# Effects of two-phase nanofluid model on MHD mixed convection in a lid-driven cavity in the presence of conductive inner block and corner heater

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## Abstract

This paper investigates a steady mixed convection in a lid-driven square cavity subjected to an inclined magnetic field and heated by corner heater with an inserted square solid block. Water–Al\(_2\)O\(_3\) nanofluid fills the cavity based on Buongiorno’s two-phase model. A corner heater is configured in the left lower corner of the cavity by maintaining 40% of the bottom and vertical walls at constant hot temperature. The top horizontal wall is moving and maintained at a constant low temperature. The remainder walls are thermally insulated. The governing equations are solved numerically using the finite element method. The governing parameters are the nanoparticles volume fraction (\(0 \le \phi \le 0.04\)), Reynolds number (\(1 \le Re \le 500\)), Richardson number (\(0.01 \le Ri \le 100\)), Hartmann number (\(0 \le Ha \le 50\)) and the size of the inner solid (\(0.1 \le D \le 0.7\)). The other parameters: the Prandtl number, Lewis number, Schmidt number, ratio of Brownian to thermophoretic diffusivity and the normalized temperature parameter, are fixed at \(Pr=4.623\), \(Le=3.5\times 10^{5}\), \(Sc=3.55\times 10^{4}\), \(N_{\mathrm{BT}}=1.1\) and \(\delta =155\), respectively. The inclination of the magnetic field is fixed at \(\gamma =\frac{\pi }{4}\). Results show that at low Reynolds number, the increase in nanoparticles loading more the 2% becomes useless. It is also found that a big size of the solid block can augment heat transfer in the case of high values of both the Reynolds and Richardson numbers.

## Keywords

Lid-driven cavity Magnetic field Thermophoresis Brownian Corner heater Buongiorno’s model## Nomenclature

- \(\overrightarrow{\mathbf{B }}\)
Applied magnetic field

- \({\mathbf B} \)
Magnitude of magnetic field

- \(C_{p}\)
Specific heat capacity

*d*Side length of inner block

- \(d_{\mathrm{f}}\)
Diameter of the base fluid molecule

- \(d_{\mathrm{p}}\)
Diameter of the nanoparticle

*D*Dimensionless side length of the inner block, \(D=d/L\)

- \(D_{\mathrm{B}}\)
Brownian diffusion coefficient

- \(D_{\mathrm{B0}}\)
Reference Brownian diffusion coefficient

- \(D_{\mathrm{T}}\)
Thermophoretic diffusivity coefficient

- \(D_{\mathrm{T0}}\)
Reference thermophoretic diffusion coefficient

- \({\mathbf {g}}\)
Gravitational acceleration

*Ha*Hartmann number

*Gr*Grashof number

*k*Thermal conductivity

- \(K_{\mathrm{r}}\)
Square wall to nanofluid thermal conductivity ratio, \(K_{\mathrm{r}}=k_{\mathrm{w}}/k_{\mathrm{nf}}\)

*L*Width and height of enclosure

*Le*Lewis number

- \(N_{\mathrm{BT}}\)
Ratio of Brownian to thermophoretic diffusivity

- \(\overline{Nu}\)
Average Nusselt number

*Pr*Prandtl number

*Re*Reynolds number

- \(Re_{\mathrm{B}}\)
Brownian motion Reynolds number

*Ri*Richardson number, \(Ri=Gr/{Re}^2\)

*Sc*Schmidt number

*T*Temperature

- \(T_0\)
Reference temperature (310 K)

- \(T_{\mathrm{fr}}\)
Freezing point of the base fluid (273.15 K)

- \({\mathbf {v}} \), \({\mathbf {V}} \)
Velocity and dimensionless velocity vector, respectively

