Researchers have become interested in convection heat transition in the driven enclosure due to its diverse range of applications, including solar thermal collectors, float glass manufacture, microelectronic devices, and electrical devices. A vast number of researchers have recognized the focus on lid-driven, various-shaped cavities, both from engineering and theoretical viewpoints. The present model’s specific engineering applications may include computer CPU heat sinks, radiators in cars, heat exchangers in power plants, heat transmission devices, and so on. Das et al. [1] examined free convection flow within the wavy frame enclosure. Their findings showed that a wavy surface’s amplitude and undulation numbers affect the components that control heat in an enclosure. The natural convection within a vertically wavy wall enclosing an unstable case was statistically modeled by Rostami [2]. Azizul et al. [3] reported the impact of a heat line concept on mixed convection indoors with a wavy surface frame and nanofluids. Free convection via a skewed design wavy porous chamber was examined by Misirlioglu et al. [4]. The impact of wavy base cover on combined convection heat augmentation in a driven chamber was documented by Al-Amiri et al. [5]. They assessed the effect of flow creation and heat removal characteristics on the number of wavy surfaces, the Richardson number, and the appropriateness of a wavy surface. Mushate [6] examined a porous enclosure with undulations using CFD to predict natural convection. The outcomes showed that the heat change rate improves as the Rayleigh number rises and decreases as the amplitude rises. Mansour et al. [7] used a heated non equilibrium model to explore a free convective porous cavity with a wavy form and the effect of thermic radiation. The coupled convection throw nanofluid flow within a wavy form cage was studied by Nada et al. [8]. They discovered that, with regard to geometry, base surface ratios, and the Richardson number, the heat replacement rate increases with an increase in the nanoparticle volume fraction. Using a nanofluid and Buongiorno’s mathematical description, Sheremet et al. [9] scrutinized free convection within a curved porous cavity with sinusoidal heat delivery on both level surfaces. Free convective movement in a non-uniformly heated curvy edge porous enclosure with nanofluid reporting the impacts of thermophoresis and Brownian dispersion was explored by Sheremet et al. [10]. Natural convection inside porous wavy structures, having sinusoidal warming and interior heat generation, was studied by Cheong et al. [11]. Alsabery et al. [12] examined the entropy-generating capabilities of a solid rotating cylinder with heat fluctuation in a warmed porous cavity underneath a curved framework. A recent heat conveyor study on convection inside a triangular wavy structure enclosure was made by Asad et al. [13].

Many engineering applications, including vacuum cleaners, washing machines, and blenders, all feature electric components that operate using magnetic principles. Ashorynejad et al. [14] explored combined nanofluid in a MHD convective wavy frame open enclosure. They followed the rule that the Nusselt number declines with an increase in a Hartmann number, but increases with a rise in Rayleigh number and nanoparticle size. Rahman et al. [15] studied the collective impacts of joule heating on magnetohydrodynamic convection within a driven frame enclosure. These authors and those of the references [16–19] emphasized that the Hartmann number significantly influences the thermal fluctuations of the flow. Additionally, Joule heating factors influence flow design and isotherms. Öztop et al. [20] provided a summary of coupled magneto-convection in a lid-driven, differentially heated, wavy-walled structure occupied with nanofluid. They showed that the rate of heat removal declines as the Hartmann number increases. Additionally, depending on Hartmann (Ha) and Richardson values, the nanoparticles might cause the heat substitution rate to climb or fall. There are more recent studies on square, wavy, and triangle cavities under the effect of a hydromagnetic field in [21–24].

