Stability of a compressed gas bubble in a viscous fluid

Abstract

The stability of the spherical shape of the free surface of a gas bubble compressed by an incompressible fluid as it appears in the inertial confinement fusion problem is considered. (i) The equations derived by Prosperetti [Accad. Naz. Lincei 62, 196 (1977)] generalizing the Plesset equation are recovered in cases when the outer fluid is nonviscous, the flow being not potential, and it is shown that vorticity may change drastically the results of the potential case. (ii) In the case of viscous external fluid, the equations derived by Prosperetti [Q. Appl. Math. 1, 399 (1977)] and other external conditions on a sphere of finite radius are derived. (iii) Assuming that the time scale of the dynamics of the spherical bubble is large with respect to the time scale of the perturbation (frozen assumption), the linear stability of the collapsing bubble is studied numerically. The parameters are here (a) an inertia force (related with acceleration R̈ of the radius of the bubble), (b) the Reynolds number built with the decaying rate of the bubble, (c) surface tension, and (d) the aspect ratio (ratio between the gap width of the viscous fluid and the radius of the bubble). It is shown that the spherical shape is always linearly unstable in the absence of surface tension. In the presence of surface tension, there is a critical inertia parameter value and the most dangerous mode is always stationary. For the case of a large surface tension, the spherical wavenumber l of the most dangerous mode, is low. Finally, it is shown that the Rayleigh–Taylor instability might only be observed for both small aspect ratio and Reynolds number, depending on the surface tension.