Magnetohydrodynamic Nanofluid Natural Convection in a Cavity under Thermal Radiation and Shape Factor of Nanoparticles Impacts : A Numerical Study Using CVFEM

Ali J. Chamkha 1,2, A.S. Dogonchi 3,* and D.D. Ganji 4 1 Mechanical Engineering Department, Prince Sultan Endowment for Energy and Environment, Prince Mohammad Bin Fahd University, Al-Khobar 31952, Saudi Arabia; achamkha@pmu.edu.sa 2 RAK Research and Innovation Center, American University of Ras Al Khaimah, P.O. Box 10021, Ras Al Khaimah, UAE 3 Department of Mechanical Engineering, Aliabad Katoul Branch, Islamic Azad University, Aliabad Katoul, Iran 4 Mechanical Engineering Department, Babol Noshirvani University of Technology, Babol 484, Iran; mirgang@nit.ac.ir * Correspondence: a.s.dogonchi@stu.nit.ac.ir


Introduction
The inherently moderate thermal conductivity of regular fluids, such as water, oil, etc., causes a deterioration in the performance of engineering systems.To overcome this barrier, researchers add metallic nanoscale particles-owing to the fact that they have a higher thermal conductivity than regular fluids-to these fluids.This combination is called a nanofluid.Nanofluids have higher thermal conductivity in comparison with regular fluids.Most recently, a vast number of studies have been carried out concerning heat transfer and the flow of nanofluids .Entropy generation on a magnetohydrodynamic nanofluid toward a stagnation-point flow over a penetrable expanding wall was investigated by Bhatti and Rashidi [1].They observed the increasing velocity of the fluid due to the pronounced effects of the magnetic field and the porosity parameter.Moreover, they noted that the entropy generation number increased with increasing Brinkman number and Reynolds number.Entropy generation for a non-Newtonian Eyring-Powell nanofluid through a penetrable expanding wall was analyzed by Bhatti et al. [2].They deduced that the velocity profile rises owing to the high impact of the suction parameter.Moreover, they concluded that Brownian motion and thermophoresis raised the temperature profile remarkably.The impacts of the flow of an electromagnetohydrodynamic nanofluid over a Riga sheet were analyzed by Abbas et al. [3].The convective Poiseuille boundary-layer flow of an ethylene glycol (C 2 H 6 O 2 )-based nanofluid with suspended aluminum oxide (Al 2 O 3 ) nanoparticles through a porous wavy channel was investigated by Zeeshan et al. [4].In the presence of thermal radiation, the heat transfer and flow of a magnetohydrodynamic Go-water nanofluid between parallel surfaces were examined by Dogonchi et al. [5].They determined that the Nusselt number and temperature profile both increased as the radiation parameter decreased and the solid volume fraction increased.Dogonchi and Ganji [6] examined nanofluid flow, mass, and heat transfer between non-parallel surfaces subject to a magnetic field.They concluded that concentration and temperature profiles along with the Nusselt number go up as the Schmidt number rises.
Recently, several researchers have considered the natural convection flow of a pure fluid and nanofluid in enclosures [25][26][27][28][29][30][31] due to the extensive scientific and engineering applications.Heat transfer and fluid flow in a wavy enclosure subject to a porous medium and nanoparticles were explored by Sheremet et al. [25].They illustrated that an increasing Rayleigh number causes a secondary vortex and temperature stratification core in the left bottom corner and center of the cavity, respectively.In a wavy enclosure containing nanoparticles and a porous medium, natural convection and the influence of thermal dispersion was studied by Sheremet et al. [26].They determined that the heat transfer was enhanced with the Rayleigh number, undulation number, and dispersion parameter.Sheremet et al. [27] analyzed entropy generation and nanofluid flow in an enclosure.The outcomes indicate that a higher wavenumber leads to greater mean entropy generation.Numerical and experimental investigations on natural convection in a cavity were conducted by Mahmoudinezhad et al. [28].
This concise review illustrates that it is still worthwhile to allocate more effort to the study of natural convection in enclosures using a combination of metallic nanoscale particles and normal fluids.Therefore, this work analyzed magnetohydrodynamic nanofluid natural convection in an enclosure and the impact of the shape factor of the nanoparticles and thermal radiation using control-volume finite element method (CVFEM) and also obtained a correlation with the average Nusselt number.The flow and heat transfer effects of diverse control parameters were explored.

