Mixed convection in lid-driven cavities has attracted significant attention due to its wide applicability in thermal engineering systems such as heat exchangers, cooling of electronic devices, and energy storage units. The complex interaction between shear-driven flow induced by the moving lid and buoyancy-driven flow due to temperature gradients governs the overall heat transfer characteristics. Early investigations primarily focused on simple square enclosures with differentially heated walls, establishing fundamental insights into flow behavior and heat transfer mechanisms [1].

With the advancement of computational techniques, researchers extended these studies to include magnetohydrodynamic (MHD) effects. The application of an external magnetic field introduces a Lorentz force that suppresses fluid motion and alters thermal transport. Rahman and Alim [2] analyzed MHD mixed convection in a lid-driven cavity with internal heating elements and reported that increasing the Hartmann number significantly dampens fluid circulation and reduces the Nusselt number. Similar findings were reported by Jani et al. [3] and Hussein et al. [4], who demonstrated that magnetic fields can effectively control convective flow structures in various enclosure geometries.

Subsequent studies incorporated internal heat sources and geometric modifications to enhance heat transfer performance. Rabbi et al. [5] investigated ferrofluid-filled cavities and showed that heater configuration plays a crucial role in determining thermal efficiency. Alam et al. [6]-[8] examined sinusoidal walls and partially heated boundaries, revealing that non-uniform thermal conditions significantly influence vortex formation and temperature distribution. Munshi et al. [9] further explored non-standard cavity geometries and highlighted the role of corner heaters in modifying flow patterns.

The influence of entropy generation and thermodynamic irreversibility in MHD mixed convection systems was studied by Chamkha et al. [10], who emphasized the importance of electrical conductivity models. Later, Chowdhury and Alim [11] and Haque et al. [12] investigated complex configurations involving wavy walls and multiple heated blocks, reporting enhanced heat transfer due to increased surface area and flow interaction. Similarly, Fayz-Al-Asad et al. [13] and Bakar et al. [14] analyzed the effects of fins and internal heat generation, demonstrating that such modifications can significantly improve thermal performance depending on their placement and intensity.

Recent research trends have focused on advanced working fluids and configurations. Farahani et al. [15] examined nanofluid flow under magnetic fields and porous media effects, while Runa et al. [16] studied semi-circular enclosures with internal obstacles using finite element methods. Keya et al. [17] explored heat exchangers embedded within lid-driven cavities, highlighting their practical engineering relevance. More recently, Mahmuda Maya et al. [18] and Ali et al. [19] investigated MHD mixed convection using nanofluids and hybrid nanofluids, demonstrating improved heat transfer performance compared to conventional fluids.

Furthermore, Sarker and Alam [20] and Theeb et al. [21] extended these analyses to complex geometries and hybrid nanofluid systems under magnetic influence, emphasizing the combined effects of internal heat sources, cavity shape, and fluid properties. These studies collectively indicate that incorporating internal structures such as fins and obstacles can enhance heat transfer by increasing the effective heat transfer area and promoting fluid mixing.

Despite these extensive investigations, limited attention has been given to the combined effects of lid-driven motion, MHD influence, internal heating, and fin-based augmentation within a single computational framework. In particular, the role of fins in modifying flow structure and heat transfer under magnetic fields remains inadequately explored. Therefore, a comprehensive computational study focusing on MHD mixed convection in a heated lid-driven cavity with fins is essential to bridge this research gap and provide deeper insights into optimizing thermal system performance.

Assumptions:

1) Laminar, incompressible, Newtonian fluid flow.

2) 2D square cavity ( x , y directions).

3) Uniform internal heat generation Q .

4) Constant physical properties ( ρ , μ , k , β , σ ).

5) Magnetic field applied, negligible induced magnetic field (low magnetic Reynolds number).

6) Boussinesq approximation: density variations only affect buoyancy term.

A two-dimensional square cavity is considered in the present investigation, as schematically illustrated in Figure 1. The upper wall of the cavity moves horizontally in its own plane with a constant velocity U 0 , while the remaining walls satisfy the no-slip boundary condition. The thermal boundary conditions are imposed such that the left vertical and bottom walls are maintained at a uniform high temperature T h , whereas the right vertical wall is kept at a constant low temperature T c . The upper moving wall is assumed to be thermally insulated. The cavity comprises six solid fins, with three affixed to the left vertical wall and the remaining three to the right vertical wall at equal spacing. All fins are assumed to be thermally insulated and have a length l and width b.

The fluid inside the cavity is electrically conducting, and all cavity walls are assumed to be electrically insulating. A uniform magnetic field of constant strength B 0 is applied parallel to the direction of the lid motion. The induced magnetic field generated by the fluid motion is assumed to be negligible in comparison with the applied magnetic field, thereby decoupling the Navier–Stokes equations from Maxwell’s equations.

The thermo-physical properties of the fluid are assumed to be constant, except for the density variation in the buoyancy term, which is modeled using the Boussinesq approximation. Viscous dissipation effects are neglected. The fluid is further assumed to be Newtonian, incompressible, and the flow is considered to be steady and laminar.

Figure 1. Schematic diagram of the physical model with six inside fins along the vertical walls.

MomentumEquations(Navier-StokeswithMHDandbuoyancy)

y-momentum: ρ ( u ∂ v ∂ x + v ∂ v ∂ y ) = − ∂ p ∂ y + μ ( ∂ 2 v ∂ x 2 + ∂ 2 v ∂ y 2 ) + ρ g β ( T − T c ) + σ B 2 v

(3)

where:

u , v = velocity components in x and y . p = pressure. μ = dynamic viscosity. ρ = density. σ = electrical conductivity. B = magnetic field strength. β = thermal expansion coefficient. g = gravity acceleration. T = temperature.

