Numerical simulations of the influence of Brownian and gravitational forces on the stability of CuO nanoparticles by the Eulerian Lagrangian approach

Transcription

1 Received: 31 January 2017 Revised: 23 April 2017 Accepted: 24 April 2017 DOI: /htj RESEARCH ARTICLE Numerical simulations of the influence of Brownian and gravitational forces on the stability of CuO nanoparticles by the Eulerian Lagrangian approach Habib Aminfar 1 Mousa Mohammadpourfard 2 Ramin Mortezazadeh 1 1 Faculty of Mechanical Engineering, University of Tabriz, Tabriz, Iran 2 Faculty of Chemical and Petroleum Engineering, University of Tabriz, Tabriz, Iran Correspondence Habib Aminfar, Faculty of Mechanical Engineering, University of Tabriz, Tabriz , Iran hh_aminfar@tabrizu.ac.ir Abstract This study investigates the influence of Brownian and gravitational forces on the velocity and transport of induced CuO nanoparticles in oil. Moreover, the concept of gravitational sedimentation and colloidal stability have been taken into consideration to give more physical insight about how these forces influence on the particle s displacement. For this purpose, the Lagrangian method has been proposed to compute the instantaneous location of the interested nanoparticle while the continuous medium behaviors are reflected by an Eulerian description. The relative concentration of the nanoparticles is measured to support the physical validity of the numerical procedure. It is observed that the predictions of the present Euler Lagrange model are in good agreement with the reported experimental data in the literature. In the absence of the gravitational force, the velocity magnitude of the nanoparticles is decreased and the Brownian motion keeps the dispersion stable. The velocity of nanoparticles Nomenclature: C c, Cunningham correction factor; D B, Brownian diffusion coefficient, (m 2 s); d p, diameter of nanoparticle, (m); F, force, (n); g, gravitational acceleration, (= 9.81 m s 2 ); H, height of vessel, (m); Kn, Knudsen number; k B, Boltzmann constant, (= j k); m, mass, (kg); n, number of nanoparticles; P, pressure, (pa); S m, pnterphase momentum transfer rate, (n m 3 ); T, temperature, (k); t, time, (s); Δt, time step, (s); u, velocity, (m s); x, axial direction, (m); y, vertical direction, (m) Greek abbreviation: μ, dynamic viscosity, (pa.s); ρ, density, (kg m 3 ); λ, molecular mean free Path of base fluid, (m); δv, volume of a computational cell, (m 3 ) Subscripts: f, fluid; p, particle; Br, Brownian motion; Buo, buoyancy; d, drift;gr, gravity Heat Transfer Asian Res. 2018;47: wileyonlinelibrary.com/journal/htj 2017 Wiley Periodicals, Inc. 72

