A Review on Non-Newtonian Nanofluid Applications for Convection in Cavities under Magnetic Field

Abstract

This review is about non-Newtonian nanofluid applications for convection in cavities under a magnetic field. Convection in cavities is an important topic in thermal energy system, and diverse applications exist in processes such as drying, chemical processing, electronic cooling, air conditioning, removal of contaminates, power generation and many others. Some problems occur in symmetrical phenomena, while they can be applicable to applied mathematics, physics and thermal engineering systems. First, brief information about nanofluids and non-Newtonian fluids is given. Then, non-Newtonian nanofluids and aspects of rheology of non-Newtonian fluids are presented. The thermal conductivity/viscosity of nanofluids and hybrid nanofluids are discussed. Applications of non-Newtonian nanofluids with magnetohydrodynamic effects are given. Different applications of various vented cavities are discussed under combined effects of using nanofluid and magnetic field for Newtonian and non-Newtonian nanofluids. The gap in the present literature and future trends are discussed. The results summarized here will be beneficial for efficient design and thermal optimization of vented cavity systems used in diverse energy system applications.

1. Introduction

For decades, heat transfer enhancement techniques have captured researchers’ attention to improve the heat transfer quality in different kind of systems. Researchers have attempted to use different kinds of techniques for this aim. Generally, heat transfer techniques are known for diminishing the effect of thermal resistance by increasing the impact of the heat transfer area or by creating turbulence as a way of forced convection. Sometimes, these techniques, which are performed in systems, create an increase in the required pumping power, and this creates higher costs for systems. This situation has led researchers to look for other techniques which are more practical and have lower costs. One of these methods is the heat transfer enhancement technique, which involves changing the thermal properties of the base fluid of the system. To achieve this, researchers started to search for methods to create new types of fluids with better thermal properties. In early studies, they started using microparticles to improve the base fluid in terms of heat enhancement. Although they gained some success in their studies, it was later observed that there were some limitations. It was seen that there were some problems in the required processes. There were usually tremendous pressure drops in the fluids used in the systems. Because the size of the microparticles was not small enough, blockages occurred during processes. There was a big possibility for sedimentation to occur. Microparticles caused corrosion in the systems as another negative effect. Although these particles were capable of increasing heat transfer in the systems, they increased the required work, due to the pressure drops they caused. As developments in material technology started to appear, new techniques were revealed, and researchers started to be able to use smaller particles in systems. These particles were called nanoparticles due to their size, which was a thousand times smaller than the size of microparticles. As these special particles started to be used in different kinds of systems, researchers saw their advantages versus microparticles. One of these basic advantages was the remarkable heat transfer amount due to the large surface areas of the particles. Due to their very small size, there was remarkable mobility in the particles, which caused a significant heat transfer improvement in the systems. Additionally, it was seen that by increasing the temperature, the thermal conductivity of the particles rose. Another advantage of using nanoparticles in a base fluid was their low weight. Thus, this caused less sedimentation in the systems. The possibility of corrosion was lower in the usage of these particles, and a remarkable reduction was seen in the required work for the systems. Due to all these basic advantages, nanofluids have been used in many applications alone or with other techniques in terms of active or passive heat transfer enhancements, or both, and they have also been playing significant roles in convective heat transfer applications.

Considering the applications of nanofluids, some examples can be given such as some industrial processes, cooling processes, microchannel systems, optical devices such as lasers, X-ray devices, optical fibers and also some large devices such as transport lorries, fuel cells, and heat exchangers. Other areas where nanofluids are used can be mentioned, such as microelectronics, pharmaceutical processes, some medical devices, home refrigerators, grinding processes, and the boiler flue temperature-lowering processes [1].

Additionally, researchers have examined nanofluids by considering other phenomena in heat transfer such as cavities and magnetohydrodynamics of the fluids. In systems with cavities, the fluids in the systems are examined in special environments where permeability is considered besides other parameters which are included in heat transfer. Since there is flow of the nanofluid in a specific system, the heat transfer is known as convection, and different types of it such as natural, forced and both at the same time have been examined with cavity effects. To understand the way nanofluids behave and the changes in their features that might occur considering all these parameters, researchers have been performing their studies for a certain time. Another parameter which is considered about nanofluids is the magnetic field effect, which leads scientists to examine the magnetohydrodynamics of the nanofluids. Exposing the fluid to a certain magnetic field, the electrical conductivity of the fluids has been considered, and some parameters such as velocity profiles, temperature, etc. have been examined.

Among the last two phenomena related to nanofluids in convection mentioned above (cavities and magnetohydro dynamics), there have been applications considering them both at the same time. The effects of a magnetic field on nanofluids in porous mediums have been examined, and some correlations have been found between these parameters including others related to studies. Due to various geometries and applications, the physical properties of these special fluids have been studied in a more detailed way to try to understand them.

Nanofluids have strong effects in different kinds of applications such as heat transfer, mass transfer, flow behavior, etc., as mentioned above. There is still a long way to go in order to understand nanofluids in a deeper way. Thus, some researchers keep studying by considering different parameters and features to understand the issue better in some ways or reveal the hidden aspects of these fluids which have not been discovered yet. In this review, the applications of non-Newtonian nanofluids in vented cavities under magnetic field effects have been analyzed. Convective heat transfer from vented cavity system is important to be considered in manty thermal applications including air conditioning, electronic cooling, drying, chemical processing and many others. The improvements in the thermal performance of vented cavity systems by using nanofluid for both Newtonian and non-Newtonian fluid cases have been considered under the presence of magnetic field effects.

2. Brief Information on Nanofluids and Non-Newtonian Fluids

2.1. Nanofluids

In a basic way, a nanofluid is a new type of fluid based on using extremely small particles (smaller than micro-structures) and adding them into a kind of base fluid; in other words, it involves creating a suspension using nano-sized particles and a specific base fluid. The term used for this special suspension was first used by Choi et al. [2]. Since every kind of fluid can be considered as having a nano-structure because of its molecular chains, this term might not be the right one to describe the suspension, but regardless, the term became plausible and it started to be used in the related areas in science.

There are at least four scale factors that nanofluids have: the molecular scale, the micro-scale, macro-scale and mega-scale. Depending on their nature, the researchers have focused on the improvement of nanofluids’ macro-scale and mega-scale properties by controlling micro-scale features of them [3,4].

Nanofluids that have one kind of nanoparticle as metallic or non-metallic have been used for heat enhancement in heat transfer, but these types are useful to some point. Although attempts to create better heat enhancements by increasing the concentration of nanoparticles have been performed, these types of nanoparticles seem to have stable thermal performance in heat transfer. Because of this limitation, researchers have discovered better ways for heat enhancement by creating hybrid nanofluids. Chemical properties, compatibility, purity, dispersibility and geometrical properties such as the size and the shape of nanoparticles affect the thermal performance of hybrid nanofluids. There are basically two methods in preparing hybrid nanofluids. One of them is the single-step method and the other one is the multi-step method. In the single-step method, the preparation and dispersion of nanoparticles are performed at the same time, and in multi-step method, the first step is to prepare nanoparticles, and the second step is to add nanoparticles into the base fluid one by one.

