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Applied Thermal Engineering

Volume 78, 5 March 2015, Pages 326–338

Laminar Forced Convection of a Nanofluid in a Microchannel: Effect of Flow Inertia and External

Forces on Heat Transfer and Fluid Flow Characteristics

Mahmoud Ahmed† and Morteza Eslamian

1‡

† Department of Mechanical Engineering, Assiut University, Assiut, 71516, Egypt

‡ University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai, 200240, China

Abstract

The multi-phase Lattice Boltzmann Method (LBM) is used to explore some unprecedented aspects of

laminar forced convection in a bottom heated rectangular microchannel. Important physical parameters,

such as forces exerted on fluid parcels as well as on the dispersed nanoparticle phase are studied, in an

attempt to elucidate the mechanism that results in establishment of a relative velocity between

nanoparticles and the continuous fluid phase (slip velocity). The significance of the external forces, such

as the gravitational, thermophoresis and Brownian forces is investigated. A recently established

expression for the estimation of thermophoresis force in nanofluids is employed to study the true effect of

thermophoresis, as other studies either neglect this effect, or are parametric or employ expressions that

overestimate this effect. The results indicate that in laminar forced convection, the Brownian force has a

significant effect on flow and heat transfer characteristics for low Re number flows (Re~1-10), but

thermophoresis may be safely neglected for all flow conditions. At low Re number flows, the nanofluid

flow is heterogeneous, and heat transfer characteristics of nanofluid compared to the base fluid, such as

Nu and convection heat transfer coefficient, significantly increase, while at higher Re numbers, such as

Re = 100, flow behaves homogeneously and therefore the application of a nanofluid is not justified.

Keywords: Laminar forced convection; Nanofluid; Thermophoresis; Brownian motion, Heated

microchannel, two phase lattice Boltzmann method (LBM)

1 Corresponding author, Email: Morteza.Eslamian@sjtu.edu.cn (M. Eslamian); Ph: +86-21-3420-7249; Fax: +86-

21-3420-6525

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1. Introduction

Microchannels have emerging microfluidic applications in thermal management, cooling of electronic

devices, biology and lab-on-a-chip devices, etc. A microchannel heat sink has the capability to dissipate a

large amount of heat from a small area with a high heat transfer rate and less fluid inventory [1]. Lack of

effective heat dissipation routes in electronic boards and digital computing is a burden for developing

faster computers and electronic devices. In the development of microchannels, the main issues are the

heat transfer rate, microchannel allowable temperature, and the pumping power. In recent years,

application of nanofluids in heated microchannels has been considered as a method for heat transfer

augmentation.

The term nanofluid denotes engineered colloids comprised of conductive particles, such as metal

and metal oxide nanoparticles, dispersed in a base fluid, usually to improve the fluid thermal

characteristics. Owing to small sizes of nanoparticles, in most cases, a stable suspension forms without

particle settlement. Most recent and reliable studies indicate an increase in fluid thermal conductivity and

heat transfer coefficient, when a nanofluid is used in lieu of pure base fluid. Heat transfer augmentation in

natural, mixed and forced convection of nanofluids is believed to be due an enhancement in thermal

conductivity, as well as, development of a relative velocity between the main fluid flow and the

suspended nanoparticles (slip/drift velocity), which enhances flow mixing and therefore heat transfer rate.

Any significant external force acting on nanoparticles may be a source of slip velocity or drift. Due to the

presence of slip velocity, homogenous single phase models and even dispersion models may provide only

a rough approximation of heat transfer rate in the system and may not be suitable for accurate and

fundamental studies on nanofluids. Thus, at least for certain flow conditions, only models that consider

the nanofluid as a multiphase flow system are reliable. The proper and general modeling approach is to

use a two phase model considering particles as a discrete phase, given that the nanoparticle concentration

is low, and the base fluid is a continuum phase. Instead of using the conventional two phase flow

modeling approach based on the Navier-Stokes equations, energy and continuity equations, and the force

balance on the particles, the two-phase lattice Boltzmann method (LBM) is employed in this work. This is

because from a microscopic point of view, the LBM can better reveal the inherent nature of the flow and

energy transport processes inside the nanofluid and can better take into account the effect of interactions

between the molecules and particles of the mixture. Although the LBM is a multiphase flow simulation

tool, the external forces that may be present in the flow need to be defined and included in the model,

separately. Below, these external forces are outlined, followed by a review of the pertinent works.

In a fluid flow, fluid parcels/molecules move as a result of gravity, shear and pressure forces.

When a second phase, such as nanoparticles, is dispersed in the continuous phase, some fluid parcels are

displaced by the newly embedded particles. This rearrangement may result in realization and creation of

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additional forces. Some of these forces include but are not limited to Brownian (random motion),

thermophoresis (thermal diffusion as a result of a temperature gradient), lift, Magnus (particle rotation),

diffusiophoresis (diffusion as a result of a concentration gradient), fluid drainage, and so on. As a result of

the presence of these external forces, particles may attain velocities different from the velocity of the

displaced fluid parcels (drift or slip velocity). The relative magnitude of external forces exerted on

particles compared to the magnitude of the forces that would otherwise exert on the displaced fluid

parcels depends on several factors, such as the nature of flow (laminar vs. turbulent), geometry and

boundary conditions, particle size and shape, etc. The effect of various forces that may cause slip velocity

in various flow conditions has been studied by several researchers, usually based on an approximate and

parametric time scale analysis, e.g. [2]. In laminar flows, turbulent eddies, which induce an abrupt change

in the flow direction, are absent. Also, forces such as diffusiophoresis, Magnus effect, fluid drainage, are

insignificant in nanofluids and will not be considered here. In-line with this argument, in our previous

works [3, 4], it was observed that, in the order of importance, the Brownian, thermophoresis, and

gravitational forces are responsible for slip velocity in, particularly at moderate Ra numbers (Ra ~ 106).

