Numerical simulation on MHD mixed convection of Cu-water nanofluid in a trapezoidal lid-driven cavity

Abstract

Optimization of the cooling processes by fluids in many industries such as power generation, transportation, machining, and electronics is very important. Heat transfer equipment to achieve higher efficiency, need to downsizing of the equipment, and increase the heat transfer rate per unit of surface. The aim of this paper is to investigate the effect of MHD mixed convection flow of Cu-water nanofluid in a trapezoidal lid-driven cavity with different tilt angles in a range of 0° to 60°. The cavity consists of the non-uniformly heated bottom wall, insulated top wall, and isothermal sidewalls. The governing equations of the flow by using the finite volume method and the SIMPLE algorithm have been numerically solved. The studies for the wide range of Richardson number (Ri) from 0.01 to 10, volume fraction of nanoparticles from 0.1% to 4% and Hartman number (Ha) from 0 to 40 in the steady-state have been done. Result in the form of streamline, vorticity and temperature contours for three geometry, local Nusselt number graphs for non-uniformly heated bottom surface and velocity profile on the vertical centerline of the cavity in different geometry have been investigated. The results show that the average Nusselt number on the non-uniformly heated bottom wall is dependent on dimensionless parameters and tilt angles. Also, applying a magnetic field, reduce the velocity profile changes, and using nanoparticles will cause increasing the Nusselt number.

Keywords

Mixed convection magneto-hydrodynamic trapezoidal cavity non-uniformly heated surface

1. Introduction

Magneto-hydrodynamics (MHD) is the study of the magnetic properties of electrically conducting fluids. The set of equations that describe MHD are a combination of the Navier–Stokes equations and Maxwell equation. A nanofluid is a fluid containing nanoparticles. These fluids are colloidal suspensions of nanoparticles in a base fluid. Several investigations have been carried out on convection especially with nanofluids and MHD [1–10].

Nemati et al. [11] investigated the effects of magnetic field on natural convection flow of a nanofluid in a rectangular cavity. They indicated that the averaged Nusselt number increases for nanofluids when increasing the volume fraction of nanoparticles, while, in the presence of a high magnetic field, this effect decreases. Sajjad Hossain and Abdul Alim [12] simulated the MHD free convection within the trapezoidal cavity with non-uniformly heated bottom wall. They found that the average and local Nusselt number at the non-uniform heating of the bottom wall of the cavity is depending on the dimensionless parameters and also tilts angles. Mejri and Mahmoudi [13] investigated MHD natural convection in a nanofluid-filled open enclosure with a sinusoidal boundary condition. They found that the heat transfer rate decreases with the increase of Hartmann number and increases with the rise of the Rayleigh number. Mahmoudi et al. [14] analyzed MHD natural convection of a nanofluid in a cavity with non-uniform boundary condition in the presence of uniform heat generation/absorption. They concluded that the effect of nanoparticles is more important for heat generation condition than absorption generation condition. Chamkha et al. [15] reviewed the recent developments in the study of MHD convection heat transfer using nanofluids in various geometries and applications. Selimefendigil and Öztop [16] analyzed MHD mixed convection in a flexible walled and nanofluids filled the lid-driven cavity with volumetric heat generation. They concluded that the local and averaged heat transfer enhances as the value of volume fraction of nanoparticles of the nanoparticle increases. Miroshnichenko et al. [17] investigated MHD natural convection in a partially open trapezoidal cavity filled with a nanofluid. They found that an increase in Hartmann number leads to the heat transfer reduction, while an increase in the nanoparticles volume fraction reflects the heat transfer enhancement. Hussain et al. [18] studied MHD mixed convection and entropy generation of water–alumina nanofluid flow in a double lid-driven cavity with discrete heating. They investigated the impacts of emerging parameters such as Reynolds number (Re), Richardson number (Ri), Hartman number (Ha), volume fraction of nanoparticles as well as the angles of inclined magnetic field on the flow. Sheremet et al. [19] investigated MHD natural convection of nanofluid flow in an inclined wavy cavity with corner heater. They found that an increase in the Hartmann number leads to a reduction of convective flow rate and heat transfer rate. Bakar et al. [20] investigated the effect of magnetic field on mixed convection heat transfer in a lid-driven square cavity. They found that the Ri and Ha significantly affect the heat transfer mechanism. Also, Heat transfer rate increases with the increase of Ri. Zhang et al. [21] analyzed MHD natural convection in a 2D cavity and 3D cavity with thermal radiation effects. Kefayati and Tang [22] simulated natural convection and entropy generation of MHD non-Newtonian nanofluid in a cavity. They found that an increase in Rayleigh number enhances heat and mass transfer for various power-law indexes. Sivaraj and Sheremet [23] studied MHD natural convection in an inclined square porous cavity with a heat-conducting solid block. Their results indicate that the inclusion of the magnetic field reduces the convective heat transfer rate in the cavity. Ghasemi and Siavashi [24] investigated MHD nanofluid free convection and entropy generation in porous enclosures with different conductivity ratios. They found that, depending on Ra and Ha values, use of nanofluid with porous media to enhance heat transfer can be either beneficial. Sheikholeslami and Shamlooei [25] investigated the convective flow of nanofluid inside a lid-driven porous cavity. They concluded that convective heat transfer improves with the increase of Darcy and Reynolds numbers while it decreases with the enhance of Ha. Shekar Balla et al. [26] studied MHD boundary layer flow and heat transfer in an inclined porous square cavity filled with nanofluids. They compared the present with the available results. Karimipour et al. [27] studied the effects of different nanoparticles of Al₂O₃ and Ag on the MHD nanofluid flow and heat transfer in a microchannel, including slip velocity and temperature jump. They observed that more slip coefficient corresponds to less Nusselt number and more slip velocity especially at larger Hartmann number. Öztop et al. [28] investigated mixed convection of MHD flow in a nanofluid filled and partially heated wavy walled lid-driven enclosure. They found that the rate of heat transfer decreases with increasing the Hartmann number.

The aim of this paper is to investigate the effect of MHD mixed convection flow of Cu-water nanofluid in a trapezoidal cavity with different tilt angles in a range of 0° to 60°. The cavity consists of the non-uniformly heated bottom wall, insulated top wall, and isothermal sidewalls. The studies for the wide range of Richardson number (Ri) from 0.01 to 10, the volume fraction of nanoparticles (φ) from 0.1% to 4% and Hartman number (Ha) from 0 to 40 in the steady-state have been done.

2. Solution method

2.1. Geometry of the problem

The effect of electrical conductivity of the nanofluid (Copper oxide-Water) was investigated in a trapezoidal cavity in the presence of a magnetic field. Figure 1 shows a cavity with an isothermal bottom wall (T _h), isothermal lateral walls (T _c = 293 K), insulated and movable upper wall with a constant velocity of U _∞ and a tilt angle of θ. A magnetic field (B _o) was applied to the cavity along the horizontal direction. Values of governing parameters are given in the below:

Hartmann number: 0 ≤ Ha ≤ 40

Richardson number: 0.01 ≤ Ri ≤ 10

The tilt angle: 0°, 30° and 60°

Volume fraction: 0.1, 0.55, 1, 2 and 4%.

