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Journal of Thermal Analysis and Calorimetry

, Volume 135, Issue 1, pp 729–750 | Cite as

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

  • A. I. AlsaberyEmail author
  • M. A. Ismael
  • A. J. Chamkha
  • I. Hashim
Article
  • 88 Downloads

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

Consider the steady two-dimensional mixed convection problem in a square cavity with length 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:
$$\begin{aligned} \nabla \cdot {\mathbf {v}}= & {} 0, \end{aligned}$$
(1)
$$\begin{aligned} \rho _{\mathrm{nf}}{} {\mathbf {v}} \cdot \nabla {\mathbf {v}}= & {} -\nabla p + \nabla \cdot (\mu _{\mathrm{nf}} \nabla {\mathbf {v}} )+(\rho \beta )_{\mathrm{nf}}(T-T_{\mathrm{c}}){\mathbf {g}} \nonumber \\ &+\,\sigma _{\mathrm{nf}}{} {\mathbf {v}} \times \overrightarrow{\mathbf{B }}, \end{aligned}$$
(2)
$$\begin{aligned} (\rho C_{\text{p}})_{\mathrm{nf}}{} {\mathbf{v}} \cdot \nabla T_{\mathrm{nf}}= & {} -\nabla \cdot (k_{\mathrm{nf}} \nabla T_{\mathrm{nf}}) - C_{{\text{p}},{\text{p}}} J_{\text{p}}\cdot \nabla T_{\mathrm{nf}}, \end{aligned}$$
(3)
$$\begin{aligned} {\mathbf {v}} \cdot \nabla \varphi= & {} - \frac{1}{\rho _{\text{p}}}\nabla \cdot J_{\text{p}}, \end{aligned}$$
(4)
where \({\mathbf {v}} \) is the velocity vector, \({\mathbf {g}}\) is the gravitational acceleration vector, \(\varphi \) is the local volume fraction of nanoparticles and \(J_{\text{p}}\) is the nanoparticles mass flux.
The energy equation of the conductive inner block is
$$\begin{aligned} \nabla ^2 T_{\mathrm{w}} = 0. \end{aligned}$$
(5)
Based on Buongiorno’s model, nanoparticles mass flux can be written as:
$$\begin{aligned} J_{\text{p}} &=\, J_{\text{p,B}}+J_{\text{p,T}}, \end{aligned}$$
(6)
$$\begin{aligned} J_{\text{p,B}}= & {} -\rho _{\text{p}} D_{\mathrm{B}} \nabla \varphi ,\quad D_{\mathrm{B}}=\frac{k_{\mathrm{b}} T}{3 \pi \mu _{\mathrm{f}} d_{\mathrm{p}}}, \end{aligned}$$
(7)
$$\begin{aligned} J_{\text{p,T}}=\, & {} -\rho _{\text{p}} D_{\text{T}} \nabla T,\quad D_{\rm T}=0.26\frac{k_{\mathrm{f}}}{2k_{\mathrm{f}}+k_{\text{p}}} \frac{\mu _{\mathrm{f}}}{\rho _{\mathrm{f}} T}\varphi . \end{aligned}$$
(8)
Fig. 1

a Physical model of convection in a square cavity together with the coordinate system and b grid-points distribution for grid size of G6