2 AMINFAR ET AL. 73 in the presence of the gravity field is in order of 10 9 while its velocity is in the order of m/s to m/s in the absence of gravity field. KEYWORDS Brownian motion, Eulerian Lagrangian, gravitational force, nanofluid, sedimentation 1 INTRODUCTION Nanofluids are a new kind of multifunctional colloids that are produced by the dispersion of nanosized particles into a liquid medium. Applications of the nanofluids have been applied in a variety of areas: from microprocessors to industrial heat exchangers, and in different practical fields such as cancer detection and for treating equipment in industrial engineering systems. 1 6 Still, the existence of the associated unresolved challenges such as high cost and colloidal instability prevents nanofluids from many micro- and macroscale applications. 7 The miniature size of the dispersed nanoparticles in the nanofluid provides a firm mixture with the base fluid at near-molecular level. 8 Thus, the nanoparticles remain homogeneously dispersed and show a robust resistance against gravity, which has a tendency to transport nanoparticles to the bottom of the containers. The small size of the nanoparticles does coincide with a high level of surface energy, which is the most important source of colloidal instability in the nanofluids. 1 Subsequently, the attractive Van der Waals force initiates nanoparticles aggregation and form the nanoparticles clusters. The aggregated nanoparticles begin settling down due to the gravitational force effects, which is known as the nanoparticles sedimentation process. 9 While nanoparticles leave the nanofluid during the sedimentation process and the nanofluid becomes a purer fluid, the effective thermal conductivity of nanofluid is decreased anomalously. Regarding the above explanations, nanoparticles aggregation decreases the level of colloidal stability; however, the effects of nanoparticles aggregation on the thermal performance of the nanofluids still remain beyond the mere. Based on the prior knowledge, it is necessary to have a better insight into the sedimentation process and effective slip mechanisms of the nanoparticles. Quantitative studies have been focused on the different slip mechanism s importance in the nanofluids, the effects of nanoparticles aggregation, and the Brownian motion on the heat transport augmentation in the nanofluid. Savithiri and colleagues 10 presented a scaling analysis to open new windows to understanding the role of different slip mechanisms on the nanoparticles movement. They showed the Brownian force becomes the most prominent force in smaller particles, below 10 nm, and achieved results for any nanoparticle and nanofluid combination. Buongiorno 11 employed a scaling analysis for turbulent flow of water alumina nanofluid to describe the relative importance of different nanoparticle transport mechanisms. He also suggested that Brownian diffusion and thermophoresis become the two most important slip mechanisms in the absence of turbulent eddies and likewise achieved results for any nanoparticle and nanofluid combination. Alexander and colleagues 12 produced the sedimentation and size histogram curves for the suspension of gold nanoparticles in water. Their findings are based on the Mason Weaver sedimentation diffusion equation (MWSD) predictions and the experimental ones. The MWSD equation results were in agreement with the experimental measurements. The nanoparticles aggregation was not observed during the experiment as the MWSD equation ignores this phenomenon. More importantly, the

3 74 AMINFAR ET AL. statistical behavior of the colloids is not considered in this formalism, which means that the Brownian motion has insignificant contribution. Shima and colleagues 13 conducted an experiment to evaluate the Brownian motion participation in the thermal conductivity enhancement for the dispersion of the magnetic nanoparticles in water. They observed that in the presence of magnetic fields, nanoparticles make chain-wise aggregation structures and the Brownian motion of nanoparticles disappears. Also, Hwang and colleagues 14 presented a scale analysis and numerical work in the Eulerian frame of reference for the laminar flow of Al 2 O 3 water suspension. They concluded that the Brownian and thermophoresis diffusion have a prominent contribution in heat transport augmentation and particles migration. Gharagozloo and colleagues 15 studied thermal conductivity and aggregation evolution over time, both through an experiment and a Brownian motion base Monte Carlo method. Nanoparticles agglomeration, due to the stochastic motion, is highlighted as the prominent mechanism in the remarkable enhancement of the thermal conductivity. They observed that the advancement of aggregation decreases over time, whenever the nanoparticles cluster grew or the initial diameters of the particles enlarged. In other words, the aggregation progress coincides with the Brownian motion reduction. Between the nanofluids flows studies, the Eulerian viewpoint is the preferred method for performing a foreseeable order of magnitude analysis. Besides that, to examine the sedimentation phenomena in the nanofluid colloids the Brownian dynamic simulation is the commonly used method. Therefore, in the present work, the Euler Lagrange approach is utilized to simulate the sedimentation process of the copper oxide nanoparticles, which are dispersed in a low viscous oil. It is noteworthy that, in the colloidal stability simulation, the nanofluids sedimentation and aggregation become two inseparable phenomena. If only aggregation was simulated and sedimentation ignored, the nanoparticle concentration would remain fixed at different heights of the container However, if the nanoparticles agglomeration is neglected appropriate simulation does not yield correct sedimentation velocity. 9 Instead of direct simulation of interparticle interactions, effects of interparticle interactions are modeled alternatively using the concept of nanoparticle clusters. Through the experiments of Hwang and colleagues, 19 nanoparticle diameter is determined to be equal to the aggregation diameter. Aggregation diameters are chosen according to the electronic microscope photos of Wang and colleagues. 20 Furthermore, we have investigated the influence of the Brownian and gravitational forces on the nanoparticles velocity distribution and displacement in a stationary medium. To support the physical validity of the interested simulation, the nanoparticles relative concentration and the mean square displacement of an individual nanoparticle is assessed. This validation step is accomplished by comparing the obtained results with the experimental results of Hwang and colleagues 19 and numerical results of Ganguly and Chakraborty 21 in the literature. The importance and influence of the gravitational and Brownian forces on the nanoparticle transport and sedimentation on the base fluid in a 2D isothermal enclosure have been studied by the Euler Lagrange method. Moreover, the problem is solved for the case where the lower wall of the enclosure is heated with considering gravity, Brownian, and thermophoresis forces on the transport and sedimentation of particles. Further details of the methodology, computational assumption, materials, and results will be discussed in the following sections. 2 THEORETICAL FORMULATION 2.1 Governing equations In this study, the Euler Lagrange model provides an explanation for the sedimentation process of the induced copper oxide nanoparticles, CuO, in a low viscous oil. Likewise, the effects of the

