Eur. Phys. J. Plus (2018) 133: 66 DOI 10.1140/epjp/i2018-11893-3

THE EUROPEAN PHYSICAL JOURNAL PLUS

Regular Article

Numerical study for forced MHD convection heat transfer of a nanofluid in a square cavity with a cylinder of constant heat flux Amin Hassanpour1 , A.A. Ranjbar2 , and M. Sheikholeslami2,a 1 2

Department of Mechanical Engineering, Mazandaran University of Science and Technology, Babol, Iran Department of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran Received: 19 January 2017 / Revised: 4 January 2018 c Societ` Published online: 21 February 2018 –  a Italiana di Fisica / Springer-Verlag 2018 Abstract. In this research, flow and forced convection heat transfer of a water-copper nanofluid in the presence of magnetic field is studied. The walls of the square ventilation cavity are insulated. The dominating equations are solved by implementing the finite-volume method (FVM) using the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm. The effects of Hartmann number, nanoparticles volume fraction and Reynolds number on the flow and heat transfer characteristics were examined. The results demonstrate that increasing Reynolds and Hartmann numbers lead to increase the average Nusselt number. By evaluating the geometrical parameters, it was found that the size and number of vortices in the flow field decrease by increasing the inlet width. Besides, the increase of the average Nusselt number occurs with the increase of the inlet width. Moreover, it has been observed that the effect of the Hartmann number is more pronounced for higher Reynolds numbers.

1 Introduction Researches about effects of magnetic field on fluid and works in this area have numerous practical and beneficial applications; a couple of these areas are procedures in metals industry, controlling the liquid metals in the process of continuous casting, plasma welding and electrolytic cells of the hall process. One of the applications of this process is in nuclear industry where a coating of liquid metal is used due to its high heat transfer characteristics and shielding feature under the effect of the magnetic field. Also, using magnetic field in medicine is one of the important topics in magnetic science. High blood viscosity is a heart disease factor; when the blood concentration increases, it is possible that blood vessels get injured and this increases the risk of heart attacks. At the moment, the only treatment method is using drugs. An alternative method is putting the individual in a magnetic field which improves blood circulation throughout the body [1]. On the other hand, ventilation pipes and cavities have many applications in industry including coolers of electronic devices, heat exchangers design, air conditioning, food industries processes, solar collectors and so on. Therefore, checking the heat transfer and the behavior of the fluid flow in these types of cavities under the effect of magnetic field seems highly essential. Researchers have always tried to help heat transfer in procedures with new methods, and, using nanofluids is one of them. Fluids with suspended nanoparticles inside them are called nanofluids. Nanofluid is a term which was suggested by Choi in 1995 in the US’s Argonne laboratory. Nanofluids could be considered as the next generation of heat transfer fluids because they provide new exciting possibilities to increase heat transfer in comparison with pure liquids. These new possibilities feature superior properties in comparison with conventional heat transferring fluids and also fluids containing particles in the microsize scale. Suspending nanoparticles in liquids with different bases can change heat transfer and flow characteristics in the base fluid [2]. Many researchers have assumed nanotechnology as an important factor for a huge industrial revolution. The low heat transfer of conventional fluids like air, water and oil is the main obstacle to improving performance of heat exchangers. Increasing nanoparticles in a pure fluid, so called nanofluid, can improve the heat conduction of the compound [3]. Nanofluids can be used for a wide range of sources including electronic systems such as microprocessors, micro electro-mechanical systems and in the biotechnology application areas [4]. a

e-mail: [email protected] (corresponding author)

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Eur. Phys. J. Plus (2018) 133: 66

