Search in book...
Toggle Font Controls
Create new playlist

Name your new playlist

Playlist description (optional)
Sign In

Email address

Password

Forgot Password?

or

Continue with Facebook

Continue with Google
Sign Up

Full Name

Email address

Confirm Email Address

Password

or

Continue with Facebook

Continue with Google

Chapter 1

Nanofluid: Definition and Applications

Abstract

The use of additives in the base fluid like water or ethylene glycol is one of the techniques applied to augment the heat transfer. Recently innovative nanometer sized particles had been dispersed in the base fluid in heat transfer fluids. The fluids containing the solid nanometer size particle dispersion are called “nanofluids.” Two main categories were discussed in detail: the single-phase modeling in which the combination of nanoparticle and base fluid is considered as a single-phase mixture with steady properties and the two-phase modeling in which the nanoparticle properties and behaviors are considered separately from the base fluid properties and behaviors. Moreover, nanofluid flow and heat transfer can be studied in the presence of thermal radiation, electric field, magnetic field, and porous media. Furthermore, several numerical and semianalytical methods can be used to simulate nanofluid hydrothermal behavior. In this chapter, definition of nanofluid and its application have been presented.

Keywords

nanofluid

natural convection

force convection

semianalytical method

numerical method

magnetic field

electric field

1.1. Introduction

Nanofluids are produced by dispersing the nanometer-scale solid particles into base liquids with low thermal conductivity, such as water, ethylene glycol (EG), oils, etc. Control of heat transfer in many energy systems is crucial due to the increase in energy prices. In recent years, nanofluids technology was proposed and studied by some researchers experimentally or numerically to control heat transfer in a process. The nanofluid can be applied to engineering problems, such as heat exchangers, cooling of electronic equipment, and chemical processes. There are two ways of simulating nanofluid: single phase and two phase. In the first method, researchers assumed that nanofluids treated as the common pure fluid and conventional equations of mass, momentum, and energy are used and the only effect of nanofluid is its thermal conductivity and viscosity which are obtained from the theoretical models or experimental data. These researchers assumed that nanoparticles are in thermal equilibrium and there aren’t any slip velocities between the nanoparticles and fluid molecules; thus they have a uniform mixture of nanoparticles. In the second method, researchers assumed that there are slip velocities between nanoparticles and fluid molecules. So the volume fraction of nanofluids may not be uniform anymore and there would be a variable concentration of nanoparticles in a mixture. There are several numerical and semianalytical methods which have been used by several authors to simulate nanofluid flow and heat transfer.

1.1.1. Definition of nanofluid

Low thermal conductivity of conventional heat transfer fluids, such as water, oil, and EG mixture is a serious limitation in improving the performance and compactness of many engineering equipment, such as heat exchangers and electronic devices. To overcome this disadvantage, there is strong motivation to develop advanced heat transfer fluids with substantially higher conductivity. An innovative way of improving the thermal conductivities of fluids is to suspend small solid particles in the fluid. Various types of powders, such as metallic, nonmetallic, and polymeric particles can be added into fluids to form slurries. The thermal conductivities of fluids with suspended particles are expected to be higher than that of common fluids. Nanofluids are a new kind of heat transfer fluid containing a small quantity of nanosized particles (usually less than 100 nm) that are uniformly and stably suspended in a liquid. The dispersion of a small amount of solid nanoparticles in conventional fluids changes their thermal conductivity remarkably. Compared to the existing techniques for enhancing heat transfer, the nanofluids show a superior potential for increasing heat transfer rates in a variety of cases [1].

1.1.2. Model description

In the literature, convective heat transfer with nanofluids can be modeled using mainly the two-phase or single approach. In the two-phase approach, the velocity between the fluid and particles might not be zero [2] due to several factors, such as gravity, friction between the fluid and solid particles, Brownian forces, Brownian diffusion, sedimentation, and dispersion. In the single-phase approach, the nanoparticles can be easily fluidized and, therefore, one may assume that the motion slip between the phases, if any, would be considered negligible [3]. The latter approach is simpler and computationally more efficient.

1.1.3. Conservation equations

1.1.3.1. Single-phase model

Although nanofluids are solid–liquid mixtures, the approach conventionally used in most studies of natural convection handles the nanofluid as a single-phase (homogenous) fluid. In fact, due to the extreme size and low concentration of the suspended nanoparticles, the particles are assumed to move with same velocity as the fluid. Also, by considering the local thermal equilibrium, the solid particle–liquid mixture may then be approximately considered to behave as a conventional single-phase fluid with properties that are to be evaluated as functions of those of the constituents. The governing equations for a homogenous analysis of natural convection are continuity, momentum, and energy equations with their density, specific heat, thermal conductivity, and viscosity modified for nanofluid application. The specific governing equations for various studied enclosures are not shown here and they can be found in different references [4]. It should be mentioned that sometimes this assumption is not correct. For example, Ding and Wen [5] showed that this assumption may not always remain true for a nanofluid. They investigated the particle migration in a nanofluid for a pipe flow and stated that at Peclet numbers exceeding 10 the particle distribution is significantly nonuniform. Nevertheless, many studies have performed the numerical simulation using single-phase assumption and reported acceptable results for the heat transfer and hydrodynamic properties of the flow.

1.1.3.2. Two-phase model

Several authors have tried to establish convective transport models for nanofluids [6]. Nanofluid is a two-phase mixture in which the solid phase consists of nanosized particles. In view of the nanoscale size of the particles, it may be questionable whether the theory of conventional two-phase flow can be applied in describing the flow characteristics of nanofluid. On the other hand, several factors, such as gravity, friction between the fluid and solid particles and Brownian forces, the phenomena of Brownian diffusion, sedimentation, and dispersion may affect a nanofluid flow. Consequently, the slip velocity between the fluid and particles cannot be neglected for simulating nanofluid flows. As the two-phase approach considers the movement between the solid and fluid molecule, it may have better prediction in nanofluid study. To fully describe and predict the flow and behavior of complex flows, different multiphase theories have been proposed and used. The large number of published articles concerning multiphase flows typically employed the Mixture Theory to predict the behavior of nanofluids [7]. A comprehensive survey of convective transport in nanofluids was made by Buongiorno [8], using a model in which Brownian motion and thermophoresis are accounted for. Buongiorno developed a two-component four-equation nonhomogeneous equilibrium model for mass, momentum, and heat transfer in nanofluids. The nanofluid is treated as a two-component mixture (base fluidþnanoparticles) with the following assumptions: no chemical reactions; negligible external forces; dilute mixture (φ = 1); negligible viscous dissipation; negligible radiative heat transfer; nanoparticle and base fluid locally in thermal equilibrium. Invoking the aforementioned assumptions, the following equations represent the mathematical formulation of the nonhomogenous single-phase model for the governing equations as formulated by Buongiorno [8]:

1.1.3.2.1. Continuity equation

$\nabla . v = 0$ $\nabla . v = 0$

(1.1)

where

v

$v$

is the velocity

1.1.3.2.2. Nanoparticle continuity equation

$\frac{\partial φ}{\partial t} + v . \nabla φ = \nabla . (D_{B} \nabla φ + D_{T} \frac{\nabla T}{T})$ $\frac{\partial φ}{\partial t} + v . \nabla φ = \nabla . (D_{B} \nabla φ + D_{T} \frac{\nabla T}{T})$

(1.2)

where φ is nanoparticle volume fraction, D_B is the Brownian diffusion coefficient given by the Einstein–Stokes’s equation:

$D_{B} = \frac{k_{B} T}{3 π μ d_{p}}$ $D_{B} = \frac{k_{B} T}{3 π μ d_{p}}$

(1.3)

where μ is the viscosity of the fluid, d_p is the nanoparticle diameter, k_B = 1.385 × 10⁻²³ is the Boltzmann constant, and D_T is the thermophoretic diffusion coefficient, which is defined as

$D_{T} = (\frac{μ}{ρ}) (0.26 \frac{k}{k + k_{p}})$ $D_{T} = (\frac{μ}{ρ}) (0.26 \frac{k}{k + k_{p}})$

(1.4)

where

k

$k$

and

k_{p}

$k_{p}$

are the thermal conductivity of the fluid and particle materials, respectively.

1.1.3.2.3. Momentum equation

$v . \nabla v = - \frac{1}{ρ_{n f}} \nabla p + \nabla . τ + g$ $v . \nabla v = - \frac{1}{ρ_{n f}} \nabla p + \nabla . τ + g$

(1.5)

where

$τ = - μ_{n f} (\nabla v + {(\nabla v)}^{t})$ $τ = - μ_{n f} (\nabla v + {(\nabla v)}^{t})$

(1.6)

where the superscript“t” indicates the transpose of

\nabla v

$\nabla v$

. Also p is pressure.

1.1.3.2.4. Energy equation

$v . \nabla T = \nabla (α_{n f} \nabla T) + \frac{ρ_{p} c_{p}}{ρ_{n f} c_{n f}} (D_{B} \nabla φ . \nabla T + D_{T} \frac{\nabla T . \nabla T}{T})$ $v . \nabla T = \nabla (α_{n f} \nabla T) + \frac{ρ_{p} c_{p}}{ρ_{n f} c_{n f}} (D_{B} \nabla φ . \nabla T + D_{T} \frac{\nabla T . \nabla T}{T})$

(1.7)

where φ and T are nanoparticle concentration and temperature of nanofluid, respectively.

This nanofluid model can be characterized as a “two-fluid” (nanoparticles + base fluid), four-equation (mass, momentum, energy), nonhomogeneous (nanoparticle/fluid slip velocity allowed) equilibrium (nanoparticle/fluid temperature differences not allowed) model. Note that the conservation equations are strongly coupled. That is,

v

$v$

depends on φ via viscosity; φ depends on T mostly because of thermophoresis; T depends on φ via thermal conductivity and also via the Brownian and thermophoretic terms in the energy equation: φ and T obviously depend on

v

$v$

because of the convection terms in the nanoparticle continuity and energy equations, respectively.

In a numerical study by Behzadmehr et al. [9] for the first time a two-phase mixture model was implemented to investigate the behavior of Cu–water nanofluid in a tube and the results were also compared with previous works using a single-phase approach. The authors claimed that the simulation done by assuming that basefluid and particles behave separately possessed results that are more precise compared to the previous computational modeling. They implemented the mixture theory for their work. It was suggested that the continuity, momentum, and energy equations be written for a mixture of fluid and a solid phase. Some assumptions were also stated for the model, such as a strong coupling between two phases and the fluid being closely followed by the particles with each phase owning a different velocity leading to a term called slip velocity of nanoparticles as in Eq. (1.8):

$V_{p f} = V_{P} - V_{f} = \frac{ρ_{p} d_{p}^{2}}{18 μ_{f} f_{drag}} \frac{(ρ_{p} - ρ_{m})}{ρ_{p}} a, a = g - (V_{m} . \nabla) V_{m}, f_{drag} = \{\begin{cases} 1 + 0.15 {Re}_{p}^{0.687} \\ 0.0183 {Re}_{p} \end{cases}$ $V_{p f} = V_{P} - V_{f} = \frac{ρ_{p} d_{p}^{2}}{18 μ_{f} f_{drag}} \frac{(ρ_{p} - ρ_{m})}{ρ_{p}} a, a = g - (V_{m} . \nabla) V_{m}, f_{drag} = \{\begin{cases} 1 + 0.15 {Re}_{p}^{0.687} \\ 0.0183 {Re}_{p} \end{cases}$

(1.8)

The conservation equations (continuity, momentum, and energy, respectively) will be written for the mixture as follows:

$\nabla . (ρ_{m} V_{m}) = 0$ $\nabla . (ρ_{m} V_{m}) = 0$

(1.9)

$\nabla . (ρ_{m} V_{m} V_{m}) = - \nabla P_{m} + \nabla . [τ - τ_{t}] + ρ_{m} g + \nabla . (\sum_{k = 1}^{n} φ {}_{k}ρ_{k} V_{dr, k} V_{dr, k})$ $\nabla . (ρ_{m} V_{m} V_{m}) = - \nabla P_{m} + \nabla . [τ - τ_{t}] + ρ_{m} g + \nabla . (\sum_{k = 1}^{n} φ {}_{k}ρ_{k} V_{dr, k} V_{dr, k})$

(1.10)

$\nabla . (φ_{p} V_{k} (ρ_{k} h_{k} + p)) = \nabla . (k_{eff} \nabla T - C_{p} ρ_{m} v t)$ $\nabla . (φ_{p} V_{k} (ρ_{k} h_{k} + p)) = \nabla . (k_{eff} \nabla T - C_{p} ρ_{m} v t)$

(1.11)

where

V_{dr, p}

$V_{dr, p}$

is the particle draft velocity that is related to the slip velocity and is defined as

$V_{dr, p} = V_{P} - V_{f} = V_{p f} - \sum_{k = 1}^{n} \frac{φ {}_{k}ρ_{k}}{ρ_{m}} V_{f k}$ $V_{dr, p} = V_{P} - V_{f} = V_{p f} - \sum_{k = 1}^{n} \frac{φ {}_{k}ρ_{k}}{ρ_{m}} V_{f k}$

(1.12)

1.1.4. Physical properties of the nanofluids for single-phase model

Base nanofluid properties have been published over the past few years in the literature. However, only recently some data on temperature-dependent properties have been provided, even though they are only for nanofluid effective thermal conductivity and effective absolute viscosity.