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

The heat capacitance of the nanofluids \((\rho C_{\text{p}})_{\mathrm{nf}}\) is given as
$$\begin{aligned} (\rho C_{\text{p}})_{\mathrm{nf}}=(1-\varphi )(\rho C_{\text{p}})_{\mathrm{f}}+\varphi (\rho C_{\text{p}})_{\text{p}}. \end{aligned}$$
(9)
The effective thermal diffusivity of the nanofluids \(\alpha _{\mathrm{nf}}\) is given as
$$\begin{aligned} \alpha _{\mathrm{nf}}=\frac{k_{\mathrm{nf}}}{(\rho C_{\text{p}})_{\mathrm{nf}}}. \end{aligned}$$
(10)
The effective density of the nanofluids \(\rho _{\mathrm{nf}}\) is given as
$$\begin{aligned} \rho _{\mathrm{nf}}=(1-\varphi )\rho _{\mathrm{f}}+\varphi \rho _{\text{p}}. \end{aligned}$$
(11)
The thermal expansion coefficient of the nanofluids \(\beta _{\mathrm{nf}}\) can be determined by:
$$\begin{aligned} {(\rho \beta )_{\mathrm{nf}}} = (1 - \varphi ){(\rho \beta )_{\mathrm{f}}} + \varphi {(\rho \beta )_{\text{p}}}. \end{aligned}$$
(12)
The dynamic viscosity ratio of water–Al\(_{2}\)O\(_{3}\) nanofluids for 33-nm particle size in the ambient condition was derived in reference [12] as
$$\begin{aligned} \frac{\mu _{\mathrm{nf}}}{\mu _{\mathrm{f}}} = 1/\left( 1-34.87\left( d_{\mathrm{p}}/d_{\mathrm{f}}\right) ^{-0.3}\varphi ^{1.03}\right) . \end{aligned}$$
(13)
The thermal conductivity ratio of water–Al\(_{2}\)O\(_{3}\) nanofluids is calculated by the Corcione et al. model [12] as:
$$\begin{aligned} \frac{k_{\mathrm{nf}}}{k_{\mathrm{f}}}= 1 + 4.4 Re_{\mathrm{B}}^{0.4} Pr^{0.66} \left( \frac{T}{T_{\mathrm{fr}}}\right) ^{10} \left( \frac{k_{\text{p}}}{k_{\mathrm{f}}}\right) ^{0.03} \varphi ^{0.66}, \end{aligned}$$
(14)
where \(Re_{\mathrm{B}}\) is defined as
$$\begin{aligned} Re_{\mathrm{B}}=\frac{\rho _{\mathrm{f}} u_{\mathrm{B}} d_{\mathrm{p}}}{\mu _{\mathrm{f}}}, \quad u_{\mathrm{B}} = \frac{2k_{\mathrm{b}} T}{\pi \mu _{\mathrm{f}} d_{\mathrm{p}}^2}. \end{aligned}$$
(15)
$$\begin{aligned} u_{\mathrm{B}} = \frac{2k_{\mathrm{b}} T}{\pi \mu _{\mathrm{f}} d_{\mathrm{p}}^2}. \end{aligned}$$
(16)
Here \(k_{\mathrm{b}}=1.380648 \times 10^{-23} ({\rm J}\,{\rm K}^{-1})\) is the Boltzmann constant, \(l_{\mathrm{f}}=0.17\) nm is the mean path of fluid particles, \(d_{\mathrm{f}}\) is the molecular diameter of water given as [12]
$$\begin{aligned} d_{\mathrm{f}} = \frac{6 M}{N \pi \rho _{\mathrm{f}}}, \end{aligned}$$
(17)
where 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:
$$\begin{aligned} d_{\mathrm{f}} = \left( \frac{6 \times 0.01801528}{6.022 \times 10^{23} \times \pi \times 998.26}\right) ^{1/3}=3.85 \times 10^{-10}\,\text {m}. \end{aligned}$$
(18)
The electrical conductivity ratio \(\frac{\sigma _{\mathrm{nf}}}{\sigma _{\mathrm{f}}}\) is defined as [43]
$$\begin{aligned} \frac{\sigma _{\mathrm{nf}}}{\sigma _{\mathrm{f}}} = 1+ \frac{3 \left( \frac{\sigma _{p}}{\sigma _{\mathrm{f}}}-1\right) \varphi }{\left( \frac{\sigma _{p}}{\sigma _{\mathrm{f}}}+2\right) - \left( \frac{\sigma _{p}}{\sigma _{\mathrm{f}}}-1\right) \varphi }. \end{aligned}$$
(19)
Now we introduce the following non-dimensional variables:
