*3.3. Bubble Rise Diagram*

Some samples from the well-known bubble rise diagram by Clift et al. [29] were numerically investigated in order to qualitatively validate the presented SC LBM model for different Reynolds and Bonds numbers. Periodic boundary condition is set on the domain's edges, while selected time stamps from each case are shown in Figure 5. The Atwood number is set as zero, while the SC interaction parameters are *G*<sup>12</sup> = *G*<sup>21</sup> = 6.0 and *G*<sup>11</sup> = *G*<sup>22</sup> = 0. The selected points have the following Bonds and Reynolds numbers:


**Figure 5.** (**a**) Shape regimes for bubble rise after Clift et al. [29]. (**b**) Four simulation samples from different points on the diagram.

The results have very good qualitative agreement with the shape regimes for bubble rise, showing the complex interfacial deformation behaviors. In Case 1, with the highest Reynolds and Bonds number, edges of the bubble are quickly split to smaller bubbles and only the cap remains. In Case 2, with lower Reynolds number, the bubble skirts and the splitting is less severe. In Case 3, with lower Bonds number, the bubble is connected, though it starts wobbling along its motion. In Case 4, with the lowest Reynolds and Bonds number, the bubble shows a slight spherical shape.

All the simulations in this section were carried out using the TRT with Λ = 1/4 for stability issues that occurred with SRT at high Reynolds and Bonds numbers. It is Worth mentioning that the Reynolds number in this simulation is based on /2*gR* according to Equation (23) instead of the terminal velocity as in [29], which will yield to ~5% difference between both Reynolds numbers according to the terminal velocity formula by Davis and Taylor [44].
