Enhancing thermal performance in enclosures filled with nanofluids subjected to sinusoidal heating: a numerical study

ABSTRACT

This study conducts a comprehensive numerical investigation into enhancing thermal transfer within square enclosures filled with water-based oxide nanoparticle suspensions, subjected to central sinusoidal heating. Further flow configuration, influenced by an inclined magnetic field, is designed with a focus on enhancing thermal efficiency for engineering applications. Key innovations include the application of sinusoidal heating elements to enhance thermal performance significantly. Computational analysis supported by finite element analysis, quantifies the impact of these parameters on flow dynamics and thermal transmission, presenting a substantial advance in the understanding of nanofluid-filled enclosure thermal management. The study reveals that the undulation of the heating element plays a crucial role in the heat transfer rate, with improvements observed as undulation increases. The introduction of magnetic fields further controls flow distribution and buoyancy effects, as demonstrated by our findings that an increase in the Rayleigh number correlates with enhanced convection, dominating the cavity's thermal dynamics. Additionally, the report outlines the conditions under which the Nusselt number increases, indicating enhanced thermal performance. These insights are pivotal for designing optimized heat transfer systems and energy-efficient applications, setting a new benchmark for thermal management strategies in practical engineering contexts.

KEYWORDS:

• Natural convection
• oxide nanofluid
• inclined Lorentz forces
• sinusoidal circular heated element
• KKL model
• numerical simulation

Nomenclature

Symbols

$x,y$	=	Cartesian coordinates $\left[\text{L}\right]$
$u$, $v$	=	Velocity components $\left[\text{L }{\text{T}}^{-1}\right]$
$T$	=	Temperature$\left[\text{K}\right]$
$p$	=	Pressure $\left[\text{M }{\text{L}}^{-1}{\text{T}}^{-2}\right]$
$\mu$	=	Dynamic viscosity $\left[\text{M }{\text{L}}^{-1}{\text{T}}^{-1}\right]$
$g$	=	Gravitational acceleration $\left[\text{L }{\text{T}}^{-2}\right]$
$\varphi$	=	Nanoparticles fraction
$F_X,F_Y$	=	Lorentz forces
$\alpha$	=	Magnetic field inclination angle
$\mathrm{Ra}$	=	Rayleigh number
$C_p$	=	Specific heat capacity $\left[\text{M}{\text{L}}^2{\text{K}}^{-1}{\text{T}}^{-2}\right]$
$d_s$	=	Nanoparticles diameter
$T_0$	=	Average temperature
$A_m$	=	Amplitude ratio

Subscripts

$s$	=	Solid particles
$\mathrm{nf}$	=	Nanofluid
$c$	=	Cold
$X,Y$	=	Dimensionless variable
$U,V$	=	Dimensionless velocity components
$\theta$	=	Dimensionless temperature
$P$	=	Dimensionless pressure
$\rho$	=	Density $\left[\text{M }{\text{L}}^{-3}\right]$
$\text{β}$	=	Thermal expansion coefficient$\left[{\text{K}}^{-1}\right]$
$\gamma$	=	Penalty parameter
$\sigma$	=	Electrical conductivity [${\text{T}}^3{\text{A}}^2{\text{M}}^{-1}{\text{L}}^{-3}$]
$\mathrm{Ha}$	=	Hartmann number
$\text{Pr}$	=	Prandtl number
$k$	=	Thermal conductivity $\left[\text{ML }{\text{K}}^{-1}{\text{T}}^{-3}\right]$
$k_b$	=	Boltzmann constant
$n$	=	Dimensionless wavelength number
$\mathrm{Nu}$	=	Nusselt number
$f$	=	fluid
$h$	=	Hot

1 Introduction

Natural convection is a widely studied phenomenon due to its importance in various industrial and engineering applications. It is the process of fluid motion driven by density differences caused by temperature variations in a fluid. The study of natural convection in different enclosures has been of interest to researchers for many years. Understanding the fluid flow and heat transfer characteristics in different types of enclosures is important for the design and optimization of various engineering systems. Numerous studies have been conducted on natural convection in different enclosures, exploring a wide range of factors such as geometry, orientation, fluid properties, heat source/sink placement, and boundary conditions. These studies have provided valuable insights into the flow patterns and heat transfer characteristics of natural convection in different enclosures. For example, Tian et al. (2022) conducted experimental and numerical analysis on natural convection heat transfer in novel sinusoidal cavities to improve thermal transportation in electronic cooling and photothermal conversion. Their exploration with Ferrite particles exhibits optimal conditions for enhanced heat transfer. While Li et al. (2022) investigated natural convective thermal distribution in an enclosure with supercritical water. The numerical experiment reported that, due to the distinct physical properties of supercritical water, pressure, temperature difference, and aspect ratio influenced the velocity and temperature within the cavity. Lu et al. (2022) studied the impact of copper foam structural features and porosity on natural convection heat transportation. They found that heat transfer drops when porosity exceeds 78%, and that foams heated on the sides, compared to those heated from the bottom, exhibited superior heat transfer. Nammi et al. (2022) investigated natural convection heat transfer in a square enclosure with four heated cylinders embedded in porous media, and they reported that convective heat transfer prevails at high $\mathrm{Ra}\left(\ge {10}^5\right)$ for certain Darcy numbers $\left(\mathrm{Da}\right)$ while conduction dominates at low Rayleigh numbers $\left(\mathrm{Ra}\le {10}^4\right)$. In a three-dimensional enclosure Atia et al. (2023) simulated free convective flow influenced by various shapes heated pin-fins. Their findings concluded that circular pin-fins offer the higher heat transfer also at various $\mathrm{Ra}$ the number of pin-fins improve heat sink performance, contributing to the technological development in heat dissipation. In their investigation, Dogonchi et al. (2019) examined the behaviour of natural convection within a cavity, incorporating an inclined elliptical heating source and subjected to both nanoparticle shape factors and magnetic fields. The study highlights the significant role of nanoparticle morphology and magnetic field application in enhancing convection efficiency, opening avenues for optimized thermal management in engineering designs. Alsabery et al. (2018) explored the mixed convection characteristics of Al2O3-water nanofluid in a uniquely configured double lid-driven square cavity, featuring a solid inner insert and utilizing Buongiorno’s two-phase model for analysis. Their findings point to the vital influence of the cavity's dual motion and the presence of the insert in advancing the nanofluid’s heat distribution capabilities, offering strategic insights for heat dissipation improvements in enclosed systems. Ghalambaz et al. (2019) conducted an investigation into the thermal behaviour of hybrid nanofluids within a bifurcated cavity. Their study identified that the introduction of a vertical partition membrane significantly bolsters thermal stratification, presenting a novel method for enhancing heat distribution in segmented enclosures. In another study, Ben-Nakhi and Chamkha (2006) explored the influence of fin adjustments on convection within a square enclosure. They discovered that the strategic inclination of a thin fin critically enhances fluid movement and thermal effectiveness, underscoring the utility of fin orientation as a straightforward, yet potent, mechanism to improve cooling techniques.

Franco et al. (2023) evaluated natural convective flow in an open cavity influenced by solid blocks. The study explored block count, and thermal conductivity ratio affect convection through interference, channelling, and fin-like diffusion and provides analytical correlations for practical design and analysis. Tang et al. (2022) experimentally investigated natural convection heat transfer in a photothermal conversion system with metal copper foam with Ferrite particles. They showed in their findings that the higher pore density in copper foams and particle mass fractions significantly enhanced heat transfer. In another study, Liu et al. (2022) examined the effects of different angles and aspect ratios on natural convection in a paraffin-filled rectangular enclosure. The study found that the angle of the rotation influenced the velocity and the maximum values along radial height. Additionally, melting time was longest at acute angles.