Benos et al. [5] studied the natural convection flow inside a shallow cavity in the presence of an external vertical steady magnetic field. Selimefendigil and Oztop [6] investigated the mixed convection of copper-water nanofluid in a lid-driven square enclosure using the finite-element method. They concluded that a magnetic field could be used to control the local heat transfer. It was found that increasing the volume fraction of nanoparticles improves heat transfer. Pekmen et al. [7] utilized the dual reciprocity boundary element method (DRBEM) to solve unsteady flow in channels under the effect of external magnetic field. They demonstrated that, by increasing magnetic Reynolds numbers, magnetic potential and the flow density circulate near the sudden changes of the wall. Rahman et al. [8] used the finite element in a numerical form to study heat transfer in an open channel in combination with a square cavity which had been warmed in the left part in partial and perfect form. It was observed that, for high values of the Hartmann number, the length of the warm part has a trivial effect on the flow field. On the contrary, in the partially warmed part, more heat transfer occurs at higher Rayleigh numbers. The results showed that in both cases the flow and the heat transfer power increase by increasing the Rayleigh number. The flow speed decreases by increasing the Hartmann number and this reduces the flow and heat transfer power. Besides, a thin boundary layer was observed for both studied samples for smaller Hartmann numbers. Luo et al. [9] carried out a numerical study of the effects of radiation heat on magneto-hydrodynamic flow and heat transfer in the presence of an external magnetic field in a dissimilar warm cavity. They found out that by increasing the Hartmann number, the isothermal surface has considerable changes and the flow structure and isothermal lines change at different levels in the cubic cavity. The conductor, radiation and total Nusselt number decrease. It means that the magnetic force can reduce the fluid flow and heat transfer. The effect of the Hartmann number in three-dimensional structure of the flow mainly exists in the center of a cubic cavity and the maximum transverse speed drops with Hartmann number. Selimefendigil and Oztop [10] simulated mixed convection heat transfer of nanofluid in a lid-driven enclosure with a flexible wall under magnetic influence. It was observed that the average heat transfer decreases with decreasing the Richardson number and increasing the Hartmann and Rayleigh number. Kasaeipoor et al. [11] conducted numerical studies on convection heat transfer of the copper-water nanofluid mixture in a T-shaped cavity in the presence of an external magnetic field. They solved dominating governing equations by the finite-volume method. Their results showed that at low Reynolds numbers, Nusselt and Hartmann numbers gradually increase but at high values of Reynolds numbers, this rise is more considerable. The results showed that the presence of nanoparticles causes raise in heat transfer, except for the situation where Re = 100, Ha < 10 and also Re = 400 and Ha < 60 in which pure water has a higher thermal rate than the nanofluid. They also concluded that by increasing the cavity proportion the amount of heat transfer increases. Sheikholeslami Kandelousi [12] studied the effect of variable magnetic field on nanofluid flow and heat transfer. He found that increasing Ha leads to an increase in the Lorentz force which tends to resist against the fluid flow and, as a result, it decreases the speed of nanofluid. Selimefendigil et al. [13] simulated a mixed convection in a square cavity with a flexible wall filled with silicon nanofluid under the influence of an adiabatic cylinder in the center and the production of volumetric heat. Their investigations indicated that the average Nusselt number increases with the volume fraction of all the various nanoparticles, spherical, brick and blade shapes, and that this increase in average Nusselt number is higher in the fluid with cylindrical particles. Sheikholeslami et al. [14] studied forced convection heat transfer in a porous semi-annulus under the effect of a uniform magnetic field. The results indicated that a nanofluid with platelet particles has a better heat transfer rate. Their results demonstrated that increasing the Hartmann number causes the temperature gradient to decrease. Azizian et al. [15] inspected the effect of external magnetic field on convection heat transfer and pressure drop of the magnetic nanofluid in condition of laminar-flow regime. In fact, the main part of their research is the effect of power and uniformity of the magnetic field on the convection heat transfer coefficient. The results indicated that the major increase in local heat transfer coefficient could be achieved by increasing the power and gradient of the magnetic field. The highest rise in local transmission coefficient develops near this regime and it occurs when a big magnetic flux gradient is applied in the radial direction. The main mechanism was taken responsible for the convection of heat transfer of the particles transmission by the magnetic field gradient. Sheikholeslami and Seyednezhad [16] simulated the effect of an electric field on the nanofluid natural convection. Sheikholeslami [17] investigated the nanofluid free convection in a porous medium. They considered the effect of an electric field on nanofluid behavior. Sheikholeslami [18] investigated water-based nanofluid flow in the mesence of magnetic field. They considered the Brownian motion effect on nanofluid behavior. Sheikholeslami and Rokni [19] investigated the thermal radiation effect on nanofluid flow in a porous medium. They applied an electric field to enhance the rate of heat transfer. Sheikholeslami [20] presented a ¨ mesoscopic approach for nanofluid flow in a porous channel. Selimefendigil and Oztop [21–23] investigated the effect of an external magnetic field on nanofluid treatment in various applications. They proved that magnetic field can be considered as a control tool for heat transfer.

