The applications of heat transfer phenomena in heat carrier fluids include cooling systems in the transportation industry and cooling and heating systems in buildings, textile, electronic cooling devices, chemical, and other processing plants, making these fluids popular among researchers. Engineers and medical scientists paid special attention toward understanding the nature of fluids and more specifically to the fluids of engineering and biomedical engineering interest. The heat transfer flow in confined containers with different geometries such as cubic, rectangular, square, and hexagonal being the commonly used geometries where a simple driving force acts on the fluid is the tangential motion of the wall bounding. Practical examples of these problems include the nuclear reactor heat exchangers [1] and solar receivers [2]. Being the simplest geometries, cavity problems are the benchmark to test the efficiency of a numerical and computational method. The type of the confined geometry and the boundary conditions at the edges of the cavity play a fundamental part in the efficiency of the considered computational scheme and, consequently, the physical phenomena involved. These confined geometries possess ample potential to solve various problems of heat and mass transfer mechanisms in the industrial sector. Manju et al. [1] examined the flow of nonlinear fluids within the obstacle in a cavity with a power-law model to describe the fluid’s behavior. The multi-relaxation time (MRT) collision model of the Lattice–Boltzmann Method (LBM) was used for simulating nonlinear fluid flow with the obstacle in a cavity. They investigated the stability in the cell-Reynolds number of the MRT-LBM algorithm for nonlinear fluids, and the physical structural features and the analysis of streamlines and turbidity contours are conducted. For one single obstacle, the effects on flow characteristics and vortex formation of different parameters, such as Reynolds number (in terms of lid-dimension), obstacle dimension, the power-law index, and obstacle shape, were investigated. In addition, the impact of two embedded square obstacles in the cavity was analyzed side-by-side. In the end, the complex fluid flow was examined as a porous block for two obstruction ratios, over the multiple square obstacles inside the cavity. Li et al. [2] studied a three-dimensional incompressible resistive magnetic dynamic equation method of finite elements, in which there is a divergence between velocity, density, and magnetic induction. It is desirable for the discrete solutions, particularly for the momentum equations, to also meet divergence-free conditions. They designed a stable, mixed finite element method that can achieve the target, inspired by the restricted transportation method. The good standing of the discrete solutions was also demonstrated.

In a lid-driven cavity, Shafqat et al. [3] analyzed the characteristics of fins and the inclining magnetic field with nanofluid. They also studied the impact of the length and distance between these fins. The magnetic field and size of the fins were found to be effective parameters in the enhancement of the heat transfer mechanism. The use of the Galerkin finite element method discretized a two-dimensional system of partial differential equations. For the computation of the velocity and temperature fields, an FEM scheme involving the cubic polynomial (P3) has been implemented, while the pressure was calculated with a quadratic (P2). With the adaptive Newton method, the system of discrete equations was optimized. An experimental study was also conducted in order to validate the finite element code. The characteristics of thermal flow in a cavity in the presence of a hybrid mesh were discussed by Khalil et al. [4]. They assumed that the box has a porous medium root and non-Newtonian fluids. Through free convection, the fluid flow is reached. The Y-shaped uniform fin was installed on the lower cavity wall. The boundary conditions for their cavity were that the right and left walls were considered cold, while the upper walls were considered adiabatic. The extreme triangular ribs in the lower walls were considered cold, while the second and third ribs were taken as Y-shaped fins. In addition, the flux field was formulated and solved mathematically using the FEM technique with complex meshing. Xiong et al. [5] numerically discussed the thermal flow features in a cavity of a triangular lid-driven wall using the finite element method with various obstacle configurations.

