Engineering Transactions, 5, 2, pp. 271-329, 1957

Rozchodzenie się Małych Zaburzeń w Mieszaninie Cieczy z Pęcherzykami Gazu

A. Szaniawski
Instytut Podstawowych Problemów Techniki PAN

The physical properties of a real gas-liquid emulsion are difficult to describe mathematically. An ideal emulsion of specified properties is therefore investigated in this paper. Such properties can be treated as an approximation only to those of a real mixture. Thus, it is assumed that the emulsion is quasi-homogeneous, implying that identical bubbles of a perfect gas are uniformly distributed in a compressible inviscid liquid, and that the transient thermodynamical state of the mixture is determined by four parameters: two parameters of state of the liquid and two of the gas (for which are here taken the mean pressure and temperature of both agents).
Equations of motion resulting from these assumptions being established they are then linearized for the particular case of propagation of small perturbations. The number of equations is, however, smaller by two than the number of unknowns. Some additional assumptions should therefore be made taking into consideration the influence of inertia of the liquid surrounding a bubble, as well as heat exchange between the agents. Three types of these additional assumptions are considered. In the first type, the emulsion is assumed to undergo only barotropic changes. Then, a perturbation is propagated with constant velocity and no damping. Wave velocities calculated for three particular cases: a1 (3.8) with no heat exchange between the agents; a2 (3.8), with infinitely rapid heat exchange, and a3 (3.9), with constant specific volume of the gas (Fig. 2) constitute limiting values for additional assumptions of other
types. In the second type, the influence of the inertia of the liquid surrounding a gas bubble is disregarded. It is assumed that heat exchange between the liquid and the gas is proportional to the difference of the temperatures and the area of bubble surface. From the equation of wave propagation thus obtained, (3.16), the velocity and the coefficient of damping are arrived at for a one-dimensional (plane or spherical) harmonic wave. The velocity a increases from dz tending asymptotically to a, with increasing w. The damping coefficient ß increases at the same time to the constant
value ßmax (Fig. 4 and 5). In the third type is considered a harmonic vibration of a gas bubble in an infinite incompressible liquid space, heat exchange being taken into consideration. The conclusions reached here for the temperature difference and heat exchange between the agents are transferred to the emulsion.
The dependence of the velocity a of a plane harmonic wave and the damping coefficient ß, thus obtained, on the frequency of vibration w is represented at Fig. 10. The appearance of the phenomenon of resonance can be observed.

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