Nanofluids, which are colloidal suspensions of nanoparticles in base fluids, have gained significant attention in recent years due to their unique thermal and fluidic properties. Natural convection with nanofluids has been an area of interest due to its potential for heat transfer enhancement. Several studies have investigated the effect of nanoparticles on natural convection, and the findings indicate that adding nanoparticles to base fluids can significantly increase their thermal conductivity and convective heat transfer coefficient, resulting in improved thermal performance. For instance, Cao et al. (2022) investigated natural convective flow within an inclined enclosure using Alumina particles nanofluid as an efficient and cost-effective cooling agent for an inclined hot plate. The study finds an improvement in the average Nusselt number by 21% with the increment in volume fraction of particles by 6% in low Reynolds number conditions. Rahmoune et al. (2022) analyzed the nanofluid natural convection in an enclosure resembles H shaped and found enhanced heat transfer performance with the addition of nanoparticles and Rayleigh number while its behaviour declines against increasing form factor of the geometry. Al-Farhany et al. (2022) studied the influence of baffle length on Cu particles nanofluid natural convection in a square enclosure subjected to sinusoidal thermal distribution. The study reveals enhanced heat transfer rate and velocity profile at higher Rayleigh numbers and baffle length of Lb = 0.3. Wu et al. (Wu et al., 2022) found that the use of Alumina-water nanofluid in a half-elliptical enclosure with hot tube significantly increased heat transfer rate additionally inclined angles also impact its magnitude at higher Rayleigh number. Lai et al. (2022) considered the flow and thermal impact of heated rotating cylinder on nanofluid convection in square cavity. The results depicted that rotation decreased the impact of gravity on flow field and reduced the heat transfer rate. Ma et al. (2023) scrutinized nanofluid heat transfer in L-shaped enclosure with varying flow dynamic resistance along its inner surface. The improvement in heat transfer is strongly observed dependent on stress-free patches and aspect ratio while less dependent on particle fraction. In their study, Tao et al. (2023) explored the enhancement of thermal efficiency in solar thermal technologies through the application of water-based binary composite nanofluids. Their research highlights the significant potential for improving renewable energy system performance while focusing on sustainability.

In another contribution by Boujelbene et al. (2023) examined the complex dynamics of nanofluid flow under the influence of magnetic fields and slip conditions. Their findings offer a deeper understanding of how these conditions affect thermal transport and fluid mechanics, presenting new avenues for optimizing industrial and engineering processes. Das et al. (2024) conducted a comprehensive examination of natural convection in recto-triangular cavities by using a CuO-water nanofluid and applying a magnetic field. Their research revealed that magnetic fields effectively manipulate the convective flow, enhancing the heat transfer which could be beneficial for cooling systems. The work by Halder et al. (2024) on entropy generation and heat transport in butterfly-shaped cavities under magnetic fields provided insights into optimizing thermal systems. The study highlighted that the strategic placement of heat sources and sinks, along with magnetic control, can significantly advance thermal management. Manna et al. (2024) analyzed the impact of enclosure shapes on the flow of nanofluids in the presence of magnetic fields. Their findings suggest that enclosure geometry is a critical factor in controlling flow patterns and thermal performance, offering valuable guidance for the design of efficient heat exchangers. Chelia et al. (2023) conducted natural convection heat transfer in a square enclosure with nanofluid for two different cases of heated boundary, Lattice Boltzmann method is utilized for simulation. It is found that an enhancement in heat transfer rate of 6.81% computed along bottom heated edge and possess lower values against side heated boundaries. Nanofluid convection has emerged as a key area of interest for researchers aiming to enhance heat transfer mechanisms, offering significant advancements in thermal management for a range of applications from electronic cooling to sustainable energy systems (Biswas et al., 2022, 2023; Chatterjee et al., 2023; Xu et al., 2023).

Magnetohydrodynamic (MHD) nanofluid natural convection has gained significant attention in recent years due to its potential for enhancing heat transfer performance in various applications, such as energy and engineering fields. MHD refers to the study of the interaction between magnetic fields and electrically conductive fluids. The addition of nanoparticles to MHD natural convection systems has been shown to further enhance heat transfer performance due to the unique thermal and fluidic properties of nanofluids. The impact of magnetic field to natural convection systems can significantly alter the fluid flow patterns and enhance heat transfer rates. MHD natural convection has been shown to be particularly effective in applications such as heat exchangers, solar collectors, and cooling systems for electronic devices, where efficient heat transfer is essential. MHD natural convection is a promising area of research with significant potential for improving the efficiency and performance of various heat transfer systems. Many researchers have studied the potential of MHD natural convection and its enhancement with the addition of nanoparticles. To illustrate Abderrahmane et al. (2022) performed finite element analysis to evaluate MHD natural convection of power law nanoliquid in a halved annulus enclosure. They reported that both the power-law index and Hartmann number reduce the heat transfer rate. Huang et al. (2022) conducted a numerical simulation by employing nonorthogonal MRT lattice Boltzmann method to investigate natural convective flow in a porous corrugated triangular cavity filled with $A{l}_2{O}_3$ particles fluid exposed to magnetic field. They reported that optimal corrugated wall parameters enhance heat transfer rate while it is reduced by increasing Hartmann number. Wang and Hai (2023) studied natural convection for nanofluid flow in a square cavity under the influence of magnetic field and the thickness of the three baffle considered on the hot wall of the cavity. It is found that the heat transfer rate declines with increasing thickness of baffles while lengthening the baffles disrupted the vortex inside the enclosure. Shi et al. (2023) investigated the effect of magnetic field and nanoparticles distribution on natural convective flow in a square enclosure and the simulation results enhancement in entropy generation against magnetic forces which least to stronger circulation and temperature gradient in the nanofluid. Saha et al. (2023) simulated numerically magneto-natural convection of Alumina-water nanofluid in a cavity with a wavy top wall and a fixed heated fin at bottom edge. The investigation evaluated the shape of nanoparticles and magnetic forces and found that as compared to spherical particles the blade-shaped particle improves the temperature transport by 7.65%. Kolsi et al. (2023) conducted the influences of Eleatic wall and magnetic effects on nanofluid convection in a cavity with porous plate. The study provided insights into enhancing thermal management in heat transfer systems and it is observed that wall elastic modulus, magnetic forces and permeability of porous plate improve heat transfer rate. Bibi and Xu (2023) provided prediction for Nusselt number utilizing machine learning and CFD simulation for natural convection nanofluid in a trapezoidal enclosure with magnetic effects. The study found that Multi expression programming results align with simulated outcomes and reported that heat transfer increased by up to 27.7% with higher nanoparticles concentration but decreased by 9.37% with higher Hartmann number. Rehman et al. (Rehman et al., 2023) investigated the effect of sinusoidal heated rods and Lorentz forces on natural convection of CuO/water nanofluid in a rectangular enclosure it is reported that heat transfer rate at the upper corrugated wall possesses maximum values.

In light of the extensive review of existing literature and the outlined objectives of our study, several pertinent questions emerge that our research seeks to address. A primary area of inquiry involves understanding the complex interplay between an inclined magnetic field and sinusoidal heating on the natural convection processes within nanofluid-filled enclosures. Specifically, we are interested in exploring how nanofluids, subjected to these combined influences, compared to traditional fluids in terms of thermal dispersion and temperature regulation capabilities. Furthermore, the investigation aims to elucidate the impacts of varying magnetic field strengths and orientations on the flow dynamics and heat transfer rates in such enclosures. Additionally, the role of nanoparticles in enhancing the thermal performance of enclosures under the combined effects of magnetic fields and oscillating heat conditions warrants detailed examination. By addressing these questions, our study intends to bridge a significant gap in the current body of knowledge, offering valuable insights that could inform the design of more energy-efficient thermal management systems, thereby contributing to the development of sustainable and ecologically responsible solutions in the field.

Central to enhancing thermal performance – defined as the enclosure's efficiency in heat transfer – is the utilization of nanofluids. Hence the key to our investigation is the role of nanofluid that has the potential to significantly improve heat dispersion and temperature regulation within the enclosed space. The filled enclosure serves as a controlled environment to simulate practical thermal management scenarios. It is subjected to central sinusoidal heating, a strategic choice that initiates real-world applications where heat sources may not uniformly distribute heat. This heating method introduces a variable thermal gradient, offering insights into how nanofluids behave under oscillating heat conditions, which is critical for applications requiring precise temperature control. Despite the nature of the geometry, similar configurations can be observed in several practical scenarios. For example, the heated and adiabatic boundaries can represent the internal surfaces of electronic equipment where certain components generate heat (heated boundary) while others are insulated or not directly exposed to cooling (adiabatic boundary). The cold wall could represent a cooling plate or heat sink where heat is actively removed from the system. The study is organized as follows.

