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Review on Boundary-layer Flow of Nanofluids Over

A Continuously Stretching Sheet

Eldwin Djajadiwinata1*

1 Department of Mechanical Engineering, College of Engineering, King Saud University,

Kingdom of Saudi Arabia.

ABSTRACT

This paper presents a review of boundary-layer flow of nanofluids over a continuously

stretching sheet. The review started with an introduction on the problem considered. Afterwards, the

main approach of solving the problem, i.e., the single-phase approach and two-phase approach, are

presented. Based on the above mentioned approaches, several literatures related to boundary-layer

flow of nanofluids over a continuously stretching sheet are reviewed. Finally, the conclusions /

summaries are given.

Keyword: Nanofluids, Stretching surface, Boundary-layer

*Corresponding author: Eldwin Djajadiwinata; email: eldwin_dj@yahoo.com

Telp: +966-530823159

1. INTRODUCTION

Flow over stretching surface is an important engineering problem such as in the

cooling of material coming out of extrusion device and hot rolling device. The heat transfer

and flow characteristics will affect the quality of the product. Therefore, one must be able to

predict these flow and heat transfer processes in order to design the best method of cooling or

heating the product.

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The typical scheme of the stretching process can be seen in Figure 1 below:

Figure 1: Scheme of continuously stretched sheet such as in polymer extrusion process.

In such processes, boundary layer flow of the ambient fluid will be generated due to the no-

slip condition on the surface of the sheet as can be seen in Figure 1.

Researchers, mainly, have two approaches in predicting the flow and heat transfer of

nanofluids, i.e., the single-phase approach and two-phase approach. In single phase approach,

the nanofluids are treated as single-phase fluid which means there is no slip velocity between

phases (solid-particles vs. base fluid). The second approach, which is the two phase flow

approach, the velocity between the solid-particles and the base fluids is not neglected [1].

Regarding the flows over stretching surface, researchers have attempted the both

aforementioned approaches to predict their flow and heat transfer characteristics.

Unfortunately, so far, experimental investigation on this subject cannot be found yet.

Regarding the two-phase approach, researchers focused on two main mechanisms that

responsible for the slip velocity to exist, namely, Brownian diffusion and thermophoresis.

These two phenomena, i.e., Brownian diffusion and thermophoresis, have been predicted to

have the largest effect on the convective heat transfer enhancement in nanofluids whenever

the turbulent effects are negligible. These phenomena have been comprehensively analyzed

by Buongiorno [2].

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Thermophoresis is defined as the migration of a colloidal particle or large molecule

in a solution in response to a macroscopic temperature gradient. The particles will move to

the direction of decreasing temperature [3]. The random motion of nanoparticles within the

base fluid is called Brownian motion, and results from continuous collisions between the

nanoparticles and the molecules of the base fluid. If the turbulent eddies are present,

however, turbulent transport of the nanoparticles dominates, i.e., the nanoparticles are carried

by the turbulent eddies and other diffusion mechanisms are negligible [2].

2. THE GOVERNING EQUATIONS

In this section, the governing equations for flow of nanofluids are presented. The

governing equations based on the single-phase approach are the standard/usual single phase

governing equations. The only difference is that the properties of the fluid are taken as the

effective properties of the nanofluids obtained either experimentally or theoretically.

On the other hand, the governing equations of nanofluids for the two-phase flow

approach will follow those developed by Buongiorno [2] which take into account the

Brownian diffusion and thermophoresis effects. The assumptions for these equations are

written below and each of them was justified very well by Buongiorno [2].

1. Incompressible flow

2. No chemical reactions

3. Negligible external forces

4. Dilute mixture

5. Negligible viscous dissipation

6. Negligible radiative heat transfer

7. Nanoparticles and base fluid locally in thermal equilibrium

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The governing equations consist of continuity equation for the nanofluid, continuity

equation for the nanoparticles, momentum equation, and energy equation. These equations

are presented below.

