MHD Natural convection and entropy generation of Fe₃O₄–MgO/water hybrid nanofluid in an inclined enclosure with flower-shaped heated fins

Abstract

Controlling both the heat removal and the associated irreversibility is a central requirement in compact energy-storage and electronic-cooling systems containing multiple localized heat sources. This study presents a comprehensive numerical investigation of magnetohydrodynamic (MHD) natural convection and entropy generation in an inclined enclosure filled with a binary hybrid nanofluid composed of magnetite (Fe₃O₄)) and magnesium oxide (MgO) dispersed in water. The dimensionless governing equations for continuity, momentum, and energy are solved using the finite volume method (FVM) implemented in a FORTRAN-based computational code. The simulation is conducted for a wide range of controlling parameters, including Rayleigh number (Ra=10³–10⁶), Hartmann number (Ha=0–75), enclosure inclination angle (γ = 0°–120°), and nanoparticle volume fraction (φ = 0–4%). In addition to the governing parameters, four geometric configurations (C1–C4) of heated flower-shaped internal fins are examined to assess the impact of heat source placement on flow circulation. The results show that increasing Ra and φ intensifies buoyancy-driven circulation and enhances the average Nusselt number, while higher Ha values suppress convection due to the Lorentz force. At Ra=10⁶ and φ = 0.04, raising Ha from 0 to 75 reduces the Nusselt number by 21.9% at γ = 0° but only 12.7% at γ = 60°, so tilting lowers the sensitivity to the field. Without a field the vertical enclosure performs best, whereas under a field the tilted one becomes superior by 5–7.2%. Further noteworthy findings and comprehensive discussions are provided throughout the remainder of this study. These findings provide important guidance for the optimal design of MHD thermal systems incorporating complex internal heat sources.

Introduction

Natural convection heat transfer under the influence of a magnetic field, commonly referred to as magnetohydrodynamic (MHD) convection, has attracted significant research interest due to its applications in advanced cooling technologies, energy systems, and material processing. In recent years, the utilization of nanofluids suspensions of nanoparticles in conventional base fluids has been proposed as an effective strategy to enhance thermal transport properties. Building on this concept, hybrid nanofluids, which combine two or more types of nanoparticles, have emerged as a promising class of working fluids offering superior thermal conductivity, stability, and energy efficiency compared to single particle nanofluids [1–4]. The coupling of MHD convection with hybrid nanofluids enables a wide range of engineering applications, including electronic cooling, nuclear reactors, solar collectors, and biomedical devices, where precise thermal management is required under magnetic field control [5–9]. Consequently, numerous studies have investigated the flow structures, heat transfer enhancement, and stability characteristics of MHD natural convection with hybrid nanofluids in different geometrical configurations.

Mondal et al. [10] explored a novel implementation of the partial magnetic field induced on a differentially heated inclined porous square enclosure containing Cu−Al₂O₃/water hybrid nanofluid. The partial magnetic field is imposed using varying effective width and the allocation impacts of this partial magnetic field on the inclined cavity have been investigated systematically. Authors found that adjusting the position and width of the imposed magnetic fields, cavity angle, and involved pertinent parameters allows less reduction in heat transfer, which is ∼15% (with partial magnetic field) and ∼30% (with whole domain magnetic field) relative to the no-magnetic field. Pandit et al. [11] investigated the complex thermal-fluid behavior within a tilted porous enclosure filled with a Cu− Al2O3-water hybrid nanofluid, subject to segmented magnetic fields, wavy cooling segments, and distributed heat sources. Results revealed that the application of a segmented magnetic field with strategic orientation resulted in up to 26% enhancement by modulating flow patterns and creating localized convection cells. In order to investigate the influence of enclosure shape on magnetohydrodynamic (MHD) nanofluidic flow, heat transfer and irreversibility in square, trapezoidal and triangular thermal systems under fluid volume constraints, with the aim of optimizing thermal behavior in diverse applications, Manna et al. [12] used numerical simulations based on a FEM to analyze the effects of pertinent parameters on heat transfer characteristics and thermodynamic entropy production. It has been found that the magnetic field intensity has a significant impact on fluid flow and heat transport in natural convection, with higher Hartmann numbers resulting in reduced flow strength and heat transfer. Souayeh [13] numerically investigated the influence of a magnetic field on natural convection flow and entropy generation in a square cavity filled with a hybrid nanofluid, where heating was induced by two isothermal semicircles located along the bottom and left cavity walls. The findings revealed that the average Nusselt number decreases with increasing Hartmann number, indicating that the entropy generation rate follows the same trend as heat transfer. In a related work, Souayeh et al. [14] carried out a detailed numerical study on inclined magnetohydrodynamic (MHD) natural convection and entropy generation within a square cavity filled with a ternary hybrid nanofluid, incorporating a thermally cooled saw-tooth-shaped inner body. Their results showed that increasing Ha suppresses convection due to magnetic damping, resulting in conduction-dominated heat transfer and lower entropy generation.

Enclosures with cooled boundaries and internally heated obstacles have been widely studied as canonical models for investigating natural convection and heat transfer in engineering systems. The introduction of internal bodies with different geometrical forms such as cylinders, elliptic shapes, triangular blocks, and more complex lobed or flower-like structures significantly alters the thermal and flow fields inside the cavity. These internal shapes act as localized heat sources and simultaneously influence fluid circulation patterns, boundary layer development, and overall heat transfer rates. The design and arrangement of such obstacles are therefore of great interest in applications ranging from electronic cooling and thermal insulation to solar collectors and energy storage devices. Recent studies have demonstrated that the geometry, size, and positioning of these heated inner bodies can either enhance or suppress natural convection efficiency, making the systematic exploration of cavity obstacle configurations an active area of research.

Souayeh and Ben-Beya [15] conducted a numerical study on transient natural convection and entropy generation within a tilted square cavity filled with a binary hybrid nanofluid (Ag/GO–water), incorporating localized cooling sources and internal heat generation/absorption. Their findings indicated that spectral analysis identified distinct frequency peaks, with increasing Rayleigh numbers amplifying oscillatory motion and fostering the formation of well-structured vortex patterns. In another work, Souayeh et al. [16] examined three-dimensional Fe₃O₄–water nanofluid flow inside a cooled cubic enclosure heated by an isothermal solid sphere positioned at various locations (P1–P10). The study revealed that the optimal placement of the sphere achieves a balance between enhancing heat transfer and reducing entropy generation, thereby improving the thermal performance of the cubic swimming pool. A numerical investigation of the natural convective flow of Cu-water nanofluid in an isosceles triangular-shaped cavity, where an air stream cools the lateral walls while its base is subjected to discrete heating has been performed by B. Saleem and al. [17]. One of their main findings is that the cooling effect from the cavity side walls is enhanced as Re∞ rises. An approach to effective cooling has been executed by Shao et al. [18]. They considered ternary hybrid nanofluid natural convection within a porous prismatic enclosure with two movable hot baffles. Authors found that amplification of Darcy number upgrades vertical and horizontal velocities up to 88.51% and 83.77%, respectively. The effects of a magnetic field on mixed convection flow and entropy generation within a double lid-driven square cavity filled with a hybrid nanofluid and two isothermally heated semi-circles located on the bottom and left walls of the cavity has been studied by Souayeh [19]. From this study, it has been inferred that optimal heat transfer is achieved for Ri = 0.04, Ha = 100, ϕ = 0%, β = 1, and λ = 0.5 with 63% enhancement.

While the geometry and arrangement of internal obstacles play a decisive role in shaping convection patterns, the orientation of the enclosure itself adds another critical dimension to heat transfer performance. Tilting an enclosure relative to gravity alters the buoyancy-driven flow structures and redistributes thermal boundary layers, leading to significant variations in natural convection behavior. Such tilted configurations are particularly relevant in practical systems where enclosures are rarely aligned perfectly with gravity, such as in solar collectors, electronic devices, underground heat exchangers, and inclined building components. Moreover, the combined effects of enclosure inclination, cooled walls, and heated internal bodies create highly complex thermal interactions that cannot be captured by considering vertical or horizontal orientations alone. Therefore, the study of tilted enclosures provides essential insights into realistic thermal management scenarios and extends the applicability of cavity-based convection research.

Salari et al. [20] investigated the 3D numerical analysis of natural convection and entropy generation within tilted rectangular enclosures filled with stratified fluids of MWCNTs/water nanofluid and air. It has been indicated that tilt angle has considerable effect on flow structure and enhances significantly the total entropy generation. Aghakhani et al. [21] performed a numerical analysis of natural convective heat transfer and entropy generation of alumina/water nanofluid in a tilted enclosure with an elliptically constant temperature with magnetic field and radiation effects. It has been concluded that augmenting the fin aspect ratio results in an augment of 85% for zero inclination angle and an augment of 93% for the inclination angle of 90. Souayeh et al. [22] studied numerically the three-dimensional natural convection in a cubic enclosure induced by an isothermally heated circular cylinder at different inclinations. Authors found that an optimal heat transfer appears when the cylinder orientation becomes vertical at 90˚ and for high Rayleigh number value reaching 10⁶ for the two cases of both inner cylinder and lateral walls of the cubical enclosure. In another research work, Souayeh et al. [23] examined numerically the three natural convection of air induced by temperature difference between a cold outer tilted cubic enclosure and a hot inner cylinder. Authors indicate that optimal average heat transfer rate is obtained for both high Rayleigh number of 10⁶ and high inclination of 90° for the two cases of the inner cylinder and cubical enclosure.