4 AMINFAR ET AL. 75 TABLE 1 Required information and material properties of the present study Properties Value Properties Value ρ f (Kg m 3 ) 915 d p (Nm) 10,50,100 μ(pa.s) 0.03 n p 8000 ρ p (Kg m 3 ) 6320 H(m) 0.05 gravitational and Brownian forces on the nanoparticles transport and velocity distribution is determined. The required information of the nanofluid and control volume is exhibited in Table 1. In this approach, the Eulerian description specifies the continuous medium condition by solving the Navier Stokes equations. Meanwhile, the particle s position is expressed within a Lagrangian frame of reference. 22,23 Regarding carrier phase as a continuum is the first step to express the base fluid in the Eulerian standpoint. Offering this assumption depends highly on the magnitude of the Knudsen number, Kn, which is introduced as follows 11 : Kn = λ d p, (1) where d p is the diameter of an individual nanoparticle and λ shows the molecular mean free path of the base fluid molecules evolved by the following equation 10 : RT λ = πd m N A P, (2) 2 where T and P are the temperature and pressure of the base fluid, N A is the Avagodro s constant, R is the universal gas constant, and d m is the diameter of the base fluid molecules. The value of λ for the studied fluid is around 0.3 nm, and average diameter of nanoparticle clusters is 100 nm. Thus, Kn is small enough, that is, Kn = If the Knudsen number meets this condition, the assumption of base fluid as a continuum is acceptable Base fluid hydrodynamics The continuous phase is expressed by the volume-averaged Navier Stokes equations, which are solved in an Eulerian manner. The general forms of the continuity, momentum, and energy equations in the Cartesian frame of reference are defined as follows:. [ u f ] =0 (3) ( ρ f u f ) t +. ( ρ f u f u f ) = (P ) +. ( μ ( uf )) + ρf g + S m (4). ( ρ f c p,f u f T f ) =. ( kf T f ) + Se. (5) The nanoparticle s presence in the base fluid is reflected by the interphase momentum transfer rate, S m, due to the interface forces between the base fluid and the particles. Moreover, the f subscript refers to the base fluid, u is velocity, ρ is density, and μ is dynamic viscosity. The Eulerian variable