2 Physical model and governing equations

The enclosure depicted in Figure 1(a) contains a centrally positioned heated element, facilitating the investigation of thermal dynamics within. The cavity is defined by a uniform side-length, L. The assumptions include the implementation of specific boundary conditions: no-slip conditions at all walls, isothermal conditions at the left and right walls, a partially heated lower boundary, and adiabatic conditions for the remaining edges. In further assumptions, the enclosure is filled with a CuO-water nanofluid, characterized by two-dimensional flow dynamics and a steady-state thermal distribution. Critical to the thermal behaviour within this system is the buoyancy-driven convection, prompted by temperature differentials across the fluid medium. This convective movement is further influenced by Lorentz forces, which are applied obliquely, adding a directional complexity to the flow dynamics. These boundary conditions, along with the assumed temperature-dependent properties for viscosity and thermal conductivity under the Boussinesq approximation, form the basis of our numerical analysis. Stability and convergence are ensured through mesh independence tests, validating the computational accuracy and reliability of our findings against established benchmarks. In light of the preceding assumptions, the governing equations can be articulated as follows (Alsabery et al., 2020). (1) $\begin{array}{rl}\mathrm{\nabla }\cdot \overrightarrow{v}& =0,\end{array}$(1) (2) $\begin{array}{rl}\overrightarrow{v}\cdot \mathrm{\nabla }\overrightarrow{v}& =\frac{-1}{{\rho }_{\mathrm{nf}}}\mathrm{\nabla p}+{\nu }_{\mathrm{nf}}{\mathrm{\nabla }}^2\overrightarrow{v}+\frac{\overrightarrow{i}}{{\rho }_{\mathrm{nf}}}{f}_x\\ & +\frac{\overrightarrow{j}}{{\rho }_{\mathrm{nf}}}\left(f_y+{\left(\rho \text{β}\right)}_{\mathrm{nf}}g\left(T-\mathrm{Tc}\right)\right),\end{array}$(2) (3) $\begin{array}{rl}\overrightarrow{v}\cdot \mathrm{\nabla T}& =\frac{k_{\mathrm{nf}}}{{\left(\mathrm{\rho Cp}\right)}_{\mathrm{nf}}}{\mathrm{\nabla }}^{2}T.\end{array}$(3)

Figure 1. Geometical discription of physical model (a) and domain discritzation (b).

Here Equation (1) ensures mass conservation within the flow field, asserting that the fluid is incompressible. Equation (2) is the conservation of momentum; this equation incorporates several forces acting on the fluid element. The term $\overrightarrow{v}\cdot \mathrm{\nabla }\overrightarrow{v}$ represents the convective acceleration, reflecting the change in velocity due to the fluid's motion. The pressure gradient $\frac{-1}{{\rho }_{\mathrm{nf}}}\mathrm{\nabla p}$ accounts for the force due to pressure changes within the fluid. Viscous forces are captured by ${\nu }_{\mathrm{nf}}{\mathrm{\nabla }}^2\overrightarrow{v}$, relating to the fluid's resistance to flow. The buoyancy force $\left({\left(\text{β}\right)}_{\mathrm{nf}}g\left(T-\mathrm{Tc}\right)\right)$ is driven by temperature differences in the fluid, causing less dense, warmer fluid to rise and denser, cooler fluid to fall. Lastly, the Lorentz force $f_x$ and $f_y$ denote the which are defined as: $\begin{array}{rl}{f}_x& ={\sigma }_{\mathrm{nf}}{B_o}^2\sin \alpha \left(\mathrm{vcos}\alpha -\mathrm{usin}\alpha \right)\end{array}$ $\begin{array}{rl}{f}_y& ={\sigma }_{\mathrm{nf}}{B_o}^2\cos \alpha \left(-\mathrm{vcos}\alpha +\mathrm{usin}\alpha \right)\end{array}$

In which $\left(\alpha ,B_o\right)$ are the inclination angle and strength of the imposed magnetic field, whereas ${\sigma }_{\mathrm{nf}}$ is the nanofluid’s electrical conductivity. These forces result from the interaction between the magnetic field and the electric currents within the fluid, affecting the flow direction and velocity.

Further $\overrightarrow{v}=\left(u,v\right)$ is the velocity profiles, $\mathrm{\nabla }=\left(\frac{\mathrm{\partial }}{\mathrm{\partial }x},\frac{\mathrm{\partial }}{\mathrm{\partial }y}\right)$ is two-dimensional differential operator, $\left({\rho }_{\mathrm{nf}},{\nu }_{\mathrm{nf}}\right)$ indicates the nanofluid density and viscosity, $\left({\text{β}}_{\mathrm{nf}},{\left(\mathrm{Cp}\right)}_{\mathrm{nf}},k_{\mathrm{nf}}\right)$ are the thermal expansion coefficient, specific heat and thermal conductivity of nanofluid. $T$ signify temperature profile while $\left(T_h,T_C\right)$ the thermal distribution on heated and cooled boundaries.

Here adopted model for thermophysical characteristics is presented by Koo–Kleinstreuer–Li (Koo & Kleinstreuer, 2005; J. Li, 2008) the properties are described as: (4) $\begin{array}{r}{k}_{\mathrm{nf}}=k_{\mathrm{static}}+k_{\mathrm{Brownion}}\end{array}$(4) The conventional static and Brownian motion part as discussed by (Koo & Kleinstreuer, 2005; Maxwell, 1873) can be described as (5) $\begin{array}{r}{k}_{\mathrm{static}}=k_f\left(1+\frac{3\left(\frac{k_s}{k_f}-1\right)\varphi }{\left(\frac{k_s}{k_f}+2\right)-\left(\frac{k_s}{k_f}-1\right)\varphi }\right)\end{array}$(5) (6) $\begin{array}{r}{k}_{\mathrm{Brownion}}=5\times {10}^4\varphi \text{β}(\rho C_p{)}_f\sqrt{\frac{k_b{T}_0}{{\rho }_s{d}_s}}f\left(T_0,\varphi \right),\end{array}$(6) Equation (6) is modified by Li (J. Li, 2008) and defined a new function $\zeta$ thus the relation reduces as (6) $\begin{array}{r}{k}_{\mathrm{Brownion}}=5\times {10}^4\varphi (\rho C_p{)}_f\sqrt{\frac{k_b{T}_0}{{\rho }_s{d}_s}}\zeta \left(T_0,\varphi ,d_s\right).\end{array}$(6)

In which, $\left(k_b=1.38\times {10}^{-23},T_0=0.5\left(T_h-T_c\right)\mathrm{\&}{d}_s\right)$ are the Boltzmann constant, average temperature and solid particles diameter whereas the modified function $\zeta$ is given by: $\begin{array}{rl}\zeta \left(T_0,\varphi ,d_s\right)& =(a_1+a_2\ln d_s+a_3\ln \varphi +a_4\ln d_s\ln \varphi \\ & +a_5\ln {d_s}^2)\ln T_0\end{array}$ (7) $\begin{array}{rl}& +a_6+a_7\ln d_s+a_8\ln \varphi +a_9\ln d_s\ln \varphi \\ & +a_{10}\ln {d_s}^2\end{array}$(7)

In above Equation (7) the coefficients $a_1-a_{10}$ and numerical values of thermophysical properties for $\mathrm{CuO}$-water are shown in Tables 1 and 2. Moreover, the electrical conductivity and effective viscosity in term of static and Brownian motion as considered in (Koo & Kleinstreuer, 2005; Maxwell, 1873) can be written as: (8) $\begin{array}{rl}{\mu }_{\mathrm{nf}}& ={\mu }_{\mathrm{static}}+{\mu }_{\mathrm{Brownion}}=\frac{{\mu }_f}{{\left(1-\varphi \right)}^{2.5}}+\frac{k_{\mathrm{Brownion}}}{k_f}\frac{{\mu }_f}{\mathrm{Pr}}\end{array}$(8) (9) $\begin{array}{rl}{\sigma }_{\mathrm{nf}}& ={\sigma }_f\left(1+\frac{3\left(\frac{{\sigma }_s}{{\sigma }_f}-1\right)\varphi }{\left(\frac{{\sigma }_s}{{\sigma }_f}+2\right)-\left(\frac{{\sigma }_s}{{\sigma }_f}-1\right)\varphi }\right)\end{array}$(9)

Table 1. Parameter values for $\mathrm{CuO}$-Water nanofluid (Haq ul & Aman, 2019).

