Heat transfer enhancement compared to entropy generation by imposing magnetic field and hybrid nanoparticles in mixed convection of a Bingham plastic fluid in a ventilated enclosure

Subhasree Dutta (Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, India)

Somnath Bhattacharyya (Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, India)

Ioan Pop (Facultatea de Matematica si Informatica, Universitatea Babes Bolyai, Cluj-Napoca, Romania)

International Journal of Numerical Methods for Heat & Fluid Flow

ISSN: 0961-5539

Article publication date: 28 March 2022

Issue publication date: 20 July 2022

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Abstract

Purpose

The purpose of this study is to analyze the nonhomogeneous model on the mixed convection of Al2O3–Fe3O4 Bingham plastic hybrid nanofluid in a ventilated enclosure subject to an externally imposed uniform magnetic field. Entropy generation and the pressure drop are determined to analyze the performance of the heat transfer. The significance of Joule heating arising due to the applied magnetic field on the heat transfer of the yield stress fluid is described.

Design/methodology/approach

The ventilation in the enclosure of heated walls is created by an opening on one vertical wall through which cold fluid is injected and another opening on the opposite vertical wall through which fluid can flow out.

Findings

This study finds that the inclusion of Fe₃O₄ nanoparticles with the Al₂O₃-viscoplastic nanofluid augments the heat transfer. This rate of enhancement in heat transfer is higher than the rate by which the entropy generation is increased as well as the enhancement in the pressure drop. The yield stress has an adverse effect on the heat transfer; however, it favors thermal mixing. The magnetic field, which is acting opposite to the direction of the inlet jet, manifests heat transfer of the viscoplastic hybrid nanofluid. The horizontal jet of cold fluid produces the optimal heat transfer.

Originality/value

The objective of this study is to analyze the impact of the inclined cold jet of viscoplastic electrically conducting hybrid nanofluid on heat transfer from the enclosure in the presence of a uniform magnetic field. The combined effect of hybrid nanoparticles and a magnetic field to enhance heat transfer of a viscoplastic fluid in a ventilated enclosure has not been addressed before.

Keywords

Citation

Dutta, S., Bhattacharyya, S. and Pop, I. (2022), "Heat transfer enhancement compared to entropy generation by imposing magnetic field and hybrid nanoparticles in mixed convection of a Bingham plastic fluid in a ventilated enclosure", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 32 No. 9, pp. 3007-3038. https://doi.org/10.1108/HFF-09-2021-0623

Publisher

:

Emerald Publishing Limited

Nomenclature

Bn: = Bingham number, τ0*H/ηU0;
c_p: = Specific heat (J kg⁻¹ K⁻¹);
D_B: = Brownian diffusion coefficient, K_BT/3πµ_fd_p;
D_T: = Thermophoretic diffusion coefficient, 0.26(K_fµ_fφ_b)/(K_p + 2K_f)ρ_f;
g: = Gravitational acceleration (m/s²);
Gr: = Grashof number, (1−φb)(βfgΔTH3)/νf2;
H: = Enclosure height, (m);
Ha: = Hartmann number, B0Hσf/μf;
J: = Joule heating parameter, HσfB02u0/(ρCp)f(Th−Tc);
k: = Thermal conductivity (W/m K);
Nr: = Buoyancy ratio, φb(ρpi−ρf)/{ρfβfΔT(1−φb)};
Nu: = Local Nusselt number, {−(knf/kf) (∂θ/∂n)};
p^*: = Pressure, (N/m²);
Pr: = Prandtl number, ν_f/α_f;
Re: = Reynolds number, ρ_fU₀H/µ_f;
Ri: = Richardson number, Gr/Re²;
S_gen: = Dimensionless total entropy generation;
S_av: = Average entropy generation;
t^*: = Time (s);
t: = Dimensionless time;
T: = Temperature (K);
(u^*, v^*): = Velocity components in x- and y-direction, respectively (m/s);
(u, v): = Dimensionless velocity components in x- and y-direction, respectively; and
U₀: = Characteristic velocity of the flow (m/s).

Greek symbols

α: = Thermal diffusivity (m²/s);
β: = Coefficient of thermal expansion;
λ: = Inclination angle of jet flow;
μ: = Dynamic viscosity (kg/m s);
ν: = Kinematic viscosity (m²/s);
θ: = Dimensionless temperature;
ρ: = Density, (kg/m³); and
φ: = Nanoparticles volume fraction.

Subscripts

av: = Average value;
b: = Bulk value;
f: = Clear fluid;
nf: = Hybrid nanofluid;
p: = Solid particle; and
i = 1, 2: = Al₂O₃/Fe₃O₄ nanoparticles.

Superscripts

*: = Dimensional quantity; and
0: = Clear fluid (φ_b = 0).

1. Introduction

Ventilated enclosures having an inlet and outlet port is one of those configurations where both the shear dominated forced convection and buoyancy driven natural convection occur simultaneously. Ventilation is a useful mechanism to maintain thermal balance between heating and cooling by minimizing the energy consumption (Bianco et al., 2018; Koufi et al., 2017). Thermal management in ventilated enclosures provides a guideline for designing computer chips, heat exchangers, refrigerator, cooling of electronic devices, room heating and solar panel (Gupta et al., 2015). Several authors (Arroub et al., 2016; Bilgen and Muftuoglu, 2008) studied the mixed convection within a ventilated enclosure by considering different thermal boundary conditions to achieve an enhanced thermal performance. The experimental and numerical studies (Ingole and Sundaram, 2016; Shiriny et al., 2019) show that the direction of the injection flow at the inlet has dependency on the thermal performance of the enclosure.

The impact of nanofluid on heat transfer and pressure drop in a ventilated cavity is analyzed by several authors (Kherroubi et al., 2020; Sourtiji et al., 2014, 2011). These studies show that the rate of heat transfer intensifies due to the inclusion of nanoparticles in the base fluid. Sourtiji et al. (2014) conclude that maximum heat transfer is achieved by placing the outlet port at the end corner, whereas the minimum heat transfer is achieved when the outlet port is fixed at the middle of the side wall. Recently, Kherroubi et al. (2020) investigated the impact of the nanofluid on the three-dimensional fluid flow and heat transfer by considering a ventilated cubic enclosure having heated side walls and concluded that the use of nanoparticles improves the thermal performance of the fluid but increases the pressure drop coefficient significantly. All those studies on heat transfer in a ventilated enclosure deals with Newtonian fluid with a single type of nanoparticle suspended in it.

In recent years, a superior category of nanofluids, the “hybrid nanofluid” composed of more than one type of nanoparticles mixed with the base liquid, has been established to provide better thermal characteristics such as higher thermal conductivity, chemical stability and mechanical resistance as compared with the individual nanofluids (Esfe et al., 2015; Ghalambaz et al., 2020b). This extended type of nanofluid is widely used in several heat transfer fields such as motor cooling, thermal management of vehicles, transformer cooling and refrigeration due to its improved efficiency over that of the single type nanofluids (Ellahi et al., 2019; Ghalambaz et al., 2020a; Khan et al., 2020b). Heat transfer augmentation in magnetohydrodynamics (MHD) forced convection within an enclosure by considering the Ag–MgO–water hybrid nanofluid is studied numerically by Ma et al. (2019). Indeed, contradictory results also exist in the literature. The agglomeration of different types of nanoparticles may cause a reduction in the thermal conductivity of the hybrid nanofluid (Sarkar et al., 2015). Mehryan et al. (2019) analyzed the impact of using Cu and Al₂O₃ nanoparticles for the hybridization and concluded that for a low Rayleigh number, hybrid nanofluid produces a higher heat transfer rate. However, for a higher range of the Rayleigh number, the hybrid nanoparticles reduce the heat transfer compared with the Al₂O₃-nanofluid due to the increased dynamic viscosity of the hybrid nanofluid. Studies on hybrid nanofluids are important to achieve enhanced thermal performance by using expensive nanoparticles with minimum quantity (Rashad et al., 2018). For example, the price of copper nanoparticles is about 10 times greater than that of alumina nanoparticles.

The hydrodynamics of nanofluid can be analyzed based on the homogeneous model in which the nanoparticles are considered to move with the fluid, and uniform distribution of nanoparticles in the fluid is assumed (Dogonchi et al., 2019a). In the nonhomogeneous model of the nanofluid, a relative flux between the fluid and nanoparticles is accounted for, which develops a nonhomogeneous distribution of the nanoparticles in the medium. Buongiorno (2006) demonstrated the nonhomogeneous model by considering that the slip velocity between nanoparticles and fluid can be generated by seven mechanisms, namely, Brownian diffusion, thermophoresis, diffusiophoresis, inertia, Magnus effect, fluid drainage and gravity, of which only the Brownian and thermophoretic diffusion have a significant contribution. In the absence of a turbulent effect, the thermophoretic force influence to migration of the nanoparticles in the opposite direction of the imposed temperature gradient, whereas Brownian diffusion increases the homogeneity in the nanoparticles distribution due to the higher concentration gradient of the nanoparticles. It has been established by several authors (Corcione, 2010; Ho et al., 2010) that the nonhomogeneous model has better agreement with the experimental data. The comparative study (Esfandiary et al., 2016) between the homogeneous and the nonhomogeneous model shows that Brownian diffusion and thermophoresis have a significant impact on the heat transfer performance of the nanofluid. The nonhomogeneous model provides a higher heat transfer rate than the homogeneous model due to the presence of slip velocity between the fluids and nanoparticles. Although the contradictory result is also being reported in the literature (Sheikhzadeh et al., 2013) i.e. the single-phase homogeneous model generates a higher heat transfer rate in comparison with the two-phase nonhomogeneous model. Garoosi et al. (2014) have shown that instead of a monotonic increment of average Nusselt number with the nanoparticles volume fraction as is seen in the homogeneous model, the nonhomogeneous model shows that there exists an optimal volume fraction at which the maximum heat transfer occurs.

