This article addresses the numerical simulations of MHD free convection flow of \((\text{MgO}/{\text{H}}_{2}\text{O}\) ) nanofluid in a hexagonal-complex-shaped cavity within an embedded heated fin. Investigators have commonly recognized these hexagonal-shaped cavities for their involvement in heat exchanger, electronic cooling, automobile industry, chemical processing and aerospace engineering. Internal fins are significant in enhancing heat transmission rate of phase change material (PCM). Phase change material has significant applications in engineering, storage of solar energy, thermal systems and conservation of energy in buildings. PCM has low thermal conductivity. So, these cavities with internal heated fins are used to raise the heat transmission rate of PCM. Hence, the current research is organized to explore the characteristics of natural convection on \((\text{MgO}/{\text{H}}_{2}\text{O}\) ) nanofluid inside the hexagonal-complex-shaped cavity. A heated fin is placed at different angles separately. Impact of MHD through Lorentz force is employed. The governing nonlinear PDEs are transformed into non-dimensional form. FEM is invoked to find out the solutions of dimensionless equations by using Galerkin approach. Impacts of key variables such as volume fraction ( \(0.01\le {\phi }_{p}\le 0.05\) ), Rayleigh number ( \({10}^{4}\le Ra\le {10}^{6}\) ) and Hartmann number ( \(0\le M\le 50\) ) on velocity and temperature distributions are visualized. Analysis of heat transfer rate is examined. Key findings reveal that heat transmission rate rises by increasing volume fraction and Rayleigh number, while opposite trend is noticed for M. An increment in length of inner fin tends to increase heat transfer rate.