Coefficients	Values	Coefficient	Values
$a_1$	$-26.593310846$	$a_6$	$48.40336955$
$a_2$	$-0.403818333$	$a_7$	$-9.787756683$
$a_3$	$-33.3516805$	$a_8$	$190.245610009$
$a_4$	$-1.915825591$	$a_9$	$10.9285386565$
$a_5$	$6.42185846658E-02$	$a_{10}$	$-0.72009983664$

Table 2. Thermophysical properties for $\mathrm{CuO}$-Water nanofluid (Haq ul & Aman, 2019).

Thermophysical properties	Water	Nanoparticles (CuO)
$\rho \left(\frac{\mathrm{kg}}{m^3}\right)$	997.1	6500
$C_p\left(\frac{J}{\mathrm{kg}}K\right)$	4179	540
$k_f\left(\frac{W}{\mathrm{mK}}\right)$	0.613	18
$\text{β}\times {10}^5\left(K^{-1}\right)$	21	29
$d_s\left(\mathrm{nm}\right)$	–	45

A dimensionless equations system can be obtained by utilizing the following variables (Hemmat Esfe et al., 2016; Ul Haq et al., 2019) (10) $\begin{array}{rl}X& =\frac{x}{L},Y=\frac{y}{L},U=\frac{\mathrm{uL}}{{\alpha }_f},V=\frac{\mathrm{vL}}{{\alpha }_f},\\ P& =\frac{p{L}^2}{{\rho }_f{\alpha }_f},\theta =\frac{T-T_c}{T_h-T_c}\\ \mathrm{Ra}& =\frac{g{\text{β}}_f\left(T-T_c\right)L^3}{{\alpha }_f{\nu }_f},\text{Pr}=\frac{{\nu }_f}{{\alpha }_f},\mathrm{Ha}={\mathrm{B}}_{o}L\sqrt{\frac{{\sigma }_f}{{\mu }_f}}\end{array}$(10)

By use of the above variables, the non-dimensional equations take the form (11) $\begin{array}{rl}\breve{V}\cdot \mathrm{\nabla }\breve{V}& =\frac{-{\rho }_f}{{\rho }_{\mathrm{nf}}}\mathrm{\nabla P}+\frac{{\nu }_{\mathrm{nf}}}{{\nu }_f}\text{Pr}{\mathrm{\nabla }}^2\breve{V}+\overrightarrow{i}\frac{{\rho }_f}{{\rho }_{\mathrm{nf}}}\text{Pr}{F}_X\\ & +\overrightarrow{j}\frac{{\rho }_f}{{\rho }_{\mathrm{nf}}}\left(\text{Pr}{F}_Y+\frac{{\left(\rho \text{β}\right)}_{\mathrm{nf}}}{{\left(\rho \text{β}\right)}_f}\text{Pr}\mathrm{Ra\theta }\right),\end{array}$(11) (12) $\begin{array}{rl}\breve{V}\cdot \mathrm{\nabla \theta }& =\frac{{\left(\rho C_p\right)}_f}{{\left(\rho C_p\right)}_{\mathrm{nf}}}\frac{k_{\mathrm{nf}}}{k_f}{\mathrm{\nabla }}^2\theta .\end{array}$(12) where $\breve{V}=\left(U,V\right)$ is the dimensionless flow velocity, $\left(\text{Pr},\mathrm{Ra},\mathrm{Ha}\right)$ are the Prandtl number, Rayleigh parameter and Hartman number, while the non-dimensional Lorentz forces $\left(F_X,F_Y\right)$ are defined as (Shehzad et al., 2020): $\begin{array}{rl}{F}_X& =\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}H{a}^2\sin \alpha \left(\mathrm{Vcos}\alpha -\mathrm{Usin}\alpha \right)\end{array}$ (13) $\begin{array}{rl}{F}_Y& =\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}H{a}^2\cos \alpha \left(-\mathrm{Vcos}\alpha +\mathrm{Usin}\alpha \right)\end{array}$(13)

The boundary constraints of physical domain can be written in non-dimensional form as:

The no-slip condition implies that $U=V=0$ on all boundaries. The thermal conditions are as follows.

At vertical portions of the setup: (14) $\begin{array}{r}\text{At}\begin{array}{c}X=0,0\le Y\le 1\\ X=1,0\le Y\le 1\end{array}\theta =0\end{array}$(14) Upper and lower adiabatic portions (15) $\begin{array}{r}\mathrm{At}\begin{array}{c}Y=0,0\le X\le a,b\le X\le 1\\ Y=1,0\le X\le 1\end{array}.\frac{\mathrm{\partial }\theta }{\mathrm{\partial }n}=0\end{array}$(15)

At outer length and inner element (16) $\begin{array}{rl}& \text{At }Y=0,a\le X\le b,\theta =1\end{array}$(16) (17) $\begin{array}{rl}& (X-0.5{)}^2+(Y-0.5{)}^2=r^2,\theta =1\end{array}$(17)

Here we have considered the inner element in the form of a parametric curve as: (18) $\begin{array}{r}C\left(t\right)=\left(r\left(t\right)\cos \left(t\right),r\left(t\right)\sin \left(t\right)\right)\end{array}$(18) where $r\left(t\right)=A_m\sin \left(\mathrm{nt}\right)+r$, in which $\left(A_m,n\right)$ signify the amplitude and element frequency.

The heat transfer rate is the quantity of physical interest, here it is defined as said by Nusselt number can be expressed as follows (19) $\begin{array}{r}\mathrm{Nu}=-\frac{k_{\mathrm{nf}}}{\mathrm{kf}}{\int }_{{\text{h}}_L}\frac{\mathrm{\partial }T}{\mathrm{\partial }n}\mathrm{dX}\end{array}$(19) where $h_L$ represents both the outer and internal heated portions while $n$ is the normal direction to the heated surfaces.

3 Numerical procedure

The Galerkin Finite Element Method (FEM) is adopted for discretization of the computational domain, relying on the penalty finite element approach to relax the incompressibility condition. This approach effectively perturbs the continuity equation by a small number to approximate pressure, facilitating the handling of higher-dimensional problems without explicitly solving for pressure. The discrete domain is interpolated using test and trial functions for velocity components and temperature profiles, allowing for the algebraic reduction of the governing differential equations. Throughout this process, mesh independence tests are conducted to verify that the numerical solution is not dependent on the mesh size, thereby affirming the reliability of the computational findings. Thus the solution of partial differential system (11), (12) with boundary conditions (14)–(17) is offered by utilizing Galerkin finite element method (FEM). Thus, in the above system (11), (12) the functions $\left(P,U,V,T\right)$ are all counted as un-knowns, this implies that numerical computation become difficult in higher dimensional problem. To beat this challenge the penalty finite element approach (Dyne & Heinrich, 1993; Heinrich & Vionnet, 1995) is utilized here which relax incompressiblility condition. The principle of penalty formulation is to perturbate continuity equation by a small number containing pressure can be represented as: (20) $\begin{array}{r}P=-\gamma \left(\frac{\mathrm{\partial }U}{\mathrm{\partial }X}+\frac{\mathrm{\partial }V}{\mathrm{\partial }Y}\right)\end{array}$(20) since $P$ is assumed to be finite and choosing $\gamma$ sufficiently large then the above Equation (20) so far so good approximation to continuity equation. The pressure term is eleminated from (11) and reduced equation takes the form. (21) $\begin{array}{rl}\breve{V}\cdot \mathrm{\nabla }\breve{V}& =\gamma \frac{{\rho }_f}{{\rho }_{\mathrm{nf}}}\mathrm{\nabla }\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }X}+\frac{\mathrm{\partial }V}{\mathrm{\partial }Y}\right)+\frac{{\nu }_{\mathrm{nf}}}{{\nu }_f}\text{Pr}{\mathrm{\nabla }}^2\breve{V}\\ & +\overrightarrow{i}\frac{{\rho }_f}{{\rho }_{\mathrm{nf}}}\text{Pr}{F}_X\\ & +\overrightarrow{j}\frac{{\rho }_f}{{\rho }_{\mathrm{nf}}}\left(\text{Pr}{F}_Y+\frac{{\left(\rho \text{β}\right)}_{\mathrm{nf}}}{{\left(\rho \text{β}\right)}_f}\text{Pr}\mathrm{Ra\theta }\right),\end{array}$(21)

