Influence of entropy on Brinkman–Forchheimer model of MHD hybrid nanofluid flowing in enclosure containing rotating cylinder and undulating porous stratum

The current article aims to discuss the natural convection heat transfer of Ag/Al2O3-water hybrid filled in an enclosure subjected to a uniform magnetic field and provided with a rotating cylinder and an inner undulated porous layer. The various thermo-physical parameters are investigated such as Rayleigh number (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$100 \le Ra \le 100000$$\end{document}100≤Ra≤100000), Hartmann number (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \le Ha \le 100$$\end{document}0≤Ha≤100), and the nanoparticles concentration (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0.02 \le \phi \le 0.08$$\end{document}0.02≤ϕ≤0.08). Likewise, the rotational speed of the cylinder (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$- 4000 \le \omega \le + 4000$$\end{document}-4000≤ω≤+4000), as well as several characteristics related to the porous layer, are examined li its porosity (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0.2 \le \varepsilon \le 0.8$$\end{document}0.2≤ε≤0.8), Darcy number (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$- 100000 \le Da \le - 100$$\end{document}-100000≤Da≤-100) which indicates the porous medium permeability and the number of undulations (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \le N \le 4$$\end{document}0≤N≤4). The calculations are carried out based on the Galerkin Finite element method (GFEM) to present the streamlines, isotherms, entropy generation, and average Nusselt numbers in details. The main results proved that increment of Rayleigh number and Darcy number enhances heat transfer convection within the enclosure. Whilst, the porosity presents a minimal impact. Also, the rotational speed in a positive direction has a favorable influence on the heat transfer dispersion across the cavity.