1.1.4.1. Density

In the absence of experimental data for nanofluid densities, constant-value temperature independent values, based on nanoparticle volume fraction, are used:

$ρ_{n f} = ρ_{f} (1 - φ) + ρ_{p} φ$ $ρ_{n f} = ρ_{f} (1 - φ) + ρ_{p} φ$

(1.13)

1.1.4.2. Specific heat capacity

It has been suggested that the effective specific heat can be calculated using the following equation as reported in [10]:

${(C_{p})}_{n f} = {(C_{p})}_{f} (1 - φ) + {(C_{p})}_{p} φ$ ${(C_{p})}_{n f} = {(C_{p})}_{f} (1 - φ) + {(C_{p})}_{p} φ$

(1.14)

Other authors suggest an alternative approach based on heat capacity concept [11]:

${(ρ C_{p})}_{n f} = {(ρ C_{p})}_{f} (1 - φ) + {(ρ C_{p})}_{p} φ$ ${(ρ C_{p})}_{n f} = {(ρ C_{p})}_{f} (1 - φ) + {(ρ C_{p})}_{p} φ$

(1.15)

These two formulations may of course lead to different results for specific heat. Due to the lack of experimental data, both formulations are considered equivalent in estimating nanofluid specific heat capacity [12].

1.1.4.3. Thermal expansion coefficient

Thermal expansion coefficient of nanofluid can be obtained as follows [1]:

${(ρ β)}_{n f} = {(ρ β)}_{f} (1 - φ) + {(ρ β)}_{p} φ$ ${(ρ β)}_{n f} = {(ρ β)}_{f} (1 - φ) + {(ρ β)}_{p} φ$

(1.16)

1.1.4.4. The electrical conductivity

The effective electrical conductivity of nanofluid was presented by Maxwell [13] as follows:

$σ_{n f} / σ_{f} = 1 + 3 (σ_{P} / σ_{f} - 1) φ / ((σ_{P} / σ_{f} + 2) - (σ_{P} / σ_{f} - 1) φ)$ $σ_{n f} / σ_{f} = 1 + 3 (σ_{P} / σ_{f} - 1) φ / ((σ_{P} / σ_{f} + 2) - (σ_{P} / σ_{f} - 1) φ)$

(1.17)

1.1.4.5. Dynamic viscosity

Various models have been suggested to model the viscosity of a nanofluid mixture that take into account the percentage of nanoparticles suspended in the base fluid. The classic Brinkman model [14] seems to be a proper one which has been extensively used in the studies on numerical simulation concerning nanofluids. Eq. (1.1) shows the relation between the nanofluid viscosity, basefluid viscosity, as well as the nanoparticle concentration in this model.

$μ_{n f} = μ_{f} / {(1 - φ)}^{2.5}$ $μ_{n f} = μ_{f} / {(1 - φ)}^{2.5}$

(1.18)

However, in some recent computational studies, other models have been selected to be used in the numerical process, like the work done by Abu-neda and Chamkha [15] to investigate the convection of CuO–EG–water nanofluid in an enclosure where Namburu correlation for viscosity [16] was applied:

$\log (μ_{n f}) = A e^{- B T}$ $\log (μ_{n f}) = A e^{- B T}$

(1.19)

where

$\begin{array}{l} A = 1.837 φ^{2} - 29.643 φ + 165.65 \\ B = 4 \times 10^{- 6} φ^{2} - 0.001 φ + 0.0186 \end{array}$ $\begin{array}{l} A = 1.837 φ^{2} - 29.643 φ + 165.65 \\ B = 4 \times 10^{- 6} φ^{2} - 0.001 φ + 0.0186 \end{array}$

(1.20)

In their study, the results were compared to that of viscosity modeled by Brinkman. It was outlined that as far as a value for normalized average Nusselt number for the fluid is concerned, for various values of Rayleigh number, the Brinkman model owns a prediction of higher value compared to that for the Namburu model showing the notable role of viscosity model used in the calculations. The authors also state that a combination of different models might as well be implemented that will show different dependence on volume concentration as well as the geometry aspect ratio yet along with the limitation that the models include only those mentioned in the study. Other studies have also shown that different models might lead to different results, like that by a number of suggested relations for viscosity models used in numerical studies are also presented in Table 1.1.

Table 1.1

Different models for viscosity of nanofluids used in simulation

Model	Equation
Einstein model [17]	$μ_{n f} = (2.5 φ + 1) μ_{f}, φ < 0.05$ $μ_{n f} = (2.5 φ + 1) μ_{f}, φ < 0.05$
Pak and Cho’s correlation [18]	$μ_{n f} = μ_{f} (1 + 39.11 φ + 533.9 φ^{2})$ $μ_{n f} = μ_{f} (1 + 39.11 φ + 533.9 φ^{2})$
Jang et al. model [19]	$μ_{n f} = (2.5 φ + 1) μ_{f} [1 + η {(d_{p} / H)}^{- 2 ɛ} φ^{2 / 3} (ɛ + 1)]$ $μ_{n f} = (2.5 φ + 1) μ_{f} [1 + η {(d_{p} / H)}^{- 2 ɛ} φ^{2 / 3} (ɛ + 1)]$
Koo and Kleinstreuer [20]	$μ_{n f} = 5 \times 10^{4} β φ ρ_{f} \sqrt{\frac{k_{B} T}{d_{p} ρ_{p}}} f (T, φ), \{\begin{cases} β = 0.0137 {(100 φ)}^{- 0.8229} for φ < 1 % \\ β = 0.0011 {(100 φ)}^{- 0.7272} for φ > 1 % \end{cases}$ $μ_{n f} = 5 \times 10^{4} β φ ρ_{f} \sqrt{\frac{k_{B} T}{d_{p} ρ_{p}}} f (T, φ), \{\begin{cases} β = 0.0137 {(100 φ)}^{- 0.8229} for φ < 1 % \\ β = 0.0011 {(100 φ)}^{- 0.7272} for φ > 1 % \end{cases}$
Maiga model [21]	$μ_{n f} = μ_{f} (1 + 7.3 φ + 123 φ^{2})$ $μ_{n f} = μ_{f} (1 + 7.3 φ + 123 φ^{2})$
Brownian model [22]	$μ_{n f} = μ_{f} (1 + 2.5 φ + 6.17 φ^{2})$ $μ_{n f} = μ_{f} (1 + 2.5 φ + 6.17 φ^{2})$
Nguyen model [23]	$μ_{n f} = μ_{f} (1 + 0.025 φ + 0.015 φ^{2})$ $μ_{n f} = μ_{f} (1 + 0.025 φ + 0.015 φ^{2})$
Masoumi et al. [24]	$μ_{n f} = μ_{f} + ρ_{p} V_{B} d_{p}^{2} / (72 C δ)$ $μ_{n f} = μ_{f} + ρ_{p} V_{B} d_{p}^{2} / (72 C δ)$
Gherasim et al.[25]	$μ_{n f} = μ_{f} 0.904 e^{14.8 φ}$ $μ_{n f} = μ_{f} 0.904 e^{14.8 φ}$

1.1.4.6. Thermal conductivity

Different nanofluid models based on a combination of the different formulas for the thermal conductivity adopted in the studies of natural convection are summarized in Table 1.2. Also Table 1.3 demonstrates values of thermo physical properties for different materials used as suspended particles in nanofluids.

Table 1.2

Different models for thermal conductivity of nanofluids used in simulation

Model	Equation
Koo and Kleinstreuer [20]	$k_{n f} = k_{f} [\frac{k_{p} + 2 k_{f} - 2 φ (k_{f} - k_{p})}{k_{p} + 2 k_{f} + φ (k_{f} - k_{p})}] + 5 \times 10^{4} β φ ρ_{f} {(C_{p})}_{f} \sqrt{\frac{k_{B} T}{d_{p} ρ_{p}}} f (T, φ)$ $k_{n f} = k_{f} [\frac{k_{p} + 2 k_{f} - 2 φ (k_{f} - k_{p})}{k_{p} + 2 k_{f} + φ (k_{f} - k_{p})}] + 5 \times 10^{4} β φ ρ_{f} {(C_{p})}_{f} \sqrt{\frac{k_{B} T}{d_{p} ρ_{p}}} f (T, φ)$
Jang and Choi model [26]	$\frac{k_{n f}}{k_{f}} = (1 - φ) + B k_{p} φ + 18 \times 10^{6} \frac{3 d_{f}}{d_{p}} k_{f} {Re}_{d p}^{2} \Pr φ$ $\frac{k_{n f}}{k_{f}} = (1 - φ) + B k_{p} φ + 18 \times 10^{6} \frac{3 d_{f}}{d_{p}} k_{f} {Re}_{d p}^{2} \Pr φ$
Bruggeman model [27]	$\begin{array}{l} k_{n f} = 0.25 k_{f} (3 φ - 1) k_{p} / k_{f} + [3 (1 - φ) - 1] + \sqrt{∆_{B}} \\ ∆_{B} = {[(3 φ - 1) k_{p} / k_{f} + (3 (1 - φ) - 1)]}^{2} + 8 k_{p} / k_{f} \end{array}$ $\begin{array}{l} k_{n f} = 0.25 k_{f} (3 φ - 1) k_{p} / k_{f} + [3 (1 - φ) - 1] + \sqrt{∆_{B}} \\ ∆_{B} = {[(3 φ - 1) k_{p} / k_{f} + (3 (1 - φ) - 1)]}^{2} + 8 k_{p} / k_{f} \end{array}$
Chon et al. model [28]	$k_{n f} = k_{f} [1 + 64.7 φ^{0.7640} {(d_{f} / d_{p})}^{0.369} {(k_{f} / k_{p})}^{0.7476} \Pr_{T}^{0.9955} {Re}^{1.2321}]$ $k_{n f} = k_{f} [1 + 64.7 φ^{0.7640} {(d_{f} / d_{p})}^{0.369} {(k_{f} / k_{p})}^{0.7476} \Pr_{T}^{0.9955} {Re}^{1.2321}]$
Charuyakorn et al.[29]	$\frac{k_{n f}}{k_{f}} = [\frac{k_{p} + 2 k_{f} - 2 φ (k_{f} - k_{p})}{k_{p} + 2 k_{f} + φ (k_{f} - k_{p})}] (1 + b φ P e_{p}^{m})$ $\frac{k_{n f}}{k_{f}} = [\frac{k_{p} + 2 k_{f} - 2 φ (k_{f} - k_{p})}{k_{p} + 2 k_{f} + φ (k_{f} - k_{p})}] (1 + b φ P e_{p}^{m})$
Stationary model [30]	$k_{n f} = k_{f} [1 + k_{p} φ d_{f} / (k_{f} (1 - φ) d_{p})]$ $k_{n f} = k_{f} [1 + k_{p} φ d_{f} / (k_{f} (1 - φ) d_{p})]$
Yu and Choi [31]	$\frac{k_{n f}}{k_{f}} = [\frac{k_{p} + 2 k_{f} - 2 φ (k_{f} - k_{p}) {(1 + η)}^{3}}{k_{p} + 2 k_{f} + φ (k_{f} - k_{p}) {(1 + η)}^{3}}]$ $\frac{k_{n f}}{k_{f}} = [\frac{k_{p} + 2 k_{f} - 2 φ (k_{f} - k_{p}) {(1 + η)}^{3}}{k_{p} + 2 k_{f} + φ (k_{f} - k_{p}) {(1 + η)}^{3}}]$
Patel et al. [32]	$\frac{k_{n f}}{k_{f}} = 1 + \frac{k_{p} d_{f} φ}{k_{f} d_{p} (1 - φ)} [1 + c \frac{2 k_{B} T d_{p}}{π α_{f} μ_{f} d_{p}^{2}}]$ $\frac{k_{n f}}{k_{f}} = 1 + \frac{k_{p} d_{f} φ}{k_{f} d_{p} (1 - φ)} [1 + c \frac{2 k_{B} T d_{p}}{π α_{f} μ_{f} d_{p}^{2}}]$
Mintsa et al. [33]	$k_{n f} = k_{f} (1.72 φ + 1.0)$ $k_{n f} = k_{f} (1.72 φ + 1.0)$

Table 1.3

The thermo physical properties of the nanofluid

	ρ (kg/m³)	C_p (j/kgk)	k (W/m.k)	β (K⁻¹)	σ (Ω·m)⁻¹
Pure water	997.1	4179	0.613	21 × 10⁻⁵	0.05
Copper (Cu)	8933	385	401	1.67 × 10⁻⁵	5.96 × 10⁻⁷
Silver (Ag)	10 500	235	429	1.89 × 10⁻⁵	3.60 × 10⁻⁷
Alumina (Al₂O₃)	3970	765	40	0.85 × 10⁻⁵	1 × 10⁻¹⁰
Titanium oxide (TiO₂)	4250	686.2	8.9538	0.9 × 10⁻⁵	1 × 10⁻¹²

1.2. Simulation of nanofluid flow and heat transfer

Several semianalytical and numerical methods have been applied successfully to simulate nanofluid flow and heat transfer. In the following sections we present these works.

1.2.1. Semianalytical methods

Forced convective heat transfer to Sisko nanofluid past a stretching cylinder in the presence of variable thermal conductivity was presented by Khan and Malik [34]. They used homotopy analysis method (HAM) to solve the governing equations. They found that the curvature parameter assisted the temperature as well as concentration profiles. Momentum and heat transfer characteristics from heated spheroids in water-based nanofluids have been investigated by Sasmal and Nirmalkar [35]. They showed that smaller the nanoparticles size the better the heat transfer at low Reynolds number and volume fraction. Hayat et al. [36] studied the effects of homogeneous–heterogeneous reactions in flow of magnetite-Fe₃O₄ nanoparticles by a rotating disk. They showed that the axial, radial, and azimuthal velocity profiles are decreasing function of Hartman number. Sheikholeslami et al. [37] utilized least square and Galerkin methods to investigate MHD nanofluid flow in a semiporous channel. They indicate that velocity boundary layer thickness decreases with increase in Reynolds number and increases with increase in Hartmann number. Sheikholeslami et al. [38] studied the squeezing unsteady nanofluid flow using Adomian decomposition method (ADM). They showed that Nusselt number increases with increase in nanoparticle volume fraction and Eckert number. Sheikholeslami and Ganji [39] applied homotopy perturbation method (HPM) to analysis heat transfer of Cu-water nanofluid flow between parallel plates. They indicated that Nusselt number has direct relationship with nanoparticle volume fraction, the squeeze number, and Eckert number when two plates are separated. Application of ADM for nanofluid Jeffery–Hamel flow with high magnetic field has been presented by Sheikholeslami et al. [40]. They proved that in greater angles or Reynolds numbers high Hartmann numbers are needed to reduce backflow.