Most of the studies on convective heat transfer are confined to the Newtonian fluid, where the dynamic viscosity of the fluid is constant throughout the flow domain. In the case of the nonNewtonian fluid, the viscosity is not constant; rather, it depends on the rate of the strain, which is governed by a constitutive equation of the strain tensor (Nazari et al., 2020). A class of nonNewtonian fluid, the viscoplastic fluid, which involves a suspension of macromolecules, can model effectively the hydrodynamics of blood, synovial fluids, saliva, bitumen, honey, paste, animal waste slurries, cement paste, greases, molten lava, magmas, paints, pharmaceutical products and wet beach sand (Elelamy et al., 2020; Kefayati and Tang, 2018). The growing applications of the viscoplastic fluid in various manufacturing and processing industries (Syrakos et al., 2013) have primed us to analyze the flow and heat transfer characteristics of such types of nonNewtonian fluid. The viscoplastic fluid possesses the characteristic that it flows when the shear stress exceeds a critical value τ₀, the yield stress. For shear stress below the yield stress, the viscoplastic fluid either remains stationary or moves like a rigid body. The Bingham viscoplastic fluid involves unyielded zones with a discontinuity in shear stress across the yield surface. To avoid such discontinuity along the yield surface, Papanastasiou (1987) proposed a regularization approach by introducing the stress growth parameter through which the constitutive equation is made to be valid throughout the flow domain without involving any discontinuity across the yield surface. This regularization approach is adopted by several authors to analyze the flow behavior of the viscoplastic fluid (Kefayati, 2017, 2018; Kefayati and Huilgol, 2016).

Studies based on the homogeneous model for the mixed convection of the Bingham plastic nanofluid inside a square enclosure is made by Kefayati and Huilgol (2016) and Kefayati (2018). In those studies, the regularization approach was adopted to analyze the mixed convection of the yield stress viscoplastic fluid. Studies on nonNewtonian nanofluids based on the two-phase model are limited to power-law fluids only. Kefayati (2017) considered the nonhomogeneous model to study the mixed convection of the shear-thinning nonNewtonian nanofluid based on the power-law model and found that the Brownian diffusion and thermophoresis decline the rate of heat transfer. Our literature survey suggests that a detailed study on the heat transfer of viscoplastic nanofluid based on the nonhomogeneous model has not been made.

Several studies established that a transverse magnetic field imposed opposite to the direction of the primary flow can produce a higher heat transfer in a channel or enclosure (Heidary et al., 2015). The magnetic field-induced Lorentz force creates a strong influence on the thermal field by diminishing the recirculating vortices in the enclosure and modifying the boundary layers along the side walls (Dogonchi et al., 2019b; Tayebi and Chamkha, 2019). Recently, Kefayati and Tang (2018) studied the MHD mixed convection in a lid-driven cavity and found that the adverse Lorentz force due to the imposed magnetic field expands the unyielded regions leading to a reduction in heat transfer. The impact of a uniform magnetic field on the heat transfer of nanofluids in a cavity or channel is considered by several researchers (Khan et al., 2020a; Rashad et al., 2018; Rasool et al., 2020). The Joule heating arises due to the imposed magnetic field on the electrically conducting fluid enhancing the fluid temperature; however, it may have a marginal effect on the formation of unyielded zones of the viscoplastic fluid. The convective heat transfer of Al₂O₃–water nanofluid considering the magnetic field together with Joule heating effect is analyzed by Mehmood et al. (2017) and concluded that the Joule heating creates an adverse effect on the heat transfer.

In this paper, we have considered mixed convection of a viscoplastic nanofluid subjected to an applied magnetic field. The MHD convection of viscoplastic fluid has relevance in the context of several physiological applications such as heart valve prosthesis, cell separation and magnetic hyperthermia. The human blood can be considered to be electrically conducting nonNewtonian fluid (Majee and Shit, 2017). The magnetic field is also used to regulate the blood flow during surgery as the flow rate is enhanced due to the magnetic properties of blood (Haik et al., 2001). Heat transfer analysis in an enclosure filled with viscoplastic fluid, influenced by an externally imposed magnetic field, has relevance in polymer industry, plasma studies, electronic package, geothermal energy extraction, MHD generators, microelectromechanical systems and many other areas of engineering (Sheremet et al., 2016).

The entropy generation, which is determined based on the second law of thermodynamics, provides a measure of the thermodynamic efficiency of the system (Bejan, 1980). The relative influence of the irreversibilities generated due to thermal radiation and viscous dissipation is illustrated by computing the Bejan number (Varol et al., 2008). Several researchers (Bejan, 1980; Ellahi et al., 2018) assessed the thermal efficiency of a system by evaluating the entropy generation. Using the homogeneous model for nanofluid, the heat transfer efficiency of the viscoplastic nanofluid based on the entropy generation is made by several authors (Kefayati, 2018; Mohammadi and Moghadam, 2015). However, the entropy generation to evaluate the thermal performance of the viscoplastic fluid in a ventilated enclosure has not been addressed in the literature.

We have studied the mixed convection of the Bingham plastic hybrid nanofluid in a ventilated enclosure that is subjected to a uniform horizontal magnetic field. An inclined jet of cold Al₂O₃−Fe₃O₄/viscoplastic hybrid nanofluid is pumped through the inlet of the enclosure of heated side walls, and an outlet port is fixed at the lower corner of the opposite side wall. A uniform magnetic field is applied opposite to the direction of the inward jet. The buoyancy force, generated by the temperature gradient due to the four heated side walls together with the shear force governed by the incoming flow from the inlet, creates the mixed convection within the enclosure. The objectives of the present study are to augment the heat transfer of the viscoplastic fluid in mixed convection through the inclusion of Al₂O₃−Fe₃O₄ hybrid nanoparticles and imposed magnetic field, as well as to investigate the thermal performance of the hybrid nanofluid compared with a nanofluid composed of a single type of nanoparticles. The nonhomogeneous model is adopted to study the mixed convection of the hybrid viscoplastic plastic nanofluid. The heat transfer augmentation of a viscoplastic fluid in a ventilated enclosure through the combined effect of the hybrid nanoparticles and magnetic field has not been addressed before. In addition, none of the existing studies on viscoplastic nanofluid in a ventilated enclosure evaluated the entropy generation to describe the heat transfer performance.

We have adopted the regularization method (Papanastasiou, 1987) to model the viscoplastic fluid. The present model for the nonNewtonian fluid flow is validated by comparing it with the existing experimental results of Hassan et al. (2020). Entropy generation along with the Bejan number, thermal mixing and pressure drop between the inlet and outlet are determined to evaluate the heat transfer performance of the vented enclosure. The irreversibility due to the imposed magnetic field is included in the entropy generation. The yield stress of the viscoplastic fluid attenuates heat transfer, which can be compensated by the magnetic field applied in the reverse direction of the incoming jet. We find that for nonzero yield stress, the results based on the present nonhomogeneous model differ significantly from the corresponding results due to the homogeneous models. Based on this study, we have established that improved thermal performance can be achieved by considering the hybrid nanofluid compared with a nanofluid composed of a single type of nanoparticles.

2. Mathematical model

We consider the laminar two-dimensional mixed convection of an electrically conducting hybrid nanofluid, modeled as Bingham plastic fluid, inside a ventilated enclosure of height H having the inlet and outlet ports at the top of the left and the bottom of the right vertical walls, respectively [Figure 1(a)]. All four walls of the enclosure are maintained at higher temperature T_h, whereas the fluid flowing in through the inlet port is considered to be at a lower temperature T_c (< T_h). The inlet fluid velocity U₀ is considered to be injected at an angle λ with the horizontal (x-axis). A magnetic field of uniform strength B₀ is applied horizontally opposite to the direction of the jet flow. The hybrid nanofluid, composed of the dilute mixture of solid spherical nanoparticles Al₂O₃ and Fe₃O₄ to the electrically conducting base fluid, is considered to exhibit a viscoplastic nature. The induced magnetic field generated due to the motion of the nanofluid is assumed to be negligible compared with the applied magnetic field. The impact of Joule heating generated due to the imposed magnetic field in the energy transport equations is considered in our study. The width of the inlet and outlet port are the same, i.e. W_r = W_l = H/3.

The constitutive equation for the Bingham plastic fluid relates the shear stress (τ*) with the shear strain rate (γ^*) (Kefayati, 2018) i.e.

(1) {τ*=(η+τ0*|γ˙*|)γ˙*,for |τ*|>τ0*γ*=0,for |τ*|≤τ0*

where τ0* is the yield stress of the viscoplastic fluid, and the constant η is the flow consistency index. The shear strain rate is |γ˙*|=[2(∂u*∂x)2+2(∂v*∂y)2+(∂u*∂y+∂v*∂x)2]12. It is apparent that in the viscoplastic fluid flow, two distinct domains, namely, yielded region ( |τ*|>τ0*) and unyielded region ( |τ*|≤τ0*) with an interface, yield surface appears. The constitutive equation (1) shows that a jump discontinuity in the shear stress occurs across the yield surface. To resolve such singularity, an exponential regularization in the viscosity function is considered by Papanastasiou (1987). This adopted regularization method consists of an additional parameter m so that the modified equation is converted into a smooth function valid for the entire domain. The relation between shear stress and rate of strain is explicated as (Papanastasiou, 1987)

(2) τ*=[η+τ0*|γ˙*|{1−exp(−m|γ˙|)}]γ˙*=μ*γ˙*

The dimensionless variables are defined by x = x^*/H, y = y^*/H, t = t^*U₀/H, h = h*/H, θ=(T−Tc)/(Th−Tc)=(T−Tc)/ΔT, u = u^*/U₀, v = v^*/U₀, p = p*/ρnf*U02. φ_i is the local concentration of the ith type nanoparticles, which is scaled by the bulk concentration φbi, i.e. φi=φi*/φbi. Here, subscripts f, p_i, nf refer to the clear fluid, particles and the hybrid nanofluid, respectively. Both the base fluid and nanoparticles are in a thermal equilibrium state.

