*2.3.2. Non-homogeneous two-component model (Buongiorno model)*

Buongiorno [72] investigated the effects of seven different slip mechanisms between the base fluid and nanoparticles: gravity, thermophoresis, Brownian diffusion, inertia, Magnus effect, fluid drainage and diffusiophoresis, in the absence of turbulent effects. It was demonstrated that thermophoresis and Brownian diffusion are the most influential mechanisms on nano‐ fluids flow and heat transfer, which can affect nanoparticle concentration variations. Under such conditions, the four coupled governing equations were proposed as follows [73, 74]:

Conservation of mass:

$$\nabla \cdot (\rho\_{\ast \circ} \stackrel{\cdot}{u}) = 0 \tag{40}$$

Conservation of momentum:

$$\nabla . (\rho\_{\eta'} \overrightarrow{\dot{\mu} \ddot{u}}) = -\nabla P + \nabla . (\mu\_{\eta'} \overrightarrow{\nabla u}) \tag{41}$$

Conservation of energy:

$$
\nabla .( (\rho c\_{\rho})\_{\eta'} \vec{u} \, T) = \nabla .(k\_{\eta'} \nabla T) - c\_{\rho\_{\eta'}} \vec{J}\_{\eta \nu} \nabla T \tag{42}
$$

Conservation of nanoparticles:

Mathematical Modeling for Nanofluids Simulation: A Review of the Latest Works http://dx.doi.org/10.5772/64154 205

$$
\overrightarrow{\mu}.\nabla\varphi = -\frac{1}{\rho\_{\text{sp}}}\nabla.\overrightarrow{J}\_{\text{sp}}\tag{43}
$$

where *J* → *np* is nanoparticles flux and is defined as

results of this study were in a fair agreement with previous studies, which shows that LBM could be utilized to simulate forced convection for the nanofluids flow inside microsized

Recently, by employing a 2-D double multiple-relaxation-time (MRT) thermal Lattice-Boltzmann model, Zhang and Che [69] simulated the magneto-hydrodynamic (MHD) flow and heat transfer of copper water in an inclined cavity with four heat sources. The governing equations were solved using D2Q9- and D2Q5-MRT models, which was validated by previous investigations. The results showed that the inclination angle has a considerable effect on flow fields, the temperature patterns, and the local Nusselt number distributions. Moreover, it was concluded that MRT Lattice-Boltzmann method is competent for solving heat transfer of

In the end, LBM has been widely used for natural, forced, and mixed convection of nanofluids, which can be found in details [70, 71]. The results of this model have higher accuracy than the results of conventional CFD approaches. However, it seems that more research may be needed in order to find out to what extent LBM is applicable in the simulation of nanofluids flow and

Buongiorno [72] investigated the effects of seven different slip mechanisms between the base fluid and nanoparticles: gravity, thermophoresis, Brownian diffusion, inertia, Magnus effect, fluid drainage and diffusiophoresis, in the absence of turbulent effects. It was demonstrated that thermophoresis and Brownian diffusion are the most influential mechanisms on nano‐ fluids flow and heat transfer, which can affect nanoparticle concentration variations. Under such conditions, the four coupled governing equations were proposed as follows [73, 74]:

> .( ) 0 *nf* Ñ = r*u*

.( ) .( ) *nf nf* Ñ = -Ñ + Ñ Ñ

.(( ) ) .( ) . *nf np p nf nf <sup>p</sup>* Ñ =Ñ Ñ - Ñ

*uu P u*

*c uT k T c J T* <sup>r</sup> ur

 m

r

r

<sup>r</sup> (40)

(42)

r r <sup>r</sup> (41)

nanofluids in enclosures affected by a magnetic field.

*2.3.2. Non-homogeneous two-component model (Buongiorno model)*

configurations.

204 Modeling and Simulation in Engineering Sciences

characteristics.

Conservation of mass:

Conservation of momentum:

Conservation of energy:

Conservation of nanoparticles:

$$
\overrightarrow{J}\_{\text{up}} = \overrightarrow{J}\_{\text{up}\_{\text{h}}} + \overrightarrow{J}\_{\text{up}\_{\text{r}}} \tag{44}
$$

The aforementioned terms can be calculated as follows [75]:

$$\overrightarrow{J}\_{\text{np}\_{\text{pr}}} = \frac{k\_{\text{Br}}T}{3\pi\mu\_f d\_{\text{np}}}\tag{45}$$

$$\overrightarrow{J}\_{\text{sp}\_f} = 0.26(\frac{k\_f}{2k\_f + k\_{\text{sp}}} \frac{\mu\_f}{\rho\_f} \varphi) \tag{46}$$

where *D* represents the diffusion coefficient.



resist the fluid flow and thus reduces the flow's velocity.

Mathematical Modeling for Nanofluids Simulation: A Review of the Latest Works http://dx.doi.org/10.5772/64154 207


**Authors Geometry of study Type of**

206 Modeling and Simulation in Engineering Sciences

Malvandi, Ganji [81]

Malvandi, Ganji [82]

Malvandi et al. [83]

**nanofluid**

Al2O3 /water nanofluid

Al2O3 /water nanofluid

Al2O3 /water nanofluid **Properties Remarks**

The heat transfer rate is enhanced by the presence of the magnetic field especially for the smaller nanoparticles. Moreover, as the magnetic field strength (Ha) intensifies, the peak of the velocity profile near the walls is increased; however, the peak of the velocity profile at the core

region is decreased.

The concentration of

nanoparticles is higher near the cold wall (nanoparticles accumulation), while it is lower near the adiabatic wall (nanoparticles depletion). Also, there is an optimum value for the bulk mean of nanoparticle volume fraction in which the heat transfer rate is maximum. This optimum value decreases for smaller nanoparticles.