- \(u_{\mathrm{B}}\)
Brownian velocity of the nanoparticle

*x, y*and*X, Y*Space coordinates and dimensionless space coordinates

## Greek symbols

- \(\alpha \)
Thermal diffusivity

- \(\gamma \)
Inclination angle of magnetic field

- \(\beta \)
Thermal expansion coefficient

- \(\delta \)
Normalized temperature parameter

- \(\theta \)
Dimensionless temperature

- \(\mu \)
Dynamic viscosity

- \(\nu \)
Kinematic viscosity

- \(\rho \)
Density

- \(\sigma \)
Electrical conductivity

- \(\varphi \)
Solid volume fraction

- \(\varphi ^*\)
Normalized solid volume fraction

- \(\phi \)
Average solid volume fraction

## subscript

*b*Bottom wall

*c*Cold

*f*Base fluid

*h*Hot

- nf
Nanofluid

*p*Solid nanoparticles

*t*Top wall

*w*Solid wall

## Introduction

Mixed convection in enclosures is a broad field in which researchers are engaged. This is because of its significant role in many applications such as in cooling of electronic components, manufacturing of float glass and convective drying process. The numerical studies of Torrance et al. [1] and Ghia et al. [2] can be considered as the earliest one in the field of lid-driven mixed convection in cavity. Experimentally, Prasad and Koseff [3] demonstrated that the convective heat transfer is weakly correlated with Grashof number within their tested range of Reynolds number. Khanafer and Chamkha [4] investigated the mixed convective heat and flow in a lid-driven cavity saturated with a porous medium. Ismael et al. [5] established sophisticated correlations for the effect of partial slip in the lid-driven square cavity. Ismael [6] described an adaptive numerical procedure to relocate the finite difference nodes to fit an arc-shaped moving wall of a square cavity.

For the sake of enhancing the thermal conductivity of traditional fluids (e.g., water, oil and ethylene glycol), a dispersed nanoparticle (\(\sim \)100 nm diameter) is followed [7]. The resulting nanofluids have different promising properties with expectancy to enhance the heat exchange. Based on primitive (single-phase) nanofluid model, Tiwari and Das [8] proved, numerically, an enhancement of heat transfer in a double-sided lid-driven cavity due to dispersing different nanoparticles in water. Ismael et al. [9] investigated the role of CuO nanoparticles in enhancing the mixed convection in a lid-driven cavity heated by a corner heater. Temperature-dependent models were accounted for the nanofluid properties. They reported different situations at which the nanoparticles can adversely affect the convective heat exchange. Sheikholeslami et al. [10] investigated the Coulomb force effects on forced convection of Fe\(_3\)O\(_4\)–ethylene glycol nanofluid in a 2D porous cavity with the use of control volume-based finite element method. Abu-Nada and Chamkha [11] focused on the inclination effect of a lid-driven square enclosure filled with a water–alumina nanofluid. Based on experimental results, Corcione [12] proposed models for predicting the effective thermal conductivity and dynamic viscosity of nanofluids. For specified concentrations of nanofluid, the heat transfer has been found to be enhanced. Sheikholeslami and Shehzad [13] conducted using the control volume-based finite element method a numerical investigation on the problem of nanofluid convective flow of CuO–water nanofluid in a porous cavity in the presence of thermal non-equilibrium model. Wen and Ding [14] found experimentally that the slip velocity between the base fluid and nanoparticles is no longer being zero, resulting in an insistence to the two-phase nanofluid model. Buongiorno [15] comprehended the slip velocity by his sophisticated non-homogeneous model, which included the effect of two important slip mechanisms, namely Brownian diffusion and thermophoresis. Posteriorly, this model was used by Sheikholeslami et al. [16] to investigate the natural convection heat transfer in a 2D cavity. Based on the same model, Sheremet and Pop [17] simulated the heat transfer in a lid-driven square cavity filled with nanofluid. Garoosi et al. [18] simulated the nanofluid in a lid-driven cavity with several pairs of heaters and coolers using the two-phase Buongiorno’s model. Natural convection inside a square cavities filled with nanofluid is also studied based on the two-phase model (Esfandiary et al. [19] and Motlagh and Soltanipour [20]).