Beyond evaluating flow structures and heat transfer rates, it is equally important to assess the thermodynamic efficiency of natural convection systems. Entropy generation analysis, grounded in the second law of thermodynamics, provides a powerful framework for quantifying irreversibilities arising from heat transfer, viscous dissipation, and magnetic field effects. In the context of MHD convection with hybrid nanofluids, entropy generation offers deeper insight into the trade-off between heat transfer enhancement and energy losses due to fluid friction or magnetic damping. This aspect becomes particularly significant in cooled enclosures containing heated internal bodies, where complex flow interactions and temperature gradients strongly influence thermodynamic irreversibility. As a result, coupling entropy generation analysis with MHD natural convection of hybrid nanofluids in tilted enclosures enables a more comprehensive evaluation of system performance and guides the design of thermally efficient enclosures for advanced engineering applications.

The entropy generation analysis of natural convective flow of hybrid nanofluid in a triangular enclosure with heated wavy wall and cold obstacle has been explored by Turabi et al. [1]. The findings demonstrate that the rate of heat transfer and total entropy decreases with the increasing Hartmann number. Biswas et al. [24] investigated the magneto-ferro-nanofluidic thermal convection and entropy generation in a circle-modified butterfly-shaped cavity with strategic protrusions at both top and bottom portions with partial heating and cooling configurations. The flow characteristics demonstrate strong dependence on both Ra and Ha, with distinct multi-vortex structures emerging within the cavity under various flow conditions. In another contribution, Biswas et al. [25] investigated the thermomagnetic convection and irreversibility production in a hybrid nanofluid-filled wavy-walled porous thermal system containing a semi-circular heated bottom. Both the sidewalls of the enclosure are cooled and undulated with varying undulation numbers. Authors found that higher Ram values intensify the irreversibilities due to stronger convection effects, while elevated magnetic fields enhance magnetic field-induced irreversibilities. To complement these discussions, numerous works have examined entropy generation phenomena in MHD natural convection of nanofluids within enclosures, as reported in [26–30].

Motivation and objectives

Despite the considerable progress achieved in the study of MHD natural convection and entropy generation in nanofluid-filled enclosures, several aspects remain insufficiently explored. In particular, the combined influence of enclosure inclination, magnetic-field-induced flow suppression, complex localized heat sources, and their spatial arrangement has received limited attention, especially for binary hybrid nanofluids. Most existing studies have primarily focused on simple internal geometries or a single arrangement of heat sources, while the role of heat-source positioning and geometric asymmetry in controlling both thermal transport and thermodynamic irreversibility remains inadequately understood. Moreover, the simultaneous assessment of heat-transfer enhancement and entropy generation under different orientations of an inclined enclosure is essential for identifying practically meaningful operating conditions.

Motivated by these gaps, the present study investigates MHD natural convection and entropy generation in an inclined rectangular enclosure filled with a Fe₃O₄–MgO/water binary hybrid nanofluid. The two nanoparticles are considered at equal volume fractions, while three internally heated flower-shaped fins are introduced as localized heat sources. Four distinct configurations (C1–C4) are examined to systematically investigate the effects of symmetric, partially symmetric, separated, and laterally shifted heat-source arrangements. The analysis further incorporates the effects of Rayleigh number, Hartmann number, enclosure inclination angle, and nanoparticle volume fraction. Unlike studies that focus primarily on heat-transfer enhancement, the present investigation simultaneously evaluates thermal performance and entropy generation to identify operating conditions that provide a favourable balance between heat-transfer enhancement and thermodynamic irreversibility. The resulting framework provides new insight into the coupled effects of magnetic control, enclosure orientation, hybrid-nanofluid properties, and localized heat-source distribution, with potential relevance to the thermal management and design of compact energy and cooling systems.

Physical model, assumptions, and governing equations

Physical model description

The physical system under investigation consists of a two-dimensional rectangular enclosure of height H, uniformly inclined at an angle γ with respect to the vertical axis. All outer walls of the enclosure are maintained at a constant cold temperature, providing a uniformly cooled boundary environment. Three identical heated flower-shaped (multi-lobed) fins are embedded inside the enclosure, acting as localized heat sources. The enclosure is filled with a hybrid nanofluid composed of Fe₃O₄–MgO/water, modeled as a Newtonian, incompressible fluid in thermal and hydrodynamic equilibrium, with no-slip and no-penetration conditions at all solid surfaces.

A uniform magnetic field of strength B is applied perpendicular to the computational plane, generating magnetohydrodynamic (MHD) effects that alter the buoyancy-driven convective motion. Heat transfer inside the enclosure arises from natural convection caused by the temperature difference between the hot internal fins and the cold enclosure walls. The present formulation therefore incorporates the combined influences of enclosure inclination, hybrid nanofluid thermophysical behaviour, internal heat sources, and magnetic field interaction to examine the resulting flow structure, heat transfer characteristics, and entropy generation.

Moreover, to assess the impact of the fin arrangement on thermal and flow behaviour, four different configurations of the three heated flower-shaped fins are considered as listed in Table 1.

Table 1.

Description of the considered configurations of the internally heated flower-shaped fins

configuration	number of heated fins	fin positions	spatial arrangement	main geometric characteristic
C1	3	bottom, middle, and top	vertically aligned along the enclosure centerline with equal spacing	fully symmetric reference configuration with three distributed heat sources
C2	2	bottom and middle	vertically aligned along the centerline; top fin removed	heat-free upper region with two vertically distributed heat sources
C3	2	bottom and top	vertically aligned along the centerline; middle fin removed	two separated heat sources with an unheated central region
C4	2	middle and top	both fins shifted laterally toward the right side	asymmetric, right-biased heating distribution

The considered hybrid nanofluid in our study is a mixture of nanoparticles of Ferric Oxide (Fe₃O₄) and Magnesium Oxide (MgO) suspended in base fluid water. knowing that the mixture contains an equal volume fraction of Fe₃O₄ and MgO. The thermophysical properties of these nanoparticles and the host fluid are summarized in Table 1. The following simplifying assumptions have been taken into account throughout this study:

• Incompressible, Newtonian, and laminar fluid.

• Boussinesq approximation is applied to incorporate the density variation.

• The properties of the fluid are constant except for the density in the body force term.

• The hybrid nanoparticles are in thermal equilibrium.

• The effect of viscous dissipation is neglected since it is insignificant compared to conduction and convection effects.

• The effects of radiation were assumed to be negligible.

Actually, the above assumptions are adopted to focus the present investigation on buoyancy-driven natural convection and its interaction with the externally applied magnetic field. Since the present system is governed by natural convection, the flow intensity is characterized by the Rayleigh number rather than an imposed Reynolds number. The induced magnetic field is neglected under the low-magnetic-Reynolds-number approximation, such that the externally applied magnetic field is considered dominant. Joule heating is neglected because the associated electrical dissipation is assumed to be small compared with the conduction and buoyancy-driven convective heat-transfer mechanisms under the considered conditions. Similarly, viscous dissipation is neglected in the energy equation because the flow remains in a laminar natural-convection regime and viscous heating is expected to provide a secondary contribution relative to conduction and convection. Thermal radiation is also neglected because heat transfer within the present liquid-filled enclosure is assumed to be dominated by conduction and natural convection. These assumptions define the scope of the present mathematical model and allow the effects of buoyancy, magnetic damping, enclosure inclination, hybrid-nanofluid properties, and internal heat-source arrangement to be isolated and systematically investigated. Thus, by taking into account all the aforementioned assumptions cited above, the governing equations describing the current problem will be detailed in the next section Fig. 1.

Fig. 1.

(a) Physical domain filled with Fe₃O₄–MgO hybrid nanofluid under uniform magnetic field B. (b). Schematic of the four investigated fin configurations (C1–C4) showing vertical placement of heated fins inside the tilted cavity

Governing equations and boundary conditions

Governing equations

In this study, the governing equations for the incompressible viscous flow and thermal fields are the continuity, momentum, and energy equations expressed in their dimensional forms.

