Dynamics of Bubble Rising in Vertical and Inclined Square Channel

A stable Lattice Boltzmann Equation (LBE) Model based on the Cahn-Hilliard diffuse interface approach is used to investigate the dynamics of a bubble rising in a vertical and inclined square channel with large density and viscosity ratios. Deformation parameter ($\Delta )$ and terminal velocity (U$_{t})$ of the bubble are interrelated quantities which depend on non-dimensional numbers such as Bond Number (Bo), Morton Number (Mo) and ratio between bubble diameter and channel width ($\kappa )$. This study confirms the relationship between $\kappa $ and $\Delta $ and film thickness ($\delta )$, as it was reported by previous experimental studies. As $\kappa $ is increased, higher $\Delta $ and smaller $\delta $ are exhibited. This finding is independent of the value of Bo and Mo. In addition, an evaluation was performed for inclined channel to relate the non-dimensional value Froude Number (Fr) and the inclination angle ($\theta )$ as function of Bo and Mo. For each set of values of Bo and Mo, there is a critical value of $\theta $ which corresponds to the highest value of Fr, consequently highest U$_{t}$. This finding is consistent previous simulation and experimental results. This study was performed using a range of Bo and Mo, (10$^{-5} \quad <$ Mo $<$ 10$^{2})$ and (1 $<$ Bo $<$ 30), and the inclination of the channel is varied from 0\r{ } to 75\r{ }.