Issue 
Eur. Phys. J. Appl. Phys.
Volume 88, Number 1, October 2019
Materials for energy harvesting, conversion, storage and environmental engineering (Icome 2018)



Article Number  11101  
Number of page(s)  10  
Section  Physics and Mechanics of Fluids, Microfluidics  
DOI  https://doi.org/10.1051/epjap/2019190239  
Published online  17 December 2019 
https://doi.org/10.1051/epjap/2019190239
Regular Article
Magnetic field impact on nanofluid convective flow in a vented trapezoidal cavity using Buongiorno's mathematical model^{★}
^{1}
Laboratory of Transport Phenomena, Faculty of Mechanical and Process Engineering, USTHB, B.P. 32, ElAlia BabEzzouar, 16111 Algiers, Algeria
^{2}
Sonatrach − Direction Centrale Recherche et Développement, Avenue du 1er novembre, Boumerdes 35000, Algeria
^{3}
Université de Lorraine, LERMAB, IUT Longwy, 54400 Cosnes et Romain, France
^{*} emails: mehdi_benzema@yahoo.fr
Received:
19
August
2019
Received in final form:
26
October
2019
Accepted:
4
November
2019
Published online: 17 December 2019
Numerical study for the effect of an external magnetic field on the mixed convection of Al_{2}O_{3}–water Newtonian nanofluid in a rightangle vented trapezoidal cavity was performed using the finite volume method. The nonhomogeneous Buongiorno model is applied for numerical description of the dynamic phenomena inside the cavity. The nanofluid, with low temperature and high concentration, enters the cavity through the upper open border, and is evacuated through opening placed at the right end of the bottom wall. The cavity is heated from the inclined wall, while the remainder walls are adiabatic and impermeable to both the base fluid and nanoparticles. After validation of the model, the analysis was carried out for a wide range of Hartmann number (0 ≼ Ha ≼ 100) and nanoparticles volume fraction (0 ≼ ϕ_{0} ≼ 0.06). The flow behavior as well as the temperature and nanoparticles distribution shows a particular sensitivity to the variations of both the Hartmann number and the nanofluid concentration. The domination of conduction mechanism at high Hartmann numbers reflects the significant effect of Brownian diffusion which tends to uniform the distribution of nanoparticles in the domain. The average Nusselt number which increases with the nanoparticles addition, depends strongly on the Hartmann number. Finally, a correlation predicting the average Nusselt number within such geometry as a function of the considered parameters is proposed.
© EDP Sciences, 2019
1 Introduction
One of the approaches to enhance the thermal performances in various engineering applications is the use of nanofluids as working fluid. This new class of fluids refers to a liquid in which nanometersized particles (1–100 nm) are dispersed. The nanoparticles can be made of metals or nonmetallic materials and are characterized by a better thermal conductivity compared to the base fluid [1–4]. It is clear that adding nanoparticles to the fluids affects their physical properties. For this purpose, rheological behavior is an effective criterion to design systems involving the nanofluid flow. The study of rheological characteristics of several nanofluids has therefore attracted the interest of many researchers. As reported in references [5–10], the nanofluids can exhibit either or both Newtonian and non Newtonian behaviors.
The mixed convection in cavities can be encountered in various technological and engineering applications such as household ventilation [11], heat exchanger, solar collectors [12], cooling of electronic devices [13], food processing [14] and many others. Many of these applications can be simplified into cavity problems with multiple ventilations ports. Due to the practical interest of the mixed convection heat transfer, a number of numerical works dedicated to the nanofluids flow within vented cavities such as [12,15–24]. The main aim of these investigations has been to understand the fundamentals of diverse cooling strategies, in order to enhance the thermal performance of applications involving limited spaces. However, in most of these works, the analyses are restricted to regular geometries, while in practice; actual cavities are often in complex or irregular shapes.
Magnetohydrodynamic (MHD) finds important application areas in physics and engineering such as for coolers of nuclear reactors, metals casting [25], turbulence control, plasma physics, boundary layer control theory and many others. The use of magnetic field as a controlling parameter for heat and mass transfer in cavities has been the subject of intensive investigations in recent years. However, a review of literature indicates that the MHD mixed convection of nanofluids within ventilated cavities remains to be investigated.
The aforementioned works are based on the singlephase approach which is the most frequently used to simulate the nanofluid behavior. Magyari [26] emphasized that, with a little rescaling effort, the singlephase model can reproduce similar results to that obtained with the clear fluid models. When a relative velocity occurs between the nanosized particles and the base fluid, the assumption of the homogeneity of nanofluids can no longer hold. The theoretical study of Buongiorno [27] shows that Brownian diffusion and thermophoresis are the most significant slip mechanisms in nanofluids. Incorporating these effects, Buongiorno [27] developed a twocomponent nonhomogeneous equilibrium model for convective transport in nanofluids. A number of numerical works have adopted the Buongiorno model to describe the convective phenomena in closed cavities. However, very few have considered this approach for ventilated cavity problems, which makes it a still fertile research topic.