$$\begin{aligned}&X=\frac{x}{L},\,\, Y=\frac{y}{L},\,\, {\mathbf{V }} = \frac{\mathbf{v }}{U_{0}}, \,\, P = \frac{p L^2}{\rho _{\mathrm{nf}} \nu _{\mathrm{f}}^2}, \,\, \varphi ^*=\frac{\varphi }{\phi }, \,\, D^*_{\mathrm{B}}=\frac{D_{\mathrm{B}}}{D_{\mathrm{B0}}}, \,\, D^*_{\text{T}}=\frac{D_{\text{T}}}{D_{\mathrm{T0}}}, \nonumber \\&\;\delta = \frac{T_{\mathrm{c}}}{{{T_{\mathrm{h}}} - {T_{\mathrm{c}}}}}, \;\; \theta _{\mathrm{nf}}=\frac{T_{\mathrm{nf}}-T_{\mathrm{c}}}{T_{\mathrm{h}}-T_{\mathrm{c}}},\;\;{\theta _{\mathrm{w}}} = \frac{{{T_{\mathrm{w}}} - {T_{\mathrm{c}}}}}{{{T_{\mathrm{h}}} - {T_{\mathrm{c}}}}},\;\; D = \frac{d}{L}. \end{aligned}$$
(20)
This then yields the dimensionless governing equations:
$$\begin{aligned} \nabla \cdot {\mathbf {V}}= & {} 0, \end{aligned}$$
(21)
$$\begin{aligned} {\mathbf {V}} \cdot \nabla {\mathbf {V}}= & {} - \nabla P + \frac{\rho _{\mathrm{f}}}{\rho _{\mathrm{nf}}} \frac{\mu _{\mathrm{nf}}}{\mu _{\mathrm{f}}} {\frac{1}{Re}} {\nabla ^2} {\mathbf {V}} + \frac{(\rho \beta )_{\mathrm{nf}}}{\rho _{\mathrm{nf}} \beta _{\mathrm{f}}} Ri \cdot \theta _{\mathrm{nf}} \nonumber \\&+\,\frac{\rho _{\mathrm{f}}}{\rho _{\mathrm{nf}}} \frac{\sigma _{\mathrm{nf}}}{\sigma _{\mathrm{f}}}{} {\mathbf {V}} \times {\mathbf {B}}* , \end{aligned}$$
(22)
$$\begin{aligned} {\mathbf {V}} \cdot \nabla \theta _{\mathrm{nf}}= & {} \frac{(\rho C_{\text{p}})_{\mathrm{f}}}{(\rho C_{\text{p}})_{\mathrm{nf}}} \frac{k_{\mathrm{nf}}}{k_{\mathrm{f}}} {\frac{1}{Re \cdot Pr}} \nabla ^2 \theta _{\mathrm{nf}} + \frac{(\rho C_{\text{p}})_{\mathrm{f}}}{(\rho C_{\text{p}})_{\mathrm{nf}}} \frac{D^*_{\mathrm{B}}}{Re \cdot Pr \cdot Le} \nabla \varphi ^*\cdot \nabla \theta _{\mathrm{nf}} \nonumber \\&+\, \frac{(\rho C_{\text{p}})_{\mathrm{f}}}{(\rho C_{\text{p}})_{\mathrm{nf}}} \frac{D^*_{\text{T}}}{Re \cdot Pr \cdot Le \cdot N_{\mathrm{BT}}} \frac{\nabla \theta _{\mathrm{nf}} \cdot \nabla \theta _{\mathrm{nf}}}{1 + \delta \theta _{\mathrm{nf}}}, \end{aligned}$$
(23)
$$\begin{aligned} {\mathbf {V}} \cdot \nabla \varphi ^*= & {} \frac{D^*_{\mathrm{B}}}{Re \cdot Sc} \nabla ^2 \varphi ^*+ \frac{D^*_{\text{T}}}{Re \cdot Sc \cdot N_{\mathrm{BT}}} \cdot \frac{\nabla ^2 \theta _{\mathrm{nf}}}{1 + \delta \theta _{\mathrm{nf}}}, \end{aligned}$$
(24)
$$\begin{aligned} \nabla ^2 \theta _{\mathrm{w}}= & {} 0, \end{aligned}$$
(25)
where \({\mathbf {V}} \) is the dimensionless velocity vector (\(U_{0},V_{0}\)) and \({\mathbf B*} \) is the dimensionless magnetic vector (\(Ha^2\sin \gamma , Ha^2\cos \gamma \)). Parameters \(D_{\mathrm{B0}}=\frac{k_{\mathrm{b}} T_{\mathrm{c}}}{3 \pi \mu _{\mathrm{f}} d_{\mathrm{p}}}\) is the reference Brownian diffusion coefficient, \(D_{\mathrm{T0}}=0.26\frac{k_{\mathrm{f}}}{2k_{\mathrm{f}}+k_{\text{p}}} \frac{\mu _{\mathrm{f}}}{\rho _{\mathrm{f}} \theta }\phi \) is the reference thermophoretic diffusion coefficient, \(Re=U_{0}L/\nu _{\mathrm{f}}\) is Reynolds number, \(Ri=Gr/{Re}^2\) is Richardson number, \(Sc=\nu _{\mathrm{f}}/D_{\mathrm{B0}}\) is Schmidt number, \(N_{\mathrm{BT}}=\phi D_{\mathrm{B0}}T_{\mathrm{c}}/D_{\mathrm{T0}}(T_{\mathrm{h}}-T_{\mathrm{c}})\) is the diffusivity ratio parameter (Brownian diffusivity/thermophoretic diffusivity), \(Le=k_{\mathrm{f}}/(\rho C_{p})_{\mathrm{f}} \phi D_{\mathrm{B0}}\) is Lewis number, \(Gr=g \beta _{\mathrm{f}}(T_{\mathrm{h}} - T_{\mathrm{c}})L^{3}/\nu _{\mathrm{f}}^2\) is the Grashof number, \(Ha={\mathbf {BL}} \sqrt{\frac{\sigma _{\mathrm{f}}}{\mu _{\mathrm{f}}}}\) is the Hartmann number and \(Pr=\nu _{\mathrm{f}}/\alpha _{\mathrm{f}}\) is the Prandtl number for the base fluid. The dimensionless boundary conditions of Eqs. (21) and (25) are:
$$\begin{aligned}&\text {On\,the\,adiabatic\,right\, vertical\, wall:} \nonumber \\&U = V = 0,\,\, \frac{\partial \varphi ^*}{\partial n}=0,\, \frac{\partial \theta _{\mathrm{nf}}}{\partial n}=0, \end{aligned}$$
(26)
$$\begin{aligned}&\text {On \,the \,heated \,part\, of\, the\, horizontal \,bottom \,wall:} \nonumber \\&U = V = 0,\,\, \frac{\partial \varphi ^*}{\partial n}= - \frac{D^*_T}{D^*_{\mathrm{B}}} \cdot \frac{1}{N_{\mathrm{BT}}} \cdot \frac{1}{1 + \delta \theta _{\mathrm{nf}}} \frac{\partial \theta _{\mathrm{nf}}}{\partial n},\, \theta _{\mathrm{nf}}=1, \end{aligned}$$
(27)
$$\begin{aligned}&\text {On\, the\, adiabatic\, parts \,of \,the \,remainder \,bottom \,wall:} \nonumber \\&U = V = 0,\,\, \frac{\partial \varphi ^*}{\partial n}=0,\, \frac{\partial \theta _{\mathrm{nf}}}{\partial n}=0, \end{aligned}$$
(28)
$$\begin{aligned}&\text {On\, the \,heated \,part \,of \,the \,vertical \,left \,wall:} \nonumber \\&U = V = 0,\,\, \frac{\partial \varphi ^*}{\partial n}= - \frac{D^*_T}{D^*_{\mathrm{B}}} \cdot \frac{1}{N_{\mathrm{BT}}} \cdot \frac{1}{1 + \delta \theta _{\mathrm{nf}}} \frac{\partial \theta _{\mathrm{nf}}}{\partial n},\, \theta _{\mathrm{nf}}=1, \end{aligned}$$
(29)
$$\begin{aligned}&\text {On\, the \,adiabatic \,parts \,of \,the \,remainder \,left \,wall:} \nonumber \\&U = V = 0,\,\, \frac{\partial \varphi ^*}{\partial n}=0,\, \frac{\partial \theta _{\mathrm{nf}}}{\partial n}=0, \end{aligned}$$
(30)
$$\begin{aligned}&\text {On \,the \,horizontal \,moving \,top \,wall }(U=1): \nonumber \\&U =1,\,\, V = 0,\,\, \frac{\partial \varphi ^*}{\partial n}= - \frac{D^*_T}{D^*_{\mathrm{B}}} \cdot \frac{1}{N_{\mathrm{BT}}} \cdot \frac{1}{1 + \delta \theta _{\mathrm{nf}}} \frac{\partial \theta _{\mathrm{nf}}}{\partial n},\, \theta _{\mathrm{nf}}=0, \end{aligned}$$
(31)
$$\begin{aligned}&\text {On\, the \,walls \,of \,the \,solid \,inner \,square \,surface:} \nonumber \\&\theta _{\mathrm{nf}} = {\theta _{\mathrm{w}}}, \end{aligned}$$
(32)
$$\begin{aligned}&U = V = 0,\,\, \frac{\partial \varphi ^*}{\partial n}= - \frac{D^*_T}{D^*_{\mathrm{B}}} \cdot \frac{1}{N_{\mathrm{BT}}} \cdot \frac{1}{1 + \delta \theta _{\mathrm{nf}}} \frac{\partial \theta _{\mathrm{nf}}}{\partial n},\;\; \frac{\partial \theta _{\mathrm{nf}}}{\partial n} = K_{\mathrm{r}}\frac{\partial \theta _{\mathrm{w}}}{\partial n}, \end{aligned}$$
(33)
where \(K_{\mathrm{r}} = {k_{\mathrm{w}}}/{k_{\mathrm{nf}}}\) is the thermal conductivity ratio.
The local Nusselt number evaluated at the left and bottom sides of the heater is defined by
$$\begin{aligned} Nu_{x} = -\frac{k_{\mathrm{nf}}}{k_{\mathrm{f}}} \left( \frac{\partial \theta _{\mathrm{nf}}}{\partial X}\right) _{X=0}, \quad Nu_{y} = -\frac{k_{\mathrm{nf}}}{k_{\mathrm{f}}} \left( \frac{\partial \theta _{\mathrm{nf}}}{\partial Y}\right) _{Y=0}. \end{aligned}$$
(34)
Finally, the average Nusselt number evaluated at the left and bottom sides of the corner heater is given by:
$$\begin{aligned} {\overline{Nu}_{\rm {x}}} = \int _0^{0.4} Nu_{\rm {x}} {} {\mathrm{{d}}}Y, \quad {\overline{Nu}_{\rm {y}}} = \int _0^{0.4} Nu_{\rm y} {} {\mathrm{{d}}}X, \end{aligned}$$
(35)
and
$$\begin{aligned} {\overline{Nu}_{\mathrm{nf}}} = {\overline{Nu}_{\rm {x}}} + {\overline{Nu}_{\rm {y}}}. \end{aligned}$$
(36)

