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Impacts of sinusoidal heat flux and embraced heated rectangular cavity on natural convection within a square enclosure partially filled with porous medium and Casson-hybrid nanofluid

  • Anil Ahlawat , Mukesh Kumar Sharma EMAIL logo , Kaouther Ghachem , Badr M. Alshammari and Lioua Kolsi EMAIL logo
Published/Copyright: May 5, 2025
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Open Physics
From the journal Open Physics Volume 23 Issue 1

Abstract

In the current research, the influence of a sinusoidal heat flux and a heated rectangular cavity in a partially porous Casson hybrid nanofluid-filled square enclosure being affected by a constant magnetic field imposed in the x-direction having magnitude B 0, on natural convection were investigated numerically. A rectangular heated cavity having width H/5 and height H/2 was positioned in the middle of the square enclosure. The sinusoidal heat flux of length H/2 is positioned centrally at the enclosure’s bottom wall, although the top wall is at a lower temperature, say T c, and the rest of the enclosure was insulated. The finite-difference method, combined with the successive over relaxation, successive under relaxation, and Gauss–Siedel techniques, is employed to address the challenge of solving the nonlinear coupled governing partial differential equations of motion and energy. Numerical calculations were carried out using the values of the relevant parameters in the range 0 ≤ Ha ≤ 15, 104 ≤ Ra ≤ 106, 0.00 ≤ ϕ hnf ≤ 0.04 , 10−5 ≤ Da ≤ 10−3, 0.1 ≤ γ ≤ 1, 4 ≤ N ≤ 12. The Kirpichev heat number ( K i = 1 ) and Prandtl number (Pr = 6.26) are fixed throughout the study. Contour plots for isotherms and streamlines were plotted to ascertain the impacts of distinct factors on them. The results of this investigation suggest that the heat transfer rate rises as ϕ hnf , Ra, and γ rise and reduce as the Ha and N rises. Finally, it has been observed that decreasing the heated rectangular cavity’s aspect ratio leads to a substantial enhancement in heat transfer from both the heat flux and the heated rectangular cavity.

Keywords: natural convection; Casson fluid; hybrid nanofluid; heated rectangular cavity; average Nusselt number

Nomenclature

B 0

magnetic field intensity (T)

C p

specific heat per unit mass ( J kg − 1 K − 1 )

Nu Local hb

local Nusselt number at inner heated rectangular cavity

Nu Local hf

local Nusselt number at heat flux

Nu avg hb

average Nusselt number at inner heated rectangular cavity

Nu avg hf

average Nusselt number at heat flux

Da

Darcy number

g →

gravitational acceleration ( m s − 2 )

H

length and width of square cavity ( m )

Ha

Hartmann number

k

thermal conductivity ( W m − 1 K − 1 )

K

permeability of porous medium

K i

Kirpichev heat number

N

periodicity parameter

P

dimensionless pressure

p

pressure ( Pa )

Pr

Prandtl number

q →

velocity vector

Ra

Rayleigh number

T

dimensional temperature ( K )

u , U

dimensional and dimensionless velocity along the x-axis ( m s − 1 )

v , V

dimensional and dimensionless velocity along the y-axis ( m s − 1 )

X , Y

dimensionless Cartesian co-ordinates

x , y

dimensional Cartesian co-ordinates (m)

ε

porosity

Greek symbols

α

thermal diffusivity ( m 2 s − 1 )

β

volumetric thermal expansion coefficient ( K − 1 )

γ

Casson fluid parameter

μ

dynamic viscosity ( k g m − 1 s − 1 )

ν

kinematic viscosity ( m − 2 s − 1 )

ρ

density ( k g m − 3 )

ϕ

volume concentration of nanoparticles

ω

dimensionless vorticity function

ψ

dimensionless stream function

θ

dimensionless temperature

Subscripts

Al2O3

aluminium oxide

Cu

copper

eff

effective

f

base fluid

hnf

hybrid nanofluid

p

porous medium

Abbreviations

AR

aspect ratio

FDM

finite difference method

HTFs

heat transfer fluids

PDEs

partial differential equations

SOR

successive over relaxation

SUR

successive under relaxation

1 Introduction

Convective heat transfer through enclosures (open or closed) is gaining significant interest in light of its numerous real-world applications, which range from everyday used thermal equipment to the cooling of buildings and electrical components. Heat storage capabilities of a thermal system affect its thermal performance. In thermal systems, heat-transfer fluids (HTFs) are being utilized for heat transportation. The design of the thermal system, the stability of the fluid, and its existence are all major factors in selecting HTFs. Conventional HTFs, such as water, air, vapor, natural oils, etc., have been used for a long time, but the poor thermal conductivity of these conventional HTFs limits them to low heat convection. Choi and Eastman [1] invented an alternate known as “nanofluids” to strengthen the thermal conductivity of these HTFs by incorporating nanoparticles of metal and metal oxides (size of particles less than or equal to 100 nm) into a base fluid. In recent decades, a significant number of studies [2–8] on the preparation and implementations of nanofluids has been documented in the published literature. In modification of the “thermal conductivity” of convectional HTFs beyond the single material type nanofluid, the hybrid nanofluids are synthesized. The augmentation in thermal conductivity of fluid utilizing hybrid nanoparticles was initially examined by Jana et al. [9] experimentally.

Heat transfer and flow properties of a nanofluid contained in an oblique square cavity were examined by Al Kalbani et al. [10] who observed that with increasing Ra and volume percentage of nanoparticles, the heat transmission rate rises and also found that the heat transfer process with blade-shaped nanoparticle is faster than any other shapes. Mansour et al. [11] and Rashad et al. [12] explored the effects of the size and positioning of a heat sink and source on natural convection within a porous enclosure filled with different types of nanofluids. Furthermore, Armaghani et al. [13] explored the intriguing effects of discrete heat source placement on heat transfer and entropy generation within an open inclined L-shaped cavity filled with Ag–water nanofluid. Influence of a stationary magnetic field on the enhancement of heat convection in a rectangular enclosure having diagonal vents, filled with copper–water nanofluid, four individual heaters mounted in the middle of the enclosure walls, and embedded with square-shaped heated block numerically examined by Ushachew et al. [14]. They noted that heat convection escalates on the surging concentration of nanoparticles, whereas attenuated with the rise in magnetic field strength. Nishad et al. [15] determined the heat convection and flow characteristics of Cu–water nanofluid flowing through a wavy enclosure and observed improvement in heat convection with increasing nanoparticles’ proportion and Rayleigh number, whereas declined with the rise in Hartmann number. The primary cause of the rise in system’s temperature is the waste of energy supplied to the system during the heat transfer process. The optimal and efficient utilization of renewable energy has constantly been the primary objective throughout both industrial and academic communities. Porous media are often used to boost heat convection. Nanofluid flow in cavities that are filled with porous materials, either partially or entirely, has been the primary objective of studies conducted by Sun and Pop [16], Chamkha and Ismael [17], and Sheremet et al. [18]. The characteristics of nanofluid flow and the transfer of heat in a foam-filled cylinder under radial injection were investigated by Sharma et al. [19]. Vedavathi et al. [20] and Venkatadri et al. [21] studied the natural convection phenomenon in a semi-trapezoidal/right-angle trapezoidal porous cavity employing “Darcy–Boussinesq approximation and Tiwari and Das’ nanofluid model.” Venkatadri et al. [22] investigated hydromagnetic laminar natural convection in a quadrant-shaped enclosure with an electrically conductive liquid under radiative effects utilizing finite difference methodology and revealed that the Nusselt number increases with higher Rayleigh number and radiative parameter, while it significantly decreases with a greater Hartmann number.