Generally, the nanoparticles used for hybrid nanofluids are SWCNT-MgO, MWCNTFe₃O₄, MgO-MWCNT, Fe₂O₃-CNT, Fe₃O₄-Graphene, Graphene- Ag, Al₂O₃-CNT, SiO₂-CNT, Al₂O₃–Cu, Al₂O₃-Ag, Cu–TiO₂, Cu-Cu₂O, Cu-Zn, Ag-SiO-carbon, Ag-TiO₂, Ag-MnO, Ag-CNT, diamond-Ni, ND Co₃O₄, Al₂O₃-MEPCM, and Al₂O₃-SiO₂ [5].

2.1.1. Thermal Conductivity of Nanofluids

Creating suspensions by using solids and fluids was an option that came to peoples’ minds a long time ago. Maxwell was one of the people who contributed to studies related to nanofluids and brought out the calculation of the effective thermal conductivity of the suspension [6]. His work was followed by Hamilton and Crosser [7] and Wasp [8].

Table 1. Nanofluids based on their preparation method.

Authors	Base Fluid	Nanoparticle (D in nm)	Volume Fraction (%)	Preparation Method for Synthesis
Wang et al. [9]	DW, EG	Al2O3 (28)	1–6	Two-step
Tseng et al. [10]	Water	Al2O3 (37)	0.01–0.16	Two-step
Putra et al. [11]	Water	Al2O3 (131.2) CuO (87.3)	1, 2, 4	Two-step
Tseng et al. [12]	Ethanol-isopropanol	BaTiO3 (580)	30–60	Two-step
Tseng et al. [13]	Terpineol	Ni (300)	3–10	Two-step
Tseng et al. [14]	Water	TiO2 (7–20)	5–12	Two-step
Kwak et al. [15]	EG	CuO (12)	0.002	Two-step
Prasher et al. [16]	PG	Al2O3 (27,40, 50)	0.5, 2, 3	Two-step
Chen et al. [17]	EG	TiO2 (25)	0.8 wt%	Two-step
Chen at al. [18]	EG	TiO2 (25)	0.1, 0.21, 0.42, 0.86, 1.86	Two-step
Chevalier et al. [19]	Ethanol	SiO2 (35, 94, 190)	1.4–7	Two-step
Chen at al. [20]	DW	TiO2 (20)	0.024, 0.6, 1.18	Two-step
Namburu et al. [21]	W-EG (40–60)	CuO (29)	0–6.12	Two-step
Namburu et al. [22]	W-EG (40–60)	SiO2 (20, 50, 100)	0–10	Two-step
Chen et al. [23]	DW, EG and glycol	CNTs (15)	0.2%, 0.6%, 1%	Two-step
Chen et al. [24]	DW	TNT ($\cong$10)	12%, 24%, 0.6	Two-step
Garg et al. [25]	EG	Cu (200)	0.4–2	Single step
Lu et al. [26]	Water, EG	Al2O3 (10 and >10)	5	Two-step
Murshed et al. [27]	DIW, EG	Al2O3 (80, 150), TiO2(15)	1–5	Two-step
Schmidt et al. [28]	Iso paraffinic PAO and Decane	Al2O3 (40)	0.25–1	Two-step
Tsai et al. [29]	Diesel oil and PDMS	Fe3O4(10)	1–4.48	Two-step
Anoop et al. [30]	Water	Al2O3 (45, 150)	1, 2, 4, 6 wt%	Two-step
Anoop et al. [31]	EG; water	Al2O3 (100, 95)	0.5, 1, 2, 4, 6	Two-step
Anoop et al. [31]	EG	CuO (152)	0.5, 1, 2, 4, 6	Two-step
Chen et al. [32]	EG	TNT ($\cong$10), L = 100 nm	0–8% (mass)	Two-step
Chen et al. [20]	EG-Water	TNT ($\cong$10), L = 100 nm, TiO2 (25)	0–1.8	Two-step
Naik et al. [33]	PG-water	CuO (<50)	0.025, 0.1, 0.4, 0.8, 1, 1.25	Two-step
Zhu et al. [34]	DW	CaCO3 (20–50)	0.12–4.11	Two-step
Phuoc et al. [35]	DW	SiC (<100)	0.1%, 0.1, 1, 2, 3	Two-step
Pastoriza et al. [36]	Water	CuO (23–37, $11\mp 3)$	5%–10 wt%	Single-step, two-step
Phuoc et al. [35]	Water	MWCNTs (20–30), L = 10–30 μm,	0.24–1.43	Two-step
Yu et al. [37]	PG;EG	AIN (50)	1–10	Two-step

shows the summary of some nanofluids’ preparation based on these methods [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37].

The Maxwell model is shown in Equation (1) [6], and the Hamilton–Crosser model is shown in Equation (2) [7].

$$\frac{k_{eff}}{k_f}=1+\frac{3\left(kp/k f-1\right) \mathsf{\phi }}{\left(kp/k f+2\right)-\left(kp/k f-1\right)\mathsf{\phi }}$$

(1)

$$\frac{k_{eff}}{k_f}=1+\frac{kp+\left(n-1\right)k f-\left(n-1\right)\mathsf{\phi }\left(k f-kp\right)}{kp+\left(n-1\right)k f+\mathsf{\phi }\left(k f-kp\right)},$$

(2)

In research studies, it was found out that nanofluids had thermal conductivity much higher than any kind base fluid had [38,39,40,41]. Basically, the factors which make a nanofluid superior to a basic fluid are larger surface areas, less particle momentum and higher mobility. In the name of using ceramic nanofluids, Lee et al. [42] performed a study by using Al₂O₃–water and CuO–water nanofluids, and they gained a greater thermal conductivity ratio, contrary to the Hamilton–Crosser model [7]. Figure 1 shows the thermal conductivity ratio, and Figure 2 shows the comparison. Wang et al. [9] examined the thermal conductivity of CuO–water and Al₂O₃–water nanofluids, but their particle size was smaller, at 23 nm for CuO and 28 nm for Al₂O₃. According to the measurements, there was a significant effect of the particle size and dispersion. Xie et al. [43] measured the thermal conductivity of Al₂O₃ nanofluids with much smaller particles at the size of 1.2–302 nm. Murshed et al. [44] examined the thermal conductivity of aqueous solutions of spherical and cylindrically shaped TiO₂ nanoparticles, and the results they obtained were much greater than the ones of the Hamilton–Crosser model.

As the studies on the whole issue continued, it was found out that much higher thermal conductivity enhancement could be gained by preparing suspensions using metals with base fluids. Eastman et al. [45] indicated that they had gained a 40% enhancement in conductivity with 0.3% concentration of 10 nm-sized copper particles suspended in ethylene glycol. Another important study came out from Patel et al. [46], who used gold and silver for preparing nanofluids, for the first time in the literature. They discovered that toluene–gold nanofluid caused an enhancement of 3–7% for a volume fraction of 0.005–0.011%, while the enhancement for water–gold nanofluid was 3.2–5% for a small volume-fraction concentration of 0.0013–0.0026%. The basic reason for such an enhancement was the small size (10–20 nm) of the particles. Once they used silver as the metal for the suspension, they found out that even though silver had a higher thermal conductivity than gold, the particles used in the experiment were larger in size than the ones of gold. It is clear that the size of the particles had a greater effect than the conductivity ability and the concentration in the mentioned study. Similarly, to the last study discussed above, there are a few studies which prove that effective thermal conductivity of nanofluids shows an enhancement as the size of particles decreases [30,47,48,49,50,51,52,53,54,55,56,57,58].