The discrete domain is interpolated by assuming test and trail functions for velocity components and temperature profile. The approximated solution is obtained by expanding the dependent variable in the form (22) $\begin{array}{r}U\approx \sum _{j=1}^n{U}_j{\mathrm{\Phi }}_j^h,V\approx \sum _{j=1}^n{V}_j{\mathrm{\Phi }}_j^h,T\approx \sum _{j=1}^n{T}_j{\mathrm{\Psi }}_j^h.\end{array}$(22) Here $U_j,V_j,T_j$ (unknowns) depict the values of functions $\left(U,V,T\right)$ at associated node $j$, $h$ represent element, ${\mathrm{\Phi }}_j^h$ are the interpolated basis function assumed at each node $j$ of element $h$ and $n$ describe the total number of nodes in an element. Make use of (20) the resulting finite element model in terms of algebraic equations reduces as (23) $\begin{array}{rl}& \sum _{j=1}^n{U}_j{\int }_{{\mathrm{\Omega }}^d}\left(\sum _{i,j=1}^n{U}_j{\mathrm{\Phi }}_j^h\frac{\mathrm{\partial }{\mathrm{\Phi }}_j^h}{\mathrm{\partial }X}{\mathrm{\Phi }}_i^h+\sum _{i,j=1}^n{V}_j{\mathrm{\Phi }}_j^h\frac{\mathrm{\partial }{\mathrm{\Phi }}_j^h}{\mathrm{\partial }Y}{\mathrm{\Phi }}_i^h\right)\\ & \times \mathrm{dXdY}+\frac{{\rho }_f}{{\rho }_{\mathrm{nf}}}\lambda {\int }_{{\mathrm{\Omega }}^d}(\sum _{i,j=1}^n{U}_j\frac{\mathrm{\partial }{\mathrm{\Phi }}_j^h}{\mathrm{\partial }X}\frac{\mathrm{\partial }{\mathrm{\Phi }}_i^h}{\mathrm{\partial }X}\\ & +\sum _{i,j=1}^n{V}_j\frac{\mathrm{\partial }{\mathrm{\Phi }}_i^h}{\mathrm{\partial }X}\frac{\mathrm{\partial }{\mathrm{\Phi }}_j^h}{\mathrm{\partial }Y})\mathrm{dXdY}+\frac{{\mu }_{\mathrm{nf}}}{{\mu }_f}\frac{{\rho }_f}{{\rho }_{\mathrm{nf}}}\\ & \times \text{Pr}{\int }_{{\mathrm{\Omega }}^d}\left(\sum _{i,j=1}^n{U}_j\frac{\mathrm{\partial }{\mathrm{\Phi }}_j^h}{\mathrm{\partial }X}\frac{\mathrm{\partial }{\mathrm{\Phi }}_i^h}{\mathrm{\partial }X}\right)\mathrm{dXdY}\\ & +\frac{{\mu }_{\mathrm{nf}}}{{\mu }_f}\frac{{\rho }_f}{{\rho }_{\mathrm{nf}}}\text{Pr}{\int }_{{\mathrm{\Omega }}^d}\left(\sum _{i,j=1}^n{U}_j\frac{\mathrm{\partial }{\mathrm{\Phi }}_j^h}{\mathrm{\partial }Y}\frac{\mathrm{\partial }{\mathrm{\Phi }}_i^h}{\mathrm{\partial }Y}\right)\mathrm{dXdY}\\ & +{\int }_{{\mathrm{\Omega }}^d}{F}_X{\mathrm{\Phi }}_i^h\mathrm{dXdY}-{\int }_{{\mathrm{\Gamma }}^d}{\mathrm{\Phi }}_i^h{n}_X\mathrm{dS}=0,\end{array}$(23) (24) $\begin{array}{rl}& \sum _{j=1}^n{V}_j{\int }_{{\mathrm{\Omega }}^d}\left(\sum _{i,j=1}^n{U}_j{\mathrm{\Phi }}_j^h\frac{\mathrm{\partial }{\mathrm{\Phi }}_j^h}{\mathrm{\partial }X}{\mathrm{\Phi }}_i^h+\sum _{i,j=1}^n{V}_j{\mathrm{\Phi }}_j^h\frac{\mathrm{\partial }{\mathrm{\Phi }}_j^h}{\mathrm{\partial }Y}{\mathrm{\Phi }}_i^h\right)\\ & \times \mathrm{dXdY}+\frac{{\rho }_f}{{\rho }_{\mathrm{nf}}}\lambda {\int }_{{\mathrm{\Omega }}^d}(\sum _{i,j=1}^n{U}_j\frac{\mathrm{\partial }{\mathrm{\Phi }}_i^h}{\mathrm{\partial }Y}\frac{\mathrm{\partial }{\mathrm{\Phi }}_j^h}{\mathrm{\partial }X}\\ & +\sum _{i,j=1}^n{V}_j\frac{\mathrm{\partial }{\mathrm{\Phi }}_j^h}{\mathrm{\partial }Y}\frac{\mathrm{\partial }{\mathrm{\Phi }}_i^h}{\mathrm{\partial }Y})\mathrm{dXdY}+\frac{{\mu }_{\mathrm{nf}}}{{\mu }_f}\frac{{\rho }_f}{{\rho }_{\mathrm{nf}}}\text{Pr}\\ & \times {\int }_{{\mathrm{\Omega }}^d}\left(\sum _{i,j=1}^n{V}_j\frac{\mathrm{\partial }{\mathrm{\Phi }}_j^h}{\mathrm{\partial }X}\frac{\mathrm{\partial }{\mathrm{\Phi }}_i^h}{\mathrm{\partial }X}\right)\mathrm{dXdY}\\ & +\frac{{\mu }_{\mathrm{nf}}}{{\mu }_f}\frac{{\rho }_f}{{\rho }_{\mathrm{nf}}}\text{Pr}{\int }_{{\mathrm{\Omega }}^d}\left(\sum _{i,j=1}^n{V}_j\frac{\mathrm{\partial }{\mathrm{\Phi }}_j^h}{\mathrm{\partial }Y}\frac{\mathrm{\partial }{\mathrm{\Phi }}_i^h}{\mathrm{\partial }Y}\right)\mathrm{dXdY}\\ & +{\int }_{{\mathrm{\Omega }}^d}{F}_Y{\mathrm{\Phi }}_i^h\mathrm{dXdY}-{\int }_{{\mathrm{\Gamma }}^d}{\mathrm{\Phi }}_i^h{n}_Y\mathrm{dS}=0,\end{array}$(24) (25) $\begin{array}{rl}& \sum _{j=1}^n{T}_j{\int }_{{\mathrm{\Omega }}^d}\left(\sum _{i,j=1}^n{U}_j{\mathrm{\Phi }}_j^h\frac{\mathrm{\partial }{\mathrm{\Psi }}_j^h}{\mathrm{\partial }X}{\mathrm{\Psi }}_i^h+\sum _{i,j=1}^n{V}_j{\mathrm{\Phi }}_j^h\frac{\mathrm{\partial }{\mathrm{\Psi }}_j^h}{\mathrm{\partial }Y}{\mathrm{\Psi }}_i^h\right)\\ & \times \mathrm{dXdY}-\frac{{\alpha }_{\mathrm{nf}}}{{\alpha }_f}{\int }_{{\mathrm{\Omega }}^d}(\sum _{i,j=1}^n{T}_j\frac{\mathrm{\partial }{\mathrm{\Psi }}_j^h}{\mathrm{\partial }X}\frac{\mathrm{\partial }{\mathrm{\Psi }}_i^h}{\mathrm{\partial }X}+\\ & \sum _{i,j=1}^n{T}_j\frac{\mathrm{\partial }{\mathrm{\Psi }}_j^h}{\mathrm{\partial }Y}\frac{\mathrm{\partial }{\mathrm{\Psi }}_i^h}{\mathrm{\partial }Y})\mathrm{dXdY}-{\int }_{{\mathrm{\Gamma }}^d}{\mathrm{\Psi }}_i^h{Q}_n\mathrm{dS}=0.\end{array}$(25)

Where unit-normal $\left(n_X,n_Y\right)$ components and normal heat flux $Q_n$ to the boundary $S$. The reduced finite element model (23)–(25) is then solved iteratively after imposing the corresponding boundary condition. Here, implementing the conditions the boundary integrals in momentum and energy Equations (23)–(25) vanish because the considered test functions are assumed to be zero on boundaries.