We consider two-dimensional mixed convection of the Bingham plastic nanofluid, which is subjected to an externally imposed magnetic field applied horizontally. Under the Boussinesq approximation, the thermal buoyancy term present in the momentum equation is expressed as (ρβ)_nf (T − T_c) g =φb1φ1ρp1+φb2φ2ρp2+(1−(φb1φ1+φb2φ2))ρf{1−β(T−Tc)} g, which can be further reduced to (1−φb)ρfβf(T−Tc)−(ρp1+ρp2−ρf)(φb1φ1+φb2φ2−φb) g under the assumption of dilute concentration of nanoparticles and a suitable choice of the reference pressure field. The nondimensional form of the governing equations under the Boussinesq approximation can be expressed as follows (Alsabery et al., 2019):

(3) ∂u∂x+∂v∂y=0

(4) ∂∂t(ρnfρfu)+u∂∂x(ρnfρfu)+v∂∂y(ρnfρfu)=−(ρnfρf)∂p∂x+1Re[∂∂x{μnfμf(2μf∂u∂x)}+∂∂y{μnfμf(μf(∂u∂y+∂v∂x))}]

(5) ∂∂t(ρnfρfv)+u∂∂x(ρnfρfv)+v∂∂y(ρnfρfv)=−(ρnfρf)∂p∂y+1Re[∂∂x{μnfμf(μf(∂u∂y+∂v∂x))}+∂∂y{μnfμf(2μf∂v∂y)}]+Ri[Nr(1−∑i=12φi)+θ]+σnfσfHa2Rev

(6) ∂∂t((ρCp)nf(ρCp)fθ)+u∂∂x((ρCp)nf(ρCp)fθ)+v∂∂y((ρCp)nf(ρCp)fθ)=1Re·Pr[{∂∂x(knfkf∂θ∂x)+∂∂y(knfkf∂θ∂y)}+∑i=121Lei{∂θ∂x∂φi∂x+∂θ∂y∂φi∂y}+∑i=121Lei·NBTi{(∂θ∂x)2+(∂θ∂y)2}]+Jσnfσfv2

(7) ∂φi∂t+u∂φi∂x+v∂φi∂y=1Re·Sci(∂2φi∂x2+∂2φi∂y2)+1Re·Sci·NBTi(∂2θ∂x2+∂2θ∂y2)

The no-slip boundary condition (u = 0, v = 0) is imposed on the heated cavity walls (θ = 1) along with the no-normal flux of nanoparticles i.e. Jpi.n=0, n being the unit outward normal to the surface. This relation implies ∂φi∂y+1NBTi∂θ∂n=0. At the inlet, a velocity profile of lower temperature is assumed i.e.

u=cosλ,v=sinλ,θ=0,∂φi∂x=−1NBTi∂θ∂x

The outflow boundary conditions are considered as follows:

∂u∂x=0,∂v∂x=0,∂θ∂x=0,∂φi∂x=0

The equation for the nanoparticle volume fraction (7) is governed by the convection, Brownian diffusion and thermophoresis. The flux due to Brownian diffusion is JP,Bi = −ρpiDBi∇φi where DBi is the Brownian diffusion coefficient, DBi = KBT3πμfdpi. The mass flux due to thermophoretic effect is JP,Ti = −ρpi DTi∇TT where DTi is the thermal diffusion coefficient, DTi = 0.26 KfKpi+2Kfμfφbiρf. Here K_f is the thermal conductivity of the base fluid and Kpi with i = 1, 2 represents, respectively, the thermal conductivity of Al₂O₃ and Fe₃O₄ nanoparticles. We have considered a sufficiently dilute mixture of two types of nanoparticles so that the interactions of different species of nanoparticles and the effect of agglomeration of nanoparticles is neglected (Huminic and Huminic, 2018). For the two-phase model, Fe₃O₄ nanoparticles with volume fraction φb2 is added with Al₂O₃-nanofluid of bulk volume fractions φb1 with the ratio δ=φb1/φb2 and sum φb=φb1+φb2 is the net volume fraction.

The dimensionless parameter Hartmann number, Ha = B0Hσf/μf measures the ratio between the electromagnetic force to the viscous force, Joule heating parameter J = Ha²Ec/Re, where Ec=U02/(Th−Tc)Cpf is Eckert number involves the magnetic field. The Schmidt number Sc_i = μf/(ρfDBi) represents the ratio of momentum diffusivity and Brownian diffusivity, whereas the Lewis number Le_i = kf/{(ρCp)fDBiφbi} measures the ratio of thermal diffusivity to the Brownian diffusivity and Nr=φb(ρpi−ρf)/{ρfβfΔT(1−φb)} is the buoyancy ratio between the nanofluid and base fluid. NBTi = φbiDBi/(DTiΔT) denotes the relative performance of the Brownian diffusion to that of the thermophoretic diffusion.

The thermophysical properties of hybrid nanofluid are expressed in terms of the properties of the base fluid and the composed nanoparticles. We consider the base fluid to be a Bingham plastic liquid and both the base fluid and nanoparticles are in a thermal equilibrium state. We consider the thermophysical properties of the base fluid and nanoparticles the same as considered in several existing studies (Garoosi et al., 2014; Hojjat et al., 2011; Ouyahia et al., 2017) and are provided in Table 1. Under the low volume fraction of nanoparticles consideration, the thermophysical properties of the hybrid nanofluids can be determined through the linear model (Shah et al., 2020; Xu, 2019). In the nondimensional form, the effective density (ρ_nf) and the effective heat capacitance ((ρc_p)_nf) of the hybrid nanofluids are expressed as (Rashad et al., 2018):

(8) ρnf=(1−∑i=12φbiφi)ρf+∑i=12φbiφiρpi

(9) (ρcp)nf=(1−∑i=12φbiφi)(ρcp)f+∑i=12φbiφi(ρcp)pi

The effective thermal conductivity of the nanofluid is determined through the Maxwell–Garnetts (MG) model (Maxwell, 1873):

(10) knfkf=∑i=12kpiφbiφi∑i=12φbiφi+2kf−2kf∑i=12φbiφi+2∑i=12φbiφikpi∑i=12kpiφbiφi∑i=12φbiφi+2kf+kf∑i=12φbiφi−∑i=12φbiφikpi

Based on the Brinkman (1952) model, the apparent viscosity of the hybrid viscoplastic nanofluid can be expressed as follows (Kefayati, 2015):

(11) μnf=1(1−∑i=12φbiφi)2.5[1+Bn|γ˙|{1−exp(−M|γ˙|)}]

Here Bn=τ0*H/ηU0 is the Bingham number, and M = mU₀/H is the stress growth parameter. The parameter M is considered to be sufficiently large so that the equation holds uniformly for yielded and unyielded regions. In our computation, we have chosen M = 10⁴ to obtain the results. We have also considered several other models for the thermal conductivity and viscosity of the nanofluid such as the Corcione model (Corcione, 2011), MG–Brinkman model (Brinkman, 1952), MG–Pak and Cho model (Pak and Cho, 1998) and Patel–Brinkman model (Patel et al., 2006). As discussed later in this article, the MG–Brinkman model is found to be the most convenient, and the result obtained by this model does not deviate much from the other well-established models. A similar observation is also made in the experiential study by Mahian et al. (2016). Based on the experimental data for the Al₂O₃−Fe₃O₄ hybrid nanofluid, Sulgani and Karimipour (2019) proposed the following correlation for the thermal conductivity:

(12) knfkf=0.113x1.001θ(φ1+φ2)0.376+0.921

We have also tested our computed results based on this correlation and compared them with the results based on the MG-Brinkman model, which has been discussed later in this article.

We consider the following Maxwell (1873) model for the electric conductivity, which is justified in the present study due to the consideration of dilute suspension of nanoparticles

(13) σnfσf=1+3(∑i=12σpiφbiφiσf−∑i=12φbiφi)(∑i=12σpiφbiφiσf∑i=12φbiφi+2)−(∑i=12σpiφbiφiσf−∑i=12φbiφi).

The average Nusselt number Nu_av is obtained by integrating the local Nusselt number along the heated side walls normalized by the length of the side walls. The local entropy generated due to heat transfer irreversibility and fluid friction irreversibility, including the effect of the imposed magnetic field, is as follows:

(14) Sgen=knfkf[(∂θ∂x)2+(∂θ∂y)2]+χ[μnfμf{2(∂u∂y)2+2(∂v∂x)2+(∂u∂x+∂v∂y)2}+Ha2σnfσfv2]

The irreversibility factor χ is determined by χ = μfT0U02kf(ΔT). Here T₀ = (T_h + T_c)/2 is the reference temperature. The total entropy generation S_av is determined by integrating S_gen over the whole computational domain V divided by the total volume of the domain.

2.1 Heat function and energy flux vector

The energy flux vectors are locally tangential to the heat lines. It provides a correct visualization of the convective heat transfer and does not require to solve the Poisson equation for heat function (Costa, 2006). The energy flux vectors are defined as follows:

−∂H∂x=RePr(ρCp)nf(ρCp)fθv−knfkf∂θ∂y, ∂H∂y=RePr(ρCp)nf(ρCp)fθu−knfkf∂θ∂x

Here H(x, y) is the heat function, which satisfies the energy equation identically. The net energy flow through the fluid is locally parallel to the heat lines H(x, y) = constant. To avoid the complexity in determining the heat function, several authors have obtained the energy flux vector E, locally tangential to the heatlines, defined as Pal et al. (2018):

E=(∂H∂y,−∂H∂x)

2.2 Thermal mixing and temperature uniformity

The intensity of the thermal mixing inside the system is measured by computing θ_cup and θ_av known as mixing-cup temperature (θ_cup) and the velocity-weighted average temperature (θ_av), respectively. A better thermal mixing due to mixed convection is observed when the value of θ_cup and θ_av is higher. The mixing-cup temperature (θ_cup) and the volume-averaged temperature (θ_av) are defined as θcup=∫∫U*θ(x,y)dxdy∫∫U*dxdy, θav=∫∫θ(x,y)dxdy∫∫dxdy, where U*=u2+v2

The significance of temperature uniformity within the nonhomogeneous system can be illustrated by computing the root mean square deviation (RMSD) of mixing-cup temperature and volume-averaged temperature are defined, respectively, as follows:

(15) θcupm=∑i=1N(θi−θcup)2N,θavm=∑i=1N(θi−θav)2N

θ_i is the temperature at data point i. The smaller values of θcupm and θavm signify lower standard deviation indicating an improved temperature uniformity inside the enclosure.