Continuity equation:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$ 1

Momentum equations:

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{1}{{\rho }_{\mathit{hnf}}}\frac{\partial p}{\partial x}+\frac{{\mu }_{\mathit{hnf}}}{{\rho }_{\mathit{hnf}}}\left(\frac{{\partial }^{2}u}{{\partial x}^2}+\frac{{\partial }^{2}u}{{\partial y}^2}\right)+\frac{(\rho {\beta )}_{\mathit{hnf}}}{{\rho }_{\mathit{hnf}}}g\left(T-T_c\right)\sin \gamma$$ 2

$$\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{1}{{\rho }_{\mathit{hnf}}}\frac{\partial p}{\partial y}+\frac{{\mu }_{\mathit{hnf}}}{{\rho }_{\mathit{hnf}}}\left(\frac{{\partial }^{2}v}{{\partial x}^2}+\frac{{\partial }^{2}v}{{\partial y}^2}\right)-\frac{{\sigma }_{\mathit{hnf}}}{{\rho }_{\mathit{hnf}}}{B_0}^{2}v+\frac{(\rho {\beta )}_{\mathit{hnf}}}{{\rho }_{\mathit{hnf}}}g\left(T-T_c\right)\cos \gamma$$ 3

$$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}={\alpha }_{hnf}\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)$$ 4

In order to transform the above governing equations from dimensional to dimensionless form, the following parameters have been introduced:

$$\begin{array}{l}X=\frac{x}{H},Y=\frac{y}{H},U=\frac{u}{U_0},V=\frac{v}{U_0},P=\frac{p}{{\rho }_{hnf}{U}_0^2},\theta =\frac{T-T_C}{T_H-T_C},U_0=\frac{{\alpha }_f}{H},\\ \Delta T=T_H-T_C,\tau =t\frac{U_0}{H}\end{array}$$

Thus, the dimensionless equations governing the fluid flow and heat transfer in the current research work are given by [31]:

$$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0$$ 5

$$\frac{\partial U}{\partial \tau }+U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=-\frac{\partial P}{\partial X}+\mathrm{Pr}\frac{{\nu }_{\mathit{hnf}}}{{\nu }_f}\left(\frac{{\partial }^{2}U}{{\partial X}^2}+\frac{{\partial }^{2}U}{{\partial Y}^2}\right)+\frac{(\rho {\beta )}_{\mathit{hnf}}}{{\rho }_{\mathit{hnf}}{\beta }_f}\mathit{Ra}\mathrm{Pr}\theta \sin \gamma$$ 6

$$\begin{array}{c}\frac{\partial V}{\partial \tau }+U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}\\ =-\frac{\partial P}{\partial Y}+\mathrm{Pr}\frac{{\nu }_{\mathit{hnf}}}{{\nu }_f}\left(\frac{{\partial }^{2}V}{{\partial X}^2}+\frac{{\partial }^{2}V}{{\partial Y}^2}\right)+\frac{(\rho {\beta )}_{\mathit{hnf}}}{{\rho }_{\mathit{hnf}}{\beta }_f}\mathit{Ra}\mathrm{Pr}\theta \cos \gamma -\frac{{\sigma }_{\mathit{hnf}}}{{\sigma }_f}\frac{{\rho }_f}{{\rho }_{\mathit{hnf}}}{\mathit{Ha}}^2\mathrm{Pr}V\end{array}$$ 7

$$\frac{\partial \theta }{\partial \tau }+U\frac{\partial \theta }{\partial X}+V\frac{\partial \theta }{\partial Y}=\frac{{\alpha }_{hnf}}{{\alpha }_f}\left(\frac{{\partial }^2\theta }{\partial X^2}+\frac{{\partial }^2\theta }{\partial Y^2}\right)$$ 8

Where (u, U) and (v, V) are the velocity components, g is the gravitational acceleration. (p, P) and T are pressure and temperature. The density, dynamic viscosity, thermal conductivity, specific heat at constant pressure, volumetric thermal expansion coefficient, and electrical conductivity are represented by ρ, μ, k, Cp, β, and σ respectively. The hybrid nanofluid is denoted by the subscripts ‘hnf’ and the plain fluid by ‘f’. The thermophysical properties of the base fluid and hybrid nanofluid are summarized in Table 2.

Table 2.

Thermophysical properties of host fluid and hybrid nanoparticles [32, 33]

Physical Property	Base Fluid (H2O)	Fe3O4	MgO
$\rho (kg/m^3)$	997.1	5180	3580
$C_p(J/kg.K)$	4179	670	955
$k(w/m.K)$	0.613	9.7	45
$\beta (K^{-1})\times {10}^{-5}$	21	0.5	3.36 × 10−5
$\Gamma (S/m)$	5.5 × 10−6	2.5 × 103	2.6 × 10−6

It is worth noting that the present MHD formulation adopts the standard low-magnetic-Reynolds-number (inductionless) approximation. Under this approximation, the magnetic field induced by the fluid motion is assumed to be negligible compared with the externally imposed magnetic field, and therefore the applied magnetic field is treated as uniform and prescribed. Consequently, the magnetic induction equation is not solved separately, and the effect of the magnetic field on the fluid motion is represented through the Lorentz-force term in the momentum equations. This approximation is widely adopted in MHD flow models for (Re_m«1), where $({\mathrm{Re}}_{\mathrm{m}}={\mu }_0{\sigma }_{hnf}{U}_c{L}_c)$ denotes the magnetic Reynolds number, L_c characteristic enclosure length, U_c representative velocity, ${\sigma }_{hnf}$ and effective electrical conductivity. The present investigation is primarily characterized by the Rayleigh number, as it concerns buoyancy-driven natural convection; (Re_m) is not considered as an independent control parameter.

Furthermore, the pertinent dimensionless parameters; Prandtl number, Hartmann number and Rayleigh number that appeared in the dimensionless governing equations are defined as follows:

$$\mathrm{Pr}=\frac{{\nu }_f}{{\alpha }_f}$$

$$Ha=B_{0}H\sqrt{\frac{{\Gamma }_f}{{\mu }_f}}$$

$$Ra=\frac{g{\beta }_f\left(T_H-T_C\right)H^3}{{\nu }_f{\alpha }_f}$$

In order to calculate the thermophysical properties of hybrid nanofluid, the following equations based on mixture relation were used based on the following models [34]:

Hybrid nanofluid density

$${\rho }_{hnf}={\phi }_{F{e}_3{O}_4}{\rho }_{MgO}+{\phi }_{F{e}_3{O}_4}{\rho }_{MgO}+(1-{\phi }_{F{e}_3{O}_4}-{\phi }_{MgO}){\rho }_f$$ 9

Hybrid nanofluid heat capacitance

$${\left(\rho C_p\right)}_{hnf}={\phi }_{F{e}_3{O}_4}{\left(\rho C_p\right)}_{F{e}_3{O}_4}+{\phi }_{MgO}{\left(\rho C_p\right)}_{MgO}+\left(1-{\phi }_{F{e}_3{O}_4}-{\phi }_{MgO}\right){\left(\rho C_p\right)}_f$$ 10

Hybrid nanofluid buoyancy coefficient

$${\left(\rho \beta \right)}_{hnf}={\phi }_{F{e}_3{O}_4}{\left(\rho \beta \right)}_{F{e}_3{O}_4}+{\phi }_{MgO}{\left(\rho \beta \right)}_{MgO}+\left(1-{\phi }_{F{e}_3{O}_4}-{\phi }_{MgO}\right){\left(\rho \beta \right)}_f$$ 11

Hybrid nanofluid thermal conductivity

$$\begin{array}{l}\frac{k_{hnf}}{k_f}=\left[\frac{{\phi }_{F{e}_3{O}_4}{k}_{F{e}_3{O}_4}+{\phi }_{MgO}{k}_{MgO}}{{\phi }_{hnf}}+2k_f+2\left({\phi }_{F{e}_3{O}_4}{k}_{F{e}_3{O}_4}+{\phi }_{MgO}{k}_{MgO}\right)-2{\phi }_{hnf}{k}_f\right]\\ \times {\left[\frac{{\phi }_{F{e}_3{O}_4}{k}_{F{e}_3{O}_4}+{\phi }_{MgO}{k}_{MgO}}{{\phi }_{hnf}}+2k_f+2\left({\phi }_{F{e}_3{O}_4}{k}_{F{e}_3{O}_4}+{\phi }_{MgO}{k}_{MgO}\right)+{\phi }_{hnf}{k}_f\right]}^{-1}\end{array}$$ 12

Dynamic viscosity

$${\mu }_{hnf}=\frac{{\mu }_f}{{\left(1-\left({\phi }_{F{e}_3{O}_4}+{\phi }_{MgO}\right)\right)}^{2.5}}$$ 13

And

$${\alpha }_{hnf}=\frac{k_{hnf}}{{(\rho C_p)}_{hnf}}$$ 14

The effective electrical conductivity of hybrid nanofluid as shown below:

$$\frac{{\sigma }_{hnf}}{{\sigma }_f}=1+\frac{3\left(\frac{\left({\phi }_{F{e}_3{O}_4}{\sigma }_{F{e}_3{O}_4}+{\phi }_{MgO}{\sigma }_{MgO}\right)}{{\sigma }_f}-\left({\phi }_{F{e}_3{O}_4}+{\phi }_{MgO}\right)\right)}{\left(\frac{{\phi }_{F{e}_3{O}_4}{\sigma }_{F{e}_3{O}_4}+{\phi }_{MgO}{\sigma }_{MgO}}{{\phi }_{hnf}{\sigma }_f}+2\right)-\left(\frac{{\phi }_{F{e}_3{O}_4}{\sigma }_{F{e}_3{O}_4}+{\phi }_{MgO}{\sigma }_{MgO}}{{\phi }_{hnf}{\sigma }_f}-\left({\phi }_{F{e}_3{O}_4}+{\phi }_{MgO}\right)\right)}$$ 15

Where ${\phi }_{hnf}={\phi }_{F{e}_3{O}_4}+{\phi }_{MgO}$

Boundary conditions

Regarding the boundary conditions of our physical model, the walls are impermeable and subject to non-slip conditions. The boundary conditions are given as follows:

$u=v=0,\theta =0$: at all the all-side walls of the enclosure.

$u=v=0,\theta =1$: at the inner fins.