5 76 AMINFAR ET AL. S m is computed from the Lagrangian particles by volume averaging the contributions from all of the individual nanoparticles in a cell volume, which is expressed as 24 : S m = n p δv n i=1 F i, (6) S e = n p δv n i=1 c p dt p dt, (7) where p subscript refers to particles, n denotes the number of nanoparticles in the computational domain, δv is the cell volume centered on each grid, point and F is assigned as the net force that acts on each nanoparticle Nanoparticles dynamics The momentum equation of Lagrangian nanoparticles includes several forces such as drag force, F Dr, gravitational force, F Gr, Brownian motion force, F Br, and buoyancy force, F buo. Newton s equation of motion is solved, and from there the nanoparticles instantaneous velocity and location are determined. The governing equations for the spherical and solid nanoparticles are written as follows: m p d u p dt = F Dr + F Gr + F Br + F buo, (8) where d r p dt = u p. (9) The term r p is the location we are interested in and where the nanoparticle s centers are assigned. The hydrodynamic drag force that nanoparticles experience from the liquid phase is calculated by applying the Stokes drag law as follows 22 : F Dr =3πd p μ ( u f u p ). (10) Here, u f is the fluid phase velocity for the cell where the nanoparticle is located. The gravitational and buoyancy forces experienced by the nanoparticles are obtained from the upcoming expressions, which is presented in Eqs. (11) and (12): F Gr = πd3 p ρ p 6 g, (11) F buo = πd3 p ρ f 6 g. (12) Finally, the continuous collisions between a single nanoparticle and the surrounding molecules of the carrier phase correspond to the stochastic displacements. This rapidly oscillating behavior of the

6 AMINFAR ET AL. 77 nanoparticle is reflected by the Brownian force. The present numerical work employed the proposed formula of Li and Ahmadi 25 for the Brownian force and is as follows: 6πμd p k B T F Br = ξ. (13) ΔtC c The term T demonstrates the isothermal medium temperature, which is equal to 300 K, ξ is called the Gaussian random number with zero-mean unit-variance-independent, k B is the Boltzmann constant, and Δt is the simulation time step. The following relation is employed to evaluate the dimensionless Cunningham correction factor 26 : C c =1+2kn ( e ( 0.55kn)). (14) It should be noted that since the Knudsen number is much less than unity, the Cunningham correction factor is nearly equal to unity. It is worth mentioning that Brownian force could be rewritten with respect to Brownian diffusion coefficient, D B, as follows 11 : D B = k BT 3πμd p, (15) 2 F Br = ξk b T D B Δt. (16) 2.2 Initial and boundary conditions Fig. 1(a) schematically demonstrates the computational domain and the initial dispersion of the studied nanoparticles. Initially, both particles and the base fluid are considered to be stationary, and it is assumed that at the beginning, particles are uniformly distributed inside the base fluid. The problem has been solved first for an isothermal system (i.e., all of the walls of the enclosure have been considered to be adiabatic), the velocity and distribution of the particles, and the base fluid inside the enclosure due to gravity and Brownian forces using Eulerian Lagrangian method. Then, the problem is solved for the case where the lower wall of the enclosure is heated including gravity, Brownian, and thermophoresis forces. The no-slip boundary conditions have also been applied over the walls of the enclosure. At the container boundaries for the continuous medium, the no-slip boundary conditions are applied. All the fluid solid boundaries are assumed perfectly smooth; thus, in the classical Stokes flow the nanoparticle s contact velocity with the control volume boundaries should meet the following condition 21 : u p,b + u p,a =0. (17) The terms u p,b and u p,a are the nanoparticle velocity vectors before contact and after contact with the walls, respectively. 2.3 Numerical method The computational domain and the grid are demonstrated in Fig. 1(b). The height and width of the enclosure are represented by the symbols H and l, respectively. To perform a numerical modeling, the

7 78 AMINFAR ET AL. FIGURE 1 Schematic of the physical models: (a) initial distribution of the nanoparticles and (b) used grid computational domain is discretized by using the structured, equally spaced grids. A network of computational grids is utilized to carry out the numerical procedure. The governing equations of the Lagrangian standpoint are advanced in time; for this purpose, the forth-order Runge Kutta scheme has been employed. This scheme enhances the numerical solution accuracy and stability through an embedded error control. Furthermore, the control volume technique is used to describe the Eulerian frame of reference. A modified form of the SIMPLE procedure is employed to solve the pressure equation, whereas the weighted average second-order upwind scheme is also used to calculate the volume-averaged Navier Stokes equations. The simulation time step, Δt, is chosen to be equal to 10 3 second and that is same for both the base liquid and dispersed phase. In order to have appropriate accuracy in the numerical modeling, in every time step the residual levels have been set to for the nanoparticles and for the liquid phase. 3 RESULTS AND DISCUSSIONS The effects of the gravitational field and Brownian motion have been investigated, and the velocity distribution of nanoparticles and their relative concentration will be presented in the paper.

