For the study of transients in gas-liquid flows, the equations of the so-called separated flow model are inadequate, because they possess, in the general case where gas and liquid move at different velocities, complex characteristics. This paper is concerned with the equations of motion for bubbly flow. The equations are discussed with emphasis on the aspects of relative motion and the characteristics are calculated. It is found that all characteristics are real. The results are used to establish a relation between gas velocity, liquid velocity, void fraction and sound velocity at critical flow. This relation agrees very well with experimental data for these quantities as measured by Muir and Eichhorn in the throat of a converging-diverging nozzle.
Prosperetti, A., & van Wijngaarden, L. (1976). On the characteristics of the equations of motion for a bubbly flow and the related problem of critical flow. Journal of engineering mathematics, 10(2), 153-162. https://doi.org/10.1007/BF01535658