Entropy generation equations

The dimensionless local entropy generation $(S_T^{⁎})$ is evaluated as the sum of three distinct sources of irreversibility: entropy generation due to fluid friction $(S_{Vis\text{co}us}^{⁎})$, entropy generation caused by thermal gradients ($S_{Thermal}^{⁎}$), and entropy dissipation resulting from the influence of the magnetic field $(S_{MHD}^{⁎})$.

$$S_T^{⁎}=S_{Vis\text{co}us}^{⁎}+S_{Thermal}^{⁎}+S_{MHD}^{⁎}$$ 16

The mathematical expression for the entropy generation associated with fluid friction thermal gradients and magnetic field are given by:

$$S_{Vis\text{co}us}^{⁎}=\eta \frac{{\mu }_{hnf}}{{\mu }_f}\left[2\left({\left(\frac{\partial U}{\partial X}\right)}^2+{\left(\frac{\partial V}{\partial Y}\right)}^2\right)+{\left(\frac{\partial U}{\partial Y}+\frac{\partial V}{\partial X}\right)}^2\right]$$ 17

$$S_{Thermal}^{⁎}=\frac{k_{hnf}}{k_f}\left({\left(\frac{\partial \theta }{\partial X}\right)}^2+{\left(\frac{\partial \theta }{\partial Y}\right)}^2\right)$$ 18

$$S_{MHD}^{⁎}=\eta H{a}^2\frac{{\sigma }_{hnf}}{{\sigma }_f}{\left(U\sin \gamma -V\cos \gamma \right)}^2$$ 19

Where $\eta$ is the irreversibility factor defined as follows:

$$\eta =\frac{{\mu }_f{T}_o}{k_f}{\left(\frac{{\alpha }_f}{H\Delta T}\right)}^2$$ 20

In all simulations, this factor is consistently set to 10^-5 [31].

The dimensionless total entropy generation $(S_T)$ is obtained by integrating the dimensionless local entropy generation $(S_T^{⁎})$ over the entire computational domain.

$$S_T=S_{Vis\text{co}us}+S_{Thermal}+S_{MHD}$$ 21

$$S_{Vis\text{co}us}={\int }_0^1{\int }_0^1{S}_{Vis\text{co}us}^{⁎}dXdY$$ 22

$$S_{Thermal}={\int }_0^1{\int }_0^1{S}_{Thermal}^{⁎}dXdY$$ 23

$$S_{MHD}={\int }_0^1{\int }_0^1{S}_{MHD}^{⁎}dXdY$$ 24

Heat transfer calculation

The thermal performance of the system can be enhanced by increasing the heat transfer rate while simultaneously minimizing entropy generation. Therefore, evaluating these two parameters is essential for identifying the conditions that yield optimal performance. In this study, the heat transfer rate is evaluated over all the cold walls of the enclosure:

$$N{u}_{loc}=-\frac{k_{hnf}}{k_f}\left(\frac{\partial \theta }{\partial n}\right)$$ 25a

$$\overline{Nu}=\left(\frac{1}{8}\right)\left[{\int }_0^1{N{u}_{loc}|}_{Y=0}dX+{\int }_0^1{N{u}_{loc}|}_{Y=3}dX+{\int }_0^3{N{u}_{loc}|}_{X=0}dY+{\int }_0^3{N{u}_{loc}|}_{X=1}dY\right]$$ 25b

Numerical approach, grid sensitivity analysis and code validation

Method of the solution

In the present numerical investigation, the simulations were carried out using a FORTRAN in-house code, designated “NASIM” [35, 36]. This solver addresses the unsteady Navier–Stokes and energy Eqs. (1)–(8), which govern the fluid flow, together with the entropy generation formulations (16)–(24), subject to the boundary conditions. The computational framework incorporates the following numerical strategies:

• Time discretization of transient terms was achieved via a second-order implicit Euler backward scheme [37].

• Lorentz force, buoyancy, and nonlinear contributions were treated explicitly, while diffusion terms were handled implicitly within the temporal discretization [38].

• Velocity–pressure coupling inherent in the continuity and momentum equations was resolved through the projection method [39].

• The Poisson equation was employed to compute the corrected velocity fields and pressure adjustments [40].

• Spatial discretization was performed using the finite volume method on a two-dimensional staggered grid arrangement [41].

• To mitigate excessive numerical diffusion in advective terms, the QUICK scheme was adopted [42].

• Computational efficiency was enhanced via a full multigrid acceleration technique to minimize the iteration count [43].

• The resulting discretized algebraic systems were solved using the red–black successive over-relaxation method [44].

Convergence at each time step was ensured by monitoring the dependent variable χ, where the iterative solution was considered converged once the following criterion was satisfied:

$${\sum _{i,j}\left|{\chi }_{i,j}^n-{\chi }_{i,j}^{n-1}\right|}^2\le {10}^{-8}$$ 26

Grid dependency

A grid independence test was carried out to validate the accuracy of the numerical results and to determine the most suitable mesh resolution. The verification relied on comparing the predicted average Nusselt number and total entropy generation values obtained from five different non-uniform mesh distributions: 100 × 50, 120 × 60, 144 × 72, 160 × 80 and 196 × 98, with a denser clustering of elements adjacent to the solid boundaries. The simulations modeled magnetohydrodynamic (MHD) convective heat transfer and entropy generation around internally heated fins placed inside a rectangular cavity filled with hybrid nanofluid, with thermophysical properties provided in Table 2.

The test was performed under the conditions Pr=6.2 (representing water as the base fluid), Ra=10⁵, φ = 4%, γ = 60˚ and Ha=25, considering the initial configuration referred to Configuration C1. The objective was to choose a mesh that maintains high solution accuracy while minimizing computational requirements. The calculated average Nusselt numbers at the east wall of the cavity, shown in Table 3, revealed that the 160 × 80 non-uniform mesh (Fig. 2) achieved a favourable compromise between precision and computational expense. Therefore, this mesh resolution was adopted for all subsequent simulations.

Table 3.

Average Nusselt number and total entropy generation values for different grid sizes at Ra=10⁵, φ = 4%, Ha=25 and γ = 60˚ for the configuration C1

Grid	$\overline{Nu}$	Error (%)	ST	Error (%)
100 × 50	2.6512	-	20.8510	-
120 × 60	2.6840	1.23	21.1120	1.25
144 × 72	2.7216	0.29	21.4110	0.31
160 × 80	2.7295	0.09	21.4428	0.15
196 × 98	2.7312	0.06	21.4550	0.06

Fig. 2.

Non-uniform grid of 160 × 80 size

Validation of the code

To verify the accuracy of the proposed numerical method used in the in-house FORTRAN code by taking into account the natural convection of hybrid nanofluids validations of average and local Nusselt numbers were done. Our results were compared with the experimental results of Yıldız et al. [45] (Fig. 3). As demonstrated in Figs. 3(a), 3, local Nusselt number profiles for Al₂O₃ (Fig. (3), local Nusselt number for the hybrid nanofluid Al₂O₃-SiO₂ (Fig. (3) and average Nusselt number for different nanoparticle volume fractions (Fig. (3) represented in this study are in good agreement with those represented by Yıldız et al. [45]. Additionally, the in-house FORTRAN solver was benchmarked against a canonical problem of two-dimensional natural convection in a differentially heated square cavity containing an Al₂O₃–water nanofluid. As presented in Table 4, the predicted average Nusselt number exhibited excellent agreement with the benchmark results reported by Souayeh and Ben-Beya [16], thereby affirming the solver’s accuracy and numerical robustness.

Fig. 3.

Comparison of present results with Yildiz et al. [45] for local Nusselt number of nanofluid Al₂O₃ (a), local Nusselt number of the hybrid nanofluid Al₂O₃-SiO₂ (b) and average Nusselt number for different nanoparticle volume fractions (c)

Table 4.

Comparison of present results of mean Nusselt number for Al₂O₃-H₂O nanofluid with Souayeh and Ben-Beya [16] for different nanoparticle volume fractions and Rayleigh numbers at Ha=30

	Souayeh and Ben-Beya [16]	Present Study	Error (%)
Ra = 103
φ = 0%	1.002	1.002	0.0
φ = 2%	1.061	1.060	0.09
φ = 4%	1.121	1.121	0.0
φ = 6%	1.184	1.184	0.0
Ra = 104
φ = 0%	1.181	1.182	0.08
φ = 2%	1.231	1.231	0.0
φ = 4%	1.283	1.283	0.0
φ = 6%	1.339	1.339	0.0
Ra = 105
φ = 0%	3.135	3.138	0.09
φ = 2%	3.207	3.200	0.21
φ = 4%	3.279	3.277	0.06
φ = 6%	3.351	3.347	0.12
Ra = 106
φ = 0%	7.815	7.817	0.02
φ = 2%	7.992	7.992	0.0
φ = 4%	8.166	8.166	0.0
φ = 6%	8.335	8.335	0.0

Results and discussion

Rationale for the choice of study parameters

The selection of governing parameters in this investigation was carefully made to capture the essential physical mechanisms influencing MHD natural convection and entropy generation in the tilted cooled enclosure with flower-shaped heat sources. The Rayleigh number (Ra=10³, 10⁴ and 10⁵) was varied to represent the transition from weak to strong buoyancy-driven convection regimes, enabling assessment of thermal and flow field evolution under different convection intensities. The Hartmann number (Ha=0, 25, 50 and 75) was introduced to quantify the influence of the externally applied magnetic field, where low values represent negligible Lorentz force effects and higher values correspond to stronger magnetic damping of the flow. The nanoparticle volume fraction (φ = 0, 0.02, 0.03 and 0.04) was chosen to examine the thermal enhancement capability of the hybrid nanofluid [Fe₃O₄–MgO/water], while maintaining physically realistic concentrations that ensure stable suspension and minimize agglomeration. The inclination angle of the cavity (γ = 0°, 30°, 60°, and 120°) was selected to explore how geometric orientation modifies the interaction between buoyancy and magnetic forces, thereby altering the thermal stratification and entropy generation within the cavity. Collectively, these parameters provide a comprehensive framework to analyze the coupled effects of magnetic field intensity, nanoparticle loading, cavity inclination, and buoyancy strength on the heat transfer performance and thermodynamic irreversibility of hybrid nanofluids in complex enclosures.

