I. The Propagation of Shock Waves in Non-Uniform Gases. II. The Stability of the Spherical Shape of a Vapor Cavity in a Liquid

Citation

Mitchell, Thomas Patrick (1956) I. The Propagation of Shock Waves in Non-Uniform Gases. II. The Stability of the Spherical Shape of a Vapor Cavity in a Liquid. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/PDMS-YT82. https://resolver.caltech.edu/CaltechETD:etd-06142004-153608

Abstract

I. The one-dimensional propagation of shock waves in a perfect gas in which the pressure and the density are not necessarily uniform is investigated by seeking similarity solutions of the equations describing the non-isentropic motion of the gas. It is shown that such solutions can be found and that they can be related to specific types of compressive piston motion. In particular, the propagation of the shock resulting from the uniform compressive motion of a piston in a non-uniform gas is studied. For this case a first order, ordinary, non-linear differential equation which determines the shock strength as a function of distance is derived. An analytic solution of this equation is obtained for a gas in which the pressure is constant but the density varies, and for which the ratio of the specific heats, [gamma], is 3/2. There is no restriction placed upon the permissible density variations. In situations in which the pressure and the density distributions are variable, and in which general values of [gamma] are allowed, numerical results are presented. It is not possible in such cases to derive analytic solutions of the equation. The discussion of the shock propagated by the non-uniform motion of a piston is more difficult. However, some details are given in the case of strong and weak shocks resulting from a decelerative piston motion. II. The stability of the spherical shape of a gas bubble in a liquid is investigated for the case in which the difference between the pressure in the bubble, P[subscript i] and the pressure in the liquid, P[subscript o], is constant. These conditions apply approximately to a vapor bubble growing, (P[subscript i] > P[subscript o]), or collapsing, (P[subscript i] < P[subscript o]), in a liquid at constant external pressure. The general solution for the behavior of a small deformation in the spherical shape of the cavity is readily determined when surface tension is neglected. For a growing bubble the deformation increases slowly and monotonically; for a collapsing bubble the deformation oscillates with small amplitude until the mean radius of the bubble approaches zero, when the magnitude of the deformation increases rapidly. The consistency and applicability of the small amplitude theory is thus demonstrated. A solution is also obtained which includes the effect of surface tension. In this case the distortion amplitude decreases with increasing radius for the expanding bubble and the singularity in the distortion amplitude for the collapsing bubble at zero radius persists.

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