Problem Description
The natural convection of a magnetohydrodynamic nanofluid in an enclosure subject to the effects of thermal radiation and the shape factor of nanoparticles is analyzed.A sample grid distribution and a physical model of this work are displayed in Figure 1.We assume the nanofluid to be Newtonian and incompressible, while we assume that the flow is laminar.Under the above-mentioned suppositions and the Boussinesq approximation, the energy, momentum, and continuity equations for the current problem can be defined as: ∂v ∂y In these equations v, u, B 0 , P, σ f , T, and q rad denote the velocity in the y-direction, velocity in the x-direction, magnetic field, pressure, electric conductivity, and temperature, respectively, with the last variable representing radiative heat flux.
In these equations v, u, B0, P, σf, T, and qrad denote the velocity in the y-direction, velocity in the x-direction, magnetic field, pressure, electric conductivity, and temperature, respectively, with the last variable representing radiative heat flux.For radiation, by applying the Rosseland approximation, we have: So, Equation (4) has been altered to ρnf, μnf, (ρCp)nf are defined as follows: ( ) Considering the influences of nanoparticle shape, the nanofluid's thermal conductivity (knf) is characterized as follows: For radiation, by applying the Rosseland approximation, we have: So, Equation (4) has been altered to ρ nf , µ nf , (ρCp) nf are defined as follows: Considering the influences of nanoparticle shape, the nanofluid's thermal conductivity (k nf ) is characterized as follows: The shape factor is represented here as m.The shape factor for diverse particle shapes is illustrated in Table 1 [32].Further, the thermophysical features of the nanofluid are listed in Table 2 [15].The vorticity and stream function are stated as: We define the following dimensionless variables: Subject to Equation ( 14), the governing equations are changed to the following non-dimensional forms: Subject to the boundary conditions: In these equations, Pr, Ra, N, and Ha denote the Prandtl number, Rayleigh number, radiation parameter, and Hartmann number, respectively, which are defined as: Along the hot wall, the local and average Nusselt number can be determined by:

Numerical Solution and Validation
We solved Equations ( 11)-( 13) subject to the conditions in Equation ( 14) via the control-volume-based finite element method (CVFEM) [33][34][35][36].To ensure mesh independence, Nu ave was obtained for diverse mesh sizes.Table 3 indicates that a grid of 141 × 141 should be applied to the present problem.Further, we measured our FORTRAN code outcomes against other works in the literature to prove the numerical solution under study [37,38].They are in excellent accordance, as is exhibited in Table 4.