Numerical method and validations

The governing dimensionless equations (21)–(25) subjected to the boundary conditions (26)–(33) are solved with Galerkin weighted residual finite element method. The computational domain is discretized into triangular elements as shown in Fig. 1b. Triangular Lagrange finite elements of different orders are used for each of the flow variables within the computational domain. Residuals for each conservation equation are obtained by substituting the approximations into the governing equations. To simplify the nonlinear terms in the momentum equations, a Newton–Raphson iteration algorithm was used. The convergence of the solution is assumed when the relative error for each of the variables satisfies the following convergence criterion:
$$\begin{aligned} \left| \frac{\varGamma ^{{\text{i}}+1}-\varGamma ^{\text{i}}}{\varGamma ^{{\text{i}}+1}}\right| \le \eta , \end{aligned}$$
where 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

To ensure the independence of the present numerical solution on the grid size of the numerical domain, we have used different grid sizes to calculate the minimum strength of the flow circulation (\(\varPsi _{\mathrm {min}}\)), maximum strength of the flow circulation (\(\varPsi _{\mathrm {max}}\)) and average Nusselt number for \(Re=100, Ri=10, \phi =0.02, Ha=15\) and \(D=0.3\). The results shown in Table 1 and Fig. 2 indicate insignificant differences for the G6 grids and above. Therefore, for all computations in this paper for similar problems to this subsection, the G6 non-uniform grid is employed.
Table 1

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

G6

12248

\(-\) 0.056863

0.0034405

10.923

G7

31258

\(-\) 0.056827

0.0034314

10.922

Fig. 2

Variation of average Nusselt number (\(\overline{Nu}_{\mathrm{nf}}\)) with grid sizes for \(Re=100\), \(Ri=10\), \(\phi =0.02, Ha=15\) and \(D=0.3\)

Comparisons with others

To assure the validity of the present code, we conducted different previously published cases, and these are the mixed convection in a lid-driven cavity with corner heater subjected to an external magnetic field by Oztop et al. [44]. The results are shown in Fig. 3 for no inserted solid block. The streamlines and isotherms show excellent agreement between the outcomes of our code and the results of Oztop et al. [44]. Moreover, the problem of natural convection inside a cavity filled with nanofluid using Buongiorno’s model is shown in Fig. 4 where the average Nusselt number was compared with experimental work of Ho et al. [45] and numerical works of Sheikhzadeh et al. [46] and Motlagh and Soltanipour [20]. It shows very good agreement with Sheikhzadeh et al. [46], which is based on the same model, and acceptable agreement when compared with the experimental work of [45]. However, overestimated results are observed when \(Ra\le 10^6\) as already demonstrated by Motlagh and Soltanipour [20]. The comparison with Corcione et al. [47] also showed very good agreement between the maps of streamlines, isotherms and nanoparticles distribution inside a free cavity as shown in Fig. 5. Figure 6 presents other comparisons regarding the enhancement in the thermal conductivity and dynamic viscosity due to the addition of the \({\mathrm {Al}}_2{\mathrm {O}}_3\) nanoparticles with two different experimental results (Chon et al. [48], Ho et al. [45]) and the numerical results of Corcione et al. [47] as well. Except at some nanoparticle loading, the predictions of the thermal conductivity enhancement are in good agreement. Additional validation has been achieved by comparing the average Nusselt number with the numerical predictions of Motlagh and Soltanipour [20] for the case of natural convection inside a square cavity filled by nanofluid and free of inside block as shown in Fig. 7. Very good agreements are obtained in low and high Rayleigh numbers, especially at low loading of nanoparticles. Based on these validations, the numerical outcomes of the present numerical code assure high accurate results.
Fig. 3