In the presence of the magnetic field, the velocity gradients near the wall grow, which increases the slip velocity at boundaries and thus the heat transfer rate intensifies. What is more? Increasing Ha (intensifying the magnetic field) leads to an increase in the Lorentz force (a retarding force to the transport phenomena), which tends to resist the fluid flow and thus reduces the flow's velocity.

1 *nm*≤*dp* ≤100 *nm*

1 *nm*≤*dp* ≤100 *nm*

1 *nm*≤*dp* ≤100 *nm*

0≤*ϕ* ≤0.1 0.1≤ *NBT* ≤10 0≤*Ha* ≤15

0≤*ϕ* ≤0.1 4≤ *NBT* ≤10

0≤*ϕ* ≤0.1 0.1≤ *NBT* ≤10 0≤*Ha* ≤10

> moderate range of NBT. Furthermore, the heat absorption boosts the pressure drops of nanofluid.



**Table 1.** Some recent studies on modified Buongiorno model.

**Authors Geometry of study Type of**

208 Modeling and Simulation in Engineering Sciences

Malvandi, Ganji [87]

Malvandi et al. [88]

Malvandi et al. [89]

**nanofluid**

Al2O3 /water nanofluid

Al2O3 /water nanofluid

Al2O3 and TiO2/waterbased nanofluids

**Properties Remarks**

It is shown that nanoparticles eject themselves from the heated walls, construct a depleted region, and accumulate in the core region, but they are more likely to accumulate toward the wall with the lower heat flux.

The non-uniform nanoparticle distribution makes the velocities move toward the wall with the higher heat flux and enhances the heat transfer rate there. In addition, it is shown that the advantage of nanoparticle inclusion is increased in the presence of a magnetic field, though heat transfer enhancement is decreased.

The heat transfer enhancement of titania-water nanofluids is completely insignificant relative to such enhancement for alumina-water nanofluid. Therefore, alumina-water nanofluid exhibits a better performance compared to titania-water nanofluids.

1 *nm*≤*dp* ≤100 *nm*

1 *nm*≤*dp* ≤100 *nm*

1 *nm*≤*dp* ≤100 *nm* 0≤*ϕ* ≤0.06 0.2≤ *NBT* ≤10

0≤*ϕ* ≤0.1 0.2≤ *NBT* ≤10 0≤*Ha* ≤10

0≤*ϕ* ≤0.1 0.2≤ *NBT* ≤10 0≤*Ha* ≤10

> Sheikhzadeh et al. [76] studied the effects of Brownian motion, thermophoresis, and Dufour (transport model) on laminar-free convection heat transfer of alumina-water nanofluid flow in a square enclosure. Variable thermophysical properties utilized for fluid characterization and the governing equations were discretized using FVM. The results illustrated that the Dufour effect on heat transfer is not significant. In addition, a comparison between experi‐ mental data and numerical results revealed that the transport model is in better agreement with experimental results, compared to single-phase model.

> Using the same method, Bahiraei et al. [77] studied the laminar convection heat transfer of alumina-water nanofluid inside a circular tube, considering particle migration effects. The results showed that with the Reynolds number or volume fraction augmentation, the average heat transfer coefficient enhances. In addition, it was reported that by considering the particle migration effect, higher heat transfer coefficient would be achieved.

> Using modified Buongiorno model, Malvandi et al. [78] investigated MHD mixed convection heat transfer for Al2O3-water nanofluid inside a vertical annular pipe. The governing equations reduced to two-point O.D.E.s, which were solved by means of the Runge-Kutta-Fehlberg scheme. The obtained results indicated that the excellence of using nanofluids for heat transfer enhancement purpose is diminished by the presence of a magnetic field. Moreover, it was noted that the imposed thermal asymmetry may change the direction of nanoparticle migra‐ tion, and, hence, alters the velocity, temperature, and nanoparticle concentration profiles. **Table 1** shows some new works on modified Buongiorno model.

#### *2.3.3. Other approaches*

In some other studies, novel numerical approaches have been employed to solve the governing equations of nanofluids. SPH method has been used by Mansour and Bakier [91] to study free convection within an enclosed cavity filled with Al2O3 nanoparticles. The left and right walls

of the cavity had a complex-wavy geometry while upper and lower walls were both flat and insulated. Complex-wavy walls were modeled as the superposition of two sinusoidal func‐ tions. The results revealed that heat transfer performance may be optimized by tuning the wavy-surface geometry parameter in accordance with the Rayleigh number. Using optimal homotopy analysis method (OHAM), Nadeem et al. [92] examined 2-D stagnation point flow of a non-Newtonian Casson nanofluid over a convective-stretching surface. The governing non-linear partial differential equations were converted into non-linear ordinary differential equations and solved analytically using OHAM. The results showed that heat transfer rate is an increasing function of the stretching parameter, Prandtl and Biot numbers and it decreases with an increase in non-Newtonian parameter, Brownian motion, and thermophoresis.

The laminar axisymmetric flow of a nanofluid over a non-linearly stretching sheet was studied by Mustafa et al. [93], both numerically and analytically. The simultaneous effects of Brownian motion and thermophoretic diffusion of nanoparticles were taken into account. The numerical solution was computed by employing implicit finite difference scheme known as Keller-Box method. The results obtained from both solutions were in excellent agreement with each other. The results demonstrated that the effect of Brownian motion on fluid temperature and wall heat transfer rate is insignificant. Moreover, it was reported that increases in Schmidt number lead to a thinner nanoparticle volume fraction boundary layer.