Rheological characteristics of non-Newtonian fluids are positively influencing their utilization in engineering, industry, biology, and other fields of science. The study of non-Newtonian fluids is problematic for researchers due to the nonlinear character of equations. There are various noteworthy non-Newtonian fluid models, but one in particular “Casson fluid model” introduced by Casson [23] is defined as “the liquid which is supposed to have an infinite viscosity at zero rate of shear and zero viscosity at infinite rate of shear.” Honey, tomato sauce, jelly, melted chocolates, etc., are different types of real liquids that can be accurately described by the Casson fluid model. Pop and Sheremet [24] performed a computational analysis to examine the phenomenon of natural convection within a square enclosure exposed to differential heating and filled using Casson fluid while considering the impacts of “thermal radiation and viscous dissipation” and observed that the Casson parameter increases heat convection whereas, with a rise in Eckert number, heat transport is reduced.

Hamid et al. [25] discussed the heat convection and flow simulations of Casson fluid contained in a trapezoidal cavity, subjected to partial heating from the bottom side and determined that the velocity of the flowing fluid declines and heat convection enhances as the Casson fluid parameter increases. Aghighi et al. [26] employed Casson fluid to examine the process of free convection within a square cavity. Aneja et al. [27] employed a porous square enclosure containing Casson fluid to perform numerical investigations on natural convection processes. Devi et al. [28] carried out a computational study to analyze the properties of natural convection and flow of fluid in a square cavity filled with Casson fluid. Sivasankaran et al. [29] performed simulations numerically in order to learn how thermal radiation and porous region affects the process of heat convection and flow pattern in a Casson fluid-filled square cavity. The Brinkman model was utilized by Yasmin et al. [30] to describe the thermal and flow properties of a magnetohydrodynamic Casson nanofluid inside a non-uniformly heated square cavity. Munir and Turabi [31] examined the effects of a non-planar heated bottom wall, hybrid nanofluid, and inclined magnetic field on the natural convection within a triangular cavity having a cooled cylinder and revealed that heat transfer rate enhanced with an increase in undulations of non-planar wall and higher wave amplitude but diminishes with an increase in magnetic field strength. In recent studies, Hussain et al. [32], Ganesh et al. [33], and Turabi and Munir [34] employed the Casson fluid model to understand heat convection in different enclosures.

Not enough is being done in this research area to meet the standards of current engineering and technology requirements. In light of the reviewed literature, the present study is inspired to extend the applications of Casson fluid in natural convection by employing a hybrid nanofluid. To the extent of our best understanding, an investigation of this kind has never been carried out in the past. So, this study aims to fill that gap by examining the dual heating mechanism, featuring both sinusoidal heat flux and an embraced heated rectangular cavity, which profoundly impacts the natural convection process. A key highlight of this work is its investigation into the interaction between partially porous media and fluid flow, shedding light on the intricate thermal and hydrodynamic behaviors within porous structures. This configuration, with its ability to introduce thermal resistance and dampen flow, significantly influences convection patterns. These findings hold critical implications for the design optimization of advanced energy storage systems and heat exchangers. The proposed model serves as a conceptual framework for simulating heat exchange systems in automobiles, where filters act as porous media, a heated shaft is immersed in lubricant, and heat flux arises from the surroundings. Also, these innovative aspects are pivotal for advancing thermal management systems across various engineering applications, such as cooling technologies and environmental engineering, by optimizing heat transfer processes in intricate fluid and geometric configurations. Therefore, this work helps to ensure efficient and sustainable thermal solutions. The primary concerns motivating this current research are delineated as follows:

  • How do variations in the porosity of the medium and the hybrid nanofluid composition impact the convective heat transfer coefficient and flow dynamics?

  • How does Casson fluid dynamics influence the efficiency of heat transfer?

  • What role does the heated rectangular cavity play in enhancing or modifying the heat transfer characteristics within the enclosure?

  • What are the effects of sinusoidal heat flux on the natural convection and heat transfer performance within a square enclosure partially filled with a porous medium and Casson-hybrid nanofluid?

2 Mathematical model

A graphical outline of the current study under investigation is depicted in Figure 1, along with the associated grid generation process. The side length of the square enclosure is H. A layer with a porous structure having H width and H/2 height is located adjacent to the lower wall of the enclosure. A rectangular cavity of sides H/2 and H/5 heated at temperature T h is positioned in the center, inside the enclosure. A sinusoidal heat flux of H/2 length is positioned in the center of the lower wall, and the top wall is kept at the temperature T c (T c < T h), whereas adjacent vertical walls are insulated. The porous stratum at the fluid interface is supposed to be permeable, allowing crossflow from fluid-to-porous medium. The Casson fluid model was integrated into this study to represent the non-Newtonian characteristics of the working fluid. Casson fluids exhibit a shear-thinning behavior with a defined yield stress. This model is particularly relevant for engineering and industrial applications involving complex fluids such as lubricants, biological fluids, and polymer solutions. By combining the Casson fluid model with the hybrid nanofluid, the study provides a framework to explore advanced thermal fluids that not only enhance thermal conductivity through nanoparticles (Cu and Al2O3) but also adapt to varying shear conditions for improved flow and heat transfer control. The current mathematical model for natural convective systems is formulated under the following assumptions:

  • The base fluid (H2O) and nanoparticles (Cu and Al2O3) are in the state of thermal stability.

  • The enclosure occupies single-phase H2O-based Casson hybrid nanofluid, and the lower half region of the enclosure is filled with a porous medium that is saturated with the same hybrid nanofluid.

  • The flow is laminar, two-dimensional, steady, and incompressible.

  • The density variation is modeled using the Boussinesq approximation, while the remaining thermophysical properties are assumed constant and provided in Table 1.

  • The influences of viscous dissipation and Joule heating are considered negligible throughout this study.

  • The enclosure is divided into two equal-sized regions, with the lower region fully occupied by a Darcian, incompressible, homogeneous, and isotropic porous medium.

  • The solid boundaries are assumed to satisfy the no-slip condition.

  • The influences of Brownian motion and thermophoresis are deemed insignificant.

  • A static magnetic field of strength B 0 is aligned horizontally.

Figure 1 Representation of physical domain (a) and view of its grid’s discretization 121 × \times 121 (b).
Figure 1

Representation of physical domain (a) and view of its grid’s discretization 121 × 121 (b).

Table 1

Thermo-physical properties of nanoparticles (Al2O3 and Cu) and base fluid (H2O) [35,36]

Property H2O Cu Al2O3
ρ (kg m−3) 997.1 8,933 3,970
c p (J kg−1 K−1) 4,179 385 765
k (W m−1 K−1) 0.613 401 40
β (K−1) 21 × 10−5 1.67 × 10−5 0.85 × 10−5

The thermophysical properties of the Casson-hybrid nanofluid were derived by incorporating empirical and theoretical formulations. The viscosity of the fluid was modeled to reflect its dependence on the shear rate, as governed by the Casson model, while the effective thermal conductivity was calculated using established hybrid nanofluid models. This integration allows for a comprehensive examination of the interaction between non-Newtonian behavior, enhanced thermal conductivity, and natural convection within the partially porous enclosure. The constitutive equation for an isotropic and incompressible Casson fluid flow is given as follows [28]:

(2.1) τ ij = 2 μ B + p y 2 π e ij , π > π c , 2 μ B + p y 2 π c e ij , π < π c .

Here, π = e ij e ij and e ij are the ( i , j ) th components of the rate of strain tensor, π c is the critical value that depends on the type of non-Newtonian model, μ B is the dynamic viscosity of Casson fluid, and p y is the yield stress.