The continuity equation for nanofluids is

(1)

The continuity equation for the nanoparticles is

[

] (2)

Where , , , and are nanofluid velocity, nanofluid temperature, nanoparticle

volumetric fraction, Brownian diffusivity and thermophoresis diffusivity, respectively.

Equation (2) states that the nanoparticles can move homogeneously with the fluid

(second term of the left-hand side), but they also possess a slip velocity relatively to the fluid

(right-hand side), which is due to Brownian diffusion and thermophoresis [2].

The momentum equation for a nanofluid takes the same form as for a pure fluid, but

it should be remembered that the viscosity is a strong function of nanoparticles volumetric

fraction. If one introduces a buoyancy force and adopts the Boussinesq approximation, then

the momentum equation can be written as

(

) (3) (3)

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Where the Nanofluid density, , is defined as

(4)

The nanofluid density can be approximated by the base fluid density, , when the volume

fraction is small. When the Boussinesq approximation is applied to take into account natural

convection, the buoyancy term is approximated as

[ { ( )}] (5) (3) (3)

Finally, the energy equation for nanofluid is written as

[

]

[

]

(6)

Detailed explanations of these governing equations can be seen in [2] and [4].

3. LITERATURE REVIEW

As has been mentioned before, there are two main approaches used to deal with

convection with nanofluid. Therefore, the literature review about nanofluid flow over

continuously stretching surface will be devided into two main catagories, i.e., the single-

phase approach and the two-phase approach.

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3.1. The single-phase approach

Vajravelu et al. [5] conducted study on convective heat transfer of Ag-water and Cu-

water nanofluids flowing over a vertical continuously stretching surface. Their focus is on the

effects of nanoparticle volumetric fraction on the flow and heat transfer characteristics having

buoyancy and internal heat generation or heat absorption. The velocity of the surface was

assumed to be a linear function of x (U(x) = b(x)). The nanofluids properties were estimated

using the equations available in the literature. The scheme of the problem considered is

shown in Figure 2.

Figure 2: Physical model and coordinate system [5].

The method used was similarity method which will transform the coupled non-linear

partial differential equations into coupled non-linear ordinary differential equations.

Afterwards, these equations solved these ordinary differential equations.

They have found that both, base fluids and nanofluids behave the same trend in terms

of its response to natural convection, i.e., the buoyancy will assist the main flow (to +x

direction) if it has the same direction and vice versa.

It was also found that the increase in nanoparticle volume fraction will decrease the

velocity profile and increase the skin friction. Vajravelu et al. [5] stated that this was because

the nanoparticles lead to further thinning of the boundary layer. On the other hand, the

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thermal boundary layer increases with respect to the increase of the volume fraction. This is

reasonable since the thermal conductivity of nanoparticles is higher than that of the base

fluid. Hence, if the volume fraction of nanoparticles increases, most likely, the nanofluids

thermal conductivity will also increase. This, in turn, will increase the thermal boundary

layer.

Study on natural convection flow of a nanofluid over a linearly stretching sheet in the

presence of magnetic field has been conducted by Hamad [6]. The magnetic field applied was

assumed constant and perpendicular to the stretching sheet/surface. The temperature of the

surface as well as the ambient was also assumed constant. The sheet velocity was taken to be

proportional to the x direction (the stretching direction) in the form of . The

governing equations were solved analytically using similarity method.

The nanoparticles data used are for Copper (Cu), Silver (Ag), Alumina (Al2O3), and

Titanium oxide (TiO2). Hamad [6] found that the magnetic field affected the momentum

boundary layer thickness inversely while, on the other hand, it affected the thermal boundary

layer proportionally. He also observed that the increase of nanoparticle volume fraction will

increase the thermal boundary layer which is consistent with the finding of Vajravelu et al.

[5]. It is also shown that the reduced Nusselt number, ⁄ , decreases as the

volume fraction of nanoparticles as well as the magnetic field increases.

