Stability Criteria of the Steady Flow of a Liquid Containing Bubbles Along a Nozzle

The steady cavitating flow through a converging-diverging nozzle is considered. A continuum model is assumed with the Rayleigh-Plesset equation to account for the bubble dynamics. A similar problem has been studied previously by Wang and Brennen, and they found that if the upstream gas volume fraction of the bubbles exceeds a critical value there is flashing flow instability. In the present work, a perturbation analysis is made introducing a small parameter, ε, that is the ratio of the initial bubble diameter to the length scale of the nozzle. As a result of this analysis, the critical value of the upstream void fraction is calculated as a function of the several parameters appearing in the problem, and turns out to be very small and proportional to $ε3.$ A correlation is proposed giving explicitly the functional dependence of this critical value.