Results and Discussion
In this research, a numerical study was carried out to analyze the influences of the Hartmann number (Ha = 0-50), Rayleigh number (Ra = 10 3 , 10 4 , and 10 5 ), radiation parameter (N = 1-3), volume fraction of the nanofluid (φ = 0-0.04),and shape factor (m = 3, 6.3698, and 16.1576) on magnetohydrodynamic nanofluid natural convection in a cavity.The influences of thermal radiation and the shape factor of nanoparticles were determined.
The impacts of the Ra and Ha on isotherms and streamlines are illustrated in Figures 2 and 3, respectively.As shown by the increasing Ra, the stream function's maximum absolute value increases; hence, the flow moves more quickly in the cavity.This phenomenon is due to the intensity of the natural convection, which has a direct relationship with the stream function's maximum absolute value.On the other hand, a higher Ra leads to a distortion of the isotherms and consequently causes the vertical stratification of isotherms, particularly at lower Ha and higher Ra.That is, the thermal boundary layer develops along two side walls.This is because, at a higher Ra and a lower Ha, the prevailing mode of heat transfer is convection.Moreover, the power of natural convection and, consequently, the stream function's maximum absolute value both diminish as Ha goes up.In addition, as Ha goes up, the central streamline expands, and the core's temperature stratification diminishes.So, the two walls' thermal boundary layers will disappear.In fact, in this case, the isotherms are parallel to each other.Since an increasing Ha leads to a decreased buoyancy force, the conduction mode of heat transfer dominates over the convection mode, especially at higher Ha.
Figure 4 demonstrates the influences of N and φ on the local and average Nusselt number.Based on Equation ( 15), the multiplication of three terms, i.e., (k nf /k f ), ((4/3)N + 1), and ∂θ/∂X, will create a non-dimensional number called the local Nusselt number.By increasing φ, the growth of the ratio (k nf /k f ) will overcome a smaller ∂θ/∂X for a specified N, so the local Nusselt number will rise as φ is increased.Conversely, by increasing N, the increased ((4/3)N + 1) term will counteract the reduced value of ∂θ/∂X for a given φ, which will result in a higher Nu loc .However, the average Nu number behaves similarly to the local Nu number.
Figures 5 and 6 illustrate the impacts of Ra, Ha, and m (sphere, column, and lamina) on the local and average Nusselt number, respectively.According to Equation ( 8) and Table 1, for a specified nanofluid (for example, Cu-water), as the shape factor increases, (k nf /k f ) also increases.For instance, for a Cu-water nanofluid with φ = 2%, the spherical nanoparticle (m = 3) results in the lowest thermal conductivity rate (k nf /k f = 1.0609), while the laminar nanoparticle (m = 16.1576)yields the highest thermal conductivity rate (k nf /k f = 1.3216) of the studied various nanoparticle shapes in this work.Hence, based on Equation (15), by increasing the shape factor, the increased ratio (k nf /k f ) will overcome the reduction in ∂θ/∂X for a specified φ, so the local Nusselt number increases with an increased shape factor.One can easily surmise, then, that both the local and average Nusselt numbers increase with increasing Ra and decreasing Ha.These fluctuations of Nu ave.and Nu loc. with Ha and Ra can be explained as follows: lower values of Ra lead to a weakened buoyancy force in the system.So, in this case, the conduction mode of heat transfer governs the system.However, the convection mode dominates over the conduction mode at higher Ra because of the increased buoyancy force.Furthermore, introducing a magnetic field can lessen the power of the buoyancy force in the cavity.So, at higher Ha, the conduction mode will govern the system.
Finally, we defined the relationship for the average Nusselt number and the influencing parameters in the current study.The relationship is defined as follows:

Conclusions
The natural convection of a magnetohydrodynamic nanofluid in an enclosure subject to the effects of thermal radiation and the shape factor of nanoparticles was examined numerically using CVFEM.The investigation of nanofluid heat transfer and flow was conducted using an extensive range of Rayleigh numbers, radiation parameters, nanofluid volume fractions, Hartmann numbers, and nanoparticle shape factors.The outcomes indicate that the local and average Nusselt numbers increase with increasing Rayleigh number, radiation parameter, and nanofluid volume fraction, while they decrease with decreasing Hartmann number.Moreover, the laminar nanoparticle has a higher local Nusselt number than the other nanoparticle shapes.Also, in this study, a correlation for the average Nusselt number and the other factors studied was obtained with an R-squared of 0.9927.

Figure 1 .
Figure 1.(a) Physical model and coordinate system (b) grid distribution.

Figure 1 .
Figure 1.(a) Physical model and coordinate system (b) grid distribution.

Figure 4 .
Figure 4. (a) Local Nusselt (Nu) numbers for different values of volume fractions of the nanofluid and radiation parameters when Ra = 10 4 , Ha = 25, and m = 3, (b) average Nu number for different values of volume fractions of the nanofluid and radiation parameters.

Figure 4 . 16 Ra = 10 5 Figure 5 .
Figure 4. (a) Local Nusselt (Nu) numbers for different values of volume fractions of the nanofluid and radiation parameters when Ra = 10 4 , Ha = 25, and m = 3, (b) average Nu number for different values of volume fractions of the nanofluid and radiation parameters.

Figure 6 .
Figure 6.Average Nu number for different values of Rayleigh and Hartmann numbers and shape factors when N = 2, φ = 2%, and m = 3.

Figure 5 . 16 Ra = 10 5 Figure 5 .
Figure 5. Local Nu number for different values of Rayleigh and Hartmann numbers and shape factors when N = 1 and φ = 2%.

Figure 6 .
Figure 6.Average Nu number for different values of Rayleigh and Hartmann numbers and shape factors when N = 2, φ = 2%, and m = 3.

Figure 6 .
Figure 6.Average Nu number for different values of Rayleigh and Hartmann numbers and shape factors when N = 2, φ = 2%, and m = 3.

Table 4 .
Comparison between present results and other works for Nu avg .