Flow and heat transfer of Cu–Water nanofluid between a stretching sheet and a porous surface in a rotating system was studied by Sheikholeslami et al. [41]. They showed that for both suction and injection, the heat transfer rate at the surface increases with increase in nanoparticle volume fraction, Reynolds number, and injection/suction parameter and decreases with power of rotation parameter. Sheikholeslami et al. [42] used HAM to describe nanofluid flow over a permeable stretching wall in a porous medium. They found that increase in the nanoparticle volume fraction will decrease momentum boundary layer thickness and entropy generation rate while this increases the thermal boundary layer thickness. Sheikholeslami and Ganji [43] utilized Galerkin optimal homotopy asymptotic method for investigating magnetohydrodynamic nanofluid flow in a permeable channel. They showed that velocity boundary layer thickness decreases with increase in Reynolds number and nanoparticle volume fraction and increases with increase in Hartmann number. Sheikholeslami et al. [44] presented an application of HPM for simulation of two phase unsteady nanofluid flow and heat transfer between parallel plates in the presence of time-dependent magnetic field. Nanofluid flow and heat transfer between parallel plates considering Brownian motion has been investigated by Sheikholeslami and Ganji [45]. They used differential transformation method (DTM) to solve the governing equations. They showed that skin friction coefficient increases with increase in the squeeze number and Hartmann number. Sheikholeslami et al. [46] studied the steady nanofluid flow between parallel plates. They indicated that Nusselt number augments with increase in viscosity parameters but decreases with augment of magnetic parameter, thermophoretic parameter, and Brownian parameter. DTM has been applied by Domairry et al. [47] to solve the problem of free convection heat transfer of non-Newtonian nanofluid between two vertical flat plates. They showed that as the nanoparticle volume fraction increases, the momentum boundary layer thickness increases. Table 1.4 shows the summary of the semianalytical method studies on nanofluid.

Table 1.4

Summary of the semianalytical method studies on nanofluid

Authors	Geometry of study	Method	Type of nanofluid	Properties	Remarks
Sheikholeslami et al. [37]		Least square and Galerkin methods	Cu–water	$\begin{array}{l} 1 \leq Re \leq 10 \\ 0 \leq φ \leq 0.06 \\ 0 \leq Ha \leq 10 \\ \Pr = 6.8 \end{array}$ $\begin{array}{l} 1 \leq Re \leq 10 \\ 0 \leq φ \leq 0.06 \\ 0 \leq Ha \leq 10 \\ \Pr = 6.8 \end{array}$	Velocity boundary layer thickness decreases with increase in Reynolds number and increases with increase in Hartmann number
Sheikholeslami et al. [38]		Adomian decomposition method	Cu–water Ag–water Al₂O₃–water TiO₂–water	$\begin{array}{l} - 1 \leq S \leq 1 \\ 0 \leq φ \leq 0.06 \\ 0.01 \leq Ec \leq 2 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} - 1 \leq S \leq 1 \\ 0 \leq φ \leq 0.06 \\ 0.01 \leq Ec \leq 2 \\ \Pr = 6.2 \end{array}$	Nusselt number increases with increase in nanoparticle volume fraction and Eckert number but it decreases with increase in the squeeze number
Sheikholeslami and Ganji [39]		HPM	Cu–water	$\begin{array}{l} - 1 \leq S \leq 1 \\ 0 \leq φ \leq 0.06 \\ 0 \leq Ec \leq 1 \\ 0 \leq δ \leq 1 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} - 1 \leq S \leq 1 \\ 0 \leq φ \leq 0.06 \\ 0 \leq Ec \leq 1 \\ 0 \leq δ \leq 1 \\ \Pr = 6.2 \end{array}$	Nusselt number has direct relationship with nanoparticle volume fraction, $δ$ $δ$ , the squeeze number, and Eckert number when two plates are separated but it has reverse relationship with the squeeze number when two plates are squeezed
Sheikholeslami et al. [40]		ADM	Cu–water	$\begin{array}{l} 100 \leq Re \leq 300 \\ 2.5 ° \leq α \leq 5 ° \\ 0 \leq φ \leq 0.06 \\ 0 \leq Ha \leq 2000 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 100 \leq Re \leq 300 \\ 2.5 ° \leq α \leq 5 ° \\ 0 \leq φ \leq 0.06 \\ 0 \leq Ha \leq 2000 \\ \Pr = 6.2 \end{array}$	In greater angles or Reynolds numbers high Hartmann numbers are needed to reduce backflow
Sheikholeslami et al. [41]		HAM	Cu–water	$\begin{array}{l} 5 \leq R \leq 20 \\ 0.5 \leq Kr \leq 6 \\ - 1 \leq λ \leq 1 \\ 0 \leq φ \leq 0.2 \\ 0 \leq Ha \leq 2000 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 5 \leq R \leq 20 \\ 0.5 \leq Kr \leq 6 \\ - 1 \leq λ \leq 1 \\ 0 \leq φ \leq 0.2 \\ 0 \leq Ha \leq 2000 \\ \Pr = 6.2 \end{array}$	For both suction and injection, the heat transfer rate at the surface increases with increase in nanoparticle volume fraction, Reynolds number, and injection/suction parameter and it decreases with power of rotation parameter
Sheikholeslami et al. [42]		HAM	Cu–water Ag–water Al₂O₃–water TiO₂–water	$\begin{array}{l} 0.1 \leq Re \leq 1.5 \\ - 0.1 \leq f_{w} \leq 1 \\ 0 \leq n \leq 5 \\ 0 \leq φ \leq 0.2 \\ 0 \leq Ha \leq 2000 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 0.1 \leq Re \leq 1.5 \\ - 0.1 \leq f_{w} \leq 1 \\ 0 \leq n \leq 5 \\ 0 \leq φ \leq 0.2 \\ 0 \leq Ha \leq 2000 \\ \Pr = 6.2 \end{array}$	Increase in the nanoparticle volume fraction will decrease momentum boundary layer thickness and entropy generation rate while this increases the thermal boundary layer thickness. Such effects are found to be more noticeable in the Ag–water solution than in the other solutions.
Sheikholeslami and Ganji [43]		Galerkin optimal homotopy asymptotic method	Cu–water	$\begin{array}{l} 0 \leq H a \leq 20 \\ 1 \leq Re \leq 20 \\ 0 \leq φ \leq 0.06 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 0 \leq H a \leq 20 \\ 1 \leq Re \leq 20 \\ 0 \leq φ \leq 0.06 \\ \Pr = 6.2 \end{array}$	Velocity boundary layer thickness decreases with increase in Reynolds number and nanoparticle volume fraction and increases with increase in Hartmann number.
Sheikholeslami et al. [44]		HPM	Al₂O₃–water	$\begin{array}{l} 0 \leq H a \leq 12 \\ 0 \leq S \leq 12 \\ 0.01 \leq E c \leq 0.4 \\ 0.5 \leq S c \leq 12 \\ \Pr = 10 \end{array}$ $\begin{array}{l} 0 \leq H a \leq 12 \\ 0 \leq S \leq 12 \\ 0.01 \leq E c \leq 0.4 \\ 0.5 \leq S c \leq 12 \\ \Pr = 10 \end{array}$	Nusselt number is an increasing function of Hartmann number, Eckert number, and Schmidt number but it is a decreasing function of squeeze number
Sheikholeslami and Ganji [45]		DTM	CuO–water, Al₂O₃–water	$\begin{array}{l} 0 \leq S \leq 10 \\ 0 \leq H a \leq 20 \\ - 1 \leq H s \leq - 10 \\ 0.01 \leq E c \leq 0.4 \\ 0.5 \leq S c \leq 12 \\ \Pr = 10 \end{array}$ $\begin{array}{l} 0 \leq S \leq 10 \\ 0 \leq H a \leq 20 \\ - 1 \leq H s \leq - 10 \\ 0.01 \leq E c \leq 0.4 \\ 0.5 \leq S c \leq 12 \\ \Pr = 10 \end{array}$	Skin friction coefficient increases with increase in the squeeze number and Hartmann number but it decreases with increase in nanofluid volume fraction. Nusselt number increases with augment of nanoparticle volume fraction, Hartmann number while it decreases with increase in the squeeze number
Sheikholeslami et al. [46]		DTM	Al₂O₃–water	$\begin{array}{l} 0 \leq S \leq 10 \\ 0 \leq Ha \leq 20 \\ - 1 \leq Hs \leq - 10 \\ 0.01 \leq Ec \leq 0.4 \\ 0.5 \leq Sc \leq 12 \\ \Pr = 10 \end{array}$ $\begin{array}{l} 0 \leq S \leq 10 \\ 0 \leq Ha \leq 20 \\ - 1 \leq Hs \leq - 10 \\ 0.01 \leq Ec \leq 0.4 \\ 0.5 \leq Sc \leq 12 \\ \Pr = 10 \end{array}$	Nusselt number augments with increase of viscosity parameters but it decreases with augment of magnetic parameter, thermophoretic parameter, and Brownian parameter
Domairry et al. [47]		DTM	Cu–water	$\begin{array}{l} 0 \leq Ha \leq 20 \\ 0 \leq δ \leq 100 \\ 0 \leq φ \leq 0.1 \\ 0 \leq Ec \leq 6 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 0 \leq Ha \leq 20 \\ 0 \leq δ \leq 100 \\ 0 \leq φ \leq 0.1 \\ 0 \leq Ec \leq 6 \\ \Pr = 6.2 \end{array}$	As the nanoparticle volume fraction increases, the momentum boundary layer thickness increases, whereas the thermal boundary layer thickness decreases

ADM, Adomian decomposition method; DTM, differential transformation method; HAM, homotopy analysis method; HPM, homotopy perturbation method.

1.2.2. Runge–Kutta method

Slip effects on unsteady stagnation point flow of a nanofluid over a stretching sheet has been investigated by Malvandi et al. [48]. They showed that Cu–water nanofluids exhibit a better thermal performance among the other considered nanofluids. Malvandi [49] investigated the unsteady flow of a nanofluid in the stagnation point region of a time-dependent rotating sphere. Ashorynejad et al. [50] studied nanofluid flow and heat transfer due to a stretching cylinder in the presence of magnetic field. They showed that choosing copper (for small values of magnetic parameter) and alumina (for large values of magnetic parameter) leads to the highest cooling performance for this problem. Heated permeable stretching surface in a porous medium was studied by Sheikholeslami and Ganji [51]. Three-dimensional nanofluid flow, heat, and mass transfer in a rotating system have been presented by Sheikholeslami and Ganji [52]. They showed that Nusselt number has direct relationship with Reynolds number while it has reverse relationship with rotation parameter, magnetic parameter.

Sheikholeslami et al. [53] studied the nanofluid flow and heat transfer in a rotating system in the presence of a magnetic field. Sheikholeslami and Ganji [54] studied two-phase modeling of nanofluid in a rotating system with permeable sheet. Unsteady nanofluid flow and heat transfer in the presence of magnetic field considering thermal radiation has been investigated by Sheikholeslami and Ganji [55]. Sheikholeslami et al. [56] studied MHD nanofluid flow and heat transfer considering viscous dissipation. They showed that the magnitude of the skin friction coefficient is an increasing function of the magnetic parameter, rotation parameter, and Reynolds number and it is a decreasing function of the nanoparticle volume fraction. Sheikholeslami et al. [57] studied the effect of thermal radiation on magnetohydrodynamics nanofluid flow and heat transfer by means of two-phase model. Sheikholeslami [58] used KKL model for simulating nanofluid flow and heat transfer in a permeable channel. Effect of uniform suction on nanofluid flow and heat transfer over a cylinder has been studied by Sheikholeslami [59]. Sheikholeslami and Abelman [60] studied two-phase simulation of nanofluid flow and heat transfer in an annulus in the presence of an axial magnetic field. Nanofluid spraying on an inclined rotating disk for cooling process has been investigated by Sheikholeslami et al. [61]. Sheikholeslami et al. [62] investigated nanofluid flow and heat transfer over a stretching porous cylinder considering thermal radiation. They showed that skin friction coefficient increases with increase in Reynolds number and suction parameter but it decreases with increase in nanoparticle volume fraction. Table 1.5 shows the summary of the Runge–Kutta method studies on nanofluid. Chamkha and Aly [63] have studied the boundary layer flow of a nanofluid past a vertical flat plate. They have considered the Brownian motion and the thermophoresis effect. They have transformed the governing equations to a nonsimilar form and used numerical techniques to solve the same. They have reported that the local skin-friction coefficient increased as either of the suction, injection parameter, thermophoresis parameter, Lewis number, or heat generation or absorption parameter increased, while it decreased as either of the buoyancy ratio, Brownian motion parameter, or the magnetic field parameter increased.