The flow was also turbulent based on obtained parameters.

Fig. 1.

Schematic diagram of the physical system.

2.2. Assumptions

Laminar flow

Continuum medium

Nanofluid properties are assumed to be only a function of volume fraction of nanoparticles.

Incompressible nanofluid

Newtonian nanofluid

Two-dimensional Cartesian system (x, y).

Viscous dissipation is ignored.

A no-slip boundary condition is applied to all cavity walls.

2.3. Governing equations

By taking into account the above assumptions, continuity equation, momentum equations and energy equation are given in the below [29], $\begin{eqnarray}\displaystyle {\displaystyle \frac{\partial }{\partial x_{i}}}(u_{i})=0 & & \displaystyle\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle {\rho}_{nf}{\displaystyle \frac{\partial }{\partial x_{j}}}(u_{i}u_{j}) & = & \displaystyle -\frac{\partial p}{\partial x}+{\displaystyle \frac{\partial }{\partial x_{j}}}\left[{\mu}_{nf}\left(\frac{\partial u_{i}}{\partial x_{j}}+{\displaystyle \frac{\partial u_{j}}{\partial x_{i}}}\right)\right]+{\displaystyle \frac{\partial }{\partial X_{j}}}(-\overline{{\rho}_{nf}u_{i}^{^{\prime }}u_{j}^{^{\prime }}})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼({\rho}{\beta})_{nf}g(T-T_{c})-{\sigma}_{nf}B_{o}^{2}\text{v}\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \frac{\partial }{\partial x_{i}}(u_{i}(E{\rho}_{nf}+P))=\frac{\partial }{\partial x_{j}}\left[\left({\lambda}+\frac{C_{P,nf}{\mu}_{t}}{Pr_{t}}\right)\frac{\partial T}{\partial x_{j}}+u_{i}\left({\tau}_{ij}\right)_{eff}\right]=0. & & \displaystyle\end{eqnarray}$ (3) Where total energy (E) and deviatoric stress tensor (τ_ij)_eff are defined in the below, $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle E=C_{P,nf}T-{\displaystyle \frac{p}{{\rho}_{nf}}}+\left({\displaystyle \frac{u^{2}}{2}}\right)\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle ({\tau}_{ij})_{eff}=\left[{\mu}_{nf}\left({\displaystyle \frac{\partial u_{i}}{\partial x_{j}}}+\frac{\partial u_{j}}{\partial x_{i}}\right)\right].\end{eqnarray}$ (5) The transport equation for k −ω shear stress transfer (k −ω SST) model is given in the below [30], $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\displaystyle \frac{\partial }{\partial x}}({\rho}_{nf}ku_{i})={\displaystyle \frac{\partial }{\partial x_{j}}}\left({\rm\Gamma}_{k}{\displaystyle \frac{\partial k}{\partial x_{j}}}\right)+G_{k}-Y_{k}+S_{k}\end{eqnarray}$ (6) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{\partial }{\partial x_{j}}({\rho}_{nf}{\omega}ku_{i})=\frac{\partial }{\partial x_{j}}\left({\rm\Gamma}_{{\omega}}{\displaystyle \frac{\partial {\omega}}{\partial x_{j}}}\right)+G_{{\omega}}-Y_{{\omega}}+D_{{\omega}}+S_{{\omega}}.\end{eqnarray}$ (7) Where G _k is production of turbulent kinetic energy and the G _ω represents the production of ω, $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle G_{k}=-\overline{{\rho}u_{i}^{^{\prime }}u_{j}^{^{\prime }}}\left({\displaystyle \frac{\partial u_{j}}{\partial x_{i}}}\right),\quad G_{k}=\min (G_{k}10{\beta}^{\ast }k{\omega})\end{eqnarray}$ (8) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle G_{{\omega}}={\displaystyle \frac{{\alpha}}{{\vartheta}_{t}}}G_{k}.\end{eqnarray}$ (9) In the above equation, 𝜗_t is the turbulent kinematic eddy viscosity, 𝛽^∗ is constant of the model and 𝛼 is obtained from the following equation, $\begin{eqnarray}\displaystyle {\alpha}={\alpha}_{\infty }{\displaystyle \frac{({\alpha}_{0}^{\ast }+Re_{t}/R_{{\omega}})}{(1+Re_{t}/R_{{\omega}})}} & & \displaystyle\end{eqnarray}$ (10) R _ω = 2.195 and the 𝛼 term is expressed in the below, $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\alpha}_{\infty }=F_{1}{\alpha}_{\infty ,1}+(1-F_{1}){\alpha}_{\infty ,2}\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\alpha}_{\infty \cdot 1}={\displaystyle \frac{{\beta}_{i.1}}{{\beta}_{\infty }^{\ast }}}-\frac{{\kappa}^{2}}{{\sigma}_{{\omega},1}\sqrt{{\beta}_{\infty }^{\ast }}}\end{eqnarray}$ (12) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\alpha}_{\infty \cdot 2}={\displaystyle \frac{{\beta}_{i,2}}{{\beta}_{\infty }^{\ast }}}-{\displaystyle \frac{{\kappa}^{2}}{{\sigma}_{{\omega},2}\sqrt{{\beta}_{\infty }^{\ast }}}}.\end{eqnarray}$ (13) Where 𝜅 = 0.41 and 𝛽_i = 0.072 and 𝛼_∞ = 1 at high Reynolds numbers. Γ_k and Γ_ω are effective diffusion of k and ω and expressed in the below, $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\rm\Gamma}_{k}={\mu}+\frac{{\mu}_{t}}{{\sigma}_{k}}\end{eqnarray}$ (14) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\rm\Gamma}_{{\omega}}={\mu}+\frac{{\mu}_{t}}{{\sigma}_{{\omega}}}.\end{eqnarray}$ (15) Where σ_k and σ_ω terms represent the Prandtl numbers of turbulence in the k −ω models as expressed in the below, $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\sigma}_{{\omega}}={\displaystyle \frac{1}{F_{1}/{\sigma}_{{\omega},1}+(1-F_{1})/{\sigma}_{{\omega},2}}}\end{eqnarray}$ (16) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\sigma}_{k}={\displaystyle \frac{1}{F_{1}/{\sigma}_{k,1}+(1-F_{1})/{\sigma}_{k,2}}}.\end{eqnarray}$ (17) μ_t is eddy viscosity as expressed in the below, $\begin{eqnarray}\displaystyle {\mu}_{t}={\alpha}^{\ast }{\displaystyle \frac{{\rho}k}{{\omega}}}. & & \displaystyle\end{eqnarray}$ (18) 𝛼^∗ and ${\alpha}_{0}^{\ast }$ are expressed by the following equation, $\begin{eqnarray}\displaystyle {\alpha}_{0}^{\ast }=B_{i}/3,\quad {\alpha}^{\ast }={\alpha}_{\infty }^{\ast }\frac{({\alpha}_{0}^{\ast }+Re_{t}/R_{k})}{(1+Re_{t}/R_{k})}. & & \displaystyle\end{eqnarray}$ (19) Where ${\alpha}_{\infty }^{\ast }=∼1$ , $Re_{t}=\frac{{\rho}k}{{\mu}{\omega}}$ and R _k = 6. The F ₁ is also given in the below, $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle F_{1}=\tan ({\Phi}_{1}^{4})\end{eqnarray}$ (20) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Phi}_{1}=\min \left[\max \left(\frac{\sqrt{k}}{0.09{\omega}y}{\displaystyle \frac{500{\mu}}{{\rho}y^{2}{\omega}}}\right){\displaystyle \frac{4{\rho}k}{{\sigma}_{{\omega}\cdot 2}D_{{\omega}}^{+}y^{2}}}\right]\end{eqnarray}$ (21) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle D_{{\omega}}^{+}=\max \left[2{\rho}{\displaystyle \frac{1}{{\sigma}_{{\omega},2}}}\frac{1}{{\omega}}{\displaystyle \frac{\partial k}{\partial x_{j}}}\frac{\partial {\omega}}{\partial x_{j}}10^{-10}\right].\end{eqnarray}$ (22) Y _k and Y _ω are defined in the blow, $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle Y_{k}={\rho}{\beta}^{\ast }k{\omega}\end{eqnarray}$ (23) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle Y_{{\omega}}={\rho}{\beta}{\omega}^{2}.\end{eqnarray}$ (24) 𝛽 and 𝛽^∗ are constant values. For two-dimensional flows, we have: $\begin{eqnarray}\displaystyle \overline{P}=\left(\frac{\partial \overline{u_{1}}}{\partial x_{2}}+\frac{\partial \overline{u_{2}}}{\partial x_{1}}\right)(\overline{u_{1}^{^{\prime }}u_{2}^{^{\prime }}})-\overline{u_{1}^{^{\prime }2}}{\displaystyle \frac{\partial \overline{u_{1}}}{\partial x_{1}}}-\overline{u_{2}^{^{\prime }2}}{\displaystyle \frac{\partial \overline{u_{2}}}{\partial x_{2}}}. & & \displaystyle\end{eqnarray}$ (25)

