Supplementary material: Three-dimensional Two-phase Smoothed Particle Hydrodynamics Simulation of Bubble/Droplet Rise and Coalescence at Moderate Reynolds Numbers

This document extends the main article with detailed analysis on bubble dynamics under varying physical parameters, including the effects of density, viscosity, Eötvös, and Reynolds numbers. It also contains a comprehensive technical appendix, which describes the numerical implementation of the 3D two-phase SPH methodology, covering the kernel function, artificial sound speed, time-stepping, boundary conditions, and the full simulation algorithm. A performance summary with computational costs is also included. In this revised version, Figures S1, S6, and S9 have been modified to remove the need for previously included background images or plots. All supplementary figures are now fully original and generated by us. The captions of these figures, as well as the corresponding text in the Supplementary Material, were updated to clarify that cited works are only used for comparison purposes and that readers should refer to the original publications for further details. These are minor revisions that do not affect the scope, content, or conclusions of the study.