In addition to the governing physical parameters, four geometric configurations (C1–C4) of the heated flower-shaped fins were considered to assess how the spatial arrangement of internal heat sources affects the convective flow, thermal transport, and entropy generation. These configurations differ in the number and vertical placement of the heated fins, as well as in their degree of geometric symmetry. Configuration C1 represents the fully symmetric case with three fins distributed uniformly along the vertical centerline. Configurations C2 and C3 each contain two fins but with different removed positions, C2 excludes the top fin, whereas C3 excludes the middle fin allowing the effect of partially interrupted heating zones to be evaluated. Configuration C4 includes only the middle and top fins, shifted laterally toward the right wall, providing an asymmetric heating pattern that enables examination of symmetry-breaking effects on flow circulation and irreversibility. Incorporating these configurations allows a systematic investigation of how fin placement modulates buoyancy strength, flow structure, and thermodynamic behavior within the enclosure.

Flow dynamics, heat distribution, and entropy generation patterns

Figure 4 shows the stream function (ψ), temperature (θ), kinetic energy (E_K) and total local entropy generation (S_T) for Ra=10⁴, 10⁵ and 10⁶ at Ha=50, φ = 0.04 and γ = 60°. When Ra increases from 10⁴ to 10⁵, the peak stream function increases exactly ten times, from |ψ|_max = 0.14 to 1.40. This linear increase shows that the flow is still controlled by a balance between the buoyancy force and the viscous and Lorentz forces, which are both linear in velocity. The inertia terms are still weak, so the flow simply follows the applied thermal forcing. For this reason, the temperature field hardly changes between the two cases. In both, the three fins are surrounded by one continuous warm column (θ ≈ 0.7–0.8) that fills the gaps between them. This means that each fin is placed inside the warm wake of the fin below it, so the upper fins are heated by fluid that is already hot. The temperature difference available to them is therefore reduced, and the three fins behave as one thermally connected group rather than as three separate heat sources. This wake effect, and not the strength of the circulation, is what limits the heat removal at low Ra.

Fig. 4.

Effect of Rayleigh numbers on flow structure, thermal distribution, kinetic energy, and entropy generation at Ha=50, φ = 0.04 and γ = 60°

At Ra=10⁶ the behaviour changes. The peak stream function reaches 11.20, which is only eight times the value at Ra=10⁵ instead of ten times. This slower increase indicates that the inertia terms are no longer negligible. Two other changes confirm this. First, the two circulation senses are no longer equal (ψ_max = +11.20 and ψ_min = −10.25, a difference of about 9%), whereas at lower Ra they were identical. The momentum of the plumes is now strong enough to break the symmetry of the fin arrangement. Second, the warm column between the fins disappears. The gaps cool down to θ ≈ 0.4–0.5, and each fin develops its own thin and inclined thermal layer. The stronger circulation therefore flushes the gaps and separates the fins from each other, which removes the wake effect described above. This is the most important consequence of increasing Ra: the fins change from a shielded group to three ventilated and independent heat sources.

The entropy field follows this change rather than the flow strength. The peak local entropy increases only from 130.3 to 149.0 ( + 14%) when Ra goes from 10⁴ to 10⁵, although the circulation becomes ten times stronger, and then rises to 275.0 ( + 85%) at Ra = 10⁶. The reason is that the local entropy is controlled by the thermal term, which depends on the square of the temperature gradient. In the shielded low-Ra cases this gradient is fixed mainly by the geometry of the fins and the walls and is almost independent of the flow. Only at Ra=10⁶, when the boundary layers become thin, does the gradient at the fin surface increase clearly, and the entropy peak follows. The S_T contours also show that the entropy is not spread over the whole fin surface. It is concentrated in the narrow corners between two neighbouring lobes, where the cold incoming stream is turned and the thermal layer is thinnest. The flower shape therefore gives extra heat transfer area, but it also creates a few small regions of very high irreversibility.

Figure 5 shows the effect of the inclination angle for γ = 0°, 30° and 120° at Ra=10⁴, Ha=50 and φ = 0.04. In the present system the inclination does not simply make the convection stronger or weaker. It changes how the buoyancy force is shared between two directions, and it also changes the shape of the cells. In Eqs. (6) and (7) the buoyancy force appears as Ra·Pr·θ·sin γ in the X-momentum equation and Ra·Pr·θ·cos γ in the Y-momentum equation, while the Lorentz term −(σ_hnf/σ_f)(ρ_f/ρ_hnf)Ha²·Pr·V acts only on the Y-component. Tilting the cavity therefore moves part of the driving force from the damped direction to the undamped one. At γ = 0° all the buoyancy acts in the Y-direction and is fully damped by the magnetic field. At γ = 120°, sin γ = 0.866, so about 87% of the driving acts in the X-direction, where the Lorentz force has no effect. This explains the values obtained. The peak stream function is almost the same for the three angles ( | ψ|_max = 0.14, 0.12 and 0.14 at γ = 0°, 30° and 120°), and the kinetic energy is in fact highest at the largest angle (E_K,max = 0.50, 0.48 and 0.52). The loss of driving in the Y-direction at γ = 120° is compensated by the part of the buoyancy that is released from the magnetic damping. In a magnetised cavity, therefore, increasing inclination does not always reduce the convection as it would without a magnetic field. It can be used to move the buoyancy force away from the damped direction.

Fig. 5.

Effect of the cavity inclination angle on flow structure, thermal distribution, kinetic energy, and entropy generation at Ha=50, φ = 0.04 and Ra=10⁴

What changes clearly is the shape of the flow. At γ = 0° the flow is perfectly symmetric about the vertical centreline and forms eight cells, arranged as four counter-rotating pairs, one pair on each side of each fin. The fluid rises along the fin surfaces and descends along the two cold side walls, so each gap between two fins forms a closed convective unit. At γ = 30° this symmetry is lost. The cell boundaries become inclined, the vortex centres move closer to the fins, and small secondary cells appear in the corners. The peak stream function drops by about 14%, because the driving in the Y-direction is reduced by cos 30° while the driving in the X-direction is still small. At γ = 120° the separate units are replaced by one large, inclined circulation that crosses the whole array and connects the gaps between the fins. The driving is now mainly in the X-direction, and since the cavity is narrow in this direction the same force acts over a much shorter distance, which produces steeper velocity gradients. This point becomes important in the entropy results of Fig. 7.

Fig. 7.

Influence of inclination angle on thermal, viscous, and magnetic entropy generation and Bejan number distributions at Ha=50, φ = 0.04, and Ra=10⁶

The peak local entropy is almost unchanged (S_T,max = 130.3, 132.0 and 130.5 at γ = 0°, 30° and 120°, a variation of only 1.3%), again because this maximum is fixed by the temperature gradient at the fin corners and not by the bulk flow.

Figure 6 shows the effect of the magnetic field for Ha = 0, 25 and 75 at Ra=10⁴, φ = 0.04 and γ = 60°. Together with the corresponding case of Fig. 4, the complete sequence Ha = 0, 25, 50 and 75 gives |ψ|_max = 0.29, 0.22, 0.14 and 0.09 and E_K,max = 1.06, 0.75, 0.50 and 0.37. The circulation and the kinetic energy are therefore reduced by 69% and 65% respectively. The effect of the field on the shape of the flow is even more important. At Ha=0 and Ha=25 the flow forms two large cells: an upper cell of positive circulation around the top fin and a lower cell of negative circulation around the bottom fin, separated by an inclined line that passes through the middle fin. These two cells transport heat over the whole height of the cavity and connect the three fins together. At Ha=75 they break into six small vortices, each one limited to the region around a single fin and separated by almost motionless zones where ψ = 0. The magnetic field therefore does not only slow down the flow; it also cuts the connection between the heat sources and brings the array back to the shielded situation described above. The isotherms confirm this, since they become more stratified and the continuous warm column between the fins appears again.

Fig. 6.

Effect of magnetic field intensity on flow structure, thermal distribution, kinetic energy, and entropy generation at Ra=10⁴, φ = 0.04, and γ = 60°

Although the circulation is reduced by 69%, the peak local entropy decreases by only 1.9% (S_T,max = 132.0, 131.3 and 129.5 for Ha = 0, 25 and 75). This shows clearly that the peak entropy and the flow strength are governed by two different mechanisms. The peak entropy depends on the temperature gradient close to the fin surface, while the flow strength depends on the momentum balance in the bulk.