Development of a slip velocity is equivalent to having a heterogeneous flow. Below is a review of recent

pertinent works, including numerical and experimental investigations. Most numerical works use the

conventional numerical schemes, while few use the LBM. In most cases, the thermophoresis and/or

Brownian forces are neglected or modeled inadequately.

Experimental data on forced convection in channels and microchannels are quite abundant, e.g.,

[5-16]. While in natural convection, particle agglomeration is a potential source of error and discrepancy

in the experimental data, in forced convection particle agglomeration and clustering is minimized due to

the fluid flow. Nevertheless, particle precipitation on channel inner surfaces has been reported as a source

of disagreement between experimental data and numerical simulations. Overall, the experimental studies

predict an increase in heat transfer rate and pumping power with an increase in nanoparticle volume

concentration and Re number. Chein and Huang [5] analyzed the performance of nanofluids in a silicon

microchannel where it was found that Nu number increases significantly with an increase in Re and

particle loading. Up to 15% reduction in thermal resistance was observed with no significant increase in

pressure drop. Jang and Choi [6] and Chein and Chuang [7] performed experimental analyses in

microchannel heat sinks, where it was concluded that utilizing nanofluids at low flow rates reduces the

thermal resistance of the heat sink and enhances the cooling performance. However, at high flow rates the

advantage of using nanofluids, as far as heat transfer is concerned, is negated. This conclusion is in-line

with the finding of the present work. Singh et al. [8] adopted a convectional Eulerian-Lagrangian

approach with Brownian and thermophoresis forces considered although, the expression used for

thermophoresis force is only valid for gases, and therefore overestimates the effect of thermophoresis up

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to several orders of magnitude. Nevertheless, their numerical results match with their experimental data

where an increase in Nu number is observed with an increase in Re number and particle volume

concentration. Asirvatha et al. [9] performed an experimental study in a small channel considering

laminar, transition and turbulent flows. A 69% increase in convective heat transfer coefficient was

achieved at particle volume concentration of 0.9%. Their results show that the Dittus-Boelter correlation

used with the nanofluid physical properties highly underestimates the Nu number. In another

experimental work, the performance of different loadings of water-based alumina nanofluids was

investigated in a commercially available cooling system of a computational processing unit [10]. An

enhancement up to 18% in the convective heat transfer coefficient was reported at particle volume

fraction of up to 1.5%. Ho et al. [11] conducted experiments to investigate forced convective cooling

performance of a copper microchannel heat sink with a nanofluid as the coolant. The nanofluid, compared

to the base fluid, had a significantly higher average heat transfer coefficient and lower thermal resistance

and wall temperature, in the expense of slightly increased pumping power. In another experimental and

numerical work, Kalteh followed the conventional two-phase flow Eulerian-Lagrangian approach;

however Brownian and thermophoresis forces were neglected [12]. Numerical predictions were in

qualitative agreement with their experimental data. In another experimental study, both an increase and

decrease in the heat transfer rate were observed by Anoop et al. [13]. Heat transfer deterioration was

attributed to nanoparticle precipitation (fouling) on microchannel surfaces. Refs. [14, 16] predict an

increase in heat transfer rate when a nanofluid is utilized, whereas in Ref. [17] for Re numbers in the

range of 500 to 2500, an increase in the heat transfer rate was observed for particle volume fractions of

0.24% and 1.03%, but a decrease for particle volume fraction of 4.5%. The pumping power in all cases

increases significantly, in such a way that the application of a nanofluid was found infeasible. In contrast

to Ref. [17] which predicts only a slight increase in heat transfer rate, in Ref. [18] a significant increase in

the heat transfer rate up to 250% is reported when a nanofluid is used to cool a microchannel in the range

of Re= 50 to 800. Murshed et al. [19] also reported a significant increase in heat transfer rate when TiO2

nanofluid was used within the range of Re=900 to 1700 in a circular channel.

Koo and Kleinstreuer [20] performed a numerical simulation where Brownian motion was

considered, as an external force. In another study, heat transfer enhancement in the cooling system of

microprocessors and electronic components was studied, where a considerable enhancement in convective

heat transfer coefficient around 40% was reported for 6.8% particle volume fraction [21]. An analytical

and numerical investigation was performed considering the mixed convection of nanofluids in a vertical

channel [22], where Brownian and thermophoresis effects were considered as two variable parameters. A

significant change in the flow and heat transfer characteristics was observed with a change in Brownian

and thermophoresis forces in the form of non-dimensional parameters. However, it is noted that real

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strength of both of these forces can be estimated using available theories and changing these parameters

within an unrealistic range may be misleading. Fan et al. [23] performed a similar parametric study in a

horizontal channel. In another study that considers thermophoresis and Brownian forces, Nield and

Kuznetsov [24] investigated forced convection in a parallel plate channel filled with a nanofluid with or

without a porous medium. Surprisingly, their results show that the combined effect of thermophoresis and