Imposing an externally applied magnetic field into fluid or nanofluids can accelerate of decelerate the convection, which is encountered in many industrial applications as reviewed by Nkurikiyimfura et al. [21]. Pirmohammadi and Ghassemi [22] conducted a numerical study to discover the effect of the magnetic field on the natural convection in an inclined square cavity. Due to the magnetic field, an enhancement and/or deterioration of convection in a nanofluid filling cavities was noticed individually by the numerical analysis of Mahmoudi et al. [23] and Ghasemi et al. [24]. Kefayati [25] addressed the effect of magnetic field on natural convection in open cavity filled with water–\({\mathrm {Al}}_2{\mathrm {O}}_3\) nanofluid utilizing the lattice Boltzmann method. He reported a reduction in heat exchange rate with the strength of the magnetic field. Following the same approach (the lattice Boltzmann method), Sheikholeslami et al. [26] investigated the magnetic field effect on the heat transfer inside a cavity filled with water–CuO nanofluid taking into account the Brownian motion. They indicated an enhancement in heat exchange with the magnetic field. Selimefendigil and Öztop [27] reported deterioration of convective heat transfer with applying a magnetic field into a square cavity filled with a nanofluid and including internal heat generation. Chamkha and Ismael [28] presented a suppressed role for the magnetic field when applied across a lid-driven trapezoidal cavity filled with water–Cu nanofluid. Sivaraj and Sheremet [29] studied the natural convection in inclined cavity saturated with a porous medium including a thermally conducting solid block. They confirmed the reduction in heat transfer with the applied magnetic field. Recently, Sheikholeslami [30] used the Lattice Boltzmann method to study the influence of magnetic field on natural convection of CuO–water nanofluid within porous cavity. Chamkha et al. [31] numerically studied the effect of partial slip condition on entropy generation and MHD convective heat transfer in a lid-driven porous cavity filled with Cu–water nanofluid. Sheikholeslami and Seyednezhad [32] and Sheikholeslami [33] simulated the problem of natural convection of Fe\(_3\)O\(_4\)–ethylene glycol nanofluid in a porous media in the presence of electric field using the control volume-based finite element method. Sheikholeslami [34] considered the Brownian motion and magnetic field effects on forced convective heat transfer of CuO–water nanofluid in a 2D porous cavity. The results indicated that the convective heat transfer improved with Reynolds number while it reduced with the increase in Hartmann number.

Convection inside cavities can be found in other mode including a solid conductive (or adiabatic) body to simulate some industrial applications, for example, cooling hot ingots, solidification and heat exchanger. House et al. [35] showed that the natural convection in a square cavity decreases with increasing the size of a centered conductive body. Ha et al. [36] studied the unsteady natural convection heat transfer in vertical cavities including a centered conductive body. Zhao et al. [37] showed that the conjugate heat transfer in a square cavity is strongly affected by the thermal conductivity of a body inserted at the center of the cavity. Mahmoodi and Sebdani [38] considered the natural convection heat transfer in a square cavity filled with nanofluid and including a centered conductive solid body. They reported that the size of the solid body could augment or deteriorate Nusselt number depending on the values of Reynolds number. Mahapatra et al. [39] investigated the entropy generation due to the conjugate heat transfer in a square cavity including adiabatic and isothermal solid blocks. Ismael and Ghalib [40] investigated the effect of position and size of a conductive square block on heat and mass transfer inside a partially layered square cavity. The cavity was heated differentially. They addressed that for maximum convective heat transfer, the position of the solid body should not be in the cavity center. Alsabery et al. [41] studied the natural convection in a square porous cavity saturated with nanofluid and containing a concentric solid. The cavity was heated in sinusoidal mode; thus, they examined the problem along the unsteady state. In other collaboration, Alsabery et al. [42] implemented Buongiorno’s two-phase model to investigate the mixed convection of water–\({\mathrm {Al}}_2{\mathrm {O}}_3\) nanofluid in a lid-driven square cavity including a conductive solid block. They predicted that for high values of Reynolds and Richardson numbers, the increase in the size of the solid block could enhance the heat exchange rate.

Although the mixed convection in enclosed cavities has attracted much researchers’ attention, it is still appropriate and convenient to think of new arrangements to utilize the thermal energy as much as possible. As such, the collaborators of Alsabery et al. [42] think that it is expedient to study the effect of the magnetic field on mixed convection of water–\({\mathrm {Al}}_2{\mathrm {O}}_3\) nanofluid in a cavity with conductive inner block heated by a corner heater based on Buongiorno’s two-phase model. This configuration has not been investigated yet. It is believed that this work is a good contribution for improving the thermal performance and the heat transfer enhancement in some engineering industries.