In addition, Khan et al. [6] analyzed the characteristics of the nonlinear fluid and thermal flow in a cavity with an embedded fin. They assumed that the fluid was used for the first time on the square cavity of the bottom surface with a fin. The Y-shaped fin plays a vital role in understanding the underlying theory and enhancing the thermal flow rate through the fin. Studies show that the embedded fins can significantly increase the surface area and, hence, the heat transfer rate during the flow inside the cavities. The cavity and the appropriate boundary conditions used by Khan et al. show an enhancement in the convective heat transfer rate. They also considered the tips of fin to be hot, cold, and adiabatic. In the dynamic and energy equations, the effects of the magnetic field and radiation were analyzed during simulation.

A computational analysis was carried out with the FEM approach by Khan et al. [7] for the thermal flow of water-based ferrofluids in a cavity. They considered that the porous medium and the trapezoidal enclosure were filled with ferrofluid. To study ethylene glycol (EG)–based Fe₃O₄ nanofluid, Nguyen et al. [8] proposed a complete thermal conductive structure in the permeable curved domain. An electric force was introduced into the porous space in order to produce an electrohydrodynamic (EHD) effect. One of the striking features of the ferrofluid is the dependence of magnetization upon the temperature gradient, and this thermomagnetic coupling makes these fluids more demanding in various fields. Fe₃O₄ nanoparts of different shapes were introduced, namely: platelets, bricks, cylinders, and spherical are suspended in the considered base fluid. The CVFEM solves the systems of partial differential equations (PDEs).

The Boltzmann method (MRT-LBM) in-house multi-relaxation time lattice was designed for resolving and examining the Cu-water nano-fluid mixed convection (MHD) within a 3D geometry with the lid-driven walls by Ghasemi [9]. A mixed 3D convection is formed by considering different velocities of the walls in opposite directions (even natural due to differences in temperature and convection effects). Different aspect ratios and physical parameters are also the reason for the different results in flow analysis; the research by Haq et al. [10] considered the thermal flow behavior of a partially heated cavity that contains a heated cylinder inside it. To investigate flow analysis and heat transfer of nanoparticles in a triangular cavity with a bottom circular wall, the magnetic field was applied in order to control the flow field [11]. The condition is not only a benchmark for the numerical methods but also for the effectiveness and efficiency of these schemes. The thermal flow mixed convection that was studied by Alsabery et al. [12] took place in an enclosure, where the upper moving wall was considered.

Furthermore, Ching [13] conducted a computational feature of a magnet inclined on the mixed convection thermal transmission and entropy generation rates of Cu–water nanofluids in a cavity. The processes of heat energy transport in the cavity were illustrated with the contours of the energy flux vectors being traced. The computations demonstrated the effects on energy stream vectors for the various parameters such as the Hartmann’s number, Richardson number, Reynolds number, nanoparticle volume fraction, irreversibility distribution rate, and direction of the applied magnetic field. The transfer involved vertical rectangular cavities with convection heating and moving walls. Bondarenko et al. [14] numerically explored the thermal flow behaviors of hybrid nanoparticles filled in the cavity. The cavity with a triangular embedded obstacle and with constant heat flux was reported by Gangawane et al. [15] to explore the influence of the thermal flow. In the study published in [16] titled “The Mixed Convection Heat Transfer in Horizontal Rectangular Cavities with Single- and Double-Sided Moving Walls,” the authors Louaraychi et al. [16] considered a cavity with moving walls, among other factors, to investigate mixed convection heat transfer. In addition to these reports, please see [15, 17] for further reading on lid-driven flows. Many numerical approaches have been used to evaluate the thermal flow in enclosed domains. A great example of the FEM approach that was applied to a different context is the use of the.