2.1 Governing equations

The equations that govern the flow in their dimensionless form, incorporating aforementioned assumptions and the dimensionless parameters from Aneja et al. [27] and defined in Eq. (2.2), are written together for the porous and fluidic regions in Eqs. (2.3)–(2.6) in accordance with Sivasankaran et al. [29]:

(2.2) X = x H , Y = y H , U p , hnf = u p , hnf H α f , V p , hnf = v p , hnf H α f , θ hnf = T hnf − T c T h − T c ,   θ p = T p − T c T h − T c , Pr = ν f α f , Da = K H 2 , Ra = g β f ( T h − T c ) H 3 ν f α f , P = p H 2 ρ f α f 2 , Ha = B 0 H σ f μ f , γ = μ B 2 π c ∂ p ∂ y ,

where Pr, Da, K, Ha, γ , and Ra are Prandtl number, Darcy number, permeability of porous medium, Hartmann number, Casson fluid parameter, and Rayleigh number, respectively, and the subscripts “f” used for fluid, “p” for porous, and “hnf” for hybrid nanofluid, and q → p , hnf = ( U p , hnf , V p , hnf ) is the velocity vector. The Rayleigh number (Ra) highlights the competition between buoyancy and thermal diffusion, influencing convection intensity. The Hartmann number (Ha) captures magnetic field effects, where higher values suppress flow through Lorentz forces. The Darcy number (Da) signifies porous medium permeability, affecting fluid flow and heat transfer. The Casson fluid parameter (γ) reflects the non-Newtonian shear-thinning behavior, which is critical for advanced fluids. The Prandtl number (Pr) links momentum and thermal diffusivity, while nanoparticle volume fraction ( ϕ hnf ) quantifies enhancements in thermal conductivity.

Continuity equation

(2.3) ∇ · q → p , hnf = 0 .

Momentum equations

(2.4) ( q → p , hnf · ∇ ) U p , hnf = − ε 2 ρ f ρ hnf ∂ P ∂ X + ε Pr μ hnf μ f ρ f ρ hnf 1 + 1 γ ∂ 2 ∂ X 2 ( U p , hnf ) + ∂ 2 ∂ Y 2 ( U p , hnf ) − δ μ hnf μ f ρ f ρ hnf ε 2 Pr Da U p , hnf ,

(2.5) ( q → p , hnf · ∇ ) V p , hnf = − ε 2 ρ f ρ hnf ∂ P ∂ Y + ε Pr μ hnf μ f ρ f ρ hnf 1 + 1 γ ∂ 2 ∂ X 2 ( V p , hnf ) + ∂ 2 ∂ Y 2 ( V p , hnf ) − δ μ hnf μ f ρ f ρ hnf ε 2 Pr Da V p , hnf + ( ρ β ) hnf ρ hnf β f ε 2 PrRa θ p , hnf − ρ f ρ hnf σ hnf σ f ε 2 Ha 2 Pr V p , hnf .

Energy equation

(2.6) ( q → p , hnf · ∇ ) θ p , hnf = α * α f ∂ 2 ∂ X 2 ( θ p , hnf ) + ∂ 2 ∂ Y 2 ( θ p , hnf ) ,

where ∇ = ∂ ∂ X i ˆ + ∂ ∂ Y j ˆ is the gradient operator. The region of hybrid nanofluid and the region of fluid-saturated porous medium are identified as

For clear hybrid nanofluid domain: ε = 1 , δ = 0 , α * = α hnf , q → p , hnf = ( U hnf , V hnf ) , ω p , hnf = ω hnf , ψ p , hnf = ψ hnf and for fluid-saturated porous regions: ε = ε , δ = 1 , α * = α eff , q → p , hnf = ( U p , V p ) , ω p , hnf = ω p , ψ p , hnf = ψ p . The porous medium’s porosity ( ε ) is constant and taken ε = 0.398 throughout the study. Thermophysical properties of the hybrid nanofluid, in relation to nanoparticle volume concentration ( ϕ Al 2 O 3 and ϕ cu ), are described as follows [37]:

(2.7) ϕ hnf = ϕ Cu + ϕ Al 2 O 3 ρ hnf = ( 1 − ϕ hnf ) ρ f + ϕ Cu ρ Cu + ϕ Al 2 O 3 ρ Al 2 O 3 ( ρ β ) hnf = ( 1 − ϕ hnf ) ρ f + ϕ Cu ( ρ β ) Cu + ϕ Al 2 O 3 ( ρ β ) Al 2 O 3 ( ρ C p ) hnf = ( 1 − ϕ hnf ) ρ f + ϕ Cu ( ρ C p ) Cu + ϕ Al 2 O 3 ( ρ C p ) Al 2 O 3 α hnf = k hnf ( ρ C p ) hnf ; α eff = k eff ( ρ C p ) hnf where k eff = ( 1 − ε ) k s + ε k hnf .

where ( ρ C p ) hnf is the heat capacity of hybrid nanofluid. In order to provide a description of the expression for the thermal conductivity of hybrid nanofluids ( k hnf ), the Maxwell [38] model is utilized, and the corresponding expression is given as follows:

(2.8) k hnf = ϕ Cu k Cu + ϕ Al 2 O 3 k Al 2 O 3 ϕ hnf + 2 k f + 2 ( ϕ Cu k Cu + ϕ Al 2 O 3 k Al 2 O 3 ) − 2 ϕ hnf k f ϕ Cu k Cu + ϕ Al 2 O 3 k Al 2 O 3 ϕ hnf + 2 k f − ( ϕ Cu k Cu + ϕ Al 2 O 3 k Al 2 O 3 ) + ϕ hnf k f k f .

The viscosity of hybrid nanofluids ( μ hnf ) is determined using the following Eq. (2.9), which is based on the Brinkman [39] model

(2.9) μ hnf = μ f ( 1 − ϕ hnf ) 2.5 .

The pressure gradient term from the momentum equations is removed with the stream function defined as U p , hnf = ∂ ∂ Y ( ψ p , hnf ) , V p , hnf = − ∂ ∂ X ( ψ p , hnf ) and − ω p , hnf = ∂ ∂ Y ( U p , hnf ) − ∂ ∂ X ( V p , hnf ) , then resultant equations in the form of stream function ( ψ ) and vorticity ( ω ) are given in the following equations:

Equation of continuity

(2.10) ∇ 2 ψ p , hnf = − ω p , hnf .

Momentum equation

(2.11) ∂ ∂ Y ( ψ p , hnf ) ∂ ∂ X ( ω p , hnf ) − ∂ ∂ X ( ψ p , hnf ) ∂ ∂ Y ( ω p , hnf ) = ε Pr μ hnf μ f ρ f ρ hnf 1 + 1 γ ∇ 2 ω p , hnf − μ hnf μ f ρ f ρ hnf δ ε 2 Pr Da ω p , hnf + Ha 2 Pr ε 2 σ hnf σ f ρ f ρ hnf ∂ 2 ∂ X 2 ( ψ p , hnf ) + ( ρ β ) hnf ρ hnf β f ε 2 PrRa ∂ ∂ X ( θ p , hnf ) .

Energy equation

(2.12) ∂ ∂ Y ( ψ p , hnf ) ∂ ∂ X ( θ p , hnf ) − ∂ ∂ X ( ψ p , hnf ) ∂ ∂ Y ( θ p , hnf ) = α * α f ( ∇ 2 θ p , hnf ) ,

where ∇ 2 = ∂ 2 ∂ X 2 + ∂ 2 ∂ Y 2 is the Laplacian operator.

The reduced non-dimensional boundary conditions (BCs) on enclosure’s wall are

(2.13) ψ hnf = 0 , θ hnf = 0 , ω hnf = − ∂ 2 ψ hnf ∂ Y 2 at Y = 1 and ψ p = 0 , ω p = − ∂ 2 ψ p ∂ Y 2 at Y = 0 ψ p , hnf = 0 , ∂ ∂ X ( θ p , hnf ) = 0 , and ω p , hnf = − ∂ 2 ∂ X 2 ( ψ p , hnf ) at X = 0 , 1 ∂ θ p ∂ Y = − K i k f k eff sin ( N π X ) , 0.25 ≤ X ≤ 0.75 , and Y = 0 ∂ θ p ∂ Y = 0 , at 0 ≤ X ≤ 0.25, 0.75 < X ≤ 1 , and Y = 0 .