Brownian diffusion reduces the Nu number. Authors of Ref. [24] were consulted regarding their results;

they recently published an erratum to address some issues in their original work and made some

corrections and clarifications and limitations to the range of applicability of their results [25]. Akbarinia et

al. [26] performed a numerical investigation on forced convection in a microchannel where slip and non-

slip boundary conditions (non-zero Kn number) and Brownian motion were considered. It was concluded

that with an increase in nanoparticle concentration, nanofluid viscosity increases and therefore the

channel inlet velocity should be increased in order to keep the Re number constant. Thus it was concluded

that the increase in the Nu number or heat transfer rate at constant Re is due to an increase in the flow

velocity and not the presence of nanoparticles. In other words, it was argued that at constant inlet velocity,

an increase in particle volume concentration has no significant effect on heat transfer rate; but it is noted

that in this case the flow Re number substantially decreases. There are many other papers that have

considered various aspects of nanofluids forced convection in a microchannel, using conventional

numerical schemes, either single phase or multiphase, e.g. [27-31]. In recent years, however, the

multiphase Lattice Boltzmann Method (LBM) has proved to be an effective tool to simulate multiphase

flows, problems with complex boundaries, and problems subjected to various external forces [32]. Below

is a brief review of works that use the LBM to simulate nanofluid laminar flow in a microchannel.

Using LBM, simulations were conducted by Yang and Lai [33, 34] at low Re numbers in a

microchannel, where it was found that the average Nu number increases with an increase in Re number

and particle volume concentration. In another work, Hung et al. [35] predicted nanoparticle optimum

concentration for obtaining maximum heat transfer coefficient. Sidik et al. [36] studied nanofluid laminar

flow in a bottom heated finned channel, where the presence of fins was found to enhance heat transfer

rate. Their LBM simulation takes into account the Brownian motion of nanoparticles. Importance of slip

velocity and temperature at the boundaries in laminar nanofluid flow was studied by Karimipour et al.

[37], where significant temperature jump was observed at the entrance, which resulted in a decrease in the

Nu number. Brownian motion was taken into account in their LBM simulation. A comprehensive review

of the LBM simulation approach as well as a review of recent works on natural and forced convection in

cavities and microchannels is given in Ref. [38]. A list of forced convection correlations and more

discussion on various aspects of forced convection in channels and microchannels may be found in Refs.

[39-41].

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The main findings of the existing literature may be summarized as follows: (1) Most experimental

studies predict an increase in heat transfer rate (or Nu number) with inclusion of nanoparticles, with some

differences in details of enhancement; (2) most numerical studies have considered a single phase fluid

approach, which may oversimplify the problem and obscure the interaction between the base fluid and

nanoparticles, at least for a range of Re number; (3) most studies have neglected the effect of

thermophoresis and Brownian forces that may be considerable in laminar flows. Very few studies have

considered thermophoresis effect, while inaccurate or unsuitable expressions have been used to

investigate the true effect of thermophoresis. Based on the forgoing summary, the objective of this work

is to explore the effect of thermophoresis and Brownian motion in fluid and heat transfer characteristics of

laminar forced convection flow in a microchannel. To this end, the two phase lattice Boltzmann method

(LBM) is employed along with the most accurate expression for thermophoresis in nanofluids. The

analysis is novel and different from existing works, in that the fluid flow and external forces are calculated

and compared to gain insight into the physics of the problem and to address and answer some

fundamental questions about the significance of external forces and validity and the range of applicability

of single phase modeling approach.

2. Modeling Approach

The main equations of the two phase LBM is given in other works e.g., [42-45], as well as in our previous

works [3, 4], and not repeated here for brevity. In the following section, discussion is limited to the

relevant external forces, only.

Major external forces that may cause slip velocity are discussed in Ref. [2]. In natural convection,

Brownian, thermophoresis and gravitational forces are identified as significant forces that may cause slip

velocity [2, 3]. In this work, low Re number forced convection is investigated wherein the aforementioned

three external forces are expected to be considerable. As a result of the slip velocity, a drag force is also

induced. Since, thermophoresis is a diffusion and slow process, the thermophoresis force FT can be

related to thermophoretic velocity UT through the Stokes equation:

FT = 3πμUT dp (1)

where UT is related to the thermophoresis or thermodiffusion coefficient DT as follows:

TDU TT (2)

The thermophoresis coefficient may be estimated based on the several approaches [43]. Here, the

hydrodynamics-based theory is used to estimate DT [46, 47]:

Tkk

kAD

pT

22 (3)

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where the physical properties are associated with the nanofluid (k, μ, ρ) and the particles (kp). Coefficient

A is a function of liquid physical properties and temperature and is derived in Ref. [47]. For water we

have shown that A is equal to 0.0085 [46]. The thermophoresis force is a strong force in particles

suspended in a gas, while it is weaker when the carrier fluid is a liquid [48]. Therefore, an expression

prescribed for the thermophoresis force in gases is not applicable to liquids and vice versa.

The drag coefficient for the drag force exerted on particles in the range of Re numbers explored

in this study is obtained from the following correlation [49]:

nmD BC Re0.1

Re

24 1.0 ≤ Re ≤ 2000 (4)

where B = 0.0665, m = 2/3, and n = 3/2.

Brownian motion is the random and fluctuating motion of particles caused by the collision of

fluid molecules with the suspended particles. The components of the Brownian force are modeled as a

Gaussian white noise process. Details of calculating the Brownian force can be found elsewhere, e.g., ([50]

and references therein). In the lattice Boltzmann method, the total force per unit volume acting on

nanoparticles of a single lattice is written as follows:

TBDHP FFFF

V

nF (5)

The force FH is the net effect of the buoyancy and gravity forces, acting on opposite directions. The sum

of forces per unit volume, acting on the base fluid Fw is written as follows:

TBDw FFF

V

nF (6)

Detailed analysis of momentum and heat transfer interchange between fluid and nanoparticles is

explained in Ref. [42]”. An empirical correlation is used for viscosity of the nanofluid normalized by the

viscosity of the base fluid [51]. This correlation has been tested against a large database. The correlation

by Maxwell is used for the effective thermal conductivity of the nanofluid, which is simple, widely used,

and has reasonable accuracy for low concentration nanofluids.