Table 1.5

Summary of the Runge–Kutta method studies on nanofluid

Authors	Geometry of study	Method	Type of nanofluid	Properties	Remarks
Malvandi et al. [48]		Runge–Kutta–Fehlberg method	Cu–water, Al₂O₃–water and TiO₂–water nanofluids	$\begin{array}{l} 1 \leq d_{p} \leq 100 nm \\ 0 \leq φ \leq 0.2 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 1 \leq d_{p} \leq 100 nm \\ 0 \leq φ \leq 0.2 \\ \Pr = 6.2 \end{array}$	Cu–water nanofluids exhibits a better thermal performance among the other considered nanofluids. Slip velocity at the walls increases the heat transfer rate.
Malvandi [49]		Runge–Kutta–Fehlberg method	Nonhomogeneous mixtures	$\begin{array}{l} 1 \leq d_{p} \leq 100 nm \\ 0 \leq φ \leq 0.2 \end{array}$ $\begin{array}{l} 1 \leq d_{p} \leq 100 nm \\ 0 \leq φ \leq 0.2 \end{array}$	Increasing the thermophoresis is found to decrease heat transfer and concentration rates. This decrease suppresses for higher thermophoresis number. In addition, it was observed that unlike the heat transfer rate, a rise in Brownian motion leads to an increase in concentration rate
Ashorynejad et al. [50]		Fourth order Runge–Kutta method	Cu–water Ag–water Al₂O₃–water TiO₂–water	$\begin{array}{l} 1 \leq Re \leq 5 \\ 0 \leq M \leq 10 \\ 0 \leq φ \leq 0.2 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 1 \leq Re \leq 5 \\ 0 \leq M \leq 10 \\ 0 \leq φ \leq 0.2 \\ \Pr = 6.2 \end{array}$	Choosing copper (for small values of magnetic parameter) and alumina (for large values of magnetic parameter) leads to the highest cooling performance for this problem
Sheikholeslami and Ganji [51]		Fourth order Runge–Kutta method	Cu–water Ag–water Al₂O₃–water TiO₂–water Cu–EG	$\begin{array}{l} 1 \leq Re \leq 5 \\ - 0.5 \leq γ \leq 1 \\ 0 \leq ɛ \leq 1 \\ 0 \leq φ \leq 0.2 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 1 \leq Re \leq 5 \\ - 0.5 \leq γ \leq 1 \\ 0 \leq ɛ \leq 1 \\ 0 \leq φ \leq 0.2 \\ \Pr = 6.2 \end{array}$	Choosing titanium oxide as the nanoparticle and EG as base fluid proved to have the highest cooling performance for this problem.
Sheikholeslami and Ganji [52]		Fourth order Runge–Kutta method	Al₂O₃–water	$\begin{array}{l} 1 \leq R \leq 10 \\ 0 \leq M \leq 12 \\ - 0.5 \leq γ \leq 1 \\ Kr = 0.5, Nt = 0.1 \\ Nb = 0.1, Sc = 0.1 \\ \Pr = 10 \end{array}$ $\begin{array}{l} 1 \leq R \leq 10 \\ 0 \leq M \leq 12 \\ - 0.5 \leq γ \leq 1 \\ Kr = 0.5, Nt = 0.1 \\ Nb = 0.1, Sc = 0.1 \\ \Pr = 10 \end{array}$	Nusselt number has direct relationship with Reynolds number while it has reverse relationship with rotation parameter, magnetic parameter, Schmidt number, thermophoretic parameter, and Brownian parameter
Sheikholeslami et al. [53]		Fourth order Runge–Kutta method	Cu–water Ag–water Al₂O₃–water TiO₂–water	$\begin{array}{l} 5 \leq R \leq 20 \\ 0 \leq M \leq 50 \\ 0.5 \leq Kr \leq 6 \\ 0 \leq φ \leq 0.2 \\ - 0.5 \leq λ \leq 1 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 5 \leq R \leq 20 \\ 0 \leq M \leq 50 \\ 0.5 \leq Kr \leq 6 \\ 0 \leq φ \leq 0.2 \\ - 0.5 \leq λ \leq 1 \\ \Pr = 6.2 \end{array}$	The highest values are obtained when titanium oxide is used as nanoparticle. Also it can be found that Nusselt number decreases with increase of magnetic parameter due to presence of Lorentz forces.
Sheikholeslami and Ganji [54]		Fourth order Runge–Kutta method	Al₂O₃–water	$\begin{array}{l} 1 \leq R \leq 10 \\ 0.5 \leq Kr \leq 6 \\ 0 \leq λ \leq 3 \\ Nt = Nb = 0.1 \\ \Pr = 10 \end{array}$ $\begin{array}{l} 1 \leq R \leq 10 \\ 0.5 \leq Kr \leq 6 \\ 0 \leq λ \leq 3 \\ Nt = Nb = 0.1 \\ \Pr = 10 \end{array}$	Nusselt number has direct relationship with Reynolds number and injection parameter while it has reverse relationship with rotation parameter, Schmidt number, thermophoretic parameter, and Brownian parameter
Sheikholeslami and Ganji [55]		Fourth order Runge–Kutta method	Al₂O₃–water	$\begin{array}{l} 0.5 \leq S \leq 12 \\ 0.01 \leq Ec \leq 0.4 \\ 0.01 \leq Nt \leq 0.4 \\ 0.5 \leq Sc \leq 20 \\ 0 \leq Rd \leq 16 \\ \Pr = 10 \end{array}$ $\begin{array}{l} 0.5 \leq S \leq 12 \\ 0.01 \leq Ec \leq 0.4 \\ 0.01 \leq Nt \leq 0.4 \\ 0.5 \leq Sc \leq 20 \\ 0 \leq Rd \leq 16 \\ \Pr = 10 \end{array}$	Concentration boundary layer thickness increases with increase of radiation parameter
Sheikholeslami et al. [56]		Fourth order Runge–Kutta method	CuO–water, Al₂O₃–water	$\begin{array}{l} 0.1 \leq R \leq 2 \\ 0 \leq M \leq 9 \\ 0 \leq φ \leq 0.04 \\ 1 \leq Kr \leq 6 \\ 0.01 \leq Ec \leq 0.04 \\ \Pr = 10 \end{array}$ $\begin{array}{l} 0.1 \leq R \leq 2 \\ 0 \leq M \leq 9 \\ 0 \leq φ \leq 0.04 \\ 1 \leq Kr \leq 6 \\ 0.01 \leq Ec \leq 0.04 \\ \Pr = 10 \end{array}$	The magnitude of the skin friction coefficient is an increasing function of the magnetic parameter, rotation parameter, and Reynolds number and it is a decreasing function of the nanoparticle volume fraction
Sheikholeslami et al. [57]		Fourth order Runge–Kutta method	Al₂O₃–water	$\begin{array}{l} 0.1 \leq R \leq 20 \\ 0 \leq Rd \leq 12 \\ 0.01 \leq Sc \leq 6 \\ 0 \leq M \leq 16 \\ 0 \leq φ \leq 0.04 \\ 0.1 \leq Kr \leq 12 \\ \Pr = 10 \end{array}$ $\begin{array}{l} 0.1 \leq R \leq 20 \\ 0 \leq Rd \leq 12 \\ 0.01 \leq Sc \leq 6 \\ 0 \leq M \leq 16 \\ 0 \leq φ \leq 0.04 \\ 0.1 \leq Kr \leq 12 \\ \Pr = 10 \end{array}$	Nusselt number has direct relationship with radiation parameter and Reynolds number while it has reverse relationship with other active parameters. Also it can be found that concentration boundary layer thickness decreases with the increase of radiation parameter
Sheikholeslami [58]		Fourth order Runge–Kutta method	Al₂O₃–water	$\begin{array}{l} - 1 \leq α \leq 2 \\ 1 \leq R \leq 4 \\ 0 \leq φ \leq 0.04 \\ 0 \leq m \leq 3 \\ \Pr = 6.8 \end{array}$ $\begin{array}{l} - 1 \leq α \leq 2 \\ 1 \leq R \leq 4 \\ 0 \leq φ \leq 0.04 \\ 0 \leq m \leq 3 \\ \Pr = 6.8 \end{array}$	Heat transfer enhancement has direct relationship with Reynolds number when power law index is equal to zero but opposite trend is observed for other values of power law index.
Sheikholeslami [59]		Fourth order Runge–Kutta method	CuO–water, Al₂O₃–water	$\begin{array}{l} 0.1 \leq Re \leq 2.7 \\ 0 \leq γ \leq 2 \\ 0 \leq φ \leq 0.04 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 0.1 \leq Re \leq 2.7 \\ 0 \leq γ \leq 2 \\ 0 \leq φ \leq 0.04 \\ \Pr = 6.2 \end{array}$	Adding nanoparticle into the base fluid of this problem is capable of changing the flow pattern. It is found that Nusselt number is an increasing function of nanoparticle volume fraction, suction parameter, Reynolds number
Sheikholeslami and Abelman [60]		Fourth order Runge–Kutta method	Al₂O₃–water	$\begin{array}{l} 1 \leq R \leq 4 \\ 1 \leq M \leq 4 \\ 0.01 \leq Ec \leq 0.1 \\ 0.01 \leq Nt \leq 0.4 \\ 0.01 \leq Nt \leq 0.6 \\ 0.001 \leq Ec \leq 0.02 \\ \Pr = 10 \end{array}$ $\begin{array}{l} 1 \leq R \leq 4 \\ 1 \leq M \leq 4 \\ 0.01 \leq Ec \leq 0.1 \\ 0.01 \leq Nt \leq 0.4 \\ 0.01 \leq Nt \leq 0.6 \\ 0.001 \leq Ec \leq 0.02 \\ \Pr = 10 \end{array}$	Nusselt number has a direct relationship with the aspect ratio and Hartmann number but it has a reverse relationship with the Reynolds number, Schmidt number, Brownian parameter, thermophoresis parameter, and Eckert number.
Sheikholeslami et al. [61]		Fourth order Runge–Kutta method	Al₂O₃–water	$\begin{array}{l} 0.2 \leq δ \leq 1 \\ 0.1 \leq N t = N b \leq 0.4 \\ 0.1 \leq S c \leq 8 \\ \Pr = 10 \end{array}$ $\begin{array}{l} 0.2 \leq δ \leq 1 \\ 0.1 \leq N t = N b \leq 0.4 \\ 0.1 \leq S c \leq 8 \\ \Pr = 10 \end{array}$	Nusselt number is an increasing function of each active parameter. Latent heat has a direct relationship with Schmidt number, Brownian parameter, and thermophoretic parameter but it has reverse relationship with normalized thickness.
Sheikholeslami et al. [62]		Fourth order Runge–Kutta method	Cu–water	$\begin{array}{l} 0.5 \leq γ \leq 2 \\ 0.5 \leq R d \leq 2 \\ 0 \leq φ \leq 0.04 \\ \Pr = 6.8 \end{array}$ $\begin{array}{l} 0.5 \leq γ \leq 2 \\ 0.5 \leq R d \leq 2 \\ 0 \leq φ \leq 0.04 \\ \Pr = 6.8 \end{array}$	Skin friction coefficient increases with increase in Reynolds number and suction parameter but it decreases with increase in nanoparticle volume fraction

1.2.3. Finite difference method

Chamkha and Rashad [64] have studied the flow of a nanofluid around a nonisothermal wedge. They have considered the Brownian movement and the thermophoresis effects. They have concluded that the local skin-friction coefficient, local Nusselt number, and the local Sherwood number reduced as either of the magnetic parameter or the pressure gradient parameter was increased. The presence of the Brownian motion and the thermophoresis effects caused the local Nusselt number to decrease and the Sherwood number to increase. Sheremet and Pop [65] used Buongiorno’s mathematical model for conjugate natural convection in a square porous cavity filled with nanofluid. They showed that high thermophoresis parameter, low Brownian motion parameter, low Lewis and Rayleigh numbers, and high thermal conductivity ratio reflect essential nonhomogeneous distribution of the nanoparticles inside the porous cavity. Sheremet et al. [66] studied the three-dimensional natural convection in a porous enclosure filled with a nanofluid using Buongiorno’s mathematical model. Sheremet et al. [67] investigated the effect of thermal stratification on free convection in a square porous cavity filled with a nanofluid using Tiwari and Das’ nanofluid model.

Ghalambaz et al. [68] studied the free convection heat transfer in a porous cavity filled with a nanofluid using Tiwari and Das’ nanofluid model. Double-diffusive mixed convection in a porous open cavity filled with a nanofluid using Buongiorno’s model has been studied by Sheremet et al. [69]. Sheremet and Pop [70] studied nanofluid free convection in a triangular porous cavity porous. Natural convection in a horizontal cylindrical annulus filled with a porous medium saturated by a nanofluid has been investigated by Sheremet and Pop [71]. Magnetic field effect on the unsteady natural convection in a wavy-walled cavity filled with a nanofluid has been studied by Sheremet et al. [72]. Khan et al. [73] studied the three-dimensional flow of nanofluid induced by an exponentially stretching sheet. They showed the existence of interesting Sparrow–Gregg-type hills for temperature distribution corresponding to some range of parametric values. Nanofluid flow with multimedia physical features for conjugate mixed convection and radiation has been studied by Hsiao [74]. Table 1.6 shows the summary of the finite difference method studies on nanofluid.

