This study presents a comprehensive numerical investigation of thermo-hydrodynamic behaviour and entropy generation in a two-dimensional square cavity filled with a ternary hybrid nanofluid (Fe3O4–Ag–TiO2/H2O) under magnetohydrodynamic (MHD) mixed convection. The system features internally heated circular fins and is driven by sinusoidally varying velocities along the left and bottom walls, introducing periodic momentum forcing. Key governing parameters are systematically examined, Hartmann number (0 ≤ Ha ≤ 80), Richardson number (Ri = 0.01–1), nanoparticle volume fraction ($0\leq \varphi \leq 0.08$), sinusoidal wavelength (0.2 ≤ Lx = Ly = LDriven ≤ 0.8), and phase deviation between moving walls. High-fidelity simulations are conducted using a custom FORTRAN solver combining the Finite Volume Method with Full Multigrid Acceleration to resolve coupled continuity, momentum, energy, and entropy generation equations. Rigorous parameterization ensures accurate representation of interactions among magnetic damping, buoyancy, and nanoparticle-enhanced thermal transport. Results show that sinusoidal velocity modulation effectively controls flow coherence and entropy production, with certain phase relationships minimizing irreversibility while maintaining favorable heat transfer. Optimal operation is achieved under moderate magnetic fields (Ha = 20), Richardson number (0.1 ≤ Ri ≤ 0.5), nanoparticle concentration ($\varphi =0.06$), and wavelength (LDriven = 0.6), illustrating the trade-off between thermal performance and entropy generation. These findings provide new physical insights into periodically driven MHD convection systems and practical guidelines for designing energy-efficient thermal systems utilizing hybrid nanofluids.
