Numerical simulation of single bubble rising in vertical and inclined square channel using lattice Boltzmann method

A 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 Δ, film thickness δ, 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 k as it was reported by previous experimental studies. As k is increased, higher Δ and smaller δ are exhibited. This finding is independent of the value of Bo and Mo. In addition, a relationship was established between δ and Δ with non-dimensional numbers such as Capillary number Ca and Weber number We. An evaluation was performed for inclined channels to relate the Froude number Fr with the inclination angle θ, where in each case there is a critical value of θ which corresponds to the highest value of Fr, consequently highest U_t. This finding is consistent with previous simulation and experimental results. Moreover, a relation was established between the critical value of θ and Ca and Bo. This three-dimensional study was performed using a range of Bo (1<Bo<12), Mo (10⁻⁴<Mo<10³) and the inclination of the channel is varied from 0 to 75°.