## Mathematical formulation

*L*and with the cavity center inserted by a solid square with side

*d*, as illustrated in Fig. 1a. The Reynolds and Richardson numbers range chosen in the study keeps the nanofluid flow incompressible and laminar. The heat source into the cavity is considered as a segment of the lower left corner occupying \(0.8\times L\). This segment is kept at a higher isothermal temperature, \(T_{\mathrm{h}}\). The top wall is moving to the right and maintained isothermal with colder temperature, \(T_{\mathrm{c}}\). The other walls are thermally insulated. An inclined uniform magnetic field is applied into the cavity. The boundaries of the annulus are assumed to be impermeable, and the fluid within the cavity is a water-based nanofluid having Al\(_{2}\)O\(_{3}\) nanoparticles. The Boussinesq approximation is applicable. The induced magnetic field in the nanofluid is ignored because it is very small compared with the applied one. In addition, the inner block is taken as fiberglass, which is permeable to the magnetic field and has poor electrical conductivity. According to these assumptions, the Joule heating effect can safely be ignored. By considering these assumptions, the continuity, momentum and energy equations for the laminar and unsteady-state natural convection can be written as follows:

The thermo-physical properties of the nanofluid can be determined as follows:

*M*is the molecular weight of the base fluid,

*N*is the Avogadro number and \(\rho _{\mathrm{f}}\) is the density of the base fluid at standard temperature (310 K). Accordingly, and basing on water as a base fluid, the value of \(d_{\mathrm{f}}\) is obtained:

## Numerical method and validations

*i*represents the iteration number and \(\eta \) is the convergence criterion. In this study, the convergence criterion was set at \(\eta =10^{-6}\)

### Grid size assessment

Grid testing for \(\varPsi _{\mathrm {min}}\), \(\varPsi _{\mathrm {max}}\) and \(\overline{Nu}_{\mathrm{nf}}\) at different grid sizes for \(Re=100\), \(Ri=10\), \(\phi =0.02, Ha=15\) and \(D=0.3\)

Grid size | Number of elements | \(\varPsi _{\mathrm {min}}\) | \(\varPsi _{\mathrm {max}}\) | \(\overline{Nu}_{{\mathrm{nf}}}\) |
---|---|---|---|---|

G1 | 792 | \(-\) 0.054851 | 0.0030588 | 10.204 |

G2 | 1308 | \(-\) 0.055778 | 0.0031997 | 10.502 |

G3 | 2262 | \(-\) 0.056333 | 0.0032983 | 10.7742 |

G4 | 3258 | \(-\) 0.056595 | 0.0033442 | 10.8605 |

G5 | 4980 | \(-\) 0.05662 | 0.0033743 | 10.922 |

| 12248 | \(-\) 0.056863 | 0.0034405 | 10.923 |

G7 | 31258 | \(-\) 0.056827 | 0.0034314 | 10.922 |

### Comparisons with others

## Results and discussion

^{−1}\(^\circ \)C) for all the cases in this study. It is worth mentioning that the ranges of these parameters were chosen according to the physical aspects reported in relative literature. The thermo-physical properties of the base fluid (water) and solid Al\(_2\)O\(_3\) phases are given in Table 2. It may be unattainable to comprehend the effect of each parameter individually; however, for better description, we will try to display the results as possible as according to the following subsections.

