Analysis of magnetohydrodynamic ferrofluid flow with internal heat generation and Joule heating in a heater-embedded square chamber

This study addresses a research gap by investigating free convection and entropy generation in a square cavity filled with Fe 3 O 4 -water ferrofluid, incorporating magnetohydrodynamic flow alongside internal heat generation and Joule heating. Previous investigations have ignored the combined effect of magnetohydrodynamic flow, Rayleigh number, and internal heat generation in ferrofluids for multiple discrete heating systems. Numerical simulations are performed using the Galerkin finite element weighted residual method to solve the two-dimensional Navier-Stokes and heat energy equations, forming the mathematical foundation of the model. An all-inclusive parametric evaluation is carried out to explore the impacts of Rayleigh number (10 3 ≤ Ra ≤ 10 7 ), Hartmann number (0 ≤ Ha ≤ 20), Joule heating variable (0 ≤ J ≤ 9.15×10 –9 ), and internal heat generation coefficient (Δ = 0, 2), while maintaining constant Prandtl and Gebhart numbers. The impact of these parameters on the system’s thermal behavior and entropy generation within the computational domain is thoroughly assessed. Results are presented through detailed visualizations of eddy formation, total Nusselt number ( Nu ), total entropy generation, and the thermal performance criterion ( TPC ) for various source-sink arrangements. It is observed that increasing Ra significantly enhances heat transfer across all configurations, transitioning from a conduction- to a convection-dominated transport mechanism. It is also qualitatively evident through streamline and isotherm profiles and quantitatively by rising Nusselt numbers as Ra increases from 10 3 to 10 7 . Including a magnetic field suppresses convective currents, particularly at higher Ha values, leading to decreased heat transfer efficiency and reduced vortex intensity. The internal heat generation parameter further contributes to chaotic flow dynamics, weakening buoyancy-driven flow and heat transfer efficiency. In terms of the Nusselt number, Case 4 exhibits the highest thermal performance within the ranges of 10 3 ≤ Ra ≤ 1.5×10 4 and 10 6 ≤ Ra ≤ 10 7 , and Case 1 shows better heat transport efficiency within the intermediate Ra values from 1.5×10 4 to 10 6 . When Δ rises from 0 to 2, the Nusselt number also decreases by around 11%, illustrating the effect of internal heat production on reducing heat transport efficiency. In the lower Rayleigh number range ( Ra < 5×10 5 ), Case 1 exhibits the lowest TPC values, indicating the best ecological thermal effectiveness with the lowest irreversibility. However, for Ra ≥ 5×10 5 , Case 2 outperforms the other arrangements, with TPC dropping to 0.260 at Ra = 10 7 , indicating superior performance driven by more effective convective circulation and moderate entropy generation. Moreover, a 1.1% reduction in TPC values occurs when Ra ranges from 10 4 to 10 7 , making the system more efficient at Ha = 0.