Except for the validation part in which boundary and operating conditions are identical to those of

the employed experimental data, the rest of the results are associated with the following conditions: Water

is used as the base fluid and 10 nm copper oxide (CuO) nanoparticles as the dispersed phase with density

of 6500 kg/m3 and specific heat capacity of 535.6 J/kg.K. Thermal conductivity of CuO nanoparticles is

variable; here an average value of 20 W/m.K is used. The temperature variation in the microchannel is

less than 10 ºC, and therefore physical properties of water are assumed temperature independent and

evaluated at 305 K. A section of a two dimensional microchannel made of two parallel plates with a

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distance of h = 100 μm and a length of 25h = 2.5 mm is considered as the computation domain (Figure 1).

A mesh of 50 × 1250 nodes is used for the computations. Table 1 shows the mesh sensitivity analysis

justifying the selection of the used mesh, which is a compromise between accuracy and computational

time. The nanofluid enters the microchannel at a temperature of TC = 300 K. The bottom wall of the

microchannel is heated and kept at TH = 310 K, while the upper wall is colder and kept at TC. The inlet

velocity and initial particle loading are changed to achieve a range of Re numbers (1-500) and particle

volume fractions (0.0 to 0.05).

3. Results and Discussion

The code is validated using three different sets of available experimental and theoretical data at constant

temperature and constant heat flux boundary conditions. The first set of results is at constant wall

temperature for pure water flowing in a parallel plate channel with the third kind boundary conditions

(one plate adiabatic and the other one at constant wall temperature). In this case, the fully developed Nu

number is equal to 4.0 [52]. The predicted Nu number based on the present developed code is 3.8 at Re =

100, which is close to the theoretical value of 4.0. For the case of constant heat flux, the fully developed

Nu number for pure water is equal to 5.385, as reported in [52], while the present work predicts Nu

number of about 5.3 at Re = 100, which is close to the theoretical value. To further validate the code, in

Figure 2 we have shown the numerical and analytical results for the pure fluid (water) velocity

distribution along the microchannel cross section at x* = x/h = 15, non-dimensionalized with the inlet

velocity at a given Re number. A good agreement between the analytical results and numerical results is

observed.

To further validate the code, Figure 3 compares the predicted (this work) and experimental values

of Nu number from Ref. [12] for pure water and for an alumina-based nanofluid (40 nm particle size) in a

microchannel heated from below at constant heat flux (20.5 kW/m2), where Re varies from 70 to 300 and

ϕ varies from 0.0 to 0.002. The channel length to width ratio is 163. Comparisons show a good agreement

between the measured and predicted values. For pure water (ϕ = 0.0), there is a very good agreement

between measured and predicted values, except for Re = 90, where the model underestimates the

measured values. At ϕ = 0.001, and 0.002, the predicted results have the same trend as the measured

values, although the model underestimates the measured values around 10-20%. This may be attributed to

uncertainties in the experimental data and the lack of details about the measured values, such as the

number of replications, the level of confidence used in estimating the mean values of Nu number, and so

on. The inadequacy of the numerical simulations, such as the accuracy of the correlations used to estimate

the nanofluid physical properties and the shortcomings of the models used to simulate the external forces,

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may be also responsible for deviation between the numerical results and experimental data. Figure 3

further shows that the experimental and numerical results at small Re numbers (Re < 100) are shifted

upward indicating the effective role of using a nanofluid in low Re numbers. This is because at low Re

numbers, the external forces, such as the Brownian and thermophoresis forces make the flow

inhomogeneous and consequently have a considerable role in heat transfer augmentation, an effect which

is mitigated as Re number increases. This effect will be extensively investigated in the rest of this paper.

Exploring and comparing the magnitude of forces induced in the system due to the fluid flow

momentum and inertia as well as external forces is central to understanding the hydrodynamic and

thermal behavior of nanofluid and nanoparticles. As mentioned before, in a laminar forced convection

flow with a nano-meter-sized dispersed phase, one can think of Brownian motion and Brownian force to

be present due to the small sizes of the particles. Thermophoresis may be also present due to the existence

of a temperature gradient across the channel walls, if the channel is heated. Gravitational forces, i.e.,

particle weight and buoyancy are also present and their magnitude may or may not be considerable. It is

safe to assume that other external forces are negligible [2]. The fluid flow force, i.e. the force exerted on

the fluid parcels, originate from the flow momentum as a result of shear forces, pressure gradient and

gravity, easily obtained from the NS equations. Here, however, since the LBM approach is used, the fluid

flow force is obtained in a different way. To make the calculation of the fluid flow force consistent with

that of the external forces exerted on 10 nm nanoparticles, it is assumed that the fluid flow force is

equivalent to the force exerted on a parcel of nanofluid in the form of a 10 nm droplet (same size as the

size of the dispersed nanoparticles). This force can be obtained from the general drag force equation:

DD AC 2

2

1VFt (7)

where Ft and V are the fluid flow force and velocity, respectively, and are in the vector form, CD is the

drag coefficient of a sphere at given Re number (Eq. 4), and AD is the effective surface area. Figure 4

shows the variation of these forces exerted on 10 nm particles/droplets along the channel cross-section at

x* = x/h = 15, for particle volume fraction of ϕ = 0.01 and for different values of Re, i.e., 1, 10, 100, and