Yacob et al. [7] did a research on mixed convection flow adjacent to a stretching

vertical sheet in a nanofluid. Three types of nanofluids are compared, i.e., Cu-water, Al2O3-

water, and TiO2-water nanofluids. The sheet velocity varies as (x is set in

vertical direction and y is in horizontal direction) where is a constant. The temperature of

the stretching surface is assumed to be where and are constants.

The nanofluids properties are obtained from the available literature.

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They used similarity method in solving the governing equations in conjunction with

the boundary conditions. Consistent with the aforementioned literatures, Yacob et al. [7]

found that for all nanofluids show increase in thermal boundary layer thickness with respect

to nanoparticle volume fraction increase. Consequently, the temperature gradient decreases

with increasing nanoparticle volume fraction. On the other hand, the velocity boundary layer

thickness decreases with nanoparticle volume fraction increase and therefore the velocity

gradient at the surface increases with volume fraction.

It should be noted that, interestingly, the author stated that a decrease in thermal

conductivity is to enhance the heat transfer rate at the surface. Thus, they arrived to a

conclusion that TiO2-water nanofluid, which has the lowest thermal conductivity, has better

heat transfer capability compared to that of Al2O3-water and CuO-water nanofluids. This

statement is based on the trend of the non-dimensional temperature gradient which is

increasing as the volume fraction (consequently the thermal conductivity) is increasing.

However, in our opinion, this statement is misleading because the heat transfer rate does not

only depend on the temperature gradient but also depend on the thermal conductivity. Thus,

to know the effect of nanofluid’s volume fraction on the heat transfer rate, one should

consider, both, the increase of the thermal conductivity and the decrease of the temperature

gradient.

3.2. The two-phase approach

Khan and Pop [8] were the first reserachers who investigated laminar fluid flow and

heat transfer of nanofluid over continuously stretching surface. They take into account

Brownian motion and thermophoresis in analyzing the problem. In solving the governing

equations in conjunction with the boundary conditions, Khan and Pop used similarity method

which resulted in dimensionless ordinary differential equations that depended on Prandtl

number, Pr, Lewis number, Le, Brownian motion number Nb, and thermophoresis

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number, Nt. The scheme of the problem can be represented by Figure 1. The velocity of the

sheet was assumed linear with x direction ( ). It was also assumed that the

temperature and the nanoparticle volume fraction at the stretching surface, and ,

respectively, were constant. Furthermore, the temperature and volume fraction of

nanoparticles at the ambient, and , respectively, were also assumed constant.

They solved the governing equations (based on the modified form of Eq.(1) - (3), and

Eq.(6)) together with the boundary conditions by means of similarity method. The following

definitions were also introduced:

,

,

Where and are the volumetric specific heat of the nanoparticle and the base

fluid, respectively. The Brownian diffusivity, thermophoresis diffusifivty, and kinematic

viscosity of the nanofluid are denoted as , , and , respectively. It was also defined

that the reduced Nusselt number and the reduced Sherwood number (both are defined based

on the gradient at the wall) to be ⁄ and

⁄ .

Khan and Pop [8] have found that the thermal boundary layer increased with respect

to the increase of Brownian motion number, , and thermophoresis number, .

Furthermore, the nanoparticle volume fraction boundary layer decreased with the increase

of .

It was also shown that the reduced Nusselt number was decreasing function of Nb

and Nt which means that the higher the Nb or Nt the lower the reduced Nusselt number. Most

likely this is due to the effect of Brownian motion and Thermoporesis which cause the

increase of the effective thermal conductivity of nanofluids and, consequently, will reduce the

Nusselt number.

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We would like to point out that Khan and Pop [8] have stated that dimensionless heat

transfer rate decreased with the increase of Nb and Nt. In our opinion, this statement is

misleading because the actually Nusselt number is the dimensionless heat transfer coefficient

or dimensionless temperature gradient and not the dimensionless heat transfer rate. The heat

transfer rate is actually a function of, both, thermal conductivity and the temperature gradient.

Figure 3: Effects of Nb, Nt and Pr on the dimensionless concentration gradient at the surface [8].