Furthermore, with regard to metal and metal oxide nanofluids, some of which are discussed above, there are other studies on this issue which have been conducted by focusing on the effect of particle volume fraction. Anoop. et al. [31] discovered an increase in k in ethylene glycol-based Cu nanofluids, which was twice that of the Maxwell predictions [4]. Their study showed an incredible k enhancement (150%) with a very low volumetric concentration (1%). There are other factors which effect the thermal conductivity of nanofluids. One of them is the effect of temperature. According to a few reports, there occurs an enhancement in k/k_f (the ratio of thermal conductivities of nanofluid and fluid) as the temperature increases [24,27,44,46,47,50,51,52,56,57,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74]. Other research shows the least effect of temperature on k/k_f [56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71]. Additionally, some studies show that there is a decrease in the k/k_f ratio as the temperature decreases [75,76]. Most studies show an enhancement in k with an increase in particle aspect ratio [44,77,78,79,80,81,82,83,84,85]. Buongiorno J. et al. found in their study that nanorods dispersed in polyalphaolefin (PAO) had greater k enhancement than Al₂O₃ nanoparticles dispersed in PAO [86]. Most of the studies show that there is a decreased thermal conductivity ratio with an increase in k of the base fluid [23,42,80,81,82,83,84,85,86].

Some researchers have performed studies focusing on the thermal conductivity of hybrid nanoparticles. Munkhbayar et al. [87] demonstrated a study with TiO₂-water nanofluids. They also used smooth Ag particles to add into the suspension. With their study being performed with the temperature ranging from 15 to 40 °C, they observed the thermal conductivity of the mono-nanoparticle suspension could be increased with Ag structures.

In another study, Esfe et al. [88] focused on the thermal conductivity of CNT-Al₂O₃/water between the temperature values of 27 and 57 °C. The volume concentration of the nanoparticles had a range of 0.02 to 1%, and they observed that the temperature had a great effect on the thermal conductivity of the nanoparticles, and its effect was clearer at higher concentrations. In other study of theirs, Esfe et al. [89] examined the suspension of Cu-TiO₂/water-ethylene glycol (60/40) with different volume concentrations of the particles: 0.1, 0.2, 0.4, 0.8, 1.0, 1.5 and 2.0%. During the study, the temperature had a range of 30 °C to 60 °C. They observed that within the increasing temperature and concentration, the thermal conductivity of the suspension increased. They obtained a thermal conductivity value of approximately 1.43 with the maximum concentration (2.0%) and the maximum temperature value (60 °C).

2.1.2. Viscosity of Nanofluids

Since viscosity is one of the important flow properties of fluids, researchers have also studied this issue regarding nanofluids. It is also important to examine viscosity when it comes to clarifying the thermal behavior of fluids in heat transfer. Naturally, it is an important issue for nanofluids too. There are some theoretical studies that were performed a long time ago. Einstein had already come up with a formula about the viscosity of nanofluids back then, but his formula was more about spherical nanoparticles that had a very low volume fraction ($\mathsf{\phi }<0.02$). Equation (3) shows the nanofluid viscosity and base fluid viscosity relationship based on the volume fraction [90]:

$$\frac{{\mu }_{nf}}{{\mu }_f}=1+2.5\mathsf{\phi }$$

(3)

This formula has some limitations. It was not sufficient when it came to structure and interaction between particles in the solution, especially in high concentrations. Mooney [91] created another formula for spherical particle but for higher concentrations, as shown below:

$$\frac{{\mu }_{nf}}{{\mu }_f}=e^{\frac{\mathsf{\epsilon }\mathsf{\phi }}{1-k\mathsf{\phi }}}$$

(4)

Here, k is a constant which is known as the self-crowding factor, whose values differ between 1.35 and 1.91, and $\mathsf{\epsilon }$ is the fitting parameter whose value is 2.5.

Krieger and Dougherty [92] came up with a model for shear viscosity for hard spherical particles monodispersed in the suspension:

$$\frac{{\mu }_{nf}}{{\mu }_f}={\left[1-\frac{\mathsf{\phi }}{{\mathsf{\phi }}_m}\right]}^{-n{\mathsf{\phi }}_m}$$

(5)

Here,${ \mathsf{\phi }}_m$ is the maximum particle packing fraction, which is between 0.495 and 0.54 and about 0.605 at higher share rates, and $n$ is the intrinsic viscosity which has a value of 2.5 for monodispersed suspension of hard spheres.

Batchelor [93] came up with an equation which was modified from one of Einstein’s viscosity formulae based on the Brownian motion effect. İt was related to isotropic suspension of rigid and spherical nanoparticles as:

$${\mu }_{nf}=\left(1+2.5\mathsf{\phi }+6.5{\mathsf{\phi }}^2\right){\mu }_{f }$$

(6)

These models are known as classical models about viscosity, and later, with mathematical improvement, these models turned into new models.

Brinkman [94] extended Einstein’s equation for use with moderate particle concentration. He thought of the effect of solute molecules added into an existing continuous medium of particle concentrations less than 4%. It received more acceptance from researchers. The equation is as shown below:

$${\mu }_{nf}={\mu }_f / {\left(1-\mathsf{\phi }\right)}^{2.5}$$

(7)

Franken and Acrivos [95] showed another formula:

$$\frac{{\mu }_{nf}}{{\mu }_f}=\frac{9}{8}\left[\frac{{\frac{\mathsf{\phi }}{{\mathsf{\phi }}_m}}^{1/3}}{\frac{{\left({\mathsf{\phi }}_m-\mathsf{\phi }\right)}^{1/3}}{{{\mathsf{\phi }}_m}^{1/3}}}\right]$$

(8)

Lundgren [96] presented another equation, which is a reduced form of Einstein’s formula having a Taylor series expansion of $\mathsf{\phi }$ as:

$${\mu }_{nf}=\left[1+2.5\mathsf{\phi }+\frac{25}{4}{\mathsf{\phi }}^2+f\left({\mathsf{\phi }}^3\right)\right]{\mathsf{\phi }}_f$$

(9)

Graham [97] developed the Franken Acrivos model by adding particle radius and inter-particle spacing. It has relevance with Einstein’s formula for small $\mathsf{\phi }$ as:

$${\mu }_{nf}={\mu }_f\left(1+2.5\mathsf{\phi }+4.5\left(\frac{1}{\left(\frac{h}{\mathrm{d}p}\left(2+\frac{h}{\mathrm{d}p}\right)\right){\left(1+\frac{h}{\mathrm{d}p}\right)}^2}\right)\right)$$

(10)

Here, $h$ is the inter-particle spacing, and $dp$ is the particle radius.