Table 1.6

Summary of the finite difference method studies on nanofluid

Authors	Geometry of study	Method	Type of nanofluid	Properties	Remarks
Sheremet and Pop [65]		Finite difference method	Porous cavity with water-based nanofluid	Ra = 100–500; Le = 1–10; Nb = 0.1–0.4; Nt = 0.1–0.4; Nr = 0.1–0.4; Kr = 0.1–10.0; D = 0.1–0.3.	High thermophoresis parameter, low Brownian motion parameter, low Lewis and Rayleigh numbers, and high thermal conductivity ratio reflect essential nonhomogeneous distribution of the nanoparticles inside the porous cavity.
Sheremet et al. [66]		Finite difference method	Porous cavity with water-based nanofluid	Ra = 30–500, Le = 1–10, Nr = 0.1–0.4, Nb = 0.1–0.4, Nt = 0.1–0.4, aspect ratio (A = 0.2–5.0).	Average Nusselt number at the hot surface is an increasing function of the Rayleigh number and Brownian motion parameter, and a decreasing function of the Lewis number, buoyancy-ratio and thermophoresis parameters. The average Sherwood number at the hot surface is an increasing function of the Rayleigh number and thermophoresis parameter, and a decreasing function of the Lewis number, buoyancy-ratio, and Brownian motion parameters.
Sheremet et al. [67]		Finite difference method	Cu–water	Ra = 10–500, ϕ = 0.0–0.05, porosity of porous medium (ɛ = 0.1–0.8), thermal stratification parameter (b = 0–1.0) solid matrix of the porous medium (aluminum foam and glass balls)	An increase in the thermal stratification parameter leads to an essential reduction of the convective core sizes, displacement of this core close to the lower left corner and rotation of this vortex along the horizontal axis. More intensive attenuation of convective flow with ɛ and b occurs for low thermal conductivity material for the solid matrix of the porous medium.
Ghalambaz et al. [68]		Finite difference method	Cu–water	−60° < α < 60°, aspect ratio 0.1 < A < 10, 10 < Ra <1000, 0 < ϕ < 0.1, ɛ = 0.3, 0.5, 0.7, k_s = 2k_bf	Presence of nanoparticles deteriorates the heat transfer in all studied cases. The decrease of the porosity increases the porous matrix thermal conductivity while the decrease of the inclination angle and of the aspect ratio would boost the deterioration of heat transfer.
Sheremet et al. [69]		Finite difference method	Porous cavity with water-based nanofluid	Re = 10–100, Pr = 0.2, Ra = 50–500, usual Lewis number (Le = 1–50), Dufour-contaminant Lewis number (Ld = 1–50), nanofluid Lewis number (Ln = 1–50), Nr = 0.1–0.4, regular double-diffusive buoyancy ratio (Nc = 0.1–0.4), Nb = 0.1–0.4, Nt = 0.1–0.4, modified Dufour parameter (Nd = 0.1–0.4)	Average Nusselt number at hot vertical wall is an increasing function of the Rayleigh and Reynolds numbers, and a decreasing function of the usual Lewis number. While the average Sherwood number at this vertical wall is an increasing function of the usual Lewis. Effects of the Rayleigh and Reynolds numbers on $\|\bar{S h}\|$ $\|\bar{S h}\|$ and the thermophoresis parameter on the average Nusselt and Sherwood numbers are nonmonotonic. It has been shown that in the present porous problem the Richardson number does not define the prevailing of the forced or natural convection modes.
Sheremet and Pop [70]		Finite difference method	Porous cavity with water-based nanofluid	Ra = 100–500; Le = 1–10; Nb = 0.1–0.4; Nt = 0.1–0.4; Nr = 0.1–0.4; A = H/L = 1.0.	Average Nusselt number is an increasing function of Ra, Le, and a decreasing function of Nr, Nb, Nt. At the same time the average Sherwood number is an increasing function of Ra, Le, Nb, Nt and a decreasing function of Nr.
Sheremet and Pop [71]		Finite difference method	Cu–water	Ra = 200–700, ɛ = 0.1–0.8, ϕ = 0.0–0.05, the annulus radius ratio (R = 1.5, 2.0, 2.5), solid matrix of porous medium (aluminum foam and glass balls)	Average Nusselt number is an increasing function of the Rayleigh number and annulus radius ratio and a decreasing function of the porosity of porous medium regardless of the solid matrix material. A decrease in the thermal conductivity of the solid matrix material leads to an attenuation of the convective heat transfer inside the annulus for high values of the porosity and solid volume fraction of nanoparticles.
Sheremet et al. [72]		Finite difference method	Water-based nanofluid	Ra = 10⁵, Le = 10, Pr = 6.26, Nr = 0.1, Nb = 0.1, Nt = 0.1, A = 1, Ha = 0–100, undulation number (κ = 1–3), wavy contraction ratio (b = 0.1–0.3), ϕ = 0–π, τ = 0–0.13.	An increase in the Hartmann number leads to an attenuation of convective flow and heat transfer and a formation of a double-core convective cell for high values of Ha. An increase in the wavy contraction ratio leads to an increase in the wave amplitude and an attenuation of the convective flow with more intensive heating of the wavy troughs.
Khan et al. [73]		Finite difference scheme known as Keller-box method	Water-based nanofluid	$\begin{array}{l} 0 \leq λ \leq 1 \\ 0.1 \leq N b \leq 0.9 \\ 0.3 \leq N t \leq 0.5 \\ 5 \leq \Pr \leq 10 \\ - 1 \leq A \leq 3 \end{array}$ $\begin{array}{l} 0 \leq λ \leq 1 \\ 0.1 \leq N b \leq 0.9 \\ 0.3 \leq N t \leq 0.5 \\ 5 \leq \Pr \leq 10 \\ - 1 \leq A \leq 3 \end{array}$	The existence of interesting Sparrow–Gregg-type hills for temperature distribution corresponding to some range of parametric values
Kai-Long Hsiao [74]		An improved finite-difference with Box method	Cu–water	K = 400 (W/mK) $ρ = 8940 (k g / m^{3})$ $ρ = 8940 (k g / m^{3})$ $C_{p} = 385 (J / k g K)$ $C_{p} = 385 (J / k g K)$ $\begin{array}{l} 10^{1} \leq Re \leq 10^{6} \\ 0 \leq \Pr \leq 1000 \end{array}$ $\begin{array}{l} 10^{1} \leq Re \leq 10^{6} \\ 0 \leq \Pr \leq 1000 \end{array}$	The results show that dimensionless heat transfer effects increase with increasing values of Pr, R0, Nb, or G but decrease with increasing M, Nt, or Ec. Those parameters Pr, R0, Nb, or G are important factors in this study for increasing the heat transfer effects. The fourth, for mass transfer that the values Sc and Nb can obtain a good mass diffusion effect, but the parameter Nt has not processed this kind of function.

1.2.4. Finite volume method

Garoosi and Hoseininejad [75] investigated the natural and mixed convection heat transfer between differentially heated cylinders in an adiabatic enclosure filled with nanofluid. Garoosi et al. [76] applied Buongiorno model for mixed convection of the nanofluid in heat exchangers. Two-phase mixture modeling of mixed convection of nanofluids in a square cavity with internal and external heating has been studied by Garoosi et al. [77]. Teamah and El-Maghlany [78] studied the augmentation of natural convective heat transfer in square cavity by utilizing nanofluids in the presence of magnetic field. They showed that weak magnetic field; the addition of nanoparticles is necessary to enhance the heat transfer but for strong magnetic field there is no need for nanoparticles because the heat transfer will decrease. Santra et al. [79] studied the heat transfer augmentation in a differentially heated square cavity using copper–water nanofluid. Das and Ohal [80] investigated natural convection heat transfer augmentation in a partially heated and partially cooled square cavity utilizing nanofluids. Oztop et al. [81] analyzed the nonisothermal temperature distribution on natural convection in nanofluid filled enclosures. They showed that an enhancement in heat transfer rate was registered for the whole range of Rayleigh numbers. Table 1.7 shows the summary of the finite volume method studies on nanofluid.

Table 1.7

Summary of the Finite volume method studies on nanofluid

Authors	Geometry of study	Method	Type of nanofluid	Properties	Remarks
Garoosi and Hoseininejad [75]		Finite volume method	Cu–water TiO₂–water Al₂O₃–water	$\begin{array}{l} 10^{4} \leq Ra \leq 10^{7} \\ 0 \leq φ \leq 0.05 \\ 25 \leq d_{p} \leq 145 \\ \Pr = 6.8 \end{array}$ $\begin{array}{l} 10^{4} \leq Ra \leq 10^{7} \\ 0 \leq φ \leq 0.05 \\ 25 \leq d_{p} \leq 145 \\ \Pr = 6.8 \end{array}$	Changing the location of the heat source/sink from bottom–top to top–bottom configuration decreases the heat transfer rate significantly.
Garoosi et al. [76]		Finite volume method	Al₂O₃–water	$\begin{array}{l} 10^{4} \leq Ra \leq 10^{7} \\ 0.01 \leq Ri \leq 1000 \\ 10^{2} \leq Gr \leq 10^{4} \\ 0 \leq φ \leq 0.05 \\ 25 \leq d_{p} \leq 145 \end{array}$ $\begin{array}{l} 10^{4} \leq Ra \leq 10^{7} \\ 0.01 \leq Ri \leq 1000 \\ 10^{2} \leq Gr \leq 10^{4} \\ 0 \leq φ \leq 0.05 \\ 25 \leq d_{p} \leq 145 \end{array}$	At low Rayleigh numbers and high Richardson numbers, the particle distribution is fairly nonuniform while at high Ra and low Ri values particle distribution remains almost uniform for free and mixed convection cases, respectively
Garoosi et al. [77]		Finite volume method	Cu–water TiO₂–water Al₂O₃–water	$\begin{array}{l} 10^{4} \leq Ra \leq 10^{7} \\ 0.01 \leq Ri \leq 1000 \\ 0 \leq φ \leq 0.05 \\ 25 \leq d_{p} \leq 145 \end{array}$ $\begin{array}{l} 10^{4} \leq Ra \leq 10^{7} \\ 0.01 \leq Ri \leq 1000 \\ 0 \leq φ \leq 0.05 \\ 25 \leq d_{p} \leq 145 \end{array}$	Thermophoretic effects are negligible for nanoparticles with high thermal conductivity. As a result, in such conditions use of homogeneous and single-phase models is valid at any Ra and Ri
Teamah et al. [78]		Finite volume method	Water-based nanofluid	$\begin{array}{l} 10^{4} \leq Ra \leq 10^{7} \\ 0 \leq φ \leq 0.05 \\ 0 \leq Ha \leq 60 \end{array}$ $\begin{array}{l} 10^{4} \leq Ra \leq 10^{7} \\ 0 \leq φ \leq 0.05 \\ 0 \leq Ha \leq 60 \end{array}$	Weak magnetic field; the addition of nanoparticles is necessary to enhance the heat transfer but for strong magnetic field there is no need for nanoparticles because the heat transfer will decrease.
Santra et al.[79]		Finite volume method	Cu–water	$\begin{array}{l} 10^{4} \leq Ra \leq 10^{7} \\ 0 \leq φ \leq 0.05 \\ A = 1 \end{array}$ $\begin{array}{l} 10^{4} \leq Ra \leq 10^{7} \\ 0 \leq φ \leq 0.05 \\ A = 1 \end{array}$	The heat transfer rate decreases with increase in solid volume fraction for a particular Rayleigh number. However, it increases with Rayleigh number for a particular solid volume fraction.
Das and Ohal [80]		Finite volume method	Cu–water	$\begin{array}{l} 10^{4} \leq Gr \leq 10^{7} \\ 0 \leq φ \leq 0.02 \\ A = 1 \end{array}$ $\begin{array}{l} 10^{4} \leq Gr \leq 10^{7} \\ 0 \leq φ \leq 0.02 \\ A = 1 \end{array}$	The rate of heat transfer increases with an increase in the nanoparticles volume fraction
Oztop et al. [81]		Finite volume method	Cu–water TiO₂–water Al₂O₃–water	$\begin{array}{l} 10^{4} \leq Gr \leq 10^{7} \\ 0 \leq φ \leq 0.01 \\ 0 \leq ϕ \leq 90^{^} \\ A = 1 \end{array}$ $\begin{array}{l} 10^{4} \leq Gr \leq 10^{7} \\ 0 \leq φ \leq 0.01 \\ 0 \leq ϕ \leq 90^{^} \\ A = 1 \end{array}$	An enhancement in heat transfer rate was registered for the whole range of Rayleigh numbers

1.2.5. Finite element method

MHD mixed convection of nanofluid filled partially heated triangular enclosure with a rotating adiabatic has been investigated by Selimefendigil and Oztop [82]. They showed that local and average heat transfer and total entropy generation enhance as the solid volume fraction of nanoparticle and angular rotational speed of the cylinder increases and Hartmann number decreases. Heat transfer enhancements around 30% are achieved for the highest volume fraction compared to base fluid. Selimefendigil and Oztop [83] studied the natural convection and entropy generation of nanofluid filled cavity having different shaped obstacles under the influence of magnetic field and internal heat generation. Selimefendigil and Oztop [84] studied pulsating nanofluids jet impingement cooling of a heated horizontal surface. They showed that the combined effect of pulsation and inclusion of nanoparticles is not favorable for the stagnation point heat transfer enhancement for some combinations of Reynolds number and nanoparticle volume fraction. Selimefendigil and Oztop [85] studied MHD mixed convection in a nanofluid filled lid driven square enclosure with a rotating cylinder. Selimefendigil and Oztop [86] investigated numerical investigation and reduced order model of mixed convection at a backward facing step with a rotating cylinder subjected to nanofluid. Effect of nanoparticle shape on mixed convection due to rotating cylinder in an internally heated and flexible walled cavity filled with SiO₂–water nanofluids has been investigated by Selimefendigil and Öztop [87]. They indicated that Nusselt number enhances with external Rayleigh number and nanoparticle volume fraction while the opposite behavior is seen as the value of internal Rayleigh number and flexibility of the wall increases. Conjugate natural convection in a cavity with a conductive partition and filled with different nanofluids on different sides of the partition has been studied by Selimefendigil and Oztop [88]. They proved that as the values of the Grashof number, thermal conductivity ratio (Kr), and nanoparticle volume fraction increase, average Nusselt number increases. Table 1.8 shows the summary of the finite element method studies on nanofluid.