In this context, Fersadou et al. [28] have recently performed a numerical study on MHD mixed convection flow and entropy generation of Cuwater nanofluid in a two interacting open rectangular cavities. They observed a nonmonotonic variation of the heat transfer rate with increasing the magnetic field strength. The average Nusselt number increases by growing the Richardson number, the nanoparticles concentration, the heat flux ratio as well as the width of the openings. They also noticed that the heat transfer irreversibility is the most dominating contribution in the entropy generation. However, the hydromagnetic irreversibility becomes the dominant one, at low Richardson numbers and large openings size. Sheremet et al. [29], studied the effects of Brownian motion and thermophoresis during the mixed convective flow of nanofluids in a vented square cavity filled with a porous medium. The authors found an improvement in heat transfer with increasing the modified Rayleigh number, the Reynolds number and the width of the inlet/outlets sections. Furthermore, the growth of the Reynolds number leads to a greater cooling of the cavity and the intensification of the thermophoretic effect. Sheremet et al. [30] used the Buongiorno model to investigate the doublediffusive mixed convection in a ventilated porous square cavity crossed by a nanofluid. The authors concluded that the average Nusselt number increases with growing both Rayleigh and Reynolds numbers, while it decreases with the usual Lewis number. The mass transfer rate is an increasing function of the usual Lewis number. Also, the average Nusselt and Sherwood numbers are nonmonotonic functions of the Rayleigh and Reynolds numbers and the thermophoresis parameter.
As discussed before and to the best of our knowledge, the use of Buongiorno's model in problem dealing with mixed convection of nanofluid in an oddshaped vented cavities, whether or not permeated by a magnetic field, has not been reported yet in the literature. This challenge can be considered as an open research area, and that's what makes the novelty of this contribution. The results of this study can be attractive for the thermal design of electronic devices, which can inadvertently be subjected to a magnetic field. Therefore, the present research intends to explore the effect of an external magnetic field on the local nanoparticles distribution and the mixed convective heat transfer characteristics of Al_{2}O_{3}water nanofluid in a twodimensional vented trapezoidal cavity. The nanofluid, which is treated as a nonhomogeneous mixture using the Buongiorno model, exhibits a Newtonian behavior according to Anoop et al. [7]. Thus, this study can be useful for heat transfer optimization with respect to various applications involving vented cavities.
2 Problem statement and model formulation
A schematic diagram of the considered domain including both dimensions and boundary condition is illustrated in Figure 1. It consists of a twodimensional vented cavity of rightangled trapezoid cross section. The cavity is crossed by Al_{2}O_{3}water nanofluid, and is under the influence of an external magnetic field of uniform strength B_{0}, oriented in the horizontal direction. The aspect ratio of the geometry W/H is fixed at 1.2 where W and H are, respectively, the width and the height of the cavity. The inclined wall is maintained at a constant temperature T_{H}, while the remainder walls are adiabatic. Except the openings, all boundaries are supposed to be impermeable to both the base fluid and nanoparticles.
A steady and laminar flow of nanofluid enters the cavity from the top at relatively low temperature (T = T_{C}) and at nanoparticles volume fraction ϕ_{0}. The fluid is evacuated through an opening placed at the right end of the bottom wall. Both openings have size l, equal to W/6. According to the rheological investigation of Anoop et al. [7], the working fluid exhibits a Newtonian behavior for nanoparticles volume fraction of 0.5–6 vol.% and a diameter of 50 nm. The fluid is assumed incompressible satisfying the OberbeckBoussinesq approximation. The effects of viscous dissipation, Joule heating, radiation and induced magnetic field are supposed to be negligible. In the present work, the nanofluid is treated as a nonhomogeneous mixture using the Buongiorno mathematical model [27]. This model incorporates the effects of Brownian motion and thermophoresis which are responsible for the particles migration in the fluid base. The nanoparticles mass flux may be formulated as:(1)
The Brownian diffusion coefficient D_{B} and the thermophoretic diffusion coefficient D_{T} can be expressed a follows:(2) (3)
The governing equations are nondimensionalized using the following reference parameters:(4)
Considering the aforementioned assumptions, the dimensionless governing equations can be written as follow:(5) (6) (7) (8)where V represents the non dimensional velocity vector (U, V), B ^{*} the magnetic field dimensionless vector (Ha ^{2}) and ϕ_{0} is the nanoparticles volume fraction at the cavity entrance.
The appropriate boundary conditions can be expressed as: at the inlet port, at the outlet port, at the heat source,
U = 0, V = 0, ∂ φ ^{∗}/∂ N = 0, ∂ θ/∂ N = 0 on all solid boundaries, where the Reynolds, Richardson, Prandtl, Hartmann and Lewis numbers as well as the diffusivity ratio parameter and the buoyancy ratio are, respectively, defined as: (9)
In this study, the empirical correlations proposed by Corcione [31] for thermal conductivity and viscosity of the nanofluid were employed:(10) (11)
The nanofluid electrical conductivity was described by [32] Maxwell as:(12)