For a two-phase mixture, Kitano et al. [98] found a new formula to estimate the viscosity as:

$${\mu }_{nf}=\frac{{\mu }_f}{{\left[1-\left(\frac{\mathsf{\phi }}{\mathsf{\phi }m}\right)\right]}^2}$$

(11)

The formula which has the volumetric effect of viscosity for nanofluids came from Bicerano et al. [99]:

$${\mu }_{nf}=\left(1+\eta \mathsf{\phi }+k_H{\mathsf{\phi }}^2\right)$$

(12)

For 35$\%$ as a maximum value in consideration of spherical particles, a model is suggested a model with exponential expressions as [100]:

$$\frac{{\mu }_{n_f}}{{\mu }_f}=[1+\eta ({\mathsf{\phi }}_{eff}+2.5\eta +{\left(2.5\eta \right)}^2+\cdots )]$$

(13)

For nickel/terpineol nanofluids, an exponential model of the effect of volume concentration upon viscosity came out from Tseng and Chen [13]:

$${\mu }_{nf}={\mu }_f\times 0.4513e^{0.6965\mathsf{\phi }}$$

(14)

Based on the formulas of Einstein [90], Avsec and Oblak [101] created a new model for the viscosity of nanofluids as:

$$\frac{{\mu }_{n_f}}{{\mu }_f}=\left[1+2.5({\mathsf{\phi }}_{eff}+2.5{\mathsf{\phi }}_{eff}+{\left(2.5{\mathsf{\phi }}_{eff}\right)}^2+\cdots \right]$$

(15)

In this formula ${\mathsf{\phi }}_{eff}$ represents the effective volume fraction derived from the model of Yu and Choi [102] as:

$${\mathsf{\phi }}_{eff}=\mathsf{\phi }{\left(1+\frac{h}{r}\right)}^3$$

(16)

Here, h represents liquid thickness.

Cheng et al. [100] considered the aggregate structure and the effects of variable packing fraction in the suspension and came up with a modified version of Krieger-Dougherty’s equation as:

$$\frac{{\mu }_{n_f}}{{\mu }_f}={\left(1-\frac{{\mathsf{\phi }}_a}{{\mathsf{\phi }}_{\mathsf{\phi }m}}\right)}^{-2\cdot 5{\mathsf{\phi }}_m}$$

(17)

$${\varphi }_a=\mathsf{\phi }{\left(\frac{a_a}{a}\right)}^{3-D}$$

(18)

where a_a = aggregates, a = primary particles and D = fractal index, which has the value of 1.8 for nanofluids, while ${\mathsf{\phi }}_m$ = maximum volume fraction. The value of ${\mathsf{\phi }}_m$ is determined with experiments.

A new viscosity model based on Brownian motion of particles and plausible for alumina/water nanofluids to determine the viscosity in a theoretical way was performed by Masoumi et al. [103]:

$${\mu }_{nf}={\mu }_f\left(1+\frac{{\rho }_N{v}_b{d}_N^2}{72c\delta {\mu }_f}\right)$$

(19)

where ${\rho }_N$ is the density, $d_N$ is for the particle diameter, $\delta$ represents the distance between the nanoparticles and c and $V_b$ indicate the functions of temperature.

For factors such as particle diameter, particle volume fraction, and particle density, this model could be considered as a useful one. Another factor which affects the viscosity of nanofluids besides volume fraction is the temperature. Thus, some studies have been conducted to determine the viscosity of nanofluids by considering the effect of temperature. In the model, T is the temperature in Kelvin. Nguyen et al. [104] came up with a model which is dependent on temperature for viscosity for a range of particle volume fractions (1–4$\%$) as:

$$\frac{{\mu }_{nf}}{{\mu }_f}=\left(2.1275-0.0215T+0.00027T^2\right)$$

(20)

Namburu et al. [105] performed a mathematical model for 1–10% of Al₂O₃ nanofluids and a temperature range over 35–50 °C, and they found a relationship between viscosity and temperature, given below as:

$$\log \left({\mu }_{nf}\right)=A{e}^{-BT}$$

(21)

Here, viscosity is in terms of centipoises (cP), the particle volume fraction functions are shown as $A$ and B, and temperature $\left(T\right)$ is in Kelvin.

Abu-Nada [106] found a new formula based on the formula of Nguyen et al. His formula was dependent on temperature and particle volume fraction to determine viscosity as:

$$\begin{array}{c}{\mu }_{nf}=-0.155-\frac{19.582}{T}+0.794\mathsf{\phi }+\frac{2094.47}{T^2}-0.192{\mathsf{\phi }}^2-8.11\frac{\mathsf{\phi }}{T}\\ -\frac{27463.863}{T^3}+0.127{\mathsf{\phi }}^3+1.6044\frac{{\mathsf{\phi }}^2}{T}+2.1754\frac{\mathsf{\phi }}{T^2}\end{array}$$

(22)

Later, it was found out that Brinkman model was fit for Nguyenet al.’s model after the model shown above had been compared with the model of Brinkman. Masud Hosseeini [107] found another formula to define viscosity of nanofluids as:

$$\frac{{\mu }_{nf}}{{\mu }_f}=\exp \left(m+\alpha \left(\frac{T}{T_0}\right)+\beta \left({\mathsf{\phi }}_h\right)+\gamma \left(\frac{d}{1-r}\right)\right)$$

(23)

According to this equation,${\mathsf{\phi }}_h$ is the hydrodynamic volume fraction of nanoparticles, d represents the nanoparticle diameter, r is the thickness of the caping layer, T₀ is a reference temperature, T is the measured temperature of nanofluid, and m is a factor which depends on some features of the suspension such as the base fluid, the solid nanoparticles and the interaction between them. There are some studies related to the viscosity of hybrid nanofluids, although they are not as abundant as the ones performed related to their thermal conductivity. However, viscosity is another important issue in nanofluids, since it has some effects on the flow of the suspensions and their heat transfer characteristics. Baghbanzadeh et al. [108] studied the viscosity of multiwall carbon nanotubes (MWCNT) and SiO₂ nanoparticles. They also studied hybrid nanofluids SiO₂-MWCNT-water in two different proportions as 80% for SiO₂-20% for MWCNT and 50% for SiO₂-50% for MWCNT. They performed their study with three different concentrations: 0.1, 0.5 and 1%. Their study took place at a temperature range of 10–40 °C. They observed that the viscosity increased within the increase in concentration, while on the other hand, it decreased with the increasing temperature.

Table 2. Summary of studies about viscosity of nanofluids dependent on volume fraction.

