Applied Mathematics and Computation 238 (2014) 149–162

Contents lists available at ScienceDirect

Applied Mathematics and Computation

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

Boundary layer flow and heat transfer due to permeablestretching tube in the presence of heat source/sinkutilizing nanofluids

http://dx.doi.org/10.1016/j.amc.2014.03.1060096-3003/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author.E-mail addresses: sameh_sci_math@yahoo.com (S.E. Ahmed), ahmedkadhim7474@gmail.com (A.K. Hussein).

Sameh E. Ahmed a, Ahmed Kadhim Hussein b,⇑, H.A. Mohammed c, S. Sivasankaran d

a Department of Mathematics, Faculty of Sciences, South Valley University, Qena, Egyptb College of Engineering, Mechanical Engineering Department, Babylon University – Babylon City, Hilla, Iraqc Department of Thermofluids, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia (UTM), 81310 UTM Skudai, Johor Bahru, Malaysiad Institute of Mathematical Sciences, University of Malaya, Kuala Lumpur 50603, Malaysia

a r t i c l e i n f o a b s t r a c t

Keywords:NanofluidStretching tubeBoundary layerSuction/injectionHeat source/sinkPermeable surface

The present study aims to identify effects due to uncertainties of thermal conductivity anddynamic viscosity of nanofluid on boundary layer flow and heat transfer characteristics dueto permeable stretching tube in the presence of heat source/sink. Water-based nanofluidcontaining various volume fractions of different types of nanoparticles is used. The nano-particles used are Cu, Ag, CuO, and TiO2. Four models of thermal conductivity and dynamicviscosity depending on the shape of nanoparticles are considered. The results are presentedto give a parametric study showing influences of various dominant parameters such asReynolds number, the suction/injection parameter, solid volume fraction of nanoparticles,type of nanoparticles, the heat generation/absorption parameter and skin friction coeffi-cient. The results indicate that the skin friction coefficient decreases as the Reynolds num-ber and the suction/injection parameter (c) increase, while the local Nusselt numberincreases as the Reynolds number and the suction/injection parameter (c) increase. Theresults are compared with another published results and it found to be in excellentagreement.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

In recent years, considerable efforts have been directed towards the study of the boundary layer flow and heat transferover a stretching tube since it is considered as an important subject in numerous industrial engineering applications such ascooling of nuclear reactors during emergency shutdown conditions, fiber technology, extrusion processes, wire drawing,glass–fiber production, manufacture of plastic and rubber sheets and polymer extrusion, etc. [1]. From the other hand, nano-fluids have been widely used in industry, because of the growing use of these smart fluids. Many studies [2–6] explained thatnanofluids clearly exhibit enhanced thermal conductivity, which goes up with increasing volumetric fraction of nanoparti-cles. Nanofluid concept is utilized to describe a fluid in which nanometer-sized particles are suspended in conventional heattransfer basic fluids. Conventional heat transfer fluids, including oil, water, and ethylene glycol mixture are poor heat trans-fer fluids, since the thermal conductivity of these fluids play an important role on the heat transfer coefficient between the

Nomenclature

Symbol Descriptiona stretching tube radius (m)Cf skin friction coefficientc positive constantcp specific heat at constant pressure (j/kg.�C)f dimensionless stream functionk thermal conductivity (W/m.�C)Nu local Nusselt numberPr Prandtl numberQ heat generation/absorption parameterQo dimensional heat generation/absorption coefficientqw heat transfer from the stretching tube wall (W/m2)Re Reynolds numberr cylindrical coordinate in radial direction (m)T temperature (�C)u velocity component in radial direction (m/s)ww velocity of the stretching tube (m/s)z cylindrical coordinate in axial direction (m)

Greek symbolsa thermal diffusivity (m2/s)b coefficient of thermal expansion (K�1)h dimensionless temperatureg similarity variablesw shear stress from the stretching tube wall (N/m2)u volume fraction of nanofluidc suction/injection parameterl dynamic viscosity of the fluid (kg/m.s)m kinamatic viscosity (m2/s)q density (kg/m3)