3.1. Implementation procedure

The analysis employs the Finite Element Method (FEM) to discretize the problem domain meticulously, enabling the accurate calculation of flow dynamics and thermal distribution. FEM's robustness in COMSOL Multiphysics is particularly advantageous for addressing complex boundary conditions and nonlinear material properties. This capability is fundamental for simulating the nuanced behaviour of nanofluids under various thermal scenarios. The algorithm of our solution process as follows:

3.1.1 Model initialization and setup

• Domain and physics Interface: The model is initialized within a two-dimensional spatial domain. The single-phase laminar flow physics interface is selected to address incompressible flow through the Navier-Stokes equations. Concurrently, the ‘Heat Transfer in Fluids’ interface is integrated to simulate thermal distribution, utilizing the convection–diffusion equation to include both fluid movement and heat conduction.

• Material properties and geometry configuration: Specific attention is devoted to defining the CuO-water nanofluid's properties, crucial for the stationary study's success. The geometry setup is then completed, incorporating these material characteristics.

3.1.2 Geometry construction and mesh generation

• Tirangular Mesh: The set ${\mathrm{\Delta }}_h$ comprising triangular elements ${\mathrm{\Delta }}_i$ where $i$ ranges from 1 to $N$, facilitates the domain descritization. Each triangular element ${\mathrm{\Delta }}_i$ contributes to a comprehensive representation of the domain's interior space, described mathematically as: $\begin{array}{r}{\mathrm{\Omega }}_d\approx {\mathrm{\Omega }}_h=\underset{i=1}{\overset{N}{\cup }}{\mathrm{\Delta }}_i\end{array}$

The collective assembly of these elements is optimized to capture the domain's geometrical and physical behaviours with high fidelity. Following Table 3 shows the mesh parameters

• Mesh Strategy: A user-controlled mesh strategy is adopted, with a particular focus on enhancing mesh density around critical heat sources, the inner heated element and the lower wall's heated section. This refinement is crucial for capturing thermal gradients and flow dynamics accurately.

3.1.3 Physics setup and boundary conditions

• Laminar Flow and Heat Transfer Interfaces: Fluid properties within the laminar flow interface are user-defined, incorporating a no-slip boundary condition across all enclosure boundaries. Whereas in heat transfer in the fluid interface, the fluid thermophysical properties are considered user defined and on the inner element and lower portion temperature is maximum, while on vertical wall it is zero. The lagrangian shape function and quadratic element size are considered for approximation for both momentum and energy equations of the descritized domain.

Table 3. Mesh distribution for the computational domain.

Mesh Characteristics	Value	Mesh Characteristics	Value
Number of Vertices	3312	Total Meshed Area	0.9m^2
Number of Triangles	8270	Number of Edge Elements	538
Minimum Element Quality	0.6423	Element Length Ratio	0.6238
Minimum Element Size	$7.5\times {10}^{-5}$	Element Growth Rate	0.02
Average Element Quality	0.9166	Length of Mesh Edge	5.365
Element Area Ratio	0.2348	Number of Vertex Elements	8

3.1.4 Solver configuration and solution algorithm

• Solver Selection: In the stationary solver setup Direct is paired with the Raphson method. The solver configuration is set to solve the equations fully coupled, meaning that the Navier-Stokes equations and energy equations are solved simultaneously.

• Iterative Solution Approach: The solution process is iteratively advanced, employing the Newton–Raphson method for nonlinearities, ensuring precise convergence to the physical reality. A convergence criterion of ${10}^{-4}$ and a tentative limit of 25 iterations directs the solver's attempts. It is to be mentioned that the CPU time for the simulation was 56 s, and the final residual error reached a value of 0.0003, indicating a convergence towards an accurate solution within the computational model.

3.1.5 Adaptive mesh refinement and validation

• Mesh Adaptation: The implementation contemplates adaptive mesh refinement based on solution gradients, ensuring that the model's resolution dynamically adjusts to capture critical phenomena accurately.

• Model Validation: To corroborate the simulation's fidelity, model predictions are validated against empirical data and established literature, ensuring the methodology's reliability and accuracy.

3.2. Mesh Independence and validation

Ensuring result accuracy and reliability in computational fluid dynamics (CFD) simulations, mesh sensitivity evaluation is essential. The analysis seeks to identify optimal mesh resolution by exploring the impact of various mesh sizes on the Nusselt number calculation along the heated element. Initiating the grid refinement study, it becomes evident that the precision and convergence of CFD simulations are tied to the mesh resolution. A coarse mesh may cause errors and loss of important flow features, while an overly fine mesh is computationally expensive and time-consuming. Hence, assessing mesh independence for the Nusselt number is essential to find the optimal mesh size that strikes a balance of accuracy and computational efficiency. Nusselt numbers were calculated across a range of mesh sizes, varying from coarse to fine. Subsequently, the simulation is repeated with successively finer mesh resolutions to observe the variation in the Nusselt number. Upon comparing the results for different mesh sizes, it is observed that beyond 8270 elements, there are no significant changes in the Nusselt number values. This indicates a mesh-independent state, where additional mesh refinement does not significantly impact the Nusselt number results as illustrated in Figure 2(a). Consequently, subsequent simulations can be confidently conducted with this chosen mesh size (8270 elements), ensuring accurate predictions for convective heat transfer in the cavity. The result of the present exploration is also validated for various Rayleigh number with Aminossadati and Ghasemi (Aminossadati & Ghasemi, 2011) and it is observed that comparison of velocity profile shows good agreement with the published results as shown in Figure 2(b). Furthermore, the present results is validated with experimental results by (Paroncini & Corvaro, 2009). The numerical simulation results for the Rayleigh number $\mathrm{Ra}=1.68\times {10}^5$ indicate isotherms that closely resemble the experimental observations, with a distinct thermal plume rising above the heated bottom surface, mirroring the experimental isotherm patterns. For the Rayleigh number $\mathrm{Ra}=1.69\times {10}^5$, the streamlines from the simulation exhibit counter-rotating vortices which align well with the experimental flow visualization, indicating reliable simulation of the convective flow features within the square cavity. These congruences affirm the numerical model's fidelity in reproducing the experimentally observed flow and thermal fields at the respective Rayleigh numbers (Figure 3).

4. Results and discussion

The simulation has been carried out based on the governing equations of the physical model, which consists of a partially heated square enclosure with a heated element inside the core. The setup is filled with $\mathrm{CuO}$-water nanofluid, herein the results obtained after numerical simulation against the range of emerging physical parameters controlling flow profile (velocity field and stream function) and thermal distribution (temperature and heat transfer rate), such as nanoparticles fraction, Rayleigh number, Hartmann number, inclination angle of applied magnetic field, frequency and amplitude of inner heated element, are presented in line graphs and contour plots. Since the operating fluid is Newtonian nanofluid therefore the Prandtl number has fixed value $\left(\text{Pr}=6.2\right)$, the other parameters are considered in the range $\left(1\text{e}3\le \mathrm{Ra}\le 1\text{e}6\right)$, $\left(10\le \mathrm{Ha}\le 1\text{e}2\right)$, $\left(\frac{\pi }{4}\le \alpha \le \frac{3\pi }{4}\right)$ $\left(0\le \varphi \le 0.04\right)$, $\left(0\le n\le 0.1\right)$. The thermophysical properties and values of the coefficients $a_i$are presented in Tables 1 and 2 (Haq ul & Aman, 2019).

Figure 2. Mesh indepency $\left(\text{a}\right)$ and velocity comparision (b) with Aminossadati and Ghasemi (Aminossadati & Ghasemi, 2011).

Figure 3. Comparison of isotherms (a),(b), and streamlines (c),(d) with (Paroncini & Corvaro, 2009) against $\mathrm{Ra}$ while $\varphi =\mathrm{Ha}=0$.