The BCs at the sides of the inner heated rectangular cavity are as follows:

(2.14) θ p , hnf ( X , Y ) = 1 , at Y = 0.25 , 0.75 where 0.4 ≤ X ≤ 0.6 θ p , hnf ( X , Y ) = 1 , at X = 0.4 , 0.6 where 0.25 ≤ Y ≤ 0.75 .

Kirpichev heat number ( K i = q ″ H / ( k f Δ T ) ) is considered to be equal to one throughout all calculations. The BCs at the interface are obtained by equating “the tangential velocities, normal velocities, shear and normal stresses, temperature, and the heat flux across the interface, and considering same dynamic viscosity ( μ p = μ hnf ) in both porous and fluid regions, as explained by Chamkha and Ismael [40].” Hence, interface BCs at Y = 0.5 reduces to

(2.15) θ hnf = θ p ; ψ hnf = ψ p ; ω hnf = ω p k hnf k eff ∂ θ hnf ∂ Y = ∂ θ p ∂ Y ; ∂ ψ hnf ∂ Y = ∂ ψ p ∂ Y ; ∂ ω hnf ∂ Y = ∂ ω p ∂ Y .

2.2 Local and average Nusselt number

Local and average Nusselt numbers are estimated together across the heat flux and at the implanted heated rectangular cavity by utilizing the formulas given in the following equations:

Local Nusselt number ( Nu Local hf ) at the heat flux:

(2.16) Nu Local hf = K i k eff k f sin ( N π X ) θ p .

Average Nusselt number ( Nu avg hf ) at the heat flux:

(2.17) Nu avg hf = 1 0.5 ∫ 0.25 0.75 Nu Local hf d X .

Local Nusselt number at the portion of inner heated rectangular cavity embraced in the porous layer:

(2.18) Nu Local hb = k eff k f ∂ θ p ∂ X or Nu Local hb = k eff k f ∂ θ p ∂ Y .

Local Nusselt number at the portion of the inner heated rectangular cavity lies in the clear fluid layer:

(2.19) Nu Local hb = k hnf k f ∂ θ hnf ∂ X or Nu Local hb = k hnf k f ∂ θ hnf ∂ Y .

The “average Nusselt number” at the inner heated rectangular cavity ( Nu avg hb ) is determined by using the formula defined in the following equation:

(2.20) Nu avg hb = 1 0.25 k eff k f ∫ 0.25 0.50 ∂ θ p ∂ X X = 0.4 d Y + 1 0.25 k hnf k f ∫ 0.50 0.75 ∂ θ hnf ∂ X X = 0.4 d Y + 1 0.25 k eff k f ∫ 0.25 0.50 ∂ θ p ∂ X X = 0.6 d Y + 1 0.25 k hnf k f ∫ 0.50 0.75 ∂ θ hnf ∂ X X = 0.6 d Y + 1 0.2 k eff k f ∫ 0.40 0.60 ∂ θ p ∂ Y Y = 0.25 d X + 1 0.2 k hnf k f ∫ 0.40 0.60 ∂ θ hnf ∂ Y Y = 0.75 d X .

3 Numerical solution methodology

Several advanced numerical techniques have been designed to decode the complexities of fluid flow and heat transfer dynamics within the cavity. This investigation utilizes the finite difference discretization method to discretize governing Eqs. (2.10)–(2.12) in association with respective BCs. The resulting discretized equations are as follows:

(3.1) 1 h 2 ( ψ p , hnf ( i + 1 , j ) + ψ p , hnf ( i − 1 , j ) − 4 ψ p , hnf ( i , j ) + ψ p , hnf ( i , j + 1 ) + ψ p , hnf ( i , j − 1 ) ) = − ω p , hnf ( i , j ) ,

(3.2) 1 4 h 2 ( ( ψ p , hnf ( i , j + 1 ) − ψ p , hnf ( i , j − 1 ) ) ( ω p , hnf ( i + 1 , j ) − ω p , hnf ( i − 1 , j ) ) ) − 1 4 h 2 ( ( ψ p , hnf ( i + 1 , j ) − ψ p , hnf ( i − 1 , j ) ) ( ω p , hnf ( i , j + 1 ) − ω p , hnf ( i , j − 1 ) ) ) = ε Pr μ hnf μ f ρ f ρ hnf 1 + 1 γ 1 h 2 ( ω p , hnf ( i + 1 , j ) + ω p , hnf ( i − 1 , j ) − 4 ω p , hnf ( i , j ) + ω p , hnf ( i , j + 1 ) + ω p , hnf ( i , j − 1 ) ) − μ hnf μ f ρ f ρ hnf δ ε 2 Pr Da ω p , hnf ( i , j ) + ( ρ β ) hnf ρ hnf β f ε 2 PrRa 1 2 h ( θ i + 1 , j − θ i − 1 , j ) + Ha 2 Pr ε 2 σ hnf σ f ρ f ρ hnf 1 h 2 ( ψ p , hnf ( i + 1 , j ) − 2 ψ p , hnf ( i , j ) + ψ p , hnf ( i − 1 , j ) ) ,

(3.3) 1 4 h 2 ( ( ψ p , hnf ( i , j + 1 ) − ψ p , hnf ( i , j − 1 ) ) ( θ p , hnf ( i + 1 , j ) − θ p , hnf ( i − 1 , j ) ) ) − 1 4 h 2 ( ( ψ p , hnf ( i + 1 , j ) − ψ p , hnf ( i − 1 , j ) ) ( θ p , hnf ( i , j + 1 ) − θ p , hnf ( i , j − 1 ) ) ) = α * α f 1 h 2 ( θ p , hnf ( i + 1 , j ) + θ p , hnf ( i − 1 , j ) − 4 θ p , hnf ( i , j ) + θ p , hnf ( i , j + 1 ) + θ p , hnf ( i , j − 1 ) ) .

Here, i and j denote the node points in x and y directions, respectively, and h is the grid space in x and y directions.

The diffusion terms are estimated using second-order central differencing, while the upwind scheme was implemented for controlling the convective terms in the temperature and linear momentum equations. These equations serve as a foundation for calculating the solution at the interior points of the computational domain. Furthermore, a detailed methodology for calculating the vorticity at the solid walls of the enclosure and the heated rectangular cavity has been included and depicted in Figure 2(a)–(c). This detailed approach ensures accuracy in capturing the BCs and enhances the overall numerical framework. The resulting difference equations (Eqs. (3.1)–(3.3)) together with BCs, as illustrated in Figure 2, are computationally solved employing finite difference method in combination with iterative techniques. Specifically, the successive over relaxation method is utilized for the Poison equation, the under relaxation method is utilized for the vorticity, and the Gauss–Seidel method is employed to solve the energy equation. Iterations continued until the condition ∑ i , j | φ i , j k + 1 − φ i , j k | ∑ i , j | φ i , j k + 1 | ≤ 10 − 7 was satisfied, where φ stands for any measured value from ψ , ω , and θ . The simulation was carried out over a maximum of 104 iterations, requiring approximately 11 min of CPU time, allowing a residual error threshold of 10−7. Furthermore, Figure 3 has been included to provide a clear visualization of the residual error trend with the number of iterations across these numerical techniques. Also, a flow chart of the computational program, offering a clear overview of the problem-solving process, is given in Figure 4. Computational calculations were carried out by utilizing the especially self-developed MATLAB code. The accuracy of self-developed codes was verified using a specific outcome of Chamkha and Ismael [41] and Jani et al. [42]. Figure 5(a) exhibits a comparison of the streamlines and isotherm contours produced by self-developed in-house MATLAB programs with the findings of Chamkha and Ismael [41] and Figure 5(b), local Nusselt number variation is examined with outcomes of Jani et al. [42]. The excellent correlation between the outcomes confirms the validity of our simulation. The grid independency of the self-developed codes was examined by computing average Nusselt number and given in Table 2. Hence, in order to obtain the required results, the 101 × 101 grids are used.

