Dynamics of Gas Bubbles in a Multi-Component Liquid

If a bubble of radius Ro was initially placed into a supersaturated (at the given pressure and temperature) mixture, then, due to the concentration difference Ap = pi — pi between the gas dissolved in the liquid far away from the bubble and the appropriate equilibrium value at the bubble surface, there arises a directional diffusion flux of the dissolved substance toward the surface. At the interface, the transition of substance from the liquid to the gaseous state takes place. The result is the increase in the bubble volume. The growth of the bubble, in its turn, results in the increase of its lift velocity, as well as the increase of convective diffusion flux. The statement of the problem and the basic dynamic equations for a bubble in a solution were described in Section 6.8. 

Consider a multicomponent solution. The distribution of concentrations of the dissolved substances is described by the equations of convective diffusion 

The dynamic balance equation for the bubble, known as Rayleigh s equation, looks like 

Here pi is the liquid phase density R - the radius of the bubble vi - kinematic viscosity of the liquid S - the coefficient of surface tension, pc - gas pressure inside the bubble poo - gas pressure far away from the bubble. 

Suppose that the gas inside the bubble represents an ideal gas mixture. Then 

---