3. Numerical methods

The governing nonlinear equations are discretized through the finite volume method over staggered grids. In the staggered grid arrangements, the computational domain is split up into a number of Cartesian cells. In each cell the scalar quantities such as pressure, temperature and the nanoparticles volume fraction are stored at the center, whereas the components of the velocity vectors are estimated at the midpoint of the corresponding sides of that cell to which they are normal. The diffusion terms in the equations are discretized by a second-order central difference type scheme based on the linear interpolation of variable values at either side of the cells. The temporal derivatives are discretized through the first-order implicit scheme. A third-order accurate upwind scheme, quadratic upwind interpolation for convective kinematics (QUICK) scheme (Hayase et al., 1992), is adopted to approximate the convective terms in the momentum, energy and mass flux equations. The QUICK scheme imparts stability to the numerical scheme in the regions where a sharp change of variables occurs. The present mass-conservative numerical scheme has an overall second-order accuracy. A coupled correction of the pressure and velocity field is implemented through the pressure correction-based iterative technique, the SIMPLE (Fletcher, 2012) algorithm. In each iteration, the resulting block tri-diagonal system of linear algebraic equations is solved by a block elimination method. A detailed discussion on the discretization method is provided in Appendix 1.

We consider that the flow starts impulsively from rest and achieves a steady state after a transient phase. The time-dependent solutions are achieved by advancing the variables through a sequence of short time steps. At each time step, the iteration process starts with a known pressure field, which is upgraded iteratively by solving the Poisson equation for pressure correction. The iteration process is continued till the divergence-free velocity field is obtained. Based on the updated velocity field, the linearized momentum equations and other scalar transport equations are solved in a coupled manner. The convergence condition for the iteration is imposed as max⁡ij |ξij(k+1)−ξij(k)| ≤ 10−5, ξ corresponds to the variables u, v, θ or φ at any time step. Here, i, j denotes the cell index, and superscript k is the index for the iteration.

4. Grid independency test and code validation

We consider a uniform grid distribution along the x- and y-direction [Figure 1(b)]. An initial estimate of the grid is determined by comparing it with several existing results. A grid independence examination is performed to analyze the stability of the present numerical scheme, and an optimal grid distribution is determined for a cost-effective computation. The variation of grid size on the local Nusselt number is presented in Figure 2(a). The nondimensional quantity such as Nu is governed by the computed solution for the thermal field in the entire domain, which is coupled with the velocity field and nanoparticles distribution. The grid independency test is performed by considering four sets of grids, namely, 90 × 90, 150 × 150, 210 × 210 and 270 × 270, with the first number corresponding to the grids along the x-direction and the second number for the grids along the y-direction. For 90 × 90 grids, the grid size is dx = dy = 0.004 along with the horizontal and vertical directions. Modifying the grids from 90 × 90 to 150 × 150 creates a maximum difference of 0.18%, which reduces to 0.037% on further refinement i.e. 210 × 210 grids. Further refinement does not produce any significant change. The grid independence test suggests that 210 × 210 is the optimum grid size for our computational domain. The computer code is executed in the IBM Power 8 server, and the typical runtime for execution varies between 2 and 3 h with 99% central processing unit as the parameters such as Re or Ri vary.

A comparative study between several thermal conductivity and viscosity models for the nanofluid is assessed in Figure 2(b) by computing Nu_av at Re = 100, Gr = 10⁴ and Bn = 1. It is evident that the MG–Brinkman model (Brinkman, 1952), Corcione model (Corcione, 2011) and the MG–Pak and Cho model (Pak and Cho, 1998) yields almost the same values, whereas the Patel–Brinkman model (Patel et al., 2006) deviates from these models by a relatively larger margin. In the present study, we adopt the widely accepted MG–Brinkman model (Brinkman, 1952) to compute the effective viscosity and the thermal conductivity of the nanofluid. An experimental study by Mahian et al. (2016) also established that MG–Brinkman is appropriate to determine the thermal conductivity and viscosity of the nanofluid at a low volume fraction and produce a higher accuracy for heat transfer coefficient compared with other models.

In Figure 2(c), we made a comparison of our computed solution for the temperature profile at different values of the yield stress parameter Bn with the experimental results of Hassan et al. (2020) for the natural convection of the Bingham plastic clear fluid (φ_b = 0) in a square enclosure with discrete heat flux at the lower wall. Hassan et al. (2020) used the Carbopol polymer, which exhibits the shear-thinning viscoplastic behavior. The Bingham number (Bn), the ratio of yield stress and viscous stress, is varied to obtain the dependency of yield stress on the heat transfer rate. Results show that our computational results are fairly in agreement with the results determined experimentally. The computed solution is also found to differ by 7% from this experimental data.

Figure 3(a)–(c) presents the validation of our code with the existing numerical results. Figure 3(a) shows the comparison of our results for the average Nusselt number (Nu_av) for the Al₂O₃–water Newtonian (Bn = 0) nanofluid in a ventilated enclosure with the corresponding numerical results of Sourtiji et al. (2014) as a function of the bulk volume fraction (φ_b) in the absence of magnetic field (Ha = 0) and established an excellent agreement with maximum percentile difference 1.23%. The results are computed by varying the Richardson number for different modes of convection, namely, shear dominated (Ri < 1) and buoyancy dominated (Ri = 10) flows. A comparison of the local Nusselt number for the viscoplastic Bingham nanofluid with the numerical results of Kefayati and Huilgol (2016) is presented in Figure 3(b) for different values of Bn. We find an excellent agreement between the present results with the numerical results of Kefayati and Huilgol (2016), with a maximum difference of 3.09%. In Figure 3(c), we compare the vertical velocity profile along the vertical central line for the MHD mixed convection of a clear Newtonian (φ_b = 0, n = 1, τ₀ = 0) fluid within a lid-driven square enclosure having two corner heaters placed along the bottom and right wall of width as half of the length of the wall with the numerical results of Oztop et al. (2011) for different values of the magnetic field (Ha). Results show an excellent agreement of our computed results with the existing numerical results.

5. Results and discussion

The hybrid nanofluid is considered to compose of δ: 1 ratio of Al₂O₃ and Fe₃O₄ nanoparticles with total bulk concentration φ_b. The governing nondimensional parameters such as Nr, Sc_i, Le_i and NBTi with i = 1, 2 corresponding to the nanoparticles Al₂O₃ and Fe₃O₄, respectively, are determined based on the intrinsic parameter values as prescribed in Table 1. For the sake of simplicity, the diameter of the two different types of nanoparticles is considered to be the same i.e. dp1=dp2=30nm. We first present the flow and thermal field for the mixed convection of the Bingham plastic hybrid nanofluid, which is followed by the results on heat transfer and entropy generation. Subsequently, the heat transfer efficiency and thermal mixing for the considered range of parameter values are illustrated.

5.1 Distribution of streamlines, heatlines and nanoparticles

The fluid pumped through the inlet placed at the top corner of the left side wall spreads inside the cavity and flows out through the lower corner of the opposite side wall. The flow pattern inside the cavity depends on the incoming fluid velocity (Re), wall temperature (Gr), bulk nanoparticles volume fraction ( φbi), Bingham number (Bn), Hartmann number (Ha) and the angle of inclination (λ) of the inlet jet. The flow, thermal field and the nanoparticles distribution along with the development of unyielded regions due to the presence of the horizontal jet of hybrid nanofluid in 1:1 ratio for different Bingham number (Bn) and Hartmann number (Ha) are presented in Figure 4(a)–(c). For the case of zero yield stress, the primary flow in the cavity constitutes of a cold fluid jet directed parallel to the diagonal direction of the cavity along with two viscous eddies of lower circulation strength developing near the lower and upper corners. For the case of nonzero yield stress (Bn ≠ 0), the cold fluid jet flowing diagonally spreads and the corner eddies contract. For larger values of Bn, unyielded zones of stagnant fluid develop at both upper and lower corners of the cavity along with an unyielded region in which fluid moves as a solid body develops in the core of the cavity. Streamline patterns for larger Bn show symmetry about the diagonal line with unyielded regions formed symmetrically at the corners. An unyielded region develops in the bulk region where the fluid is not motionless rather moves like a solid body. The magnetic field interacts with the fluid at the bulk region in which the fluid experiences Lorentz force. In the presence of magnetic field unyielded regions, which develop at the corners shrinks and the fluid velocity in the vicinity of the walls become stronger. At a large value of Ha the streamlines spread throughout the domain so as to reduce the effect of the yield stress of the viscoplastic nanofluid.

The impact of the Bingham number, as well as the Hartmann number on the energy flux vector, is illustrated in the second row of Figure 4(a)–(c). The energy flux vectors are locally parallel to the direction of net energy flow in the convection field. It is seen that the vectors are directed along the diagonal in the core of the ventilated enclosure, which signifies that the heat transfer is strong in this region and is governed by the strong jet emanating from the inlet toward the outlet port. Away from the diagonal of the cavity, slower heat convection occurs, which leads to choose a longer path for the energy flux vectors to reach the outlet port. An increment of Bn diminishes the fluid convection due to the increment of viscosity of the nanofluid; hence, the heat transfer is dominated by conduction. For this, the circulation zones in the energy flux vectors reduce, and the flux vectors become parallel near the heated lower wall [Figure 4(b)]. The Lorentz force generated due to the imposed magnetic field affects the convective heat transfer phenomena within the bulk of the domain. It is evident from Figure 4(c) that as Ha increases, the heatlines follow a similar pattern like the streamlines. The heatlines are clustered near the heated walls so that the temperature gradient becomes steeper near the walls and the heat transportation between the cold fluid and the heated walls occurs at a higher rate.