Figure 2 Calculation procedure for vorticity at solid walls as explained in steps (a), (b) and (c).
Figure 2

Calculation procedure for vorticity at solid walls as explained in steps (a), (b) and (c).

Figure 3 Residual error plotted against the number of iterations for various numerical techniques.
Figure 3

Residual error plotted against the number of iterations for various numerical techniques.

Figure 4 Flow chart of the solution procedure.
Figure 4

Flow chart of the solution procedure.

Figure 5 Validation of self-developed MATLAB codes with specific outcomes of Chamkha and Ismael [41] (a) at Ra = 800 and Kr = 1 and with Jani et al. [42] (b), at different Rayleigh number (Ra) with Ha = 0.
Figure 5

Validation of self-developed MATLAB codes with specific outcomes of Chamkha and Ismael [41] (a) at Ra = 800 and Kr = 1 and with Jani et al. [42] (b), at different Rayleigh number (Ra) with Ha = 0.

Table 2

Average Nusselt number at heat flux ( Nu avg hf ) and heated rectangular block ( Nu avg hb ) to check the grid independency of codes when Da = 10−3, Ra = 105, Ha = 5, γ = 0.5 , ε = 0.398 , K i = 1.0 , N = 4 , Pr = 6.26 , ϕ hnf = 0.04

Grid size 81 × 81 101 × 101 121 × 121 141 × 141
Nu avg hf 1.0328 1.0591 1.0585 1.0584
Nu avg hb 19.6635 19.9259 19.9248 19.9242

4 Results and discussion

In the present study, the impacts of a sinusoidal heat flux and a heated rectangular cavity in a partially porous Casson hybrid nanofluid-filled square enclosure being affected by a constant magnetic field imposed in the x-direction having magnitude B 0 on natural convection were investigated numerically. Numerical calculations were carried out using the values of the calibrated parameters in the range 0 ≤ Ha ≤ 10, 104 ≤ Ra ≤ 106, 0.00 ≤ ϕ hnf ≤ 0.04 , 10−5 ≤ Da ≤ 10−3, 0.1 ≤ γ ≤ 1, and 4 ≤ N ≤ 12.

4.1 Effects of pertinent parameters on streamlines and isotherms

In Figure 6, the influence of buoyancy force on the thermal and flow profiles is depicted when the other parameters were fixed at Da = 10 − 3 , Ha = 5 , ϕ hnf = 0.04 , γ = 0.5 , K i = 1 , N = 4 . The results show that the appearance of a centrally positioned rectangular heated cavity within the enclosure results in the development of two symmetrical circulation cells. These cells exhibit clockwise and anticlockwise rotation in all situations. Physically, the Rayleigh number measures the balance between thermal buoyancy and viscous forces. A low Rayleigh number indicates weak buoyancy relative to viscous and thermal diffusion effects, resulting in more orderly, subdued streamlines and smoother, less pronounced isotherms. However, an increase in Ra improves buoyancy-assisted flow that leads to an intensification of fluid circulation in the clear-fluid and porous regions. Therefore, as Ra rises from 104 to 106, the clockwise and counterclockwise vortex cells are both magnified in size and strength. The ψ max increases from 0.28 to 22.4 as Ra upsurges from 104 to 106. The contours of the isotherms depicted in Figure 6 illustrate the impacts of Ra on a natural convection phenomenon. The dominance of heat transfer through conduction is demonstrated by the smoothness of the isotherms at small Rayleigh numbers, i.e., Ra = 104. Whereas, at Ra = 106 the isotherms become more irregular and skewed, reflecting increased thermal gradients and complex convective patterns. Hence, buoyancy force is clearly dominant over viscous forces, as evidenced by strong isotherms, and heat transfer occurs via convection. As Ra rises from 104 to 106, convective heat transfer continues to dominate, and the higher temperature zone has squeezed in the enclosure around the rectangular heated cavity, enhancing the temperature differences and the average Nusselt number on the rectangular heated cavity. In the porous layer, the increase of Ra manifests a diminution in the hot region and amplification of the convective heat transfer. Whereas in the fluid region, the isotherms form a plum-like structure above the heated cavity, and more bending toward the implanted heated rectangular cavity is observed as Ra rises to a value of 106. Natural convection intensification caused the isotherms to become symmetrical about the vertical mid-line of the square cavity at Ra = 106.

Figure 6 Impact of Ra on a pattern of streamlines and isotherms when Da = 10−3, Ha = 5, ϕ hnf = 0.04 , ε = 0.398 , γ = 0.5 , K i = 1.0 , N = 4 , Pr = 6.26 {\phi }_{{\rm{hnf}}}=0.04,\varepsilon =0.398,\gamma =0.5,{K}_{i}=1.0,N=4,{\rm{Pr}}=6.26 .
Figure 6

Impact of Ra on a pattern of streamlines and isotherms when Da = 10−3, Ha = 5, ϕ hnf = 0.04 , ε = 0.398 , γ = 0.5 , K i = 1.0 , N = 4 , Pr = 6.26 .

Figure 7 depicts flow and temperature variation profiles within the cavity with respect to the Ha. The streamlines were substantially influenced by the Ha. If the magnetic field is applied, the velocity profiles experience a type of resisting force known as the Lorentz force. As a result, the resisting force increases simultaneously with Ha, leading to a consistent decrease in velocity. This suppression appears in the streamline profiles as more aligned, subdued flow patterns with reduced circulation and weaker vortices. At low Ha values, the streamlines exhibit more pronounced and dynamic circulation, indicating stronger convection currents. However, as Ha increases, the flow becomes more damped, and the streamlines shift toward a more uniform, layered structure, reflecting the dominant influence of the magnetic field in stabilizing the flow and reducing fluid motion across the domain. This damping effect is crucial in applications controlling or reducing fluid movement, such as in magnetic damping systems or flow control in engineering processes. These findings are consistent with the outcomes of Malvandi et al. [43]. So, as the magnetic field rises, the flow as well as circulation decelerated and ψ max falls from 4.8 to 4.4 as Ha rises from 0 to 10. Furthermore, the isotherms close to the heat flux at centrally located bottom wall are slightly affected by magnetic field strength, whereas in the clear fluid region of the cavity, Ha has no significant effect on isotherms lines. It can be determined quantitatively in light of heat transfer from heat flux and embraced heated rectangular cavity. Enhancement in Ha increases isotherm concentration, which in turn decreases natural convection and increases fluid temperature.

Figure 7 Impact of Ha on a pattern of streamlines and isotherms when Da = 10−3, Ra = 105, ϕ hnf = 0.04 , ε = 0.398 , γ = 0.5 , K i = 1.0 , {\phi }_{{\rm{hnf}}}=0.04,\varepsilon =0.398,\gamma =0.5,{K}_{i}=1.0, N = 4 , Pr = 6.26 N=4,{\rm{Pr}}=6.26 .
Figure 7

Impact of Ha on a pattern of streamlines and isotherms when Da = 10−3, Ra = 105, ϕ hnf = 0.04 , ε = 0.398 , γ = 0.5 , K i = 1.0 , N = 4 , Pr = 6.26 .

