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Research Papers

# Magnetohydrodynamic CuO–Water Nanofluid in a Porous Complex-Shaped EnclosureOPEN ACCESS

[+] Author and Article Information
M. Sheikholeslami

Department of Mechanical Engineering,
Babol Noshirvani University of Technology,
P.O. Box 484,
Babol 47148-71167, Iran
e-mail: mohsen.sheikholeslami@yahoo.com

Houman B. Rokni

Department of Mechanical
and Materials Engineering,
Tennessee Technological University,
Cookeville, TN 38505

1Corresponding authors.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF THERMAL SCIENCE AND ENGINEERING APPLICATIONS. Manuscript received September 18, 2016; final manuscript received January 8, 2017; published online April 19, 2017. Assoc. Editor: Ali J. Chamkha.

J. Thermal Sci. Eng. Appl 9(4), 041007 (Apr 19, 2017) (6 pages) Paper No: TSEA-16-1269; doi: 10.1115/1.4035973 History: Received September 18, 2016; Revised January 08, 2017

## Abstract

Steady nanofluid convective flow in a porous cavity is investigated. Darcy and Koo–Kleinstreuer–Li (KKL) models are considered for porous media and nanofluid, respectively. The solutions of final equations are obtained by control volume-based finite element method (CVFEM). Effective parameters are CuO–water volume fraction, number of undulations, and Rayleigh and Hartmann numbers for porous medium. A correlation for Nuave is presented. Results depicted that heat transfer improvement reduces with the rise of buoyancy forces. Influence of adding nanoparticle augments with augment of Lorentz forces. Increasing Hartmann number leads to decrease in temperature gradient.

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## Introduction

Free convection utilized for various uses such as ventilation control, aircraft cabin insulation, cooling of electronic equipment, etc. Magnetic field has been used for metallurgy, polymer industry, cooling of continuous strips, etc. Nanofluid was proposed as innovative way to enhance heat transfer. Chamkha and Ismael [1] examined the combined conduction–convection nanofluid heat transfer in a permeable enclosure. They showed that the triangular wall thickness is an effective parameter. Ismael and Chamkha [2] reported the nanofluid flow in a composite cavity. Sheikholeslami and Ganji [3] presented various applications of nanofluid in their review paper. Sheremet et al. [4] simulated the unsteady magnetohydrodynamic (MHD) flow in an enclosure. Ismael et al. [5] investigated the entropy production of nanofluid in a permeable media. Bondareva et al. [6] utilized the heatline analysis for a titled permeable enclosure. Hayat et al. [7] utilized a two-phase model for nanofluid under the influence of radiation. They showed that the temperature gradient reduces with augment of thermal radiation. Suganthi et al. [8] used ZnO–ethylene glycol as coolants fluid. Chen et al. [9] studied the performance of solar collectors by using a silver nanoparticle. Selimefendigil and Oztop [10] examined nanofluid conjugate conduction–convection mechanism in a titled cavity. Sheikholeslami and Ellahi [11] selected an lattice Boltzmann method to simulate Lorentz forces influence on nanofluid convective heat transfer. They depicted that the temperature gradient reduces with the augment of magnetic strength. Sheremet et al. [12] reported nanofluid natural convection in a wavy porous enclosure. MHD nanofluid free convective hydrothermal analysis in a tilted wavy enclosure was presented by Sheremet et al. [13]. Influence of nonuniform Lorentz forces on nanofluid flow style has been studied by Kandelousi [14]. Influence of Coulomb forces on ferrofluid convection was analyzed by Sheikholeslami and Chamkha [15]. They concluded that augmenting Coulomb force has more profit in low Reynolds number. Andreozzi et al. [16] studied the impact of nanofluid and ribs on heat transfer in a channel. The MHD nanofluid thermal radiation was presented by Sheikholeslami et al. [17]. Sheikholeslami et al. [18] reported the impact of inconstant Lorentz force on forced convection. Several researches have been carried out in recent years [1931].

The goal of this article is to investigate the impact of constant magnetic field on nanofluid natural convection in a porous media with sinusoidal hot cylinder. CVFEM is chosen to simulate this paper. Impacts of CuO volume fraction, Hartmann and Rayleigh numbers for permeable media on heat transfer treatment are considered.

