Applied Mathematics and Computation 219 (2012) 3308–3315

Contents lists available at SciVerse ScienceDirect

Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Heat transfer in nanofluids: A computational evaluation of the effectsof particle motion

Mauricio Giraldo ⇑, Daniel Sanín, Whady F. FlórezGrupo de Energía y Termodinámica, Ingeniería Mecánica, Universidad Pontificia Bolivariana, Circ. 1 N 73-34, Medellín, Colombia

a r t i c l e i n f o

Keywords:NanofluidsConvectionAluminaBoundary elementsBoundary integral methods

0096-3003/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.amc.2011.08.056

⇑ Corresponding author.E-mail address: mauricio.giraldo@upb.edu.co (M

a b s t r a c t

Nanofluids are a novel strategy to improve heat transfer characteristics of fluids by theaddition of solid particles with diameters below 100 nm. Experimental measurements haveshown that this approach can greatly increase heat conductivity, even above that predictedby Maxwell’s theory. However, it is still not clear what mechanism accounts for thisincrease, mainly in thermal conductivity, thus great effort is being put into investigatingthe different phenomena occurring inside the fluid. This paper shows a direct numericalsimulation of the flow and thermal behaviour of a nanofluid loaded with alumina nanopar-ticles which considers the effects of particle–particle and particle–fluid interactions. Theboundary element method is used given its capabilities to deal with moving boundaryproblems. This formulation was employed to simulate the behaviour and time evolutionof a 30 nm alumina/water nanofluid at six different particle concentrations, looking toobserve flow fields, temperature distributions and total heat flow through the domain.Results showed strong convective currents caused by the presence of the nanoparticles,which in time increased total heat flow in the cavity. As expected, particle concentrationincreased the total heat flow affecting not only the conductive part of heat transfer, butthe convective part as well.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

Every physical process involves heat generation to some scale, be it an objective of such process as in boilers, or abyproduct as in internal combustion engines. Fluids are the preferred means to handle and transport this heat, howeverlow heat conductivity requires high velocities or complex geometries in order to effectively transfer this heat from a gi-ven surface to the fluid, therefore increasing manufacturing and operational costs [1]. Solid particles have long been usedfor the enhancement of the heat transfer characteristics of fluids and thus facilitate heat flow from surfaces, howeverfluid stability and pipe erosion have hindered its development at an industrial scale [2]. In recent years an alternativehas been developed: by using particles under 100 nm in diameter (hence the name nanofluids), it is possible not onlyto avoid these problems, but the resulting increase in thermal conductivity is higher than that expected from Maxwell’stheory [3,4].

Several theories have been drawn to try and explain the reasons for this higher conductivity [5], but so far predictionshave failed to reproduce experimental data. This leads to the necessity of correctly determining different flow and thermalparameters [6,7]. Most works available in the literature treat the nanofluid as a homogeneous solid–liquid mixture, and eval-uate the corresponding bulk flow and thermal properties using theoretical or empirical formulae [8–10].

. All rights reserved.

. Giraldo).

M. Giraldo et al. / Applied Mathematics and Computation 219 (2012) 3308–3315 3309

This work studies the effect of fluid motion and particle advection caused by Brownian motion and buoyancy, van derWals and double layer forces, on the heat transfer behaviour of a fluid loaded with 30 nm alumina nanoparticles. In orderto evaluate the different phenomena occurring inside the particle–fluid system, a boundary integral formulation is employedto simulate particle motion and fluid velocities; the resulting velocity field is used as input for the heat transfer evaluation.The heat transfer equation is solved via a traditional boundary integral method combined with the dual reciprocity methodto deal with the convective terms. Fluid and particle domains are coupled by no slip boundary conditions for the fluid motionsimulation, while temperature and heat continuity through the boundaries is used to solve the heat equation. Even thoughthe selected system is a moving boundary problem, the resulting implementation showed adequate simulation times byavoiding all remeshing given the lack of domain elements.