In an attempt to understand how heat transfers in a cavity, with a bottom wall that is only partially heated, a study was conducted by Mahalakshmi et al. [18]. In addition, different positions of the inner heater were also reported to affect the characteristics of heat and flow transfer. One previous study investigated a square cavity that had openings on only two sides and found that the transfer of heat was aided by nanofluid as illustrated by Hassanpor et al. [19]. The cavity was exposed to a magnetic field, which had an effect on the interior. A cylinder inside the cavity held a constant amount of heat. A partially heated C-shaped cavity was demonstrated in the study by Haq et al. [20] for the flow and heat transition mechanism. Their results clearly enhanced the heat transfer rate with an increase in the heated length. The cylindrical cavity is an important and interesting cavity. To their credit, the authors of Guestal et al. [21] calculated the parameters of a cylindrical cavity that is partially heated at the bottom and neglected the wall sections that were considered to be cold. It was concluded that a cavity with wavy walls on the top and bottom would be useful in order to examine the characteristics of porous medium and corrugation on the heat transfer phenomenon. Heat transfer analysis for the rhombus cavity with an inner embedded square object in different thermal conditions was carried out by Haq et al. [22, 23]. A natural convection model to simulate an upside-down T-formed cavity filled with a heating element was analyzed in a previous study by Izadi et al. [24]. An inclined rectangular cavity with a big rectangular element was examined using the lid-driven heat transfer analysis in a [25] study. In a lid-driven trapezoidal cavity filled with alumina–water nanofluid, Fatih et al. [26] discussed the modeling and optimization and the effects of electric convection models using the FEM method of electric conductance modeling. In a 3D porous cavity, Sheikholeslami et al. [27] discussed with help of the Boltzmann methods the water-based nanofluid in the existence of the Lorenz force. Rizwan et al. [29] considered a numerical study on a lid-driven hexagonal cavity using Newtonian fluid. They noticed that the circular obstacle plays an important role in forming isotherms. They found that the Reynolds number and Richardson number are strongly related to heat transfer. Bourantas and Loukopoulos [30] introduced a meshless numerical scheme for accurate and efficient computational results of velocity and pressure profiles by using the backward-facing step. Recently, Toudja et al. [31] used an irregular hexagonal cavity to study mixed convection using the hybrid nano power-law fluid; they found an enhancement in heat and mass transfer when the Richardson number decreases. In contrast, the n has the opposite effect on heat and mass transfer. In this direction, some recent additions addressing significant physical features and computational schemes have been mentioned in [32–39].

As suggested by the aforementioned literature review, the heat transfer analysis within the lid-driven cavity has not been conducted in a lid-driven hexagon cavity using the power-law fluid. In this study, the power-law fluid has been used to conduct heat exchange analysis in a hexagonal cavity with several thermal condition square obstacles positioned at the center of the cavity. The upper and lower lids are supposedly heated and move with velocities u = 1 and u = -1, respectively. The weak form of a system of PDEs is obtained with the help of the finite element method. The article is constructed as follows: in Sections 2, 3, flow configuration and physical quantities of the problem along with mathematical modeling are developed. In Sections 4, 5, validation of the solution is discussed and the Result and Discussion sections contain the discussion based on the outcome of the solution. In the last section, Section 6, the main findings based on the outcome are discussed.

For the description of the mathematical formulations, we have considered several constraints that must be met in order to evaluate the theoretical assumptions inside the hexagonal cavity. In this section, equations with each constraint and boundary condition and the geometry of the problem are discussed (see Figure 1A). The vertices of the cavity are defined as

Let V = ( u ∗ , v ∗ , 0 ) is defined as the velocity field for two-dimensional flow. The initially defined pressure p exerted by the fluid particles in the horizontal, x-direction and vertical, y-direction and afterward the Navier–Stokes equation is simplified as [40]. ∂ u ∗ ∂ x ∗ + ∂ v ∗ ∂ y ∗ = 0. (1) ρ ( u ∗ ∂ u ∗ ∂ x ∗ + v ∗ ∂ v ∗ ∂ y ∗ ) = − ∂ p ∗ ∂ x ∗ + ( ∂ τ x ∗ x ∗ ∂ x ∗ + ∂ τ x ∗ y ∗ ∂ y ∗ ) . (2) ρ ( u ∗ ∂ v ¯ ∂ x ∗ + v ∗ ∂ v ∗ ∂ y ∗ ) = −   ∂ p ∗ ∂ y ∗ + ( ∂ τ y ∗ x ∗ ∂ x ∗ + ∂ τ y ∗ y ∗ ∂ y ∗ ) + g β ρ ( T ∗ − T c ∗ ) . (3) ρ c p ( u ∗ ∂ T ∗ ∂ x ∗ + v ∗ ∂ T ∗ ∂ y ∗ ) = k ( ∂ 2 T ∗ ∂ x ∗ 2 + ∂ 2 T ∗ ∂ y ∗ 2 ) . (4)