Numerical values of the Nusselt number along inner heated element for various involved parameters are shown in Table 4, i.e. the impact of nanoparticles volume fraction, Rayleigh number, Hartman number and frequency of inner heated element. The results depict that an increase in the particle volume fraction $\left(\varphi \right)$ leads to an increase in the Nusselt number. This suggests that the presence of particles augments the convective heat transfer. The Nusselt number increases by approximately 26.64% as the volume fraction of nanoparticles increases from 0.0 to 0.04. The numerical values also characterize the increases in the convective heat transfer rate with the Rayleigh number. This relationship suggests that as the buoyancy forces (as characterized by the $\mathrm{Ra}$) increase, the convective heat transfer becomes more efficient, resulting in higher Nusselt numbers. The data shows a nonlinear relationship where the Nusselt number doesn't increase proportionally with Ra. Initially, the rate of increase in Nu with $\mathrm{Ra}$ is more moderate (between $\mathrm{Ra}$ of ${10}^4$ and $5\times {10}^4$), but as $\mathrm{Ra}$ reaches ${10}^6$, the increase in Nu is more pronounced. This nonlinear trend indicates the transition from a laminar to a turbulent flow regime, where heat transfer efficiency significantly increases due to the enhanced mixing of the fluid. The percentage increase in the Nusselt number as the Rayleigh number increases from ${10}^4$ to ${10}^6$ is $46.4\mathrm{\%}$. When examining the effect of the $\mathrm{Ha}$, the Nusselt number increases with the increase of $\mathrm{Ha}$ from 20 to 100 for constant $\varphi$, $\mathrm{Ra}$, and $A_m$. This trend is due to the suppression of convective currents by the magnetic field represented by $\mathrm{Ha}$, leading to higher thermal gradients and thus increased heat transfer rates. The analysis reveals a 0.53% change in the Nusselt number as the Hartmann number ($\mathrm{Ha}$) varies from 20 to 100. For constant values of $\varphi$, $\mathrm{Ra}$, and $\mathrm{Ha}$, as $A_m$ increases from 0.02 to 0.08, there is a clear decrease in the Nusselt number. This might indicate that larger amplitudes in the system lead to changes in the flow structure or boundary layer thickness, which in turn reduces the efficiency of heat transfer. It is indicated that there was a 17.41% decrease in the Nusselt number as the amplitude ($A_m$) varied from 0.02 to 0.08.

Table 4. Numerical values for the Nusselt number against various parameters.

$\varphi$	$\mathit{Ra}$	$\mathit{Ha}$	${\mathit{A}}_{\mathit{m}}$	$\mathit{N}{\mathit{u}}_{\mathbf{inner}}$
$0.0$	${10}^4$	$50$	$0.02$	$1.648206$
$0.01$	${10}^4$	$50$	$0.02$	$1.756543$
$0.02$	${10}^4$	$50$	$0.02$	$1.865735$
$0.03$	${10}^4$	$50$	$0.02$	$1.976002$
$0.04$	${10}^4$	$50$	$0.02$	$2.087522$
$0.04$	${10}^4$	$50$	$0.02$	$2.087559$
$0.04$	$5\times {10}^4$	$50$	$0.02$	$2.443501$
$0.04$	${10}^6$	$50$	$0.02$	$3.05655$
$0.04$	${10}^4$	$20$	$0.02$	$4.114023$
$0.04$	${10}^4$	$40$	$0.02$	$4.123149$
$0.04$	${10}^4$	$60$	$0.02$	$4.12824$
$0.04$	${10}^4$	$80$	$0.02$	$4.130762$
$0.04$	${10}^4$	$100$	$0.02$	$4.135935$
$0.04$	${10}^4$	$50$	$0.02$	$2.087559$
$0.04$	${10}^4$	$50$	$0.04$	$1.961884$
$0.04$	${10}^4$	$50$	$0.06$	$1.832135$
$0.04$	${10}^4$	$50$	$0.08$	$1.724072$

Figures 4(a–d) present the line graphs for Nusselt numbers, temperature and velocity profile at the defined tracks against nano-particle concentrations. From Figures 4(a, b) it is examined that the Nusselt number taken along inner heated element up-surges in wavy manner by the increment of solid particles fraction illustrated in Figure 4(a), however, the heat transport rate computed at bottom heated source length decline in response to addition of nanoparticles fraction as shown in Figure 4(b). The Nusselt number's behaviour can be recognized to the thermal boundary layer's thickness and thermal conductivity changes due to nanoparticle addition. Increasing the nanoparticle volume fraction improves the nanofluid's effective thermal conductivity, enhancing heat transfer away from the heated surface. This leads to a steeper temperature gradient at the inner heated element (hence the increase in $N{u}_{\mathrm{inner}}$). On the other hand, the local convective heat transfer can decrease as the overall heat transfer improves. This is due to the alteration of the flow regime by the nanoparticles, affecting the local fluid velocity and thermal boundary layer development, resulting in the temperature gradient decreases across the layer, reflected by the decrease in $N{u}_{\mathrm{out}}$ along the lower heated portion. The thermal transmission and flow distribution along the vertical mean path are depicted in Figures 4(c,d). It is reported in Figure 4(c) that increasing $\varphi$ the thermal field has maximum values between $\left(0.0-0.3\right)$ and decreasing nature in the interval $\left(0.7-1.0\right)$ the later diminishing behaviour illustrates the influence of upper cooled wall. The flow profile $V$ in Figure 4(d) depicts that at lower concentration it has maximum values in both interval $\left(0.0-0.3\right)$ and $\left(0.7-1.0\right).$ Its behaviour declines as the fluid becomes more concentrated by adding nano-particles fraction. To be more specific when $\varphi$ augments it implies a more viscous regime thereby the flow dynamics fall along the vertical mean path.

Figure 4. Impact $\varphi$ on $\left(\text{a}\right),\left(\text{b}\right)$ Nusselt numbers $\copyright$ temperature and $\left(\text{d}\right)$ velocity along the defined paths.

The isotherms variations and streamline distribution against various nano-particles fractions are demonstrated in Figures 5(a–f). From the portrayed figures it can be observed that when concentration of nanoparticles is zero the natural convection is stronger and thereby the contour maps for stream function and isotherms are stronger in the regime. See Figures 5(a, b). On the other hand, when particle fraction grows Figures 5(c–f) ensure that buoyancy forces get weaker and, in the output, secondary circulations (clockwise) get stronger while the heat transmission switches from convection to conduction mode. Since the addition of nanoparticles increases the fluid's viscosity as well as thermal conductivity, which can dampen the convective currents. In natural convection, the buoyancy forces driving the flow arise from temperature differences within the fluid; at lower concentration these forces are insufficient to overcome the increased viscous resistance, the flow becomes weaker. as evident in Figures 5(a–f) below.

Figure 5. Variations in Streamlines (left) and Isotherms (right) at various $\varphi$.

The variations in flow and heat configuration influenced by the element frequency number $n$ are presented in line graphs and contour maps. It is portrayed in Figure 6(a) that heat transfer rate computed at inner heated element propagates in wave form in the regime. It should be noted that undulation number $n$ really affect the heat flux rate which possess higher values as $n$ ascends. Heat transfer rate estimated at outer heated-portion is found to decrease as considered in Figure 6(b), this is because of the increasing number of undulations suppresses the thermal transmission in the lower regime. Figures 6(c, d) depict the thermal profile and $y-$velocity component computed along the mean vertical path. The graph displayed in Figure 6(c) describes that temperature in the enclosure increases as the element frequency $n$ grows. Since as, $n$ multiply the element strength increases which leads to maximum thermal transmission in the enclosure. The velocity field has minimum values by increasing the undulation of the heated element. It can be noticed that maximum rolls of heated element possess resistance to the flow profile, hence decreasing flow field in $y-$ direction as illustrated in Figure 6(d).

Figure 6. Influence of element frequency $n$ on (a), (b) Nusselt numbers, (c) temperature and (d) velocity at shown paths.

