Impacts of Heat Flux Distribution, Sloping Magnetic Field and Magnetic Nanoparticles on the Natural Convective Flow Contained in a Square Cavity

: In the present paper, the effect of the heat ﬂ ux distribution on the natural convective ﬂ ow inside a square cavity in the presence of a sloping magnetic ﬁ eld and magnetic nanoparticles is explored numerically. The nondimensional governing equations are solved in the framework of a ﬁ nite element method implemented using the Galerkin approach. The role played by numerous model parameters in in ﬂ uencing the emerging thermal and concentration ﬁ elds is examined; among them are: the location of the heat source and its length H (cid:1) , the mag-nitude of the thermal Rayleigh number, the nanoparticles shape and volume fraction, and the Hartmann number. It is found that the nano ﬂ uid velocity becomes higher when the thermal source length, the nanoparticles volume fraction and/or the thermal Rayleigh number are increased, while it decreases as the Hartmann number Ha grows and the position of the heat source moves toward the center of the lower wall of the cavity. Moreover, the temperature of the nano- ﬂ uid grows with the extension of the thermal source and decreases slowly when the heat ﬂ ux position moves toward the center of the lower wall. The outcomes of the research also indicate that the average Nusselt number becomes smaller on increasing Hartmann number Ha and heat source length H (cid:1) . The addition of Fe 3 O 4 to engine oil leads to a higher rate of heat transfer with respect to the addition of SiO 2 particles. Blade-shaped nanoparticles generate the highest value of the Nusselt number compared to all the other considered shapes.

Abstract: In the present paper, the effect of the heat flux distribution on the natural convective flow inside a square cavity in the presence of a sloping magnetic field and magnetic nanoparticles is explored numerically. The nondimensional governing equations are solved in the framework of a finite element method implemented using the Galerkin approach. The role played by numerous model parameters in influencing the emerging thermal and concentration fields is examined; among them are: the location of the heat source and its lengthH Ã , the magnitude of the thermal Rayleigh number, the nanoparticles shape and volume fraction, and the Hartmann number. It is found that the nanofluid velocity becomes higher when the thermal source length, the nanoparticles volume fraction and/or the thermal Rayleigh number are increased, while it decreases as the Hartmann number Ha grows and the position of the heat source moves toward the center of the lower wall of the cavity. Moreover, the temperature of the nanofluid grows with the extension of the thermal source and decreases slowly when the heat flux position moves toward the center of the lower wall. The outcomes of the research also indicate that the average Nusselt number becomes smaller on increasing Hartmann number Ha and heat source length H Ã . The addition of Fe 3 O 4 to engine oil leads to a higher rate of heat transfer with respect to the addition of SiO 2 particles. Blade-shaped nanoparticles generate the highest value of the Nusselt number compared to all the other considered shapes.