Figure 7 separates the local entropy generation into its thermal (S_thermal), viscous (S_viscous) and magnetic (S_MHD) parts, together with the Bejan number, for γ = 0°, 30° and 120° at Ha=50, φ = 0.04 and Ra=10⁶. The three components differ by several orders of magnitude, and they do not respond to the inclination in the same way. The thermal part decreases continuously with the tilt: S_thermal,max = 598.8, 430.1 and 372.1 at γ = 0°, 30° and 120°, that is a reduction of 38%. The contours explain why. At γ = 0° the two cold streams that descend along the side walls are symmetric and meet exactly at the lower corner of the bottom fin. This gives a clean impingement point at a narrow corner, where the thermal boundary layer is thinnest, and this single point carries the highest value of S_thermal. Weaker crescent-shaped zones appear at the side corners of each fin. When the cavity is tilted, the incoming stream rotates while the fin geometry stays fixed, so the impingement point moves away from the corner and onto the side of a lobe, where the boundary layer is thicker and the temperature gradient is smaller. The decrease of the thermal entropy with the inclination is therefore caused by this change of the impingement position and not by a general weakening of the convection. This is confirmed by Fig. 5, which showed that the circulation strength is almost the same for the three angles.

The viscous part behaves in the opposite way. It increases by a factor of 3.2, from S_viscous,max = 0.51 at γ = 0° to 1.14 at γ = 30° and 1.65 at γ = 120°. This is the direct result of the change in the direction of the buoyancy force. As the tilt increases, the flow is pushed across the narrow width of the cavity, so a similar velocity is reached over a much shorter distance. Since the viscous entropy depends on the square of the velocity gradient, the shorter distance increases the friction irreversibility even though the maximum velocity does not change.

The Bejan number stays close to unity (Be=0.94–1) over more than 99% of the cavity for the three angles, which agrees with the ratio S_thermal/S_viscous, which decreases only from 1174 to 226. Lower values appear only at a few isolated points: at the upper and lower corners of the fins, and, for γ = 30° and 60°, along a thin inclined line in the gap between the top and the middle fin, where Be falls to 0.3–0.6. These points are not regions of strong friction. They are stagnation points where the temperature gradient tends to zero, so the thermal term in the denominator of Be almost disappears and the viscous and magnetic terms become relatively important.

Figure 8 gives the same decomposition for Ha = 25, 50 and 75 at γ = 60°, φ = 0.04 and Ra = 10⁶. The three components vary monotonically with the field strength, but the thermal and viscous parts decrease while the magnetic part increases. The thermal part decreases from S_thermal,max = 303.8 to 275.0 and then 236.9, that is a reduction of 22%. This is due to the thickening of the boundary layers when the impinging flow is slowed down by the field. The viscous part decreases from 1.71 to 1.35 and then 1.13, a reduction of 34%, in agreement with the lower velocity gradients caused by the weaker circulation and by the break-up of the two large cells shown in Fig. 6. The viscous part decreases faster than the thermal part, which is expected, because it depends on the square of the velocity gradient while the thermal part depends on a temperature gradient that is partly maintained by conduction and cannot fall below its purely conductive value.

Fig. 8.

Influence of Hartmann number on thermal, viscous, and magnetic entropy generation and Bejan number distributions at γ = 60˚, φ = 0.04, and Ra=10⁶

In absolute terms the order of the three mechanisms does not change over the whole range studied: S_thermal is much larger than S_viscous, which is itself larger than S_MHD. The viscous part is about two orders of magnitude below the thermal part, and the magnetic part is about one order below the viscous part. A useful check is obtained by comparing Fig. 4 and Fig. 8 at the common case (Ra = 10⁶, Ha = 50, γ = 60°, φ = 0.04). The peak total entropy of Fig. 4 (S_T,max = 275.0) is equal to the peak thermal entropy of Fig. 8 (S_thermal,max = 275.0). More than 99.5% of the peak local entropy generation is therefore of thermal origin, which is consistent with a Bejan number close to unity, and the flow returns to a conduction-controlled state at Ha = 75.

Figures 9 and 10 show the horizontal and vertical velocity components for the four nanoparticle volume fractions at γ = 60° and Ha=50, for Ra=10⁴ and Ra=10⁶. In both figures the two components have comparable magnitudes, which is a direct consequence of the inclination. At γ = 60° the buoyancy force is divided between the two directions as sin 60° = 0.866 and cos 60° = 0.5, so both components are driven at the same time. The horizontal component is the larger of the two, exceeding the vertical one by about 20% at Ra=10⁴ and by about 44% at Ra=10⁶. This is expected, since the Lorentz term in Eq. (7) acts only on the vertical component, so the horizontal motion is not damped by the magnetic field. The highest velocities are concentrated in the narrow passages between the fins and the side walls, where the fluid is accelerated, while large regions of the cavity remain almost motionless. The two Rayleigh numbers differ by about two orders of magnitude in velocity, the extrema increasing from about 1 at Ra=10⁴ to about 90 at Ra=10⁶. This is consistent with the increase of the stream function reported in Fig. 4 and confirms the change from a weak, viscosity-controlled motion to a strong convective flow.

Fig. 9.

Distributions of horizontal and vertical U- and V- velocities inside the enclosure for different nanoparticle volume fractions at γ = 60°, Ha=50, and Ra=10⁴

Fig. 10.

Distributions of horizontal and vertical U- and V- velocities inside the enclosure for different nanoparticle volume fractions at γ = 60°, Ha=50, and Ra=10⁶

The effect of the nanoparticles is different at the two Rayleigh numbers. At Ra=10⁴ the velocities decrease steadily as the volume fraction increases, both components losing about 13% between φ = 0 and φ = 4%. Adding nanoparticles raises the viscosity and the density of the mixture, which slows the circulation, and at this Rayleigh number the flow is too weak for the additional buoyancy to compensate. It is important to note that the Nusselt number nevertheless increases with φ under the same conditions. The improvement obtained with the hybrid nanofluid at low Rayleigh number therefore comes entirely from the higher thermal conductivity of the mixture and not from a stronger flow, since the flow is in fact weaker. At Ra=10⁶ the velocities no longer follow a clear trend with φ and remain within about 10% of each other, because the buoyancy force is now strong enough to balance the increased viscous resistance.

Heat transfer and entropy response under varying parameters

Figures 11 and 12 present the average Nusselt number as a function of the inclination angle for Ha = 0, 25, 50 and 75 and for four nanoparticle volume fractions. Three separate effects can be identified. At every inclination angle, $\overline{Nu}$ decreases when the Hartmann number increases. At γ = 0° and φ = 0.04, the value falls from 4.39 at Ha = 0 to 4.16, 3.77 and 3.40 at Ha = 25, 50 and 75, which is a total reduction of 23%. The Lorentz force opposes the vertical velocity component, so the circulation that carries the heat from the fins to the cold walls is weakened and a larger part of the transport must be achieved by conduction alone.

Fig. 11.

Average Nusselt number profiles versus the inclination angles of the enclosure for different Hartmann numbers and nanoparticle volume fractions at Ra=10⁵

Fig. 12.

Average Nusselt number profiles versus the inclination angles of the enclosure for different Hartmann numbers and nanoparticle volume fractions at Ra=10⁶

For the effect of the magnetic field, it appears that the curves are not flat, and their shape depends on the magnetic field. Without a field, the highest Nusselt number is obtained close to γ = 0°, and the curve shows a shallow minimum near γ = 60°. As soon as a field is applied, the maximum moves to γ = 120°. At φ = 0.04, for example, $\overline{Nu}$ at γ = 120° exceeds the value at γ = 0° by 4.8% at Ha=25, by 5.0% at Ha=50 and by 7.4% at Ha=75, whereas at Ha=0 the tilted case is 3.2% lower. This behaviour follows directly from the form of the momentum equations. The buoyancy force is divided between the two directions as sinγ and cosγ, while the Lorentz term in Eq. (7) acts only on the Y-component. Tilting the enclosure to γ = 120° places about 87% of the driving force in the X-direction, where the magnetic damping has no effect. When no field is applied there is nothing to avoid, so the aligned configuration remains the best; when a field is applied, the inclination provides a way of protecting part of the buoyancy force from the damping, and the benefit grows with the field strength.

A second consequence of the same mechanism is visible in the spacing of the curves. At γ = 0° and φ = 0.04 the four Hartmann numbers cover a range of 29% between the highest and the lowest value, while at γ = 120° the same range is reduced to 19%. The inclined configuration is therefore less sensitive to the applied field, which is a useful property when the field strength cannot be controlled precisely.

Regarding the effect of the nanoparticle volume fraction, the Nusselt number increases with φ for all the cases examined. At γ = 0° and Ha=0, $\overline{Nu}$ rises from 4.07 at φ = 0 to 4.25, 4.34 and 4.39 at φ = 0.02, 0.03 and 0.04, that is an increase of 7.9%. The gain comes mainly from the higher thermal conductivity of the hybrid nanofluid, which improves the conduction across the boundary layers on the fins and on the cold walls. The increments become smaller as φ increases, because the higher viscosity of the mixture also reduces the circulation, and this partly cancels the benefit of the higher conductivity.