Physical properties | Fluid phase (water) | Al\(_{2}\)O\(_{3}\) |
---|---|---|

\(C_{p}/{\mathrm {J\,kg^{-1}\,K^{-1}}}\) | 4178 | 765 |

\(\rho/{\mathrm {kg/m^3}}\) | 993 | 3970 |

\(k/{\mathrm {W m^{-1} K^{-1}}}\) | 0.628 | 40 |

\(\beta \times 10^{5}/{\mathrm {K^{-1}}}\) | 36.2 | 0.85 |

\(\mu \times 10^{6}/{\mathrm {kg\,ms^{-1}}}\) | 695 | – |

\(d_{\mathrm{p}}/\text {nm}\) | 0.385 | 33 |

### Effect of nanoparticles loading

### Effects of Reynolds and Richardson numbers

### Effect of Hartmann number

*X*and

*Y*direction. These two drag components restrict the nanofluid circulation and retard the nanofluid motion within the right lower corner of the cavity as shown in Fig. 15. The strength of the circulation decreases considerably with increasing the value of the Hartmann number. The isotherms pattern is stratified and semi-horizontal within the bulk of the cavity when \(Ha=0\) (Fig. 15a). Applying the magnetic field, the drag forces opposite the buoyancy and inertia forces; as a result, the isotherms become quasi-vertical with less heat transfer within the right part of the cavity as shown in Fig. 15c–d. The deteriorated natural convection with Ha permits the conduction mode to be announced; thus, the nanoparticles of \(Ha=50\) (Fig. 15d) aggregated in an irregular distribution.

*Re*), the deterioration of \(\overline{Nu}_{\mathrm{nf}}\) is faster than that seen when \(Re=100\) as shown in Figs. 17a and 18a. Figure 18b presents that at higher values of

*Re*and

*Ri*, the magnetic field has less drag effect on \(\overline{Nu}_{\mathrm{nf}}\).

### Effect of solid block size

*D*. In the same time, when

*D*increased, the effect of the shear force imposed by the moving cold will be limited in the groove formed between the moving wall and the solid wall. This issue leads to symmetric streamlines as shown in Fig. 19d. On the other hand, when

*D*is increased to a high value (\(D=0.7\)), the solid block becomes nearer to the corner heater and much heat transfer takes place by conduction and thus the temperature gradients extend to exists in the core of the cavity as can be seen from the isotherms. The corresponding distribution of the nanoparticles manifests homogeneous pattern with the bigger block as depicted in Fig. 19d.

*Re*increased to 100, this circulation becomes stronger and easily penetrates the annulus formed by the inner block; thus, efficient contact between the nanofluid circulation and the walls of the cavity takes place, which in turn augments the Nusselt number. In addition, increasing the nanoparticles loading beyond 0.02 becomes useless at \(D=0.7\) and \(Re=100\), where the enhancement in \(\overline{Nu}_{\mathrm{nf}}\) is 6.8 and 1.8% at \(D=0.7\) compared to the case of \(D=0.1\) when \(\phi =0.01\) and 0.04, respectively. On the other hand, the enhancements at \(D=0.5\) compared to \(D=0.1\) are 4.7 and 5.3% when \(\phi =0.01\) and 0.04, respectively. This is due to the adverse effect of the Brownian diffusion raised in these conditions. Eventually, it should be pointing that the magnetic field beyond \(Ha=15\) restrains the augmentation of the Nusselt number gained by enlarging the solid block to \(D=0.7\) as shown in Fig. 22b.

## Conclusions

The current paper studies the steady laminar mixed convection of Al\(_2\)O\(_3\)–water nanofluid in a lid-driven cavity with inner block and corner heater using Buongiorno’s two-phase model, which takes into consideration the slip between the base fluid and the nanoparticles. It is believed that the results obtained using this model are reliable compared with those obtained by other single-phase models. The governing equations are solved using Galerkin weighted residual finite element method. The pertinent parameters: loading of the nanoparticles, Hartmann number, size of the inner solid block, Reynolds and Richardson numbers, were varied in the study.

The average Nusselt number augments with nanoparticles loading with the aid of the combined effect of the buoyancy and the shear forces. Nevertheless, at low Reynolds number, the increase in nanoparticles loading more the 2% becomes useless. At low Reynolds number, the solid block deteriorates the convective heat transfer while when *Re* increased to 100, the larger block size augments the Nusselt number. For strong buoyancy force, the shape of corner heater does not affect the nanofluid recirculation and is symmetric around the solid block, while for weak buoyancy force, the isotherms experience wavy front due to the heater geometry. At higher values of Reynolds and Richardson numbers, the applied magnetic field has less drag effect on the average Nusselt number. The magnetic field beyond \(Ha=15\) restrains the augmentation of the Nusselt number gained by enlarging the solid block.

## Notes

### Acknowledgements

The work was supported by the Universiti Kebangsaan Malaysia (UKM) research Grant DIP-2017-010.

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