On the fulfillment of the energy conservation law in mathematical models of evolution of single spherical bubble

Abstract

The problem of the evolution of a single spherical bubble in an infinite liquid is considered, as the result of a variation of the pressure in the liquid at an infinite distance from the bubble. It has been assumed the bubble is filled with vapor from the surrounding liquid or insoluble gas. The question of the fulfillment of the integral energy conservation law is investigated using different ways of describing the hydrodynamic and heat and mass exchange processes in both the bubble and surrounding liquid and at the bubble interface. Kinetic and internal energy of vapor (gas) in the bubble, kinetic and internal energy of the liquid, and energy of surface tension are taken into account in the energy balance. The liquid is assumed to be incompressible, viscous and heat-conducting, the vapor (gas) to be nonviscous, heat-conducting and obeying the Clapeyron equation. Thermal–physical properties, exclusive of specific heats, are allowed to be temperature-dependent. For the above suppositions and assumptions, a mathematical model ensuring exact fulfillment of the integral energy conservation law has been developed. It has been shown that the conservation integral can be fulfilled by the given model. As simplified variants of the principal model, models of the uniform bubble and pressure uniform bubble, have been proposed which ensure the exact fulfillment of the integral energy balance disregarding the relatively small vapor kinetic energy. A relation defining the imbalance in the integral energy conservation law for some often-used extra simplifications has been derived.