Some recent studies on modified Buongiorno model.
Abstract
Exploiting nanofluids in thermal systems is growing day by day. Nanofluids having ultrafine solid particles promise new working fluids for application in energy devices. Many studies have been conducted on thermophysical properties as well as heat and fluid flow characteristics of nanofluids in various systems to discover their advantages compared to conventional working fluids. The main aim of this study is to present the latest developments and progress in the mathematical modeling of nanofluids flow. For this purpose, a comprehensive review of different nanofluid computational fluid dynamics (CFD) approaches is carried out. This study provides detailed information about the commonly used formulations as well as techniques for mathematical modeling of nanofluids. In addition, advantages and disadvantages of each method are rendered to find the most appropriate approach, which can give valid results.
Keywords
 nanofluid
 CFD
 numerical simulation
 mathematical modeling
 single and twophase methods
1. Introduction
In general, the assessment of the thermal performance of a system through numerical simulations is much affordable compared to experimental studies with high expenses of material and equipment. The significance of a numerical study is highlighted when a nanofluid is utilized as the working fluid. High costs for the production of nanofluids and difficulties in preparing stable nanofluids are the main barriers to perform experiments with nanofluids. Therefore, numerical modeling of nanofluids, where a suitable approach is selected to simulate the flow, could be the best solution for problems involved with nanoparticle suspensions.
However, in spite of considerable developments in computing power and methods, literature review reveals that there is no comprehensive study to conclude the best technique for the modeling of nanofluids. In particular, due to the ultrafine size of nanoparticles, the governing terms in multiphase models are still not entirely identified. In the present work, latest studies on numerical simulations of nanofluid flow are reviewed with a particular focus on different multiphase schemes.
2. Numerical methods for nanofluids’ flow simulation
Nanofluid computational fluid dynamic (CFD) modeling can be classified into two main groups: singlephase and twophase models. However, there are few other models that may not be included in these categories, such as LatticeBoltzmann method (LBM). Moreover, different numerical approaches have been employed to solve models mentioned above to predict thermal and hydraulic characteristics of nanofluids flow. Finite volume method (FVM) and finite element method (FEM) are two main approaches for solving the governing equations of nanofluid problems. However, finite difference method (FDM), control volumebased finite element method, and some novel numerical approaches such as homotopy analysis method (HAM) and smoothed particle hydrodynamics (SPH) methods have also been utilized in the previous studies. In this study, a comprehensive review of various numerical methods for the simulation of nanofluids is accomplished.
2.1. Singlephase approaches
Although suspension of a nanofluid is inherently a twophase fluid, if some proper assumptions are made, it can be considered as a homogeneous liquid. Due to the existence of ultrafine nanoparticles, it is assumed that these particles can be easily dispersed in the host fluid. For this purpose, both the nanoparticles and base fluid are considered to be in thermal balance without any slip between their molecules. Therefore, under such assumptions, in many studies, nanofluids have been assumed as a singlephase fluid.
2.1.1. Conventional singlephase model
Mass, momentum, and energy equations, which are used for conventional liquids, could also be applied to singlephase flow with the above assumptions. In this case, only thermophysical properties of nanofluids should be determined. Therefore, the governing equations in the steady state can be expressed as below [1]:
Conservation of mass:
Conservation of momentum:
Conservation of energy:
There are numerous publications simulating nanofluids characteristics as a singlephase fluid. Mixed convection heat transfer in a Tshaped liddriven cavity was examined numerically by Mojumder et al. [2]. A residual FEM model was applied for the numerical simulation. They validated their simulation code against data of AbuNada, Chamkha [3]. The results revealed that higher Grashof number causes rise in the heat transfer rate.
Turbulent nanofluid flow for different nanoparticles such as alumina, cupric oxide, and titania was investigated by Rostamani et al. [4] at various concentrations in a long horizontal duct under constant heat flux. They employed the control volume approach and temperaturedependent thermophysical properties. It was found that increasing the number of dispersed nanoparticles in base fluid increases the pumping power and heat transfer rate. In addition, the predicted Nusselt numbers in some cases demonstrated good agreements to the obtained results by Pak, Cho [5], and Maiga et al. [6] correlations.
AbuNada [7] considered the RayleighBenardfree convection heat transfer in CuOwater nanofluids using FVM, where the effect of temperaturedependent properties was the primary objective. The obtained results by temperaturedependent models were also compared with classic models (MaxwellGarnett (MG) and Brinkman). Results displayed that for
Bouhalleb and Abbassi [8] employed Control VolumeFinite Element method to study the free convection in an inclined cavity. They solved the free convection problem using Boussinesq approximation and employing the SIMPLER algorithm for velocitypressure coupling. The results showed that the variation of inclination angel highlights a hysteresis behavior of the nanofluid flow. Also, increasing the diameter of solid nanoparticles led to a strong decay in heat transfer. Furthermore, they concluded that the efficiency of heat transfer strongly depends on the diameter of nanoparticles, not its concentration in base fluid.
While a considerable number of recent studies on numerical simulation of nanofluids have been employed by FVM [9–11] and FEM [12–14] for their studies, in several studies Finite Difference Method has been used for different applications of nanofluids. For instance, Buddakkagari and Kumar [15] studied laminarforced convection of a nanofluid over a vertical cone/plate. The nondimensional governing equations were solved using FDM model, CrankNicolson type. The results illustrated that the Prandtl number affects the boundary layer dramatically. Furthermore, the momentum boundary layer was more affected by higher values of Lewis number. Finally, it was concluded that the boundary layer growth depends on Brownian motion and thermophoresis force.
However, applying the singlephase model for nanofluids has some limitations. For instance, the results obtained from this model were strongly dependent on adopted thermophysical properties and in some cases using the singlephase model may underestimate the Nusselt number, compared to models adopting temperaturedependent properties [16, 17]. The reason behind this could be due to various factors such as gravity, friction forces, Brownian motion, and solid/liquid interface. Ding and Wen [18] showed that the primary assumption of homogeneous fluid is not always acceptable. However, the review of the previous works illustrates that choosing the appropriate thermophysical property correlations in the singlephase method results in a reasonable estimation of nanofluids properties [19, 20]. Therefore, selecting suitable thermophysical properties such as variable properties, and considering the chaotic movement of nanoparticles (dispersion model) may compensate, to some extent, the limitations of singlephase model.
