On pressure and temperature waves within a cavitation bubble

Abstract

The presented work is about the detailed pressure, temperature and velocity distribution within a plane, cylindrical and spherical cavitation bubble. The review of Plesset & Prosperetti (1977) and more recently the review of Feng & Leal (1997) describe the time behavior of the gas within a spherical bubble due to forced harmonic oscillations of the bubble wall. We reconsider and extend those previous works by developing from the conversation laws and the ideal gas law a boundary value problem for the distribution of temperature and velocity amplitude within the bubble. This is done for a plane, cylindrical, or spherical bubble. The consequences due to shape differences are discussed. The results show that an oscillating temperature boundary layer is formed in which the heat conduction takes places. With increasing dimensionless frequency, i.e. Péclet number, the boundary-layer thickness decreases and compression modulus approaches its adiabatic value. This adiabatic behaviour is reached at lower frequencies for the plane geometry in comparison with cylindrical and spherical geometry. This is due to the difference in the volume specific surface, which is 1, 2, 3 times the inverse bubble height/radius for the plane, cylindrical and spherical bubble respectively. For the plane bubble the analysis ends up in an eigenvalue problem with four eigenvalues and modes. The analytical result is not distinguishable from the numerical result for the plane case gained by a finite element solution. Interestingly if the diffusion time for the temperature distribution is of the order of the traveling time of a pressure wave no adiabatic behavior is observed. A parameter map for the different regimes is given. Since only the behavior of the gas within the bubble is considered the analysis is independent of the surface tension coefficient and the inertia of the surrounding liquid. For the plane bubble since there is no curvature there is no pressure change over the free surface. Despite of this a plane bubble is manly academic, since due to inertia the pressure within the fluid would have to be infinity if the liquid volume around the bubble is unbounded.