The flow patterns and thermal contours in terms of streamlines and isotherms against various element frequency number $n$ are presented in Figures 7(a–c) and (d–f). It seems that in the flow pattern, the clockwise circulations are increased whereas, the counterclockwise tends to decline as $n$ changes from $0.02$ to $0.08$. See Figures 7(a–c). The isotherms in Figures 7(d–f) depict that at the default $n$ $\left(=0.02\right)$ the thermal distribution is of convective nature; it switches to conductive mode when the element undulations increase up to $0.08$. At smaller element undulations the isotherms indicate that temperature is distributed through convection while at large element undulations isotherms distributed uniformly, and its intensity decreases due to predominant conduction effect.

Figure 7. Streamlines (left) and Isotherms (right) at different heated element frequency $n.$

Another important parameter that arise due to the impact of Lorentz forces is the Hartmann number $\mathrm{Ha}$, and its influences on flow dynamics and energy distribution are considered in Figures 8 and 9. From the line graphs shown in Figures 8(a–d) it can be noticed that the stronger Lorentz forces translated by maximizing Hartmann number $\mathrm{Ha}$, the attenuation of flow profile occurs due to the resistive nature of the Lorentz forces. Consequently, the $y-$velocity component decrease computed along the vertical mean position as described in Figure 8(a). The temperature profile has a transverse effect against $\mathrm{Ha}$. As plotted in Figure 8(b) it is examined that along the vertical path temperature has escalating behaviour between $\left(0.0-0.3\right)$ and reduces in the interval $\left(0.7-1.0\right)$. The heat transmission rate both at the inner element and outer heated length is a decreasing function of $\mathrm{Ha}$ as illustrated in Figures 8(c, d). It can be noticed that an increase of Hartmann number attenuates the convective heat transfer mechanism, implies that the applied magnetic field possess negative effects on buoyancy forces resulting in the dominant conductive heat transfer in the regime thus Nusselt numbers decline.

Figure 8. Hartman number impact on (a), (b) heat transfer rate, (c) temperature and (d) $\mathit{y}-$velocity at indicated paths.

Figure 9. Streamlines (a-c) and Isotherms (d-f) variations with respect to Hartman number $\mathrm{Ha}$.

Figures 9(a–c) and (d–f) present the streamlines and isotherms for various $\mathrm{Ha}$. The flow patterns indicate that streamlines rarefy at higher Hartmann number. It ensures that magnetic forces oppose the flow dynamics and can be confirmed from the isolines in Figures 9(a–c) that intensity of the form-formed circulations (clockwise and anti-clockwise) reduces against $\mathrm{Ha}$ in the range $\left(1-100\right)$. As in the above explanation we have discussed that increase of Hartmann number the convective heat transmission in the enclosure reduces and the conductive transportation remains dominant. Hence, from the isotherms contour in Figures 9(d–f) clarifies that at lower $\mathrm{Ha}$ there is still convective heat transmission occurs but when its values multiply the mode of thermal transport converted to conduction mechanism and the isothermal lines spread uniformly in the cavity.

The angle of inclination of applied magnetic field also influenced the flow and thermal profiles; its variations are presented in Figure 10. For $\alpha =0$ i.e. when the magnetic field is parallel to the flow direction there is no significant impact on the flow pattern; two circulations in streamlines distribution with uniform density are formed, as shown in Figure 10(a). Varying inclination angle from $\frac{\pi }{4}$ to $\frac{\pi }{2}$ and $\frac{3\pi }{4}$ the flow patterns change, at $\alpha =\frac{\pi }{4}$ the contours of stream function indicate that anti-clockwise circulations are pushed upward while the clockwise circulations squeezed downward. See Figure 10(b). When the Lorentz forces impact occur perpendicular $\left(\text{i}.\text{e}.,\alpha =\frac{\pi }{2}\right)$ to the flow both streamline circulations tend to reduce as exhibited in Figure 10(c). The magnetic force applied at $\alpha =\frac{3\pi }{4}$ the orientations of the clockwise circulation changes form downward to upward whereas, the counterclockwise patterns are pushed downward as presented in Figure 10(d). On the other hand, thermal profile depicts that there is no significant impact of the angle of inclination of the magnetic field. The isotherms are spread uniformly in the cavity just its orientation is reversed when $\alpha$ changes from $\frac{\pi }{4}$ to $\frac{3\pi }{4}$ i.e. at angle $\alpha =\frac{\pi }{4}$ isotherms are pushed slightly upward while at $\alpha =\frac{3\pi }{4}$ isolines are pressed downward as given in Figure 10(e–h).

Figure 10. Variations in Streamlines (a-d) and Isotherms (e-h) at different inclination angle $\alpha$.

The stem-parameter involved in natural convection transportation is Rayleigh number $\mathrm{Ra}$ and its impact on flow configuration cannot be ignored. The outcomes of the proposed solution against different $\mathrm{Ra}$ are reported in line graphs and contour maps as portrayed in Figures 11 and 12. By nature, $\mathrm{Ra}$ is related to buoyancy forces, which maintain natural convection in the regime. Hence, strengthening $\mathrm{Ra}$ boosts buoyancy effects which leads to stronger convection regime. The heat transfer rate against improving Rayleigh number indicates maximum values computed at both heated sources (i.e. inner element and outer portion). See Figures 11(a, b). The temperature distribution given in Figure 11(c) exhibits that in the interval $\left(0.0-0.3\right)$ it depicts decreasing value while in interval $\left(0.7-1.0\right)$ it has maximizing behaviour. This can be interpreted as at lower half the heat transmitted from heated lower length and inner element opposes each other thus thermal profile has lower values. Whereas, in the upper half heat propagated from inner element in the upward cooled region thus its values increase as $\mathrm{Ra}$ augments. The $y-$ velocity component computed along the vertical mean path illustrates higher values at various $\mathrm{Ra}$ as given in Figure 11(d).

Figure 11. Impact of Raleigh number on (a), (b) heat transfer rate, (c) temperature and (d) $\mathit{y}-$velcoity at indicated paths.

Figure 12. Variations in Streamlines (a-d) and Isotherms (e-h) at different inclination angle $\alpha$.

The flow distribution demonstrated by stream function and isotherms is also controlled by Rayleigh number. It can be observed that increasing impact of $\mathrm{Ra}$ possesses stronger buoyancy forces. The stream function depicts that the flow pattern involves two circulations and with growing $\mathrm{Ra}$ both the vertices getting stronger. This implies that convection transportation is dominant in the region see Figures 12(a-c). The isotherms also report that thermal transmission take place through convection remains dominant, since as $\mathrm{Ra}$ is directly proportional to stronger buoyancy effects. This explanation can be confirmed from the following figure Figures 12(d–f).

5. Closing remarks

In response to the critical exploration of nanofluid dynamics within an enclosed system, our study through finite element simulations conclusively demonstrates how nanoparticle integration, alongside strategic manipulation of heating element undulations and magnetic forces, critically influences convective heat transfer processes. Our findings directly address the posed research questions, underscoring the significant enhancement in thermal performance and flow efficiency due to nanoparticle addition, the pivotal role of element undulation, and the buoyancy and magnetic force's regulatory effect on fluid dynamics. The key findings are concluded as:

• Nanoparticles addition rarefies the streamlines and velocity profile.

• Then magnitude of isotherms and Nusselt number along the heated element enhances against increasing nanoparticles concentration.

• Increasing undulation of heated element heat transport rate possesses higher values.

• The streamlines and velocity profile reduce against maximum element undulation.

• The flow distribution is suppressed by the applied magnetic force.

• The Nusselt number escalates against higher magnetic field intensity.

• The orientation of the flow pattern changes at various inclination angle of the applied magnetic force.

• At higher Rayleigh parameters not only the heat transportation rates augments

• The streamlines and velocity field also rise because of the stronger buoyancy regime.

These findings highlight the potential of using nanoparticle-enhanced fluids and magnetic fields for superior flow and thermal management. While offering a solid foundation for understanding nanofluid behaviour in magnetohydrodynamic settings, it prompts further investigation into nanoparticle dispersion stability, magnetic field optimization, and scalability for industrial applications. Future research should also explore the environmental and economic implications of these technologies, ensuring sustainable development within thermal management systems.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

This manuscript did not utilize external datasets. All data generated or analyzed during this study are included in the published article.

Additional information

Funding

This work was supported by Natural Science Foundation of China [grant number 11872189]; Natural Science Foundation of China [grant number 12102148].