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Fig. 1. Geometry and boundary conditions of the problem.

In this study, the flow and forced convection heat transfer of the nanofluid in a square ventilated cavity in the presence of magnetic field has been investigated. The finite-volume method has been utilized to simulate this problem. Effects of active methods on nanofluid hydrothermal behavior are investigated.

2 Problem formulation Figure 1 shows the geometry of the problem. A hot cylinder is inserted inside the square cavity. The input and output of the square ventilation are Wi and Wo , respectively (Wi = Wo = 0.2H). The nanofluid enters the cavity with speed ui and temperature Ti and exits from that. The cavity walls are insulated. The magnetic field is employed vertically. 2.1 Governing equations Governing equations are as follows. Continuity: ∂u ∂v + = 0. ∂x ∂y Momentum: ∂v 1 ∂u +v =− ∂x ∂y ρnf ∂v 1 ∂u +v =− u ∂x ∂y ρnf

u

 ∂2u ∂2u σnf B02 − + u, 2 2 ∂x ∂y ρnf  2  ∂ ν ∂2ν ∂p + υnf . + ∂y ∂x2 ∂y 2 ∂p + υnf ∂x

Thermal energy: u

(1)

∂T ∂T +v = αnf ∂x ∂y





∂2T ∂2T + 2 ∂x ∂y 2

(2) (3)

 ,

(4)

where u and v are the speed components in x and y axis directions. T is the fluid temperature, p is the pressure and B0 is the strength of the magnetic field and σ is electrical conductivity of the fluid. The nanofluid used in this study is a water-copper nanofluid where the thermo physical properties of water and copper are being presented in table 1 according to ref. [24]. A low magnetic Reynolds number is employed [25]. The nanoparticle properties have been defined as [26] ρnf = (1 − φ)ρf + φρs , (ρcp )nf = (1 − φ)(ρcp )f + φ(ρcp )s , σnf = (1 − φ)σf + φσs .

(5) (6) (7)

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Eur. Phys. J. Plus (2018) 133: 66 Table 1. Thermo physical properties of water and Cu [24]. Water

Cu

cp (J/kg K)

4179

385

ρ (kg/m3 )

997.1

8933

μ (kg/ms)

9.09 × 10−4

–

k (W/mK)

0.613

401

−1

−5

)

21 × 10

σ (S/m)

0.05

β (K

1.67 × 10−5 5.97 × 107

To calculate the effective thermal conductivity of the nanofluid, the Maxwell-Garnett model has been used [27]: ks + (n − 1)kf − (n − 1)(kf + ks )φ knf , = kf ks + (n − 1)kf + (kf − ks )φ

(8)

where n is the shape coefficient of nanoparticles and it has been determined equal to 3 for spherical nanoparticles [16]. Dynamic viscosity could also be estimated by using Brinkman formula [28] as μnf = μf (1 − φ)−2.5 . The Hartmann number which shows the ratio of the Lorenz to the viscous forces is given by  σnf . Ha = B0 H ρnf vf

(9)

(10)

The local Nusselt number on the hot wall of the cylinder is defined as Nu = −

knf (∂T /∂n)S . kf Ts − Ti

(11)

The average Nusselt number is achieved through integrating the local Nusselt number over the hot surface of the cylinder:  1 s Nu = Nu ds. (12) S 0 Calculations are performed for a range of Reynolds numbers: Re =

ui H . vf

(13)