The current article aims to discuss the natural convection heat transfer of Ag/Al 2 O 3 -water hybrid filled in an enclosure subjected to a uniform magnetic field and provided with a rotating cylinder and an inner undulated porous layer. The various thermo-physical parameters are investigated such as Rayleigh number ( 100 ≤ Ra ≤ 100000 ), Hartmann number ( 0 ≤ Ha ≤ 100 ), and the nanoparticles concentration ( 0.02 ≤ φ ≤ 0.08 ). Likewise, the rotational speed of the cylinder ( −4000 ≤ ω ≤ +4000 ), as well as several characteristics related to the porous layer, are examined li its porosity ( 0.2 ≤ ε ≤ 0.8 ), Darcy number ( −100000 ≤ Da ≤ −100 ) which indicates the porous medium permeability and the number of undulations ( 0 ≤ N ≤ 4 ). The calculations are carried out based on the Galerkin Finite element method (GFEM) to present the streamlines, isotherms, entropy generation, and average Nusselt numbers in details. The main results proved that increment of Rayleigh number and Darcy number enhances heat transfer convection within the enclosure. Whilst, the porosity presents a minimal impact. Also, the rotational speed in a positive direction has a favorable influence on the heat transfer dispersion across the cavity. In recent years, the natural conjugate heat transfer magnetohydrodynamics (MHD) has been an inspiring topic for researchers due to its wide use in various sectors. Such as Boilers and cooling systems, thermal energy, and several engineering applications. Additionally, the focus of the research was on the mechanism of nanofluid heat transfer [1][2][3][4][5][6][7][8] . Pordanjani et al. 9 studied the free convection of the alumina-water nanofluid in a cavity. They examined the effect of the magnetic flux on heat transfer efficiency. Alnaqi et al. 10 studied the magnetic field impact of the Al 2 O 3 -water nanofluid in the inclined square cavity. It was exposed from their findings, the Nusselt number increase for an upper Rayleigh number and a lower Hartmann number. In an additional study, Pordanjani et al. 11 investigated the nanofluid heat transfer and total entropy generation S gen in a cavity. In the presence of magnetic flux, they retained distinct temperature profiles on the left side of the cavity divider. The upper and lower sides were insulated and the right wall was kept at low temperatures. It was found that S gen was increasing for a higher Ra and a lower Ha . Fares et al. 12  www.nature.com/scientificreports/ noticed a better response of heat transfer with a greater grand a lower Ha . Moreover, Selimefendigil and Öztop 16 studied the properties of the convection and S gen in a divided cavity filled with carbon nanotube (CNT)/water nanofluid under a magnetic flux using the finite element calculation method. The parameters were modified and the effect of the pack in the chamber was considered. Investigators found that the normal Nu rises with a greater Ra and a reduced Ha . In the study conducted by Gangawane and Bharti 17 , In a partly differently heated box, a cooler size effect was tested on MHD natural convection using the Boltzmann grid approach. It has been shown that the Nu avg is proportionally influenced by the cooler length and Ra , while it has an opposite relationship with Ha . In addition, Abbassi and Orfi 18 conducted a simulation analysis using lattice-Boltzmann method (LBM) on MHD free convection of a heated block situated on the base of an enclosure filled with nanofluids. They demonstrate that the magnetic flow has the reverse effect on heat transfer efficiency, the fluid flow, and the total S gen . The maximum heat transfer is obtained when the angle of inclination is equal to π/2 . Additionally, through the study done by Esmaeil 19 , The results of thermophysical properties on natural convection of laminar in containers in which nanofluid works is measured by using the technique of finite difference. Nanoparticles are subject to Brownian movement and thermophoresis physical transport processes. It is found that the key parameter that influences the flow of heat from nanofluids is the nanofluid viscosity. A theoretical analysis was carried out in a porous cavity filled with Cu/water nanofluid with LBM by Hoseinpour et al. 20 . In this analysis, the total S gen is found to be decreased and the Nu avg increases as the nanofluid volume fraction is raised. Also, the total S gen . is strongly affected by porous porosity. It is found that as the porosity is greater, the total S gen improves. Abu-Libdeh et al. 21 studied the impact of the deferent thermos-physical parameters on the hydrothermal and the entropy generation inside a novel porous cavity. Kasaeipoor et al. 22 conducted a free convection heat transfer and entropy generation analysis in an enclosure with refrigerant solid elements loaded with MWCNT-MgO/water hybrid nanofluid. It was inferred that the larger refrigerant solid body improves the Nu avg and S gen . In comparison, the Nu avg greatly increases improving φ and then decreases. With the rise of Ra , the S gen is enhanced and decreases with φ . Rahimi et al. 23 performed a numerical study investigating natural convection and entropy generation within an enclosure. Partially active walls, charged with two walls of carbon nanotubes, nanofluid water and fitted with cold, and hot barriers are provided, while the device is subject to LBM. The obtained results revealed that the Nu avg improves when Ra and φ increase. However, the S gen increase with Ra and decreases when φ rises up. additionally, Rahimi et al. 24 normal convection heat transfer and water-CuO nanofluid entropy generation in a square chamber supplied with fins. They noticed that by improving Ra and φ , the Nu avg enhances. Also, they found that S gen reduced by the increasing of φ . Fares et al. 25 discussed the optimization of the entropy generation inside a square cavity loaded with Ag/water nanofluid. Mainly, they investigated the effect of inclined magnetic field on S gen and Nu avg responses. Alsabery et al. 26 investigated the Impact of the use of two-phase hybrid nanofluid on mixed convection within a wavy lid-driven enclosure and equipped with a solid block. They found that the position of the solid block and surface undulation are significant in controlling heat transmission and the concentration distribution of the composite nanoparticles. Tayebi and Chamkha 27 studied MHD heat transfer within a nanofluid filled-square chamber separated by a solid conductive wall. Their findings showed that the combined impacts of the varied vertical conducting wall designs and other relevant factors can be an efficient way of regulating flow characteristics and heat transfer rate inside the system. Mebarek-Oudina et al. 28 studied the heat transfer and the entropy generation in case of magnetized hybrid nanoliquid flow involved in a trapezoidal enclosure. They demonstrated that increasing the Rayleigh number and reducing the Hartmann number enhances the thermal efficiency of the chamber. Belhadj et al. 29 investigated the nanofluid natural convection response inside a triangular cavity with an inner partial porous media installed at the right-angled corner. They noticed that the increase in Darcy number and the porosity has a boosting effect on the heat transfer efficiency. This infuence is more intensified whith greater Rayleigh and lower Hartmann numbers. Alsabery et al. 30 studied the entropy generation and mixed Convection heat transfer inside a cavity equipped with wavy walls and rotating Solid Cylinder. They found thatthe flow can be controlled by adjusting the cylinder's angular velocity. Moreover, they noticedthat the clockwise rotation around the solidcylinder intensifies the convective flow cell inside the wavy container as the Rayleigh number rises. At the top portion of the heated surface, the local Nusselt number peaks. Brahimi et al. 31 conducted a numerical study of thermal and streamline analysis inside a cavity filled with (Ag-MgO/Water) hybrid nanofluid. They determined that because the box's structure causes the flow to meander over the cliff bars, this unique flux tends to slow the flow around it, allowing the particles to thermally transfer. The LBM and modified LBM in an enclosure has been studied by [32][33][34] in conjugate natural convection and diffusion. They found that the heat transfer and the total S gen increase when Ra decreased the number of Bejan. It has been observed that as the thermal conductivity ratio rises, the Nu and the total S gen rate rise, however, be reduced. In addition, it is important to note that several recent researches have been performed to examine rarefied flow activity and heat transfer response along with S gen within enclosures 35,36 . It is noted that Knudsen number improves, the transfer efficiency reduces. Furthermore, LBM has later been used as an integrated approach focusing on kinetic theory and as a novel solution to discretization approaches 37 . Additionally, LBM presents second-order precision in both time and space; it is simple to code and can be used to manage a wide variety of flow patterns varying from microscopic to continuous levels.