Streamlines (a), Oztop et al. [44] (left), present study (right), isotherms (b), Oztop et al. [44] (left), present study (right) for \(Ha=0\) (top) and \(Ha=50\) (bottom) at \(Re=100\), \(Ri=1\), \(\phi =0\) and \(D=0.0\)

Fig. 4

Comparison of the mean Nusselt number obtained from present numerical simulation with the experimental results of Ho et al. [45], numerical results of Sheikhzadeh et al. [46] and numerical results of Motlagh and Soltanipour [20] for different values of Rayleigh numbers

Fig. 5

Corcione et al. [47] (left), present study (right) for streamlines (a), isotherms (b) and nanoparticle distribution (c) at \(Ra=3.37\times 10^{5}\), \(\phi =0.04\) and \(D=0\)

Fig. 6

Comparison of a thermal conductivity ratio with Chon et al. [48] and Corcione et al. [47] and b dynamic viscosity ratio with Ho et al. [45] and Corcione et al. [47]

Fig. 7

Comparison of average Nusselt number with Motlagh and Soltanipour [20] for a\(Ra=10^2\) and b\(Ra=10^6\) at \(D=0\)

Results and discussion

In this section, we will present the results of altering the non-dimensional parameters and their effects on the streamlines, isotherms, nanoparticle distribution, local and average Nusselt numbers. The dimensionless parameters are the nanoparticle 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 length of the conductive inner block (\(0.1 \le D \le 0.7\)). The values of Prandtl number, Lewis number, Schmidt number, inclination angle of magnetic field, ratio of Brownian to thermophoretic diffusivity and normalized temperature parameter are fixed at \(Pr=4.623\), \(Le=3.5\times 10^{5}\), \(Sc=3.55\times 10^{4}\), \(\gamma =\frac{\pi }{4}\), \(N_{\mathrm{BT}}=1.1\) and \(\delta =155\), respectively. The inner block is fiberglass with low thermal conductivity (\(k_{\mathrm{w}}=0.045\) W m−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.
Table 2

Thermo-physical properties of water with Al\(_{2}\)O\(_{3}\) nanoparticles at \(T=310\) K [49, 20]

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

The effect of nanoparticles loading (the volume fraction of nanoparticles) on the streamlines, isotherms and the distribution of nanoparticles is shown in Fig. 8 for \(Re=100\), \(Ri=10\), \(Ha=15\) and \(D=0.3\). Based on the imposed boundary condition, heat arises from the heat source (the left corner) upward toward the upper wall, which is moving to the right. As such, by shear effect, the nanofluid turns to the right and forms a clockwise circulation confining the solid block. Two secondary recirculation are formed close to the upper corners of the solid block. Close to the hot walls and the moving one, the nanofluid is intensified while it is stagnant at the lower right corner that is distant from these sources. Streamlines look unaffected with increasing the nanoparticles loading except with slight increase in the circulation intensity that refers to the Brownian diffusion of the nanoparticles. The isotherms (middle row) experience thin boundary layers close to the corner heater and isothermal zone at the distant corner. Due to the drag action of the magnetic field, the core of the cavity looks quasi-isothermal while close to the moving upper wall the shear action is predominant and forming a horizontal plume-like isotherms. Similar to the streamlines, the isotherms are also unaffected by the nanoparticles loading. The right row of Fig. 8 depicts that the distribution of the nanoparticles is segregated from the entire cavity boundaries and aggregated at the lower left corner of the solid block, which appear as a perturbation. However, within the upper part of the cavity, the nanoparticles look with high distribution. This behavior can be attributed to that the thermophoresis force and the Brownian diffusion are insufficient to take its role because the drag force exerted by the magnetic field. The nanoparticles tend to be aggregated above the solid block at higher loading values.
Fig. 8

Variation of the streamlines (left), isotherms (middle) and nanoparticle distribution (right) evolution by solid volume fraction (\(\phi \)) for \(Re=100\), \(Ri=10\), \(Ha=15\) and \(D=0.3\)