Introduction
In most of the industrial applications upsurge in heat transfer is demanded. One of the vital mechanisms enhancing heat transfer is natural convection utilizing nanofluids which are a notable subject to most recent research papers as a result of the developing need of engineering and nanotechnology applications. Nanometer-sized particles are added to regular fluids to produce nanofluids Choi [1]. Many industrial and mechanical applications have used nanofluids due to their significant advantages in augmenting heat transfer with less clogging and reduced pumping power. Types of geometry play a significant role in the augmentation of heat transfer in nanofluids. Every cavity has its characteristics that improve the heat transfer rate. To determine the influences of heat flux on the heat transfer improvement in unsteady buoyancy-driven flow inside a square cavity filled with nanofluid, a study was conducted by Nguyen et al. [2]. They found that the thermal transport properties of nanofluid significantly affect the cooling performance of a system and the rate of heat transfer declines at higher local thermal Rayleigh number. A mixed convection flow, with heat flux fixed at the bottom wall using nanofluids in a square enclosure, was simulated numerically by Mansour et al. [3]. They observed that an increase of the heat flux length, leads to a decrease in the Nusselt number whereas an increase of the nanoparticle concentration intensifies the rate of heat transfer. Oztop et al. [4] worked computationally to investigate the mixed convection hydromagnetic flow of nanofluids inside a partially heated lid-driven cavity. They claimed that the heat transfer rate decreases when the Hartmann number strengthens. They also noticed that the increase in nanoparticle concentration enhances the Nusselt number. Recently, Esfe et al. [5] conducted a numerical experiment to study the mixed convective flow in a lid-driven enclosure occupied by various nanofluids. They noticed that Richardson and Rayleigh's numbers are two important parameters enhancing the heat transfer rate. They also showed that the geometry inclination angle has extreme effects on the total entropy generation as well as on the heat transfer rate. Moreover, they found that mixing nanoparticles to the regular fluids, decreases the entropy generation and augments the heat transfer rate. Furthermore, Bondarenko et al. [6] investigated convective phenomena to determine the rate of heat transfer between two-adherent porous blocks in a lid-driven enclosure utilizing nanofluids. Their outcomes showed that a higher size of a porous block intensifies the Nusselt number. Astanina et al. [7] carried out the MHD free convection flow of nanofluids in a square cavity whose walls are partially heated under an externally applied magnetic field. Their outcomes exhibited that heat transfer rate inversely varied with the nanoparticle concentration, the strength of the applied magnetic field and the position of the heater. Free convection phenomena in a square cavity along with a circular cylinder whose side walls are adiabatic and the bottom wall temperature is variable, were investigated by Park et al. [8]. They noticed that isotherms are powerfully influenced by the temperature change of the bottom wall. Jahanbakhshi et al. [9] considered the effect of an external magnetic field and investigated the free convection flow of non-Newtonian fluid inside an L-shape enclosure. They found that Nusselt numbers are inversely related to the shear-thinning fluids and directly correlated with the shear-thickening fluids. Sheremet et al. [10] numerically simulated the free convection flow inside an inclined square cavity taking into consideration the time-sinusoidal variation of temperature. Their results showed that a growing oscillating frequency of the boundary tends to intensify the oscillation amplitude of the Nusselt number. Cho et al. [11] studied buoyancy-driven phenomena in a square cavity using an array of perpendicular circular and elliptical cylinders. They found that the solution reaches an unsteady state except for the value of the aspect ratio AR ¼ 2 and AR ¼ 4 for both lower and upper elliptical cylinders. In recent years, the rate of heat transfer for the buoyancy-driven flow is considered to be heavily and affected by the nanoparticle's shape, uniform sinusoidal roughness elements and nanoparticle random motion [12,13]. Mehryan et al. [14] studied the convection mechanism in a square enclosure occupied by solid, porous and free fluid layers. The outcomes of the study showed that the rate of heat transfer increases with the increase of the buoyancy ratio parameter Nr as well as with the increase of the Lewis number Le. Similarly, the numerical investigation for buoyancy-driven mechanism and total entropy generation in a square enclosure filled with nanofluids, using various distributions of temperature fields along with a concentric solid, was conducted by Alsabery et al. [15]. Their study showed that the inner solid size and thermal conductivity ratio respectively, control the heat transfer rate. Alsabery et al. [16] further studied numerically the buoyancy-driven flow in the presence of a corner heater and conducting block using a non-homogenous model in a square enclosure filled with nanofluids. They showed that an increase in the number of similar size blocks having the same thermal conductivity effectively intensify the Nusselt number. Sezai et al. [17] performed numerical investigation for buoyancy-driven flow induced by a discrete heat flux which was located at the bottom of a rectangular cavity. They displayed that the maximum of the Rayleigh number falls, as the heat source aspect ratio is increased. Free convection, with heat source fixed at the bottom wall of the cavity utilizing nanofluids, was examined by Aminossadati et al. [18]. They claimed that heater length and its location, vastly affect the heat transfer rate.
Saravanan et al. [19] studied numerically the buoyancy convection with two mutually orthogonal heatgenerating baffles in a closed square cavity. Their investigation showed that the length of any of the baffles, which was mounted inside the cavity, directly affected the rate of heat transfer whereas there is no effect for changing the positions of the baffles. Numerical computation for natural convection in a square cavity due to two mutually orthogonal isothermal baffles and the effect of the baffles sizes and positions inside the cavity was conducted by Kandaswamy et al. [20]. Their outcomes showed that the vertical baffle length regardless of its position enhanced the rate of heat transfer. Saravanan et al. [21] investigated numerically the buoyancydriven flow due to two mutually perpendicular heated thin plates with deferent boundary conditions in a square cavity. They found that the rate of heat transfer intensifies when one plate placed far away from the center of the cavity for isothermal boundary condition. Furthermore, buoyancy induced convection with discretely heat-generating baffles in a square cavity was examined numerically by Hakeem et al. [22]. They claimed that when the vertical baffle was located very close to the sidewall or the horizontal baffle was located very close to the bottom wall, the average Nusselt number reached its maximum.
Due to the potential applications of nanofluid in different fields of science and engineering such as solar energy, chemical, microchannel, bioengineering and automotive, it has received considerable attention by many researchers like Alsabery et al. [23,24], Al Kalbani et al. [25], Uddin et al. [26][27][28], Astanina et al. [29], Sathiyamoorthy et al. [30], Mehryan et al. [31] and Rahman et al. [32] who have studied the dynamics of nanofluids in various cavities. Mathematical modelling of nanofluids provides a representation of the parameters governing the problem that can be examined with the least possible cost, and thereby provide a good understanding of the situation. This can then be followed by experiments on the points of major interest. So far, the models that have been developed for nanofluids are one component model (Tiwari et al. [33]), two-component model (Buongiorno [34]) and nonhomogeneous dynamics model (Uddin et al. [35]) have been developed. Al-Balushi et al. [36] researched to investigate the free convection flow and heat transfer utilizing magnetic nanoparticles inside a square enclosure with a non-homogeneous dynamic model. They have explored the influences of the local thermal Rayleigh number, nanoparticles shape and nanoparticles concentration on the streamlines, isotherms, isoconcentrations, and the rate of heat transfer. Recently, Al-Balushi et al. [37] extended their previous work [36] taken into account the effects of an applied magnetic field and four different thermal boundary conditions which are uniform, parabolic in space, sinusoidal in space, and sinusoidal in time. They have also considered 12 different types of magnetic nanofluids. But both of their studies neglect the effects of heat flux on the flow and thermal fields. To the best of the authors' knowledge, no research has been carried out to investigate the free convection flow inside a square enclosure having constant heat flux at the bottom wall, using magnetic nanoparticles and sloping magnetic field, following non-homogeneous dynamic model. Therefore, this study aims to extend the work of Al-Balushi et al. [37] to study the natural convection flow of magnetic nanofluids considering heat flux, having variable length and position at the bottom wall of the square cavity, applying inclined magnetic field using nonhomogeneous dynamic model. In any real life engineering applications, the geometry is more complicated than an enclosure. However, the important insights of heat transfer of a very complex design can be provided by the simple geometry. The results of this study can be helpful in analyzing heat transfer in industrial applications such as heat exchangers, solar thermal collectors and cooling of electronic equipment.