List of symbols
As shown in the relevant literature above, convection heat transfer and entropy generation in such complex geometry subjected to a uniform magnetic field with the presence of an inner corrugated porous layer has never been studied previously. Indeed, the purpose of the current research is to analyze numerically the MHD convective heat transfer and the entropy generation for a partially differentially heated divided enclosure filled by Ag/Al 2 O 3 -water. The computational accuracy is based on LBM for a two-dimensional approximation of the governing equations. Experimental correlations for thermal and physical characteristics of nanofluids are used in this study like thermal conductivity and dynamic viscosity. The heat transfer and S gen within the partitioned cavity is investigated using dimensionless independent parameters such as Ra , nanoparticles fraction volume, Ha , and the magnetic field tilting angle. Furthermore, the heat generating rod bundle for nuclear applications, www.nature.com/scientificreports/ which may be considered as a heat generating anisotropic porous medium, is an example of particular real-world uses of the current situation.

Problem description
The geometry studied is seen in Fig. 1. Representation shown in 2D (a) and 3D (b). The top and the bottom walls are insulated. In addition, both sidewalls are equipped with hot and cold parts while the temperatures are upheld as T h and T c on the left and right walls respectively. The heater and cooler parts have been used as ( 0.4H ) for all cases and it is symmetrical on the center of the vertical walls. Whereas, all remaining walls are insulated. An undulated vertical porous layer with a thickness ( a = 0.1L ) has been placed in the cavity at ( b = 0.35L ). A rotating circular cylinder with a diameter ( d m = 0.2L ) is placed in the middle of the cavity at a distance of ( c = 0.3L ) on the x− axis. A uniform magnetic field is applied to the cavity in x− direction. The cavity is filled with the (Ag/Al 2 O 3 -water) hybrid nanofluid, which are considered Newtonian, incompressible with no viscous dissipation, and laminar flow. The thermophysical properties of hybrid nanofluid Ag/Al 2 O 3 -water are listed in Table 1. The water Prandtl number is specified to be Pr = 6.2.
The principal thermophysical characteristics of the base fluid (water) and the added nanoparticles are given in the following Table 1.

Formulation of mathematical model
Governing equations and boundary conditions:. Via a novel shape of a porous cavity for hybrid nanofluid, the stationary natural MHD convective flow is investigated. The Darcy-Brinkman-Forchheimer model 39 is being used for the numerical modelling of the porous media. To be formulated in a dimension model, Navier-Stokes and heat equations, expressed in Cartesian coordinates for the present study with the above assumptions in mind, can be given as follows: In 2D cavity fluid domain, the main conservation equations in the hybrid-nanofluid region are the followings 40 :   12,38 .
√ aε 3/2 (where a = 150 and b = 1.75 ) represents the operative thermal conductivity of porous media saturated with nanofluid, where K is the porous medium permeability and ε is its porosity, described as follows 38,41 : To reformulate the previous governing equations into non-dimensional ones, the following variables are used: Dimensionless numbers are given as follow: The non-dimensional equations in the hybrid-nanofluid region can be written as: The non-dimensional equations in the porous region can be written as 42 : ∂u ∂x + ∂v ∂y = 0, 1