2.1.2. Thermal dispersion model
Brownian and gravity forces, the friction force between the base fluid and nanoparticles, sedimentation, and dispersion may coexist in a nanofluid flow. In fact, slip motion between liquid molecules and solid particles is not negligible, and the random movement of nanoparticles ameliorates the thermal dispersion in nanofluids, which reinforces heat transfer. For the first time, the thermal dispersion model was suggested by Xuan and Roetzel [21]. They assumed that nanoparticles move randomly, causing small chaos in velocity and temperature magnitudes (
By assuming that the boundary layer between the liquid and solid phases is negligible, unsteadystate, energy equation can be written as
The thermal dispersion generates a heat flux in the flow that is equal to
By using dispersion model, Zeinali Heris et al. [26, 27] investigated convective heat transfer of different nanofluids. They studied heat transfer augmentation due to nanofluids flow through the tubes with different crosssection geometries. The numerical results were validated against experimental data [28, 29] and good agreement was observed. The results showed that the Nusselt number would enhance with increasing particle loading and decreasing particle size.
Thermal behavior of nanofluids in a cavity was analyzed by Kumar et al. [30]. The dimensionlessgoverning equations were resolved via semiexplicit FVM solver. The Grashof number, volume concentration, and nanoparticles shape effects on heat transfer rate were assessed. The results showed that the dispersed thermal conductivity is more intensive in the vicinity of walls in comparison with pipe center. Also, it was found that dispersed thermal conductivity and hydraulic diameter of the particles are strongly dependent to each other.
Akbaridoust et al. [31] examined laminar steadystate nanofluid flow through helically shaped tubes, both numerically and experimentally. The governing equations in threedimensional (3D) form were solved by finite difference approach, using a FORTRAN code. Dispersion model was modified in order to be applicable for helical tubes. This modification resulted in minimized difference between numerical results and experimental data. The results showed that higher curvature ratios cause more heat transfer rates.
2.2. Twophase approaches
Due to some factors such as Brownian force, Brownian diffusion, friction force, thermophoresis, and gravity, nanofluids may be considered twophase fluids by nature. Therefore, the classic theory of solidliquid mixture can be applied to nanofluids. In such models, nanoparticles and base fluid are considered as two separate phases with different temperatures and velocities. Although twophase approaches may obtain realistic results, they have high computational cost in comparison to singlephase models. Twophase approaches can be categorized into two general models: LagrangianEulerian and EulerianEulerian.
2.2.1. LagrangianEulerian model
In the LagrangianEulerian or discrete phase model, the fluid phase is considered as a continuum by solving the NS equations in timeaveraged form, while the dispersed phase was solved by tracking the particles in the Lagrangian frame. Also, in this model, the interaction between fluid and particles presented as a source term in both momentum and energy equations.
Mathematical formulations of the LagrangianEulerian in twophase model can be written as follows [32]:
Conservation of mass:
Conservation of momentum:
Conservation of energy:
In Lagrangian reference frame, the particle motion and energy equations are as follows:
In Eq. (22),
Turbulent nanofluid flow in helical tubes was investigated numerically and experimentally by Bahremand et al. [36]. The numerical simulation was performed by both singlephase model and LagrangianEulerian approach in connection with renormalization group (RNG) kε model. ANSYS CFX software was used for solving the governing equations. The results indicated that nanofluids with a higher concentration exhibit a greater heat transfer coefficient and pressure drop. Also, it was found that the twophase model yields more accurate results compared to singlephase model.
Aluminawater nanofluid flow and heat transfer in a long tube with uniform heating at the walls were investigated by Moraveji and Esmaeili [37]. The simulations were conducted in both single and twophase methods where the governing equations were solved by FVM. Both temperaturedependent and constant thermophysical properties were considered in the study. The results of the modeling revealed that the temperaturedependent properties are more sensitive to the Reynolds number variations and led to higher values of the Nusselt number. Comparison between singlephase and twophase (discrete phase) models showed the maximum difference of 11% for the average heat transfer coefficient.
Tahir and Mital [38] studied the laminarforced convection of Al_{2}O_{3}water nanofluid in a tube numerically. They analyzed the impacts of the Reynolds number, particle diameter, and volume fraction of the particles in their study. A good agreement was achieved between the simulation and experimental data using discrete phase method. The results of the survey demonstrated that the heat transfer coefficient increased linearly with both the Reynolds number and volume fraction of nanoparticles. However, there was a nonlinear parabolic decrease with increasing nanoparticle size. It was concluded that the Reynolds number and volume fraction have the maximum and the minimum effects on heat transfer coefficient, respectively.
A comprehensive simulation of turbulentforced convection for Cuwater was carried out by Behroyan et al. [39].The Reynolds number of the flow was chosen between 10,000 and 25,000, where the volume fraction of copper nanoparticles was taken in the range of 0.0–2.0%. Two singlephase models (Newtonian and nonNewtonian) and three twophase models were employed in this study. The ANSYS commercial CFD package was utilized to solve the governing equations. The obtained results showed that the Newtonian singlephase method as well as discrete phase method is in better agreement with experimental data, compared to other numerical approaches.
The present literature survey reveals that using LagrangianEulerian approach for modeling the heat transfer of nanofluids is in early stages. Therefore, more studies are required to be conducted to determine the capability of the LagrangianEulerian model, especially in turbulent regime [40]. Also, in some other studies such as Safaei et al. [32] and Xu et al. [41], it is emphasized that LagrangianEulerian is a suitable model just for low concentration twophase suspensions (
2.2.2. EulerianEulerian model
The other important branch of twophase models is the EulerianEulerian model. Since the EulerianEulerian model is suitable for mixtures with a high amount of particles, applying this model to nanofluids consisting of an extremely large number of nanoparticles is recommended. The main models of EulerianEulerian available in the literature are three models including Mixture, Eulerian, and VOF (volume of fluid).
2.2.2.1. Volume of fluid model
In the VOF model, the volume fractions of all phases are obtained for the entire domain of study, by solving the continuity equation for the secondary phases. A single set of momentum equations is solved for all the phases to find the velocity components. The sum of all employed phases’ volume fractions is equal to unity. Accordingly, the primary phase volume fraction magnitude is achieved. In addition, all the physical properties are calculated by using an average weighted of different phases according to their volume fraction on each control volume. Finally, a shared temperature is calculated from a single energy equation [42].
