Mechanical Energy Formulation of the Rayleigh-Plesset Equation: A Generalized Derivation

A new formulation of the energy equation corresponding to the known Rayleigh-Plesset (RP) equation is presented, thus directly linking bubble dynamics with energy considerations. An energy balance is formulated by leveraging the fact that the RP equation is a sum of distinct kinematic terms, where each system property, namely density, pressure, viscosity, and surface tension is associated with one or more kinematic terms. By differentiating the energy equation and matching its units to those of the known RP, energy functions-based RP equation is recovered, this function has unknown energy terms. Comparing the two equations gives these terms, resulting in an explicit, time-dependent energy equation. The derivation reported herein can be applied to modified RP equations as well, which are similar to the classical form of RP equation but include modified or additional factors (such as oscillating pressure, liquid compressibility. Proof of the energies' uniqueness and physical integrity is demonstrated. Lastly, an example using a modified RP equation and the Rayleigh-Plesset-Keller (RPK) equation, which accounts for liquid compressibility, is presented. This example demonstrates a more complex modification, where system properties cannot always be explicitly isolated, thus, demonstrating its wider applicability.