Formulation of the Problem 2.1 Physical Model
Let us consider the two-dimensional free convection flow of nanofluids which is unsteady, laminar and incompressible, energized by a heat flux having length H inside a square cavity of length L. The flow and thermal fields are enforced by a uniform magnetic field making an angle c ð Þ to the positive xÀaxis. Fig. 1 shows the coordinates system and the geometry of the problem. Every mathematical model is based on some assumptions. The assumptions of our model are as follows: The heat flux q 00 ¼ Àj nf T y is embedded within the middle part of the lower wall of the cavity whereas the upper wall is reserved at low temperature T ¼ T C . Two vertical walls and the remaining part of the lower wall are kept insulated. Thermophoresis, Brownian diffusion, and gravity effects are taken into account. There are no chemical reactions and negligible thermal radiation within the flow domain.

Conservation Equation for Nanofluids
The governing continuity, momentum, energy and concentration equations are as follows (see Uddin et al. [35], Al-Balushi et al. [36,37]): where the density, effective viscosity, thermal diffusivity, volumetric thermal expansion, volumetric mass expansion, heat capacitance, thermal conductivity, electric conductivity of nanofluid are defined as follows (see Uddin et al. [35]): Figure 1: Physical model and coordinates system The nanoparticle shape factor n is known by n ¼ 3=É, where É is the sphericity, d p is the diameter of nanoparticle, is the numerical value of D T and j B is the Boltzmann constant. In the right hand side in Eq. (12), additional term comes from the random motion of nanoparticles.

Initial and Boundary Conditions
The initial and the boundary conditions for the current problem are as follows: For t > 0 ; At the top wall ð0 x L; y ¼ LÞ : At the left wall ðx ¼ 0; 0 y LÞ : At the right wall ðx ¼ L; 0 y LÞ : At the bottom wall : y ¼ 0

Thermophysical Properties
The properties of base fluids and nanoparticles built the nanofluid's thermophysical properties. For the present investigation, the magnitudes of density, viscosity, thermal conductivity, specific heat capacity, and volumetric thermal expansion coefficients have been considered. Tab. 1 (see Uddin et al. [38]) and Tab. 2 (Hafezisefat et al. [39], Khan et al. [40] and Al-Balushi et al. [37]) represent the thermophysical properties of base fluids and solid nanoparticles respectively.