Subscriptsf fluidnf nanofluid particlew conditions at tube walls solid1 conditions far away from tube surface

150 S.E. Ahmed et al. / Applied Mathematics and Computation 238 (2014) 149–162

heat transfer medium and the heat transfer surface. Therefore numerous methods have been taken to improve the thermalconductivity of these fluids by suspending nano/micro or larger-sized particle materials in liquids [7]. The nanofluid conceptwhich was firstly utilized by Choi [8]. Choi et al. [9] showed that the addition of a small amount (less than 1% by volume) ofnanoparticles to conventional heat transfer liquids increased the thermal conductivity of the fluid up to approximately twotimes. Wang [10] simulated the steady flow in a viscous and incompressible fluid outside of a stretching hollow cylinder inan ambient fluid at rest. The problem was governed by a third-order nonlinear ordinary differential equation that led to exactsimilarity solutions of the Navier–Stokes equations. Li and Xuan [11] investigated experimentally for 35 nm copper nanofluidflowing in a circular tube under constant heat flux condition. They observed an enhancement of heat transfer coefficient from1.05 to 1.14 times of volume concentration in the range of 0.5–1.2% under the same flow of velocity. Ishak et al. [12] pre-sented a numerical solution of flow and heat transfer outside a stretching permeable cylinder. The governing system of par-tial differential equations was converted to ordinary differential equations by using similarity transformations, which werethen solved numerically using the Keller-box method. The main purpose of their study was to investigate the effects of thegoverning parameters, namely the suction/injection parameter, Prandtl number, and Reynolds number on the velocity andtemperature profiles as well as the skin friction coefficient and the Nusselt number. The results were shown graphically. Thevalues of the skin friction coefficient and the Nusselt number were presented in tables. Sundar and Sharma [13] studiednumerically heat transfer and friction factor characteristics of Al2O3 nanofluid in a circular tube. At Reynolds number of10,000, Nusselt number for 0.1% Al2O3 nanofluids increased by 1.24 times over the base fluid. At Reynolds number of30,000, Nusselt number for 0.1% Al2O3 nanofluids increased by 1.22 times over the base fluid. While, at Reynolds numberof 30,000, friction factor for 4% Al2O3 nanofluid increased by 1.42 times over the base fluid. Moreover, heat transfer coeffi-cient of nanofluid increased with an increase in the volume concentration of nanofluid and Reynolds number. They observed

S.E. Ahmed et al. / Applied Mathematics and Computation 238 (2014) 149–162 151

that higher temperature operation of the nanofluid caused a higher percentage increase in the heat transfer rate. Also, theyconcluded that Prandtl number of nanofluid increased with decrease in operating temperature, because the viscosity playeda dominant role. Ho et al. [14] identified the effects due to uncertainties in effective dynamic viscosity and thermal conduc-tivity of nanofluid on laminar natural convection heat transfer in a square enclosure. Numerical simulations had been under-taken incorporating a homogeneous solid–liquid mixture formulation for the two-dimensional buoyancy-driven convectionin the enclosure filled with alumina–water nanofluid. Two different formulas from the literature were each considered forthe effective viscosity and thermal conductivity of the nanofluid. Simulations had been carried out for the pertinent param-eters in the following ranges: the Rayleigh number, Ra = 103–106 and the volumetric fraction of alumina nanoparticles,u = 0–4%. Significant difference in the effective dynamic viscosity enhancement of the nanofluid calculated from the twoadopted formulas, other than that in the thermal conductivity enhancement, was found to play as a major factor, by leadingto contradictory results concerning the heat transfer efficacy of using nanofluid in the enclosure. Ishak et al. [15] investigateda steady two-dimensional flow of an electrically conducting incompressible fluid due to a stretching cylindrical tube. Sim-ilarity solutions were obtained for a linearly stretching tube with a constant surface temperature. Effects of the magneticparameter, Prandtl number and Reynolds number on the flow and heat transfer characteristics had been examined. Theyconcluded that the transverse magnetic field could be used to suppress the velocity field, which in turn caused the enhance-ment of the temperature field. Also, they observed that the Nusselt number increased with Reynolds number and Prandtlnumber but decreased with magnetic parameter. Ishak and Nazar [16] investigated the axisymmetric laminar boundarylayer flow along a continuously stretching cylinder immersed in a viscous and incompressible fluid. The governing partialdifferential boundary layer equations in a cylindrical form were first transformed into ordinary differential equations beforebeing solved numerically by a finite-difference method. The problem under consideration reduced to the flat plate case whenthe curvature parameter was absent. They concluded that the surface shear stress and the heat transfer rate at the surfaceincreased as the curvature parameter increased. Khan and Pop [17] investigated numerically the problem of laminar fluidflow resulting from the stretching of a flat surface in a nanofluid. The model used for the nanofluid incorporated the effectsof Brownian motion and thermophoresis. A similarity solution was presented which depended on the Prandtl number (Pr),Lewis number (Le), Brownian motion number (Nb) and thermophoresis number (Nt). The variation of the reduced Nusseltand reduced Sherwood numbers with Nb and Nt for various values of Pr and Le was presented in tabular and graphical forms.Linear regression estimations of the reduced Nusselt and reduced Sherwood numbers were also obtained in terms of Brown-ian motion and thermophoresis parameters. It was found that the reduced Nusselt number was a decreasing function, whilethe reduced Sherwood number was an increasing function of each values of the parameters Pr, Le, Nb and Nt considered.Sundar and Sharma [18] studied experimentally turbulent convective heat transfer and friction factor behavior of Al2O3