Figure 8 explores how the permeability of the porous medium, as characterized by the Darcy number (Da), influences fluid flow and heat transfer. Physically, a high Darcy number indicates high permeability, allowing for easier fluid flow through the porous medium. This results in streamlines that appear more streamlined and less obstructed, with isotherms showing smoother, more uniform thermal distribution. The value of ψ max decreases from 4.8 to 4.4 as the value of Da decreases from 10−3 to 10−5. It is clear from Figure 8 that a higher value for Da increases the permeability of the porous layer, allowing more fluid to penetrate the porous layer and thus intensifying the vortex cell’s strength. The results obtained are in proportionate with the findings reported by Chamkha and Ismael [40]. In contrast to the outcomes at a lower value of Da = 10−5, isotherm contours around the embraced rectangular heated cavity are denser compared to when Da = 10−3. The porous medium’s permeability was reported to decrease as the Da decreases; therefore, with the decrease in Da, a higher temperature zone has formed near the rectangular heated cavity, i.e., fluid retains more heat, leading to a rise in fluid’s temperature in the porous region and a significant decline in heat convection from both heat flux and embraced rectangular heated cavity. Enhancing Da improves heat convection rate, as evidenced by the relatively flat isotherms for the region of clear fluid and supported by the quantitative data tabulated in Table 3. It is worth noting that the current Casson fluid model simplifies to the Newtonian fluid model as “Casson fluid parameter, i.e., γ → ∞ .” For γ = 0.1, 0.5, and 1, the streamline and isotherms are shown in Figure 9.

Figure 8 Impact of Da on the pattern of streamlines and isotherms when Ra = 105, Ha = 5, ϕ hnf = 0.04 , ε = 0.398 , γ = 0.5 , K i = 1.0 , N = 4 , Pr = 6.26 {\phi }_{{\rm{hnf}}}=\hspace{.3em}0.04,\varepsilon =0.398,\gamma =0.5,\hspace{.3em}{K}_{i}=1.0,\hspace{.3em}N=4,{\rm{Pr}}=6.26 .
Figure 8

Impact of Da on the pattern of streamlines and isotherms when Ra = 105, Ha = 5, ϕ hnf = 0.04 , ε = 0.398 , γ = 0.5 , K i = 1.0 , N = 4 , Pr = 6.26 .

Table 3

Effect of parameters and AR of a rectangular heated cavity on average Nusselt number calculated at heat flux ( Nu avg hf ) and at a heated rectangular cavity ( Nu avg hb )

Ra Da Ha γ ϕ hnf N Nu avg hf (AR = 2/5) Nu avg hb (AR = 2/5) Nu avg hf (AR = 1/6) Nu avg hb (AR = 1/6)
105 10−3 5 0.5 0.04 4 1.0591 19.9259 1.1616 22.3020
103 10−3 5 0.5 0.04 4 0.8511 12.9091 0.8720 16.2572
104 10−3 5 0.5 0.04 4 0.8578 12.9335 0.8851 16.4556
106 10−3 5 0.5 0.04 4 1.5017 44.0016 1.5249 44.6247
105 10−1 5 0.5 0.04 4 1.1278 20.2330 1.2463 23.2339
105 10−2 5 0.5 0.04 4 1.1184 20.1908 1.2352 23.1062
105 10−4 5 0.5 0.04 4 0.9842 19.4858 1.0571 21.1820
105 10−3 0 0.5 0.04 4 1.0645 20.1762 1.1672 22.5222
105 10−3 10 0.5 0.04 4 1.0435 19.2288 1.1450 21.6883
105 10−3 15 0.5 0.04 4 1.0193 18.2225 1.1180 20.7978
105 10−3 5 0.01 0.04 4 0.8527 12.9072 0.8752 16.2129
105 10−3 5 0.1 0.04 4 0.8874 13.5078 0.9472 16.7219
105 10−3 5 1 0.04 4 1.1189 23.0881 1.2185 24.7418
105 10−3 5 0.5 0.00 4 1.0500 19.5626 1.1462 21.4740
105 10−3 5 0.5 0.02 4 1.0550 19.7623 1.1545 21.8929
105 10−3 5 0.5 0.06 4 1.0625 20.0599 1.1675 22.7051
105 10−3 5 0.5 0.08 4 1.0722 21.4299 1.1721 23.1090
105 10−3 5 0.5 0.04 8 1.0550 19.9128 1.1107 21.6158
105 10−3 5 0.5 0.04 12 1.0548 19.9078 1.1483 22.1571
Figure 9 Effect of γ \gamma on streamlines and isotherms when Da = 10−3, Ha = 5, ϕ hnf = 0.04 , ε = 0.398 , Ra = 10 5 , K i = 1.0 , N = 4 , Pr = 6.26 {\phi }_{{\rm{hnf}}}=0.04,\varepsilon =0.398,{\rm{Ra}}={10}^{5},{K}_{i}=1.0,N=4,{\rm{Pr}}=6.26 .
Figure 9

Effect of γ on streamlines and isotherms when Da = 10−3, Ha = 5, ϕ hnf = 0.04 , ε = 0.398 , Ra = 10 5 , K i = 1.0 , N = 4 , Pr = 6.26 .

Higher values of γ indicate an abatement in the effective viscosity of the Casson nanofluid, causing an escalation in the velocity; as a result, the amplitude of the stream function is enhanced. Also, it is noticed that a surge in γ supports to densification of the streamlines. The value of ψ max increases from 0.8 to 6.4 as the value of Casson parameter (γ) increases from 0.1 to 1. The outcomes of this investigation are supported by the findings of Ganesh et al. [33]. On isotherm profiles, the effects of γ are precisely identical to the effects of the Ra. At γ = 0.1, the isotherms are flattening out, suggesting that most of the heat is transferred by conduction. The hot region in the porous layer reduces with the rise in Casson parameter γ, which leads to an improvement in heat transfer through convection. As it is well known, increasing the value of the Casson parameter, i.e., γ, causes the fluid’s viscosity to decrease, resulting in a rise in heat convection.

Figure 10 explores the impact of solid nanoparticle volume concentration ( ϕ hnf ) on the streamlines as well as on isotherms. Incorporation of Cu–Al2O3 hybrid nanoparticles into the base fluid, profoundly affects both the streamlines and the isotherms. There is a considerable reduction in the stream function ( ψ ) strength as ϕ hnf increases up to 8%. This reduction in ψ is triggered by a rise in flow resistance which is a consequence of the incorporation of nanoparticles to the base fluid. The ψ max is reduced from 5.2 to 4.4 as the value of ϕ hnf rises from 0 to 8%. The isotherms experience significant changes with an upsurge in ϕ hnf , as illustrated in Figure 10. It is discovered that enhancing in ϕ hnf leads to a decrease in the level of compactness exhibited by the isotherms surrounding the encapsulated rectangular heated cavity; therefore, heat convection is improved. The incorporation of nanoparticles causes a reduction in the temperature difference in the fluid, particularly just above the rectangular heated cavity. In order to explore the impact of N on the heat convection process and fluid flow properties, Figure 11 is plotted for N = 4, 8, and 12. Figure 11 shows that varying the value of N, there were no significant changes on the streamlines, while isotherms are changed significantly near the heat flux in the porous region but nearly unchanged in the clear fluid region. An increase in N slightly enlarges the hot zone in the porous region; consequently, the temperature difference decreases in the porous region, especially just below the rectangular heated cavity, and more heat is trapped in the porous region, resulting in a decrease in the heat convection rate.

Figure 10 Impact of ϕ hnf {\phi }_{{\rm{hnf}}} on the pattern of streamlines and isotherms when Da = 10−3, Ha = 5, γ = 0.5 , ε = 0.398 , Ra = 10 5 , \gamma =0.5,\varepsilon =0.398,{\rm{Ra}}={10}^{5}, K i = 1.0 , {K}_{i}=1.0, N = 4 , Pr = 6.26 N=4,{\rm{Pr}}=6.26 .
Figure 10

Impact of ϕ hnf on the pattern of streamlines and isotherms when Da = 10−3, Ha = 5, γ = 0.5 , ε = 0.398 , Ra = 10 5 , K i = 1.0 , N = 4 , Pr = 6.26 .