The nanofluid properties are evaluated via the following mixture law:(13)where ζ stands for the following properties: the density ρ_{nf}, the specific heat (ρc_{p})_{nf} and the thermal expansion (ρβ)_{nf} of the nanofluid. Table 1 summarizes the thermophysical properties of the base fluid and Al_{2}O_{3} nanoparticles
The local and average Nusselt number on the heated surface can be expressed as follows:(14) where N is vector normal to the inclined active wall: .
Fig. 1 Schematic description of the studied cavity. 
3 Numerical procedure and CFD code verification
The set of flow governing equations (5)–(8) along with specified boundary conditions is numerically solved using the finite volume method. Integration of each equations of the system over the corresponding control volume leads to the following algebraic equation for every nodal point(15)where n is the iteration level, n_{b} are neighbor coefficients, b is the source term and φ refers to the discrete value of the dependent variable over the control volume. All space discretizations are performed applying the secondorder accuracy hybrid scheme. A staggered grid system together with the SIMPLER algorithm is adopted to avoid the checkerboard pressure problem. It amounts to storing the scalar variables (pressure, temperature and concentration) at the cell center while the velocity components are stored on the staggered control volumes. The resulting system of linear algebraic equations is iteratively solved using a linebyline method, combining the tridiagonal matrix algorithm (TDMA). The underrelaxation method has been required to ensure the convergence, which is reached when the following criterion is satisfied
For this study, a regular grid distribution in X and Ydirections is used. Figure 2 depicts the used mesh treatment for the inclined surface which is approached with staircaselike zigzag lines. This method is based on earlier numerical works [12,18].
The grid independence study for the considered problem is performed using the average Nusselt number as a sensitive parameter for Gr = 10^{5}, Re = 100, ϕ_{0} = 0.02 and Ha = 0, 80. The corresponding results are presented in Table 2. Accordingly, a grid size of 156 × 130 has been selected for this study. This grid size is fine enough to improve the accuracy of the normal gradient on the inclined surface, and to hold the nanoparticles mass flux condition zero.
The simulations were achieved on the basis of an inhouse FORTRAN computer code. To check its reliability, a comparison with selective experimental and numerical data from the published literature was performed. The first validation concerns the experimental study conducted by Minaei et al. [33], which deals with the case of mixed convection in an airfilled ventilated square cavity. The right vertical wall of the cavity is equipped with a discrete heat source. The experimental uncertainty was estimated by the authors to be 6.2% at most, for the local Nusselt number. Figure 3 shows the comparison of the average Nusselt number for five heat source locations at Re = 955 and three values of Rayleigh number. The maximum gap found between both result does not exceed 4.39%, which is acceptable taking into account the experimental error of 6.2%. Another validation is made by comparing the streamlines, the isotherms and the average Nusselt number with those obtained by Kaseipoor et al. [20] for the MHD convection flow in a vented Tshaped cavity crossed by Cuwater nanofluid. The results are presented in Figure 4, which shows a very good agreement. Further, Figure 5 depicts a comparison made between the present nanoparticles concentration field and those obtained numerically by Celli et al. [34], for the case of the natural convection of Al_{2}O_{3}water nanofluid, in a differentially heated square cavity using the Buongiorno model. The results are found to be in a good agreement with [34].
Fig. 2 Finitevolume grid for the computational domain. 
Grid independence study.
Fig. 3 Average Nusselt number according to the heat source location for different Rayleigh numbers. 
Fig. 4 Comparison of the present results with the numerical work of [20]. (a) The average Nusselt number at ϕ = 0.04. (b) Streamlines and isotherms at Re = 50, Ha = 80 for ϕ = 0 and 0.04. 
Fig. 5 Comparison of the nanoparticles distribution obtained by [34] and that of the present code at Ra = 5 and ϕ = 0.1. 
4 Results and discussion
A numerical study has been performed to investigate the effect of an external magnetic field on mixed convection of Al_{2}O_{3}water nanofluid flow in a trapezoidal ventilated cavity, using the nonhomogeneous Buongiorno model. The computations are performed for a wide range of Hartmann number (0 ≼ Ha ≼ 100), and nanoparticle volume fraction (0 ≼ ϕ_{0} ≼ 0.06). The predicted results are discussed through streamlines, isotherms, nanoparticles isoconcentrations, velocity profiles as well as local and average Nusselt number. A base set of conditions was defined as Gr = 10^{5}, Re = 100, Pr = 6.2 and Δ T = 5 K. The non dimensional parameters: Le, N_{BT} and N_{R} will vary according to the nanoparticle volume fraction ϕ_{0}.
The flow variations observed inside the vented cavity are the result of interaction between the forced convection due to the nanofluid injection and buoyancyinduced convection. Thus, the flow structure is characterized by a main flow (open streamlines) which can set in motion one or more induced secondary cells. Many investigations are devoted to analyze the effect of a magnetic field in cavities due to its ability to control the heat transfer [35]. The Lorentz force acts in a way to promote the fluid velocity in one direction at the expense of the other, in order to conserve the flow rate. It is expected in the present study, that the orientation of the magnetic field in the horizontal direction controls the downward flow intensity and the size of the recirculations.