Figures 13 and 14 show the total entropy generation for the same conditions at Ra=10⁵ and Ra=10⁶. The trends with Ha and φ are the same as those of the Nusselt number, but their amplitude depends strongly on the Rayleigh number. The magnetic field reduces the entropy generation in all cases, since the weaker circulation produces smaller velocity and temperature gradients. The size of this reduction, however, is very different at the two Rayleigh numbers. At Ra=10⁵, γ = 0° and φ = 0.04, S_T decreases only from 22.92 at Ha=0 to 21.04 at Ha=75, a reduction of 8%. At Ra=10⁶, γ = 0° and φ = 0.03, the same increase of the field reduces S_T from 33.00 to 25.74, a reduction of 22%. The reason is that at Ra=10⁵ the irreversibility is produced mainly by the temperature gradients around the fins, which are maintained by conduction and are almost independent of the flow, whereas at Ra=10⁶ a significant part of the irreversibility is associated with the convective motion itself and can therefore be suppressed by the field.

Fig. 13.

Total entropy generation profiles versus the inclination angles of the enclosure for different Hartmann numbers and nanoparticle volume fractions at Ra=10⁵

Fig. 14.

Total entropy generation profiles versus the inclination angles of the enclosure for different Hartmann numbers and nanoparticle volume fractions at Ra=10⁶

The entropy generation increases with the nanoparticle volume fraction at both Rayleigh numbers, by about 10% at Ra=10⁵ and about 5% at Ra=10⁶ over the range φ = 0 to 0.04. The higher conductivity and viscosity of the mixture increase both the thermal and the viscous contributions.

The effect of the inclination angle is the opposite of that observed for the Nusselt number. At Ra=10⁶ and Ha=75 the entropy generation is lowest near γ = 0° to 30° (25.7) and increases towards γ = 120° (27.7), an increase of about 8%. This is expected from the same mechanism: the inclination restores part of the circulation by moving the buoyancy force away from the damped direction, and the restored motion produces additional irreversibility. The inclination that is most favourable for heat transfer is therefore also the one that produces most entropy.

The highest average Nusselt number, $\overline{Nu}=4.39$, is obtained at Ra=10⁶, φ = 0.04, Ha 0 and γ = 0°. When a magnetic field is present, the optimum angle changes and the highest value become $\overline{Nu}=4.36$ at Ha=25 and γ = 120°.

On the other hand, the lowest total entropy generation, S_T = 18.9, is obtained at Ra=10⁵, φ = 0, Ha=75 and γ = 0°. This is, however, the condition of weakest driving, and the corresponding Nusselt number is the lowest of the whole study. Thus, the two objectives are therefore in direct conflict, and the extreme values are obtained at opposite ends of the parameter range, as summarized in Table 5.

Table 5.

Operating conditions giving the extreme values of the two objectives

Objective	Ra	φ	Ha	γ	$\overline{Nu}$	ST
Maximum heat transfer (no field)	10⁶	0.04	0	0°	4.39	33
Maximum heat transfer (with field)	10⁶	0.04	25	120°	4.36	33
Minimum entropy generation	10⁵	0	75	0°	2.4	18.9

Because the two quantities vary in almost the same proportion, no intermediate value of the inclination angle or of the Hartmann number gives a better compromise than the end points of the range. Moving from γ = 0° to γ = 120° at Ra=10⁶, Ha=75 and φ = 0.03 increases the Nusselt number by 6.9% and the entropy generation by 7.5%, so the gain and the penalty cancel almost exactly. The same is true for the magnetic field, since raising Ha from 0 to 75 reduces both quantities by about the same percentage.

The selection of the operating point should therefore be based on the objective rather than on a compromise between the two. The Rayleigh number and the volume fraction should be taken as high as the application allows, since both increase the heat transfer. The inclination should be close to γ = 120° when a magnetic field is applied and the heat transfer is to be maximized, and close to γ = 0° when the irreversibility is to be minimized. The angle γ = 30° behaves almost identically to γ = 0° in every case examined, with differences of less than 0.5% in both quantities, and therefore offers no separate advantage, whereas γ = 60° should be avoided, because it gives the lowest Nusselt number without any corresponding reduction of the entropy generation. Finally, the Hartmann number should be selected from the cooling capacity that is actually required, because it reduces the heat transfer and the irreversibility together and does not provide a genuine thermodynamic optimum.

Figure 15 presents the average Nusselt number and the total entropy generation as a function of the inclination angle for four Hartmann numbers, at Ra=10⁶ and φ = 0.04. For every angle, the Nusselt number decreases when the Hartmann number increases. At γ = 0°, $\overline{Nu}$ falls from 4.25 at Ha=0 to 4.04, 3.67 and 3.32 at Ha=25, 50 and 75, a total reduction of 21.9%. The Lorentz force opposes the vertical velocity component and therefore weakens the circulation that transports the heat from the fins towards the cold walls, so that a larger part of the transfer must be achieved by conduction alone. The magnitude of this reduction, however, depends strongly on the orientation of the enclosure. The difference between the highest and the lowest bar of each group falls from 21.9% at γ = 0° to 12.7% at γ = 60° and 16.0% at γ = 120°.

Fig. 15.

Average Nusselt number (a) and total entropy generation (b) versus the inclination angle of the enclosure for Hartmann numbers Ha=0, 25, 50 and 75, at Ra=10⁶ and φ = 0.04

The same mechanism explains the effect of the inclination itself. Without a magnetic field the vertical arrangement gives the highest Nusselt number, and the value at γ = 120° is 2.1% lower than at γ = 0°. When a field is applied the situation is reversed, and the tilted enclosure becomes the more favourable one, by 5.0%, 4.6% and 7.2% at Ha=25, 50 and 75 respectively. Since there is nothing to avoid in the absence of a field, the aligned configuration remains the best; once a field is applied, the inclination protects part of the buoyancy force from the damping, and the benefit increases with the field strength. A related effect is visible at γ = 60° and γ = 120°, where the Nusselt number obtained at Ha=25 slightly exceeds the value obtained without a field (4.03 against 4.01, and 4.24 against 4.16). A weak field organises the circulation into a more coherent cell and suppresses the secondary vortices, which improves the transport towards the walls, but this is observed only when part of the driving force is already shielded from the damping.

The total entropy generation shown in Fig. 15 follows exactly the same trends. It decreases with the Hartmann number, from 33 at Ha=0 to 25.96 at Ha=75 for γ = 0°, and the corresponding curves become closer to each other as the enclosure is tilted. The maximum values of the two quantities are also obtained at the same conditions, $\overline{Nu}=4.25$ and S_T = 33 at γ = 0° and Ha=0, while the minimum entropy generation, S_T = 25.85, is obtained at γ = 30° and Ha=75, where the Nusselt number is also among the lowest. Thus, the most important result of Fig. 15 is the close correspondence between the two panels. Increasing the Hartmann number from 0 to 75 reduces the Nusselt number by 21.9% and the entropy generation by 21.3% at γ = 0°, by 12.2% and 11.9% at γ = 60°, and by 14.4% and 13.1% at γ = 120°.

Effect of the fin configuration

Figure 16 compares the four fin arrangements at the optimum operating point identified above, Ra=10⁶, Ha=25, γ = 120° and φ = 0.04. The reference configuration C1, which contains three fins, gives the highest average Nusselt number, $\overline{Nu}=4.24$, and also the highest entropy generation, S_T = 33.01. Removing one fin reduces both quantities, but by amounts that depend strongly on which fin is removed. Configuration C2, in which the top fin is removed, retains 67.5% of the heat transfer, whereas C3 and C4 retain only 46.9% and 49.3% respectively, although all three contain the same number of heated fins. The difference is therefore not caused by the reduced heated surface alone but by the position of the remaining fins. In C2 the two fins remain adjacent and continue to drive a single coherent circulation, while in C3 the unheated central region separates the two sources and the large-scale cell breaks into two weaker local vortices. In C4 the lateral displacement of the fins produces the same loss of coherence. The entropy generation follows the same order but decreases more rapidly. Relative to C1, the reductions are 34.9%, 62.8% and 55.5% for C2, C3 and C4, compared with reductions of 32.5%, 53.1% and 50.7% for the Nusselt number. The consequence is that the ecological coefficient of performance increases as fins are removed, from 0.1284 for C1 to 0.1331, 0.1619 and 0.1420 for C2, C3 and C4, that is an improvement of 3.7%, 26.1% and 10.6% respectively.

Fig. 16.

Bar charts of average Nusselt number, total entropy generation and ecological coefficient of performance for the four considered configurations at (Ra=10⁶, Ha=25, γ = 120˚ and φ = 4%)

To sum up, the choice of configuration consequently depends on the objective. C1 should be selected when the maximum heat removal is required, since it transfers more than twice the heat of C3. C3 should be selected when the irreversibility is to be minimised, since it produces the highest ECOP, although at less than half the cooling capacity. C2 offers the best compromise, retaining two thirds of the heat transfer of C1 with a slightly higher ECOP and the highest heat transfer per fin of all four arrangements.

Figure 17 presents the same results normalised with respect to configuration C1. Normalization is performed with respect to the average Nusselt number at the configuration C1, which is defined by the following equation:

$${\overline{Nu}}^{⁎}=\frac{\overline{Nu}(C_i)}{\overline{Nu}(C_1)}\mathrm{where}i=1\to 4.$$ 26

Fig. 17.