The Average Nusselt Number
The significant quantity for the numerical simulation is the average Nusselt number ðNu ave Þ along the heated bottom wall. The ratio of convective to conductive heat transfer across the boundary is welldefined by the Nusselt number as follows:  (32) where h Ã is coefficient of the convective heat transfer of the nanofluid and can be expressed by Using non-dimensional variables defined in (21) we have The local Nusselt number for nanofluid for the present investigations is calculated as At the bottom heated wall, the average Nusselt number can be expressed as
In FEM, we need to generate proper meshes or grids to solve the boundary value problems effectively. On the generated networks, the physical variables such as velocity, temperature and pressure are determined. In Fig. 2, we have created the meshes of the current flow domain measuring their quality.

Grid Independency Test
In finite element method, it is essential to determine the grid size considering sufficient number of elements in the resolution field to obtain a converged solution. In the present analysis we have done so considering CoFe 2 O 4 -EO nanofluid when Pr ¼ 6:84, Ra T ¼ 10 6 , Ra C ¼ 10 4 , f Ã ¼ 0:1, D Ã ¼ 0:5, H Ã ¼ 0:4, Ha ¼ 10, c ¼ 30 , n ¼ 3, d p ¼ 10nm and s ¼ 10. Grid independency is tested considering six dissimilar non-uniform grids such as coarse, normal, fine, finer, extra fine and extremely fine having 1024, 1537, 2527, 6518, 17036 and 26436 number of elements in the resolution field. Fig. 3 displays the rate of heat transfer Nu ave ð Þ against the number of elements for the above-said grid sizes. This figure shows that value of Nu ave changes slightly for 17036 and 26436 elements. Thus, for the numerical simulation, 17036 elements for the grid independent solutions have been used.

Code Validity
The precision of the current code is checked for a base fluid with the outcomes of Nguyen et al. [2], Aminossadati et al. [18], and Cheikh et al. [47] when Ra T ¼ 10 4 , Pr

Results and Discussion
The numerically simulated results are obtained for an unsteady 2D free convection flow of magnetic nanofluids in the presence of a sloped magnetic field inside a square cavity for different model parameters especially the thermal Rayleigh number 10 5 Ra T 10 6 , nanoparticles solid volume fraction  Þ . Ferrite-water was considered nanofluid as default unless otherwise specified. Fig. 4 displays the influences of increasing the nanoparticles loading (or volume fraction) f Ã ð Þ and thermal buoyancy parameter on the streamlines for Pr ¼ 6:84, Ra T ¼ 10 6 , Ra C ¼ 10 4 , Ha ¼ 10,   is an indication of a high level of convection. Adding nanoparticles to a regular fluid reduces the temperature rate which is an indication of enhanced performance of cavity cooling. At Ra T ¼ 10 5 , as nanoparticles volume fraction intensifies, the isotherm lines become parallel to each other and become parabolic in shape at heat flux which signposts that conduction is the dominant mode of heat transfer. The influences of varying Ra T and f Ã on isoconcentrations loop when Pr ¼ 6:84, Ra T ¼ 10 6 , Ra C ¼ 10 4 , Ha ¼ 10, D Ã ¼ 0:5, H Ã ¼ 0:4, c ¼ 30 , n ¼ 3, d p ¼ 10 nm and s ¼ 10 are exhibited in Fig. 6. Remarkably, the loops of isoconcentrations have similar behavior of streamlines but their appearances are skinny and feeble due to the advanced Brownian diffusion. Also, levels of isoconcentrations, as well as their loops, are distributed in the entire domain of the enclosure and condensed on the cooled and adiabatic wall where temperatures are low due to thermophoresis phenomenon which intensifies buoyancy-driven force and concentrations of nanoparticles.