Figure 6 shows the impact of Hartmann number on the distribution of streamlines and isotherms inside cavity at ϕ_{0} = 0.02. In absence of magnetic field, the flow pattern is characterized by a downward forced flow, along the inlettooutlet axis, setting in motion a large secondary clockwise (CW) rotating cell. For this case, the combined effects of inertia forces due the incoming flow, and buoyancy forces from the heat source lead to the expansion and acceleration of the CW cell which invades the major part of the cavity. By striking the bottom wall in the vicinity of the outlet, the cell causes a rise of the fluid within it. This is reflected by a curvature of the closed streamlines. The temperature field regarding this case, shows an important distortion of isotherms around the interaction zone between the forced flow and the convective cell, where high temperature gradients can be observed. However, the low momentum of the cell at the vicinity of the hot source favors the conduction heat transfer mechanism, confirmed by the stratification of the isotherms in this region. For a moderate value of Hartmann number (Ha = 10), the results have relatively the same characteristics as those observed previously. However, the closed streamlines are more regular, which indicates the weakening of the convective circulation. One can also notice a slight detachment of isotherms close to the hot wall, indicating the thickening of the thermal boundary layer. As notified before, the Lorentz force that exerts a resistance to the fluid motion, acts vertically against the downward flow and the buoyancy forces. Indeed, the analysis of the momentum equations shows that applying a magnetic field weakens the momentum in the Ydirection, and by virtue of flow rate conservation, promotes the Xdirection one. For higher values of Hartmann number (50 and 100), significant changes in the flow structure occur. A deployment of the forced flow can be observed, which squeezes the CW cell towards the bottom left corner. This trend increases with the intensification of magnetic field. At the same time with increasing Ha the hydrodynamic boundary layer along the hot wall is thinned, which is confirmed by a compacted distribution of open streamlines near this wall. A similar behavior is also observed for the thermal boundary layer. The temperature field reflects the suppression of the convective heat transfer mode, dominated by the conduction one. However, more intensive cooling of the cavity occurs in these cases.
Figure 7 is presented to highlight the effect of the Lorentz force on the vertical velocity profiles along the horizontal cross section (Y = 0.5, ϕ_{0} = 0.02). For Ha = 0, a high flow acceleration, ensured by forced flow, can be observed near the right adiabatic wall due to the continuing effect of inertia forces. One can also note an acceleration of the cell motion when interacting with the main flow, while it is attenuated in the left part of the cavity. As expected from the trend of streamlines, the flow structure when Ha = 10 remains unchanged, however, a reduction of V is observed on almost the entire cross section. The positive values of the velocity profiles when Ha reaches the value 20, indicate the widening of the main flow over the entire cross section. In this case, nanofluid motion tends to accelerate near the inclined wall, and to decelerate elsewhere. Indeed, the buoyancy forces, acting in the direction of the downward flow, lead the acceleration of the fluid motion near the hot wall further to the increase of its temperature. In addition, the effect of buoyancy decreases by increasing the Hartmann number beyond the value of 20, resulting in an almost flat velocity profile at Ha = 100, and strong velocity gradients near the inclined wall.
The distribution of nanoparticles inside the cavity for the different Hartmann number is illustrated in Figure 8. A general view of this figure shows that increasing the magnetic field strength tends to uniform the distribution of nanoparticles inside the domain. When Ha = 0, a compacted distribution of nanoparticles is encountered around the interaction zone between the forced flow and the CW rotating cell, where an intensive mass transfer occurs. The acceleration of the fluid motion in this zone generates strong velocity gradients that allow a migration of nanoparticles into the cell which subsequently are trapped within it. The major part of the main flow is then depleted of nanoparticles, and is characterized by a lower concentration than imposed at the cavity entrance i.e., ϕ ^{*} < 1. It should be noted that the mixing due to the strong convective cell favors the nonuniformity of the nanoparticles distribution. This trend attenuates slightly by increasing Ha to 10 further to the weakening of the cell. However, the continuous effect of the forced convection leads a more uniform distribution of the nanoparticles within the main flow. One can also find that the thickness of the nanoparticle boundary layer close to the interface of the hot wall and the cell is thinner than the thermal boundary layer. This is due to the fact that the Brownian diffusivity is smaller than the thermophoretic one, reflected by a relatively low value of N_{BT}, which in this case, equals to 0.16. Based on the present data, it can be deduced that the Brownian motion and the thermophoresis effects are negligible in mixed convection of Al_{2}O_{3}water nanofluid. For higher magnetic field intensities, the nanoparticles distribution obtained is dramatically different with respect to the previous cases. The nanoparticles are