MULTIPLE-SCALE AND NUMERICAL ANALYSES FOR THE NONLINEAR OSCILLATIONS OF A GAS BUBBLE SURROUNDED BY A MAXWELL'S FLUID

Abstract

In this work, we have revisited theoretically the linear and nonlinear oscillations of a single bubble immersed in a Maxwell's fluid under the action of an acoustic pressure field. We adopt the above rheological model to identify in a simple manner the viscoelastic influence and the impact of the external acoustic field on the motion of the bubble. The governing equations are reduced to a modified Rayleigh–Plesset equation, which is solved together with an ordinary differential equation needed to predict the rheological influence of the normal stresses on the interface of the bubble. The resulting dimensionless governing equations were solved numerically; however, for small deviations from the equilibrium radius of the bubble, we add a frequency analysis by using the multiple-scale method to find the influence of the viscoelastic parameters for those conditions near to resonance, together with the estimation of the bending curve, which characterizes the well-known bending phenomenon. For a single value of the dimensionless Weber number, we have identified that for low Reynolds and Deborah numbers, the oscillations are periodic, with some harmonic ones well defined. However, as we increase the above dimensionless parameters, strong nonlinearities appear, and they are more notable when the effect of the viscous damping is reduced. Furthermore, the effect of the nonlinear terms of the governing equations depends strongly on the amplitudes of the oscillations: when the multiple scale analysis is used and we consider small deviations of the dimensionless equilibrium radius, we obtain that the resonance conditions for the amplitude of the bubble are reduced if the Deborah number is increased. On the other hand, for moderate values of the deviations of the equilibrium radius and retaining the validity of the multiple scale analysis, the foregoing behavior is also conserved. In this last case, stronger amplitudes of the radius of the bubble are obtained also for increasing values of the Deborah number.