Shock waves in a liquid containing gas bubbles

Abstract

Considerations of continuity, momentum and energy together with an equation of state are applied to the propagation of plane shock waves in a gas + liquid mixture. The shock-wave relations assume a particularly simple form when the temperature rise across a shock, which is shown to be small for a very wide range of conditions, is neglected. In particular, a simple relation emerges between the shock propagation speed and the pressure on the high-pressure side of the shock, the density of the liquid and the relative proportions, by mass and volume, of gas and liquid in the mixture. It is shown from entropy considerations that a rarefaction wave cannot propagate itself without change of form, and it is argued that a compression wave can be expected to steepen into a shock wave. Consideration of the collision between two normal shock waves, moving in opposite directions, suggests that the strengths of the two shocks are unaltered by the interaction between them. This implies, in particular, that, when a shock impinges normally on a plane wall, the pressure ratio across the reflected shock is equal to that across the incident shock. When the mass ratio of gas to liquid in the mixture is allowed to tend to infinity, the various shock-wave relations for a mixture, derived with the temperature rise across the shock neglected, assume the same limiting form as the corresponding relations for a perfect gas when the ratio of specific heats tends to unity. The theoretical discussion has been illustrated by experiments with a small gas + liquid mixture shock tube. Samples of the records, obtained when the passage of a shock changes the amount of light transmitted through the mixture to a photoelectric cell, illustrate the steepening of a compression wave and the flattening of a rarefaction wave. Measurements confirm the theoretical relation for the propagation speed of shock waves. Reasonably good experi­mental confirmation is also reported of the theoretical predictions for the pressure which arises following the normal impact of a shock wave on a plane wall.