The effects of Nb, Nt and Pr on the dimensionless concentration gradient at the

surface were also presented by Khan and Pop [8]. It can be seen in Figure 3.a which is for

Prandtl number of one such as gas, the dimensionless concentration gradient decreases with

Nb and Nt. On the other hand for Prandtl number of 10 (Figure 3.b), such as water at certain

temperature, the trend is the opposite. In our opinion, the reason of this opposite phenomena

is due to the difference in nature between gas and liquid. Such opposite phenomena can also

be found on the viscosity of gas and liquid, i.e., the viscosity of gas increases with increase of

temperature while the viscosity of liquid decreases as the temperature increases.

Makinde and Aziz [9] conducted research on boundary layer flow of a nanofluid past

a continuously stretching sheet under convective boundary condition (Figure 4).

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Figure 4: Geometry of the problem [9].

In this study Makinde and Aziz [9] assumed that the velocity of the stretching sheet

was , where is a real positive number. The sheet surface temperature, ,

was the result of convective heating process which was characterized by a temperature, ,

and a heat transfer coefficient, h. The nanoparticle volume fraction at the surface and far

away from the surface are denoted by and . In order to solve this problem, they used

Buongiorno model (Eq.(1) - (6)) with some modifications.

Makinde and Aziz [9] solved the governing equations using similarity method as

done by Khan and Pop [8]. The difference was that, in this study, heat convection boundary

condition was applied. Thus, a new parameter introduced, i.e., Biot number defined as

⁄ ⁄ where , were the heat transfer coefficient, kinematic viscosity, and

thermal conductivity of the base fluid. They compared their results with the results of Khan

and Pop [8] by applying which will represent a constant temperature of the

stretching surface. As we know, Khan and Pop [8] did the analysis based on constant surface

temperature. Makinde and Aziz [9] found that their results matched very well with those of

Khan and Pop [8] for the reduced Nusselt and Sherwood numbers at Le = 10 and Pr = 10.

The interconnected effect of Nt and Nb on the reduced Nusselt number had been

observed. They found that the when the Brownian motion was weak, the change of

thermophoretic strength/thermophorsis number had little impact on the reduced Nusselt

number and Sherwood number. However, when the Brownian motion was relatively strong /

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Brownian number was relatively high, thermophoresis affected the reduced Nusselt number

strongly that can reach up to 50% reduction as the thermoporesis number, Nt, increased from

0.1 to 0.5. For the reduced Sherwood number, the increase of thermophoresis number, Nt,

from 0.1 to 0.5, increased the reduced Sherwood number up to 8%.

Makinde and Aziz [9] had also found that when the Pr, Le, and Bi were hold

constant, the thermal boundary layer as well as the local temperature increased as the

Brownian number, Nb, and thermophoresis number, Nt increased. The same trend was also

true for the thermal boundary layer when the Pr, Nb, Nt, and Le were kept constant and the

Bi was increased. However, the opposite trend was true for the thermal boundary layer

thickness when all parameters were kept constant while the Prandtl number was increase.

Furhtermore, they also concluded that the concentration boundary layer thickened as the Biot

number, Bi, increased.

4. CONCLUSION

Literature review has been conducted on a topic related to flow and heat transfer of

nanofluids over continuously stretching surface. The result of the review can be summarized

as follows:

1. There are two main approaches in dealing with convection in nanofluids including

convection of nanofluids flowing over continuously stretching surface. These two main

approaches are (1) the single-phase approach where it is assumed that there is no

relative motion between the nanoparticles and the base fluid and (2) the two-phase

approach where this relative movement of nanoparticles (slip velocity) is taken into

account.

2. Regarding the two-phase approach, Buongiorno [2] had conducted a comprehensive

study on convection heat transfer enhancement of nanofluids. He concluded that

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Brownian diffusion and thermophoresis have the largest effect on the convective heat

transfer enhancement in nanofluids whenever the turbulent effects are negligible.