Figure 11 Impact of N on the pattern of streamlines and isotherms when Da = 10−3, Ha = 5, γ = 0.5 , \gamma =0.5, ε = 0.398 , Ra = 10 5 , \varepsilon =0.398,{\rm{Ra}}={10}^{5}, K i = 1.0 , {K}_{i}=1.0, ϕ hnf = 0.04 , Pr = 6.26 {\phi }_{{\rm{hnf}}}=0.04,{\rm{Pr}}=6.26 .
Figure 11

Impact of N on the pattern of streamlines and isotherms when Da = 10−3, Ha = 5, γ = 0.5 , ε = 0.398 , Ra = 10 5 , K i = 1.0 , ϕ hnf = 0.04 , Pr = 6.26 .

Figure 12 illustrates the effect of the aspect ratio (AR) of a rectangular heated cavity on streamlines and isotherms within the partially porous enclosure, with all other parameters set at Ha = 5, Da = 10−3, N = 4, γ = 0.5 , ε = 0.398 , Ra = 10 5 , K i = 1.0 , ϕ hnf = 0.04 , Pr = 6.26 . The AR significantly influences the behavior of streamlines and isotherms within the porous cavity. For instance, when the aspect ratio is AR = 1/6, we observe a dense arrangement of isotherms, forming a plume-like structure in the fluidic region adjacent to the heated rectangular cavity. This configuration indicates enhanced flow circulation and heat transfer due to the increased fluid entry into the porous region. In contrast, as the aspect ratio increases to AR = 3/4, the curvature and compactness of the isotherms diminish. This reduction is indicative of restricted flow circulation and a reduced entry of fluid into the porous region, leading to diminished heat transfer. Consequently, the thermal boundary layer thickness also increases, which results in more heat being trapped near the heated wall and within the porous medium, further contributing to decreased heat transfer efficiency. The changes in the NuLocal hf with respect to the calibrated parameters Ha, Ra, Da, N, ϕ hnf , and γ are depicted in Figure 13. The NuLocal hf has a sinusoidal curve profile.

Figure 12 Impact of AR on the pattern of streamlines and isotherms when Ha = 5, Da = 10−3, N = 4, γ = 0.5 , \gamma =0.5, ε = 0.398 , Ra = 10 5 , \varepsilon =0.398,{\rm{Ra}}={10}^{5}, K i = 1.0 , {K}_{i}=1.0, ϕ hnf = 0.04 , {\phi }_{{\rm{hnf}}}=0.04, Pr = 6.26 {\rm{Pr}}=6.26 .
Figure 12

Impact of AR on the pattern of streamlines and isotherms when Ha = 5, Da = 10−3, N = 4, γ = 0.5 , ε = 0.398 , Ra = 10 5 , K i = 1.0 , ϕ hnf = 0.04 , Pr = 6.26 .

Figure 13 Variation in local Nusselt number ( Nu Local hf {{\rm{Nu}}}_{{{\rm{Local}}}^{{\rm{hf}}}} ) at heat flux with various parameters.
Figure 13

Variation in local Nusselt number ( Nu Local hf ) at heat flux with various parameters.

4.2 Effects of pertinent parameters on average Nusselt number

In Figures 14–17, the average Nusselt number at centrally located heat flux (Nuavg hf) at the bottom wall and the rectangular heated cavity (Nuavg hb) is examined. Figure 14 illustrates the impact of Da and ϕ hnf on Nuavg hf and Nuavg hb with varying γ.

Figure 14 Influence of Da and ϕ hnf {\phi }_{{\rm{hnf}}} on Nu avg hf {{\rm{Nu}}}_{{{\rm{avg}}}^{{\rm{hf}}}} and Nu avg hb {{\rm{Nu}}}_{{{\rm{avg}}}^{{\rm{hb}}}} with variation in ‘ γ \gamma ’.
Figure 14

Influence of Da and ϕ hnf on Nu avg hf and Nu avg hb with variation in ‘ γ ’.

Figure 15 Effect of Ra and Da on Nu avg hf {{\rm{Nu}}}_{{{\rm{avg}}}^{{\rm{hf}}}} and Nu avg hb {{\rm{Nu}}}_{{{\rm{avg}}}^{{\rm{hb}}}} with variations in γ \gamma and Ha.
Figure 15

Effect of Ra and Da on Nu avg hf and Nu avg hb with variations in γ and Ha.

Figure 16 Influence of γ \gamma and ϕ hnf {\phi }_{{\rm{hnf}}} on Nu avg hf {{\rm{Nu}}}_{{{\rm{avg}}}^{{\rm{hf}}}} and Nu avg hb {{\rm{Nu}}}_{{{\rm{avg}}}^{{\rm{hb}}}} with variation in Ha.
Figure 16

Influence of γ and ϕ hnf on Nu avg hf and Nu avg hb with variation in Ha.

Figure 17 Influence of Ra {\rm{Ra}} on Nu avgl hf {{\rm{Nu}}}_{{{\rm{avgl}}}^{{\rm{hf}}}} and Nu avg hb {{\rm{Nu}}}_{{{\rm{avg}}}^{{\rm{hb}}}} with variation in Ha and ϕ hnf {\phi }_{{\rm{hnf}}} .
Figure 17

Influence of Ra on Nu avgl hf and Nu avg hb with variation in Ha and ϕ hnf .

It is observed that with increasing γ, both Nuavg hf and Nuavg hb improve significantly for all the considered values of Da and ϕ hnf . Also, it has been found that Nuavghf and Nuavg hb drop substantially as the Da decreases; however, as γ > 0.5, this decline becomes consistent. With the reduction in Da, there is an observable attenuation in Nuavghf in comparison to Nuavg hb. Whereas the average Nusselt number increases with a growing ϕ hnf , but this effect is more substantial for a lower value of γ compared to large ones, as depicted in Figure 14. According to the data summarized in Table 3, decreasing the Da from 10−1 to 10−4, Nuavg hf reduces by 12.73%, whereas a 3.69% reduction is observed in Nuavg hb; on the other hand, 1.19 and 2.54% increment in Nuavg hf and Nuavg hb is reported as ϕ hnf rises from 0 to 6%, at a fixed value of γ = 0.5 and AR = 2/5. Buoyancy-assisted flow is enhanced as the Ra rises. Consequently, the average Nusselt number at heat flux and rectangular heated cavity is improved with the improvement of Ra, as depicted in Figure 15.

It is found that for any given value of Ra, raising γ results in a considerable improvement in both Nuavg hf and Nuavg hb. It has also been noted that the Nuavg hf and Nuavg hb strongly increase with increasing Ra. As the magnitude of the magnetic field intensifies, the Lorentz forces experience a corresponding augmentation in their strength, and heat convection slows down for any value of Da, as shown in Figure 15. In addition, for any given value of Ha, both Nuavg hf and Nuavg hb decrease as the K decreases. At fixed Da = 10−3 and AR = 2/5, Table 3 shows that Nuavg hf and Nuavg hb decrease by 4.24 and 9.69%, respectively, as Ha increases from 0 to 15. Figure 16 exhibits the effect of γ and ϕ hnf on Nuavg hf and Nuavg hb with variation in Ha. Both Nuavg hf and Nuavg hb increase as the Casson fluid parameter (γ) and ϕ hnf rises. It has been found that the impact of Ha is negligible on Nuavg hf and Nuavg hb for small values of γ = 0.01, 0.1. As the γ and ϕ hnf rise, both Nuavg hf and Nuavg hb increase for a fixed Ha; additionally, a reduction is observed with the rise in Ha. Also, Figure 16 shows that the attenuation in Nuavg hf and Nuavg hb is more noticeable with an increase in ϕ hnf than with a rise in γ . In Figure 17, the effects of buoyancy force, measured in terms of Ra, on Nuavg hf and Nuavg hb, versus Ha and ϕ hnf are expressed. As the buoyancy force rises, a substantial rise in Nuavg hf and Nuavg hb is noticed for the fixed value of Ha and ϕ hnf . The incorporation of nanoparticles enhances the thermal conductivity of hybrid nanofluids; consequently, an improvement in the process of heat transfer has been observed. While an appreciable reduction is noticed in Nuavg hf and Nuavg hb with the rise in Ha, as shown in Figure 17. The results presented in Table 3 show that Nuavg hf and Nuavg hb are affected by the AR of the rectangular heated cavity. Results demonstrate that as the AR is reduced, both Nuavg hf and Nuavg hb boost up significantly. Comparatively, the increase in Nuavg hb is more obvious than in Nuavg hf. Table 4 shows the results of an estimation of Nuavg hf and Nuavg hb utilizing two different fluids, i.e., Micropolar and Casson, having the same type of hybrid nanoparticles at various concentrations. Particularly in comparison to micropolar fluid, Nuavg hf reduces while Nuavg hb improves, when Casson fluid is utilized to fill the enclosure. For both fluids, i.e., Micropolar and Casson, Nuavg hf and Nuavg hb are found to decrease as the porous medium’s porosity increases. Also, a reduction in Nuavg hf and Nuavg hb is observed as ϕ hnf reduces. The findings in Table 4 indicate that hybrid nanoparticles are found to be superior to single nanoparticle in regard to their effectiveness. The results presented in Table 4 show that Nuavg hf and Nuavg hb are affected by the AR of the heated rectangular cavity. Results demonstrate that as the AR is reduced, both Nuavg hf and Nuavg hb boost significantly. Comparatively, the increase in Nuavg hb is more obvious than in Nuavg hf.