Table 1.8

Summary of the finite element method studies on nanofluid

Authors	Geometry of study	Method	Type of nanofluid	Properties	Remarks
Selimefendigil and Oztop [82]		Finite element method	Cu–water	$\begin{array}{l} 10^{4} \leq Gr \leq 10^{6} \\ 0 \leq φ \leq 0.05 \\ 0 \leq Ha \leq 50 \\ - 20 \leq ω \leq 20 \\ \Pr = 7.1 \end{array}$ $\begin{array}{l} 10^{4} \leq Gr \leq 10^{6} \\ 0 \leq φ \leq 0.05 \\ 0 \leq Ha \leq 50 \\ - 20 \leq ω \leq 20 \\ \Pr = 7.1 \end{array}$	Local and average heat transfer and total entropy generation enhance as the solid volume fraction of nanoparticle and angular rotational speed of the cylinder increases and Hartmann number decreases. Heat transfer enhancements around 30% are achieved for the highest volume fraction compared to base fluid.
Selimefendigil and Oztop [83]		Finite element method	Cu–water	$\begin{array}{l} 10^{4} \leq R a_{E} \leq 10^{6} \\ 10^{4} \leq R a_{I} \leq 10^{6} \\ 0 \leq φ \leq 0.05 \\ 0 \leq Ha \leq 50 \\ \Pr = 6.9 \end{array}$ $\begin{array}{l} 10^{4} \leq R a_{E} \leq 10^{6} \\ 10^{4} \leq R a_{I} \leq 10^{6} \\ 0 \leq φ \leq 0.05 \\ 0 \leq Ha \leq 50 \\ \Pr = 6.9 \end{array}$	As the value of the external Rayleigh number decreases, internal Rayleigh number and Hartmann number increase, average heat transfer enhance. Average Nusselt number enhances by about 40%–60% at the highest solid volume fraction when compared to base fluid with different obstacles installed within the cavity.
Selimefendigil and Oztop [84]		Finite volume method	Al₂O₃–water	$\begin{array}{l} 100 \leq Re \leq 400 \\ 1 \leq f \leq 10 \\ 0 \leq φ \leq 0.06 \\ \Pr = 7.1 \end{array}$ $\begin{array}{l} 100 \leq Re \leq 400 \\ 1 \leq f \leq 10 \\ 0 \leq φ \leq 0.06 \\ \Pr = 7.1 \end{array}$	The combined effect of pulsation and inclusion of nanoparticles is not favorable for the stagnation point heat transfer enhancement for some combinations of Reynolds number and nanoparticle volume fraction.
Selimefendigil and Oztop [85]		Finite element method	Cu–water	$\begin{array}{l} 0.001 \leq Ri \leq 10 \\ 0 \leq Ha \leq 50 \\ 0 \leq φ \leq 0.05 \\ - 10 \leq Ω \leq 10 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 0.001 \leq Ri \leq 10 \\ 0 \leq Ha \leq 50 \\ 0 \leq φ \leq 0.05 \\ - 10 \leq Ω \leq 10 \\ \Pr = 6.2 \end{array}$	Average heat transfer enhances with Richardson number, nanoparticle volume fraction, and cylinder rotation while magnetic field reduces the convection. The combined effect of magnetic field and cylinder rotation acts in a way to enhance local heat transfer for some locations along the heated wall.
Selimefendigil and Oztop [86]		Finite element method	Al₂O₃–water	$\begin{array}{l} 50 \leq Re \leq 200 \\ Gr = 10^{4} \\ 0 \leq φ \leq 0.05 \\ - 4 \leq Ω \leq 4 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 50 \leq Re \leq 200 \\ Gr = 10^{4} \\ 0 \leq φ \leq 0.05 \\ - 4 \leq Ω \leq 4 \\ \Pr = 6.2 \end{array}$	Cylinder rotation affects the flow and thermal patterns behind the step. The addition of the nanoparticles enhances the averaged heat transfer along the bottom wall downstream of the step. There is almost a linear relation between heat transfer enhancement and nanoparticle volume fraction.
Selimefendigil and Öztop [87]		Finite element method	SiO₂–water	$\begin{array}{l} 10^{3} \leq R a_{E} \leq 5 \times 10^{5} \\ 10^{4} \leq R a_{I} \leq 10^{6} \\ 5 \times 10^{2} \leq E \leq 10^{6} \\ 0 \leq φ \leq 0.05 \\ - 2000 \leq Ω \leq 2000 \end{array}$ $\begin{array}{l} 10^{3} \leq R a_{E} \leq 5 \times 10^{5} \\ 10^{4} \leq R a_{I} \leq 10^{6} \\ 5 \times 10^{2} \leq E \leq 10^{6} \\ 0 \leq φ \leq 0.05 \\ - 2000 \leq Ω \leq 2000 \end{array}$	Average Nussetl number enhances with external Rayleigh number and nanoparticle volume fraction while the opposite behavior is seen as the value of internal Rayleigh number and flexibility of the wall increases. Among different nanoparticle types cylindrical ones show the best performance in terms of heat transfer enhancement.
Selimefendigil and Oztop [88]		Finite element method	Al₂O₃–water and CuO–water	$\begin{array}{l} 10^{3} \leq G r \leq 10^{6} \\ 0^{o} \leq θ \leq 270^{o} \\ 0.01 \leq K r \leq 10 \\ 0 \leq φ \leq 0.04 \end{array}$ $\begin{array}{l} 10^{3} \leq G r \leq 10^{6} \\ 0^{o} \leq θ \leq 270^{o} \\ 0.01 \leq K r \leq 10 \\ 0 \leq φ \leq 0.04 \end{array}$	As the value of the Grashof number, thermal conductivity ratio (Kr) and nanoparticle volume fraction increase, average Nusselt number increase. When nanoparticles with low thermal conductivity on the right cavity are added it is more effective for the heat transfer enhancement compared to adding nanoparticles with high thermal conductivity.

1.2.6. Control volume–based finite element method

Heatline analysis has been used by Sheikholeslami et al. [89] to investigate two-phase simulation of nanofluid flow and heat transfer. They found that Nusselt number decreases as buoyancy ratio number increases until it reaches a minimum value and then starts increasing. As Lewis number increases, this minimum value occurs at higher buoyancy ratio number. Natural convection heat transfer in a cavity with sinusoidal wall filled with CuO–water nanofluid in the presence of magnetic field has been studied by Sheikholeslami et al. [90]. Effects of a magnetic field on natural convection in different enclosures filled with naofluids have been examined by Sheikholeslami et al. [91–93]. Soleimani et al. [94] studied the natural convection heat transfer in a nanofluid filled semiannulus enclosure. They found that there is an optimum angle of turn in which the average Nusselt number is maximum for each Rayleigh number. Moreover, the angle of turn has an important effect on the streamlines, isotherms, and maximum or minimum values of local Nusselt number. Effects of MHD on Cu–water nanofluid flow and heat transfer have been studied by Sheikholeslami et al. [95]. Constant temperature and heat flux boundary condition for Al₂O₃–water nanofluid filled enclosure have been examined by Sheikholeslami et al. [96–99]. Sheikholeslami et al. [100] studied free convection heat transfer in a nanofluid filled inclined L-shaped enclosure. Ferrohydrodynamic and magnetohydrodynamic effects on ferrofluid flow and convective heat transfer have been investigated by Sheikholeslami and Ganji [101]. They found that Nusselt number increases with augment of Rayleigh number and nanoparticle volume fraction but it decreases with increase in Hartmann number. Magnetic number has different effect on Nusselt number corresponding to Rayleigh number.

Sheikholeslami et al. [102] considered the effect of thermal radiation on ferrofluid flow and heat transfer in a semiannulus enclosure in the presence of magnetic source. Sheikholeslami Kandelousi [103] studied the effect of spatially variable magnetic field on ferrofluid flow and heat transfer considering constant heat flux boundary condition. Free convection heat transfer in a nanofluid filled enclosure with elliptic inner cylinder has been presented by Sheikholeslami et al. [104]. Sheikholeslami and Rashidi [105] studied the effect of space-dependent magnetic field on free convection of Fe₃O₄–water nanofluid. Effect of nonuniform magnetic field on forced convection heat transfer of nanofluid has been studied by Sheikholeslami et al. [106]. Electrohydrodynamic nanofluid hydrothermal treatment in an enclosure with sinusoidal upper wall has been investigated by Sheikholeslami and Ellahi [107]. Sheikholeslami et al. [108] utilized two phase model for nanofluid flow and heat transfer in existence of magnetic field. Sheikholeslami et al. [109] investigated forced convection heat transfer in a semiannulus under the influence of a variable magnetic field. Sheikholeslami et al. [110] studied effect of electric field on hydrothermal behavior of nanofluid in a complex geometry. They found that effect of electric filed on heat transfer is more pronounced at low Reynolds number. Sheikholeslami and Rashidi [111] investigated nonuniform magnetic field effect on nanofluid hydrothermal treatment considering Brownian motion and thermophoresis effects. Sheikholeslami and Rashidi [112] studied ferrofluid heat transfer treatment in the presence of variable magnetic field. They found that Nusselt number has direct relationship with Richardson number, nanoparticle volume fraction while it has reverse relationship with Hartmann number and magnetic number. Table 1.9 shows the summary of the control volume–based finite element method studies on nanofluid.