Mass conservation for VOF model can be expressed as
In this model, Eqs. (2) and (3) are used as momentum and energy equations.
According to literature, a few studies have been done on using the VOF model for the simulation of nanofluids. Akbari et al. [43] studied turbulentforced convective heat transfer of Al_{2}O_{3}water as well as Cuwater nanofluids inside a horizontal tube under uniform heat flux. The governing equations were solved implementing different numerical approaches, for example, singlephase, VOF, mixture, and Eulerian models, using FLUENT software. The results showed that the thermal field forecasting by multiphase models was different from the results of experimental data and singlephase approach. However, singlephase and twophase models predicted almost same hydrodynamic results. It was concluded that unlike the results of previous studies [17, 44], twophase models overestimate the thermal field. Under similar conditions, however, Hejazian et al. [45] found different results when investigating the turbulent convection of TiO_{2}water nanofluid in a horizontal tube using FVM method. The results of this study showed that the mixture and VOF models are more appropriate to predict the heat transfer field, compared to singlephase model.
Turbulent heat transfer of nanofluids flow through a minichannel heat sink was analyzed by Naphon and Nakharintr [46]. The kε turbulence model with singlephase, mixture, and VOF approaches was employed to analyze the heat transfer and flow characteristics. Also, some experiments conducted to verify the predicted results and reasonable agreements were achieved. It was concluded that the singlephase model cannot predict the Nusselt number with accuracy as good as mixture and VOF models because the impacts of Brownian motion and nonuniform distribution of nanoparticles in the solution domain are not considered in the singlephase model. In addition, under similar conditions, VOF and mixture models present more appropriate results compared to the singlephase model.
Hanafizadeh et al. [47] carried out a study to compare single and twophase approaches for Fe_{3}O_{4}water nanofluid in both developing and fully developed regions in a circular tube under constant heat flux. The study was conducted for 0.5–2 vol. % and 300 ≤ Re ≤ 1200. The results showed that higher values of both Reynolds number and volume fraction would augment the average heat transfer coefficient, while just increasing the number of dispersed nanoparticles does not have a considerable impact on heat transfer enhancement. Also, in the fully developed region, a higher number of dispersed nanoparticles in base fluid would reduce the error of studied numerical methods. On the other hand, in developing region of a tube and for low Reynolds numbers, increase in nanofluid volume fraction would decrease the accuracy of numerical methods, while this trend was reversed for moderate and high Reynolds numbers.
In total, since a limited number of studies have used this numerical approach, further studies are needed to evaluate the capability of the VOF model.
2.2.2.2. Mixture model
The mixture model is one of the most popular methods for modeling of multiphase slurry flows. The main feature of this approach is that only one set of velocity elements is solved for the mixture momentum conservation equations. The velocities of dispersed phases are extracted from the algebraic formulations [48]. Moreover, since the primary phase affects the secondary phase through drag force and turbulence, the effect of secondary phase on the primary phase could be found through mean momentum reduction and turbulence. The basic assumptions of mixture model are as follows [49, 50]:
All phases share a single pressure.
The interaction between different dispersed phases is assumed to be negligible.
Nanoparticles in the secondarily dispersed phase are spherical in shape, with a uniform size.
The concentration of the secondarily dispersed phases is solved by a scalar equation, considering the correction made by phase slip.
The governing equations of the nanofluids’ mixture model can be written as follows [51]:
Continuity:
Momentum:
Energy:
As a pioneer, Behzadmehr et al. [44] employed singlephase approach as well as twophase mixture model to study the turbulent heat transfer of copperwater nanofluid inside a circular tube. The results of their study revealed that the obtained results from mixture model are much closer to experimental data, compared to the results of singlephase model. In a similar study, Bianco et al. [52] analyzed the steadystate turbulent convection heat transfer of Al_{2}O_{3}water nanofluid in a circular tube under constant heat flux. FLUENT commercial software was used to solve the governing equations. The results showed that singlephase and mixture models give approximately the same results at low concentrations (i.e.,
Shariat et al. simulated aluminawater nanofluid in an elliptic tube [53]. The impacts of nanoparticles mean diameter and buoyancy force on the nanofluid flow behaviors were investigated in that study. The threedimensional equations of the mixture model were solved by using FVM. The results showed that at a specified value of Reynolds and Richardson numbers, an increase in nanoparticles size diminishes the Nusselt number while it does not have a remarkable effect on the friction factor. A nonlinear relation between the nanoparticles size and heat transfer characteristics of nanofluid was also observed.
Laminarfree convection heat transfer of aluminawater nanofluid inside a cavity was studied by Corcione et al. [54]. The governing equations were solved by a CFD code based on a twophase mixture model. Temperaturedependent effective properties considering the Brownian motion and thermophoresis were employed and different nanoparticles volume fractions were analyzed. It was found that the heat transfer trend reached a peak value at maximum particle loading. Using these results, new correlations were developed for different parameters such as the optimal particle loading and a maximum value of the heat transfer augmentation.
Goodarzi et al. [51] investigated both laminar and turbulent mixed convection of Cuwater nanofluid inside a rectangular shallow cavity. The upper movable lid of the cavity was considered at a lower temperature, compared to the bottom wall. FLUENT commercial code was utilized to solve the problem, along with some modifications in governing equations by developing userdefined function (UDF) codes. The results showed that the impact of the volume fraction on turbulent kinetic energy, turbulence intensity, skin friction, and wall shear stress is insignificant. However, under similar conditions, lower Richardson number leads to higher wall shear stress and turbulence kinetics energy.
The single and twophase models were employed by Naphon and Nakharintr [55] to investigate the 3D laminar convection heat transfer of nanofluids inside a minichannel heat sink. Some experiments were also carried out for validation purpose. The research outcomes demonstrated that twophase mixture model is in better agreement with experimental results, compared to singlephase model.
Recently, Siavashi et al. [56] investigated the application of nanofluids and porous media to enhance the heat transfer inside an annular pipe. The simulation was conducted to investigate the effects of different parameters such as the Darcy and Reynolds numbers as well as porous medium radius and its position on heat transfer enhancement, heat loss, and entropy generation. Twophase mixture model along with DarcyBrinkmanForchheimer equation was employed for nanofluid flow simulation in porous media. A FVM code was developed to solve the governing equations. The results showed that the geometry, nanoparticle concentration, and magnitude of the Reynolds number have considerable effects on both the performance and entropy generation numbers.