Authors	Base Fluid	Nanoparticle	Particle Size (nm)	Volume Fraction (%)	Enhancement in Viscosity (%)
Nguyen et al. [104,109]	Water	Al2O3	36	2.1–13	10–210
Nguyen et al. [109]	Water	Al2O3	47	1–13	12–430
Wang et al. [9]	DW	Al2O3	28	1–6	9–86
Wang et al. [9]	EG	Al2O3	28	1.2–3.5	7–39
Prasher et al. [16]	PG	Al2O3	27	0.5–3	7–29
Prasher et al. [16]	PG	Al2O3	40	0.5–3	6–36
Prasher et al. [16]	PG	Al2O3	50	0.5–3	5.5–24
Murshed et al. [27]	DIW	Al2O3	80	1–5	4–82
Anoop et al. [30]	Water	Al2O3	45	2–8 wt%	1–6
Anoop et al. [30]	Water	Al2O3	150	2–8 wt%	1–3
Anoop et al. [31]	Water	Al2O3	95	0.5–6	3–77
Anoop et al. [31]	Water	Al2O3	100	0.5–6	3–57
Anoop et al. [31]	EG	Al2O3	100	0.5–6	5.5–30
Chen et al. [17,18,20]	EG	TiO2	25	0.1–1.86	0.5–23
He et al. [58]	DW	TiO2	95, 145, 210	0.024–1.18	4.11
Chen et al. [20]	Water	TiO2	25	0.25–1.2	3–11
Duangthongsuk and Wongwises [75]	Water	TiO2	21	0.2–2	4–15
Anoop et al. [31]	EG	CuO	152	0.5–6	8–32
Pastoriza-Gallego et al. [36]	Water	CuO	27–37	1–10 wt%	0.5–11.5
Pastoriza-Gallego et al. [36]	Water	CuO	11$\mp 3$	1–10 wt%	2.5–73
Chevalier et al. [19]	Ethanol	SiO2	35	1.2–5	15–95
Chevalier et al. [19]	Ethanol	SiO2	94	1.4–7	12–85
Chevalier et al. [19]	Ethanol	SiO2	190	1–5.6	5–44
Chen et al. [20,32]	EG	TNT	$\cong 10 \mathrm{L}=100 \mathrm{nm}$	0.1–1.86	3.3–70.96
Chen et al. [20,24]	Water	TNT	$\cong 10 \mathrm{L}=100 \mathrm{nm}$	0.12–0.6	3.5–82
Garg et al. [25]	EG	Cu	200	0.4–2	5–24
Zhu et al. [34]	DW	CaCO3	20–50	0.12–4.11	1–69

shows the summary of studies performed on the viscosity of nanofluids, and

Table 3. Summary of studies about viscosity of hybrid nanofluids.

Author	Hybrid Nanofluid	Nanoparticle Size	Volume or Weight Fraction	Temperature Range	Rheological Behavior	Maximum İncrease in Viscosity
[110]	SiO2-graphene 2/naphthenic mineral oil	Graphene nanoparticles:12 nm	1%, 4%, 0.08% (weight fraction)	20–100 °C	-	For 4% water 29.7%
[111]	MWCNT-CuO- lubricant-(10W40)	MWCNT outer diameter:5–15 nm CuO: 40 nm	0–0.01 (volume fraction)	5–55 °C	-	49% for 1 vol.% at 5 °C
[112]	Fe3 O4–MWCNT/ Ethylene glycol	Fe3O4: 20–30 nm–outer diameter of MWCNT-5–15 nm	0.1–0.018 (volume fraction)	25–50 °C	-	63% for 0.8 vol.%
[113]	MWCNTs - SiO2/engine oil (SAE 20W50)	SiO: 40 nm- MWCNTs mean diameter: 20 nm	0.01–0.05 (volume fraction)	40–100 °C	Newtonian	171% for 1 vol.% at 100 °C
[114]	Fe3O4-CNTs - /water	MWCNTs outer diameter: 10–30 nm	0.0135–0.05 (volume fraction for CNT s), 0.009–0.1(volume fraction for CNTs),	25–55 °C	Newtonian Shear rate range:10–100 s−1	29.62% for 0.9% Fe3O4–1.35% CNT
[115]	TiO2–SiO2/water and ethylene glycol (60:40)	SiO2-22 nm TiO2-50 nm	0.03–0.5 (volume fraction)	30–80 °C	Newtonian Shear rate range: 25–187.5 s−1	62.5% for 3 vol.% at 80 °C
[116]	ND-Co3O4/water, EG, water and EG	ND: 4–5 nm	0.0015–0.05 (weight fraction)	20–60 °C	-	51% for 0.15 wt.%, at 60 °C (60%water: 40% EG)
[117]	MWCNTs - ZnO/engine oil (SAE40)	MWCNTs inner diameter: 3–5 nm, ZnO: 10–30 nm	0.01–0.05 (volume fraction)	25–60 °C	Newtonian Shear rate range:1333–13,333 s−1	33.3% for 1 vol.% at 40 °C
[118]	Fe3O4 -Ag/EG Fe3O4 -Ag/EG	Fe3O4: 20–30 nm-Ag:30–50 nm	0.012–0.0375 (volume fraction)	25–50 °C	Newtonian up to 0.3 vol.% and non-Newtonian for vol.$>0.003$	27 mPa.s for 0.3 vol.% at 25 °C
[119]	MWCNTs - MgO/EG	MWCNTs outer diameter: 5–20 nm MgO: 40 nm	0.01–0.1 (volume fraction)	30–60 °C	Newtonian Shear rate range:24.46–122.3 s−1	168% for 1 vol.% at 60 °C
[120]	TiO2–SiO2/water and EG(60:40)	TiO2: 50 nm SiO2: 22 nm	0.01 (volume fraction) Suspension ratios of TiO2–SiO2 = (20:80, 40:60, 50:50, 60:40, 80:20)	30–80 °C	Newtonian Shear rate range:61.15–122.3 s−1	52% for 1 vol.% of (50:50) suspension ratio at 80 °C
[121]	Graphene Nanoplatelets/Pt-water	GNP particle diameter: 2 μm	0.001–0.02 (weight fraction)	20–40 °C	Newtonian Shear rate range: 500 s−1	33% for 0.1 wt.% at 40 °C
[122]	Cu-Zn/SAE oil, vegetable oil, paraffin oil	Cu-Zn: 25 nm	0.005–0.1 (volume fraction)	-	Newtonian for nanofluids with base fluid of vegetable oil Shear rate range: 0–100 s−1	~37% for 0.5 vol.% SAE oil base nanofluid
[123]	CuO-MWCNTs /SAE 5w-50	Outer diameter of CuO: 40 nm MWCNT outer diameter: 5–15 nm	0.01–0.05 (volume fraction)	5–55 °C	-	35.52% at 5 °C and 12.92% at 55 °C for 1 vol.%
[124]	Fe3O4- MWCNTs /water	Outer diameter of MWCNTs: 10–30 nm	0.003–0.1 (volume fraction)	20–60 °C	-	50% for 0.3 vol.% at 60 °C
[125]	Carbon graphene oxide/ EG	-	0.0006–0.02 (weight fraction)	20–45 °C	Newtonian Shear rate range: 20–500 s−1	4.16% for 0.06 wt.%
[126]	Cu-Zn (75:25, 50:50, 25:75) - vegetable oil	Cu–Zn (75:25): 19 nm Cu–Zn (50:50): 25 nm Cu–Zn (25:75): 23 nm	0.005–0.1 (volume fraction)	30–60 °C	Newtonian	46.5 mPa.s for 0.5 vol.% (50:50) at 30 °C
[127]	Ag–MgO/water	Ag: 25 nm MgO: 40 nm	0.02–0.5 (volume fraction)	-	-	For 2 vol.% 38.1%

shows the studies of hybrid nanofluids based on viscosity.

