Homotopy perturbation method for unsteady motion of a single bubble in a highly viscous liquid

Abstract

In this study, the dynamics of the accelerated and steady-state motion of a single bubble in a quiescent highly viscous Newtonian liquid was investigated theoretically and experimentally. The presented mathematical model was based on Newton's second law of motion and a balance of buoyancy, drag, history, and added-mass forces. Due to the presence of non-linear terms in the equation of motion, homotopy perturbation method was used as a powerful analytical method to calculate the velocity analytically. To obtain accurate results in the experiments, a high-speed camera was used to record the bubble motion from the moment of detachment to the time at which the terminal velocity is reached. Moreover, based on the dimensional analysis, an empirical relationship was established for the initial velocity of the bubble at the moment of detachment from the needle tip and was used as an initial condition for solving the equation of motion. It was shown that the analytical and numerical results are in good agreement and are closely related to the experimental data. Using the analytical correlation obtained for bubble velocity, an analytical relationship was proposed for the total drag coefficient of the bubble. In addition, the study of the effect of various parameters on the dynamics of bubble motion revealed that the difference between the initial velocity of the bubble and the terminal velocity increases with increasing bubble equivalent diameter and with decreasing the ambient fluid viscosity, and as a result, the duration of the accelerated motion will rise.