found to be uniformly distributed in almost all the domain. The weakening of the downward flow in the right parts of the cavity reflects the significant effect of Brownian diffusion which tends to homogenize this region. However, the forced convection becomes a prevailing mass transfer mode as one proceeds toward the inclined wall. This is related to the acceleration of the nanofluid motion close to the heat source. One can find a significant tightening of the nanoparticles isoconcentrations by approaching the active wall, where an increase of ϕ ^{*} occurs along this direction. Thus, the fluid layers adjacent to the inclined wall correspond to the most concentrated regions of the cavity. It is important to emphasize that, in this case, the inertial effects in the vicinity of the inclined wall restricts the thermophoresis contribution in this region and this, despite the strong temperature gradients occurring there.
Figure 9a shows the local Nusselt number distribution alongside AB for various Hartmann numbers at ϕ_{0} = 0.05. Regardless of the Hartmann number value, the heat transfer rate is locally enhanced at the vicinity of the inlet, due to the interaction of the incoming cold flow with the hot wall. Lowest heat transfer rate is found near the bottom left corner due to the weak convective circulation in this region. Furthermore, the magnetic field acts in a way to improve the heat transfer rate when S ≥ 0.16. It appears, however, that the heat transfer performances are provided at Ha = 20. For an easier understanding of this behavior, the hydrodynamic and thermal fields for ϕ_{0} = 0.05 are presented (Fig. 9b). A complex and multiple recirculation flow pattern can be observed inside the cavity when Ha = 0. Due to higher fluid viscosity, both size and intensity of the CW cell decrease. A flow separation occurs at the vicinity of the vertical wall leading to the formation of a two counterclockwise (CCW) rotating cell, a large and a small one. The increase of Ha to 20 leads the disappearance of the small CCW cell and the stretching of the large one towards the left, which squeezes the main flow on the inclined wall. This provides strong velocity and temperature gradients close to the heater, and hence, a better cooling. This behavior attenuates at Ha = 100 when all the cells vanished.
Figure 10 shows the effects of Hartmann number and nanoparticles volume fraction ϕ_{0} on the average Nusselt number. It is clearly observed that increasing the nanoparticles concentration at the cavity entrance enhances the heat transfer rate due to the increase in the nanofluid thermal conductivity. Increment of nanoparticles volume fraction from ϕ_{0} = 0 to ϕ_{0} = 0.02 results in the decrease of the average Nusselt number up to a minimum value, depending on ϕ_{0}, and then increases. This occurs as a result of deceleration of the CW cell motion at moderate Ha values, which leads to increase the thickness of the boundary layer close to the active wall. Beyond the critical values of Ha, the magnetic field intensity becomes large enough to create strong velocity and temperature gradients close to the heat source, which allows an efficient cooling of the cavity. Different trend is observed when the nanoparticles volume fraction rage from 0.03 to 0.06. One can observe that the average Nusselt number remains almost constant as long as Ha ≤ 17.5. Beyond this value, Nu_{m} increases until reaching a maximum at a critical Hartmann number (Ha = 20), and subsequently decreases. By reexamining the flow pattern of Figure 9b, one can see that a large portion of thermal energy is transferred from the heater to forced flow. The behavior for low Hartmann numbers is related to the prevailing effect of inertia forces which restricts the action of Lorentz force. The heat transfer enhancement at Ha = 20 is attributed to the effect of the CCW cell, as previously explained. However, increasing Ha beyond the value of 20 reduces the beneficial effect of this cell due to the decrease of its size. In this range of ϕ_{0}, the contribution of magnetic field in the improvement of heat transfer is affected by adding nanoparticles. The increasing fluid viscosity tends indeed to depress the wall shear caused by the Lorentz force.
Summarizing the results of Figure 10, a predictive correlation relating the heat transfer rate within such cavity to the Hartmann number and nanoparticles volume fraction ϕ_{0} is proposed . The correlation can be formulated as follow:(16)
The correlation is obtained with a linear regression coefficient R ^{2} = 0.95. The expressions of the correlation coefficients are given in Table 3.
Fig. 6 Streamlines and isotherms for different Hartmann numbers at ϕ_{0} = 0.02. 
Fig. 7 Effect of Hartmann number on the velocity profile along the horizontal cross section of the cavity (Y = 0.5) for ϕ_{0} = 0.02. 
Fig. 8 Effect of Hartmann number on the local nanoparticles distribution at ϕ_{0 = }0.02. 
Fig. 9 Profiles of local Nusselt number along the heat source (a), distribution of streamlines and isotherms (b) at ϕ_{0} = 0.05, for various Hartmann numbers. 
Fig. 10 Average Nusselt number versus Hartmann number for several values of ϕ_{0}. 
Expressions of the correlation coefficients.
5 Conclusion
The MHD mixed convective heat transfer in a trapezoidal vented cavity crossed by Newtonian nanofluid (Al_{2}O_{3}water) has been numerically performed. In order to investigate the effect of a magnetic field on the local nanoparticles distribution inside the cavity, the nonhomogeneous Buongiorno model is taken into account. Simulations have been conducted for a wide range of Hartmann number and the nanoparticles volume fraction. The study discloses following salient observations.