nanofluid in a circular tube with different aspect ratios of longitudinal strip inserts in it. Experiments were conducted withwater and nanofluid in the range of Reynolds number (Re) varied as 3000 < Re < 22,000, particle volume concentration (u)varied as 0 < u < 0.5% and longitudinal strip aspect ratios (AR) varied as 0 < AR < 18. The agreement between the values ofNusselt number obtained with water was satisfactory when compared with another published results. Results indicated thatheat transfer coefficients increased with nanofluid volume concentration and decreased with aspect ratio. Hojjat et al. [19]investigated experimentally laminar convection heat transfer behavior of three different types of nanofluids flowing througha uniformly heated horizontal circular tube. Nanofluids were made by dispersion of Al2O3, CuO, and TiO2 nanoparticles in anaqueous solution of carboxymethyl cellulose (CMC). All nanofluids as well as the base fluid exhibited shear-thinning behav-ior. Results of heat transfer experiments indicated that both average and the local heat transfer coefficients of nanofluidswere larger than that of the base fluid. The enhancement of heat transfer coefficient increased by increasing nanoparticleloading. Also, they concluded that the thermal entry length of nanofluids was greater than the base fluid and became longeras nanoparticle concentration increased. Bachok and Ishak [20] studied the flow and heat transfer along a stretching cylinderwith prescribed surface heat flux. They observed that the governing parameters such as the curvature parameter and Prandtlnumber influenced the boundary layer flow and heat transfer characteristics on the surface of a horizontal cylinder. More-over, the investigation on the effects of curvature parameter on the skin friction coefficient and the local Nusselt numberproved that both of them increased as curvature parameter increased. Also, they concluded that the surface shear stressand the heat transfer rate at the surface increased as the curvature parameter increased. Very recently, Mansour et al.[21] performed a numerical solution of flow and heat transfer in micropolar fluid outside a stretching permeable cylinderwith thermal stratification and suction/injection effects. The governing system of partial differential equations was con-verted to ordinary differential equations by using similarity transformations, which were then solved using numerical tech-nique. The cases of strong concentration and weak concentration of micro-element were considered. The main purpose oftheir study was to investigate the effects of the governing parameters, namely the suction/injection parameter, thermalstratification parameter, Prandtl number, vortex viscosity parameter and Reynolds number on the velocity profiles, pressuredistributions, angular velocity profiles and temperature profiles as well as the skin friction coefficient, dimensionless wallcouple stress and the Nusselt number. The numerical results were validated by favorable comparisons with previously pub-lished results. The results were shown graphically. The values of the skin friction coefficient, dimensionless wall couplestress and the Nusselt number were presented in tables. Other useful researches have been conducted to simulate boundarylayer flow of nanofluid under different conditions and geometries [22–26]. A literature review refers that no studies havebeen done to simulate boundary layer flow and heat transfer due to permeable stretching tube in the presence of heatsource/sink utilizing nanofluids when the effect due to uncertainties of thermal conductivity and dynamic viscosity has beenconsidered. To reach this goal, the objective of the present work is to investigate the boundary layer flow and heat transfer