8 AMINFAR ET AL. 79 FIGURE 2 Nanoparticles relative concentration over time 3.1 Validation A schematic representation of the nanoparticles initial dispersion is given by Fig. 1(a). As observed, at the initial state the nanoparticles are uniformly dispersed within the modeled area. After a period of time, the nanoparticles start to settle down due to the effects of gravitational force. If we write down the mean sedimentation velocity of an individual nanoparticle by symbol u y,i, and the mean distance which the nanoparticle passes in the gravitational field direction with symbol S g, their relative concentration in the time t 1 is calculated by the following expression developed by Jiang and colleagues 9 : C = H S g C 0 H = H + t 1 1 n 0 n i=1 u y,i dt. (18) H The terms C and C 0 represent nanoparticles relative concentration in the nanofluid in the specified time and initial time, respectively, the term H is the height of the container, and n is the number of the nanoparticles. Finally, the physical validity of the present Euler Lagrange model is evaluated by the relative concentration values. The relative concentration is an important parameter to express the dispersion quality and the colloidal stability of the nanofluids. Fig. 2 compares the obtained results for the nanoparticles relative concentration by using our numerical method with experimental measurements, 19 and Langevin dynamics simulation, 21 for CuO oil nanofluid. As seen, our numerical results agree well with the experimental measurements and also the molecular dynamics simulation results. The cited Molecular Dynamics (MD) simulation was developed according to the Brownian dynamics method, which considered the aggregation and di-aggregation kinetics of the dispersed phases. It should be noted that significant difference is not observed between the curves of Euler Lagrange model and MD simulation, which is consistent with the negligible effects of the di-aggregation phenomena. In the absence of gravity, over a long period of time, the velocity and displacement of an individual particle must follow the Maxwell Boltzmann distribution. 27 This condition is provided while the mean-square displacement of a single nanoparticle, r r 0 2, obeys the following relationship which is developed by Einstein for the first time 27 : r r 0 2 =6D B t. (19)

9 80 AMINFAR ET AL. FIGURE 3 Individual nanoparticle mean-square displacements variation with time The terms r and r 0 refer to positions of an individual nanoparticle in time t and initial time. Furthermore, the numerical results of this study are compared with the above equation predictions. As seen in Fig. 3, the predicted curve by the Euler Lagrange model, no gravity condition, agrees well with the expected Maxwell Boltzmann distribution. Also, a designated curve with all forces considered on the nanoparticles movement has been shown here. After a while, the nanoparticles are transported mainly by the gravitational force instead of the Brownian motion. It can be said that, over a long time, the gravitational force becomes the dominant slip mechanism of the nanoparticles transport. Consequently, the stochastic displacement or the Brownian motion loses its applicability to keep the nanoparticles dispersion uniform. 3.2 Velocity distribution for isothermal case The nanoparticles velocity magnitude contours, with the effects of the gravitational and Brownian forces, are illustrated by Fig. 4. To have more information about the velocity distribution in different conditions, the gravitational acceleration is regularly reduced until its effect vanishes. As Fig. 4(a) shows, while the gravitational force is neglected, the nanoparticles velocity magnitude is in the order of m/s to m/s. Moreover, it is observed that the velocity distribution is not uniform over time and this distribution describes the chaotic displacement of nanoparticles in the condition without gravity. As Fig. 4(c) depicts, velocity of nanoparticles within the gravitational field is about 1nm s and its distribution then becomes uniform. Fig. 5 shows the vertical velocity, or the sedimentation velocity, of the nanoparticles at different points during the gravitational acceleration. As Figs. 5(a) to (c) depicted, enhancing the magnitude of the gravitational acceleration strengthens the sedimentation velocity magnitude and makes it more negative. Also, a comparison between Figs. 4(c) and 5(c) reveal that the main portion of the velocity is attributed to the sedimentation velocity. While the gravitational acceleration is neglected, see Figs. 4(a) and 5(a), the particles movements become chaotic and their velocity does not converge to a specific value and direction. The gravitational field influences the nanoparticles velocity vector as demonstrated by Fig. 6. As Fig. 6(a) shows, while the gravitational force is uninvolved, the nanoparticles remain uniformly distributed and the Brownian motion keeps the colloid stable. As the gravitational field is applied on the medium, see Figs. 6(b) and (c), the gravitational diffusion enhances and the particles start to move