Simulations were performed on a square cavity of side 1 lm insulated on the sides and given a 0.1 K temperature differ-ence between the top and bottom boundaries; an equal base fluid (water) was loaded with six different particle concentra-tions ranging from 1.77% to 7.07%. Results showed how the motion of the particles creates strong convective currents insidethe fluid, greatly influencing thermal exchange between surfaces compared to what is obtained from a purely conductiveapproach. As expected, higher concentrations yielded higher heat transfer through the nanofluid.

2. Governing equations

The physical system that is being simulated is composed of a fluid and a number of particles suspended inside. The fluid issurrounded by four rigid impermeable walls, each 1 lm long. The side walls are considered insulated and motionless, and a0.1 K temperature difference is imposed between the top and bottom walls. Under such conditions, two distinct domains aresimulated, Xe is the fluid domain, and Xi which is the combination of all the particles (A complete list of symbols is includedin Table 1).

The fluid must obey the Navier–Stokes system of equations. Given the small characteristic lengths involved, and the smalltemperature difference induced, the Rayleigh number is of the order Oð�9Þ, making it is possible to assume creeping flow,thus the Stokes system of equations is solved for the motion of the fluid:

@ui

@xi¼ 0; ð1Þ

� @p@xiþ g

@2ui

@xj@xj¼ 0: ð2Þ

Boundary conditions for these equations are non slip velocities on the walls, and velocity continuity on the particle sur-faces. As for the heat flow in this domain, both conductive and convective terms must be taken into account, leading to:

kf@2T@xj@xj

þ uiqf Cp�f@T@xi¼ qf Cp�f

@T@t; ð3Þ

where subindex f indicates that the properties are evaluated for the base fluid. The required velocity field to solve this equa-tion is that obtained from the solution of Eqs. 1 and 2, which are evaluated separately due to the fact that no thermal densityvariations are being considered. Heat flow inside the particles follows a similar equation:

ks@2T@xj@xj

þ uiqsCp�s@T@xi¼ qsCp�s

@T@t: ð4Þ

Here subindex s indicates that the properties are evaluated for the solid domain (particles). The thermal boundary con-dition on the particle surfaces is given by the continuity of heat through the surface, i.e.

k@T@n

����s

¼ �k@T@n

����f

: ð5Þ

Table 1Variables and parameters used.

Sp: Surface of particle p Xe: Domain external to the particlesSw: Cavity walls Xi: Domain internal to the particlesuj: Fluid velocity g: fluid viscosityq: Density k: Thermal conductivityT: Temperature xi: Vector location of a given pointp: Pressure Cp: Specific Heatn: normal vector Kijðx; yÞ: Double layer kernel or Stressleta: Thermal diffusivity uj

iðx; yÞ: Single layer kernel or Stokeslet/j: Double layer potential u�: particular solution for potential problemsq⁄: normal derivative of u⁄ Fj: Force vector acting on the particles

3310 M. Giraldo et al. / Applied Mathematics and Computation 219 (2012) 3308–3315

2.1. Forces acting on the particles

Particles immersed in a fluid experience a number of effects that determine its motion, some from the interaction be-tween the particles, some caused by external body forces. In this paper particles are simulated considering three types ofinteractions: Brownian motion, van der Wals and double layer forces; as for body forces effects, buoyancy caused by the dif-ference in density from the base fluid to the particle is also included. Each of these effects can be transformed into an equiv-alent force for simulation purposes as shown in [11]. Once these forces have been determined and calculated, they can beincluded into the boundary integral formulation for the motion of the solid particles (Eq. 6) shown in the following section.

3. Numerical formulation

Following the procedure developed in [11,12], the velocity field is defined as a function of a double layer potential with nophysical meaning, plus a Stokeslet to account for external forces acting on the n particles:

uiðxÞ ¼Z

Sw

Kijðx; yÞ/jðyÞdSy þXn

l¼1

ujiðx; y

cÞFj þZ

Sp

Kijðx; yÞ/jðyÞdSy; ð6Þ

where x is any point in the domain, y is a point on surface Sy and yc is the mass center of particle l. This equation is completedby a rigid body motion condition on the internal surfaces given that the velocity of the particles is not known beforehand[11]. The integral boundary representation of the heat transfer problem on the base fluid can expressed as:

TðxÞ ¼Z

Sw

q�ðx; yÞTðyÞdSy �Z

Sp

q�ðx; yÞTðyÞdSy �Z

Sw

u�ðx; yÞ @T@nðyÞdSy þ

ZSp

u�ðx; yÞ @T@nðyÞdSy þ

ZXe

u�ðx; yÞ 1a@T@xi

����Xe

uiðyÞdSy:

ð7Þ

While for the particles, it is given by:

TðxÞ ¼Z

Sp

q�ðx; yÞTðyÞdSy �Z

Sp

u�ðx; yÞ @T@nðyÞdSy þ

ZXi

u�ðx; yÞ 1a@T@xi

����Xi

uiðyÞdSy: ð8Þ

Eqs. (6)–(8) are solved by using the boundary element method, which has been widely used to evaluate diffente types ofproblems, including inverse heat transfer problems [13]. Several alternatives exist to handle convective terms in Eqs. (7) and(8), however, given it robustness, the dual reciprocity method (DRBEM) [14–17] is employed in this work. The resulting nonlinear system of equations arising from using the use of DRBEM in Eqs. (7) and (8) is solved using Newton’s methods as ex-plained in [15].

Finally, the heat transferred through a given surface can be calculated by evaluating the following integral:

ZS

kf@T@nðyÞdSy: ð9Þ

4. Fluid-particle pair selection and other issues

The literature presents a number of options concerning the fluid-particle pair [7,18–20]. Among these, the combinationwater-alumina is one of the most used, both by ease and cost of production, as for its thermal properties [21]. Given theabove considerations, this paper will evaluate a water based nanofluid loaded with 30 nm alumina nanoparticles at differentconcentrations. The physical properties of particles and base fluid are presented in Table 2.

The computational domain selected is a 1 lm by 1 lm square cavity, allowing at the same time to have a relatively largenumber of particles (100 in the case of 7.07% concentration) while maintaining a relatively low number of nodes in the sim-ulation (760 for the same case). In order to maintain a standard approach, all concentrations where evaluated with the sameexternal mesh, and number of nodes per particle. As mentioned earlier, the boundary conditions for the velocity field are noslip on the particle surfaces and cavity walls. All cavity walls are stationary. Thermally, a 0.1 K temperature gradient was

Table 2Physical properties of particles and base fluid.

Alumina density 1.5 g/cm3

Alumina specific heat 0.765 kJ/(kg K)Alumina thermal conductivity 25 W/(m K)Water density 0.99 g/cm3

Water viscosity 8:9� 10�4 Pa sWater specific heat 4.180 kJ/(kg K)Water thermal conductivity 0.58 W/(m K)

M. Giraldo et al. / Applied Mathematics and Computation 219 (2012) 3308–3315 3311

imposed between the bottom and top walls, the bottom being hotter, and the side walls are considered perfectly insulated. Atotal of 10000 time steps were calculated for each case, using Dt ¼ 5:0� 10�4 s.

Finally it should be mentioned that even though this paper includes some important effects such as Brownian motion, vander Waals forces and double layer repulsion, it does not include the effects of heat transferred by particles in contact due tocollisions; when particles enter into contact with each other the surfaces deform slightly allowing for a conduction of heatbetween the two. The duration of this contact can be quite small; however given that the particle size in the case of nano-fluids is well below 1 lm, its effect can be important as shown by [22]. Even further nanoparticle clustering arising fromthese collisions could have a significant role on the effective heat conductivity of the nanofluid, as presented in [23]. Inthe model employed in this paper, particles in close proximity will effectively be able to transfer heat between them moreeasily by conduction before bouncing off and moving freely at the next time steps. The separation of the particles occurs be-cause of the strength of the Brownian forces and the lack of an adequate implementation of an agglomeration model in thecode. An interesting extension of this work would be to include an agglomeration model such as that presented in [24] and acontact heat transfer model in the present implementation code in other to more accurately simulate the thermal behavior ofthe nanofluid.