## Problem Definition

Figure 1 shows the geometry, boundary condition, and sample element. The formula of the inner cylinder is

Display Formula

(1)$r=rin+A cos (N(ζ−ζ0)), A=0.5$

in which $rin,rout$ are radius of the base circle and outer cylinder, respectively.

## Governing Equation and Simulation

###### Governing Formulation.

Two-dimensional steady convective flow of nanofluid in a porous medium is considered in existence of the constant horizontal magnetic field. The partial differential equations are Display Formula

(2)$∇⋅V=0$
Display Formula
(3)$μnfKV=(∇p+I×B+ρnf g )$
Display Formula
(4)$(ρCp)nf(V⋅∇)T=knf ∇2T$
Display Formula
(5)$∇⋅I=0$
Display Formula
(6)$σnf(V×B−∇φ)=I$

In Eq. (2), Darcy model is used for porous medium. Equations (5) and (6) reduce to $∇2φ=0$ [32]. So the electric field can be neglected [33], and hence, the above equations turn into Display Formula

(7)$∂v∂y+∂u∂x=0$
Display Formula
(8)$−Kμnf∂p∂x−σnfKB02μnf(−u sin2γ+v sin γ cos γ)=u$
Display Formula
(9)$−Kμnf∂p∂y−σnfKB02μnf(−v cos2γ+u sin γ cos γ)+g(ρβ)nfKμnf(T−Tc)=v$
Display Formula
(10)$(ρCp)nf(u∂T∂x+v∂T∂y)=knf (∂2T∂x2+∂2T∂y2)$

$(ρCp)nf,(ρβ)nf$, $ρnf$ and $σnf$ are defined as Display Formula

(11)$(ρCp)nf=(ρCp)f(1−ϕ)+(ρCp)sϕ$
Display Formula
(12)$(ρβ)nf=(ρβ)f(1−ϕ)+(ρβ)sϕ$
Display Formula
(13)$ρnf=ρf(1−ϕ)+ρsϕ$
Display Formula
(14)$σnf/σf=1+3(σ11−1)ϕ/[(σ11+2)−(σ11−1)ϕ],σ11=σs/σf$

$kn f,μn f$ are obtained according to Koo–Kleinstreuer–Li (KKL) model [34] Display Formula

(15)$knf=1+3(kpkf−1)ϕ(kpkf+2)−(kpkf−1)ϕ+5×104g′(ϕ,T,dp)ϕρfcp,fκbTρpdp,Rf=dp/kp,eff−dp/kp,Rf=4×10-8km2/Wg′(ϕ,T,dp)=(a1+a2Ln(dp)+a3Ln(ϕ)+a4Ln(ϕ)ln(dp)+a5Ln(dp)2)Ln(T)+(a6+a7Ln(dp)+a8Ln(ϕ)+a9ln(dp)Ln(ϕ)+a10Ln(dp)2)$
Display Formula
(16)$μnf=μf(1−ϕ)2.5+kBrowniankf×μfPr$

All required coefficients and properties are illustrated in Tables 1 and 2 [34].

Introducing dimensionless quantities Display Formula

(17)$Ψ=ψ/αnf, (X,Y)=(x,y)/L, θ=T−TcTh−Tc$

By discarding the pressure, the final equations are Display Formula

(18)$∂2Ψ∂X2+∂2Ψ∂Y2=−A6A5Ha[∂2Ψ∂Y2( sin2γ)+∂2Ψ∂X2( cos2γ)+2∂2Ψ∂X ∂Y(sin γ) (cos γ)]−A3 A2A4 A5∂θ∂XRa$
Display Formula
(19)$(∂2θ∂Y2+∂2θ∂X2)=−∂θ∂Y∂Ψ∂X+∂Ψ∂Y∂θ∂X$

where $Ra=g K (ρβ)fL ΔT/(μf αf)$ and $Ha=σfK B02/μf$ are the Rayleigh and Hartmann numbers for the porous media. Also $Ai(i=1..6)$ are constants parameters which are obtained as Display Formula

(20)$A1=ρnf/ρf, A3=(ρβ)nf/(ρβ)f, A5=μnf/μf, A2=(ρCP)nf/(ρCP)f,A4=knf/kf, A6=σnf/σf$

and boundary conditions are Display Formula

(21)$θ=1.0 on inner wall; θ=0.0 on outer wall; Ψ=0.0 on all walls$

Local and average Nusselt over the cold cylinder can calculate as Display Formula

(22)$Nuloc=A4∂Θ∂r,Nuave=π−1∫π/23π/2Nuloc(ζ) dζ$

###### Numerical Procedure.

Linear interpolation is utilized for approximation of variables in the triangular element which is considered as a building block (Fig. 1(b)). Algebraic equations are solved via Gauss–Seidel method. More details exist in reference book.