2.4. The nanofluid properties

Nanofluid properties are calculated using the following equations [31], $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\rho}_{nf}=(1-{\varphi}){\rho}_{f}+{\varphi}{\rho}_{s}\end{eqnarray}$ (26) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle ({\rho}{\beta})_{nf}=(1-{\varphi})({\rho}{\beta})_{f}+{\varphi}({\rho}{\beta})_{s}\end{eqnarray}$ (27) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle ({\rho}C_{p})_{nf}=(1-{\varphi})({\rho}C_{p})_{f}+{\varphi}({\rho}C_{p})_{s}\end{eqnarray}$ (28) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\alpha}_{nf}={\displaystyle \frac{k_{nf}}{({\rho}C_{p})_{nf}}}\end{eqnarray}$ (29) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\gamma}=\frac{{\sigma}_{s}}{{\sigma}_{f}}\frac{{\sigma}_{nf}}{{\sigma}_{f}}=1+\frac{3({\gamma}-1){\varphi}}{({\gamma}+2)-({\gamma}-1){\varphi}}.\end{eqnarray}$ (30) In these equations, φ is the volume fraction of nanoparticles and the f, s and nf refer to the base fluid, nanoparticles and nanofluid, respectively. The following equation is used to calculate the dynamic viscosity of the nanofluid [32], $\begin{eqnarray}\displaystyle {\mu}_{nf}={\mu}_{f}(123{\varphi}^{2}+7.3{\varphi}+1)^{-2.5}. & & \displaystyle\end{eqnarray}$ (31) Where k _nf is thermal conductivity of the nanofluid. Patel [33] has proposed a model for this coefficient. This model is given in the below for spherical particles, $\begin{eqnarray}\displaystyle k_{nf}=k_{f}\left(1+{\displaystyle \frac{k_{\text{s}}A_{s}}{k_{f}A_{f}}}+ck_{s}Pe{\displaystyle \frac{A_{s}}{k_{f}A_{f}}}\right). & & \displaystyle\end{eqnarray}$ (32) Where k _s and k _f are respectively thermal conductivities of Copper nanoparticles and pure fluid. For water - Copper oxide nanofluid c = 36000 is recommended and, $\begin{eqnarray}\displaystyle {\displaystyle \frac{A_{s}}{A_{f}}}={\displaystyle \frac{d_{f}}{d_{s}}}{\displaystyle \frac{{\varphi}}{1-{\varphi}}}. & & \displaystyle\end{eqnarray}$ (33) Also, d _s = 100 nm. Molecular size of water base fluid is given in the below, $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle d_{f}=2A_{f}\end{eqnarray}$ (34) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle Pe={\displaystyle \frac{u_{s}d_{s}}{{\alpha}_{f}}}.\end{eqnarray}$ (35) Where u _s is Brownian velocity of nanoparticles, $\begin{eqnarray}\displaystyle u_{s}={\displaystyle \frac{2k_{b}T}{{\pi}{\mu}_{f}{d_{s}}^{2}}}. & & \displaystyle\end{eqnarray}$ (36) The value of Boltzmann constant (k _b) is 1.380 × 10⁻²³ m² kg s⁻² K⁻¹. The properties of water and nanoparticles are summarized in Table 1 [34].

Table 1

Thermophysical properties of pure water and Copper nanoparticles [34]

Thermophysical properties	Copper nanoparticles	Pure water
𝜌 (kgm⁻³)	8933	997.1
C _p (Jkg⁻¹ K⁻¹)	385	4179
k (Wm⁻¹ K⁻¹)	401	0.613
𝛽× 10⁻⁵ (K⁻¹)	67.1	21
σ (m⁻¹𝛺⁻¹)	107 × 96.5	0.05

After determining the volume fraction, nanofluid properties are determined. Since volume fraction varies from 0.1% to 4%, five different volume fractions are considered in this problem. These values are shown in Table 2.

Table 2

Properties of nanoluid

Name	Property	Unit	Volume fraction of nanoparticles
			0.001	0.0055	0.01	0.02	0.04
Density	𝜌	kg/m³	1005.935	1041.642	1077.35	1156.7	1315.4
Specific heat	C _p	J/kg ⋅ K	4148.281	4002.905	3867.166	3595.528	3150.569
Thermal conductivity	k	W/m ⋅ K	0.602	0.61	0.618	0.636	0.675
Dynamic viscosity	μ	kg/m ⋅ s	0.001005512	00.0010169	0.0010285	0.001055	0.00111
Kinematic viscosity	𝜗	m² /s	9.99 × 10⁻⁰⁷	9.76 ×10⁻⁰⁷	9.54 × 10⁻⁰⁷	9.12 × 10⁻⁰⁷	8.44 × 10⁻⁰⁷
Electric conductivity	σ	𝛺⁻¹ m⁻¹	0.025075	0.02541	0.02575	0.0265	0.0281
Thermal diffusivity coefficient	𝛼	m² /s	1.44 × 10⁻⁰⁷	1.46 × 10⁻⁰⁷	1.48 × 10⁻⁰⁷	1.53 × 10⁻⁰⁷	1.63 × 10⁻⁰⁷
Volumetric expansion coefficient	𝛽	K⁻¹	0.0002082	0.0002008	0.0001939	0.0001801	0.0001525