500. At x*

= 15, the flow is hydrodynamically and thermally developed. Figure 5a shows the two

components of the Brownian force in x and y directions. The Brownian force exerted on particles is an

oscillatory and sporadic force, and the net effect is to agitate the particles in the flow and to improve

fluid-particle mixing. Now back to Figure 4, since a log scale is employed, positive values can be plotted

only. Therefore, what shown in Figure 4 is the magnitudes of resultant forces, regardless of directions. As

a result, the displayed Brownian force is the absolute value of the resultant Brownian force acting on one

particle which is constant along the channel cross section, but we note that the direction of this force

randomly changes, and therefore the net effect of this force is local agitation rather than translation in a

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particular direction. The particle Brownian diffusion coefficient is a function of local temperature and

viscosity, and since these two aforementioned variables only slightly change across the channel height,

the resultant Brownian force is almost constant. The net gravitational force on a 10 nm particle is small

(2.8 × 10-20

N), which is the smallest external force in this nanofluid system. The thermophoresis force

also appears to be small and is on the order of 10-17

N and is larger in areas where there is a stronger

temperature gradient, i.e., near the bottom wall. Since, the temperature gradient and thermophoresis

coefficient vary from the bottom wall to the top wall, the thermophoresis force also changes. Later it is

shown that the temperature gradient is smaller close to the top wall and so is the thermophoresis force.

The drag force shown in Figure 4 is induced as a result of a difference between the fluid and

particle velocities (slip velocity). The drag force is sporadic mainly because of the sporadic behavior of

particle velocities due to the random Brownian force. The drag force decreases as the Re number

increases, indicating that at higher Re numbers, external forces become less significant and slip velocity

vanishes. Figure 4 also displays the fluid flow driving force, i.e., the fluid flow force or momentum,

exerted on 10 nm droplets, droplets which have been displaced by 10 nm nanoparticles of the dispersed

phase. To investigate the effect of slip velocity, it is best to compare the particle drag force, which is

induced due to slip velocity, and the fluid flow force. At low Re numbers (Re = 1), these two forces are

comparable indicating that forces that cause slip are significant, whereas at Re numbers of about 10~100

and higher the slip velocity effect vanishes.

Given that the Brownian force has a random and unsteady behavior, a sensitivity analysis was

performed on the numerical method to confirm that the numerical results are stable. To this end, velocity

profiles for the base fluid and nanofluid and nanofluid bulk temperature profile were calculated with and

without the Brownian force to confirm that without the Brownian force the velocity and temperature

profiles are steady and not sporadic. Figure 5b shows the velocity profile of the nanofluid at Re = 10 and

ϕ = 0.01, by which the stability of the numerical results is verified.

The slip velocity is further investigated in Figure 6, where the non-dimensional velocity profiles

of nanoparticles, base fluid and nanofluid are plotted at various Re numbers for a nanofluid with particle

volume concentration of ϕ = 0.01. Velocity profiles with and without the thermophoresis effect were

compared where it was found that thermophoresis had no effect (results not shown), even in the creeping

flow (Re = 1). Therefore, the Brownian force is the only external force that has a considerable effect on

the flow and thermal characteristics. Figure 6 shows that at high enough Re numbers (Re = 100 and

above), there is zero slip velocity on the particles, indicating that the flow behaves as a single component

homogenous flow. Therefore, one may conclude that provided that the external forces are small compared

to the flow momentum, the dispersed nanoparticles essentially follow the main flow leading to a

homogenous flow for which a single-phase simulation approach is sufficient. Yang and Lai [33]

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compared the results of numerical simulation of laminar forced convection in a microchannel (Re ≤ 16),

where it was concluded that at both the LBM and the conventional Eulerian-Lagrangian momentum

equations predict similar results. In their analysis, however, the effect of external forces that may cause a

relative velocity between particles and the flow was neglected.

Our results indicate that the Brownian force is the most or only important external force, which

particularly at low Re numbers can make the nanofluid behave heterogeneous. This may be in-line with

the thermal conductivity model of Koo-Kleinstreuer-Li (KKL) [53, 54], in which the effective thermal

conductivity is comprised of two terms, the static term and the Brownian motion term. As a result, it may

be adequate to use a homogenous and single phase numerical approach together with the KKL thermal

conductivity model, which accounts for the Brownian motion. Using the KKL model linked to a

commercial NS Equation solver for a single phase flow, good agreement between experimental and

numerical data was observed [53, 54]; this finding, however, cannot be independently confirmed here.

Figure 6 shows some interesting flow and nanoparticle behaviors. At sufficiently high Re numbers, the

velocity profile of nanoparticles, the base fluid and the nanofluid are nearly identical, indicating that the

fluid flow is homogenous, as argued before. At low Re numbers, however, there is a distinct difference

between the nanofluid and nanoparticles velocities. At Re = 1, the velocity profiles are highly sporadic

indicating that the magnitude of the external forces is comparable to that of the fluid flow force. Even,

adjacent to the bottom and top walls, a region of reverse nanofluid flow is realized.

The effect of Re and ϕ on nanofluid non-dimensional temperature profile is shown in Figure 7.