5. Numerical implementation

Simulations are carried out in a two dimensional space for computational reasons, and thus the concentration is mea-sured based on the area of the particles over the total area of the simulation cell. By considering a two dimensional problem,the simulations presented in this section can be assimilated as the behaviour of a number of aligned particles, or infinite cyl-inders, increasing the total volume occupied by the particles in the corresponding three dimensional case; however, thehydrodynamic and thermal behaviours calculated here can be extrapolated given that proper considerations are used. Itis important to stress that given the simulation parameters employed, there is no natural convection in the base fluid; if suchphenomena were to occur, the corresponding instabilities involved in the case of fluid heated from the bottom can greatlyalter behaviours both for two and three dimensions.

The mesh can be divided into three parts: An outer surface, the particle surfaces and internal nodes. All surfaces are dis-cretized using quadratic elements which are constituted by three nodes (For more details on the discretization see [15]); theexternal surface is composed of 80 boundary elements (160 nodes) and each particle is discretized by three boundary ele-ments (6 nodes). Finally, 961 internal nodes were used; given that all the simulations are done using a BIM approach, theseinternal nodes are only used for visualizing the flow field in the case of the velocity calculations, and as collocation nodes forthe interpolation matrices used in DR-BEM calculation of the temperature field. Even though there are different ways to ac-count for the internal nodes in the calculation, this work solved simultaneously for the temperature in the external and inter-nal nodes in order to achieve convergence of the nonlinear iterative system in a faster way.

Taking into account that no bouyancy effects are considered, the system is one-way coupled, meaning the velocity field iscalculated independently, and then used as input for the temperature calculations. This approach and the mesh size usedmeans that for the velocity calculations, the linear system was 620� 620 for the smallest case (1.77% concentration) and1520� 1520 for the largest case (7.07% concentration). In the case of temperature calculations, given that the internal nodesare solved simultaneously, the resulting system of equations solved at each iteration ranges from 1271� 1271 to1721� 1721 for the same cases.

Simulations were performed in a desktop computer running under Windows XP, with a IntelPentium DuoCore 2.4 GHzprocessor with 2 GB RAM. Simulations times varied according with system size. For the 7.07% concentration, both velocityand temperature simulations lasted close to 40 h, while the 1.77% concentration required only 9 h for the velocity calculationand 28 h for the temperature calculations. It is worth mentioning that velocity and temperature calculations can be runsimultaneously in most cases reducing the total simulation time.

6. Results and discussion

Results presented in this section are divided into two parts. First the velocity and temperature fields for two different con-centrations are presented in order to show the effects of particle motion on the creation of convective currents inside thedomain. Secondly, the total heat transfered between the walls by the fluid is presented and compared to that of the samefluid and particle distribution when the thermal effect of fluid motion is neglected.

6.1. Velocity and temperature fields

In order to introduce the problem, Fig. 1 shows the temperature field in a cavity with no particles. Since this is a closedcavity and there are no particles, along with the forementioned issue of the problem’s Rayleigh number, the fluid remainsmotionless and the temperature distribution is what could be expected of a homogeneous medium with conduction beingthe only heat transfer mechanism present.

Fig. 2 shows the temperature distribution and the velocity vectors for the case of 2.54% particle concentration for twodifferent instants of the simulation. Taking into account that the cavity walls are motionless and the Rayleigh number is

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Fig. 1. Temperature distributions inside a cavity with no particles.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

(a) (b)

Fig. 2. Temperature distributions for particle concentration 2.54% at times (a) 2.5 s and (b) 5 s.

3312 M. Giraldo et al. / Applied Mathematics and Computation 219 (2012) 3308–3315

of an order of magnitude so small that there no natural convection can be expected, the fluid should not be moving. How-ever, given that Brownian motion, van der Waals forces, double layer repulsion and buoyancy are being taken into consid-eration, there is a motion of the particles, which in turn will cause the fluid to move as can be seen from the velocity fields inFig. 2 and further ahead in Fig. 3. The relative magnitudes of the different forces used in the simulation, along with theimportance of Brownian motion regarding particle behaviour was studied in more depth in a previous paper [11].

It can be easily seen how the internal temperature distribution shown in Fig. 2 varies significantly from what is expectedof a homogeneous and static medium such as that present in Fig. 1. The inherently random motion of the particles (repre-sented here by the circles in the domain) caused by the magnitude of the Brownian effects, is influenced up to a point by thefluid, and a coordinated motion appears to occur, causing strong convective currents which aid in the transport of heat fromthe bottom to the top of the cavity.