A fin like computer CPU heat sinks, a radiator in a car, heat exchangers in power plants, and heat shifting devices are only a few examples of engineering applications that also incorporate flow construction and heat substitution inside the enclosure. In addition, cutting-edge technologies like fennec canines and hydrogen fuel cells work by discharging heat. Nag et al. [25] reviewed free convective flow in a differential thermal enclosure amidst a horizontal block on a hot wall. Tasnim et al. [26] investigated how free convection heat changed when a baffle was joined to a heated wall. They discovered that fin length and Rayleigh number have a massive effect on the influence of fin position on the rate of heat removal. Sun et al. [27] reported of mix-convection applying triangular conductive fins in a driven chamber. A triangular fin was recommended as a useful parameter for flow design and heat transport speed. The effect of baffles and their length on free convection within transition enclosures was examined by Xu et al. [28]. They determined that at a critical (Ra) level sensitive to fin length, the stream pattern near the fin surface changes from a steady to intermittently unstable flow. By affixing a vertical fin to the lower wall of the standard cage by Asad et al. [29]. Laminar natural convection was generated by Elatar et al. [30] in a square chamber with a special horizontal fin enclosed at various positions and heights connected to a heated surface. They investigated how flow design and heat removal components were affected by the length and position of the fins. Siddiqui et al. [31] studied combined convection in a micro-polar liquid sliding wall cavity. In open enclosures, the effects of parallel insulated baffles were examined by Palaniappan et al. [32]. According to the literature, further activities related to this study are [33–35]. Asad et al. [36] investigated the analytical modeling of the MHD boundary layer circulation of a chemically reacting upper convective Maxwell fluid via a vertical surface exposed to many stratifications with different characteristics. The performance of free convection and heat transfer in a curve-shaped enclosure was scrutinized by Asad et al. [37].

To the best of the researcher’s knowledge, this indicates that no question about the enclosure with a vertical fin and a wavy form on both sides has been examined. This surviving study will numerically scrutinize the influence of fin size on magneto-combined convective heat transition in a moving-wall, wavy-shaped wall enclosure attached to a vertical fin.

For the current investigation, a physical wavy model with boundary conditions estimates a wavy frame enclosure, as shown in Fig. 1. The thermo-physical characteristics of a fluid are maintained constant, with the exception of density variations in a buoyancy expression, which are controlled using the Boussinesq approximation. The dominance of viscous diffusion and radiation are also disregarded at the same time. The Newtonian, incompressible, steady, and laminar flow is generally assumed to represent the enclosing liquid.

In accordance with the presumptions, the relevant governing equations (see reference [17]) are as follows: (1)Continuity equation: ∂ψ∂X+∂φ∂Y=0, (2)Momentum equation inx-direction: ψ∂ψ∂X+φ∂ψ∂Y=−∂P∂X+1Re∇2ψ, (3)Momentum equation iny-direction: ψ∂φ∂X+φ∂φ∂Y=−∂P∂Y+1Re∇2φ+GrRe2θ−Ha2φRe, (4)Energy equation: ψ∂θ∂X+φ∂θ∂Y=1PrRe∇2θ.

The topical variables of the Eqs. (1)–(4) described above are, Pr, Ha, Gr, Re, and Ri, each of which is represented as (5)Pr=να,Re=ψlidWν,Gr=gβ(Th−Tc)W3ν2,Ri=GrRe2,Ha=B0Wσμand θ=Th−TcΔT

The pertinent governing Eqs. (1)–(4) jointly amidst Eqs. (6a)–(6e) are constructed computationally using a Galerkin finite element approach. First, using the γ (penalty variable) and the incompressibility review of Eq. (1) as follows: (8)P=−γ(∂ψ∂X+∂φ∂Y)

The following conservations of momentum Eqs. (2), (3) using Eq. (8), we get (9)ψ∂ψ∂X+φ∂ψ∂Y=γ(∂ψ∂X+∂φ∂Y)+1Re∇2ψ (10)ψ∂φ∂X+φ∂φ∂Y=γ(∂ψ∂X+∂φ∂Y)+1Re∇2φ+Riθ−Ha2φRe

Secondly, the progression of momentum and energy in Eqs. (9), (10) and (4) sequentially utilizing Eqs. (6a)–(6e) are organized by selecting the Galerkin finite element policy [38–40].

The engagement function estimates flow (ψ, φ), and heat qualities (θ) using a main set: {ϕi}i=1k as (11)ψ≈∑i=1kψiϕi(X,Y);φ≈∑i=1kφiϕi(X,Y)andθ≈∑i=1kθiϕi(X,Y)

A Galerkin weighted residual approach and finite element techniques are employed within the inner node domain (Ω), integrating them across the computational domain to derive the nonlinear Eqs. (9), (10), and (4), which are then discretized using small free triangular meshes as illustrated in Fig. 2.