By reviewing the literature, it can be seen that among different multiphase approaches, the mixture model is the most popular model for nanofluids modeling. This popularity can be due to some facts such as accuracy, simplicity in both theory and implementation, and low computational cost. However, for using this model there are some limitations and requirements, which were addressed in detail by Moraveji and Ardehali [49], Bahiraei [50], and Goodarzi et al. [51].
2.2.2.3. Eulerian model
In this model, pressure is assumed to be equal for all the phases, while other governing equations are solved separately for primary and secondary phases. The volume of the two phases is estimated by integrating the volume fraction on solution domain, where the aggregate of volume fractions totality becomes one [50]. The Eulerian model corresponding equations can be expressed as follows [42]:
Continuity:
Conservation of momentum (
The nanoparticle Reynolds number (Re_{p}) in Eq. (36) and lift force in Eq. (34) [58] are, however, based on particlefluid relative velocity, which is extremely small for nanoparticles.
Considering Eq. (37), the first two terms on the right side of Eq. (34) should be ignored.
Conservation of energy:
Where
Kalteh et al. [59] investigated the laminarforced convection heat transfer of Cuwater nanofluid inside a microchannel. The Eulerian model utilized for flow simulation and governing equations was solved by FVM. The results demonstrated that the nanoparticles are distributed uniformly inside the solution domain. The twophase model also presented a higher heat transfer augmentation compared to the singlephase model.
Laminar and turbulentforced convection of nanofluids inside small tubes were investigated by Chen et al. [60]. The multiphase flow was simulated using both mixture and Eulerian models and the results were compared with experimental data as well as the correlations from the literature. The obtained results for two models were quite similar, although mixture approach results showed better agreement with experimental results.
Thermal behavior and nanofluid flow at the entrance region of a pipe under constant heat flux were modeled by Göktepe et al. [61]. The results demonstrated that twophase models predict heat transfer coefficient and friction factor with a higher accuracy at the entrance region, compared to the singlephase model. The authors also suggested that more suitable relations for nanoparticles are required to enhance the forecast accuracy of the Eulerian model.
Recently, Sadeghi et al. [62] studied nanoparticle aggregation effect on laminar convection heat transfer of aluminawater nanofluid in a circular tube. The Eulerian model was implemented according to nanoclusters Brownian motion and their fractal structure. The governing equations were solved using ANSYS CFX commercial software. The results revealed that nanoparticles size and concentration as well as fractional structure have undeniable effects on heat transfer phenomenon of nanofluid. Also, it was noted that Brownian motion can affect the convective heat transfer of nanofluids significantly.
All in all, it can be concluded that the main advantage of the Eulerian model in comparison to singlephase model is that there is no need to apply effective property models for the nanofluids [59]. However, it may not be as precise as the mixture model [17, 60].
2.3. Other approaches
2.3.1. LatticeBoltzmann Method
Lately, LatticeBoltzmann method or Thermal LatticeBoltzmann method has become an attractive alternative to simulate the nanofluids flow. The gap between microscopic and macroscopic phenomena is removed by employing LatticeBoltzmann method since it considers molecular dynamics [50]. In LatticeBoltzmann method, the conservation equations are resolved by the assumption that the nanoparticles are microscopically located in a chain of lattices where their distributions are determined based on Boltzmann method. In the paper of Succi [63], microscopic interaction between the nanoparticles was numerically modeled utilizing a collision model and microscopic and macroscopic quantities of components were joined together. Also in [64, 65], two more different methods were employed, namely D2Q9 (twodimensional and 9velocity) square and D3Q19 (threedimensional and 19velocity) cube lattice structures. In LatticeBoltzmann method, it is easy to deal with the complex boundaries; also, the other advantages of this method include physical representation of microscopic interactions and the existence of uniform algorithms to solve the multiphase flows [65].
For the first time, Xuan and Yao [66] proposed LBM for simulating flow and energy transport of the nanofluids. After this study, the use of this method was rapidly increased. Considering interaction forces such as Brownian, gravitybuoyancy, drag, and interaction potential forces between two phases, Qi et al. [67] studied the free convection of nanofluid using a twophase LatticeBoltzmann model. It was found that while Brownian, gravitybuoyancy, and interaction potential forces have positive impacts on the augmentation of free convection, drag force has a negative impact.
Karimipour et al. [68] studied laminarforced convective heat transfer of copperwater nanofluid inside a microchannel using doublepopulation LBMBGK method. The obtained 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 configurations.
Recently, by employing a 2D double multiplerelaxationtime (MRT) thermal LatticeBoltzmann model, Zhang and Che [69] simulated the magnetohydrodynamic (MHD) flow and heat transfer of copper water in an inclined cavity with four heat sources. The governing equations were solved using D2Q9 and D2Q5MRT 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 LatticeBoltzmann method is competent for solving heat transfer of nanofluids in enclosures affected by a magnetic field.
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 characteristics.
2.3.2. Nonhomogeneous twocomponent 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 nanofluids 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:
Conservation of momentum:
Conservation of energy:
Conservation of nanoparticles:
The aforementioned terms can be calculated as follows [75]:
Sheikhzadeh et al. [76] studied the effects of Brownian motion, thermophoresis, and Dufour (transport model) on laminarfree convection heat transfer of aluminawater 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 experimental data and numerical results revealed that the transport model is in better agreement with experimental results, compared to singlephase model.
Using the same method, Bahiraei et al. [77] studied the laminar convection heat transfer of aluminawater 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 Al_{2}O_{3}water nanofluid inside a vertical annular pipe. The governing equations reduced to twopoint O.D.E.s, which were solved by means of the RungeKuttaFehlberg 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 migration, 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 Al_{2}O_{3} nanoparticles. The left and right walls of the cavity had a complexwavy geometry while upper and lower walls were both flat and insulated. Complexwavy walls were modeled as the superposition of two sinusoidal functions. The results revealed that heat transfer performance may be optimized by tuning the wavysurface geometry parameter in accordance with the Rayleigh number. Using optimal homotopy analysis method (OHAM), Nadeem et al. [92] examined 2D stagnation point flow of a nonNewtonian Casson nanofluid over a convectivestretching surface. The governing nonlinear partial differential equations were converted into nonlinear 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 nonNewtonian parameter, Brownian motion, and thermophoresis.
