## 1 Introduction  
Mixed convection occurs when the buoyancy force and shear force affect the fluid flow and heat transfer. This phenomenon is significant for various convection applications and has been extensively studied in lid-driven cavities in several engineering and technology fields. These include the cooling of electronic devices, food processing, chemical processing equipment, polymer processing, energy extraction, manufacturing processes, and material processing [1]. The mixed convection of Newtonian fluids under various scenarios within cavities has been widely studied in research [2, 3, 4]. Researchers have found that adding nanoparticles to a base fluid can enhance heat transfer [5; 6]. [PERSON]'s two-phase model, introduced in 2006, has been widely used to study nanofluid heat transfer. [PERSON] et al. [7] employed [PERSON]'s nonhomogeneous model to analyze mixed convection in a two-dimensional inclined cavity filled with nanofluid, considering different aspect ratios. Their results showed that the cavity angle and aspect ratio significantly influenced the flow, heat and mass transfer, and nanoparticle concentration. A recent review by [PERSON] et al. [8] compared one-phase and two-phase models for nanofluid research. While only 19% of the studies adopted the two-phase model, it provides valuable insights into nanoparticle accumulation on surfaces. Many studies have considered using porous media inside enclosures to improve the mixed convection coefficient of heat transfer [9; 10; 11; 12; 13]. [PERSON] et al. [14] numerically studied the free convection of a nanofluid in a square cavity saturated with a porous medium and addressed the impact of the presence of the porous medium on the nanofluid system cooling efficiency. [PERSON] and [PERSON] [15] investigated the nanofluid convection heat transfer over a vertical channel filled with a porous medium and studied the effect of the main parameters, such as \(\widetilde{\mathrm{Ri}}\), Brinkman, \(\widetilde{\mathrm{Da}}\), and Lewis numbers, on velocity, flow, and temperature. The generation of entropy and the Bejan Number in incompressible laminar mixed convective fluid within a cavity has been addressed by [PERSON] et al. [16]. Their study was performed on two configurations of tilted lid-driven trapezoidal cavities filled with water as the base fluid along with Al\({}_{2}\)O\({}_{3}\) nanoparticles. Recent studies have demonstrated that the use of a magnetic field can enhance heat transfer and reduce irreversibility (entropy) within an enclosure [17; 18; 19; 20; 21]. [PERSON] et al. [22] performed a thermal performance analysis study on a hexagonal enclosure that was filled with TiO\({}_{2}\)-H\({}_{2}\)O nanofluid. Their results confirmed that the effect of the nanofluid on the thermal performance is enhanced by increasing the values of \(\widetilde{\mathrm{Re}}\) and \(\widetilde{\mathrm{\Phi}}\), and increasing the value of \(\mathrm{Ha}\). Based on their results, the optimal average Nusselt number was achieved at \(\widetilde{\mathrm{Re}}\) = 200 and \(\widetilde{\mathrm{\Phi}}\) = 0.1 at \(\widetilde{\mathrm{Ha}}\) = 0. [PERSON] et al. [23] studied mixed convection in an odd-shaped cavity. The upper and right walls of the enclosure were heated partially, and various heating scenarios were considered based on the positions of the active parts. The main conclusions of their study indicated that for optimal mixed convection, it is recommended to place the sources of heat in the middle of the exterior orthogonal walls. [PERSON] et al. [24] investigated the mixed convection numerically in an enclosure filled with nanofluid by using a finite element method. Their results show that heat transfer is enhanced as the Reynolds number, nanoparticle volume fraction, and Richardson number increase. From another point, the heat transfer decreased with increasing Hartmann number, and the average Nusselt number and streamfunction strength increased with an increase in the Darcy number. [PERSON] et al.