3 Numerical details The dominating equations have been solved and simulated by the commercial code of FLUENT 6.3.26. We used the finite-volume method with structure mesh. We considered the residuals under 1e-6. To choose the appropriate mesh, an investigation had been conducted on number of grids. For this process, the effect of the number of grids on the average Nusselt number was verified for Re = 500, Ha = 40, ϕ = 0.04. The shape of fig. 2 demonstrates the study of independence from the mesh. As can be observed, with more than 3500 numbers, there is no significant change in Nusselt number, because of that, meshing has been chosen with 3500 mesh numbers. The current simulation results has been compared to the analytical model of the perfectly developed laminar flow of magneto-hydrodynamic [29,30] and the results have been validated. Results have been shown in fig. 3 and as can be seen, there is a good conformity and correlation between the current simulation results and the analytical model:   Ha [cosh(Ha/2) − cosh((Ha/2)(2Y − 1))] . (14) u(y) = 2 [(Ha/2) cosh(Ha/2) − sinh(Ha/2)] For better verification of the current simulation results for the nanofluid, regarding the discussion in [29], the convection heat transfer of a water-aluminum oxide nanofluid in a micro channel under the effect of simulated magnetic field and the average Nu number at Re = 100 was achieved for the nanofluid for ϕ = 0.01, 0.02, 0.03, 0.04 in three different Hartmann numbers. Results have been presented in table 2. Also, isothermal and streamlines have been shown in fig. 4 at Re = 10 and ϕ = 0.04 under the effect of Ha = 20 by comparing the results, it is observed that the current study has a good accuracy.

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Fig. 2. Grid independency study for different grid, Re = 500, Ha = 20, ϕ = 0.04.

Fig. 3. Validation of the current work with analytical solution, ref. [29, 30]. Table 2. Comparison of the average Nusselt number of the present work with ref. [30]. Re = 100 Ha = 0 Num

Ha = 20 Num

Ha = 40 Num

ϕ = 0.01

ϕ = 0.02

ϕ = 0.03

ϕ = 0.04

Present work

4.363

4.409

4.448

4.488

ref. [30]

4.282

4.394

4.506

4.618

Error (%)

1.89

0.03

1.29

2.81

Present work

5.485

5.539

5.607

5.633

ref. [30]

5.446

5.573

5.700

5.827

Error (%)

0.72

0.61

1.63

2.81

Present work

5.958

6.030

6.096

6.159

ref. [30]

5.985

6.117

6.248

6.378

Error (%)

0.45

1.42

2.43

3.43

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Fig. 4. Comparison of present work with the results obtained from Ha = 0 (. . . ), Ha = 20 (—).

Fig. 5. Streamlines for pure-water (. . . ) and nanofluid 0.04 (—) for various Re and Ha, D = 0.3H, wi = 0.2H, wo = 0.2H.

4 Results and discussion 4.1 Effects of Reynolds and Hartmann numbers Figure 5 shows the streamlines of the nanofluid ϕ = 0.04 and the pure water in the cavity for Re = 50, 250, 500 under the effect of Ha = 0, 20, 40. As can be seen by increasing the Reynolds number, the transfer of the fluid inside the cavity increases and this causes some vortexes inside the cavity and by increasing the Re number, the number of

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Fig. 6. Isotherms for pure-water (. . . ) and nanofluid 0.04 (—) for various Re and Ha, D = 0.3H, wi = 0.2H, wo = 0.2H.

these vortexes has also increases. But, by increasing the Hartmann number the Lorentz force increases and this causes reduction of flow convection of the fluid in the cavity and this is obvious from the reduction of vortexes due to the rise of the Ha number. The greater number of vortexes has been seen at Re = 500 and Ha = 0. Also by increasing the Ha number, the flow lines are extended towards the walls. At Re = 50, Ha = 20 and Re = 50, 250 and Ha = 40 the flow near the cylinder becomes two branches and with connecting the input lines at the output of the cavity, the flow lines become almost symmetrical. Figure 6 shows that isothermal lines move from the hot inside cylinder towards the exit of the cavity. By increasing the Re number, the thickness of the thermal boundary layer around the hot cylinder decreases and the temperature gradient increases. By comparing the isothermal lines of nanofluid and the pure water, it is observed that the effect of heat on nanofluid is greater than that on pure water. Figure 7 shows local Nusselt numbers at different Re. The values of the local Nusselt number are increased with increase in Reynolds number. The local Nusselt numbers are almost uniform at Re = 50, although they have more variations at higher Re, and in the area where the thermal boundary layer has reached its maximum value, the local Nusselt number is the lowest. Figure 8 shows the local Nusselt numbers at different Ha. It can be seen from the isothermal lines (fig. 6) that the thermal boundary layer has reduced by increase in Hartmann number. This is also clearly evident in the values of local Nusselt numbers, so that the range of local Nusselt numbers which have small values, have been reduced with the increase in Hartmann number. Figure 9 presents average Nusselt numbers at different Re and Ha. Results show that for each constant value of Ha, the average Nusselt number increases by increasing Reynolds number and this rise is more significant at low Reynolds numbers. In fact, the distribution of heat on the nanofluid at low speeds is better done. Also, at each Reynolds by increasing Ha, the average Nusselt number increases an increase which is considerable at higher Re. Table 3 shows the effect of the Hartmann number on the average Nu at different Re. The results show that at all of the Ha numbers, the average Nu increases. Also, the greater effect of Ha on heat transfer, at Re = 500 and Ha = 40 is an almost 26.74% increase in the average Nusselt number and has the least effect at Re = 50 and Ha = 20 with a 1.87% increase.