## Grid Independent Test and Validation

Different grids are studied to access the grid independent out puts. As demonstrated in Table 3, a grid size of $71×211$ should be chosen for the further study. Figure 2 shows the accuracy of fortran code for free convection and nanofluid heat transfer [35,36]. Table 4 depicts that our code is validated for magnetohydrodynamic heat transfer [37].

## Results and Discussion

Lorentz forces influence on nanofluid flow in a porous enclosure with complex hot cylinder is investigated. Darcy model is taken into account to present governing equations. Numerical simulations are examined for various amount of Rayleigh number for porous medium ($Ra=102,250$ and $103$), number of undulations ($N=3,4,5$ and $6$), volume fraction of CuO–water (ϕ = 0 and 0.04), and Hartmann number for porous medium (Ha = 0 to $20$). Figures 35 demonstrate the impacts of number of undulations, Rayleigh and Hartmann numbers for porous medium on streamlines and isotherms. As nanofluid temperature augments, the nanofluid begins moving from the warm surface to the outer one and dropping along the circular cylinder, afterward mounting again at the inner cylinder, generating a clockwise revolving vortex inside the cavity. In dominance of conduction mode, isotherms follow the shape of cylinders. $|Ψmax|$ augments as buoyancy force augments and it reduces as Lorentz force enhances. For odd values of N, very small eddy generates around $ζ=90 deg$. This phenomenon is observed because of opposition of the crest in contradiction of the flow movement between the hot and cold walls. But there is no secondary eddy near $ζ=90 deg$ for even values of N because the crest is parallel to the gravity force. As buoyancy forces enhances, eddies become stronger and thermal plume generate near the center line. Increasing Hartmann number causes the thermal plume to diminish and change the heat transfer mechanism from convection to conduction. So, Nu reduces with the rise of Lorentz forces. In the absence of Lorentz force when N = 3, Ra = 1000, two vortexes which rotates in reverse direction appear near the vertical center line. So strong thermal plume appears in this region. As Lorentz force augments, these two vortexes merge together, and the thermal plume vanishes. Figure 6 depicts the influence of $N, ϕ ,Ra$, and $Ha$ for porous media on $Nuave$. The correlation for $Nuave$ is

Display Formula

(23)$Nuave=0.98+5.5×10−4Ra−0.016N+0.51ϕ−0.021Ha−3.2×10−6Ra N−1.71×10−5Ra ϕ−3.6Ra Ha+0.15 Nϕ+3.15×10−4N Ha+0.016ϕ Ha+1.4×10−7Ra2+7.9×10−3N2+46.5ϕ2+1.16×10−3Ha2$

The root mean-squared error of this formula is equal to 0.98. As buoyancy force augments, temperature gradient augments and in turn $Nuave$ enhances with the rise of buoyancy forces. As nanofluid volume fraction augments, temperature of the fluid enhances and $Nuave$ augments with the enhance of thermal conductivity of the fluid. Enhancing Lorentz force causes the nanofluid flow to retard, and the Nusselt number reduces. Impacts of number of undulations, buoyancy and Lorentz forces on heat transfer improvement are demonstrated in Table 5 and Fig. 7. Adding CuO nanoparticle into water can enhance the thermal conductivity of the fluid. So this passive method is more effective when conduction mechanism is dominant. According to this fact, heat transfer improvement augments with the rise of Lorentz forces, and it reduces with the enhance of buoyancy forces. As the number of undulations augments, E enhances. Maximum value of E is 12.43 which is obtained at N = 6, Ha = 20, Ra = 100.

## Conclusions

Magnetohydrodynamic nanofluid convection in a porous cavity is investigated. The Darcy model is utilized for porous media, and the KKL model is applied to estimate the properties of nanofluid. Vorticity stream function formulation is utilized to discard the pressure source term. The obtained equations are solved via CVFEM. Isotherms, streamlines, and Nusselt number are presented for various amounts of CuO–water volume fraction, number of undulations, and Rayleigh, Hartmann numbers for the porous medium. The results depict that adding nanoparticles are more effective for higher $Ha$. $Nuave$ augments with the enhance of buoyancy forces, but it attenuates with the increase of Lorentz forces.