Non-dimensional Entropy Generation.
Local entropy in hybrid nanofluid region production measurement was obtained from totaling the conjugated fluxes and the forces developed. The non-dimensional local entropy production is given by Woods 44 in a convective process: For boundary conditions related to the walls of the studied cavity, the dimensional former presented as following: • The hot wall: • The cold Wall: • The insolated walls: • Over the rotating cylinder Thermophysical characteristics of the hybrid-nanofluid. The density and the thermal conductivity as well as the heat capacity of the hybrid nanofluid can be given as the following 29,31 where, φ signifies the nanoparticle concentration factor. µ f , ρ f , (C p ) f and κ f are fluid viscidness, consistency, operative heat capacitance and thermally conductance of the basefluid, correspondingly. µ hnf , ρ hnf , ρ(C p ) hnf and κ hnf are hybrid nanofluid dynamical viscidness, intensity, specific heat capacitance and thermal conductance. µ f , ρ f , (C p ) f , κ f and σ f are dynamical viscidness, density, specific heat capacitance and thermal conductance of the basefluid. ρ p 1 , ρ p 2 , (C p ) p 1 , (C p ) p 2 , κ p 1 and κ p 2 are the intensity, specific heat capacitance and thermal conductance of the nanomolecules.

Validation and grid independence analysis
Seven different grids were used to confirm that the results were not dependent on the grid. The independence of flow and heat transfer from the number of grids is determined using Nu avg , the stream function, and general entropy (see Table 2). Due to the different results, the sixth grid was preferred as the final grid for all cases as shown in Fig. 2. Ensuring the numerical solution method is one of the primary criteria for achieving results, previous studies of Kaluri et al. 45 were used to validate our model, as shown in Fig. 3.
It is important to remember that the above governing equations along with the limits are solved by Galerkin finite element approach. Galerkin weighted residual finite element formula solves the equations numerically (We used the COMSOL Multiphysics® software to perform the modeling 46 ). In triangular elements, the code environment is separated. On any of the flow variables inside the code domain, triangular Lagrange finite elements are used from various orders. Residue is generated by replacing the approximations with the governing equations where it represents the iteration number and η is the convergence criterion. In this study, the convergence criterion was set at η = 10 −6 .

Results and discussion
The Effect of Rayleigh number. Figure 4 shows the streamlines and isotherms inside the studied cavity filled with Ag/Al 2 O 3 -water nanofluid for varying Rayleigh numbers. It is noted that the streamlines formed two contours near the heated vertical wall and a single contour beside the cooled vertical wall of the cavity. For lower values of the Rayleigh number, the heated streamline contours dominate the cooled single contour. Increasing Rayleigh number ( Ra ) shifts the domination towards the cooling end. This may be due to the convectional transport across the cavity. Isotherms for smaller Ra were distributed vertically from both ends of thermal spots. For increasing values of Ra , isotherms were pulled towards each other. Interestingly, the heated isotherms extend up to the cooled wall occupying the top of the cavity, whereas the cooled isotherms creep through the bottom. This shows the density reduction due to heat in the cavity fluid.
Effect of Darcy number. Effect of Darcy number becomes significant around the porous structure of the cavity. As Darcy's number increases, the permeability of the medium is increased to allow the flow into it. It was visualized by the streamline accumulation on either side of a porous medium and flows into that for higher values of Darcy number Da . Compare to the hotter side, the cooler side possesses intense contour, this may be due to the slower permeability of cooled fluid.
The obtained results in Fig. 5 depict that as more fluid enters through the porous medium for increasing values of Darcy number Da , temperature around the porous medium and middle part of the cavity exhibits smoother isotherms for higher values of Darcy number Da.
The state of average Nusselt number Nu avg across the cavity for various physical parameters were portrayed in Figs. 6 and 7. When the Rayleigh number Ra increased, the heat transfer rate also gets elevated especially after 10 2 for all physical parameters involved in the problem. It is found that those physical parameters like Porosity ε , Solid Volume Fraction φ , and Darcy number Da were to boosts the heat transfer rates evident through increasing average Nusselt number Nu avg while the Hartmann number Ha tends to reduce.
Kaluri et al. [44] present study  Figure 8 illustrates the streamlines, isotherms inside the studied cavity with temperature differences on either side that were subjected to the transverse magnetic field B . In absence of magnetic field influence ( Ha = 0 ), Streamlines formed a single contour near the cooler side along with the two minor contours in the heated side. Due to the increasing resistance to the flow, for higher Hartmann numbers the intensity of those contours got shifted more towards the bottom of the cavity. Magnetically restricted stream of the fluid inside the cavity assists the slower dissipation of temperature from both ends towards each other. For higher values of Ha , the isotherms start to spread away from the surface. As the heated isotherms claim up the cooler isotherms cover the lower cavity regions.
It is noticed through Figs. 9 and 10 that for increasing Ha , the average Nusselt number Nu avg was reduced due to flow restrictions experienced by the magnetic field intensity.
Effect of porosity. Compare to normal fluid flow problems, the porosity becomes a vital parameter in nanofluid flows involving porous medium. This may be due to the suspended nanoparticles and their nature. Similar to the effects of Darcy number Da , the porosity ε also possesses clustered streamlines around porous media but in some smaller way. As the increased porosity ε facilitates the flow across the cavity, the streamlines are altered limitedly.
Regarding isotherms in Fig. 11, the effect of porosity seems to be minimal. This was observed through the graphs that no such significant alterations happened.
Effect of solid volume fraction. The concentration of solid volume fraction reflects in the quality of nanofluid in terms of its fluidity and thermal efficiency is considered. The influence of solid volume fraction over the streamlines can be noted in either side of the rotating cylinder and behind the undulated region. As the www.nature.com/scientificreports/ volume fraction gets increased the streamline contours seems to be getting faded due to slowness developed in the cavity by the added particle fraction. It also reflects in the isotherms plots that slower flow grasp more heat from the hotter side which can be seen in Fig. 12.
Effect of rotating speed. The rotation speed study has been carried out for two cases such as positive and negative values of Rotation speed ω . The findings in Fig. 13 show that for negative rotation speed, the fluid from the hotter wall side rotates over the cavity, which seems to be the maximum coverage of the cavity with hotter fluid. As the rotation speed tends towards positive, the cooler streams spread across the cavity. The isotherms also support the claim considered above. Hotter isotherms tend to rotates from the clockwise direction, while the cooler isotherms tend to rotate in an anti-clockwise direction for higher values of Rotation speed ω.