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Eur. Phys. J. Plus (2018) 133: 66

Fig. 7. Variation of local Nusselt numbers at different Reynolds numbers.

Fig. 8. Variation of local Nusselt numbers at different Hartmann numbers.

Fig. 9. Variation of the average Nusselt number with Re for various Ha (ϕ = 0.02).

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Table 3. Average Nusselt number at various Re and Ha (ϕ = 0.02). Ha = 0

Ha = 20

Ha = 40

Re = 50 Num

10.469

10.655

11.303

Increase (%)(a)

0.0

1.87

7.97

Re = 250 Num

21.740

24.488

24.817

Increase (%)

0.0

12.64

14.15

Re = 500 Num

28.459

33.457

36.070

Increase (%)

0.0

17.56

26.74

hN

m −Nm,Ha=0 Nm,Ha=0

(a)

i

× 100.

Table 4. Average Nusselt number at various Re, Ha and ϕ. Re

Ha

ϕ

Num

Increase (%)

50

0

0

8.989

0.0

0.04

12.068

34.25

0

9.670

0.0

0.04

13.064

35.09

0

18.797

0.0

0.04

24.887

32.40

0

21.112

0.0

0.04

28.779

36.41

0

24.690

0.0

0.04

32.474

31.52

0

30.809

0.0

0.04

41.739

35.48

40 250

0 40

500

0 40

4.2 Effect of particle volume fraction Table 4 shows the average Nusselt values for different volume fraction of nanoparticles at Re = 50, 250, 500 and Ha = 0, 20, 40. By increasing the volume fraction of the nanoparticles, the heat transfer rate increases for all of the Reynolds and Hartmann numbers. For example, increasing the average Nu at Re = 50 for the volume fraction of 0.04 is equal to 34.25% and for this volume fraction of nanoparticles at Re = 250, the average Nusselt has a decrease of about 36.41%. Figure 10 shows the average Nusselt numbers, regarding the volume fraction of nanoparticles for different Reynolds and Hartmann numbers. Because of the used model for the nanofluid, the change in volume fraction of nanoparticles, changes the thermophysical properties of the nanofluid. Since the nanoparticles have high heat conductivity, the average Nusselt number increases at all Reynolds and Hartmann numbers which shows that nanoparticles have caused improvement of heat transfer, it can also be observed that by increasing Reynolds and Hartmann numbers, the average Nusselt number increases. 4.3 Effect of geometry on the flow and heat transfer 4.3.1 Effect of cylinder diameter Figure 11 shows the streamlines in the cavity with an internal cylinder diameter of 0.4H. By comparing the flow contours with the cavity with internal cylinder diameter D = 0.3H (fig. 5), it could be observed that the size of the formed vortices have been decreased with increase in the diameter of the cylinder. The formed vortex at Re = 250 and Ha = 20 in the cavity with D = 0.4H, is removed.

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Fig. 10. Average Nusselt number at various Re, Ha and ϕ.

Fig. 11. Streamlines for various Re and Ha, D = 0.4H, ϕ = 0.04.