10 AMINFAR ET AL. 81 FIGURE 4 Velocity magnitudes of the nanoparticles FIGURE 5 Vertical velocities of the nanoparticles toward the direction of the gravitational field. It is seen that gravitational field tends to make the velocity vector of the nanoparticles uniform. Also, nanoparticles will follow the direction of the gravitational field like chain. Consequences of the fluid particle interactions are demonstrated through Fig. 7, which shows the velocity magnitude and vertical velocity of the base fluid. It can be observed that if only the effect of the gravitational field is neglected, as Fig. 7(a) shows, the velocity of the base fluid will be much less than the case which the gravitational field has been considered, see Fig. 7(b). Also, in condition that neglects gravity, the random motion of the nanoparticles causes the surrounding fluid to show

11 82 AMINFAR ET AL. FIGURE 6 Velocity vectors of the nanoparticles [Color figure can be viewed at wileyonlinelibrary.com] FIGURE 7 (a) Velocity magnitudes of the base fluid in the absence of gravitational field, (b) velocity magnitudes of the base fluid in the presence of the gravitational field, (c) vertical velocities of the base fluid in the absence of gravitational field, and (d) vertical velocities of the base fluid in the presence of the gravitational field an irregular behavior. Finally, Fig. 7(d) shows the vertical velocity of the base fluid considering all mentioned forces. The liquid phase moves in the opposite direction of the nanoparticles, which provides a condition to introduce the drift velocity. As the nanoparticles are influenced by the external field, they leave their stationary state and their concentration varies over the time. The drift velocity for settling in a classical Stokes flow for spherical solid particles can be calculated by using a force balance between the gravitational, buoyancy, and drag forces from the below expression 28,29 : 6πμ f d p 2 u d = ( ρ p ρ f ) vp g, (20)

12 AMINFAR ET AL. 83 FIGURE 8 Stream lines of the base fluid where v p is the volume of a particle and u d is the drift velocity of settling, which is known as sedimentation velocity: ( ) u d = d2 p ρp ρ f g. (21) 18μ f According to the negligible effects of the random fluctuation of the nanoparticles, Eq. (21) can be used for calculating the nanoparticles sedimentation velocity in the nanofluid with respect to the nanoparticles aggregation dimension. It is observed that our proved data for nanoparticle sedimentation velocity in Fig. 5(c) are close enough to the value that is given by Eq. (21). The streamlines of the base fluid have been shown in Figs. 8(a) to (c) to discuss the fluid particles interactions in detail. Also, the base fluid streamlines guide us to the concept of diffusion and nanoparticle sedimentation. As the magnitude of the gravitational acceleration is increased, the base fluid flow becomes regular. These figures confirm our previous conclusion about the negligible effects of the Brownian force on the nanoparticles transport, in comparison with the gravitational force. The velocity vectors of the particles and also the streamlines of the base fluid have been demonstrated in Figs. 9 and 10 for three different diameters of nanoparticles and considering the Brownian and gravity forces. For the particles with diameter of 10 nm, the contribution of the Brownian motion is dominant, and the gravity field has negligible effect on the motion of the particles. For the particles with diameter of 50 nm, the share of the gravity force in the displacement of the particles increases. By further increase in the diameter of the particles (100 nm), the share of the Brownian force becomes negligible in comparison with gravity force, and the particles will move due to the gravitational force. 3.3 Applying heat flux In this case, a constant heat flux of 2000 kw/m 2 is applied to the lower wall of the enclosure, and other walls of the enclosure are considered to be adiabatic. Particles move from the regions with high