## Nomenclature

• A =

amplitude

• B =

magnetic field

• E =

heat transfer enhancement

• $g$ =

gravitational acceleration vector

• Ha =

Hartmann number

• $I$ =

electric current

• K =

permeability of the porous medium

• Nu =

Nusselt number

• Ra =

Rayleigh number

• T =

fluid temperature

• V, U =

vertical and horizontal dimensionless velocity

• Y, X =

vertical and horizontal space coordinates

Greek Symbols
• α =

thermal diffusivity

• β =

thermal expansion coefficient

• ζ =

rotation angle

• Θ =

dimensionless temperature

• μ =

dynamic viscosity

• ρ =

fluid density

• σ =

electrical conductivity

• Ω and Ψ =

dimensionless vorticity and stream function

Subscripts
• c =

cold

• f =

base fluid

• loc =

local

• nf =

nanofluid

## References

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Ismael, M. A. , and Chamkha, A. J. , 2015, “ Conjugate Natural Convection in a Differentially Heated Composite Enclosure Filled With a Nanofluid,” J. Porous Media, 18(7), pp. 699–716.
Sheikholeslami, M. , and Ganji, D. D. , 2016, “ Nanofluid Convective Heat Transfer Using Semi Analytical and Numerical Approaches: A Review,” J. Taiwan Inst. Chem. Eng., 65, pp. 43–77.
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Ismael, M. A. , Armaghani, T. , and Chamkha, A. J. , 2016, “ Conjugate Heat Transfer and Entropy Generation in a Cavity Filled With a Nanofluid-Saturated Porous Media and Heated by a Triangular Solid,” J. Taiwan Inst. Chem. Eng., 59, pp. 138–151.
Bondareva, N. S. , Sheremeta, M. A. , Oztopc, H. F. , and Abu-Hamdeh, N. , 2016, “ Heatline Visualization of MHD Natural Convection in an Inclined Wavy Open Porous Cavity Filled With a Nanofluid With a Local Heater,” Int. J. Heat Mass Transfer, 99, pp. 872–881.
Hayat, T. , Nisar, Z. , Yasmin, H. , and Alsaedi, A. , 2016, “ Peristaltic Transport of Nanofluid in a Compliant Wall Channel With Convective Conditions and Thermal Radiation,” J. Mol. Liq., 220, pp. 448–453.
Suganthi, K. S. , Leela Vinodhan, V. , and Rajan, K. S. , 2014, “ Heat Transfer Performance and Transport Properties of ZnO–Ethylene Glycol and ZnO–Ethylene Glycol–Water Nanofluid Coolants,” Appl. Energy, 135(15), pp. 548–559.
Chen, M. , He, Y. , Zhu, J. , and Wen, D. , 2016, “ Investigating the Collector Efficiency of Silver Nanofluids Based Direct Absorption Solar Collectors,” Appl. Energy, 181, pp. 65–74.
Selimefendigil, F. , and Öztop, H. F. , 2016, “ Conjugate Natural Convection in a Cavity With a Conductive Partition and Filled With Different Nanofluids on Different Sides of the Partition,” J. Mol. Liq., 216, pp. 67–77.
Sheikholeslami, M. , and Ellahi, R. , 2015, “ Three Dimensional Mesoscopic Simulation of Magnetic Field Effect on Natural Convection of Nanofluid,” Int. J. Heat Mass Transfer, 89, pp. 799–808.
Sheremet, M. A. , Oztop, H. F. , Pop, I. , and Al-Salem, K. , 2016, “ MHD Free Convection in a Wavy Open Porous Tall Cavity Filled With Nanofluids Under an Effect of Corner Heater,” Int. J. Heat Mass Transfer, 103, pp. 955–964.
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Andreozzi, A. , Manca, O. , Nardini, S. , and Ricci, D. , 2016, “ Forced Convection Enhancement in Channels With Transversal Ribs and Nanofluids,” Appl. Therm. Eng., 98, pp. 1044–1053.
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Sheikholeslami, M. , 2016, “ Influence of Coulomb Forces on Fe3O4-H2O Nanofluid Thermal Improvement,” Int. J. Hydrogen Energy, 42(2), pp. 821–829.
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Sheikholeslami, M. , Hayat, T. , and Alsaedi, A. , 2016, “ MHD Free Convection of Al2O3–Water Nanofluid Considering Thermal Radiation: A Numerical Study,” Int. J. Heat Mass Transfer, 96, pp. 513–524.
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## References