The Effects of Sloped Magnetic Field
Effects of magnetic field Ha ð Þon the flow, thermal and concentration fields when Pr ¼ 6:84, Ra T ¼ 10 6 ,  increased, the convection flow is affected by the magnetic field. The maximum surface velocity falls with the increase of Ha caused by the Lorentz force. Consequently, one bigger vortex rotates anticlockwise along with a small vortex that appears at the bottom right corner of the cavity. As Hartmann's number increases this small vortex becomes bigger and rotates clockwise. This also intensifies the strength and density of the streamlines. Fig. 7 further reveals that a stronger magnetic field (Ha ! 30) forced the isotherm lines to become almost parallel to each other and suppresses the convection currents. As expected, the isoconcentrations loops become lean and delicate because of the higher Brownian diffusion.
As the heat flux length the fluid motion within the cavity also intensifies due to the enhancement of the buoyancy force. Additionally, there appears a stronger anticlockwise circulation in streamlines within the entire cavity when H Ã strengthens. As the length of heat flux H Ã upsurges, the generated amount of heat

The Effects of Heat Flux Location
The evolutions of streamlines, isotherms and isoconcentrations with different heat flux locations D Ã = 0.2, 0.3, 0.4 and 0.5 are displayed in Fig. 9 when Pr ¼ 6:84, Ra T ¼ 10 6 , Ra C ¼ 10 4 , Ha ¼ 10, H Ã ¼ 0:4, f Ã ¼ 0:1, c ¼ 30 , d p ¼ 10 nm, n ¼ 3 and s ¼ 10. For streamlines, we detected a bigger vortex circulating clockwise whereas two small vortices located at the left (top) and the right (bottom) corners of The effects of D Ã on the isoconcentrations are also quite noticeable. As D Ã increases, the inner shape of the isoconcentrations loops changes from oblate to almost circular and the small vortices appear at corners, top left and bottom right, vanish.  Figs. 10a-10c illustrate that for a spherical shape nanoparticles, the Nu ave declines with the escalation of Ha, H Ã and D Ã while it rises with the increase of f Ã . In Fig. 10d, we demonstrates the variation of average Nusselt number considering spherical, brick, cylindrical, platelet and blade shape Fe 3 O 4 nanoparticles in water. This figure reveals that irrespective of the nanoparticles loading, the highest average Nusselt number is found for the blade shape nanoparticles and lowest for the spherical shape. It is significant that blade shape nanoparticles provide a 9.64% increase of the average Nusselt number as compared to the spherical shape.  Fig. 11. From the comparative analysis, we found that kerosene-based nanofluids provide the highest rate of heat transfer. Mn-Zn ferrite offers highest increase, 3.62% of the average Nusselt number compared to the SiO 2 nanoparticles which provide the lowest rate of heat transfer among magnetic nanoparticles having kerosene as base fluid at nanoparticles concentration level f Ã ¼ 0:05.

Conclusions
The unsteady buoyancy-driven heat transfer flow, in a square cavity energized by the heat flux at the bottom of the lower edge of the cavity, using a nonhomogeneous dynamic model under the influence of a sloping magnetic field, utilizing magnetic nanoparticles, has been simulated numerically. Comparisons are Figure 11: The average Nusselt number for (a) water, (b) engine oil, (c) kerosene with different nanoparticles when Pr ¼ 6:84, Ra T ¼ 10 5 , Ra C ¼ 10 3 , Ha ¼ 10, D Ã = 0.5, H Ã ¼ 0:2, c ¼ 30 , d p ¼ 10 nm, n ¼ 3 and s ¼ 2 made between current outcomes with the earlier published data and very good agreement is achieved. Influences of numerous model parameters especially the thermal Rayleigh number, the Hartmann number, the heat flux length, the heat flux location, the solid volume fraction and nanoparticles shape on the flow, thermal and concentration fields have been studied numerically. Our simulations have led the following major conclusions: The flow, thermal and concentration fields are modified remarkably with the increase of the thermal Rayleigh number, Hartmann number, heat flux length and position, and the nanoparticle loading. The heat transfer rate Nu ave decreases with the increase of Ha, H Ã , D Ã and f Ã . Kerosene based nanofluids noticeably provide the highest rate of heat transfer, a 4.33% increase compared to the engine oil which provides the lowest rate. Mn-Zn ferrite-Ke shows the highest heat transfer rate among magnetic nanofluids. It gives a 3.62% increase compared to the SiO 2 -Ke; which provides the lowest heat transfer rate. Blade, platelet, cylinder and brick shape nanoparticles provides almost 10%, 5.8%, 3.9%, and 1.6%, respectively, an increase in the average Nusselt number compared to the spherical shape.