(12)R1j≈∑i=1kψi∫Ω[(∑i=1kψiϕi)∂ϕi∂X+(∑i=1kφiϕi)∂ϕi∂Y]ϕjdXdY+γ∑i=1kψi∫Ω∂ϕj∂X∂ϕi∂XdXdY+γ∑j=1kφi∫Ω∂ϕj∂X∂ϕi∂YdXdY+1Re∑i=1kψi∫Ω[∂ϕj∂X∂ϕi∂X+∂ϕj∂Y∂ϕi∂Y]dXdY (13)R2j≈∑i=1kφi∫Ω[(∑i=1kψiϕi)∂ϕi∂X+(∑i=1kφiϕi)∂ϕi∂Y]ϕjdXdY+γ∑i=1kψi∫Ω∂ϕj∂Y∂ϕi∂XdXdY+γ∑i=1kφi∫Ω∂ϕj∂Y∂ϕi∂YdXdY+1Re∑i=1kψi∫Ω[∂ϕj∂X∂ϕi∂X+∂ϕj∂Y∂ϕi∂Y]dXdY+Ri∫Ω(∑i=1kθiϕi)ϕjdXdY−Ha2Re∫Ω(∑i=1kφiϕi)ϕjdXdY (14)R3j≈∑i=1kθi∫Ω[(∑i=1kψiϕi)∂ϕi∂X+(∑i=1kφiϕi)∂ϕi∂Y]ϕjdXdY+1PrRe∑i=1kθi∫Ω[∂ϕj∂X∂ϕi∂X+∂ϕj∂Y∂ϕi∂Y]dXdYwhere the iteration, residual, and node numbers k, j, and i are listed consecutively. The procedures that came after Eqs. (12)–(14) were carried out using the Gaussian quadrature technique.

Lastly, Newton Raphson’s iteration scheme was engaged to find out residual equations iteratively. The elaborate clarification may be exposed in a literature review [13,37]. After a convergence inquiry, the computational approach’s convergence policies are realized and regarded as (15)|Πn+1−ΠnΠn+1|≤10−5where n is an N-R iteration loop and Π is a function of velocity and temperature.

Considering the current investigation at Pr = 0.71, L = 0.45, Ri = 1, λ = 2, and H = 0.50, identify the appropriate grid size. Table 1 displays the average Nusselt number of a fin, while Fig. 3 illustrates it. We examined a grid sensitivity assessment with various types of meshes and found an appropriate answer for the current research. Table 1 and Fig. 3 show that the grid size of 6844 nodes and 13,273 elements provided a satisfactory solution for the present numerical investigation.

The mean Nusselt number on the right surface was compared between Elatar et al. [30] and Nag et al. [25] values at Ra = 10⁶ and L = 0.20 to confirm the accuracy of the current model’s analytical results. With the largest derivation of less than 3.0%, the mean Nusselt number examined in Table 2 demonstrates the outstanding acceptance of those queries. Besides, a matching of local Nusselt numbers adjusted for the instant results with Elatar et al. [30] variant Ra (10³, 10⁴, 10⁵, and 10⁶) and H (0.25, 0.50, and 0.75) at L = 0.5 and B = 0.01 as exhibited in Fig. 4. The local Nusselt number can highlight the significant adjustment of instant results by Elatar et al. [30], as viewed in Fig. 4.

In this research, our laptop configuration was Intel (R), Core (TM) i5-8350U, CPU @ (1.70–1.90) GHz, 8.00 GB (RAM), 64-bit operating system, x64-based processor.

The present numerical outcomes have been executed by employing the weighted residual tactics finite element scheme for exploring combined magneto-convection temperature substitution and flow field into a two-sided wavy surface enclosure to dominate the uniform magnetic field. The numerical findings of variant parameters like Richardson number (Ri), fin size (L), and Hartmann number are performed to determine the properties of flow patterns and temperature transport using streamlines, isotherms, and the average Nusselt number. The mean fluid temperature removal effectiveness inside the enclosure for three variant fin sizes of thermal cases is presented in graphical and tabulator form.

The overall destination of this inquiry is to determine the impact of fin size with Richardson number and Hartmann number variance on hydromagnetic mixed convective flow with heat removal performance within a two-side wavy surface enclosure. The numerical method was verified by contrasting the mean Nusselt number and local Nusselt number obtained by the code with previously published data for existing inquiries. Based on the results, the following will be stated:

These outcomes confirmed that the fin length difference greatly influences the flow features and temperature field within the cavity.