The laminar axisymmetric flow of a nanofluid over a nonlinearly 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 KellerBox 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.
3. Conclusion
A comprehensive review of popular methods in the simulation of the nanofluids was carried out. Different CFD approaches including singlephase, multiphase, and other methods were reviewed. For each model, the governing equations and recent literature were studied.
Conventional singlephase model was the most common method to study the convective heat transfer of nanofluids. This can be due to the fact that this model simplifies the simulation and in comparison to other models has the lowest computational cost. However, the results obtained from this model may have some deviation from the experimental data. For instance, it was reported in many studies that homogeneous model underestimates the heat transfer coefficient and Nusselt number, when compared to the dispersion and twophase models. However, it was also revealed that using the temperaturedependent thermophysical properties in homogeneous model can lead to more realistic results. On the other hand, dispersion model for both constant and temperaturedependent properties showed promising results, compared with experimental data. This model requires less computational time compared to twophase model. In addition, the model takes into account thermal dispersion effect, which leads to more reliable results in comparison with the homogeneous model.
Nanofluids are inherently multiphase fluids; therefore, employing twophase model taking into account the slip velocity, Brownian motion, thermophoresis, and so forth, can lead to more appropriate results. Most of the publications confirmed that different twophase models predict more accurate results than the homogeneous model. Also, higher values of the heat transfer coefficient were reported for twophase models, compared to conventional singlephase model. A vast number of studies utilized the mixture and Eulerian models, and to smaller extent VOF and LagrangianEulerian models. Some publications noted that among all twophase models, mixture model predicts more precise results compared with experimental data. However, this model has some limitation and cannot be applied in some cases. On the other hand, since VOF and LagrangianEulerian models are employed less than other twophase models, it seems that further research might be needed to assess their precision in nanofluids simulation.
In the end, LBM and nonhomogeneous twocomponent models are rather novel approaches, used in several cases. The results predicted by these approaches showed a promising accordance with the results obtained from previous studies. Moreover, according to literature, these methods may present some wellknown advantages in the modeling of nanofluids. Obviously, more attempts should be made to find the flow characteristics of nanofluids in various systems and in the presence of different modes of heat transfer in order to examine the aforementioned approaches.
Acknowledgments
The authors gratefuly acknowledge High Impact Research Grant UM.C/HIR/MOHE/ENG/23 and the Faculty of Engineering, University of Malaya, Malaysia, for support in conducting this research work.
Diameter (m)  
Drag coefficient  
V  volume (m^{3}) 
F  Force (kg m/s^{2}) 
m  mass (kg) 
Flow velocity in 

Fluid pressure (Pa)  
Gravity acceleration (m/s^{2})  
Heat capacity (J/kg K)  
Nu  Nusselt number 
Re  Reynolds number 
h  Sensible enthalpy (J/Kg) 
Temperature (K)  
Thermal conductivity (W/m K)  
t  Time (s) 
Velocity components in 
thermal diffusivity (m^{2}/s)  
Density (kg/m^{3})  
Dynamic viscosity (Pa s)  
Nanoparticle volume fraction  
Thermal expansion coefficient (1/K) 
Br  Brownian motion 
eff  Effective 
f  Fluid 
Z, q  Indices 
np  nanoparticle 
m  Mixture 
nf  Nanofluid 
p  Particle 
T  Thermophoresis 
References
 1.
Yarmand H, Gharehkhani S, Kazi SN, Sadeghinezhad E, Safaei MR. Numerical investigation of heat transfer enhancement in a rectangular heated pipe for turbulent nanofluid. The Scientific World Journal. 2014;2014(Article ID 369593):1–9. doi:10.1155/2014/369593.  2.
Mojumder S, Sourav S, Sumon S, Mamun M. Combined effect of Reynolds and Grashof numbers on mixed convection in a liddriven Tshaped cavity filled with waterAl_{2}O_{3} nanofluid. Journal of Hydrodynamics, Ser B. 2015;27(5):782–94.  3.
AbuNada E, Chamkha AJ. Mixed convection flow in a liddriven inclined square enclosure filled with a nanofluid. European Journal of MechanicsB/Fluids. 2010;29(6):472–82.  4.
Rostamani M, Hosseinizadeh S, Gorji M, Khodadadi J. Numerical study of turbulent forced convection flow of nanofluids in a long horizontal duct considering variable properties. International Communications in Heat and Mass Transfer. 2010;37(10):1426–31.  5.
Pak BC, Cho YI. Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles. Experimental Heat Transfer an International Journal. 1998;11(2):151–70.  6.
Maiga SEB, Palm SJ, Nguyen CT, Roy G, Galanis N. Heat transfer enhancement by using nanofluids in forced convection flows. International Journal of Heat and Fluid Flow. 2005;26(4):530–46.  7.
AbuNada E. RayleighBénard convection in nanofluids: effect of temperature dependent properties. International Journal of Thermal Sciences. 2011;50(9):1720–30.  8.
Bouhalleb M, Abbassi H. Numerical Investigation Of Heat Transfer By CuO–Water Nanofluid In Rectangular Enclosures. Heat Transfer Engineering. 2016;37(1):13–23.  9.
Togun H, Safaei MR, Sadri R, Kazi SN, Badarudin A, Hooman K et al. Numerical simulation of laminar to turbulent nanofluid flow and heat transfer over a backwardfacing step. Applied Mathematics and Computation. 2014;239:153–70.  10.
Yarmand H, Ahmadi G, Gharehkhani S, Kazi SN, Safaei MR, Alehashem MS et al. Entropy generation during turbulent flow of zirconiawater and other nanofluids in a square cross section tube with a constant heat flux. Entropy. 2014;16(11):6116–32.  11.
Togun H, Ahmadi G, Abdulrazzaq T, Shkarah AJ, Kazi S, Badarudin A et al. Thermal performance of nanofluid in ducts with double forwardfacing steps. Journal of the Taiwan Institute of Chemical Engineers. 2015;47:28–42.  12.