As can be seen in the fig. 12, the formed thermal boundary layer around the cylinder has been reduced in comparison to the cavity with internal cylinder diameter D = 0.3H. Figure 13 shows the streamlines in the cavity with an internal cylinder diameter of 0.2H. compared to the larger cylinder in diameter, the number and size of vortices have been increased. As can be seen in fig. 14, the isothermal lines for all cases except for Re = 250 and 500, Ha = 0, are identical to the isothermal contours for cavities with D = 0.3H and 0.4H.

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Fig. 12. Isotherms for various Re and Ha, D = 0.4H, ϕ = 0.04.

Fig. 13. Streamlines for various Re and Ha, D = 0.2H, ϕ = 0.04.

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Eur. Phys. J. Plus (2018) 133: 66

Fig. 14. Isotherms for various Re and Ha, D = 0.2H, ϕ = 0.04.

Fig. 15. Variation of Num for different sizes of the cylinder diameter.

Figure 15 shows the variation of Nusselt number for different sizes of the cylinder diameter inside the cavity. By increasing the cylinder diameter, the average of Nusselt number is increased for all values of Ha and Re. At Re = 50, the average amount of Nusselt number is increased for all Hartmann values with increase in cylinder diameter, however, the Nusselt numbers at a constant diameter have nearly equal values. The Nusselt values in Re = 250, and Ha = 20 and 40, have a little difference at a given diameter, although, at Re = 500, the difference of the average Nusselt number at a constant diameter has been increased for all Hartmann values.

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Fig. 16. Variation of Num for different inlet width.

Fig. 17. Variation of Num for different outlet width.

4.3.2 The effects of inlet width Figure 16 shows the variation of Nusselt number for different inlet widths. It is clearly obvious that the average Nusselt number has been increased with increase in the inlet width.

4.3.3 The effects of outlet width In fig. 17, the variation of Nusselt number is shown for different outlet widths of the cavity. The average Nusselt number has not been changed significantly, although, at Re = 250 and 500, the average Nusselt number decreased with the increase in wo to 0.2H, but by increasing wo to 0.3H, the values of the average Nusselt is almost equal to the case with wo = 0.2H.

5 Conclusion In this study, forced convection heat transfer of the water-copper nanofluid inside a square ventilated cavity in which there is a cylinder with constant heat flux, was numerically investigated and the effect of Reynolds and Hartmann numbers and volume fraction of nanoparticles was checked on the heat transfer and the fluid flow and the following results were generally achieved:

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Eur. Phys. J. Plus (2018) 133: 66

– Increase of Reynolds number causes increase of vortexes in the flow field; these vortexes decrease with the magnetic field rising and they are not observed at Re = 50, Ha = 20, 40 and Re = 250, Ha = 40. – Increasing the Reynolds and Hartmann numbers, increases the average Nusselt on the hot cylinder surface; this increase is more significant at low Reynolds number at a constant Hartmann number. – Also, at a constant Reynolds number, the increase of Ha has a greater effect on thermal performance and it enhances the average Nusselt increase. – Adding nanoparticles to water and increasing the volume fraction of the nanoparticles causes increase of heat transfer at all Reynolds and Hartmann numbers. – By evaluating the geometrical parameters, it was found that the size and number of vortices in the flow field, decrease with increase in the inlet width, and the increase of average Nusselt has a direct relation with the increase in the inlet width. Moreover, by increasing the cylinder diameter in the cavity, the average Nusselt number increases and the impact of the increase in the Hartmann value on the average Nusselt number, mostly shows as higher Reynolds numbers. We would like to thank the National Elites Foundation of Iran (http://www.bmn.ir) for their moral and financial support throughout this project.

Nomenclature B0 cp D Ha H k Nu Nu, Num q  Re S Ti To u, v

Magnetic field strength Specific heat of fluid Diameter Hartmann number Height of the cavity Thermal conductivity Local Nusselt number on the heated surface Average Nusselt number Uniform heat flux Reynolds number ui L/vf Perimeter of the cylinder Inlet flow temperature Temperature of the inner heated surface Velocity components in the x-and y-directions

Wi Wo Greek α μ ρ σ φ

Width of cavity inlet Width of cavity outlet symbols Thermal diffusivity Dynamic viscosity Density Electrical conductivity Solid volume fraction

Subscripts f Pure fluid nf Nanofluid s Nanoparticle

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