The MHD Lorentz force term is usually − σ B 2 u for the x-direction, but depending on orientation of the magnetic field, it can appear in both x and y directions.

where:

k = thermal conductivity. c p = specific heat at constant pressure. Q = volumetric internal heat generation.

BoundaryConditions:

Dimensionless Formulation:

Introduce dimensionless variables:

Dimensionlessequations:

Dimensionlessnumbers:

1) Reynoldsnumber: R e = U 0 L ν

2) Prandtlnumber: P r = ν α = μ c p k

3) Grashofnumber: G r = g β ( T h − T c ) L 3 ν 2

4) Hartmannnumber: H a = B L σ μ

5) Internalheatgenerationparameter: Q * = Q L 2 k ( T h − T c )

DimensionlessBoundaryConditions:

Since the convective heat transfer coefficient: h and Nusselt number Nu, depend on the temperature gradient at a flat surface, ( − ∂ T ∂ n ) where n is normal direction to the ∂ n surface.

Therefore, we can obtain the rate of heat flux from each of the walls. The corresponding average Nusselt number at the heated surface is defined as:

where Re, Pr, Gr, Ha and Q* are denote Reynold number, Prandtl number, Hartmann number, Grashof number and heat generation coefficient respectively.

The dimensionless governing equations describing the magnetohydrodynamic (MHD) mixed convection flow with internal heat generation in a lid-driven square cavity, together with the corresponding boundary conditions, are solved numerically using the Galerkin weighted residual finite element method (FEM). The theoretical formulation and numerical implementation of this method are based on the standard procedures reported by Taylor and Hood and Dechaumphai.

The physical domain is discretized into a finite number of uniformly distributed quadrilateral elements using a mapped mesh technique. Within each element, the dependent variables—namely velocity components, temperature, and pressure—are approximated by appropriate interpolation (shape) functions. The Galerkin approach is then employed to derive the weak form of the nonlinear governing equations by selecting the weighting functions identical to the interpolation functions. The resulting integral equations are evaluated numerically using Gaussian quadrature, leading to a system of coupled nonlinear algebraic equations. The prescribed boundary conditions, including the lid-driven velocity, thermal conditions on the cavity walls, and magnetic field effects, are incorporated directly into the algebraic system. To solve the nonlinear system, the equations are linearized using the Newton–Raphson iterative technique. At each iteration, the resulting linearized system is solved using the Triangular Factorization method, which ensures computational efficiency and numerical stability.

The iterative solution process is continued until the following convergence criterion is satisfied for all dependent variables:

where ϕ represents the dimensionless velocity components, temperature, and pressure, and n denotes the iteration level.

To ensure the accuracy and reliability of the numerical solutions, a grid independence test is performed using progressively refined meshes. The final computational mesh is selected such that further refinement produces negligible changes in the key flow and thermal characteristics. Additionally, the numerical results are validated by comparison with previously published benchmark solutions available in the literature, showing excellent agreement.

In order to obtain grid independent solution, a grid refinement study is performed for a square cavity with 6 fins while Pr = 0.71, Ha = 50 and Ri = 6. Figure 2 illustrates the convergence of the average Nusselt number (Nu) at the heated surface as the grid is refined. It can be observed that grid independence is achieved at 15,956 elements, beyond which any further increase in the number of elements results in only minimal changes to Nu, as shown in Table1. For the grid refinement tests, six different non-uniform grids were evaluated with varying numbers of nodes and elements: 1213 nodes and 2464 elements; 1778 nodes and 3452 elements; 2489 nodes and 4852 elements; 3698 nodes and 7364 elements; 7986 nodes and 15,956 elements; 27,334 nodes and 54,680 elements. Based on these results, the grid with 7986 nodes and 15,956 elements is suitable for achieving an optimal balance between accuracy and computational efficiency in the simulation.

(a)

(b)

Figure 2. (a) Grid sensitivity test. (b) Mesh configuration of the cavity with fins.

Table1.Nu_av at different mesh system.

The validation of the numerical code was carried out by comparing streamlines, isotherms, and the vertical velocity component along the horizontal centerline of the cavity with the results presented in Figures 3-5. As observed from these figures, the present results show good agreement with the reference data, confirming the accuracy and reliability of the numerical model.

Figure 3. Code validation for Ra = 10⁵ and Pr = 0.71

Figure 4. Code validation for Ra = 10⁵ and Pr = 0.71.

Figure 5. Code validation for vertical velocity component along the horizontal central line of the cavity with Hartmann number at Ra = 10⁵ and Pr = 0.71.

The numerical results obtained from the present finite element simulations are analyzed to elucidate the effects of the Richardson number (Ri), Hartmann number (Ha), and internal heat generation (Q*) on the flow structure and thermal behavior within the lid-driven cavity. The discussion is supported by streamline contours, isotherm distributions, velocity and temperature profiles, and local Nusselt number variations.

The results demonstrate that heat transfer in the cavity is governed by a complex interplay of forced convection, natural convection, magnetic damping, and internal heat generation. Key observations include:

Increasing Richardsonnumber(Ri) shifts the regime from forced to natural convection, reducing heat transfer efficiency. Increasing Hartmannnumber(Ha) suppresses fluid motion and significantly reduces convective heat transfer.Optimal heat transfer occurs under moderateRiandlowHa, where mixed convection is strongest.