Selimefendigil F, Öztop HF, Chamkha AJ. Fluidstructuremagnetic field interaction in a nanofluid filled liddriven cavity with flexible side wall. European Journal of Mechanics – B/Fluids. doi: http://dx.doi.org/10.1016/j.euromechflu.2016.03.009 .  13.
Selimefendigil F, Öztop HF. Conjugate natural convection in a cavity with a conductive partition and filled with different nanofluids on different sides of the partition. Journal of Molecular Liquids. 2016;216:67–77. doi:http://dx.doi.org/10.1016/j.molliq.2015.12.102.  14.
Selimefendigil F, Öztop HF, Chamkha AJ. Mhd mixed convection and entropy generation of nanofluid filled lid driven cavity under the influence of inclined magnetic fields imposed to its upper and lower diagonal triangular domains. Journal of Magnetism and Magnetic Materials. 2016;406:266–81. doi:http://dx.doi.org/10.1016/j.jmmm.2016.01.039.  15.
Buddakkagari V, Kumar M. Transient boundary layer laminar free convective flow of a nanofluid over a vertical cone/plate. International Journal of Applied and Computational Mathematics. 2015;1(3):427–48.  16.
Jahanbin AH, Javaherdeh K. Numerical investigation of CuO nanoparticles effect on forced convective heat transfer inside a minichannel: comparison of different approaches. Life Science Journal. 2013;10(8s):183–189.  17.
Lotfi R, Saboohi Y, Rashidi A. Numerical study of forced convective heat transfer of nanofluids: comparison of different approaches. International Communications in Heat and Mass Transfer. 2010;37(1):74–8.  18.
Ding Y, Wen D. Particle migration in a flow of nanoparticle suspensions. Powder Technology. 2005;149(2):84–92.  19.
Nikkhah Z, Karimipour A, Safaei MR, ForghaniTehrani P, Goodarzi M, Dahari M et al. Forced convective heat transfer of water/functionalized multiwalled carbon nanotube nanofluids in a microchannel with oscillating heat flux and slip boundary condition. International Communications in Heat and Mass Transfer. 2015;68:69–77.  20.
Safaei M, Togun H, Vafai K, Kazi S, Badarudin A. Investigation of heat transfer enhancement in a forwardfacing contracting channel using FMWCNT nanofluids. Numerical Heat Transfer, Part A: Applications. 2014;66(12):1321–40.  21.
Xuan Y, Roetzel W. Conceptions for heat transfer correlation of nanofluids. International Journal of heat and Mass transfer. 2000;43(19):3701–7.  22.
Plumb O, editor. The effect of thermal dispersion on heat transfer in packed bed boundary layers. ASME/JSME Thermal Engineering Joint Conference; 1983.  23.
Hunt M, Tien C. Effects of thermal dispersion on forced convection in fibrous media. International Journal of Heat and Mass Transfer. 1988;31(2):301–9.  24.
Khaled AR, Vafai K. Heat transfer enhancement through control of thermal dispersion effects. International Journal of Heat and Mass Transfer. 2005;48(11):2172–85.  25.
Mojarrad MS, Keshavarz A, Shokouhi A. Nanofluids thermal behavior analysis using a new dispersion model along with singlephase. Heat and Mass Transfer. 2013;49(9):1333–43.  26.
Zeinali Heris S, Noie SH, Talaii E, Sargolzaei J. Numerical investigation of Al_{2}O_{3}/water nanofluid laminar convective heat transfer through triangular ducts. Nanoscale Research Letters. 2011;6(1):1.  27.
Zeinali Heris S, KazemiBeydokhti A, Noie S, Rezvan S. Numerical study on convective heat transfer of Al_{2}O_{3}/water, CuO/water and Cu/water nanofluids through square crosssection duct in laminar flow. Engineering Applications of Computational Fluid Mechanics. 2012;6(1):1–14.  28.
Heris SZ, Etemad SG, Esfahany MN. Experimental investigation of oxide nanofluids laminar flow convective heat transfer. International Communications in Heat and Mass Transfer. 2006;33(4):529–35.  29.
Zeinali Heris S, Esfahany MN, Etemad SG. Experimental investigation of convective heat transfer of Al_{2}O_{3}/water nanofluid in circular tube. International Journal of Heat and Fluid Flow. 2007;28(2):203–10.  30.
Kumar S, Prasad SK, Banerjee J. Analysis of flow and thermal field in nanofluid using a single phase thermal dispersion model. Applied Mathematical Modelling. 2010;34(3):573–92.  31.
Akbaridoust F, Rakhsha M, Abbassi A, SaffarAvval M. Experimental and numerical investigation of nanofluid heat transfer in helically coiled tubes at constant wall temperature using dispersion model. International Journal of Heat and Mass Transfer. 2013;58(1):480–91.  32.
Safaei M, Mahian O, Garoosi F, Hooman K, Karimipour A, Kazi S et al. Investigation of microand nanosized particle erosion in a 90° pipe bend using a twophase discrete phase model. The Scientific World Journal. 2014;2014(Article ID 740578):1–12. doi:10.1155/2014/740578.  33.
Minkowycz W, Sparrow EM, Murthy J. Handbook of numerical heat transfer. Wiley Online Library; New York 1988.  34.
Mirzaei M, Saffar Avval M, Naderan H. Heat transfer investigation of laminar developing flow of nanofluids in a microchannel based on Eulerian–Lagrangian approach. The Canadian Journal of Chemical Engineering. 2014;92(6):1139–49.  35.
Oosthuizen PH, Carscallen WE. Introduction to compressible fluid flow. Boca Raton, FL: CRC Press; 2013.  36.
Bahremand H, Abbassi A, SaffarAvval M. Experimental and numerical investigation of turbulent nanofluid flow in helically coiled tubes under constant wall heat flux using Eulerian–Lagrangian approach. Powder Technology. 2015;269:93–100.  37.
Moraveji MK, Esmaeili E. Comparison between singlephase and twophases CFD modeling of laminar forced convection flow of nanofluids in a circular tube under constant heat flux. International Communications in Heat and Mass Transfer. 2012;39(8):1297–302.  38.
Tahir S, Mital M. Numerical investigation of laminar nanofluid developing flow and heat transfer in a circular channel. Applied Thermal Engineering. 2012;39:8–14.  39.