Effect of undulation. Variation in undulation values increases the flow fluctuations across the cavity. It is
clear through the streamlines and isotherms for both decreased ( N = 1 ) and increased ( N = 4 ) values of undulations. It is shown in Fig. 14, that the formation of contour in the cavity gets increased. Especially around the porous structure and thermally varied sidewalls. Exceptionally for Rotation speed ω , it is observed that the average Nusselt number Nu avg gets reduces from the initial state around the values of Ra = 10 4 , and then it hikes continuously as it is shown in Fig. 15. It is noticed through Fig. 16 the average Nusselt number Nu avg got reduced due to flow restrictions experienced by the magnetic field intensity at any rotational speed. Fig. 17, it can be seen that the entropy generation S gen seems to be triggered from both the thermal ends of the cavity. Pair of contours formed in both ends along with the increased entropy changes in porous region for lower Rayleigh number Ra . This state gets reversed for increasing values of Ra . www.nature.com/scientificreports/ The dual contours become single in both ends, the heat end contour drags down while the other claims up. The porous part of the cavity gets away with the entropy variation for a higher value of Ra. Figure 18 indicates that for the lower Darcy number Da , the fluid struggles to enter into the porous medium caused accumulation around it. The graph of entropy generation S gen reflects the state of entropy loss in fluid due to its permeability restrictions through porous media. For higher values of Da , the flow gets into the other side to have only entropy losses in two spots of heat variations.

Entropy generation. Through
Entropy generation acts opposite to that for the Rayleigh number variations compared to the Hartmann number Ha . As it is clearly shown in Fig. 19, the contours start to build from the two rear sides and as the magnetic influence increases, they simultaneously move towards the middle part of the cavity and splits into two contours on each side.
Respective to the rotation direction of the fluid inside the cavity, Fig. 20 illustrates that the entropy generation S gen also shifted clockwise to anticlockwise for increasing values of Rotation speed ω.
Similar to the average Nusselt number, the entropy generation S gen increased for Rayleigh number greater than 10 2 . Entropy control for this problem under Rayleigh number manipulation can be done using parameters like Hartmann number Ha and solid volume fraction φ while the other parameters can assist entropy as it is shown in Fig. 21.
Regarding undulation parameter N impact in Fig. 22, both the average Nusselt number Nu avg and the entropy generation S gen gets increased to the value of undulation N = 2 . Later both tend to get reduced, especially the Nusselt number drops more than that of entropy generation. This reflects the fact that, more undulations makes the flow and its nanoparticle suspension more difficult which reduces the heat transfer process and simultaneously the entropy generation and total entropy across the system. www.nature.com/scientificreports/

Conclusions
Through the parametric study based on finite element method towards the Ag/Al 2 O 3 -water hybrid nanofluid which filled inside the porous layered enclosure influenced by magnetic field over the rotating cylinder, the following conclusion is made:    www.nature.com/scientificreports/

Data availability
The results of this study are available only within the paper to support the data.