots when Ap = p - px- is less than a critical value, 

Substituting this value for RE into (4-213) leads to the critical value of Ap given by (4-214). Finally, using the expression (4-214), we can express Rcrit solely in terms of the material parameters(7 and y, namely, 

Although two steady solutions exist for all Ap Apcrit, an important question is whether either of these solutions is stable. Thus, if we were to consider an arbitrarily small change in the bubble volume from one of the predicted equilibrium values, we ask whether the bubble radius will return to the equilibrium value or continue to either grow or collapse.20 In the former case, the corresponding equilibrium state is said to be stable to infinitesimal perturbations in the bubble volume, while it is said to be unstable in the latter case. A steady, equilibrium solution that is unstable to infinitesimal perturbations will not be realizable in any real physical system because it is impossible to eliminate disturbances of arbitrarily small magnitude. 

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