2.2. Non-Newtonian Fluids

An incompressible Newtonian fluid is totally a viscous fluid which has a linear equation as [128]:

$$\tau =\mu ·{\gamma }^{.}$$

(24)

For non-Newtonian fluids in simple shear flow, we can determine the viscosity function as $n\left({\gamma }^{.}\right)$ as [128]:

$$\eta \left({\gamma }^{.}\right)=\frac{\tau }{{\gamma }^{.}},$$

(25)

$$\tau =\eta \left({\gamma }^{.}\right)\left({\gamma }^{.}\right)$$

(26)

The viscosity function can be shown in some different models, one of which being the power law [128]:

$$\eta \left({\gamma }^{.}\right)=K·{\left|{\gamma }^{.}\right|}^{n-1}$$

(27)

where K for the consistency parameter and the power-law index both depend on temperature. According to the power law [128]:

$$\begin{array}{c}{\eta }_0\equiv \eta \left(0\right)=\infty \mathrm{when} n<1 \mathrm{and}=0 \mathrm{for} n>1,\\ {\eta }_{\infty }\equiv \eta (\infty )=0 \mathrm{when} n<1 \mathrm{and}=\infty \mathrm{for} n>1 , \end{array}$$

(28)

What is given above is contrary to the one found in experiments with non-Newtonian fluids [128]:

$$\begin{array}{c}{\eta }_0\equiv \eta \left(0\right)= \mathrm{finite} \mathrm{value}>0,\\ {\eta }_{\infty }\equiv \eta (\infty )=\mathrm{finite} \mathrm{value}>0\end{array}$$

(29)

The parameter ${\eta }_0$ is the zero-shear-rate-viscosity, and ${\eta }_{\infty }$ is called the infinite shear-rate viscosity. In

Table 4. Consistency parameter K and power-law index n for some fluids [128].

Fluid	$\mathbf{Region} \mathbf{for} {\mathit{\gamma }}^{.}\left[{\mathbf{s}}^{-1}\right]$	$\mathbf{K} = \left[\mathbf{N}{\mathbf{s}}^{\mathbf{n}}/{\mathbf{m}}^2\right]$	n
54.3% cement rock in water, 300 K	100–200	2.51	0.153
23.3% Illinois clay in water, 300 K	1800–6000	5.55	0.229
Polystyrene, 422 K	0.03–3	1.6 × 105	0.4
Tomato Concentrate, 90 °F 30% solids		18.7	0.4
Applesauce, 80 °F 11.6% solids		12.7	0.4
Banana puree, 68 °F		6.89	0.28

, some K and n values are seen.

Time-independent fluids, time-dependent fluids and viscoelastic fluids are basic fluid models [128].

2.2.1. Time-Independent Fluids

Time-independent fluids are considered in two groups as viscoplastic fluids and totally viscous fluids. Viscoplastic materials have the characteristic behavior of the function $\tau \left({\gamma }^{.}\right)$, as shown in Figure 3. The material models are solids when the yield shear stress ${\tau }_y$ is more than the shear stress, and the behavior turns out to be elastic.

If $\tau >{\tau }_y$ the material models are fluids. When the material is considered as a fluid, it is usually enumerated that the fluid is an incompressible one and that the material is rigid, with no deformations, if $\tau <{\tau }_y.$ Professor Bingham, who created the name rheology, also has his name on the basic viscoplastic fluid model. Drilling fluids, sand in water, granular materials, margarine, toothpaste, some kinds of paint, some polymer melts and fresh concrete are fluids which show a yield shear stress. In simple share flow, purely viscous fluids have Equations (24) and (26). A totally viscous fluid shows shear-thinning or pseudoplastic behavior if the viscosity shown in viscosity Equation (25) decreases inversely proportionally to shear rate (Figure 4 and Figure 5).

Most non-Newtonian fluids show shear-thinning behavior, for instance, almost all polymer melts, polymer solutions, biological fluids, and mayonnaise. In consideration of a small group of real liquids, ‘‘the apparent viscosity’’ $\raisebox{1ex}{$\tau $}\!\left/ \!\raisebox{-1ex}{${\gamma }^{.}$}\right.$ increases inversely proportionally to shear rate. These types of fluids exhibit shear-thickening behavior. When shear stresses are applied to these fluids, their volume expands. The power law (27) describes the shear-thickening fluid when n is greater than 1. A well-known issue for this is cornstarch and water mixture [128].

2.2.2. Time-Dependent Fluids

The modelling of these fluids is difficult. For a stable shear rate ${\gamma }^{.}$ at a constant temperature, the shear stress $\tau$ either increases or decreases over time, towards an asymptotic value $\tau \left({\gamma }^{.}\right)$ (Figure 6).

The fluids regain their original properties when the shear rate value is zero. There are two different groups of time-dependent fluids. Thixotropic fluids: With a constant share value, the shear stress decreases in a monotonic way. Rheopectic fluids: With a constant shear rate, the shear stress increases in a monotonic way. In other words, these fluids are known as antithixotropic fluids [128].

2.2.3. Viscoelastic Fluids

A viscous component and an elastic one form these kinds of fluids. A solvent and some polymers are fundamentals of these fluids. Paints, DNA suspensions, and some biological fluids can be given as examples [128].

3. The Non-Newtonian Nanofluids

Researchers have also conducted some studies about the rheological characteristics of nanofluids based on the way they are prepared. In experiments, some nanofluids are prepared with base fluids which show non-Newtonian behavior, and some of them are prepared with non-Newtonian base fluids. Studies show that the dispersion of nanoparticles used in Newtonian base fluids show Newtonian behavior, while most nanofluids showed non-Newtonian behavior and mostly shear-thinning behavior [10,11,12,13,14,15,129,130,131]. Hojjat et al. [130] performed an experiment using γ-Al₂O₃, TiO2 and CuO nanoparticles as non-Newtonian nanofluids. They conducted their experiment at different temperature values for different concentrations of the nanoparticles. The concentration range of the nanoparticles was 0.1–4%. For the temperature values ranging from 5 to 45 °C, they examined the rheological characteristics of the nanofluid with the concentration range by applying a shear rate of 2.5–3.1. They observed that the consistency index (K) was dependent on both temperature and concentration. From the results, they observed that the viscosity of nanofluids and base fluid showed a rheological behavior of non-Newtonian fluids showing a remarkable consistency with power law with indices (n) lower than 1. Figure 7 illustrates power-law indices for each nanofluid based on the concentration range.

Tseng and Lin [14] conducted an experiment to examine the rheological and colloidal structure of the nanofluids, which consisted of TiO₂ nanoparticles scattered in water. The concentration of the particles had a range of 0.05–0.12%. The shear rate ($\gamma )$ to which the nanofluid was exposed had a range of 10¹–10³ s⁻¹. The nanofluids showed a non-Newtonian behavior, indicating that the rheology belonged to a thixotropic kind of fluid, meaning that the shear stress showed a monotonic decrease with a constant share rate value. Figure 8 demonstrates the yield stress of TiO₂ colloids due to different concentrations with the comparison of Bingham, Casson and Herschel–Bulkley models.

Tseng and Chen [13] performed a study to examine the dispersive effect in the suspension of nickel and terpineol, a nanofluid which consisted of nickel particles dispersed in some kind of terpene modified by alcohol. The volume concentration of the Ni particles had a range of 3–10%, while the dispersive concentration in the suspension was 0.5–10%. They applied a shear rate of 1–1000 s⁻¹ during their study. Among the seven groups they used in their experiment, the surfactants which had an interaction with nonionic functional groups showed the best dissolution effect. Due to the share rate all seven groups of suspension showed a non-Newtonian behavior as pseudoplastic fluids, meaning that the viscosity decreased as the share rate increased; in other words, they behaved as shear-thinning fluids. The concentration of the dispersive substance was kept constant for 2 wt.% of the solids.