One of the visible effects of these currents is the relative homogenization of the temperatures in large zones inside thecavity when comparing Figs. 1 and 2, leaving the stronger temperature gradients limited to the proximity of the bottomand top walls. It is interesting to see how certain motions of the particles, as those occurring in the lower right quarter ofFig. 2 b), tend to create zones in the fluid with little inward or outward fluid flow, where temperature gradients behave moreclosely to what is expected of a purely conductive field. This motion of the particles on the right hand side of the cavity in-duces a larger thermal gradient on the opposite side, which will then be responsible for most of the heat being transportedfrom the bottom to the top of the cavity.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

(a) (b)

Fig. 3. Temperature distributions for particle concentration 5.7% at times (a) 2.5 s and b) 5 s.

M. Giraldo et al. / Applied Mathematics and Computation 219 (2012) 3308–3315 3313

Another important factor to point out, is the rapid change in the temperature profiles observed in the simulations. Eventhough the particles in both Fig. 2(a) and (b) have been allowed to move freely for more than 5000 time steps and could thenbe considered not to the affected by the initial conditions of the simulation, the temperature distributions vary, showing aslightly colder core for a simulation time of 5 s than for the same case at 2.5 s. Particle distribution also varies, although onlyslightly, suggesting that the location of the particles has little effect on the temperature gradients.

Looking at a higher concentration, namely 5.7% (See Fig. 3), the larger number of particles present in the simulation havethe effect of pushing some of them closer to the walls. This phenomena can be explained by the increasing repulsive forcesfrom the particles in the centre of the domain overcoming to some extent the repulsive force of the walls. This increase in theproximity of the particles to the wall make the hot zone adjacent to the lower wall to decrease in size, even if taking intoaccount that the fluid velocity does not have any significant normal velocity component.

A second effect that the greater number of particles in the system has is the strengthening of the convective patterns. Thestrength of these currents is such that hot and cold fluid jets mix in a swirling motion, as can be seen on the right hand side ofFig. 3(b); this effect is also present on Fig. 2 although it is less evident. This phenomena is also an indication of the compar-ative strength of the two heat transfer mechanisms, conduction playing a secondary role to convection; this is suggested bythe fact that in the swirling motion of the fluid the temperature increases in some points moving in the vertical direction forsome horizontal positions, contrary to what could be expected if conduction had a more important role, i.e. x ¼ 0:45 lm inFig. 3(a), or x ¼ 0:6 lm in Fig. 3(b).

6.2. Heat flow

The particle motion shown in the above section and the resulting convective currents have a very strong influence on thetotal heat transfered from the bottom to the top wall. Fig. 4 shows the time evolution of heat flowing from the bottom to thetop wall obtained by evaluating Eq. 9 on the top and bottom surfaces and averaging the results. Two evaluations are per-formed, the first considering conductive and convective terms and the second considering only the conductive component.The resulting evaluation shows that the purely conductive scheme has a much lower total heat transfered independent of theparticle concentration.

Another important issue that can be extracted from Fig. 4 is the oscillatory behaviour of heat transfered when convectionis taken into consideration. These variations are due to the random motion of the particles. When a number of them are mov-ing with important vertical components, the total heat increases, while a more or less balanced horizontal motion leads to asomewhat lower value. However, the total difference between these two instances is not so great given that from mass con-servation inside the fluid, a large portion of fluid moving in a given vertical direction will create a corresponding motion ofthe fluid in the horizontal direction, thus balancing out the effects.

Particle concentration has a strong effect on the total heat transfered as can be seen from comparing Fig. 4(a) and (b). In Fig. 5the results from calculating time averaged heat flow through the domain are presented for six different particle concentrations.As expected, the results when convection is introduced are much higher that for pure conduction in all the concentrations eval-uated, however, in both instances, total heat does increase with particle concentration as reported in the literature.

The rate of increase with particle concentration is also different for the two cases, being higher for convection-conduction.This can be explained by a greater particle-particle interaction as the concentration increases, which in turn would make thecontribution of fluid motion greater.