The slope of the line ( *y ) on each end of the curve is proportional to the heat flux from the bottom

wall (y* = –0.5) or to the top wall (y

* = 0.5). Several interesting observations are made. Heat is added to

the flow from the bottom wall, while a portion of this heat is removed from the top wall. Generally, as the

Re number increases, *y increases on the bottom wall, while it decreases on the top wall. Therefore,

an increase in Re number results in an increase in the incoming heat flux from the bottom wall to the flow,

and a decrease from the outgoing heat flux from the flow to the top wall. At no flow condition, i.e., pure

conduction, one would expect an equal heat flux from the bottom and top walls. At a fluid flow condition,

as the Re number increases, more heat is carried away by the flow downstream of the microchannel. At

high enough Re numbers, such as at Re = 10 and above, *y on the top wall and therefore the outgoing

heat flux approach zero (Figure 7), as most of the incoming heat flux from the bottom wall is swept

downstream. To further investigate the temperature change in the microchannel, the variation of the non-

dimensional bulk temperature along the microchannel and the temperature contours are calculated, using

the definition of the bulk temperature for a developing flow, and are shown in Figures 8 and 9,

respectively. Figure 8a shows that at low Re numbers, within a short distance from the entrance, the bulk

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fluid temperature rapidly increases and then only slightly continues to increase along the channel

thereafter. This indicates that for low Re number flows, for the majority of the channel length, most of the

heat added from the bottom wall is removed from the top wall (existence of strong conduction across the

channel cross section). These results are better interpreted at ϕ = 0, because at ϕ = 0.01, the Brownian

force makes the temperature profiles sporadic. Figure 8b excluded the results for Re = 1, given that a

meaningful average velocity and therefore bulk temperature cannot be defined for this case (c.f. Figure 6a

for velocity profiles at Re = 1). Both Figures 8a and 8b show that at Re = 100, the trend is very different

from the cases with Re =1 and 10, in that the temperature continuously increases from the inlet to the

outlet, as a result of the heat supplied from the bottom heated wall. Interesting findings are observed when

Figures 8a (ϕ = 0.0, pure fluid) and 8b (ϕ = 0.01, nanofluid) are compared. The temperature profiles for

the nanofluid are sporadic at low Re numbers, indicating the presence of inhomogeneity, an effect which

is negated at higher Re numbers. It is noted that although close to the microchannel entrances, the bulk

temperature associated with a higher Re number is small compared to that for a lower Re number, this

trend will reverse far from the inlet. With an increase in Re number, the heat transfer from the bottom

wall and therefore the Nu number increases and therefore far from the inlet (not shown), the fluid bulk

temperature associated with Re = 100 is expected to be greater than that for the case with a lower Re

number such as Re = 10 or 1.

Figure 9 shows the temperature contours in the microchannel from the inlet to x* = 15 for various

Re numbers and for pure water as well as for ϕ = 0.01. This figure shows that a large portion of the

channel is unaffected by the bottom heated wall, and the width of the unaffected area increases with an

increase in Re number. An important finding is that the smooth streamlines associated with pure water

become somewhat disturbed by the presence of the dispersed nanoparticle phase at ϕ = 0.01. This flow

disturbance as a result of external forces and slip velocity lead to heat transfer enhancement (to be

discussed later on in this paper).

Figure 10 shows the effect of particle volume concentration on non-dimensional temperature

profiles at Re = 10, the Re number at which peculiarities are observed in flow characteristics. It is seen

that with increase of ϕ profiles’ curvature decreases first when ϕ increases from 0 to 0.01, and then it

increases with further increase in ϕ up to 0.05. This peculiarity is observed in Figure 11, as well, where

the averaged Nu number on the bottom heated wall is plotted against the particle volume concentration

for various Re numbers. While in general, Nu increase with an increase in particle concentration and Re

number, at Re = 10, a slight decrease in Nu is observed for ϕ = 0.01 after which Nu increases with further

increase in ϕ. At Re = 10 and ϕ = 0.05, a 30% increase in Nu number is observed.

Another important parameter of interest that affects the pumping power and the energy budget is

the pressure drop in the channel. Pressure drop per unit width of microchannel is shown in Figure 12,

13

where it is observed that an increase in this parameter is predicted with an increase in the particle volume

concentration. To gain a better insight and to better analyze and interpret the results shown in Figures 11

and 12, the relative percentage of changes in Nu number, heat transfer coefficient and pressure drop of a

nanofluid with 1% particle volume concentration is calculated and listed in Table 2. About 5% increase in

pressure loss is predicted at Re = 1, and 10, when a 1% nanofluid is used. At Re = 100, pressure drop

increases to 8.86%. Interesting results are obtained for the heat transfer characteristics: At Re = 1,

application of a 1% nanofluid results in a 64% increase in Nu and 113% increase in the convection heat

transfer coefficient. At Re = 10, peculiarities are observed here again and an adverse effect is observed.

At Re = 100, application of nanofluid has negligible effect on heat transfer characteristics, while the

pressure loss increases. These results indicate the effectiveness of nanofluid at low Re numbers, in-line

with other previous results that showed a heterogeneous flow behavior at low Re numbers. And as

discussed earlier, the flow inhomogeneity is due to the significance of the external forces, particularly the

Brownian force, compared to the flow force or momentum. Therefore, one may conclude that for Re of

the order of 1, application of nanofluid is quite effective. These results are consistent and in accord with

those reported in [6] and [7], although compared to those works we are predicting much higher

enhancement in heat transfer characteristics, perhaps because we have considered external forces, which

are responsible for better mixing. Our results are also in good agreement with the experimental data of

Asirvatha et al. [9] who utilized water-based silver nanofluid at particle volume concentration of 0.9%

and measured 69% increase in the convective heat transfer coefficient. Unfortunately, systematic

experimental data for low Re number flows do not exist. Moreover, the observations made in the

experimental studies are inconclusive and scattered, dependent on the geometry, boundary conditions,

operating parameters, and so on. Some works predict a significant increase in the heat transfer parameters

when a nanofluid used, e.g. [18], whereas some other works show no considerable change, e.g. [17].