The flow pattern is sensitive to the variations of the Lorentz force intensity and nanoparticles volume fraction. Thus, when ϕ_{0} = 0.02 a unicellular flow structure is formed inside the cavity, where increasing Ha affects the size and intensity of the recirculation zone. However, at ϕ_{0 }= 0.05, the increase in Hartmann number from 0 to 20 makes the fluid flow pass from a multicellular to a bicellular structure. Moreover, the intensification of magnetic field leads to suppress convection heat transfer dominated by the conduction mechanism.

It is found that for the lowest Ha values, the mixing due to the convective cell favors the nonuniformity of the nanoparticles distribution. In contrast, the continuous effect of the forced convection leads a more uniform distribution of the nanoparticles within the main flow. The domination of conduction mechanism at the higher magnetic field intensities reflects the significant effect of Brownian diffusion which tends to uniform the distribution of nanoparticles in almost all the domain.

For a constant Hartmann number, the average Nusselt number increases with the nanoparticles addition. Increment of nanoparticles volume fraction from ϕ_{0} = 0 to ϕ_{0} = 0.02 results in the decrease of Nu_{m} up to a minimum value, depending on ϕ_{0}, and then increases. However, for 0.03 ≼ ϕ_{0} ≼ 0.06, the average Nusselt number remains almost constant as long as Ha ≤ 17.5. Beyond this value, Nu_{m} increases until reaching a maximum at a critical Hartmann number (Ha = 20), and subsequently decreases.
As an extension to this research, we suggest to investigate the effect of the outlet port position to get the best configuration. It is also interesting to examine the second law of thermodynamics using the concept of entropy generation minimization to find the optimal thermal performances of the cavity.
List of symbols
B_{0} : Magnetic induction (Tesla)
c_{p} : Specific heat capacity (J kg^{−1} K^{−1})
d : Dimensional length of the heat source (m)
D_{B} : Brownian diffusion coefficient (m^{2}/s)
D_{T} : Thermal diffusion coefficient (m^{2}/s)
g : Gravitational acceleration (m s^{–2})
k : Thermal conductivity (W m^{−1} K^{−1})
k_{b} : Boltzmann's constant (1.38066 × 10^{−23 }J K^{−1})
M : Molecular weight of the base fluid (kg mol^{−1})
N_{BT} : Ration Brownian and thermophoretic diffusivities
Nu_{m} : Average Nusselt number
S : Segment along the inclined wall line
T_{fr} : Freezing point of the base fluid (K)
u, v : Velocity components (m s^{−1})
U_{0} : Velocity of the flow at the inlet (m s^{−1})
U, V : Dimensionless velocity components
x, y : Dimensional cartesian coordinates (m)
X, Y : Dimensionless Cartesian coordinates
Greek letters
α : Thermal diffusivity(m^{2} s^{−1})
β : Thermal expansion coefficient (K^{−1})
ϕ : Nanoparticle volume fraction
µ : Dynamic viscosity (kg m^{−1} s^{−1})
υ : Kinematic viscosity (m^{2} s^{−1})
σ : Fluid electrical conductivity (Ω^{1}m^{−1})
ψ : Dimensionless stream function
Subscript
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Cite this article as: Mahdi Benzema, Youb Khaled Benkahla, Ahlem Boudiaf, SiefEddine Ouyahia, Mohammed El Ganaoui, Magnetic field impact on nanofluid convective flow in a vented trapezoidal cavity using Buongiorno's mathematical model, Eur. Phys. J. Appl. Phys. 88, 11101 (2019)
All Tables
All Figures
Fig. 1 Schematic description of the studied cavity. 