Average Nusselt number, total entropy generation and ecological coefficient of performance for the four fin configurations, normalised with respect to configuration C1, at Ra=10⁶, Ha=25, γ = 120° and φ = 0.04

This normalization allows us to compare the relative heat transfer augmentation or reduction at various fin configurations with respect to the reference position C1.

Likewise, normalization is reperformed with respect to the entropy generation and ecological coefficient of performance at the configuration C1, which are defined by the following equations:

$${S_T}^{⁎}=\frac{S_T(C_i)}{S_T(C_1)}\mathrm{where}i=1\to 4.$$ 27

$$ECO{P}^{⁎}=\frac{ECOP(C_i)}{ECOP(C_1)}\mathrm{where}i=1\to 4.$$ 28

The three quantities behave in two clearly different ways. The normalised Nusselt number and the normalised entropy generation both decrease as fins are removed, from unity to 0.675 and 0.651 for C2, to 0.469 and 0.372 for C3, and to 0.493 and 0.445 for C4. The normalised ECOP, in contrast, increases for all three reduced arrangements, reaching 1.037, 1.261 and 1.106 respectively.

The reason is contained in the vertical distance between the first two series. For each configuration the entropy generation falls below the Nusselt number, and the size of this gap determines the gain in ECOP. The gap is small for C2 (0.024), which explains its modest improvement of 3.7%, and largest for C3 (0.097), which gives the highest improvement of 26.1%. Wherever S_T* lies below $\overline{Nu}⁎$, the arrangement removes irreversibility faster than it removes heat transfer, and the ECOP exceeds that of the reference case.

It is notable that the ECOP of the three reduced configurations is higher than that of C1 in every case, so the fully populated arrangement is the least efficient of the four in thermodynamic terms even though it transfers the most heat. This indicates that each additional fin in a stacked array contributes less heat than the irreversibility it introduces, because it is placed in fluid that has already been warmed by its neighbours and therefore operates with a reduced effective temperature difference while still producing steep local gradients at its surface. Increasing the number of heat sources consequently gives diminishing thermodynamic returns, and the number of fins should be selected from the cooling capacity actually required rather than maximised.

Conclusion

The present numerical investigation has provided a comprehensive assessment of magnetohydrodynamic natural convection, heat transfer, and entropy generation in an inclined enclosure filled with a Fe₃O₄–MgO/water hybrid nanofluid and containing internally heated flower-shaped fins. By systematically varying the Rayleigh number (Ra), Hartmann number (Ha), nanoparticle volume fraction (φ), and inclination angle (γ), the complex interactions among buoyancy-driven flow, magnetic damping, nanoparticle-enhanced thermal transport, and heat-source arrangement have been clarified. Beyond the fundamental thermofluid insights, the results provide useful design guidance for thermal-management systems involving multiple localized heat-generating components, where heat removal can be regulated through the combined selection of nanofluid concentration, magnetic-field strength, enclosure orientation, and heat-source positioning. In particular, the present configuration can serve as an idealized representation of compact thermal-management systems containing multiple localized heat sources, such as closely packed electronic components or battery cells, where the heated flower-shaped fins represent distributed heat-generating regions and the cold enclosure walls represent a cooled surrounding environment. The use of a magnetically responsive hybrid nanofluid further provides an additional means of controlling buoyancy-driven circulation and thermal transport through an externally applied magnetic field. Thus, the findings provide a numerical framework for the design and optimization of compact thermal systems, including electronic cooling and battery thermal management, where controlling temperature distribution and thermodynamic irreversibility is essential for maintaining thermal performance and operational reliability.

Key findings can be summarized as follows:

The main findings can be summarised as follows.

1. The circulation intensity increases almost linearly with the Rayleigh number between Ra=10⁴ and 10⁵ and more slowly at Ra=10⁶, where inertia becomes significant. At the two lower Rayleigh numbers the three fins are joined by a continuous warm column and behave as a single thermally coupled group, whereas at Ra=10⁶ the gaps between them are ventilated and the fins become independent heat sources.

2. The magnetic field reduces the circulation by 69% between Ha=0 and 75 and it also modifies the flow topology: the two large cells observed at low field strength break into six small vortices confined to the neighbourhood of each fin, which suppresses the long-range transport between the heat sources.

3. Because the Lorentz force acts only on the axial velocity component, the inclination angle transfers part of the buoyancy force into the undamped direction. In the absence of a field the vertical enclosure gives the highest Nusselt number, whereas under a field the tilted enclosure becomes superior, the advantage increasing from 5% at Ha=25 to 7.2% at Ha=75. The sensitivity to the field is correspondingly reduced, the spread between the Hartmann numbers falling from 21.9% at γ = 0° to 12.7% at γ = 60°.

4. A weak magnetic field can slightly improve the heat transfer when the enclosure is tilted. At γ = 60° and 120° the values obtained at Ha=25 exceed those obtained without a field, as the field organises the circulation into a more coherent cell.

5. Thermal irreversibility is dominant under all conditions, the local peak of the total entropy generation being equal to that of its thermal component, and the Bejan number remaining close to unity. The viscous contribution exceeds the magnetic one by about one order of magnitude, and both remain negligible compared with the thermal contribution.

6. The maximum heat transfer, $\overline{Nu}=4.25$, is obtained at Ra=10⁶, φ = 0.04, Ha=0 and γ = 0°, with $\overline{Nu}$= 4.24 at Ha=25 and γ = 120° when a magnetic field is applied. The minimum entropy generation, S_T = 25.85, is obtained at Ha=75 with little dependence on the inclination angle.

7. The ratio $\overline{Nu}/S_T$ remains between 0.128 and 0.130 for all the values of the Rayleigh number, Hartmann number, inclination angle and volume fraction considered. The magnetic field, the inclination and the nanoparticle concentration therefore reduce or restore the heat transfer and the irreversibility in the same proportion, and none of them provides a genuine thermodynamic optimum.

8. The fin arrangement is the only parameter that modifies this ratio appreciably. The ECOP increases from 0.1284 for C1 to 0.1331, 0.1619 and 0.1420 for C2, C3 and C4, an improvement of up to 26%. Configuration C1 gives the highest heat transfer, C3 the highest thermodynamic efficiency at less than half the cooling capacity, and C2 the best compromise, retaining 67.5% of the heat transfer of C1 with the highest heat transfer per fin of all four arrangements.

Model limitations and future perspectives

Although the present numerical framework provides useful insights into MHD natural convection and entropy generation in Fe₃O₄–MgO/water hybrid nanofluids, several limitations should be acknowledged. The hybrid nanofluid is modeled using a homogeneous single-phase approach with constant thermophysical properties, and therefore particle migration, agglomeration, sedimentation, and detailed particle–fluid interactions are not explicitly considered. In addition, the induced magnetic field, Joule heating, viscous dissipation, and thermal radiation are neglected within the assumptions of the present model. The results are also obtained for a two-dimensional configuration and should therefore be interpreted within the considered geometric and operating ranges. Future studies could extend the present work through two-phase or multiphase modeling, temperature-dependent and concentration-dependent thermophysical properties, and consideration of nanoparticle stability and agglomeration effects. Further investigations could also examine three-dimensional configurations, alternative magnetic-field orientations, and multi-objective optimization of heat-transfer enhancement and entropy generation to provide additional guidance for practical thermal-management applications.

Nomenclature

Latin Symbols

▓

B

Magnetic field intensity (T)

Cp

Specific heat capacity (J/kg·K)

g

Gravitational acceleration (m/s²)

k

Thermal conductivity (W/m·K)

L

Characteristic length of the cross-shaped cavity (m)

Ha

Hartmann number

$\overline{Nu}$

Average Nusselt number

n

Normal direction

Pr

Prandtl number

P

Reference pressure scale (Pa)

q

Heat flux vector (W/m²)

Ra

Rayleigh number

Tc

Cold wall temperature (K)

Th

Hot wall temperature (K)

T

Temperature (K)

t

Dimensional time (s)

(U,V)

Dimensionless velocity components

(X,Y)

Dimensionless Cartesian coordinates

Greek Symbols

▓

φ

Solid volume fraction

α

Thermal diffusivity (m²/s)

θ

Dimensionless temperature

ν

Kinematic viscosity (m²/s)

μ

Dynamic viscosity (kg/(m·s))

ρ

Fluid density (kg/m³)

β

Thermal expansion coefficient (K⁻¹)

Ψ

Irreversibility coefficient

Ω

Global domain

$\sigma$

Electrical conductivity (µS/cm)

τ

Dimensionless time parameter

S

Entropy generation (Wm^-2K^-1)

$\eta$

Irreversibility factor

Subscripts

▓

c

Cold

h

Hot

loc

Local

tot

Total

Author contributions

Project administration; Basma Souayeh; Writing—review and editing: Basma Souayeh and Najib Hdhiri; Investigation; Basma Souayeh; Methodology; Basma Souayeh; Supervision: Basma Souayeh; Validation: Basma Souayeh and Najib Hdhiri.

Funding

This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [Grant No. KFU264784].

Data availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

Ethics approval

Not applicable.

Consent to participate

Not applicable.

Consent for publication

Not applicable.