Kwak and Kim [15] used CuO particles dispersed in ethylene glycol to examine the viscosity and thermal conductivity of the suspension. They used transmission electron microscopy (TEM) to examine the rheological characteristics and thermal conductivity of the nanofluid. They observed that the concentration at the subtilized limit was 0.002 rheologically. For viscosity values due to the applied share rate, which had a range of 10⁻³ and 10⁴, the researchers found that the suspension showed a shear-thinning behavior. Figure 9 shows the viscosity values due to the share rate for different concentrations of the suspension (10⁻⁵–10⁻²). According to Figure 9, the nanofluids based on different concentrations have a very intense shear-thinning behavior.

Kamali and Binesh [131] examined nanofluids with multi-wall carbon nanotubes (MWCNTs) to characterize the rheological behavior of the suspension. They examined the nanofluid by placing it into a tube which had a constant wall heat flux. They used power-law finite model. Navier–Stokes equations were used with finite volume method for the numerical analysis. They observed that the suspension showed a non-Newtonian behavior, a shear-thinning one like the ones mentioned in the previous studies [13,14,15]. They compared their results related to thermal performance and the ones of traditional fluids. Figure 10 shows the viscosity of the MWCNT-based nanofluid in different locations for Re = 900.

4. Magnetohydrodynamics with Non-Newtonian Nanofluids for Convection in Cavities

Researchers performed studies in cavities considering magnetic field effects, which could lead different parameters in the systems. As mentioned before, since nanofluids have taken a significant role in the heat transfer phenomenon, some researchers have applied studies using these special fluids in convection by considering both magnetohydrodynamics of the flow and cavity effects. They have tried to observe the heat enhancement or changes in velocity profiles etc. by considering these two cases. In this part, studies performed relating to convection by using nanofluids showing non-Newtonian behavior in cavities considering magnetohydrodynamics are given. Some of these studies include both convection types (natural (free) and forced convection) at the same time, which is known in the literature as mixed convection in applications.

Kefayati and Tang [132] performed a simulation study based on natural convection and generation of entropy of a non-Newtonian nanofluid considering magnetohydrodynamics (MHD) in a cavity. By performing an analysis using the lattice Boltzmann method of finite difference, they examined various parameters in their study. The power-law index value of the nanofluid they used was between 0.4 and 1. They found some negative or positive effects of the parameters on each other. Figure 11 shows some contours of the local entropy, which occurred due to some values of some parameters such as power-law index and magnetic field.

Ali et al. [133] conducted a study in order to examine the natural convection of a nanofluid which showed non-Newtonian behavior based on the power law, applying their experiment in a U-shaped cavity and considering the magnetic effect in the system. The authors examined various parameters, such as the slope angle, volume fraction of particles, power-law index, the baffles’ side ratio, etc., and obtained some results. In terms of heat enhancement, the slope angle of magnetic effect was clearer on the average Nusselt number with a baffles’ side ratio of 0.2 for power-law index (n) values 1 and 1.4. The greatest value of the average Nusselt number was seen at a slope angle of 30$°$ in their study. Acharya and Dash [134] performed a study using a nanofluid whose base fluid was water. As nanoparticles, copper oxide particles were used for the suspension. The nanofluid showed non-Newtonian behavior based on the power-law model. The nanofluid was in a wavy cavity, and the environment was exposed to a magnetic field. The aim of the study was to examine natural convection of the nanofluid under these circumstances. Volume fraction of nanoparticles (φ) were kept between the values of 0 and 0.12 during the experiment. The power-law index (n) of the non-Newtonian nanofluid was between 0.6 and 1.4. The magnetic field effect was shown by using various Hartmann numbers from 0 to 90. Rayleigh number was kept between the values of 10⁵ and 10⁶. In the study, it was shown that for all Rayleigh numbers and for Hartmann numbers lower than 60, the nanofluid showing shear-thinning behavior improved the heat transfer as the volume fraction increased between the taken values. It was also seen that for the average Nusselt number, the wavy cavity had a lesser effect than that of a plane wall. The same finding was not plausible for the local Nusselt number. Kherroubi et al. [135] used nanolubricants (non-Newtonian nanofluids) using zinc oxide as a nanoparticle. They conducted their study in two- and three-dimensional cavities with ventilation considering magnetic effect in order to examine pressure drop and heat transfer of the fluids. The non-Newtonian nanofluids showed compatibility with Bingham model, which meant that it showed viscous behavior at high stress values after behaving in a rigid way at low stress values. The magnetic field effect in the study was clearly seen on heat exchange and pressure drop by causing increases except when the slope angle (ω) was 135° and heat exchange decreased. The best performance for heat exchange was obtained when the slope angle (ω) was 0°. Figure 12 shows volume fraction effects of the nanoparticles on the pressure drop due to different slope angles of the magnetic field in the two-dimensional study, and Figure 13 demonstrates isotherms based on different Reynolds numbers and volume fractions of nanoparticles.

Aboud et al. [136] examined magnetohydrodynamic effect in a circular space which was filled with a nanofluid that showed non-Newtonian behavior. The flow type in the study was laminar and incompressible, and the nanofluid was a water-based one with copper nanoparticles. Based on various parameters such as the power-law index (n), Hartmann number (Ha), Richardson number (Ri), Prandtl number (Pr), volume fraction (φ), etc., only Grashorf and Prandtl numbers had constant values in the study, while the others had different values at certain ranges. The flow function showed an increase within an increase in the power-law index (n), almost to n = 1. It was also observed that the magnetic field had a critical effect on the fluid behavior. Figure 14 illustrates the streamlines and isotherms of the study.

Abderrahmane et al. [137] used aluminum oxide nanoparticles and water as the base fluid for their study. The nanofluid was a non-Newtonian-type fluid. They examined magnetohydrodynamics of the flow in free convection exposing the system to a half-circular poriferous medium using the finite element method. Depending on various parameters such as the power-law index ($\mathrm{n}$) ranging from 0.6 to 1, Rayleigh number with values of 103–106, slope angle ($\alpha$) ranging from 0$°$ to 90$°$, and Hartmann number with values of 0–100, they obtained some results. With the magnetic effect, velocity profiles slowed down, and there was a negative effect on the Nusselt number and heat transfer, and the increase in the power-law index caused a decrease in heat transfer. Figure 15 illustrates the effect of the parameters on average values of the Nusselt numbers.

Kefayati [138] performed a study based on mixed convection in a cavity with two taps on both of its sides. In the study, laminar flow and mixed convection of non-Newtonian nanofluids were examined considering the effect of the magnetic field. The non-Newtonian nanofluids had a power-law index (n) in the value range of 0.2–1. Aluminum oxide particles with water were used to prepare the suspension, and the nanofluid showed shear-thinning behavior in the study. Based on other parameters such as Hartmann number, Richardson number, and volume fraction with constant or ranging values, the results were obtained. For various Richardson number values, the decrease in the value of power-law index caused a negative effect on heat transfer. The rise in Hartmann number had negative effects on heat transfer. For different power-law index values, Richardson and Hartmann numbers, the addition of nanoparticles caused heat enhancement. It was also seen that the effect of the magnetic field made the average Nusselt number rise at certain values of Richardson number (0.001, 0.01). Figure 16 demonstrates the isotherms and streamlines for the lowest and highest power-law index and nanofluid concentration values with the comparison of base fluid and nanofluid.