Here,   u ∗ and v ∗ are the velocities of the fluid along x ∗ and y ∗ direction, respectively, and T ∗ − T c ∗ is the temperature difference of fluid molecules. For the power-law fluid, τ = K γ ˙ n , and other constants are defined as.

The boundary conditions for the problem under consideration are defined as follows:

For the upper and lower walls of the hexagonal cavity u ∗ = v ∗ = 0 ,   T ∗ = T h ∗ . (5)

For all the other walls of the cavity u ∗ = v ∗ = 0 ,   T ∗ = T c ∗ . (6)

Introducing the nondimensional variables x = x ∗ L ,   y = y ∗ L ,   u = u ∗ U ,   v = v ∗ U ,   p = p ∗ ρ U 2 ,   θ = T ∗ − T c ∗ T h ∗ − T c ∗ . (7)

Combining Equation 8 into Equations 2–7, we have ∂ u ∂ x + ∂ v ∂ y = 0. (8) ( u ∂ u ∂ x + v ∂ u ∂ y ) = − ∂ p ∂ x + 1 R e ( ∂ τ x x ∂ x + ∂ τ x y ∂ y ) . (9) ( u ∂ v ∂ x + v ∂ v ∂ y ) = − ∂ p ∂ y + 1 R e ( ∂ τ y x ∂ x + ∂ τ y y ∂ y ) + G r R e 2 θ . (10) ( u   ∂ θ ∂ x + v ∂ θ ∂ y ) = 1 R e P r ( ∂ 2 θ ∂ x 2 + ∂ 2 θ ∂ y 2 ) . (11)

The important physical quantities such as local and average Nusselt number are defined as N u l o c a l = ( ∂ θ ∂ y ) N u a v e r a g e = ∫ ( N u l o c a l ) w a l l } . (12) Here, Reynolds number. R e = U ∞ D υ Grashof number. G r = g β ( T h − T 0 ) D 3 υ 2 Prandtl number. P r = υ α

The system of governing Equations 8–11 does not contain analytical solution. It is, therefore, solved with the help of the FEM explained below.

1) Weak formulation

The weak form of Equations 8–11 is given as ∫ Ω ( ∂ u ∂ x + ∂ v ∂ y )   q   d Ω = 0 , (13) R e ∫ Ω ( u ∂ u ∂ x + v ∂ u ∂ y )   w   d Ω + R e ∫ Ω ∂ p ∂ x   w   d Ω − ∫ Ω ( ∂ τ x x ∂ x + ∂ τ x y ∂ y )   w   d Ω = 0 , (14) R e 2 ∫ Ω ( u ∂ v ∂ x + v ∂ v ∂ y )   w   d Ω + R e 2 ∫ Ω ∂ p ∂ y   w   d Ω − R e ∫ Ω ( ∂ τ y x ∂ x + ∂ τ y y ∂ y )   w   d Ω − G r ∫ Ω θ   w   d Ω = 0 , (15) P r R e ∫ Ω ( u ∂ θ ∂ x + v ∂ θ ∂ y )   w   d Ω − ∫ Ω ( ∂ 2 θ ∂ x 2 + ∂ 2 θ ∂ y 2 )   w   d Ω = 0 , (16)