Chamkha, A. J. , and Ismael, M. A. , 2013, “ Conjugate Heat Transfer in a Porous Cavity Filled With Nanofluids and Heated by a Triangular Thick Wall,” Int. J. Therm. Sci., 67, pp. 135–151.
Ismael, M. A. , and Chamkha, A. J. , 2015, “ Conjugate Natural Convection in a Differentially Heated Composite Enclosure Filled With a Nanofluid,” J. Porous Media, 18(7), pp. 699–716.
Sheikholeslami, M. , and Ganji, D. D. , 2016, “ Nanofluid Convective Heat Transfer Using Semi Analytical and Numerical Approaches: A Review,” J. Taiwan Inst. Chem. Eng., 65, pp. 43–77.
Sheremet, M. A. , Pop, I. , and Roşca, N. C. , 2016, “ Magnetic Field Effect on the Unsteady Natural Convection in a Wavy-Walled Cavity Filled With a Nanofluid: Buongiorno's Mathematical Model,” J. Taiwan Inst. Chem. Eng., 61, pp. 211–222.
Ismael, M. A. , Armaghani, T. , and Chamkha, A. J. , 2016, “ Conjugate Heat Transfer and Entropy Generation in a Cavity Filled With a Nanofluid-Saturated Porous Media and Heated by a Triangular Solid,” J. Taiwan Inst. Chem. Eng., 59, pp. 138–151.
Bondareva, N. S. , Sheremeta, M. A. , Oztopc, H. F. , and Abu-Hamdeh, N. , 2016, “ Heatline Visualization of MHD Natural Convection in an Inclined Wavy Open Porous Cavity Filled With a Nanofluid With a Local Heater,” Int. J. Heat Mass Transfer, 99, pp. 872–881.
Hayat, T. , Nisar, Z. , Yasmin, H. , and Alsaedi, A. , 2016, “ Peristaltic Transport of Nanofluid in a Compliant Wall Channel With Convective Conditions and Thermal Radiation,” J. Mol. Liq., 220, pp. 448–453.
Suganthi, K. S. , Leela Vinodhan, V. , and Rajan, K. S. , 2014, “ Heat Transfer Performance and Transport Properties of ZnO–Ethylene Glycol and ZnO–Ethylene Glycol–Water Nanofluid Coolants,” Appl. Energy, 135(15), pp. 548–559.
Chen, M. , He, Y. , Zhu, J. , and Wen, D. , 2016, “ Investigating the Collector Efficiency of Silver Nanofluids Based Direct Absorption Solar Collectors,” Appl. Energy, 181, pp. 65–74.
Selimefendigil, F. , and Öztop, H. F. , 2016, “ Conjugate Natural Convection in a Cavity With a Conductive Partition and Filled With Different Nanofluids on Different Sides of the Partition,” J. Mol. Liq., 216, pp. 67–77.
Sheikholeslami, M. , and Ellahi, R. , 2015, “ Three Dimensional Mesoscopic Simulation of Magnetic Field Effect on Natural Convection of Nanofluid,” Int. J. Heat Mass Transfer, 89, pp. 799–808.
Sheremet, M. A. , Oztop, H. F. , Pop, I. , and Al-Salem, K. , 2016, “ MHD Free Convection in a Wavy Open Porous Tall Cavity Filled With Nanofluids Under an Effect of Corner Heater,” Int. J. Heat Mass Transfer, 103, pp. 955–964.
Sheremet, M. A. , Oztop, H. F. , and Pop, I. , 2016, “ MHD Natural Convection in an Inclined Wavy Cavity With Corner Heater Filled With a Nanofluid,” J. Magn. Magn. Mater., 416, pp. 37–47.
Kandelousi, M. S. , 2014, “ Effect of Spatially Variable Magnetic Field on Ferrofluid Flow and Heat Transfer Considering Constant Heat Flux Boundary Condition,” Eur. Phys. J. Plus, 129, p. 248.
Sheikholeslami, M. , and Chamkha, A. J. , 2016, “ Electrohydrodynamic Free Convection Heat Transfer of a Nanofluid in a Semi-Annulus Enclosure With a Sinusoidal Wall,” Numer. Heat Transfer, Part A, 69(7), pp. 781–793.
Andreozzi, A. , Manca, O. , Nardini, S. , and Ricci, D. , 2016, “ Forced Convection Enhancement in Channels With Transversal Ribs and Nanofluids,” Appl. Therm. Eng., 98, pp. 1044–1053.
Sheikholeslami, M. , Ganji, D. D. , Javed, M. Y. , and Ellahi, R. , 2015, “ Effect of Thermal Radiation on Magnetohydrodynamics Nanofluid Flow and Heat Transfer by Means of Two Phase Model,” J. Magn. Magn. Mater., 374, pp. 36–43.
Sheikholeslami, M. , Vajravelu, K. , and Rashidi, M. M. , 2016, “ Forced Convection Heat Transfer in a Semi Annulus Under the Influence of a Variable Magnetic Field,” Int. J. Heat Mass Transfer, 92, pp. 339–348.
Sheikholeslami, M. , 2016, “ Influence of Coulomb Forces on Fe3O4-H2O Nanofluid Thermal Improvement,” Int. J. Hydrogen Energy, 42(2), pp. 821–829.
Sheikholeslami, M. , Hayat, T. , and Alsaedi, A. , 2017, “ Numerical Study for External Magnetic Source Influence on Water Based Nanofluid Convective Heat Transfer,” Int. J. Heat Mass Transfer, 106, pp. 745–755.
Sheikholeslami, M. , and Chamkha, A. J. , 2016, “ Flow and Convective Heat Transfer of a Ferro-Nanofluid in a Double-Sided Lid-Driven Cavity With a Wavy Wall in the Presence of a Variable Magnetic Field,” Numer. Heat Transfer, Part A, 69(10), pp. 1186–1200.
Sheikholeslami, M. , Hayat, T. , and Alsaedi, A. , 2016, “ MHD Free Convection of Al2O3–Water Nanofluid Considering Thermal Radiation: A Numerical Study,” Int. J. Heat Mass Transfer, 96, pp. 513–524.
Basak, T. , and Chamkha, A. J. , 2012, “ Heatline Analysis on Natural Convection for Nanofluids Confined Within Square Cavities With Various Thermal Boundary Conditions,” Int. J. Heat Mass Transfer, 55, pp. 5526–5543.
Chamkha, A. J. , Jena, S. K. , and Mahapatra, S. K. , 2015, “ MHD Convection in Nanofluids: A Review,” J. Nanofluids, 4(3), pp. 271–292.
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## Figures