Table 1.9

Summary of the control volume–based finite element method studies on nanofluid

Authors	Geometry of study	Method	Type of nanofluid	Properties	Remarks
Sheikholeslami et al. [89]		Control volume–based finite element method	Al₂O₃–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0.1 \leq N r \leq 4 \\ 1 \leq L e \leq 8 \\ N b = N t = 0.5 \\ \Pr = 10 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0.1 \leq N r \leq 4 \\ 1 \leq L e \leq 8 \\ N b = N t = 0.5 \\ \Pr = 10 \end{array}$	Average Nusselt number decreases as buoyancy ratio number increases until it reaches a minimum value and then starts increasing. As Lewis number increases, this minimum value occurs at higher buoyancy ratio number.
Sheikholeslami et al. [90]		Control volume– based finite element method	CuO–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0.1 \leq a \leq 0.3 \\ 0 \leq H a \leq 100 \\ 0 \leq φ \leq 0.04 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0.1 \leq a \leq 0.3 \\ 0 \leq H a \leq 100 \\ 0 \leq φ \leq 0.04 \\ \Pr = 6.2 \end{array}$	Nusselt number is an increasing function of nanoparticles volume fraction, dimensionless amplitude of the sinusoidal wall, and Rayleigh number while it is a decreasing function of Hartmann number
Sheikholeslami et al. [91]		Control volume– based finite element method	Cu–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0^{^} \leq λ \leq 90^{^} \\ 0 \leq φ \leq 0.06 \\ 0 \leq H a \leq 100 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0^{^} \leq λ \leq 90^{^} \\ 0 \leq φ \leq 0.06 \\ 0 \leq H a \leq 100 \\ \Pr = 6.2 \end{array}$	Hartmann number and the inclination angle of the enclosure can be control parameters at different Rayleigh numbers. In the presence of magnetic field velocity field retarded and hence convection and Nusselt number decreases
Sheikholeslami et al. [92]		Control volume–based finite element method	Cu–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0^{^} \leq γ \leq 90^{^} \\ 0 \leq H a \leq 100 \\ 0 \leq φ \leq 0.06 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0^{^} \leq γ \leq 90^{^} \\ 0 \leq H a \leq 100 \\ 0 \leq φ \leq 0.06 \\ \Pr = 6.2 \end{array}$	At Ra = 10³, maximum value of enhancement for low Hartmann number is obtained at γ = 0°, but for higher values of Hartmann number, maximum values of E occur at γ = 90°
Sheikholeslami et al. [93]		Control volume– based finite element method	Cu–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 3 \leq N \leq 6 \\ 0 \leq H a \leq 100 \\ 0 \leq φ \leq 0.06 \\ A = 0.5, \Pr = 6.2 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 3 \leq N \leq 6 \\ 0 \leq H a \leq 100 \\ 0 \leq φ \leq 0.06 \\ A = 0.5, \Pr = 6.2 \end{array}$	Nusselt number is an increasing function of nanoparticle volume fraction, the number of undulations, and Rayleigh numbers while it is a decreasing function of Hartmann number.
Soleimani et al. [94]		Control volume– based finite element method	Cu–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 45^{^} \leq γ \leq 180^{^} \\ 0 \leq φ \leq 0.06 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 45^{^} \leq γ \leq 180^{^} \\ 0 \leq φ \leq 0.06 \\ \Pr = 6.2 \end{array}$	There is an optimum angle of turn in which the average Nusselt number is maximum for each Rayleigh number. Moreover, the angle of turn has an important effect on the streamlines, isotherms, and maximum or minimum values of local Nusselt number
Sheikholeslami et al. [95]		Control volume– based finite element method	Cu–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0^{^} \leq γ \leq 90^{^} \\ 0 \leq φ \leq 0.06 \\ ɛ = 0.9, a = 0.8 L \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0^{^} \leq γ \leq 90^{^} \\ 0 \leq φ \leq 0.06 \\ ɛ = 0.9, a = 0.8 L \\ \Pr = 6.2 \end{array}$	Increasing Rayleigh number leads to decrease in heat transfer enhancement while opposite trend is observed with augment of Hartmann number
Sheikholeslami et al. [96]		Control volume– based finite element method	Al₂O₃–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0^{^} \leq H a \leq 100 \\ 0 \leq φ \leq 0.04 \\ 0.2 \leq r_{i n} / L \leq 0.4 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0^{^} \leq H a \leq 100 \\ 0 \leq φ \leq 0.04 \\ 0.2 \leq r_{i n} / L \leq 0.4 \\ \Pr = 6.2 \end{array}$	Domination of conduction mechanism causes heat transfer enhancement to increase. So enhancement in heat transfer increases with increase in Hartmann number and aspect ratio while it decreases with augment of Rayleigh number
Sheikholeslami et al. [97]		Control volume– based finite element method	Al₂O₃–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ - 90^{^} \leq γ \leq 0^{^} \\ 0 \leq φ \leq 0.04 \\ 0 \leq H a \leq 100 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ - 90^{^} \leq γ \leq 0^{^} \\ 0 \leq φ \leq 0.04 \\ 0 \leq H a \leq 100 \\ \Pr = 6.2 \end{array}$	The heat transfer between cold and hot regions of the enclosure cannot be well understood by using isotherm patterns so heatline visualization technique is used to find the direction and intensity of heat transfer in a domain.
Sheikholeslami et al. [98]		Control volume– based finite element method	Al₂O₃–water	$\begin{array}{l} 30^{^} \leq γ \leq 90^{^} \\ 0.1 \leq N r \leq 4 \\ 2 \leq L e \leq 8 \\ R a = 10^{5} \\ N b = N t = 0.5 \\ \Pr = 10 \end{array}$ $\begin{array}{l} 30^{^} \leq γ \leq 90^{^} \\ 0.1 \leq N r \leq 4 \\ 2 \leq L e \leq 8 \\ R a = 10^{5} \\ N b = N t = 0.5 \\ \Pr = 10 \end{array}$	Lewis number has no significant effect on Nusselt number at low values of buoyancy ratio number
Sheikholeslami et al. [99]		Control volume– based finite element method	Al₂O₃–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq H a \leq 100 \\ 30^{^} \leq γ \leq 90^{^} \\ 0.1 \leq N r \leq 4 \\ 2 \leq L e \leq 8 \\ N b = N t = 0.5 \\ \Pr = 10 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq H a \leq 100 \\ 30^{^} \leq γ \leq 90^{^} \\ 0.1 \leq N r \leq 4 \\ 2 \leq L e \leq 8 \\ N b = N t = 0.5 \\ \Pr = 10 \end{array}$	As buoyancy ratio number increases the effects of other active parameters are more pronounced
Sheikholeslami et al. [100]		Control volume– based finite element method	Cu–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ - 45^{^} \leq ζ \leq 45^{^} \\ 0 \leq φ \leq 0.06 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ - 45^{^} \leq ζ \leq 45^{^} \\ 0 \leq φ \leq 0.06 \\ \Pr = 6.2 \end{array}$	The results show that for Ra = 10⁴ the maximum and minimum average Nusselt number are corresponding to ζ = −45° and 45° respectively, whereas opposite trend is observed for Ra = 10⁵.
Sheikholeslami and Ganji [101]		Control volume– based finite element method	Fe₃O₄–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq H a \leq 5 \\ 0 \leq M n_{F} \leq 500 \\ 0 \leq φ \leq 0.04 \\ \Pr = 6.8 \\ E c = 10^{- 5} \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq H a \leq 5 \\ 0 \leq M n_{F} \leq 500 \\ 0 \leq φ \leq 0.04 \\ \Pr = 6.8 \\ E c = 10^{- 5} \end{array}$	Nusselt number increases with augment of Rayleigh number and nanoparticle volume fraction but it decreases with increase in Hartmann number. Magnetic number has different effect on Nusselt number corresponding to Rayleigh number
Sheikholeslami et al. [102]		Control volume– based finite element method	Fe₃O₄–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq H a \leq 10 \\ 0 \leq M n_{F} \leq 100 \\ 0 \leq N r \leq 0.027 \\ 0 \leq φ \leq 0.04 \\ \Pr = 6.8 \\ E c = 10^{- 5} \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq H a \leq 10 \\ 0 \leq M n_{F} \leq 100 \\ 0 \leq N r \leq 0.027 \\ 0 \leq φ \leq 0.04 \\ \Pr = 6.8 \\ E c = 10^{- 5} \end{array}$	Nusselt number is an increasing function of Rayleigh number, nanoparticle volume fraction, magnetic number while it is a decreasing function of Hartmann number and radiation parameter.
Sheikholeslami Kandelousi [103]		Control volume– based finite element method	Fe₃O₄–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq M n_{F} \leq 100 \\ 0 \leq H a \leq 10 \\ 0 \leq φ \leq 0.04 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq M n_{F} \leq 100 \\ 0 \leq H a \leq 10 \\ 0 \leq φ \leq 0.04 \\ \Pr = 6.2 \end{array}$	Enhancement in heat transfer decreases with increase in Rayleigh number and magnetic number but increases with increase in Hartmann number.
Sheikholeslami et al. [104]		Control volume– based finite element method	CuO–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0^{^} \leq γ \leq 90^{^} \\ 0 \leq φ \leq 0.04 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0^{^} \leq γ \leq 90^{^} \\ 0 \leq φ \leq 0.04 \\ \Pr = 6.2 \end{array}$	Increase in Rayleigh number leads to decrease in ratio of heat transfer enhancement. For high Rayleigh number the minimum heat transfer enhancement ratio occurs at γ = 90°.
Sheikholeslami and Rashidi [105]		Control volume– based finite element method	Fe₃O₄–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq M n_{F} \leq 100 \\ 0 \leq H a \leq 10 \\ 0 \leq φ \leq 0.04 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq M n_{F} \leq 100 \\ 0 \leq H a \leq 10 \\ 0 \leq φ \leq 0.04 \\ \Pr = 6.2 \end{array}$	Nusselt number is an increasing function of magnetic number, Rayleigh number, and nanoparticle volume fraction while it is a decreasing function of Hartmann number
Sheikholeslami et al. [106]		Control volume– based finite element method	Fe₃O₄–water	$\begin{array}{l} 10 \leq Re \leq 10^{3} \\ 0 \leq H a \leq 20 \\ 0 \leq φ \leq 0.04 \\ \Pr = 6.2 \end{array}$ $\begin{array}{l} 10 \leq Re \leq 10^{3} \\ 0 \leq H a \leq 20 \\ 0 \leq φ \leq 0.04 \\ \Pr = 6.2 \end{array}$	Nusselt number has direct relationship with Reynolds number, nanoparticle volume fraction while it has reverse relationship with Hartmann number
Sheikholeslami and Ellahi [107]		Control volume– based finite element method	Fe₃O₄–EG	$\begin{array}{l} 3000 \leq Re \leq 6000 \\ 0 \leq ∆ ϕ \leq 10 \\ 0 \leq φ \leq 0.04 \\ \Pr = 149.54 \end{array}$ $\begin{array}{l} 3000 \leq Re \leq 6000 \\ 0 \leq ∆ ϕ \leq 10 \\ 0 \leq φ \leq 0.04 \\ \Pr = 149.54 \end{array}$	Heat transfer rises with augment of supplied voltage and Reynolds number
Sheikholeslami et al. [108]		Control volume– based finite element method	Fe₃O₄–water	$\begin{array}{l} 10 \leq Re \leq 500 \\ 2 \leq L e \leq 8 \\ 0 \leq H a \leq 20 \end{array}$ $\begin{array}{l} 10 \leq Re \leq 500 \\ 2 \leq L e \leq 8 \\ 0 \leq H a \leq 20 \end{array}$	Nusselt number has direct relationship with Reynolds number while it has reverse relationship with Hartmann number and Lewis number
Sheikholeslami et al. [109]		Control volume– based finite element method	Fe₃O₄–water	$\begin{array}{l} 10 \leq Re \leq 600 \\ 0 \leq M n_{F} \leq 10 \\ 0 \leq H a \leq 10 \\ 0 \leq φ \leq 0.04 \\ \Pr = 6.8 \end{array}$ $\begin{array}{l} 10 \leq Re \leq 600 \\ 0 \leq M n_{F} \leq 10 \\ 0 \leq H a \leq 10 \\ 0 \leq φ \leq 0.04 \\ \Pr = 6.8 \end{array}$	Effects of Kelvin forces are more pronounced for high Reynolds number. Heat transfer enhancement has direct relationship with the Reynolds number and the magnetic number while it has inverse relationship with the Hartmann number
Sheikholeslami et al. [110]		Control volume– based finite element method	Fe₃O₄–EG	$\begin{array}{l} 3000 \leq Re \leq 6000 \\ 0 \leq ∆ ϕ \leq 10 \\ 0 \leq φ \leq 0.04 \\ \Pr = 149.54 \end{array}$ $\begin{array}{l} 3000 \leq Re \leq 6000 \\ 0 \leq ∆ ϕ \leq 10 \\ 0 \leq φ \leq 0.04 \\ \Pr = 149.54 \end{array}$	Effect of electric field on heat transfer is more pronounced at low Reynolds number.
Sheikholeslami and Rashidi [111]		Control volume– based finite element method	Fe₃O₄–water	$\begin{array}{l} 10^{3} \leq Ra \leq 10^{5} \\ 0.1 \leq Nr \leq 4 \\ 0 \leq Ha \leq 10 \\ 2 \leq Le \leq 4 \\ \Pr = 6.85 \end{array}$ $\begin{array}{l} 10^{3} \leq Ra \leq 10^{5} \\ 0.1 \leq Nr \leq 4 \\ 0 \leq Ha \leq 10 \\ 2 \leq Le \leq 4 \\ \Pr = 6.85 \end{array}$	Nusselt number has direct relationship with Rayleigh number, buoyancy ratio number, and Lewis number while it has reverse relationship with Hartmann number
Sheikholeslami, and Rashidi [112]		Control volume– based finite element method	Fe₃O₄–water	$\begin{array}{l} 0.001 \leq Ri \leq 10 \\ 0 \leq δ^{} \leq 0.6 \\ 0 \leq M n_{F} \leq 10 \\ 0 \leq Ha \leq 10 \\ Re = 100 \end{array}$ $\begin{array}{l} 0.001 \leq Ri \leq 10 \\ 0 \leq δ^{} \leq 0.6 \\ 0 \leq M n_{F} \leq 10 \\ 0 \leq Ha \leq 10 \\ Re = 100 \end{array}$	Nusselt number has direct relationship with Richardson number, nanoparticle volume fraction while it has reverse relationship with Hartmann number and magnetic number

1.2.7. Lattice Boltzmann method

Investigation of nanofluid flow and heat transfer in the presence of magnetic field using KKL model has been studied by Sheikholeslami et al. [113]. Nanofluid hydrothermal behaviors in an enclosure with curve boundaries have been studied by Sheikholeslami et al. [114–117]. Ashorynejad et al. [118] studied magnetic field effects on natural convection flow of a nanofluid in a horizontal cylindrical annulus. They found that flow oscillations can be suppressed effectively by imposing an external radial magnetic field. MHD effects on nanofluid flow and heat transfer in a semiannulus enclosure have been studied by Sheikholeslami et al. [119,120]. They showed that the enhancement in heat transfer increases as Hartmann number increases but it decreases with increase in Rayleigh number. Free convection of ferrofluid in a cavity heated from below in the presence of an external magnetic field has been investigated by Sheikholeslami and Gorji [121]. They found that particles with a smaller size have better ability to dissipate heat, and a larger volume fraction would provide a stronger driving force which leads to increase in temperature profile.

Sheikholeslami et al. [122] studied MHD free convection in an eccentric semiannulus filled with nanofluid. Sheikholeslami et al. [123] simulated MHD CuO–water nanofluid flow and convective heat transfer considering Lorentz forces. Entropy generation of nanofluid in the presence of magnetic field was studied by Sheikholeslami and Ganji [124]. Sheikholeslami et al. [125] simulated magnetohydrodynamic natural convection heat transfer of Al₂O₃–water nanofluid in a horizontal cylindrical enclosure with an inner triangular cylinder. Sheikholeslami and Ellahi [126] studied ferrofluid flow for magnetic drug targeting. They showed that back flow occurs near the region where the magnetic source is located. Sheikholeslami et al. [127] simulated the magnetic field effect on hydrothermal behavior of nanofluid in a cubic cavity. Three-dimensional mesoscopic simulation of magnetic field effect on natural convection of nanofluid has been studied by Sheikholeslami and Ellahi [128]. Nanofluid heat transfer enhancement and entropy generation has been studied by Sheikholeslami et al. [129]. Effect of a magnetic source on free convection in a cavity subjugated to nanofluid has been investigated by Kefayati [130–133]. Al₂O₃–water nanofluid flow and heat transfer in an open cavity and closed enclosure have been investigated by Mahmoudi et al. [134–136]. Table 1.10 shows the summary of the lattice Boltzmann method studies on nanofluid. Recently several authors apply new methods for simulation of hydrothermal behavior [137–167].