13 84 AMINFAR ET AL. FIGURE 9 particles Velocity vector of the particles due to Brownian and gravity forces for different diameters of the FIGURE 10 particles Streamlines of the base fluid due to Brownian and gravity forces for different diameters of the temperatures to the regions with low temperatures due to temperature gradient. 30 Figs. 11 and 12 show the variations of the velocity magnitude and vertical velocity of the particles, respectively, with distance from the heated wall due to thermophoresis and drag forces. As seen in the figures, the velocity of particles in the vicinity of the heated wall has the maximum value, and then it is decreased with an increase in the distance from the heated wall. As seen in Fig. 12, the velocity of the particles is against the direction of the gravity field, so that the application of an appropriate heat flux can prevent particles from sedimentation. Fig. 13 depicts the variations of the velocity magnitude of the particles with the distance from the heated wall due to gravity, thermophoresis, and drag forces. The velocity of the particles near the heated wall is in the order of 10 6, and it is the same as the case where gravity force was not considered. While the lower share of the gravity force is in the regions near the heated wall, the share of the gravity force will be increased in the regions away from the heated wall.

14 AMINFAR ET AL. 85 FIGURE 11 Velocity magnitude of the particles with distance from the heated wall due to thermophoresis and drag forces in x/l = 0.5 FIGURE 12 forces in x/l = 0.5 Vertical velocity of the particles with distance from the heated wall due to thermophoresis and drag 4 CONCLUSIONS This paper presents a numerical simulation for the nanoparticles sedimentation phenomenon in a 2D enclosure by the Euler Lagrange approach. Furthermore, the importance and influence of the gravitational and Brownian forces on the nanoparticles transport and the base fluid have been discussed. The following conclusions were obtained: The nanoparticles concentration, velocity, and displacement have been mainly influenced by the gravitational field and the contribution of the Brownian forces is negligible. In the absence of the gravitational force, the velocity magnitude of the nanoparticles is decreased. Due to the rapid fluctuation in the Brownian force, the nanoparticles velocity magnitude and direction become nonuniform and do not converge to a specific value and direction. In the absence of the gravitational force, the Brownian motion keeps the dispersion stable. By increasing the magnitude of gravitational field, the velocity amplitude of the nanoparticles increases and they follow the gravitational field in a chain-wise pattern.

15 86 AMINFAR ET AL. FIGURE 13 Velocity magnitude of the particles with distance from the heated wall due to gravity, thermophoresis, and drag forces in x/l = 0.5 As the effect of only the Brownian motion is considered, the surrounding fluid of the nanoparticles shows a chaotic behavior. By gradually increasing the gravitational field, this behavior is suppressed until the effects of the Brownian motion disappear. REFERENCES 1. Ghadimi A, Saidur R, Metselaar HSC. A review of nanofluid stability properties and characterization in stationary conditions. Int J Heat Mass Transfer. 2011;54: Jain PK, Lee KS, El-Sayed IH, El-Sayed MA. Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: applications in biological imaging and biomedicine. JPhysChemB. 2006;110: Kokura S, Handa O, Takagi T, Ishikawa T, Naito Y, Yoshikawa T. Silver nanoparticles as a safe preservative for use in cosmetics. Nanomed Nanotechnol Biol Med. 2010;6: Li J, Kleinstreuer C. Thermal performance of nanofluid flow in microchannels. Int J Heat Fluid Flow. 2008;29: Sergis A, Hardalupas Y. Anomalous heat transfer modes of nanofluids: a review based on statistical analysis. Nanoscale Res Lett. 2011;6: Shahmoradi Z, Etesami N, Nasr Esfahany M. Pool boiling characteristics of nanofluid on flat plate based on heater surface analysis. Int Commun Heat Mass Transfer. 2013;47: Saidur R, Leong KY, Mohammad HA. A review on applications and challenges of nanofluids. Renewable Sustainable Energy Rev. 2011;15: Wen D, Lin G, Vafaei S, Zhang K. Review of nanofluids for heat transfer applications. Particuology. 2009;7: Jiang W, Ding G, Peng H, Hu H. Modeling of nanoparticles aggregation and sedimentation in nanofluid. Curr Appl Phys. 2010;10: Savithiri S, Pattamatta A, Das S. Scaling analysis for the investigation of slip mechanisms in nanofluids. Nanoscale Res Lett. 2011;6: Buongiorno J. Convective transport in nanofluids. JHeatTransfer. 2006;128: Alexander CM, Dabrowiak JC, Goodisman J. Gravitational sedimentation of gold nanoparticles. J Colloid Interface Sci. 2013;396: Shima PD, Philip J, Raj B. Role of microconvection induced by Brownian motion of nanoparticles in the enhanced thermal conductivity of stable nanofluids. Appl Phys Lett. 2009;94;