Fig. 1

(a) Geometry and the boundary conditions with (b) the mesh of geometry considered in this work

Fig. 2

Comparison of the present solution with the previous work [35] for different Rayleigh numbers when Ra = 105, Pr = 0.7 and (b) comparison of the temperature on axial midline between the present results and numerical results obtained by Khanafer et al. [36] for Gr=104, ϕ=0.1 and Pr=6.2(Cu-Water)

Fig. 3

Isotherms (left) and streamlines (right) contours for different values of number of undulations and Hartmann number for porous medium when ϕ=0.04,Ra=100

Fig. 4

Isotherms (left) and streamlines (right) contours for different values of number of undulations and Hartmann number for porous medium when ϕ=0.04,Ra=250

Fig. 5

Isotherms (left) and streamlines (right) contours for different values of number of undulations and Hartmann number for porous medium when ϕ=0.04,Ra=1000

Fig. 6

Effects of the number of undulations, nanoparticle volume fraction, Rayleigh number, and Hartmann number for porous medium on average Nusselt number

Fig. 7

Effects of the Hartmann number and Rayleigh number for porous medium on the ratio of heat transfer enhancement due to the addition of nanoparticles when N = 6

## Tables

Table1 The coefficient values of CuO–water nanofluid [34]
Table 2 Thermo physical properties of water and nanoparticles [34]
Table 3 Comparison of the average Nusselt number $Nuave$ for different grid resolutions at $Ra=103$, $n=6,A=0.5,Ha=20$, and $ϕ=0.04$
Table 4 Average Nusselt number versus different Grashof numbers under various strengths of the magnetic field at Pr = 0.733
Table 5 Effects of the number of undulations and Hartmann number on the ratio of heat transfer enhancement when Ra = 102

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