In the text 
Fig. 2 Finitevolume grid for the computational domain. 

In the text 
Fig. 3 Average Nusselt number according to the heat source location for different Rayleigh numbers. 

In the text 
Fig. 4 Comparison of the present results with the numerical work of [20]. (a) The average Nusselt number at ϕ = 0.04. (b) Streamlines and isotherms at Re = 50, Ha = 80 for ϕ = 0 and 0.04. 

In the text 
Fig. 5 Comparison of the nanoparticles distribution obtained by [34] and that of the present code at Ra = 5 and ϕ = 0.1. 

In the text 
Fig. 6 Streamlines and isotherms for different Hartmann numbers at ϕ_{0} = 0.02. 

In the text 
Fig. 7 Effect of Hartmann number on the velocity profile along the horizontal cross section of the cavity (Y = 0.5) for ϕ_{0} = 0.02. 

In the text 
Fig. 8 Effect of Hartmann number on the local nanoparticles distribution at ϕ_{0 = }0.02. 

In the text 
Fig. 9 Profiles of local Nusselt number along the heat source (a), distribution of streamlines and isotherms (b) at ϕ_{0} = 0.05, for various Hartmann numbers. 

In the text 
Fig. 10 Average Nusselt number versus Hartmann number for several values of ϕ_{0}. 

In the text 
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