The third row of Figure 4 shows the variation of nanoparticles distribution due to the increment of yield stress as well as the presence of Lorentz force at a fixed nanoparticle size dp1=dp2=30 nm and the sum of bulk volume fractions φ_b = 0.01, with 1:1 volume fraction ratio of two types of nanoparticles. Because the nonhomogeneous model considers a slip velocity between the base fluid and the nanoparticles, the impact of Brownian diffusion and thermophoresis is taken under consideration. Because the walls of the enclosure are maintained at a higher temperature, whereas the cold fluid flows along the diagonal, a comparatively higher concentration of nanoparticles is created near the inlet than the outlet port and a thin layer of low volume fraction nanoparticles is formed in the vicinity of the heated walls. Figure 4(b) shows that as the Bingham number enhances, the shape of the isoconcentration of nanoparticles is distorted. This is due to the fact that an increment in Bn causes a drop in the temperature gradient. For a higher value of Bn, the distribution of nanoparticles in the core becomes uniform as the increased yield stress reduces the convective transport of nanoparticles by enhancing the fluid viscosity and formation of the unyielded region. It is evident from Figure 4(c) that the Lorentz force affects the nanoparticles distribution as Ha increases. The isoconcentration lines are contracted toward the wall of the enclosure leading to a faster change in the nanoparticle distribution near the wall.

5.2 Augmentation in heat transfer for hybrid nanofluid

A comparative study between the viscoplastic nanofluids composed of either Al₂O₃ (δ = ∞) or Fe₃O₄ (δ = 0) nanoparticles with the corresponding hybrid nanofluid (δ = 1) consists of an equal proportion of Al₂O₃, and Fe₃O₄ nanoparticles are illustrated in Figure 5(a)–(c). It is evident from the results that the combination of Al₂O₃ and Fe₃O₄ nanoparticles produces higher heat transfer than the nanofluid composed of a single type of nanoparticles. The thermophysical properties such as thermal conductivity, viscosity, density and electric conductivity are being modified due to the inclusion of Fe₃O₄ nanoparticles with the Al₂O₃-nanofluid. It also improves the chemical stability of the hybrid nanofluid. Figure 5(a) shows that the enhancement in Nu_av for the hybrid nanofluid is pronounced at the higher range of Reynolds number. The presence of Fe₃O₄ particles in the nanofluid increases the shear rate, which becomes prominent for fluid at higher inertia i.e. higher Reynolds number. For higher values of the magnetic field, the hybrid nanofluid produces a substantial enhancement in Nu_av. This is due to the fact that the electrical conductivity of the Fe₃O₄ nanoparticles is higher, which manifests the impact of the magnetic field when Fe₃O₄ nanoparticles are considered i.e. δ > 0. For the hybrid nanofluid, the increment of thermal conductivity is due to the chain formation of the magnetic nanomaterials with the base fluid, which intensifies with an increase of φ_b (Rashad et al., 2018). It is evident that three types of nanofluids, i.e. δ = 0,1,∞ exhibits a similar pattern, however, a significant increment in Nu_av is observed for the case of hybrid nanofluid and the impact is pronounced for a higher value of Re due to strong convection.

Figure 5(b) depicts the impact of the inclination angle of the inlet jet on Nu_av for different values of the Bingham number. It is evident from the results that the maximum heat transfer is obtained at λ = 0^∘ for all types of nanofluids. We find that the increment in yield stress (Bn) diminishes the rate of heat transfer. An increase in Bn enhances the fluid viscosity and also creates unyielded regions in the flow field. For this, the heat convection attenuates as Bn is increased. The discrepancy between the single nanofluid and the hybrid nanofluid reduces with the increase of Bn as the impact of hybrid nanoparticles attenuates at higher yield stress, creating a large reduction in the convective heat transfer.

It is evident from Figure 5(c) that the rate of heat transfer augments due to the inclusion of Fe₃O₄ with the Al₂O₃ nanofluid i.e. δ = 1. As mentioned before, the thermal conductivity of the hybrid nanofluid becomes higher than the nanofluid composed of a single type of particle. A significant influence of hybrid nanofluid compared with single nanofluids is observed for higher φ_b. We find that for all three types of fluids, an increment in Ri reduces Nu_av due to the formation of a secondary vortex, as discussed earlier. From Figure 5(a)–(c), it can be concluded that the hybrid nanofluid (δ = 1) achieves a higher heat transfer rate than any of the viscoplastic nanofluids made of a single type of nanoparticles. For this, we consider the hybrid nanofluid in the rest of our computation to investigate the thermal performance of the system.

5.3 Heat transfer and entropy generation

Figure 6(a) shows the dependence of the average rate of heat transfer (Nu_av) on Ri at different values of the particle volume fraction composed of equal proportion of Al₂O₃ and Fe₃O₄ nanoparticles i.e. δ = 1. We find that an increment in Ri causes a reduction in Nu_av. Here Ri is increased by increasing the temperature difference at a fixed Re. As observed from the flow and thermal field (Figure 4), an increase in Ri makes the region of primary flow squeeze, and as a result, the corner eddies expand. These vortices entrap fluid and refrain cold jet to spread near the heated walls, which, in turn, creates a monotonic decrement in the Nu_av with the increase of Ri. In the buoyancy-dominated regime i.e. at Ri = 10, two corner eddies appear with a sharp zone of a primary jet emanating along the diagonal direction. We find that an increment of the bulk volume fraction causes a notable enhancement in heat transfer determined by both the homogeneous and nonhomogeneous models. This inclusion of nanoparticles enhances the thermal conductivity of the nanofluid and modifies the thermal boundary layers. These effects manifest as φ_b is increased.

It is evident from Figure 6(a) and 6(b) that the nonhomogeneous model produces a higher heat transfer rate than the single-phase homogeneous model, and both models follow a similar trend as Ri or Re is varied. The Brownian diffusion and thermophoresis of the nanoparticles lead to an increment in heat transfer. The difference between the homogeneous and the nonhomogeneous model is 10.64% and 9.71% for φ_b = 0.01 and 0.05, respectively, at a fixed Ri (=10). The coefficient of the thermal buoyancy term in the nonhomogeneous model reduces as φ_b is increased, which leads to a reduction in Nu_av by a small margin compared with the homogeneous model. However, the discrepancy between the nonhomogeneous model and the results based on the single-phase model becomes pronounced as the Reynolds number is increased [Figure 6(b)]. At a higher Re, the dispersion of nanoparticles creates a higher thermal conductivity of the fluid, which is not accounted in the homogeneous model.

Figure 6(b) shows that an increment of the yield stress parameter Bn attenuates Nu_av for both the homogeneous and the nonhomogeneous models. An increase in yield stress creates a larger region of unyielded zones within the flow domain, which attenuates the fluid convection. We find that the reduction in Nu_av with the increment of Bn occurs at a slower rate for the lower range of Re. The maximum reduction in Nu_av when Bn is varied from 1 to 10 is 8.64% at Re = 200 while it is 7.23% at Re = 50. The disparity between the homogeneous and the nonhomogeneous models increases with the augmentation of the yield stress parameter Bn. The difference is 9.12% for Bn = 1, 9.94% for Bn = 5 and 10.65% for Bn = 10 at Re = 100. The variation of Nu_av at different Re with the inclination angle of the jet shows that Nu_av for any choice of Re achieves a local maximum at λ = 0° [Figure 6(c)]. Because the inclined cold fluid jet creates vortices in the upper and lower corners and contracts the region of primary flow, it inhibits the interactions of the cold fluid with the heated sidewalls of the enclosure. At λ = 0^∘ (horizontal jet), the main flow covers the entire domain, leading to an augmented heat transfer. A similar conclusion has also been established by Shiriny et al. (2019) based on the Newtonian fluid. The results show that the rate of enhancement in Nu_av, with the increment in Re, is faster for a positive inclination angle. The imposed transverse magnetic field, acting in the opposite direction of the injected jet, increases Nu_av, and this augmentation in Nu_av is pronounced at higher Re. At a higher Reynolds number, the heat transfer is dominated by forced convection. For this, the impact of Ha on Nu_av becomes pronounced for a higher range of Re. Heat transfer augmentation by imposing an external magnetic field in the mixed convection of a ZnO-viscoplastic nanofluid in ventilated cavities is also found by Kherroubi et al. (2020). The horizontal injection of the cold fluid provides maximum heat transfer at any value of Ha as the fluid convection in the bulk region becomes stronger for λ = 0°.

The dependency of Nu_av on the magnetic field for different values of Ri and Bn is presented in Figure 7(a). The average Nusselt number enhances monotonically with the increment of Ha and achieves a saturation at a large Ha i.e. Ha > 80. As is seen from the streamline pattern, the imposed magnetic field induced Lorentz force acts in the supporting direction to the cold fluid emerging from the inlet. The corner eddies are diminished, and the cold fluid emanating from the inlet spreads throughout the domain. This leads to an increment in heat transfer as Ha is increased. It is observed that the impact of Ha is pronounced for lower values of Bn as the fluid convection attenuates at a higher Bn. The reduction of Nu_av with the increase of Bn in MHD mixed convection of viscoplastic fluids in a lid-driven cavity is also found by Kefayati and Tang (2018). With the increment of bulk volume fraction, the thermal conductivity manifests, and the thermal boundary layer modifies as well, which leads the Nu_av to increase monotonically with the increase of φ_b.

Figure 7(b) presents the effect of Joule heating on Nu_av at a fixed Ha. The Joule heating is generated due to the interaction of the electric current produced by the imposed magnetic field on the conducting fluid medium. It converts the kinetic energy into thermal energy, which leads to an increment in the fluid temperature. Because Joule heating enhances the fluid temperature leading to a thickening of the thermal boundary layer, and hence, it attenuates the wall temperature gradient i.e. a decrease of Nu_av. We find that the impact of Joule heating on Nu_av is pronounced for a higher range of Bn. The Joule heating parameter causes a decrement in conduction, whereas the convection is reduced due to the increment of yield stress (Bn). Both in combination creates an overall reduction in the heat transfer rate. The fact that the impact of Ha is pronounced for J = 0 is also evident from the figure. We find that the effect of the magnetic field is pronounced for a higher nanoparticles concentration as the electrical conductivity of the nanofluid manifests with φ_b.

