The Lattice Boltzmann Method (LBM), rooted in kinetic theory with formal links to the Navier-Stokes equations, offers a versatile platform for simulating complex fluid dynamics and multiphysics phenomena. This work explores the application of various LBM models to a specific class of such problems. After reviewing the traditional LBM and efficient simplified variants for momentum and continuity, we focus on extensions necessary for multiphysics coupling. Key among these is the LBM for generalized advection-diffusion equations (ADE). We then demonstrate the utility of these methods through several case studies: analyzing the distinct turbulent mixing characteristics in miscible and immiscible Rayleigh-Taylor instabilities; modeling magnetohydrodynamic (MHD) flows where fluid motion interacts with magnetic fields; capturing the pattern formation in the viscous fingering phenomenon; and simulating complex particle-laden flows using an Eulerian-Eulerian LBM framework, exemplified by systems involving limescale formation. At the end, we provide some concluding remarks offering perspectives on future research directions, including potential algorithmic enhancements, extensions to other coupled physics, and further validation efforts.

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