Behroyan I, Ganesan P, He S, Sivasankaran S. Turbulent forced convection of Cu–water nanofluid: CFD model comparison. International Communications in Heat and Mass Transfer. 2015;67:163–72.  40.
Vanaki SM, Ganesan P, Mohammed H. Numerical study of convective heat transfer of nanofluids: a review. Renewable and Sustainable Energy Reviews. 2016;54:1212–39.  41.
Xu P, Wu Z, Mujumdar A, Yu B. Innovative hydrocyclone inlet designs to reduce erosioninduced wear in mineral dewatering processes. Drying Technology. 2009;27(2):201–11.  42.
Akbari M, Galanis N, Behzadmehr A. Comparative analysis of single and twophase models for Cfd studies of nanofluid heat transfer. International Journal of Thermal Sciences. 2011;50(8):1343–54.  43.
Akbari M, Galanis N, Behzadmehr A. Comparative assessment of single and twophase models for numerical studies of nanofluid turbulent forced convection. International Journal of Heat and Fluid Flow. 2012;37:136–46.  44.
Behzadmehr A, SaffarAvval M, Galanis N. Prediction of turbulent forced convection of a nanofluid in a tube with uniform heat flux using a two phase approach. International Journal of Heat and Fluid Flow. 2007;28(2):211–9.  45.
Hejazian M, Moraveji MK, Beheshti A. Comparative numerical investigation on TiO_{2}/water nanofluid turbulent flow by implementation of single phase and two phase approaches. Numerical Heat Transfer, Part A: Applications. 2014;66(3):330–48.  46.
Naphon P, Nakharintr L. Turbulent two phase approach model for the nanofluids heat transfer analysis flowing through the minichannel heat sinks. International Journal of Heat and Mass Transfer. 2015;82:388–95.  47.
Hanafizadeh P, Ashjaee M, Goharkhah M, Montazeri K, Akram M. The comparative study of single and twophase models for magnetite nanofluid forced convection in a tube. International Communications in Heat and Mass Transfer. 2015;65:58–70.  48.
ElBatsh H, Doheim M, Hassan A. On the application of mixture model for twophase flow induced corrosion in a complex pipeline configuration. Applied Mathematical Modelling. 2012;36(11):5686–99.  49.
Moraveji MK, Ardehali RM. Cfd modeling (comparing single and twophase approaches) on thermal performance of Al_{2}O_{3}/water nanofluid in minichannel heat sink. International Communications in Heat and Mass Transfer. 2013;44:157–64.  50.
Bahiraei M. A comprehensive review on different numerical approaches for simulation in nanofluids: traditional and novel techniques. Journal of Dispersion Science and Technology. 2014;35(7):984–96.  51.
Goodarzi M, Safaei M, Vafai K, Ahmadi G, Dahari M, Kazi S et al. Investigation of nanofluid mixed convection in a shallow cavity using a twophase mixture model. International Journal of Thermal Sciences. 2014;75:204–20.  52.
Bianco V, Manca O, Nardini S. Numerical investigation on nanofluids turbulent convection heat transfer inside a circular tube. International Journal of Thermal Sciences. 2011;50(3):341–9.  53.
Shariat M, Moghari RM, Akbarinia A, Rafee R, Sajjadi S. Impact of nanoparticle mean diameter and the buoyancy force on laminar mixed convection nanofluid flow in an elliptic duct employing two phase mixture model. International Communications in Heat and Mass Transfer. 2014;50:15–24.  54.
Corcione M, Cianfrini M, Quintino A. Twophase mixture modeling of natural convection of nanofluids with temperaturedependent properties. International Journal of Thermal Sciences. 2013;71:182–95.  55.
Naphon P, Nakharintr L. Numerical investigation of laminar heat transfer of nanofluidcooled minirectangular fin heat sinks. Journal of Engineering Physics and Thermophysics. 2015;88(3):666–75.  56.
Siavashi M, Bahrami HRT, Saffari H. Numerical investigation of flow characteristics, heat transfer and entropy generation of nanofluid flow inside an annular pipe partially or completely filled with porous media using twophase mixture model. Energy. 2015;93:2451–66.  57.
Schiller L, Naumann Z. A drag coefficient correlation. Vdi Zeitung. 1935;77(318):51.  58.
Drew D, Lahey R. Analytical modeling of multiphase flow. In: Particulate twophase flow. Butterworth–Heinemann Boston 1993:509–66.  59.
Kalteh M, Abbassi A, SaffarAvval M, Harting J. Eulerian–Eulerian twophase numerical simulation of nanofluid laminar forced convection in a microchannel. International journal of heat and fluid flow. 2011;32(1):107–16.  60.
Chen Yj, Li Yy, Liu Zh. Numerical simulations of forced convection heat transfer and flow characteristics of nanofluids in small tubes using twophase models. International Journal of Heat and Mass Transfer. 2014;78:993–1003.  61.
Göktepe S, Atalık K, Ertürk H. Comparison of single and twophase models for nanofluid convection at the entrance of a uniformly heated tube. International Journal of Thermal Sciences. 2014;80:83–92.  62.
Sadeghi R, Haghshenasfard M, Etemad SG, Keshavarzi E. Theoretical investigation of nanoparticles aggregation effect on wateralumina laminar convective heat transfer. International Communications in Heat and Mass Transfer. 2016;72:57–63.  63.
Succi S. Lattice Boltzmann equation: failure or success? Physica A: Statistical Mechanics and its Applications. 1997;240(1):221–8.  64.
Karimipour A, Esfe MH, Safaei MR, Semiromi DT, Jafari S, Kazi S. Mixed convection of copper–water nanofluid in a shallow inclined lid driven cavity using the Lattice Boltzmann method. Physica A: Statistical Mechanics and its Applications. 2014;402:150–68.  65.
Zhang J. Lattice Boltzmann method for microfluidics: models and applications. Microfluidics and Nanofluidics. 2011;10(1):1–28.  66.
Xuan Y, Yao Z. Lattice Boltzmann model for nanofluids. Heat and mass transfer. 2005;41(3):199–205.  67.
Qi C, He Y, Yan S, Tian F, Hu Y. Numerical simulation of natural convection in a square enclosure filled with nanofluid using the twophase Lattice Boltzmann method. Nanoscale Research Letters. 2013;8(1):1–16.  68.