The local Nusselt number along the horizontal and vertical heater legs is shown in Fig. 9, which indicates heat transfer augmentation with increasing the nanoparticles loading. This result is trustworthy because we have considering the Brownian diffusion and the thermophoresis effects that take into account the slip between the base fluid and the nanoparticles. The local Nusselt number is minimum at the vertex of the heater and maximum at its two edges. The overall heat transfer can be clearly understood by the average Nusselt number (\(\overline{Nu}_{\mathrm{nf}}\)) which is depicted in Fig. 10. It shows that \(\overline{Nu}_{\mathrm{nf}}\) augments with nanoparticles loading with aid of the combined effect of the buoyancy and the shear forces, where when \(\phi \) increases from 0.0 to 0.04, the percentage increase in \(\overline{Nu}_{\mathrm{nf}}\)\(Re=10\) and \(Re=100\) are 8.8 and 12%, respectively. Nevertheless, it is clear from Fig. 10a that at \(Re=10\) and \(Ri > 10\), the increase in nanoparticles loading more the 0.02 becomes useless as the buoyancy force overcomes the improvement gained from the enhanced thermal conductivity and thermophoresis.
Fig. 9

Variations of local Nusselt numbers interface with Xa and Yb for different \(\phi \) at \(Re=100\), \(Ri=10\), \(Ha=15\) and \(D=0.3\)

Fig. 10

Variation of average Nusselt number with Ri for different \(\phi \) when a\(Re=10\) and b\(Re=100\) at \(Ha=15\) and \(D=0.3\)

Effects of Reynolds and Richardson numbers

The effect of varying Reynolds number on the three contour maps is shown in Fig. 11 at \(\phi =0.02\), \(Ri=10\), \(Ha=15\) and \(D=0.3\). For balanced inertia and viscus forces, \(Re=1\) (Fig. 11a), the effect of the moving wall is limited to the neighboring nanofluid. This in turn motivates a single recirculation close to the upper moving wall. When \(Re=10\) (Fig. 11b), this circulation strengthens and extends to confine the solid block. When \(Re\ge 100\), the inertia forces augments more and more, the circulation strengthens, becomes multi-cellular and extends to a substantial zone leading to diminishing the stagnant zone at the lower part of the cavity as shown in Fig. 11c–d. The isotherms of \(Re\le 10\) show a dominant conduction within the cavity and even in the solid block. When \(Re\ge 100\), the strong circulation leads to uniform temperature in the core of the cavity and the temperature gradient is limited in a thin boundary layer close to the heater legs as shown in Fig. 11c–d. The nanoparticles distribution, which follows the local temperature gradient, depicts that at low Reynolds numbers (\(Re\le 10\)), the Brownian diffusion is prominent which is distinguished by low distribution on the bottom and vertical walls, while high distribution is seen close to the upper moving cold wall. When \(Re=100\), the nanoparticles concentrate around the solid block as shown in Fig. 11c while at higher Reynolds number (\(Re=500\)), the uniform temperature distribution produces a homogeneous distribution of nanoparticles within the cavity as shown in Fig. 11d.
Fig. 11

Variation of the streamlines (left), isotherms (middle) and nanoparticle distribution (right) evolution by Reynolds number (Re) for \(\phi =0.02\), \(Ri=10\), \(Ha=15\) and \(D=0.3\)

On the other hand, the effect of Richardson number is shown in Fig. 12 for \(Re=100\) that depicts at dominant forced convection (\(Ri=0.01\)) and balanced forced and free convections (\(Ri=1\)), the nanofluid circulation is intensified close to the upper moving wall (Fig. 12a–b). When \(Ri=10\), the predominant buoyancy force strengthens the recirculation and switches the multi-cellular pattern as shown in Fig. 12c, while a stagnant zone still exists at the lower right corner of the cavity. At \(Ri=100\), the strong buoyancy force induces the symmetric intensified recirculation around the solid block, i.e., the geometry of corner heater has minor effect on the recirculation pattern. The isotherms at \(Ri\le 1\) experience a predominant conduction mode with wavy front due to the heater geometry as portrayed in the middle row of Fig. 12a–b. For dominant free convection (\(Ri\ge 10\)), the isotherms are stratified within the bulk of the cavity and the steep temperature gradient is characterized adjacent to the heater in the absence of the wavy isotherm pattern as shown in Fig. 12c–d. The distribution of the nanoparticles for \(Ri\le 1\) experiences asymmetric pattern around the solid block, and this refers to the dominance conduction heat transfer which switches the thermophoresis effect as shown in Fig. 12a–b. The thermophoresis effect becomes insufficient compared with the buoyancy force associated when \(Ri\ge 10\); thus, the distribution looks homogeneous and symmetric around the solid block. However, a sever concentration as perturbation-like is seen on the vertical wall of the solid block beside the heater edge.
Fig. 12