2.5. Dimensionless forms of governing equations

The following dimensionless parameters are used to rewrite the above equations in dimensionless form, $\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}X={\displaystyle \frac{x}{L}},\quad Y={\displaystyle \frac{y}{L}},\quad U={\displaystyle \frac{uL}{{\alpha}_{f}}},\quad V={\displaystyle \frac{{\vartheta}L}{{\alpha}_{f}}}\\ {\Theta}={\displaystyle \frac{T-T_{C}}{T_{H}-T_{C}}},\quad P={\displaystyle \frac{pL^{2}}{{\rho}_{nf}{\alpha}_{nf}^{2}}}.\end{array} & & \displaystyle\end{eqnarray}$ (37) In these equations, Richardson, Prandtl and Hartman numbers are defined in the below, $\begin{eqnarray}\displaystyle \text{Ri}=\frac{Gr}{Re^{2}},\quad \text{Gr}={\displaystyle \frac{g{\beta}_{nf}L^{3}(T_{h}-T_{c})}{{\vartheta}_{nf}^{2}}},\quad \Pr ={\displaystyle \frac{{\vartheta}_{\text{nf}}}{{\alpha}_{\text{nf}}}},\quad \text{Ha}=B_{0}L\sqrt{{\displaystyle \frac{{\sigma}_{\text{nf}}}{{\rho}_{\text{nf}}{\vartheta}_{\text{nf}}}}}. & & \displaystyle\end{eqnarray}$ (38) Where 𝛼_f = k _f∕(𝜌C _p)_f is thermal diffusivity coefficient of the base fluid. The dimensionless forms of continuity, momentum and energy equations are obtained by substituting dimensionless parameters in Eq. (1) to (3) and simplifying them.

- Continuity equation $\begin{eqnarray}\displaystyle {\displaystyle \frac{\partial U}{\partial X}}+{\displaystyle \frac{\partial V}{\partial Y}}=0. & & \displaystyle\end{eqnarray}$ (39) - Momentum equation in the X-direction $\begin{eqnarray}\displaystyle U{\displaystyle \frac{\partial U}{\partial X}}+V{\displaystyle \frac{\partial U}{\partial Y}}=-\frac{\partial P}{\partial X}+{\displaystyle \frac{{\rho}_{f}}{{\mu}_{f}{\rho}_{nf}Re}}\times \left[{\displaystyle \frac{\partial }{\partial X}}\left({\mu}_{\text{nf}}\frac{\partial U}{\partial X}\right)+\frac{\partial }{\partial Y}\left({\mu}_{\text{nf}}\frac{\partial U}{\partial Y}\right)-{\displaystyle \frac{2}{3}}{\displaystyle \frac{\partial }{\partial X}}\left(\frac{\partial U}{\partial X}\right)\right]. & & \displaystyle\end{eqnarray}$ (40) - Momentum equation in the Y-direction $\begin{eqnarray} \displaystyle U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y} & = & \displaystyle -\frac{\partial P}{\partial Y}+\frac{{\rho}_{f}}{{\mu}_{f}{\rho}_{nf}Re}\times \left[\frac{\partial }{\partial X}\left({\mu}_{\text{nf}}\frac{\partial V}{\partial X}\right)+\frac{\partial }{\partial Y}\left({\mu}_{\text{nf}}\frac{\partial V}{\partial Y}\right)-\frac{2}{3}\frac{\partial }{\partial Y}\left(\frac{\partial V}{\partial Y}\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼{\displaystyle \frac{\partial \overline{u_{1}}}{\partial Y}}(-\overline{u_{1}^{^{\prime }}u_{2}^{^{\prime }}})+{\displaystyle \frac{({\rho}{\beta})_{nf}}{{\rho}_{nf}{\beta}_{f}}}Ri\Theta-{\displaystyle \frac{Ha^{2}}{Re}}\times {\displaystyle \frac{{\sigma}_{nf}}{{\sigma}_{n}}}\times \frac{{\rho}_{f}}{{\rho}_{nf}}\times V. \end{eqnarray}$ (41) - Energy Equation $\begin{eqnarray} \displaystyle U\frac{\partial\Theta }{\partial X}+V\frac{\partial\Theta }{\partial Y}=\frac{{\alpha}_{nf}}{{\alpha}_{f}}\times \frac{1}{k_{nf}RePr}\times \left[\frac{\partial }{\partial X}\left(k_{nf}\frac{\partial\Theta }{\partial X}\right)+\frac{\partial }{\partial Y}\left(k_{nf}\frac{\partial\Theta }{\partial Y}\right)\right]=0. & & \displaystyle \end{eqnarray}$ (42)

2.6. Boundary conditions

On the bottom wall: 0 ≤ X ≤ 1, $\Theta$ = sin(πX), V = 0, U = 0

On the side walls: 0 ≤ Y ≤ 1, 𝛩 = 0, V = 0, U = 0

On the upper wall: − tan θ ≤ X ≤ (1 + tan θ), ${\displaystyle \frac{\partial {\Theta}}{\partial Y}}=∼0$ , V = 0, U = U _∞.

Given the boundary condition on the bottom wall, local Nusselt numbers will be negative if X is not defined in the 0 < X <1 interval.

2.7. Other equations [35]

The local Nusselt number can be expressed as follows, $\begin{eqnarray}\displaystyle Nu_{y}=\frac{hL}{k_{f}}. & & \displaystyle\end{eqnarray}$ (43) Where k _f is thermal conductivity of the base fluid and h is the heat transfer coefficient obtained from the following equation, $\begin{eqnarray} \displaystyle h=\frac{qw}{T_h-T_{c}}. & & \displaystyle \end{eqnarray}$ (44) The local Nusselt number for heated or cooled walls are obtained as follows, $\begin{eqnarray}\displaystyle Nu_{Y}=-\frac{k_{nf}}{k_{f}}\left(\frac{\partial {\Theta}}{\partial X}\right)_{X=0\text{ or }1}. & & \displaystyle\end{eqnarray}$ (45) The average Nusselt number in the heated wall is obtained by integrating the above equation by the heated wall as follows, $\begin{eqnarray}\displaystyle Nu_{m,h}=\int _{0}^{1}Nu_{Y}(Y)dY=\frac{k_{nf}}{k_{f}}\int _{0}^{1}\left(\frac{\partial {\Theta}}{\partial X}\right)_{X=0}dY. & & \displaystyle\end{eqnarray}$ (46)

3. Results and discussion

In this study, the mixed convection inside a trapezoidal cavity filled with CuO-water nanofluid was investigated. The upper wall of the cavity is insulated. The temperature at lateral walls is T _c, and the temperature in the bottom wall is T _h. A magnetic field (in the opposite direction of the flow) is applied to the cavity (Fig. 1)). The Streamlines, the temperature contour, and the Nusselt number are calculated. The problem was simulated using the given assumptions. To solve this problem, a pressure-based solver is used in a two-dimensional mode. The gravity acceleration is considered as g = 9.81 ms⁻².