Conclusions

Some unprecedented aspects of the flow and heat transfer characteristics of laminar forced

convection in a heated microchannel utilizing a nanofluid was numerically analyzed by multi-phase

Lattice Boltzmann Method (LBM). The fluid flow force due to viscous and pressure forces was estimated

and compared with the external forces acting on the dispersed nanoparticle phase. The following major

conclusions are made:

For a range of Re numbers from 1 to 500, the fluid flow force, external forces and the drag force

exerted on the dispersed phase were estimated and compared. It was observed that, at Re numbers higher

than about 10 or so, the fluid flow force is the dominant force. For lower Re numbers, the oscillatory

Brownian force is the largest external force and plays a major role, while the thermophoresis and

14

gravitational forces are rather insignificant. It was argued that the net effect of the Brownian force is to

create a local oscillatory motion rather than a translational motion. At low Re numbers (Re <10 or so),

external forces, particularly the Brownian force, create a relative drift velocity (slip velocity) between the

nanoparticles and the main fluid flow. This slip velocity results in sporadic temperature and velocity

profiles (heterogeneous flow), whereas at higher Re numbers, the profiles follow smooth patterns

observed in homogenous laminar flows. In other words, the presence of the dispersed nanoparticle phase

makes the flow inhomogeneous or heterogeneous for small Re numbers. Based on the current results and

in the range of parameters considered here, for small Re numbers which are typically used in

microchannels, a multi-phase heterogeneous model has to be used.

And finally, it is concluded that for small Re numbers (Re ~ 1), the introduction of the dispersed

nanoparticles, makes the fluid flow heterogeneous through creating a drift velocity between nanoparticles

and the base fluid. This results in better mixing, enhancing the heat transfer characteristics of the fluid

significantly, doubling the heat transfer coefficient, in the expense of a 5% increase in pressure loss at

particle volume concentration of 1%. The effect is reversed or is insignificant at high Re numbers (Re ~

10, 100, and higher), mainly because the nanoparticles are entrained in the main flow stream and cannot

cause mixing. In summary, application of nanofluids for low Re numbers (Re~1) seems to be highly

beneficial.

Nomenclature

A coefficient in Eq. (3)

B coefficient in Eq. (4)

Abs absolute value (used for the net gravitational forces)

AD Droplet reference (cross sectional) area

CD Drag coefficient

dP Nanoparticle diameter

DT thermodiffusion (thermophoresis) coefficient

Fb buoyancy force

FB Brownian force

FD drag force

F H net gravitational force as a result of buoyancy and particle weight

FP

sum of forces acting on a particle per unit volume

FT thermophoresis force

15

Ft representing fluid force acting on a fluid parcel, in this paper, drag force acting on a 10

nm liquid droplet

Fw sum of forces acting on the base fluid

g gravitational acceleration

h microchannel height, also convection heat transfer coefficient

k nanofluid thermal conductivity

kp particle thermal conductivity

L microchannel length

m a coefficient in Eq. (4)

n number of particle in a given lattice; also a coefficient in Eq. (4)

Re Reynolds Number

T fluid local temperature

TH temperature of the hot wall (bottom wall)

TC temperature of the cold wall (top wall); also inlet flow temperature

UT thermophoresis velocity

u horizontal velocity component

u* horizontal velocity non-dimensionalized with respect to inlet velocity

U0 microchannel inlet velocity

v vertical velocity component

V lattice volume

V fluid flow velocity vector

VP velocity vector of a particle

wp particle weight

x x coordinate

x*

non-dimensional channel distance in x direction (=x/h)

y y coordinate

y*

non-dimensional coordinate in y direction (=y/h)

Greek Symbols

ρ nanofluid overall density

ρP particle density

ϕ particle volume concentration or particle loading

μ nanofluid viscosity

16

θ dimension-less temperature 𝑇−𝑇𝐶

𝑇𝐻−𝑇𝐶

Superscript

* non-dimensional parameters

17

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21

Table 1: Grid independence test at Re=100. The 50 × 1250 grid was used.

Table 2: Percentage of increase of pressure loss, Nu number, and convection heat transfer coefficient, h, at particle

volume concentration of ϕ=0.01.

Figure 1: Schematic of the physical domain with thermal boundary conditions and the coordinate system. Velocities

are zero at top and bottom walls. The domain is a two dimensional microchannel made of two parallel plates kept at

different temperatures.

Figure 2: Code validation: Dimension-less velocity distribution along the channel cross section at x*= 15, and for

ϕ=0.0. The figure shows the LBM numerical results as well as analytical results for laminar flow for the purpose of

code validation.

Figure 3: Code validation: Qualitative comparison between experimental data and numerical results of the present

work for Nu number vs. Re number. Experimental data were taken from Ref. [12]. Note that the problem geometry

and boundary conditions (constant heat flux) for experimental data and numerical results are the same.

Figure 4: Variation of magnitude of the resultant forces exerted on nanoparticles or nanodroplets (for the case of Ft)

of 10 nm size along the channel cross section x*= 15 for ϕ=0.01 and for different values of Re number.

22TyTxT FFF ,

22ByBxB FFF ,

22DyDxD FFF , bpH FwAbsF . (a) Re=1; (b) Re=10; (c) Re=100; (d)

Re=500.

Figure 5: (a) Spatial variation of components of the Brownian force acting on 10 nm nanoparticles. Solid lines

represent the horizontal component in x direction and dotted lines represent the vertical component in y direction. (b)

Sensitivity analysis of the numerical method: Nanofluid velocity profile with and without the Brownian force.

Plotted along the microchannel vertical direction at x* = 15 for Re = 10 and ϕ = 0.01.

