Perturbation method on the modified Rayleigh-Plesset equation in pseudo-compressible bubbly viscoelastic liquid

Abstract

In bubbly-liquid flow, a transient phenomenon is happened due to the dynamics of the bubble, pressure, or temperature at any location in the flow. It is well known that shock wave propagation in liquid media is strongly affected by the presence of bubbles that interact with the shock wave, and the effects of the gas bubbles. The presence of interfacial interactions between the bubble and the liquid, bubbles interaction and compressibility of the viscoelastic liquid flow inevitably make the problem a difficult one. Due to these factors that a mathematical model of bubblyliquid flow becomes more complex than the transient flow encountered in singlephase flow. The mathematical model of a transient pseudo-compressible two-phase gas bubble in viscoelastic liquid flow and heat transfer is discussed. Specific models are derived to describe the shock wave propagation behaviour of the bubbly-liquid flow. The gas behaviour inside a spherical bubble under the shock wave is analysed using the polytropic models. The modified Rayleigh-Plesset equation that described the bubble dynamics in a pseudo-compressible viscoelastic liquid is derived using the idea of conservation of kinetic energy equation, incorporating the effect of a bubble to bubble interaction. Kelvin-Voigt (linear viscoelastic) liquid and second grade (nonlinear viscoelastic) liquid are the two specific liquids considered. The governing equations are approximately solved using reduction perturbation method, and then Kortewegde- Vries-Burger (KdVB) equations are derived. Adomian decomposition method is applied to solve the equations numerically. The result shows that the combination of acoustic, thermal, and viscous damping effect causes rapid damping in the wave propagation in the bubbly viscoelastic liquid flow. For Kelvin-Voigt liquid, the size of the bubble radius has no influence on the amplitude of the shock wave from zero up to 40s, while for second-grade liquid, the bigger the bubble, the larger the amplitude of the wave propagation at t = 8 s. For the thermal variation, for both Kelvin-Voigt and second grade liquids, it is observed that a higher polytropic index gives rise to the lower amplitude of the shock wave. This implies that an isothermal process will give rise to a shock with the highest amplitude and dissipate faster. A highly viscous Kelvin-Voigt liquid dissipates faster, while for second-grade liquid, the result indicates no effect in either the amplitude or the steepness on the shock wave. It is observed in the case of Kelvin-Voigt liquid, that there is a significant effect of the modulus of elasticity on the shock wave, and no difference in the amplitude of the shock wave as a result of variation of relaxation parameter for second-grade liquid. For both Kelvin-Voigt and second-grade liquids, the variation in the number of bubbles and cluster size has relatively no effect on the shock propagation in the parameter values under consideration. Global and local stability analysis of the KdVB equations are carried out, and many novel wave solutions to the equations are derived. This study is useful for the performance analyses of bubbly viscoelastic liquid flow in predicting shock propagation in the liquid, and provides useful information for the effects of heat transfer phenomenon and applied pressure on transient pseudo-compressible bubblyliquid flow.