3.1. Verification

Figure 2 shows the comparison of Nusselt number obtained from numerical solution and Nusselt number obtained from Ref. [36]. Table 3 shows the comparison between results.

Table 3
Comparison of the average Nusselt number in the present study and the Ref. [36] for 45° geometry

Ra Present study Ref. [26] Difference percent (%)

Ra = 10⁴ 2.914 2.871 1.4

Ra = 10⁵ 4.743 4.702 0.8

Ra	Present study	Ref. [26]	Difference percent (%)
Ra = 10⁴	2.914	2.871	1.4
Ra = 10⁵	4.743	4.702	0.8

Fig. 2.

Comparison of the local Nusselt number obtained from the numerical method and the Ref. [36] for 45° geometry at Ra = 10⁵.

3.2. Trapezoidal cavity (θ = 90°)

3.2.1. Streamlines and vorticity contours

Figure 3 shows the streamline contours for different Hartmann numbers, Richardson numbers, and volume fractions of nanoparticles. When the cavity is under the influence of a horizontal magnetic field, the Lorentz force is applied to the nanofluid from the bottom wall while free convection heat transfer moves the fluid upwards. Therefore, the Lorentz force opposes with free convection. Free convection can be ignored, and forced convection can be considered in the case that Ri < 1. This is because the inertial force dominates the Buoyance force. Free convection cannot be ignored as the Richardson number increases since velocity decreases at the bottom wall [37–45]. Mixed convection occurs at Ri = 1. Therefore, no significant difference can be detected in streamlines as Hartmann number increases. Free convection moves the fluid upwards at Ri = 10 that leads to the formation of vortices in the streamline contours. Figure 4 shows vorticity contours for different Hartmann numbers, Richardson numbers and volume fraction of nanoparticles. The figure shows the largest vortex at the upper wall in all Richardson numbers. This is due to the movable upper wall. The vortex is also larger at Ri = 0.01 because the wall velocity is higher at this number compared to other Richardson numbers.

Fig. 3.

Streamline contours.

Fig. 4.

Vorticity contours.

3.2.2. Temperature contours

Figure 5 shows the temperature contours for different Hartmann numbers, Richardson numbers, and volume fractions of nanoparticles. The figure shows that the temperature contours show a higher temperature near heated walls. The temperature decreases as the distance from the heated wall increase. The Buoyancy force increases, and the density of isothermal lines increases near the cooled walls with increasing the Richardson number in a constant Hartman number. Therefore, convection heat transfer enhances and consequently, Nusselt number increases by increasing the Richardson number in a constant Hartmann number. Temperature decreases at Ri = 0.01 because velocity increases compared to the two other Richardson numbers given that velocity and temperature are inversely related to each other. The temperature contour is more uniform at Ri = 0.01 and Ri = 1 because the forced convection dominates the free convection. However, the temperature contour is non-uniform at Ri = 10 with increasing the Hartmann number because the electromagnetic force increases, and consequently, the Buoyance force decreases. Therefore, convection heat transfer decreases.

Fig. 5.

Temperature contours.

3.2.3. Variation of the local Nusselt number in the bottom wall

Figure 6 shows the variations in the local Nusselt number at different Richardson numbers, Ha = 40 and φ = 1%. This figure shows a sinusoidal increase in the local Nusselt number at Ri = 0.01 since the velocity increases at the upper wall and consequently forced convection enhances. Therefore, the Nusselt number increases. As Richardson increases, the Nusselt number decreases until the Nuselt number minimizes at Ri = 10. Therefore, forced convection has a greater effect on heat transfer compared to free convection. Figure 7 shows variations in the local Nusselt number at different volume fractions, Ha = 40 and Ri = 1. This figure shows that the Nusselt number in the heated wall does not significantly change at the entrance and the end of the cavity as nanoparticle volume fraction increases. However, a significant change can be detected in the Nusselt number in the center of the cavity. Therefore, the Nusselt number increases in the heated wall by increasing the volume fraction. This is because of the thermal conductivity increases by adding nanoparticles to the base fluid. Figure 8 shows variations in the local Nusselt number at different Hartmann numbers, the Richardson number Ri = 0.01 and φ = 0. 1%. The figure shows that the Nusselt number decreases as the Hartman number increases. The Nusselt number is higher at Ha = 0 and Ha = 40 due to high velocity at the upper wall.

Fig. 6.

Variations of the local Nusselt number at the bottom wall in different Richardson numbers (Ha = 40 and φ = 1%).

Fig. 7.

Variations of the local Nusselt number at the bottom wall in different volume fractions (Ha = 40 and Ri = 1).

Fig. 8.

Variations in the local Nusselt number at the bottom wall in different Hartman numbers (Ri = 0.01 and φ = 0. 1%).

3.2.4. Velocity profile in the central vertical line of the cavity

Figure 9 shows the velocity profile in the central vertical line of the cavity in different Richardson numbers, Ha = 40 and φ = 1%. This figure shows that the velocity profile decreases as the Richards number increases. This is because velocity decreases at the upper wall as the Richardson number increases. The velocity profile is more uniform at Richardson number Ri = 10. The range of variations in the velocity profile widens by decreasing the Richardson number and vortices appear in the cavity. Figure 10 shows the velocity profile in the central vertical line of the cavity in different Hartmann numbers, Ri = 10 and φ = 1%. The figure shows that the range of variations in the velocity profile narrows as the Hartman number increases.

Fig. 9.

The velocity profile in the central vertical line of the cavity in different Richardson numbers (Ha = 40 and = φ = 1%).

Fig. 10.

The velocity profile in the central vertical line of the cavity in different Hartmann numbers (Ri = 10 and φ = 1%).

3.3. Trapezoidal cavity (θ = 30°)

3.3.1. Streamline and vorticity contours

Figure 11 shows the streamline contours for different Hartmann numbers, Richardson numbers and volume fractions of nanoparticles. This figure shows that formation of a small vortex on the left side of the cavity at Ri = 0.01. This indicates enhancement in forced convection. Larger vortices are formed in the cavity as Richardson increases. This is because the velocity decreases at bottom wall of the cavity, forced convection is ignored, and free convection is considered. Free convection heat transfer moves the fluid upwards at Ri = 10. The Lorentz force opposes the free convection at Ha = 40, which decreases the velocity of displacement of flow lines in the cavity. Figure 12 shows the vorticity contours at different Hartmann numbers, Richardson numbers and volume fractions of nanoparticles. Similar to square geometry, vortex lines are gathered on the upper wall. This is because the upper wall is movable. On the contrary, small vortices are formed at Ri = 10 because the velocity decreases at the upper wall and the upper wall in this geometry is larger than the square geometry.

Fig. 11.

Streamline contours.

Fig. 12.

Vorticity contours.