Mansour et al. [139] performed a study based on the radiative convection of a nanofluid which had a macropolar and hybrid structure. They considered magnetohydrodynamics in their study, which consisted of mixed convection of the nanofluid. The flow took place in a curved cavity with a tap which had a strange shape. The local slide effect was also considered in the study. The equations relating to the flow control were solved, and different issues such as temperature, angular velocity, streamlines, and Nusselt number were evaluated for different cases. An enhancement in angular velocity was seen due to the change in whirlpool parameter. Due to the growth of slope angle, an improvement was seen in the local Nusselt number values. When the Darcy number had a value of 0.01 and the nanoparticle factor rose from 0 to 10%, an enhancement was seen in the average Nusselt number up to the value of 9.9%.

Selimefendigil and Chamkha [140] performed their study in a tapped cavity. They used a non-Newtonian nanofluid showing a shear-thinning behavior. In their experiment, they also used the effect of magnetic field to observe the mixed convection in a cavity which had a grooved bottom-wall with a wavy triangle shape. The fluid in the study behaved according to power law. They considered different parameters such as the slope angle of the magnetic field and some dimensionless numbers such as Richardson and Hartmann numbers. From their study, they obtained some remarkable results. One of the most significant ones was that a heat transfer enhancement was seen for a shear-thinning fluid in a clearer way. Another remarkable result was that average Nusselt number showed an enhancement with the increment of the slope angle of the magnetic field. Selimefendigil and Chamkha [141] performed a numerical simulation in another study. They performed the simulation for a non-Newtonian nanofluid which showed behavior according to the power law. They considered the magnetic field effect by keeping their experiment in a cavity which had the shape of a triangle. They used a specific nanofluid model for their study. Considering different parameters such as dimensionless numbers, e.g., Richardson and Hartmann numbers, slope angle, and volume fraction of the nanoparticles, they generally examined the heat transfer enhancement in addition to some other parameters. They observed that the slope angle of the magnetic field had a connection with the fluid type for the minimum value of the average Nusselt number. One of the most remarkable results of their study was that the heat transfer enhancement had its greatest value for the highest value of the opening rate. Figure 17 illustrates the effect of the Richardson number and power-law indices on the streamlines due to some other fixed values.

Jahanbakhshi et al. [142] examined natural convection in a cavity which had the shape of the letter L by using a non-Newtonian nanofluid due to the power-law model. They considered the effect of the magnetic field in heat transfer in a numerical way. For a power-law index lower than 1, i.e., for fluids showing shear-thinning behavior, the heat transfer ratio was observed to decrease. On the other hand, for a power-law index higher than 1, i.e., for fluids showing shear-thickening behavior, the heat transfer ratio was observed to increase. The non-Newtonian behavior, whether shear-thinning or shear-thickening, did not have a significant role in the thermal behavior of the fluids when the surrounding angle was lower and higher than 60$°$. Kefayati [143] performed a study using nanofluid based on water and copper nanoparticles which showed non-Newtonian behavior in a cavity. The MHD effect in natural convection was considered in the study. Heat transfer and entropy output were simulated in the study. Considering certain parameters at some fixed values such as a power-law index in the range of 0.6–1, Rayleigh number from 10⁴ to 10⁵, Hartmann number between the values of 0 and 90, and volume fraction of 0 to 4%, results were gained. One of the most significant results was the observation of the heat transfer enhancement due to the increase in the power-law index with the existence of a magnetic field. The nanoparticle addition caused an increase in heat transfer considering different parameters in the study. For entropy output, the increase in the nanoparticle volume fraction and Rayleigh number had positive effects. The increment in Hartmann number had negative effects on both the entropy output and heat transfer. Benos et al. [144] used carbon nanotubes with water as a nanofluid for their study. The nanofluid showed changing behavior from Newtonian to non-Newtonian during the experiment, and during this behavior change, the stack of nanoparticles was examined. There was also the effect of the magnetic field in the experiment. The convection in the study was natural, and the case was examined in a fordable cavity. From the study, some remarkable results were obtained. The stack of nanoparticles was dependent on the rheological behavior change in the nanofluid. The increase in the nanoparticles’ pile was observed to have effects on thermal conductivity and viscosity of the nanofluid. The heat transfer was affected by the changes in viscosity and conductivity. A slow-down in nanofluid and degradation in heat transfer were observed due to the increase in nanoparticle fraction.

5. Conclusions

In this article, theoretical, experimental, and numerical studies on convection in vented cavities with non-Newtonian nanofluids under magnetic field have been discussed in detail. The following conclusions can be drawn:

(1)

Nanofluid type and solid volume fraction are important in the overall thermal performance enhancement of vented cavities. When considered with other parameters of vented cavity such as port location, thermal performance variations have been reported by using a nanofluid and varying its type.

(2)

Hybrid nanofluids are found to be promising as compared to nanofluids with mono-particles in the thermal efficiency of vented cavities.

(3)

The magnetic field suppresses multi-recirculations in vented cavities, and depending upon the operating parameters, thermal performance improves with the application of an external magnetic field. Magnetic field inclination also influences the thermal and flow field features, while it can be considered another important parameter in MHD applications for vented cavities.

(4)

Most of the studies for convection in cavities under magnetic field effects are numerical ones. Experimental studies are needed to support the numerical studies.

(5)

Most of the studies in vented cavities with a magnetic field are for uniform magnetic field cases. Non-uniform and spatially varying magnetic field effects should be considered.

(6)

Including nanoparticles in the base fluid under magnetic field for convection in vented cavities, thermal performance improvement has been shown. Even though reliable correlations exist for thermal conductivity and viscosity of nanofluids, there is a lack of experimentally supported correlations for electrical conductivity of nanofluids. Most correlations for electric conductivity of nanofluids use a Maxwell relation, while for accurate modeling of a nanofluid under a magnetic field in vented cavities, more experimental studies are needed for developing accurate relations for electric conductivity.

(7)

There are a few studies that considered the non-Newtonian aspects of fluid or nanofluids in vented cavities. Most of the studies considered Newtonian fluid behavior even when using higher nanoparticle loading in the base fluid. Some studies considered theoretical models (e.g., the power-law model) of using nanofluid non-Newtonian behavior in vented cavities.

6. Future Recommendations

(1)

In numerical analysis, it is recommended to use available models that fits into one of the existing non-Newtonian models with empirical constants from experimental data.

(2)

As the thermal performance changes considerably by using non-Newtonian nanofluids, pressure drop features should also be considered.

(3)

More numerical or experimental studies are recommended for using hybrid nanofluids and considering its non-Newtonian behavior with its impact on hydrothermal efficiency.

(4)

Vented cavity-flow applications can be extended to be used in renewable energy system technologies, and further performance enhancements and efficiency improvement can be obtained by using hybrid nanofluids. Magnetic field and non-Newtonian fluid behavior parameters can be considered to control the hydrothermal performance by including pumping power features.

(5)

Finally, machine learning based optimization techniques may be used alongside a cost analysis of using nanofluids to obtain a cost-effective energy-efficient vented-cavity system design.