0 1 2 3 4 50.05

0.1

0.15

0.2

0.25

0.3

Time [s]

Hea

t flo

w [W

]

Convection−ConductionPure conduction

0 1 2 3 4 50.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time [s]

Hea

t flo

w [W

]

Convection−ConductionPure conduction

(a) (b)

Fig. 4. Heat flow through the cavity for particle concentration (a) 2.54% and (b) 5.7%.

0 2 4 6 80.05

0.1

0.15

0.2

0.25

0.3

Particle concentration [%]

Tim

e av

erag

ed H

eat f

low

[W]

Convection−ConductionPure conduction

Fig. 5. Time averaged heat flow through the cavity.

3314 M. Giraldo et al. / Applied Mathematics and Computation 219 (2012) 3308–3315

Fig. 5 also shows the effects of convection, increasing the total heat transfered by almost 400% compared to pure conduc-tion. It is important to emphasize that this increase is well above that reported by experimental measurements which is be-low 20% [21,25]. Other than the approximation to a two dimensional flow employed here, one possible cause for thisdifference could come from the fact that at these scales the velocity continuity condition on the particle surfaces is not totallyaccurate [26]. In order to assure closer proximity to experimental results, adequate slip boundary conditions must be im-posed on the particle surfaces [27]; this work is currently underway.

7. Concluding remarks

This paper presents a boundary integral formulation for the simulation of hydrodynamic and thermal behaviour of nano-fluids considering the effects of particle interactions at a nanoscale level, and the role of particle induced fluid motion on thetotal heat transfered. This formulation was them employed to simulate the behaviour of a alumina/water nanofluid inside atwo dimensional 1 lm � 1 lm square cavity with a 0.1 K temperature gradient between the bottom and top walls. Resultsobtained showed a great influence of particle motion on the total heat transferred by the nanofluid in qualitative accordancewith experimentally obtained reported data.

Particle interactions and Brownian motion cause the particles to move about the cavity, inducing fluid motion in a sometimes coordinated manner, which has strong effects on the total heat transfered. Under this scenario, the convective com-ponent overshadows pure conductive heat transfer effects that are present in the domain. As particle concentration in-creases, so does the total heat flowing through the cavity, having a larger effect on convective heat transfer than on thecase of pure conduction.

M. Giraldo et al. / Applied Mathematics and Computation 219 (2012) 3308–3315 3315

Work is currently being undertaken in order to include other surface effects, namely fluid slip, in order to increase thecorrelation between experimentally reported data and results obtained via the present formulation.

Acknowledgments

The authors are thankful for the support from the Fundación para la Promoción de la Investigación y la Tecnología – Bancode la República de Colombia project 2.484.

References

[1] X. Wang, A.S. Mujumdar, Heat transfer characteristics of nanofluids: a review, Int. J Therm. Sci. 46 (2007) 1–19.[2] Yuanqiao Rao, Nanofluids: stability, phase diagram, rheology and applications, Particuology 8 (2010) 549–555.[3] H. Xie, J. Wang, T. Xi, Y. Liu, F. Ai, Q. Wu, Thermal conductivity enhancement of suspensions containing nanosized alumina particles, J. Appl. Phys. 91

(2002) 4568–4572.[4] S. Lee et al, Measuring thermal conductivity of fluids containing oxide nanoparticles, J. Heat Transfer 121 (1999) 280–289.[5] P. Keblinski et al, Mechanisms of heat flow in suspensions of nano-sized particles (nanofluids), Int. J. Heat Mass Transfer 45 (2002) 855–863.[6] Bachok Norfifah, Ishak Anuar, Pop Ioan, Boundary-layer flow of nanofluids over a moving surface in a flowing fluid, Int. J. Therm. Sci. 49 (2010) 1663–

1668.[7] H.A. Mohammed, G. Bhaskaran, N.H. Shuaib, H.I. Abu-Mulaweh, Influence of nanofluids on parallel flow square microchannel heat exchanger

performance, Int. Commun. Heat Mass Transfer 38 (2011) 1–9.[8] Corcione Massimo, Heat transfer features of buoyancy-driven nanofluids inside rectangular enclosures differentially heated at the sidewalls, Int. J.