Karimipour A, Nezhad AH, D'Orazio A, Esfe MH, Safaei MR, Shirani E. Simulation of copper–water nanofluid in a microchannel in slip flow regime using the Lattice Boltzmann method. European Journal of MechanicsB/Fluids. 2015;49:89–99.  69.
Zhang T, Che D. Double Mrt thermal Lattice Boltzmann simulation for MHD natural convection of nanofluids in an inclined cavity with four square heat sources. International Journal of Heat and Mass Transfer. 2016;94:87–100.  70.
Kamyar A, Saidur R, Hasanuzzaman M. Application of computational fluid dynamics (CFD) for nanofluids. International Journal of Heat and Mass Transfer. 2012;55(15–16):4104–15.  71.
Sidik NAC, Razali SA. Lattice Boltzmann method for convective heat transfer of nanofluids–a review. Renewable and Sustainable Energy Reviews. 2014;38:864–75.  72.
Buongiorno J. Convective transport in nanofluids. Journal of Heat Transfer. 2006;128(3):240–50.  73.
Malvandi A, Ganji D. Effects of nanoparticle migration on water/alumina nanofluid flow inside a horizontal annulus with a moving core. Journal of Mechanics. 2015;31(03):291–305.  74.
Hedayati F, Malvandi A, Kaffash M, Ganji D. Fully developed forced convection of alumina/water nanofluid inside microchannels with asymmetric heating. Powder Technology. 2015;269:520–31.  75.
Malvandi A, Moshizi SA, Ganji DD. Twocomponent heterogeneous mixed convection of alumina/water nanofluid in microchannels with heat source/sink. Advanced Powder Technology. 2016;27(1):245–54. doi:http://dx.doi.org/10.1016/j.apt.2015.12.009.  76.
Sheikhzadeh GA, Dastmalchi M, Khorasanizadeh H. Effects of nanoparticles transport mechanisms on Al_{2}O_{3}–water nanofluid natural convection in a square enclosure. International Journal of Thermal Sciences. 2013;66:51–62.  77.
Bahiraei M, Mostafa Hosseinalipour S, Hangi M. Prediction of convective heat transfer of Al_{2}O_{3}water nanofluid considering particle migration using neural network. Engineering Computations. 2014;31(5):843–63.  78.
Malvandi A, Safaei M, Kaffash M, Ganji D. Mhd mixed convection in a vertical annulus filled with Al_{2}O_{3}–water nanofluid considering nanoparticle migration. Journal of Magnetism and Magnetic Materials. 2015;382:296–306.  79.
Malvandi A. The unsteady flow of a nanofluid in the stagnation point region of a timedependent rotating sphere. Thermal Science. 2015;19(5):1603–12.  80.
Malvandi A, Moshizi S, Soltani EG, Ganji D. Modified Buongiorno's model for fully developed mixed convection flow of nanofluids in a vertical annular pipe. Computers & Fluids. 2014;89:124–32.  81.
Malvandi A, Ganji DD. Magnetohydrodynamic mixed convective flow of Al_{2}O_{3}–water nanofluid inside a vertical microtube. Journal of Magnetism and Magnetic Materials. 2014;369(0):132–41. doi:http://dx.doi.org/10.1016/j.jmmm.2014.06.037.  82.
Malvandi A, Ganji D. Effects of nanoparticle migration on force convection of alumina/water nanofluid in a cooled parallelplate channel. Advanced Powder Technology. 2014;25(4):1369–75.  83.
Malvandi A, Moshizi SA, Ganji DD. Effect of magnetic fields on heat convection inside a concentric annulus filled with Al_{2}O_{3}–water nanofluid. Advanced Powder Technology. doi: http://dx.doi.org/10.1016/j.apt.2014.07.013 2016;25 (6):18171824..  84.
Malvandi A, Ganji DD. Magnetic field effect on nanoparticles migration and heat transfer of water/alumina nanofluid in a channel. Journal of Magnetism and Magnetic Materials. 2014;362:172–9. doi:http://dx.doi.org/10.1016/j.jmmm.2014.03.014.  85.
Malvandi A, Ganji DD. Brownian motion and thermophoresis effects on slip flow of alumina/water nanofluid inside a circular microchannel in the presence of a magnetic field. International Journal of Thermal Sciences. 2014;84:196–206. doi:http://dx.doi.org/10.1016/j.ijthermalsci.2014.05.013.  86.
Moshizi SA, Malvandi A, Ganji DD, Pop I. A twophase theoretical study of Al_{2}O_{3}–water nanofluid flow inside a concentric pipe with heat generation/absorption. International Journal of Thermal Sciences. 2014;84:347–57. doi:http://dx.doi.org/10.1016/j.ijthermalsci.2014.06.012.  87.
Malvandi A, Ganji D. Effects of nanoparticle migration on hydromagnetic mixed convection of alumina/water nanofluid in vertical channels with asymmetric heating. Physica E: Lowdimensional Systems and Nanostructures. 2015;66:181–96.  88.
Malvandi A, Ganji D, Kaffash M. Magnetic field effects on nanoparticle migration and heat transfer of alumina/water nanofluid in a parallelplate channel with asymmetric heating. The European Physical Journal Plus. 2015;130(4):1–21.  89.
Malvandi A, Heysiattalab S, Ganji D. Thermophoresis and Brownian motion effects on heat transfer enhancement at film boiling of nanofluids over a vertical cylinder. Journal of Molecular Liquids. 2016;216:503–9.  90.
Malvandi A. Film boiling of magnetic nanofluids (MNFs) over a vertical plate in presence of a uniform variabledirectional magnetic field. Journal of Magnetism and Magnetic Materials. 2016;406:95–102.  91.
Mansour M, Bakier M. Free convection heat transfer in complexwavywall enclosed cavity filled with nanofluid. International Communications in Heat and Mass Transfer. 2013;44:108–15.  92.
Nadeem S, Mehmood R, Akbar NS. Optimized analytical solution for oblique flow of a cassonnano fluid with convective boundary conditions. International Journal of Thermal Sciences. 2014;78:90–100.  93.
Mustafa M, Khan JA, Hayat T, Alsaedi A. Analytical and numerical solutions for axisymmetric flow of nanofluid due to nonlinearly stretching sheet. International Journal of NonLinear Mechanics. 2015;71:22–9.