We have compared Nu_av determined by the present model with the results based on the experimental correlation for thermal conductivity (12) in Figure 7(c) for a different proportion of the hybrid nanoparticles i.e. δ = 0.5, 1, 2. We find that the results based on these two models are qualitatively similar. However, the results based on the correlation (12) produce a higher Nu_av. It is evident that Nu_av for the hybrid nanofluid augments with the reduction of δ i.e. as the percentage of Fe₃O₄ becomes higher than the Al₂O₃ nanofluid. The inclusion of Fe₃O₄ nanoparticles enhances the electrical conductivity of the medium, leading to an enhanced thermal performance when subjected to an imposed magnetic field. This impact of the applied magnetic field enhances as the relative volume fraction of Fe₃O₄ is increased. As a result, the hybrid composition with a 1:2 ratio (δ = 0.5) achieves a higher rate of heat transfer.

The variation of the total entropy generation with the governing thermophysical parameters are presented in Figure 8(a)–(c). The entropy generation is governed by the combined contribution of heat transfer irreversibility and fluid friction irreversibility due to the shear and buoyancy force for the present mixed convection flow of hybrid nanofluid. Because the average Bejan number is higher than 0.5, the heat transfer irresistibility dominates in generating the entropy. Figure 8(a) demonstrates that the total entropy generation enhances with the increment of Ha as the Lorentz force but declines with the increase of the Bingham number. An enhancement in Bingham number leads to a drop in entropy generation as both the fluid friction and heat transfer irresistibility reduces as Bn is increased. An increase in yield stress of the fluid elongates the unyielded regions in the flow domain where shear stress is low and also reduces the heat transfer due to convection. These, in combination, reduce S_av as Bn is increased. We have seen in Figure 7(a) that the average heat transfer reduces with the increase of Ri; however, S_av is increased as Ri increases. This implies that the fluid friction irreversibility due to shear stress contributes significantly to entropy generation for the buoyancy-dominated flow (Ri > 1). When Joule heating is considered, the temperature gradient attenuates due to an increase in fluid temperature, which leads to a reduction in entropy generation.

Figure 8(b) shows that an enhancement in Re increases S_av. An increase in Re increases the convective heat transfer as well velocity gradient due to faster change in velocity. Figure 8(c) shows that the variation of S_av with λ follows a similar pattern as the Nu_av variation, and the optimal entropy generation is obtained at λ = 0°, at which the maximum heat transfer is determined. The impact of λ is pronounced for higher values of Re, for which the inertia force dominates.

The dependency of entropy generation on Richardson number, nanoparticles concentration and Joule heating parameter at different Reynolds numbers is shown in Figure 9(a)–(c). Figure 9(a) shows that S_av increases as the volume fraction of the hybrid nanofluid are increased. An increment in φ_b enhances the temperature gradient as well as fluid friction, which in turn enhances S_av. As expected, an increase in Ri decreases S_av, and the variation of S_av with Ri is similar for any nonzero Bn. It is depicted from Figure 9(b) that in the presence of a magnetic field, the heat transfer irreversibility is higher for the Fe₃O₄ nanofluid (δ = 0), and it becomes lower for Al₂O₃ nanofluid (δ = ∞). We find that the addition of Al₂O₃ nanoparticles with the Fe₃O₄-nanofluid reduces the S_av compared with the case of Fe₃O₄-nanofluid. For the hybrid nanofluid (δ = 1), the irreversibility in comparison with the Fe₃O₄-nanofluid (δ = 0) is reduced due to the fact that the irreversibility attenuates as the electrical conductivity of the hybrid nanofluid becomes lower with the increment of volume relative fraction (δ) of the Al2O3 nanoparticles. The Joule heating decreases S_av by the same mechanisms for which Nu_av is reduced. A monotonic increment in S_av with the enhancement of φ_b is observed [Figure 9(c)]. We find that the variation of S_av with φ_b is similar for any choice of Ha.

5.4 Enhancement of heat transfer with respect to entropy generation and pressure drop

Enhancement in heat transfer ( Ehφ) and the entropy generation ( Esφ) due to the addition of Al₂O₃ and Fe₃O₄ nanoparticles in equal proportion (δ = 1) are determined by Ehφ = Nu_av/Nu_av(φ_b = 0) and ESφ = S_av/S_av(φ_b = 0), respectively. The dependency of Ehφ and ESφ on the governing parameters, e.g. bulk volume fraction ( φbi), Richardson number (Ri), Reynolds number (Re) and Bingham number (Bn), are presented in Figure 10(a)–(c). Our results show that both the heat transfer and entropy generation augments due to the inclusion of two different types of nanoparticles for all the cases considered i.e. both Ehφ, ESφ>1. We also find that Ehφ>Esφ [inset of Figure 10(a)], which implies that the augmentation in heat transfer is higher than the loss due to entropy generation for all the cases considered. This is an important conclusion, which provides a guideline for designing optimal heat transfer for the ventilated enclosure.

A comparative study on the enhancement factors of heat transfer and entropy generation ( Ehφ, Esφ) between the hybrid nanofluid and Al₂O₃-nanofluid (δ = ∞), as well as Fe₃O₄-nanofluid (δ = 0), is presented in Figure10(a). It is evident from the results that the enhancement in heat transfer is much higher than the increment factor of entropy generation. An inclusion of Fe₃O₄ nanoparticles within the Al₂O₃-viscoplastic nanofluid manifests thermal performance ( Ehφ>Esφ). The Fe₃O₄ nanoparticles being electrically conducting increases the heat transfer rate of the hybrid nanofluid for a higher value of Ha. For this, the applied magnetic field produces an augmentation in heat transfer as well as entropy generation for the hybrid nanofluid with Ehφ much larger than Esφ. The physical reason for Ehφ>Esφ in hybrid nanofluid is due to the fact that the inclusion of Fe₃O₄ into the Al₂O₃-nanofluids does not change the viscosity of the medium significantly compared with the enhancement in thermal conductivity. This characteristic of the hybrid nanofluid in the presence of the magnetic field may provide the guideline to design a thermodynamically efficient composition of nanofluid, resulting in intensified heat transfer with diminished entropy generation. We find from Figure 10(b) that an increment in yield stress creates an adverse effect on both Ehφ, Esφ. The rate of increment in both the average Nusselt number and entropy generation attenuates as Bn is increased due to the adverse effect of yield stress.

Figure 10(c) shows that both the ratios Ehφ and ESφ amplify as Re is increased. The development of unyielded regions in the Bingham plastic fluid attenuates the heat transfer, and this impact becomes pronounced at a larger Re. For the hybrid nanofluid, an imposition of the external magnetic field leads to an increment in both the enhancement factor for heat transfer and entropy generation; however, at a higher Re, the increment in Ehφ is higher in comparison with the increment of Esφ.

The dimensionless pressure drop between the inlet and outlet port is defined as follows:

Pd=(Pin−Pout)/(1/2)U02ρf

P_in and P_out are the average pressure at inlet and outlet ports, respectively. We denote the enhancement in pressure drop due to the inclusion of nanoparticles as EPdφ i.e. EPdφ=Pd/Pd0, where Pd⁰ is the corresponding pressure drop for the case of φ_b = 0 (clear fluid). We define the ratio χPdh as χPdh=Ehφ/ EPdφ , which is the ratio in heat transfer enhancement compared with the enhancement in pressure drop due to the addition of nanoparticles. The results presented in Figure 11(a)–(c) show the variation of χPdh for different Ri and Re. Our results show that this computed ratio for all the cases is higher than 1. This implies that the inclusion of nanoparticles creates the rate of augmentation in heat transfer much higher than the enhancement of pressure drop between the nanofluid and clear fluid within the enclosure. The impact of the volume fraction of the nanoparticles in hybrid nanofluid on the ratio of enhancement χPdh is presented in Figure 11(a) at different Ri. An increment in bulk volume fraction of nanoparticles enhances the viscosity of the nanofluid, which causes a larger pressure drop compared with the clear fluid. However, the rate by which the heat transfer is enhanced is higher than the rate of increment in pressure drop, i.e. Ehφ>EPdφ. Although the enhancement in heat transfer diminishes with the increment of buoyancy force [inset of Figure 11(a)], the ratio χPdh of enhancement between the heat transfer and pressure drop compared with the clear fluid enhances with the increment of Ri [Figure 11(a)].

The effect of Re on the ratio of relative increment in heat transfer compared with the pressure drop due to the addition of nanoparticles is shown in Figure 11(b). An enhancement in Re leads to an increase in the Nu_av but declines the pressure drop. It is evident from the flow field that for lower values of Re, the primary flow expands in the core of the enclosure but contracts at the outlet port. This contraction of streamlines creates a significant enhancement in pressure drop near the outlet. However, the relative augmentation in heat transfer for nanofluid compared with the pressure drop is higher for the considered range of Reynolds number, and this ratio manifests with the increase of Re. The imposed magnetic field creates a larger pressure drop than the clear fluid i.e. EPdφ becomes higher as Ha is increased. As the magnetic field pulls the nanoparticles toward the wall where the velocity is comparatively lower, the friction becomes higher near the wall, leading to an increment in pressure drop. This impact of the magnetic field on the pressure drop is suppressed in comparison with the heat transfer augmentation. For this, increment in Ha creates an enhancement in χPdh.

The ratio χPdh between the enhancement in heat transfer and pressure drop compared with the clear fluid for hybrid nanofluid as well as a single type of nanofluids is depicted in Figure 11(c). It is evident from the result that at the same volume fraction, the rate of heat transfer is higher than the pressure drop for the hybrid nanofluid compared with the corresponding single type of nanofluids. For the hybrid nanofluid, the enhancement in thermal conductivity is comparatively higher than the increment in viscosity. As a result χPdh>1 and it follows a similar pattern as that of Nu_av when varied as a function of Re.