3.3.2. Temperature contours

Figure 13 shows the temperature contours for different Hartmann numbers, Richardson numbers, and volume fractions of nanoparticles. Similar to square geometry, the temperature contour shows a higher temperature at the heated wall. The temperature decreases as the distance from this wall increases. The electromagnetic force increases by increasing the Hartmann number at Ri = 0.01, which consequently decreases the Buoyance force. Therefore, convection heat transfer is expected to decrease. Mixed convection occurs at Ri = 1. In this case-scenario, the density of isothermal lines (i.e., temperature gradient) in the heated source decreases by increasing the Hartmann number. Free convection occurs in the temperature contour at Ri = 10 as the Richardson number increases and the density of isothermal lines decreases in the heated wall as the Hartmann number increases.

Fig. 13.

Temperature contours.

3.3.3. Variations in the local Nesslet number at the bottom wall

Figure 14 shows the variations in the local Nusselt number in different Richardson numbers, Ha = 40 and φ = 1%. This figure shows that the Nusselt numerical value does not exceed 70, and the Nusselt number has decreased slightly in this geometry compared to square geometry at Ri = 0.01. This is because the upper wall is enlarged in 30-degree geometry and the upper wall is larger than the square geometry. In this case, the Nusselt number decreases by increasing the number of Richardson. This is due to the reduction in inertial forces, which decreases the Nusselt number.

Fig. 14.

Variations in the local Nusselt number at the bottom wall in different Richardson numbers (Ha = 40 and φ = 1%).

Figure 15 shows the variation in the local Nusselt number in different volume fraction of nanoparticles, Ha = 40, and Ri = 1. This figure shows that the Nusselt number increases in the heated wall by increasing the volume fraction of nanoparticles. The results indicated that adding nanoparticles to the base fluid in the presence of a magnetic field decreases the convection heat transfer.

Fig. 15.

Variations in the local Nusselt number at the bottom wall in different volume fractions (Ha = 40 and Ri = 1).

Figure 16 shows the variations in the local Nusselt number in different Hartmann numbers, Ri = 0.01 and φ = 0. 1%. This figure shows that the Nusselt number increases by increasing the Hartman number. This is because the Lorentz force applied to the flow field has decreased; the Hartmann number has increased in this Richardson number. These issues enhance heat transfer. According to momentum equations, Buoyancy force opposes the Lorentz force. Therefore, forced convection heat transfer occurs at Ri = 0.01, which increased the intensity of the magnetic field and consequently increased the local Nusselt number. On the other hand, forced convection heat transfer dominates at lower Richardson numbers. As a result, the intensity of the magnetic field increases, which strengthens the flow near the walls. Therefore, the heat transfer enhances and consequently the local Nesselt number increases. In other words, the Buoyance force decreases at lower and average Richardson numbers. Therefore, the Lorentz forces increase that strengthens the flow near the heated walls and enhances the heat transfer. Therefore, higher Hartmann number does not have a great effect on the heat transfer. In other words, the strengthened magnetic field has a great effect on the heat transfer at a certain threshold. If the intensity of the magnetic field exceeds this threshold, heat transfer is not enhanced much. Also, it is not cost-effective to increase the intensity of the magnetic field more than this threshold.

Fig. 16.

Variations in the local Nusselt number at the bottom wall in different Hartman numbers (Ri = 0.01 and φ = 0. 1%).

3.3.4. Velocity profile in the central vertical line of the cavity

Figure 17 shows the velocity profile in the central vertical line of the cavity in different Richardson numbers, Ha = 0 and φ = 1%. This figure shows that the velocity profile decreases as the Richardson number increases. The velocity profile is also not symmetrical at Ri = 0.01 compared to square geometry. This is due to the trapezoidal shape of the cavity and asymmetrical upper and bottom walls. The velocity profile is more uniform as the Richardson number increases. This is due to the formation of vortices in the cavity. Figure 18 shows the velocity profile in the central vertical line of the cavity in different Hartmann numbers, Ri = 10 and φ = 1%. Figure 18 shows high fluctuations in the velocity profile due to the trapezoidal shape of the cavity. The velocity profile decreases in the center of the cavity at Ha = 0 due to larger and asymmetrical 30-degree geometry compared to square geometry.

Fig. 17.

The velocity profile in the central vertical line of the cavity in different Richardson numbers (Ha = 40 and = φ = 1%).

Fig. 18.

The velocity profile in the central vertical line of the cavity in different Hartmann numbers (Ri = 10 and φ = 1%).

3.4. Trapezoidal cavity (θ = 45°)

3.4.1. Streamline and vorticity contours

Figure 19 shows the Streamline contours for different Hartmann numbers, Richardson numbers, and volume fraction of nanoparticles. This figure shows no significant difference in the contour of flow lines as Hartmann number increases at Ri = 0.01, but the Streamlines accumulate at the upper wall. Several vortices are formed on the left side of the cavity at Ri = 1, but the vortices disappear, and a large vortex is formed in the cavity as the Richardson number increases and the Buoyance force increases. The Streamlines are compressed together at Ri = 10 in the absence of a magnetic field. The Streamline aligns as parallel lines with cavity walls as the magnetic field is intensified. Figure 20 shows the vorticity contours for different Hartmann numbers, Richardson numbers, and volume fraction of nanoparticles. The figure shows more visible vortices in the cavity compared to 30-degree geometry due to the larger upper wall in the 60-degree geometry. The vortices in the cavity become asymmetrical by increasing the Richardson number and decreasing the velocity at the upper wall. The vortices accumulated on the upper wall will disperse throughout the cavity at Ri = 10.

Fig. 19.

Streamline contours.

Fig. 20.

Vorticity contours.

3.4.2. Temperature contours

Figure 21 shows the temperature contours for different Hartmann numbers, Richardson numbers, and volume fraction of nanoparticles. This figure shows that the convection heat transfer decreases by increasing the Hartmann number at all Richardson numbers and various volume fractions. The temperature contours accumulate near the heated wall as the Richardson number increases and the heat transfer enhances. A vortex is formed on the right side of the cavity at Ha = 0 by increasing the Richardson number. This is due to enhancement in increase free convection heat transfer in contrast to forced convection.

Fig. 21.

Temperature contours.

3.4.3. Variations in local Nusselt number at the bottom wall

Figure 22 shows variations in local Nessult number in different Richardson numbers, Ha = 40, and φ = 1%. Similar to previous case-scenarios, the local Nusselt number decreases as the Richardson number increases. The number of Nusselt number does not exceed 40 at Ri = 0.01. This Nusselt number is lower than other geometries. This is because the fluid should pass through a lager bottom wall to increase the Nusselt number. The geometry of the two previous geometries is less. Therefore, the Nusselt number is lower on the bottom wall compared to previous geometries. The two graphs align at Ri = 1 and Ri = 10 to L = 0.4 m but the two graphs largely separate from each other at Ri = 1. This is due to a reduction in the inertial force and decreased Nusselt number.

Fig. 22.

Variations in the local Nusselt number at the bottom wall in different Richardson numbers (Ha = 40 and φ = 1%).