Figure 6: Velocity profiles of the base fluid (liquid), nanoparticles, and nanofluid along the channel cross section at

x* = 15 for ϕ = 0.01 and for different values of Re. (a) Re = 1; (b) Re = 10; (c) Re = 100; (d) Re = 500. All external

forces were considered, although only the Brownian force had a considerable effect.

Figure 7: Effect of Re and ϕ on temperature distribution across the microchannel cross section at x*=15. (a) ϕ=0.0

and (b) ϕ=0.01. All external forces are considered.

Figure 8: Variation of the nanofluid bulk temperature versus dimensionless axial distance at various Re numbers for

(a) ϕ = 0 and (b) ϕ = 0.01. At ϕ = 0.01, results for Re = 1 are not shown due to the change in the direction of flow

velocity by the Brownian force and difficulty in defining the bulk temperature.

Figure 9: Temperature contours within the channel at various Re numbers for ϕ=0.0 and ϕ=0.01.

Figure 10: Effect of particle volume concentration on nanofluid temperature profile for Re=10.

Figure 11: variation of the Nu number and convection heat transfer coefficient h averaged along the bottom heated

wall of the microchannel versus ϕ at different values of Re. At Re=1, computations were possible only up to ϕ=0.01.

Figure 12: Variation of pressure drop per unit channel width along the microchannel versus particle volume

concentration at Re = 1, 10, 100, and 500. At Re=1, computations were possible only up to ϕ=0.01.

22

Table 1: Grid independence test at Re = 100. The 50 × 1250 grid was used.

l

Table 2: Percentage of increase of pressure loss, Nu number, and convection heat transfer coefficient, h,

at particle volume concentration of ϕ=0.01.

Re = 1 Re = 10 Re = 100

% increase in pressure loss +5.15% +5.55% +8.86%

% increase in Nu number +64% -12% +2.7%

% increase in heat transfer coefficient h +113% -9.7% +5.5%

Number of grids Nu

40 × 10000 3.9

50 × 1250 3.99

60 × 1500 4.0

23

Figure 1: Schematic of the physical domain with thermal boundary conditions and the coordinate system

used in this work, except for Figure 12 in which a constant wall heat flux was employed. Velocities are

zero at top and bottom walls. The domain is a two dimensional microchannel made of two parallel plates

kept at different temperatures.

24

Figure 2: Code validation: Dimension-less velocity distribution along the channel cross section at x* = 15,

and for ϕ = 0.0. The figure shows the LBM numerical results as well as analytical results for laminar flow

for the purpose of code validation.

25

Figure 3: Code validation: Qualitative comparison between experimental data and numerical results of the

present work for Nu number vs. Re number. Experimental data were taken from Ref. [12]. Note that the

problem geometry and boundary conditions (constant heat flux) for experimental data and numerical

results are the same.

26

(a) (b)

(c) (d)

Figure 4: Variation of magnitude of the resultant forces exerted on nanoparticles or nanodroplets (for the

case of Ft) of 10 nm size along the channel cross section x* = 15 for ϕ = 0.01 and for different values of

Re number. 22TyTxT FFF , 22

ByBxB FFF , 22DyDxD FFF , bpH FwAbsF . (a) Re = 1; (b) Re =

10; (c) Re = 100; (d) Re = 500.

27

(a) (b)

Figure 5: (a) Spatial variation of components of the Brownian force acting on 10 nm nanoparticles. Solid

lines represent the horizontal component in x direction and dotted lines represent the vertical component

in y direction. (b) Sensitivity analysis of the numerical method: Nanofluid velocity profile with and

without the Brownian force. Plotted along the microchannel vertical direction at x* = 15 for Re = 10 and ϕ

= 0.01.

28

(a) (b)

(c) (d)

Figure 6: Velocity profiles of the base fluid (liquid), nanoparticles, and nanofluid along the channel cross

section at x* = 15 for ϕ = 0.01 and for different values of Re. (a) Re = 1; (b) Re = 10; (c) Re = 100; (d) Re

= 500. All external forces were considered, although only the Brownian force had a considerable effect.

29

(a) (b)

Figure 7: Effect of Re and ϕ on temperature distribution across the microchannel cross section at x* = 15.

(a) ϕ = 0.0 and (b) ϕ = 0.01. All external forces are considered.

30

(a) (b)

Figure 8: Variation of the nanofluid bulk temperature versus dimensionless axial distance at various Re

numbers for (a) ϕ = 0 and (b) ϕ = 0.01. At ϕ = 0.01, results for Re = 1 are not shown due to the change in

the direction of flow velocity by the Brownian force and difficulty in defining the bulk temperature.

31

(a) Re = 1.0, ϕ = 0.0

(b) Re = 1.0, ϕ = 0.01

(c) Re = 10, ϕ = 0.0

(d) Re = 10, ϕ = 0.01

(e) Re = 100, ϕ = 0.0

(f) Re = 100, ϕ = 0.01

Figure 9: Temperature contours within the channel at various Re numbers for ϕ=0.0 and ϕ=0.01.

32

Figure 10: Effect of particle volume concentration on nanofluid temperature profile for Re = 10.

33

Figure 11: Variation of the Nu number and convection heat transfer coefficient h averaged along the

bottom heated wall of the microchannel versus ϕ at different values of Re. At Re = 1, computations were

possible only up to ϕ = 0.01.

34

Figure 12: Variation of pressure drop per unit channel width along the microchannel versus particle

volume concentration at Re = 1, 10, 100, and 500. At Re = 1, computations were possible only up to ϕ =

0.01.