5.5 Thermal mixing and temperature uniformity

The impact of the Bingham number on the buoyancy-dominated convection (Ri = 10) on the thermal mixing and temperature uniformity is shown in Figure 12(a) and 12(b). We find that both θ_cup and θ_av decrease with the increment of Bn [Figure 11(a) and 11(b)], which implies a reduction in thermal mixing for the viscoplastic nanofluid. The fluid convection reduces as the yield stress (Bn) is increased, which attenuates the thermal mixing. As shown before, an increment in Ri also reduces the primary flow domain and elongates the corner eddies. This creates an adverse effect on thermal mixing. We find that thermal mixing is enhanced due to the inclusion of nanoparticles due to enhanced thermal conduction. It is evident from Figure 12(a) and 12(b) that both θ_cup and θ_av increases with the increment of Ha. This implies a better thermal mixing due to the externally applied magnetic field for buoyancy-dominated convection.

The variation of θcupm and θavm Figure 12(a) and 12(b) indicates an augmentation in RMSD of mixing-cup temperature and volume-averaged temperature with the increment of Bingham number. This implies that the presence of viscoplastic nanofluid lessens the thermal mixing but enhances the temperature uniformity inside the ventilated enclosure. It is evident from the results that the temperature nonuniformity is pronounced as the Hartmann number is increased. We find that the increment of nanoparticles concentration in the hybrid nanofluid has a marginal impact on the variation θcupm and θavm. Figure 12(c) depicts the impact of several nanofluids on the thermal mixing of the system. It is evident from the results that in the presence of a magnetic field, the thermal mixing is augmented in hybrid nanofluid (δ = 1) in comparison with the single-particle nanofluids (δ = 0, ∞).

6. Conclusion

A numerical investigation on the MHD mixed convection of a viscoplastic hybrid nanofluid in a ventilated enclosure with heated side walls is performed. Cold fluid is injected from the inlet at an angle λ with the horizontal, and an outlet is placed at the bottom corner of the opposite sidewall of the enclosure. The system is subjected to a uniform magnetic field imposed horizontally, acting opposite to the inlet jet. A nonhomogeneous model is adopted, which takes into account the relative velocity between the nanoparticles and the fluid. The governing equations are solved through the control volume-based algorithm. Comparison of the present model with the existing experimental results is encouraging. The impact of yield stress and magnetic field along with the Joule heating effect on the heat transfer performance of the hybrid nanofluid is analyzed. The main findings of the present study can be highlighted as follows:

The difference between the nonhomogeneous and homogeneous models manifests as the yield stress is increased. This discrepancy between the two models becomes higher as the Reynolds number, Richardson number and nanoparticles volume fraction is increased.
The hybrid nanofluid produces enhanced heat transfer with a lower rate of entropy generation and pressure drop than a nanofluid composed of a single type of nanoparticles. The inclusion of the Al₂O₃ − Fe₃O₄ hybrid nanoparticles on heat transfer performance manifests in the presence of the externally imposed magnetic field. Augmentation of heat transfer by 7.38% is achieved by 1:1 mixing of Al₂O₃ and Fe₃O₄ nanoparticles compared with the single Al₂O₃-nanofluid at Reynolds number Re = 50, whereas it is 16.61% at Re = 200.
The rate of heat transfer diminishes with the increment of Richardson number as the region of the primary diagonal flow contracts, and the corner vortices elongate as the Richardson number is increases. The heat transfer becomes higher when the cold fluid is injected horizontally, and it reduces as the angle of inclination of the inlet jet deviates from horizontal.
The yield stress of the fluid attenuates heat transfer by creating unyielded regions within the enclosures and increasing the viscosity of the fluid. The magnetic field acting opposite to the direction of the inward cold jet contracts the unyielded regions and produces an augmentation in heat transfer of the viscoplastic fluid.
The Lorentz force generated due to the imposed magnetic field makes the fluid spread throughout the domain. This results in an enhanced heat transfer as well as a higher thermal mixing. The adverse effects on heat transfer due to the yield stress can be compensated by the applied magnetic field. The Joule heating induced by the imposed magnetic field increases the fluid temperature and consequently reduces the Nusselt number. Though the entropy generation increases with the increase of the strength of the magnetic field, the rate of enhancement remains lower than the rate by which the heat transfer is enhanced. Present results can serve as a useful tool in enhancing the heat transfer performance of a viscoplastic fluid in a ventilated enclosure.

Figures

Figure 1.

(a) Schematic diagram of the physical domain with boundary conditions; (b) uniform mesh distribution in the computational domain

Figure 2.

(a) Grid independency test on Nu along the lower wall at different grid sizes when Ri = 1, Bn = 10, Re = 100 and φ_b = 0.01; (b) comparison of Nu_av obtained by several thermal conductivity and viscosity models at Gr = 10⁴, Re = 100, Ca = 1, Pn = 1 for φ_b = 0.01; (c) comparison of temperature profile along the vertical axis at x = 0.2 and Ra = 2.3 × 10⁵ with experimental results of Hassan et al. (2020) for natural convection with φ_b = 0, Bn = 0.244, 0.664

Figure 3.

(a) Comparison of Nu_av as a function of φ_b with Sourtiji et al. (2014) at different Ri (= 0.1, 1, 10) for mixed convection at Re = 100; (b) comparison of Nu along the lower wall for Bn = 0 and 1 at Re = 100, Pr = 1 with Kefayati and Huilgol (2016) for Bn = 0 and 1 at Re = 100, Pr = 1; (c) Comparison of vertical velocity profile along the y-axis i.e. x = 0.5 when Re = 100, Gr = 10⁴, φ_b = 0 for Ha = 0, 10, 30 with numerical results of Oztop et al. (2011) for MHD mixed convection in a lid-driven square enclosure with corner heaters of length 0.5

Figure 4.

Contour plot for streamlines and formation of unyielded regions (first row), heatlines (second row) and the nanoparticles distribution (third row) for (a) Bn = 0, Ha = 0; (b) Bn = 10, Ha = 0; (c) Bn = 10, Ha = 50 at Ri = 1, δ = 1, φ_b = 0.01, Re = 100, λ = 0^∘ and δ = 1

Comparative study on Nuav for Fe3O4-nanofluid (δ = 0, dashed lines), Al2O3-nanofluid (

δ=∞, solid lines), Al2O3−Fe3O4 hybrid nanofluid (δ = 1, dashed-dot lines) as a function of (a) Re at Ri = 1, λ = 0∘, Bn = 10, φb = 0.01; (b) λ at Re = 100, Ri = 1, Ha = 50, φb = 0.01; (c) Ri at Re = 100, λ = 0∘, Bn = 10, Ha = 50

Figure 5.

Comparative study on Nu_av for Fe₃O₄-nanofluid (δ = 0, dashed lines), Al₂O₃-nanofluid ( δ=∞, solid lines), Al₂O₃−Fe₃O₄ hybrid nanofluid (δ = 1, dashed-dot lines) as a function of (a) Re at Ri = 1, λ = 0^∘, Bn = 10, φ_b = 0.01; (b) λ at Re = 100, Ri = 1, Ha = 50, φ_b = 0.01; (c) Ri at Re = 100, λ = 0^∘, Bn = 10, Ha = 50

Variation of Nuav for the 1:1 Al2O3−Fe3O4-hybrid nanofluid (δ = 1) with (a) Ri for different values of Bn = 1 (red lines), Bn = 10 (blue lines) and φb = 0.01, 0.05 at δ = 1, Re = 100, Ha = 0 for the nonhomogeneous (solid line) and homogeneous (dashed line) model; (b) Re for Bn = 1, 5, 10 at Ri = 1, δ = 1, Ha = 0 for the nonhomogeneous (solid line) and homogeneous (dashed line) model; (c) λ for different values of Ha = 10 (dashed lines), Ha = 50 (solid lines), and Ha = 100 (dashed-dotted lines) for the hybrid nanofluid at Ri = 1, Bn = 10, δ = 1 and φb = 0.01

Figure 6.

Variation of Nu_av for the 1:1 Al₂O₃−Fe₃O₄-hybrid nanofluid (δ = 1) with (a) Ri for different values of Bn = 1 (red lines), Bn = 10 (blue lines) and φ_b = 0.01, 0.05 at δ = 1, Re = 100, Ha = 0 for the nonhomogeneous (solid line) and homogeneous (dashed line) model; (b) Re for Bn = 1, 5, 10 at Ri = 1, δ = 1, Ha = 0 for the nonhomogeneous (solid line) and homogeneous (dashed line) model; (c) λ for different values of Ha = 10 (dashed lines), Ha = 50 (solid lines), and Ha = 100 (dashed-dotted lines) for the hybrid nanofluid at Ri = 1, Bn = 10, δ = 1 and φ_b = 0.01

Figure 7.

Variation of Nu_av with (a) Ha for Bn = 0 (dashed-dotted lines), Bn = 1 (solid lines) and Bn = 10 (dashed lines) and Ri (= 0.1, 10) at φ_b = 0.01, Re = 100, δ = 1; (b) φ_b for Ha = 50 (blue lines), and Ha = 100 (red lines) at Re = 100, Ri = 1, Bn = 1, 10, φ_b = 0.01 with J = 0 (solid lines) and different J (dashed lines) at Ec = 0.1; (c) φ_b for different thermal conductivity model i.e. the MG-model and correlation (19) for δ = 0.5 (dashed-dotted lines), δ = 1 (solid lines) and δ = 2 (dashed lines) at Ha = 50, Bn = 10, Ri = 1

Figure 8.