Table 4

Comparative study of Nu avg hf and ( Nu avg hb ) when Ra = 105, Da = 10−5, and a constant heat flux of length H/2, centrally positioned at the bottom wall of the enclosure and filled with Casson hybrid nanofluid (current study) and micropolar hybrid nanofluid (Ahlawat and Sharma [44])

ϕ cu ϕ Al 2 O 3 ϕ hnf ε Nu avg hf (Casson fluid (current study)/micropolar fluid [44] Nu avg hb (Casson fluid (current study)/micropolar fluid [44]
0.00 0.00 0.00 0.398 1.0959/1.2107 09.9708/09.6153
0.05 0.05 0.10 0.260 1.2268/1.3481 13.8803/11.3845
0.05 0.05 0.10 0.398 1.2200/1.3406 13.8866/11.3727
0.05 0.05 0.10 0.476 1.2161/1.3364 13.8890/11.3678
0.025 0.025 0.05 0.398 1.1532/1.2746 11.8487/10.4722
0.02 0.03 0.05 0.398 1.1530/1.2742 11.8502/10.4586
0.03 0.02 0.05 0.398 1.1534/1.2749 11.8466/10.4853
0.04 0.06 0.10 0.398 1.2197/1.3396 13.8878/11.3457
0.06 0.04 0.10 0.398 1.2203/1.3414 13.8840/11.3989
0.10 0.00 0.10 0.398 1.1068/1.3447 13.3004/11.5003
0.00 0.10 0.10 0.398 1.1062/1.3326 13.5910/11.1852

The findings of this study have very important physical implications in the applications involving natural convection in enclosures with porous media and hybrid nanofluids. For example, the enhancement in heat transfer with the increase of Ra highlights the critical role of buoyancy-driven flow in the promotion of convective heat transfer, which is vital for optimizing cooling processes in electronic devices and solar thermal collectors. The impact of Da on the flow structure and thermal performances indicates that the selection of porous materials with appropriate permeability can lead to a significant influence on the effectiveness of thermal insulation systems and porous heat exchangers. The suppression of the fluid motion with Ha provides valuable insights for designing MHD systems where the control of the flow and thermal characteristics is required, such as in nuclear reactors and MHD generators. In addition, the results present a dependence of the heat transfer on γ and ϕ hnf which is important for fluid rheology and thermal conductivity enhancements allowing the achievement of efficient energy management. Finally, the sensitivity of the heat transfer to the AR emphasizes the need for careful geometric optimization in applications such as energy storage devices and compact heat exchangers.

5 Conclusions

This study corresponds to a numerical investigation of the impacts of sinusoidal heat flux and an embraced heated rectangular cavity on natural convection in a square enclosure partially filled with a porous medium and containing a Casson-hybrid nanofluid under the influence of a horizontal magnetic field. The findings revealed the interplay of key parameters such as Ra, Da, Ha, γ, and ϕ hnf on the heat transfer and flow dynamics. The main results include the following:

  • The increase of Ra leads to an intensification of the fluid circulation and an enhancement of the convective heat transfer, evidenced by higher average Nusselt number values and pronounced isotherm distortions.

  • The rise of Ha suppressed the convection currents, reduced flow velocity, and dampened heat transfer due to the Lorentz force, which led to a stabilization of the flow.

  • Higher Da causes better promotion of the fluid flow through the porous layer, which boosts the heat transfer, while lower Da restricts flow and traps heat near the source.

  • The increase of γ causes a reduction of the fluid viscosity, facilitating stronger convection currents and better thermal performance.

  • The enhancement of ϕ hnf leads to an improved thermal conductivity of the hybrid nanofluid and results in a marked increase in heat transfer efficiency.

  • Reducing the AR of the heated cavity significantly boosted heat transfer, with the influence being more pronounced in the porous region.

Limitations of the study:

The limitations of the research problem could include the following points:

  1. The focus on a square enclosure and specific heating mechanisms limits the exploration of other geometries, heating patterns, and cavity shapes, which may be more relevant for certain engineering applications.

  2. The absence of experimental data to validate the numerical findings may raise questions about the accuracy and reliability of the conclusions when applied to real-world systems.

  3. The study may not address the environmental impact or cost-effectiveness of using hybrid nanofluids and porous materials in real-world applications, which are critical for sustainable development.

Addressing these limitations in future studies could enhance the reliability, applicability, and broader impact of the findings.

6 Future scope

This work unveils the effects of sinusoidal heat flux and heated rectangular cavity on natural convection within a square enclosure, partially filled with porous medium, offering novel insights into heat transfer dynamics. However, numerous possibilities remain open for further investigation to improve the understanding and application of this research problem:

  1. Incorporating temperature-dependent thermal properties of the hybrid nanofluid enhances model accuracy, making it more reflective of real-world conditions.

  2. Employing optimization techniques to identify the best configurations for sinusoidal heat flux amplitude, porous medium properties, and nanoparticle volume fractions would aid in designing efficient thermal systems.

  3. Experimental validation of numerical results would enhance reliability and establish a benchmark for future theoretical research.

  4. Enhancing heat transfer efficiency through diverse fluids and nanoparticles.

Acknowledgments

Anil Ahlawat would like to express gratitude to the Council for Scientific and Industrial Research (CSIR), New Delhi, India, for the financial support through letter no. 09/752(0073)-EMR-I. We extend our thanks to DST, New Delhi, for providing computational facilities under DST-FIST. Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2025R41), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

  1. Funding information: Anil Ahlawat would like to express gratitude to the Council for Scientific and Industrial Research (CSIR), New Delhi, India, for the financial support through letter no. 09/752(0073)-EMR-I. Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2025R41), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

  2. Author contributions: Anil Ahlawat: Methodology, Formal analysis, Software, Writing – original draft, Writing – review & editing. Mukesh Kumar Sharma: Methodology, Formal analysis, Conceptualization, Validation, Writing – review & editing. Kaouther Ghachem: Methodology, Formal analysis, Writing – review & editing. Badr M. AlShammari: Methodology, Formal analysis, Writing – review & editing. Lioua Kolsi: Methodology, Formal analysis, Writing – original draft, Writing – review & editing, Data curation. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2024-09-14
Revised: 2025-01-22
Accepted: 2025-03-18
Published Online: 2025-05-05

© 2025 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2025-0142
Keywords for this article
natural convection; Casson fluid; hybrid nanofluid; heated rectangular cavity; average Nusselt number
Creative Commons
BY 4.0

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