Table 1.10

Summary of the lattice Boltzmann method studies on nanofluid

Authors	Geometry of study	Method	Type of nanofluid	Properties	Remarks
Sheikholeslami et al. [113]		Lattice Boltzmann method	Al₂O₃–water	$\begin{array}{l} 10^{4} \leq R a \leq 10^{6} \\ 0 \leq φ \leq 0.04 \\ 0 \leq H a \leq 60 \\ R / L = 0.5 \\ \Pr = 6.8 \end{array}$ $\begin{array}{l} 10^{4} \leq R a \leq 10^{6} \\ 0 \leq φ \leq 0.04 \\ 0 \leq H a \leq 60 \\ R / L = 0.5 \\ \Pr = 6.8 \end{array}$	Enhancement in heat transfer increases with increase in Hartmann number except for Ra = 10⁴ in which Ha = 40 roles as a critical Hartmann number
Sheikholeslami et al. [114]		Lattice Boltzmann method	Cu–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{6} \\ 0 \leq φ \leq 0.06 \\ 0.65 \leq ɛ \leq 0.95 \\ a = 0.4 \\ \Pr = 6.8 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{6} \\ 0 \leq φ \leq 0.06 \\ 0.65 \leq ɛ \leq 0.95 \\ a = 0.4 \\ \Pr = 6.8 \end{array}$	The minimum value of enhancement of heat transfer occurs at ɛ = 0.95 for Ra = 10⁵ but for other values of Rayleigh number it is obtained at ɛ = 0.65.
Sheikholeslami et al. [115]		Lattice Boltzmann method	Cu–water	$\begin{array}{l} 10^{4} \leq R a \leq 10^{6} \\ 0 \leq φ \leq 0.06 \\ 0 \leq H a \leq 60 \\ \Pr = 6.8 \end{array}$ $\begin{array}{l} 10^{4} \leq R a \leq 10^{6} \\ 0 \leq φ \leq 0.06 \\ 0 \leq H a \leq 60 \\ \Pr = 6.8 \end{array}$	As the nanoparticle volume fraction and Rayleigh number increase average Nusselt number increases but opposite trends are observed when the Hartmann number increases.
Sheikholeslami et al. [116]		Lattice Boltzmann method	Cu–water	$\begin{array}{l} 10^{4} \leq R a \leq 10^{6} \\ 0 \leq φ \leq 0.04 \\ - 60^{^} \leq γ \leq 60^{^} \\ R / L = 0.5 \\ \Pr = 6.8 \end{array}$ $\begin{array}{l} 10^{4} \leq R a \leq 10^{6} \\ 0 \leq φ \leq 0.04 \\ - 60^{^} \leq γ \leq 60^{^} \\ R / L = 0.5 \\ \Pr = 6.8 \end{array}$	The change of inclination angle has a significant impact on the thermal and hydrodynamic flow fields
Sheikholeslami et al. [117]		Lattice Boltzmann method	Cu-water Ag–water Al₂O₃–water TiO₂–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{6} \\ 0 \leq φ \leq 0.04 \\ 1.5 \leq λ \leq 4.5 \\ \Pr = 6.8 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{6} \\ 0 \leq φ \leq 0.04 \\ 1.5 \leq λ \leq 4.5 \\ \Pr = 6.8 \end{array}$	Choosing copper as the nanoparticle leads to obtain the highest enhancement for this problem.
Ashorynejad et al. [118]		Lattice Boltzmann method	Ag–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{6} \\ 0 \leq φ \leq 0.04 \\ 0 \leq H a \leq 60 \\ \Pr = 6.8 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{6} \\ 0 \leq φ \leq 0.04 \\ 0 \leq H a \leq 60 \\ \Pr = 6.8 \end{array}$	Flow oscillations can be suppressed effectively by imposing an external radial magnetic field.
Sheikholeslami et al. [119]		Lattice Boltzmann method	Al₂O₃–water	$\begin{array}{l} 10^{4} \leq R a \leq 10^{6} \\ 0 \leq φ \leq 0.04 \\ 0 \leq H a \leq 100 \\ \Pr = 6.8 \end{array}$ $\begin{array}{l} 10^{4} \leq R a \leq 10^{6} \\ 0 \leq φ \leq 0.04 \\ 0 \leq H a \leq 100 \\ \Pr = 6.8 \end{array}$	The enhancement in heat transfer increases as Hartmann number increases but it decreases with increase in Rayleigh number
		Lattice Boltzmann method
Sheikholeslami et al. [120]		Lattice Boltzmann method	Cu–water	$\begin{array}{l} 10^{4} \leq R a \leq 10^{6} \\ 0 \leq φ \leq 0.06 \\ 0 \leq H a \leq 40 \\ 1.5 \leq λ \leq 4.5 \\ \Pr = 6.8 \end{array}$ $\begin{array}{l} 10^{4} \leq R a \leq 10^{6} \\ 0 \leq φ \leq 0.06 \\ 0 \leq H a \leq 40 \\ 1.5 \leq λ \leq 4.5 \\ \Pr = 6.8 \end{array}$	Enhancement ratio increases with decrease in Rayleigh number and it increases with augment of Hartmann number
Sheikholeslami and Gorji [121]		Lattice Boltzmann method	Cobalt–kerosene	$\begin{array}{l} 10^{4} \leq R a \leq 10^{6} \\ 0 \leq φ \leq 0.04 \\ 0.675 \leq A \leq 1.325 \\ 0.2 \leq ɛ \leq 0.8 \\ \Pr = 6.8 \end{array}$ $\begin{array}{l} 10^{4} \leq R a \leq 10^{6} \\ 0 \leq φ \leq 0.04 \\ 0.675 \leq A \leq 1.325 \\ 0.2 \leq ɛ \leq 0.8 \\ \Pr = 6.8 \end{array}$	Particles with a smaller size have better ability to dissipate heat, and a larger volume fraction would provide a stronger driving force which leads to increase in temperature profile
Sheikholeslami et al. [122]		Lattice Boltzmann method	Cu–water	$\begin{array}{l} 10^{4} \leq R a \leq 10^{6} \\ 0 \leq φ \leq 0.06 \\ 0 \leq H a \leq 40 \\ 0.2 \leq \frac{δ}{L} \leq 0.8 \\ λ = 3.5 \\ \Pr = 6.8 \end{array}$ $\begin{array}{l} 10^{4} \leq R a \leq 10^{6} \\ 0 \leq φ \leq 0.06 \\ 0 \leq H a \leq 40 \\ 0.2 \leq \frac{δ}{L} \leq 0.8 \\ λ = 3.5 \\ \Pr = 6.8 \end{array}$	Nusselt number has direct relationship with nanoparticle volume fraction and Rayleigh number but it has inverse relationship with Hartmann number and position of inner cylinder at high Rayleigh number
Sheikholeslami et al. [123]		Lattice Boltzmann method	CuO–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq φ \leq 0.04 \\ 0 \leq H a \leq 100 \\ 0.4 \leq ɛ \leq 0.8 \\ \Pr = 6.8 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq φ \leq 0.04 \\ 0 \leq H a \leq 100 \\ 0.4 \leq ɛ \leq 0.8 \\ \Pr = 6.8 \end{array}$	Enhancement in heat transfer increases as Hartmann number and heat source length increase but it decreases with increase in Rayleigh number. Also it can be found that effect of Hartmann number and heat source length is more pronounced at high Rayleigh number.
Sheikholeslami and Ganji [124]		Lattice Boltzmann method	CuO–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq φ \leq 0.04 \\ 0 \leq H a \leq 100 \\ \Pr = 6.8 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq φ \leq 0.04 \\ 0 \leq H a \leq 100 \\ \Pr = 6.8 \end{array}$	Heat transfer rate and dimensionless entropy generation number increase with increase in the Rayleigh number and nanoparticle volume fraction but it decreases with increase in the Hartmann number
Sheikholeslami et al. [125]		Lattice Boltzmann method	Al₂O₃–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq φ \leq 0.04 \\ 0 \leq H a \leq 100 \\ \Pr = 6.8 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq φ \leq 0.04 \\ 0 \leq H a \leq 100 \\ \Pr = 6.8 \end{array}$	Lattice Boltzmann method with double population is a powerful approach for the simulation of natural convection heat transfer in nanofluids in regions with curved boundaries
Sheikholeslami and Ellahi [126]		Lattice Boltzmann method	Fe₃O₄–plasma	$\begin{array}{l} 50 \leq Re \leq 400 \\ 0 \leq M n_{F} \leq 100 \\ φ = 0.04 \\ H a = 20 \end{array}$ $\begin{array}{l} 50 \leq Re \leq 400 \\ 0 \leq M n_{F} \leq 100 \\ φ = 0.04 \\ H a = 20 \end{array}$	Back flow occurs near the region where the magnetic source is located. Also it can be found that skin friction coefficient is a decreasing function of Reynolds number and magnetic number
Sheikholeslami et al. [127]		Lattice Boltzmann Method	Al₂O₃-water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq φ \leq 0.04 \\ 0 \leq H a \leq 100 \\ \Pr = 6.8 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq φ \leq 0.04 \\ 0 \leq H a \leq 100 \\ \Pr = 6.8 \end{array}$	Nusselt number increases with increase of nanoparticle volume fraction and Rayleigh number while it decreases with increase in Hartmann number.
Sheikholeslami and Ellahi [128]		Lattice Boltzmann Method	Al₂O₃-water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq φ \leq 0.04 \\ 0 \leq H a \leq 100 \\ \Pr = 6.8 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq φ \leq 0.04 \\ 0 \leq H a \leq 100 \\ \Pr = 6.8 \end{array}$	Applying magnetic field results in a force opposite to the flow direction that leads to drag the flow and then reduces the convection currents by reducing the velocities
Sheikholeslami et al. [129]		Lattice Boltzmann method	Cu–water Ag–water Al₂O₃–water TiO₂–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq φ \leq 0.04 \\ 2 \leq H / t \leq 9 \\ \Pr = 6.8 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq φ \leq 0.04 \\ 2 \leq H / t \leq 9 \\ \Pr = 6.8 \end{array}$	The effect of nanoparticle volume fraction is found to be more pronounced for low Rayleigh number as compared to high Rayleigh number
Kefayati [130]		Lattice Boltzmann method	Cu–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq φ \leq 0.06 \\ 0 \leq H a \leq 90 \\ 0.5 \leq A \leq 2 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq φ \leq 0.06 \\ 0 \leq H a \leq 90 \\ 0.5 \leq A \leq 2 \end{array}$	The heat transfer decreases by increment of Hartmann number. Heat transfer decreases with growth of the aspect ratio but this growth causes the effect of nanoparticles to increase. Magnetic field augments the effect of nanoparticles at high Rayleigh numbers (Ra = 10⁵). The effect of nanoparticles rises for high Hartmann numbers when the aspect ratio increases.
Kefayati [131]		Lattice Boltzmann method	Cu–water	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq φ \leq 0.06 \\ - 60^{°} \leq θ \leq 60^{°} \\ 0 \leq H a \leq 30 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq φ \leq 0.06 \\ - 60^{°} \leq θ \leq 60^{°} \\ 0 \leq H a \leq 30 \end{array}$	The heat transfer is decreased by the increment of Hartmann number for various Rayleigh numbers and the inclined angles. Magnetic field augments the effect of nanoparticles at high Rayleigh numbers.
Kefayati [132]		Lattice Boltzmann method	Kerosene/ Cobalt	$\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq φ \leq 0.04 \\ 0.25 \leq Y_{p} \leq 0.75 \end{array}$ $\begin{array}{l} 10^{3} \leq R a \leq 10^{5} \\ 0 \leq φ \leq 0.04 \\ 0.25 \leq Y_{p} \leq 0.75 \end{array}$	Heat transfer decreases by the increment of the nanoscale ferromagnetic volume fraction. The external magnetic field influences the nanoscale ferromagnetic at Ra = 10⁴ more than other Rayleigh numbers as the least values were observed at Ra = 10³. Heat transfer obtains the most value at Yp = 0.5 H for multifarious Rayleigh numbers, the most effect of the nanoscale ferromagnetic for Ra = 10⁴ and 10⁵ was perceived at Yp = 0.75H.
Kefayati [133]		Lattice Boltzmann method	Cu–water	$\begin{array}{l} 10^{3} \leq Ra \leq 10^{5} \\ 0 \leq φ \leq 0.15 \\ 0 \leq H a \leq 90 \end{array}$ $\begin{array}{l} 10^{3} \leq Ra \leq 10^{5} \\ 0 \leq φ \leq 0.15 \\ 0 \leq H a \leq 90 \end{array}$	As the nanoparticle volume fraction and Rayleigh number increased average Nusselt number increased but opposite trends were observed when the Hartmann number increases.
Mahmoudi et al. [134]		Lattice Boltzmann method	Al₂O₃–water	$\begin{array}{l} 10^{3} \leq Ra \leq 10^{6} \\ 0 \leq φ \leq 0.06 \\ 0 \leq Ha \leq 60 \\ - 10 \leq q \leq 10 \end{array}$ $\begin{array}{l} 10^{3} \leq Ra \leq 10^{6} \\ 0 \leq φ \leq 0.06 \\ 0 \leq Ha \leq 60 \\ - 10 \leq q \leq 10 \end{array}$	The nanoparticles effect is more important at a high Rayleigh number. Also, the nanoparticles effect is more important for heat generation condition (q < 0) than absorption generation condition (q < 0).
Mahmoudi et al. [135]		Lattice Boltzmann method	Al₂O₃–water	$\begin{array}{l} 10^{3} \leq Ra \leq 10^{5} \\ 0 \leq φ \leq 0.06 \\ 0 \leq Ha \leq 60 \end{array}$ $\begin{array}{l} 10^{3} \leq Ra \leq 10^{5} \\ 0 \leq φ \leq 0.06 \\ 0 \leq Ha \leq 60 \end{array}$	The heat sinks positions greatly influence the heat transfer rate depending on the Hartmann number, Rayleigh number, and nanoparticle solid volume fraction.
Mejri and Mahmoudi [136]		Lattice Boltzmann method	Al₂O₃–water	$\begin{array}{l} 10^{3} \leq Ra \leq 10^{6} \\ 0 \leq φ \leq 0.06 \\ 0 \leq Ha \leq 60 \\ γ = 0, π / 4, π / 2, \\ 3 π / 4, π \end{array}$ $\begin{array}{l} 10^{3} \leq Ra \leq 10^{6} \\ 0 \leq φ \leq 0.06 \\ 0 \leq Ha \leq 60 \\ γ = 0, π / 4, π / 2, \\ 3 π / 4, π \end{array}$	The heat transfer rate decreases with the increase in Hartmann number and increases with the increase in Rayleigh number.