Figure 23 shows the variations in the local Nusselt number in different volume fractions, Ha = 40 and Ri = 1. This figure shows that the Nusselt number increases by increasing the volume fraction. This is due to increased thermal conductivity by adding nanoparticles to the base fluid. In this case, the upper wall is larger than the previous two geometries, and the fluid should pass through a larger space to increase the local Nusselt number at the on the bottom wall at different volume fractions. Figure 24 shows variation in the local Nusselt number in different Hartmann numbers, Ri = 0.01 and φ = 0.1%. This figure shows that the Nusselt number increases as the Hartmann number increases. As previously mentioned, heat transfer enhances by decreasing the Lorentz force applied to the flow field and increasing the Hartmann number in this Richardson number.

Fig. 23.

Variations in the local Nusselt number at the bottom wall in different volume fractions (Ha = 40 and Ri = 1).

Fig. 24.

Variations in the local Nusselt number at the bottom wall in different Hartman numbers (Ri = 0.01 and φ = 0. 1%).

3.4.4. Velocity profile in the central vertical line of the cavity

Figure 25 shows the velocity profile in the central vertical line of the cavity in different Richardson number, Ha = 0, and φ = 1%. This figure shows that the velocity profile has decreased by 0.05 at Ri = 0.01 compared to 30-degree geometry. This is due to larger 60-degree geometry compared to 30-degree geometry. Figure 26 shows the velocity profile in the central vertical line of the cavity at different Hartmann numbers, Ri = 10, and φ = 1%. The graph is similar to the previous section, but the velocity of the fluid is higher at Ha = 40 and L = 0.4 m compared to Ha = 0. This is due to the strengthened magnetic field and Lorentz force. The velocity profile has also increased in this geometry compared to other geometries. This can be due to a lower velocity at the upper wall, which is larger compared to the previous two geometries.

Fig. 25.

The velocity profile in the central vertical line of the cavity in different Richardson numbers (Ha = 40 and = φ = 1%).

Fig. 26.

The velocity profile in the central vertical line of the cavity in different Hartmann numbers (Ri = 10 and φ = 1%).

4. Conclusion

In this study, we investigated the effect of MHD mixed convection flow of Cu-water nanofluid in a trapezoidal cavity with different tilt angles in a range of 0° to 60°. The studies for the wide range of Richardson number, the volume fraction of nanoparticles and Hartman number have been done. The following results can be deduced from this study:

The average Nusselt number on the bottom wall is dependent on dimensionless parameters and tilt angles.

Also, applying a magnetic field, reduce the velocity profile changes.

Using nanoparticles will cause increasing the Nusselt number.

5. Compliance with ethical standards

Disclosure of potential conflicts of interest: Not applicable

Research involving Human Participants and/or Animals: : Not applicable

Informed consent: : Not applicable

Footnotes

Acknowledgements

This article is part of a research project sponsored by the Young Researchers and Elite Club.

References

MHD natural convection in a nanofluid-filled cavity with linear temperature distribution, Journal of Computational Methods in Sciences and Engineering 14(4-5) (2014), 291–313.

Combined effects of internal heat generation and mass effects on MHD Casson fluid with convective boundary conditions, International Journal of Applied Electromagnetics and Mechanics 54(2) (2017), 211–221.

Numerical simulation and stability analysis on MHD free convective heat and mass transfer unsteady flow through a porous medium in a rotating system with induced magnetic field, International Journal of Applied Electromagnetics and Mechanics 41(2) (2013), 121–141.

Modeling of fully coupled mhd flows in annular linear induction pumps, International Journal of Applied Electromagnetics and Mechanics 44(2) (2014), 155–162.

Three-dimensional MHD modeling of railgun plasma armature with the CE/SE Method, International Journal of Applied Electromagnetics and Mechanics 46(1) (2014), 207–216.

The analysis of MHD blood flows through porous arteries using a locally modified homogenous nanofluids model, Bio-Medical Materials and Engineering 27(1) (2016), 15–28.

Quasi-two-dimensional case studies of MHD flow and heat transfer behind a square cylinder in a duct, International Journal of Applied Electromagnetics and Mechanics 49(1) (2015), 123–132.

Effects of convective conditions and chemical reaction on peristaltic flow of Eyring- Powell fluid, Applied Bionics and Biomechanics 11(4) (2014), 221–233.

Dehghani

M.S.

Toghraie

and Mehmandoust

, Effect of MHD on the flow and heat transfer characteristics of nanofluid in a grooved channel with internal heat generation, International Journal of Numerical Methods for Heat & Fluid Flow 29(4) (2019), 1403–1431.

10.

Barnoon

Toghraie

Dehkordi

R.B.

and Abed

, MHD mixed convection and entropy generation in a lid-driven cavity with rotating cylinders filled by a nanofluid using two phase mixture model, Journal of Magnetism and Magnetic Materials 483:224–248.

11.

Nemati

Farhadi

Sedighi

Ashorynejad

H R.

and Fattahi

, Magnetic field effects on natural convection flow of nanofluid in a rectangular cavity using the Lattice Boltzmann model, Scientia Iranica B 19(2) (2012), 303–310.

12.

Hossain

M.S.

and Alim

M.A.

, MHD free convection within trapezoidal cavity with non-uniformly heated bottom wall, International Journal of Heat and Mass Transfer 69 (2014), 327–336.

13.

Mejri

and Mahmoudi

, MHD natural convection in a nanofluid-filled openenclosure with a sinusoidal boundary condition, Chemical Engineering Research and Design 98 (2015), 1–16.

14.

Mahmoudi

Mejri

Abbassi

M.A.

and Omri

, Analysis of MHD natural convection in a nanofluid-filled open cavity with non-uniform boundary condition in the presence of uniform heat generation/absorption, Powder Technology 269 (2015), 275–289.

15.

Chamkha

A.J.

Jena

S.K.

and Mahapatra

S.K.

, MHD convection of nanofluids: a review, Journal of Nanofluids 4 (2015), 271–292.

16.

Selimefendigil

and Öztop

H.F.

, Analysis of MHD mixed convection in a flexible walled and nanofluids filled lid-driven cavity with volumetric heat generation, International Journal of Mechanical Sciences 118 (2016), 113–124.

17.

Miroshnichenko

I.V.

Sheremet

M.A.

Oztop

H.F.

and Al-Salem

, MHD natural convection in a partially open trapezoidal cavity filled with a Nanofluid, International Journal of Mechanical Sciences 119 (2016), 294–302.

18.

Hussain

Mehmood

and Sagheer

, MHD mixed convection and entropy generation of water–alumina nanofluid flow in a double lid driven cavity with discrete heating, Journal of Magnetism and Magnetic Materials 419 (2016), 140–155.

19.

Sheremet

M.A.

Oztop

H.F.

and Pop

, MHD natural convection in an inclined wavy cavity with corner heater filled with a nanofluid, Journal of Magnetism and Magnetic Materials 416 (2016), 37–47.

20.

Bakar

N.A.

Karimipour

and Roslan

, Effect of magnetic field on mixed convection heat transfer in a lid-driven square cavity, Journal of Thermodynamics 2016: Article ID 3487182, 14.

21.

Zhang

J.-K.

B.-W.