For numerical approximation, we compute the continuous solutions with the discrete ones in the finite-dimensional sub-spaces u ≈ u ℏ ∈ W ℏ , v ≈ v ℏ ∈ W ℏ , θ ≈ θ ℏ ∈ W ℏ , p ≈ p ℏ ∈ Q ℏ , (17) where the basis functions are defined as u ℏ ≈ ∑ k = 1 ndof u k φ k ( x , y ) , v ℏ ≈ ∑ k = 1 ndof v k φ k ( x , y ) , p ℏ ≈ ∑ k = 1 ndof p k ψ k ( x , y ) , θ ℏ ≈ ∑ k = 1 ndof θ k φ k ( x , y ) . (18) Here, W = [ H 1 ( Ω ) ] 3 test subspaces for u ,   v , θ Q = L 2 ( Ω ) test subspace for pressure. Ω = entire flow domain.Using (17) in (14–16), the following discrete version is obtained R e ∫ Ω ( u ℏ ∂ u ℏ ∂ x + v ℏ ∂ u ℏ ∂ y )   w ℏ   d Ω + R e ∫ Ω ∂ p ℏ ∂ x   w ℏ   d Ω − ∫ Ω ( ∂ ( τ x x ) ℏ ∂ x + ∂ ( τ x y ) ℏ ∂ y )   w ℏ   d Ω = 0 , (19) R e 2 ∫ Ω ( u ℏ ∂ v ℏ ∂ x + v ℏ ∂ v ℏ ∂ y )   w ℏ   d Ω + R e 2 ∫ Ω ∂ p ℏ ∂ y   w ℏ   d Ω − ∫ Ω ( ∂ ( τ y x ) ℏ ∂ x + ∂ ( τ y y ) ℏ ∂ y )   w ℏ   d Ω − G r   ∫ Ω θ ℏ   w ℏ   d Ω = 0 , (20) ∫ Ω ( ∂ u ℏ ∂ x + ∂ v ℏ ∂ y )   q ℏ   d Ω = 0 , (21) R e P r ∫ Ω ( u ℏ ∂ θ ℏ ∂ x + v ℏ ∂ θ ℏ ∂ y )   w ℏ   d Ω − ∫ Ω ( ∂ 2 θ ℏ ∂ x 2 + ∂ 2 θ ℏ ∂ y 2 )   w ℏ   d Ω = 0 , (22)

In the matrix form [ R e .   L ℏ + N ( u ℏ , v ℏ , θ ℏ ) 0 R e . B 1 0 0 R e 2 .   L ℏ + N ℏ ( u ℏ , v ℏ , θ ℏ ) R e 2 . B 2 − G r . M ℏ B 1 T B 2 T 0 0 0 0 0 P r . R e . L ℏ + N ℏ ( u ℏ , v ℏ , θ ℏ ) ] [ U ¯ V ¯ P ¯ θ ¯ ] = [ F 1 F 2 F 3 F 4 ] , (23) which can be written as follows: Aξ=F

Here,

L h = discrete Laplace matrix.

N h = convection matrix (Non-linear)

M h = mass matrix.

ξ = solution vector.

F = RHS after implementation of boundary condition.

This nonlinear system is iterated until a given convergence condition is fulfilled in order to compute the solution. The nonlinear iterations are terminated after the residual has been reduced by 10 − 6 points. The steps carried out during the implementation of the scheme are disclosed in Figure 2C.

2) Validation of the numerical method

Before the simulation is performed, the verification of the code is carried out by reproducing the existing results. In this regard, the research work of H.F Oztop and E. Abu-Nada [28] is considered. In Figure 2A, the comparison of the results shows the code compatibility. The computational grid on a coarse level is shown in Figure 1B, where a hybrid meshing is carried out to capture the flow dynamics accurately near the boundaries. The second-order elements from the space P 2 are considered for velocity and temperature approximations, while the pressure uses linear approximations using the space P 1 .