Variation of the streamlines (left), isotherms (middle) and nanoparticle distribution (right) evolution by Richardson number (Ri) for \(\phi =0.02\), \(Re=100\), \(Ha=15\) and \(D=0.3\)

Figures 13 and 14 both address the augmentation of the local Nusselt number with Reynolds and Richardson numbers. The convection harvested from the vertical part of the heater is much greater than the horizontal one. In addition, Fig. 10 depicts the augmentation of the average Nusselt number with Richardson number.
Fig. 13

Variations of local Nusselt numbers interface with X (a) and Y (b) for different Re at \(\phi =0.02\), \(Ri=10\), \(Ha=15\) and \(D=0.3\)

Fig. 14

Variations of local Nusselt numbers interface with X (a) and Y (b) for different Ri at \(\phi =0.02\), \(Re=100\), \(Ha=15\) and \(D=0.3\)

Effect of Hartmann number

The inclined magnetic field imposes two components of drag forces in negatives 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.
Fig. 15

Variation of the streamlines (left), isotherms (middle) and nanoparticle distribution (right) evolution by Hartmann number (Ha) for \(\phi =0.02\), \(Re=100\), \(Ri=10\) and \(D=0.3\)

The applied magnetic field does not only restrict the natural convection, but the Brownian diffusion of the nanoparticles is also restricted. Therefore, the local and average Nusselt numbers (Figs. 1618) deteriorate with Hartmann number. For weak shear-driven force (low values of 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 \(\overline{Nu}_{\mathrm{nf}}\).
Fig. 16

Variations of local Nusselt numbers interface with X (a) and Y (b) for different Ha at \(\phi =0.02\), \(Re=100\), \(Ri=10\) and \(D=0.3\)

Fig. 17

Variation of average Nusselt number with Ha for different \(\phi \) when a\(Re=10\) and b\(Re=100\) at \(Ri=10\) and \(D=0.3\)

Fig. 18

Variation of average Nusselt number with Ha for different Ri when a\(Re=10\) and b\(Re=100\) at \(\phi =0.02\) and \(D=0.3\)

Effect of solid block size

The existence of the solid block reduces the available area for the circulation of the nanofluid and also exerts a thermal resistance due to its low thermal conductivity compared with nanofluid. Nevertheless, it can serve in augmenting the convective heat transfer in some circumstances. A such, Fig. 19 demonstrates the reduction in the circulation strength with increasing the 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.
Fig. 19

Variation of the streamlines (left), isotherms (middle) and nanoparticle distribution (right) evolution by the length of the conductive inner block (D) for \(\phi =0.02\), \(Re=100\), \(Ri=1\) and \(Ha=15\)

The favorable effect of the solid block can be recognized by pursing the variations of the average Nusselt number, which is depicted in Figs. 2022. These figures explain that at low Reynolds number, the solid block deteriorates the convective heat transfer. The attribution of this issue refers to that the larger block size prevents the weak recirculation (which is generated due to the movement of the upper cold wall), to spread out through the cavity. However, when 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.
Fig. 20

Variation of average Nusselt number with \(\phi \) for different D when a\(Re=10\) and b\(Re=100\) at \(Ri=10\) and \(Ha=15\)

Fig. 21

Variation of average Nusselt number with Ri for different D when a\(Re=10\) and b\(Re=100\) at \(\phi =0.02\) and \(Ha=15\)

Fig. 22

Variation of average Nusselt number with Ha for different D when a\(Re=10\) and b\(Re=100\) at \(\phi =0.02\) and \(Ri=10\)

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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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Refrigeration & Air-conditioning Technical Engineering Department, College of Technical EngineeringThe Islamic UniversityNajafIraq
  2. 2.School of Mathematical Sciences, Faculty of Science & TechnologyUniversiti Kebangsaan MalaysiaBangiMalaysia
  3. 3.Mechanical Engineering Department, Engineering CollegeUniversity of BasrahBasrahIraq
  4. 4.Department of Mechanical Engineering, Prince Sultan Endowment for Energy and the EnvironmentPrince Mohammad Bin Fahd UniversityAl-KhobarSaudi Arabia
  5. 5.RAK Research and Innovation CenterAmerican University of Ras Al KhaimahRas Al KhaimahUnited Arab Emirates

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