Therm. Sci. 49 (2010) 1536–1546.[9] M. Muthtamilselvan, P. Kandaswamy, J. Lee, Heat transfer enhancement of copper–water nanofluids in a lid-driven enclosure, Commun. Nonlinear Sci.

Numer. Simulat. 15 (2010) 1501–1510.[10] J. Ravnik, L. Skerget, M. Hribersek, Analysis of three-dimensional natural convection of nanofluids by BEM, Eng. Anal. Bound. Elem. 34 (2010) 1018–

1030.[11] M. Giraldo, Y. Ding, R.A. Williams, Boundary integral study of nanoparticle flow behaviour in the proximity of a solid wall, J. Phys. D Appl. Phys. 41

(2008) 1–9.[12] H. Power, G. Miranda, Second kind integral equation formulation of Stokes flow past a particle of arbitrary shape, SIAM J. Appl. Math. 4 (1987) 689–

698.[13] T.T.M. Onyango, D.B. Ingham, D. Lesnic, Inverse reconstruction of boundary condition coefficients in one-dimensional transient heat conduction, Appl.

Math. Comput. 207 (2009) 569–575.[14] C.P. Rahain, A.J. Kassab, Pressure correction DRBEM solution for heat transfer and fluid flow in incompressible viscous fluids, Eng. Anal. Bound. Elem. 18

(1996) 265–274.[15] M. Giraldo, H. Power, W.F. Flórez, Mobility of shear thinning viscous drops in a shear Newtonian carrying flow using DR-BEM, Int. J. Numer. Methods

Fluids 59 (2009) 1321–1349.[16] W.F. Flórez, H. Power, F. Chejne, Numerical solution of thermal convection problems using the multi-domain boundary element method, Numer.

Methods Partial Differential Equations 18 (2002) 469–489.[17] P. Satravaha, S. Zhu, An application of the LTDRM to transient diffusion problems with nonlinear material properties and nonlinear boundary

conditions, Appl. Math. Comput. 87 (1997) 127–160.[18] W. Yu, M. France, D.S. Smith, D. Singh, E.V. Timofeeva, L. Routbort, J. Heat, transfer to a silicon carbide/water nanofluid, Int. J. Heat Mass Transfer 52

(2009) 3606–3612.[19] H. Xie, L. Chen, Adjustable thermal conductivity in carbon nanotube nanofluids, Phys. Lett. A 373 (2009) 1861–1864.[20] Santra A. Kumar, S. Sen, N. Chakraborty, Study of heat transfer due to laminar flow of copper–water nanofluid through two isothermally heated parallel

plates, Int. J. Therm. Sci. 48 (2009) 391–400.[21] D. Zhu, X. Li, Nan Wang, Wang, X. Wang, J. Gao, H. Li, Dispersion behavior and thermal conductivity characteristics of Al2O3H2O nanofluids, Curr. Appl.

Phys. 9 (2009) 131–139.[22] J. Sun, M.M. Chen, A theoretical analysis of heat transfer due to particle impact, Int. J. Heat Mass Transfer 31 (1988) 969–975.[23] W. Yu, D.M. France, S.U.S. Choi, J.L. Routbort, Review and assessment of nanofluid technology for transportation and other applications, Energy Systems

Division, Argonne National Laboratory (2007) Report.[24] P. Kosinski, A.C. Hoffmann, An extension of the hard-sphere particle–particle collision model to study agglomeration, Chem. Eng. Sci. 65 (2010) 3231–

3239.[25] S. Mirmasoumi, A. Behzadmehr, Effect of nanoparticles mean diameter on mixed convection heat transfer of a nanofluid in a horizontal tube, Int. J. Heat

Fluid Flow 29 (2008) 557–566.[26] C. Neto, D.R. Evans, E. Bonaccurso, H. Butt, V. Craig, Boundary slip in Newtonian liquids: a review of experimental studies, Rep. Progr. Phys. 68 (2005)

2859–2897.[27] C. Nieto, M. Giraldo, H. Power, Boundary integral equation approach for stokes slip flow in rotating mixers, Discrete Contin. Dyn. Syst. Ser. B 15 (4)

(2011) 1019–1044.