The discrete nonlinear system of equations has been addressed by the Newton method and the linearized inner system with a direct solver, PARDISO. There are many benefits of using the PARDISO solver (see [41–47] for further information). The PARDISO solver uses LU factorization and reduces the number of cycles required for the desired level of convergence. The kinetic energy is calculated for different grid levels (see Table 1). The percentage of errors decreases as the refinement level increases. During the study, the ninth refinement level is used as the error between levels 8 and 9 is minimum.

The non-Newtonian power-law fluid along with the heat is studied in a hexagonal cavity having a square obstacle inside. The upper and lower wall and the obstacle are considered heated. The remaining walls of the obstacles are assumed cold. The suitable scales are used to non-dimensionalize the equations. A mathematical model containing Navier–Stokes and heat energy equations is obtained and handled with the help of a finite element system. Quadratic and triangular elements are used in the development of the FEM. On the nonlinear algebraic equations that have been obtained from the FEM formulation, Newton’s Raphson iteration scheme is applied. The range of the parameters involved in the study is defined as ( 100 ≤ R e ≤ 400 ) , ( 1 ≤ P r ≤ 10 ) , ( 10 3 ≤ G r ≤ 10 6 ) , and ( 0.5 ≤ n ≤ 1.5 ) .

In Figures 3–5, the streamlines (left) and isotherms (right) are plotted for the variation of R e ,   P r , and G r . The streamlines and isotherms plots for increasing Reynolds number by fixing n = 0.5 and P r = 1 are shown in Figure 3. The strength of the flow field increases with increasing values of R e . Due to the dominancy of conduction heat transfer, the isotherms at high R e are parallel and smooth. The effects of P r on streamlines and isotherms are depicted through Figure 4. The streamlines are almost the same for the variation of P r , while P r has significant effects on isotherms and can be seen in Figure 4A. Figure 5 is plotted to see the impact of Grashof number on streamline and isotherms, keeping n = 0.5 , R e   = 200 , and P r = 1 . There is significant change in streamline patterns as the Grashof number increases. For large values of G r , it starts becoming symmetrical across the obstacle.

When it comes to heat transfer, the Nusselt number is a key parameter that can influence the rate of heat exchange. The line graphs for local and average Nusselt number have been displayed in Figures 6–11. Figure 6 displays the impact of n on N u local at the upper wall. The N u local increases as the value of n increases. The same trend can be seen for the values of Re as shown in Figure 7. Figures 8, 9 display the average Nusselt number plotted against n for the variation of Gr and Re number, respectively. The N u a v g increases as the value of G r and Re increases. In Figure 10, the impact of n on u and v components of velocities is plotted. It can be seen from the figure that velocity increases as the n moved from shear thinning to shear thickening. Reynolds number has opposite effects on the velocity profile for x from 0 to 0.5 (see Figure 11).

Table 2 displays the impact of n on kinetic energy for different Re. When n increases, the kinetic energy inside the cavity increases. The kinetic energy has opposite effects for increasing values of Reynolds numbers. In Table 3, the effects of the Grashof number on the kinetic energy are calculated. It is evident from the table that an increase in Gr increases the kinetic energy.

A lid-driven hexagonal cavity with heated upper and lower walls with square obstacles has been discussed for the power-law fluid model. It is supposed that upper and lower walls have sliding velocities u = 1 and u = -1, respectively. The governing equations are converted into dimensionless form with the help of suitable scales. Then, the weak form of the equations is generated and solved with the Newton iterative scheme. The results have been depicted through the graphs to illustrate the physical consequences. By sketching streamlines and isothermal patterns against involved parameters, deviations in velocity and temperature fields are discussed. Effects of the involved parameter on velocity, temperature, and kinetic energy have been discussed. Local and average Nusselt numbers have been determined at various wall places to observe the heat transmission phenomenon. The major findings that are achieved through the abovementioned work are listed below.

• For streamlines, Prandtl number has almost the same effects as the Reynolds number, but for isotherms, P r effects are enhanced. There are strong isotherms near the heated object.