16 AMINFAR ET AL Hwang KS, Jang SP, SUS C. Flow and convective heat transfer characteristics of water-based Al2O3 nanofluids in fully developed laminar flow regime. Intl J Heat Mass Transfer. 2009;52: Gharagozloo PE, Goodson KE. Aggregate fractal dimensions and thermal conduction in nanofluids. J Appl Phys. 2010;108; Li XF, Zhu DS, Wang XJ. Evaluation on dispersion behavior of the aqueous copper nano-suspensions. J Colloid Interface Sci. 2007;310: Choi C, Yoo HS, Oh JM. Preparation and heat transfer properties of nanoparticle-in-transformer oil dispersions as advanced energy-efficient coolants. Curr Appl Phys. 2008;8: Li X, Zhu D, Wang X. Evaluation on dispersion behavior of the aqueous copper nano-suspensions. J Colloid Interface Sci. 2007;310: Hwang Y, Park HS, Lee JK, Jung WH. Thermal conductivity and lubrication characteristics of nanofluids. Curr Appl Phys. 2006;6:e67 e Wang BX, Zhou LZ, Peng XF. A fractal model for predicting the effective thermal conductivity of liquid with suspension of nanoparticles. Intl. J Heat Mass Transfer. 2003;46: Ganguly S, Chakraborty S. Sedimentation of nanoparticles in nanoscale colloidal suspensions. Phys Lett A. 2011;375: Kondaraju S, Jin EK, Lee JS. Direct numerical simulation of thermal conductivity of nanofluids: the effect of temperature two-way coupling and coagulation of particles. Int J Heat Mass Transfer. 2010;53: Subramaniam S. Lagrangian Eulerian methods for multiphase flows. Prog Energy Combust Sci. 2013;39: Minkowycz WJ, Sparrow EM, Murthy JY. Handbook of Numerical Heat Transfer. New Jersey: Wiley; Li A, Ahmadi G. Dispersion and deposition of spherical particles from sources in a turbulent channel flow. Aerosol Sci Technol. 1992;16: Kleinstreuer C. Two Phase Flow: Theory & Application. New York: Taylor & Francis Press; McQuarrie DA. Statistical Mechanics. New York: Harper and Row; Gharagozloo PE, Goodson KE. Temperature-dependent aggregation and diffusion in nanofluids. Int J Heat Mass Transfer. 2011;54: Witharana S, Palabiyik I, Musina Z, Ding Y. Stability of glycol nanofluids The theory and experiment. Powder Technol. 2013;239: Eslamian M, Saghir MZ. Novel thermphoretic particle separators: numerical analysis and simulation. Appl Therm Eng. 2013; How to cite this article: Aminfar H, Mohammadpourfard M, Mortezazadeh R. Numerical simulations of the influence of brownian and gravitational forces on the stability of CuO nanoparticles by the Eulerian Lagrangian approach. Heat Transfer Asian Res. 2018;47: doi.org/ /htj.21291