Nonlinear Bubble Behavior due to Heat Transfer

Abstract

Previous studies of the forced oscillation of a spherical bubble in solution have been investigated by using the Rayleigh equation to obtain the time dependent bubble radius and a polytropic relation to obtain the gas pressure inside the bubble depending bubble volume (Lauterborn, 1976). In fact, the polytropic approximation with proper index values has been widely used for the gas undergoing quasi-equilibrium process in which there is heat transfer. However, the polytropic pressure-volume relationship fails to account the thermal damping due to heat transfer through the bubble wall because Pb dVb is a perfect differential and consequently its integral over a cycle vanishes (Prosperiretti et al., 1988) where Pb is the gas pressure inside the bubble and Vb is the bubble volume. Furthermore, the polytropic approximation assumes the uniform temperature for the gas intrinsically, which is valid only for a particular case and it is hard to tell whether the gas inside the bubble oscillating under ultrasound behaves isothermally or adiabatically (Loefstedt et al., 1993). In this study, we have formulated a general bubble dynamics model, which is as follows. The density, velocity and pressure distributions for the gas inside a spherical bubble were obtained by solving the continuity and momentum equations analytically (Kwak et al., 1995, Kwak and Yang, 1995). With the set of analytical solutions for the conservation equations, the temperature distribution for the gas inside the bubble was also obtained by solving the energy equation for the gas. The heat transfer through the bubble wall was considered to obtain the instantaneous thermal boundary layer thickness from the mass and energy conservations for the liquid layer adjacent to the bubble wall by the integral method. The mass and momentum equations for the liquid outside the bubble wall provided the well known equation of motion for the bubble wall, the Rayleigh-Plesset equation in an incompressible limit or the Keller-Miksis equation in a compressible limit. The bubble dynamics model was applied to an evolving bubble formed form the fully evaporated droplet at the superheat limit (Kwak et al., 1995) and phenomena of sonoluminescence which is light emission associated with the catastrophic collapse of a micro-bubble oscillation under ultrasound (Young, 2005). With uniform density, temperature and pressure approximations which are valid for the characteristic time scale of ms, the calculated values of the far field pressure signal from the evolving the bubble formed form the fully evaporated droplet at its superheat limit (Kwak et al., 1995) are in good agreement with the experimental results (Shepherd and Sturetevant, 1982). With uniform pressure approximation which is valid for the characteristic time scale