Dong

Luo

X.-H.

and Lin

, Analysis of Magneto-hydrodynamics (MHD) natural convection in 2D cavity and 3D cavity with thermal radiation effects, International Journal of Heat and Mass Transfer 112 (2017), 216–223.

22.

Kefayati

G.H.R.

and Tang

, Simulation of natural convection and entropy generation of MHD non-Newtonian nanofluid in a cavity using Buongiorno’s mathematical model, International Journal of Hydrogen Energy 42 (2017), 17284–17327.

23.

Sivaraj

and Sheremet

M.A.

, MHD natural convection in an inclined square porous cavity with a heat conducting solid block, Journal of Magnetism and Magnetic Materials 426 (2017), 351–360.

24.

Ghasemi

and Siavashi

, MHD nanofluid free convection and entropy generation in porous enclosures with different conductivity ratios, Journal of Magnetism and Magnetic Materials 442 (2017), 474–490.

25.

Sheikholeslami

and Shamlooei

, Convective flow of nanofluid inside a lid driven porous cavity using CVFEM, Physica B 521 (2017), 239–250.

26.

Balla

C.S.

Kishan

Gorla

R.S.R.

and Gireesha

B.J.

, MHD boundary layer flow and heat transfer in an inclined porous square cavity filled with nanofluids, Ain Shams Engineering Journal 8 (2017), 237–254.

27.

Karimipour

D’Orazio

and Safdari Shadloo

, The effects of different nano particles of Al₂O₃ and Ag on the MHD nano fluid flow and heat transfer in a microchannel including slip velocity and temperature jump, Physica E 86 (2017), 146–153.

28.

Öztop

H.F.

Sakhriehb

Abu-Nadad

and Al-Salem

, Mixed convection of MHD flow in nanofluid filled and partially heated wavy walled lid-driven enclosure, International Communications in Heat and Mass Transfer 86 (2017), 42–51.

29.

Manca

Nardini

and Ricci

, A numerical study of nanofluid forced convection in ribbed channels, App. Therm. Eng. 37 (2012), 280–292.

30.

Menter

F.R.

, Two equation eddy-viscosity turbulence models for engineering Applications, AIAA J 32 (1994), 1598–1605.

31.

Ghasemi

Aminossadati

S.M.

and Raeisi

, Magnetic field effect on natural convection in a nanofluid-filled square enclosure, International Journal of Thermal Science 50 (2011), 1748–1756.

32.

Patel

H.E.

Sundararajan

Pradeep

Dasgupta

and Das

S.K.

, A micro-convection model for thermal conductivity of nanofluids, Pramana J. Phys. 65 (2005), 863–869.

33.

Santra

A.K.

Sen

and Chakraborty

, Study of heat transfer due to laminar flow of copper-water nanofluid through two isothermally heated parallel plates, Int. J. Therm. Sci. 48 (2009), 391–400.

34.

Soleimani

Sheikholeslami

Ganji

D.D.

and Gorji-Bandpay

, Natural convection heat transfer in a nanofluid filled semi-annulus enclosure, International Communications in Heat and Mass Transfer 39(4) (2012), 565–574.

35.

Aminossadati

S.M.

and Ghasemi

, Natural convection cooling of a localized heat source at the bottom of a nanofluid-filled enclosure, European Journal of Mechanics B/Fluids 28 (2009), 630–640.

36.

Basak

Roy

and Pop

, Heat flow analysis for natural convection within trapezoidal enclosures based on heat line concept, International Journal of Heat and Mass Transfer 52 (2009), 2471–2483.

37.

Parsaiemehr

Pourfattah

Akbari

O.A.

Toghraie

and Sheikhzadeh

, Turbulent flow and heat transfer of Water/Al₂O₃ nanofluid inside a rectangular ribbed channel, Physica E: Low-Dimensional Systems and Nanostructures 96 (2018), 73–84.

38.

Toghraie

Abdollah

M.M.D.

Pourfattah

Akbari

O.A.

and Ruhani

, Numerical investigation of flow and heat transfer characteristics in smooth, sinusoidal and zigzag-shaped microchannel with and without nanofluid, Journal of Thermal Analysis and Calorimetry 131(2) (2018), 1757–1766.

39.

Arabpour

Karimipour

and Toghraie

, The study of heat transfer and laminar flow of kerosene/multi-walled carbon nanotubes (MWCNTs) nanofluid in the microchannel heat sink with slip boundary condition, Journal of Thermal Analysis and Calorimetry 131(2) (2018), 1553–1566.

40.

Sarlak

Yousefzadeh

Akbari

O.A.

Toghraie

and Sarlak

, The investigation of simultaneous heat transfer of water/Al2O3 nanofluid in a close enclosure by applying homogeneous magnetic field, International Journal of Mechanical Sciences 133 (2017), 674–688.

41.

Arabpour

Karimipour

Toghraie

and Akbari

O.A.

, Investigation into the effects of slip boundary condition on nanofluid flow in a double-layer microchannel, Journal of Thermal Analysis and Calorimetry 131(3) (2018), 2975–2991.

42.

Foroutan

Haghshenas

Hashemian

Eftekhari

S.A.

and Toghraie

, Spatial buckling analysis of current-carrying nanowires in the presence of a longitudinal magnetic field accounting for both surface and nonlocal effects, Physica E: Low-Dimensional Systems and Nanostructures 97 (2018), 191–205.

43.

Barnoon

Toghraie

Eslami

and Mehmandoust

, Entropy generation analysis of different nanofluid flows in the space between two concentric horizontal pipes in the presence of magnetic field: single-phase and two-phase approaches, Computers & Mathematics with Applications 77(3) 662–692.

44.

Mousavi

S.M.

Darzi

A.A.R.

Ali Akbari

Toghraie

and Marzban

, Numerical study of biomagnetic fluid flow in a duct with a constriction affected by a magnetic field, Journal of Magnetism and Magnetic Materials 473 (2019), 42–50.

45.

Barnoon

and Toghraie

, Numerical investigation of laminar flow and heat transfer of non-Newtonian nanofluid within a porous medium, Powder Technology 325 (2018), 78–91.

Numerical simulation on MHD mixed convection of Cu-water nanofluid in a trapezoidal lid-driven cavity

Abstract

Keywords

1. Introduction

2. Solution method

2.1. Geometry of the problem

2.3. Governing equations

2.7. Other equations [35]

3.1. Verification

Table 3 Comparison of the average Nusselt number in the present study and the Ref. [36] for 45° geometry Ra Present study Ref. [26] Difference percent (%) Ra = 104 2.914 2.871 1.4 Ra = 105 4.743 4.702 0.8

3.2.1. Streamlines and vorticity contours

3.3.1. Streamline and vorticity contours

3.4.1. Streamline and vorticity contours

5. Compliance with ethical standards

Footnotes

Acknowledgements

References

Table 3
Comparison of the average Nusselt number in the present study and the Ref. [36] for 45° geometry

Ra Present study Ref. [26] Difference percent (%)

Ra = 10⁴ 2.914 2.871 1.4

Ra = 10⁵ 4.743 4.702 0.8