Governing / conservation equations for nanofluids were also developed. If the turbulent

eddies are present, however, turbulent transport of the nanoparticles dominates, i.e., the

nanoparticles are carried by the turbulent eddies and other diffusion mechanisms are

negligible [2]

3. For the single phase approach, the concentration is assumed uniform all over the

domain of interest. Thus the study is focused on the effect of concentration and Prandtl

number of the nanofluids on the thermal boundary layer, velocity boundary layer,

reduced Nusselt number and skin friction.

4. For the two-phase approach, things are more complicated. Besides the three

conventional conservation equations, there is another equation, i.e., continuity equation

for nanoparticles which will result in concentration gradient/boundary layer near the

surface of the surface. The main parameters under consideration are Prandtl number, Pr,

Lewis number, Le, Brownian number, Nb, and thermophoresis number, Nt and their

effects on the reduced Nusselt number, ⁄ , and the reduced Sherwood

number, ⁄ .

5. All of the studies taking the two-phase approach based their governing equations on

those developed by Buongiorno [2] with or without modifications.

6. Generally, in the single phase approach it is found that the increase in nanoparticle

volume fraction will decrease the velocity profile and increase the skin friction. On the

other hand, the thermal boundary layer increases with respect to the increase of the

volume fraction. It also can be concluded that reduced Nusselt number, ⁄ ,

decreases as the volume fraction of nanoparticles increases.

7. For the two-phase approached it can be concluded that, in general, when the Prandtl

number, Pr, and Lewis number, Le, were hold constant, the thermal boundary layer as

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well as the local temperature increased as the Brownian number, Nb, and

thermophoresis number, Nt increased. However, the opposite trend was true for the

thermal boundary layer thickness when all parameters were kept constant while the

Prandtl number was increased.

8. Also the two-phase approached it can be concluded that, for fixed Pr and Le, the

reduced Nusselt number decreases while, on the other hand, the reduced Sherwood

number increases as the Brownian number and thermophoresis number increase.

REFERENCES

[1] Haddad, Z., Oztop, H. F., Abu-Nada, E., and Mataoui, A., 2012, "A Review on Natural

Convective Heat Transfer of Nanofluids," Renewable and Sustainable Energy Reviews, 16(7),

pp. 5363-5378.

[2] Buongiorno, J., 2005, "Convective Transport in Nanofluids," Journal of Heat Transfer,

128(3), pp. 240-250.

[3] Http://Aerosols.Wustl.Edu/Education/Thermophoresis/Section01.Html, last accessed

January 6, 2014,

[4] Nield, D. A., and Kuznetsov, A. V., 2009, "Thermal Instability in a Porous Medium Layer

Saturated by a Nanofluid," International Journal of Heat and Mass Transfer, 52(25–26), pp.

5796-5801.

[5] Vajravelu, K., Prasad, K. V., Lee, J., Lee, C., Pop, I., and Van Gorder, R. A., 2011,

"Convective Heat Transfer in the Flow of Viscous Ag–Water and Cu–Water Nanofluids over

a Stretching Surface," International Journal of Thermal Sciences, 50(5), pp. 843-851.

[6] Hamad, M. a. A., 2011, "Analytical Solution of Natural Convection Flow of a Nanofluid

over a Linearly Stretching Sheet in the Presence of Magnetic Field," International

Communications in Heat and Mass Transfer, 38(4), pp. 487-492.

[7] Yacob, N. A., Ishak, A., Nazar, R., and Pop, I., 2013, "Mixed Convection Flow Adjacent

to a Stretching Vertical Sheet in a Nanofluid," Journal of Applied Mathematics, 2013(pp. 6.

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[8] Khan, W. A., and Pop, I., 2010, "Boundary-Layer Flow of a Nanofluid Past a Stretching

Sheet," International Journal of Heat and Mass Transfer, 53(11–12), pp. 2477-2483.

[9] Makinde, O. D., and Aziz, A., 2011, "Boundary Layer Flow of a Nanofluid Past a

Stretching Sheet with a Convective Boundary Condition," International Journal of Thermal

Sciences, 50(7), pp. 1326-1332.