Variation of S_av for hybrid nanofluid with δ = 1 as a function of (a) Ha for different Ri = 0.1 (dashed lines), Ri = 1 (solid lines) and Ri = 10 (dashed-dotted lines) and Bn (0, 1, 10) at φ_b = 0.01, Re = 100, δ = 1; (b) Re for Bingham number Bn = 1, 5, 10 at Ri = 1, δ = 1, Ha = 0; (c) λ for different values of Ha = 10 (dashed lines), Ha = 50 (solid lines) and Ha = 100 (dashed-dotted lines) at Ri = 1, Bn = 10 and φ_b = 0.01

Figure 9.

Variation of S_av for hybrid nanofluid with δ = 1 as a function of (a) Ri for different values of Bn = 1 (dashed lines), Bn = 10 (solid lines) for φ_b = 0.01, 0.05, Re = 100, δ = 1, Ha = 0; (b) φ_b for different δ = 0, 1, ∞ and Bn = 1 (solid lines), 10 (dashed lines) at Re = 100, Ri = 1, Ha = 50; (c) φ_b for different values of Ha = 10 (dashed lines), Ha = 50 (solid lines), and Ha = 100 (dashed-dotted lines) at Re = 100, Ri = 1, Bn = 10, φ_b = 0.01 with J = 0 (blue lines) and different J (Red lines) at Ec = 0.1 using the nonhomogeneous model

Variation of

Ehφ and

ESφ as a function of (a) Ha at a fixed Ri = 1, Bn = 10, φb = 0.01 for Fe3O4 nanofluid (δ = 0, dashed lines), Al2O3 nanofluid (

δ=∞, solid lines) and Al2O3 − Fe3O4 hybrid nanofluid (δ = 1, dashed-dotted lines); (b) Bn for Ri = 0.1 (dashed lines) and Ri = 10 (solid lines) and φb = 0.01 (Blue lines) and φb = 0.05 (Red lines) at a fixed Re = 100, Ha = 50; (c) Re at φb = 0.01, Ri = 1 for Bn = 0, Ha = 0 (dashed-dotted lines), Bn = 10, Ha = 0 (dashed lines) and Bn = 10, Ha = 50 (solid lines), respectively

Figure 10.

Variation of Ehφ and ESφ as a function of (a) Ha at a fixed Ri = 1, Bn = 10, φ_b = 0.01 for Fe₃O₄ nanofluid (δ = 0, dashed lines), Al₂O₃ nanofluid ( δ=∞, solid lines) and Al₂O₃ − Fe₃O₄ hybrid nanofluid (δ = 1, dashed-dotted lines); (b) Bn for Ri = 0.1 (dashed lines) and Ri = 10 (solid lines) and φ_b = 0.01 (Blue lines) and φ_b = 0.05 (Red lines) at a fixed Re = 100, Ha = 50; (c) Re at φ_b = 0.01, Ri = 1 for Bn = 0, Ha = 0 (dashed-dotted lines), Bn = 10, Ha = 0 (dashed lines) and Bn = 10, Ha = 50 (solid lines), respectively

Variation of

χPdh for Fe3O4/Al2O3 hybrid nanofluid (delta = 1) as a function of (a) φb for Ri (=0.1, 1, 10) at Re = 100, Ha = 0, Bn = 10. Inset shows the variation of

Ehφ as a function of φb for Ri (=0.1, 1, 10); (b) Re for φb = 0.01, 0.03, 0.05 with Ha = 0 (dashed lines), Ha = 50 (solid lines) at Ri = 1, Bn = 10. Variation of

χPdh with (c) Re for Ha = 0, 50 for Fe3O4-nanofluid (δ = 0, dashed lines), Al2O3-nanofluid (δ = ∞, solid lines) and Fe3O4/Al2O3 hybrid nanofluid (δ = 1, dashed-dotted lines) at Ri = 1, Bn = 10 and φb = 0.01

Figure 11.

Variation of χPdh for Fe₃O₄/Al₂O₃ hybrid nanofluid (delta = 1) as a function of (a) φ_b for Ri (=0.1, 1, 10) at Re = 100, Ha = 0, Bn = 10. Inset shows the variation of Ehφ as a function of φ_b for Ri (=0.1, 1, 10); (b) Re for φ_b = 0.01, 0.03, 0.05 with Ha = 0 (dashed lines), Ha = 50 (solid lines) at Ri = 1, Bn = 10. Variation of χPdh with (c) Re for Ha = 0, 50 for Fe₃O₄-nanofluid (δ = 0, dashed lines), Al₂O₃-nanofluid (δ = ∞, solid lines) and Fe₃O₄/Al₂O₃ hybrid nanofluid (δ = 1, dashed-dotted lines) at Ri = 1, Bn = 10 and φ_b = 0.01

Figure 12.

Variation of (a) θ_cup and θcupm; (b) θ_av and θavm Ri=10 as a function of Bn with φ_b = 0.01, Ha = 0 (dashed lines), φ_b = 0.05, Ha = 0 (solid lines), φ_b = 0.05, Ha = 50 (dashed-dotted lines) at a fixed at Ri = 10, Re = 100, δ = 1; (c) θ_cup and θ_av with Bn for δ = 0 (dashed lines), δ = ∞ (solid lines), δ = 1 (dashed-dotted lines) at φ_b = 0.05, Ha = 50, Ri = 10, Re = 100

Figure A1.

Schematic diagram for control volume V_P where e, w, n and s are the cell faces of the cell-centered at P

Table 1.

Thermophysical properties of Al₂O₃ and Fe₃O₄ nanoparticles and base fluid at T = 290K

Parameter	Al₂O₃	Fe₃ O₄	Base fluid
c_p(J/kgK)	765	670	4179
ρ(kg/m³)	3970	5200	997.1
k(W/mK)	40	6	0.6
β(K^{– 1})	0.85 × 10^{– 5}	1.67 × 10^{– 5}	2.1 × 10^{– 4}
µ × 104 (kgm⁻¹ s⁻¹)	–	–	695
dp (nm)	0.385	47	–

Appendix. Discretization of the transport equations

The general form of the convective-diffusion equations for the generic variable C is follows:
(16) ∂C∂t+∂∂x(uc)+∂∂y(vc)=∇2C+SC

The computational domain is divided into a number of elementary sub-cells (Figure A1) with area V_p whose sides are dx_P and dy_P. Equation (16) is integrated over a cell V_p and the discretized form to advance the solution from nth time step to (n + 1)th time step is as follows:
(17) CPn+1−CPndt dxP dyP+[ueCe−uwCw]n+1dyP+[vnCn−vsCs]n+1dxP=[∂C∂x|e−∂C∂x|w]n+1dyP+[∂C∂y|n−∂C∂y|s]n+1dxP+(SC)P dxP dyP

Here n, s, e and w refer to the northern, southern, eastern, western face of the cell (Figure A1). An implicit first-order scheme is used to discretize the time derivatives.

The diffusion flux at interfaces “e” and “w” are evaluated as follows:
∂C∂x|e=CE−CP0.5(dxP+dxE)     and     ∂C∂x|w=CP−CW0.5(dxP+dxW)

A similar procedure is adopted to estimate the diffusion flux at the other cell faces “n” and “s”. Note that the big letter subscripts denote the cell centers in which variables are stored, and small letter subscripts denote the corresponding cell faces.

To evaluate the value of C at the cell faces in the convective term, the QUICK scheme (Hayase et al., 1992) has been used. The QUICK scheme is a three-point upstream-weighted quadratic interpolation for the convected property C at the cell faces. Thus, C_e and C_w at the cell faces can be obtained as follows:
Ce={38CE+34CP−18CW,ue>0,34CE+38CP−18CEE,ue<0;Cw={38CP+34CW−18CWW,uw>0,34CP+38CW−18CE,uw<0;

Similarly, the values of C at interfaces “n” and “s” can be expressed. Therefore, the discretized equation (17) can be written as follows:
(18) aPCPn+1=aECEn+1+aWCWn+1+aNCNn+1+aSCSn+1+aP0CPn
where the coefficients a_i (i = E, W, P, N and S) are as follows:
aE=De−min⁡(ue,0),  aW=Dw+max⁡(uw,0),aN=Dn−min⁡(vn,0),  aS=Ds+max⁡(vs,0),aP=De+max⁡(ue,0)+Dw−min⁡(uw,0)+ Dn+max⁡(vn,0)+Ds−min⁡(vs,0)+aP0
with aP0=dxPdyP/dt. The contribution of the diffusion terms is included in D_i (i = E, W, P, N and S). Equation (18) is the discretized form of equation (16).

After discretization, the x- and y-momentum equations at the (n + 1)th time step for u and v, respectively, yields
(19) aiuui−1,jn+1+biuui,jn+1+ciuui+1,jn+1=diu−(pi+1,jn+1−pi,jn+1)△yaivvi−1,jn+1+bivvi,jn+1+civvi+1,jn+1=div−(pi,j+1n+1−pi,jn+1)△x}

The temperature and the nanoparticles concentration equations for at the (n + 1)th time step over the (i,j)th scalar control volume, we get
(20) aiffi−1,jn+1+biffi,jn+1+ciffi+1,jn+1=difaiggi−1,jn+1+biggi,jn+1+ciggi+1,jn+1=dig}

Here f stands for the temperature and g for nanoparticle concentration. For a fixed value of n, the system of equations can be expressed together in a matrix form as follows:
(21) AiXi−1n+1+BiXin+1+CiXi+1n+1=Di,           i=1,2,….,N
where [Xin+1]T=(ui,jn+1 vi,jn+1 fi,jn+1 gi,jn+1) is the variable vector, A_i, B_i and C_i are the 4 × 4 coefficient matrices, and D_i is the known vector.

With these, the system of equations can be expressed in a block matrix form as follows:
(22) AX=D
where
A=(B1C1A1B1C1⋱⋱⋱An−1Bn−1Cn−1AnBn)
is a block tri-diagonal matrix, each element being a square matrix of order four, X being the vector of unknowns, and D is the vector of knowns. The linear system of equations is solved by using a block elimination algorithm.

Figure A1

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Corresponding author

Ioan Pop can be contacted at: ipop@math.ubbcluj.ro