Analytical Studies of Steady and Non-Steady Motions of a Bubbly Liquid

Abstract

A consistent set of continuum-like equations which describe, under certain limitations, the flow of bubbly gas-liquid mixtures is applied in the solution of a few problems that bear on technological issues of nuclear reactor safety. The solutions of these problems illustrate the significance of the ratio between the viscous relaxation time of the bubbles and the characteristic time of the flow, in scaling experimental results. The choked flow of a bubbly mixture through a contraction in a one-dimensional duct is treated. It is found that in many cases the ratio of the contraction residence time to the viscous relaxation time is small, indicating the motion of the bubbles will be dictated largely by the dynamic forces on them. The one-dimensional equations are solved approximately for small values of this ratio. A rudimentary experiment on choked bubbly flow through a contraction was conducted using a contraction with gradual changes in area, making the experimental situation amenable to a one-dimensional analysis. Distributions of pressure and mass flow rates of liquid and gas were measured. The results compare favorably with theoretical calculations. The rise through a liquid of a uniform cloud of bubbles is also analyzed. Self-preserving wave solutions of the non-linear equations are found to exist and have the form of transitions between a region of high void fraction below and a region of lower void fraction above. These waves are unstable to small disturbances in response to which they will steepen, developing into clumps of bubbles above which are regions of low void fraction. The fact that the bubbles in these clumps may coalesce presents a mechanism for a change in flow regime from bubbly to some other, perhaps slug or annular flow. The effect of bubble-bubble interactions i.n impeding the formation of these clumps i.s speculated upon. Finally, the flow of a bubbly mixture over a wavy wall is analyzed. The solution illustrates some of the important deviations from one-dimensional flow and shows the manner in which phase separation tends to make use of the strict one-dimensional flow assumption more limited than in single phase flow. The solution is incomplete in the sense that the effect of interactions between bubbles and solid boundaries is lacking.