152 S.E. Ahmed et al. / Applied Mathematics and Computation 238 (2014) 149–162

characteristics due to stretching tube when the tube surface is permeable in the presence of heat source/sink utilizing nano-fluids. The effects due to uncertainties of thermal conductivity and dynamic viscosity have been also investigated.

2. Mathematical analysis

2.1. Governing equations and geometrical configuration

Fig. 1 shows a schematic diagram of the laminar, steady, incompressible flow of a nanofluid generated by the effect of apermeable stretching tube of radius (a) in the axial direction in a fluid at rest. It is assumed that the tube surface is main-tained at a constant temperature (Tw). The surrounding fluid temperature is (T1) where the tube surface temperature is con-sidered greater than the surrounding fluid temperature. The r-axis is measured along the radial direction while the z-axis ismeasured along the tube axis direction. The viscous dissipation effect is neglected during the analysis due to its slight effect.The fluid outside the stretching tube is water (H2O) based nanofluid containing four different types of solid spherical nano-particles: Cu, Ag, CuO, and TiO2. The nanofluid is assumed incompressible and the flow is assumed to be Newtonian, laminarand steady. It is assumed that the base fluid (i.e., water) and the nanoparticles are in thermal equilibrium state and no slipoccurs between liquid and nanofluid phases in terms of both velocity and temperature. Table 1 explains different models ofnanofluid based on different formulas for thermal conductivity and dynamic viscosity while Table 2 shows thermophysicalproperties of water and nanoparticles. The solid volume fractions (u) have been varied from 0% to 5% with an increment of1%. The suction/injection parameter (c) is varied as �0.5, 0 and 0.5. For, c = 0 the effect of the body curvature is neglected soan impermeable surface or a flat surface is obtained. The range of Reynolds number is taken as (0.5 6 Re 6 10) while the heatgeneration/absorption parameter (Q) is varied as �0.5, 0 and 0.5. The continuity, momentum and energy governing equa-tions for this problem can be expressed as follows:

@ðrwÞ@z

þ @ðruÞ@r¼ 0 ð1Þ

w@w@zþ u

@w@r¼ 1

qnflnf

@2w@z2 þ

1r@w@r

!" #ð2Þ

w@u@zþ u

@u@r¼ 1

qnf� @p@rþ lnf

@2u@r2 þ

1r@u@r� u

r2

!" #ð3Þ

w@T@zþ u

@T@r¼ 1ðqcpÞnf

knf@2T@r2 þ

1r@T@r

!þ Q oðT � T1Þ

" #ð4Þ

The dimensional boundary conditions of the present problem are given by:

1- At the surface of the stretching tube

at r ¼ a; u ¼ Uw; w ¼ ww; T ¼ Ts ð5Þ

2- Far away from the surface of the stretching tube

at r !1 w! 0; T ! T1 ð6Þ

Fig. 1. Physical model and coordinate system.

Table 1Models of nanofluid based on different formulas for thermal conductivity and dynamic viscosity.

Model Shape of nanoparticles Thermal conductivity Dynamic viscosity

1 Spherical knf

kf¼ ksþ2kf�2uðkf�ksÞ

ksþ2kfþuðkf�ksÞlnflf¼ 1ð1�uÞ2:5

2 Spherical knf

kf¼ ksþ2kf�2uðkf�ksÞ

ksþ2kfþuðkf�ksÞlnf

lf¼ 1þ 7:3uþ 123u2

3 Cylindrical (nanotubes) knf

kf¼ ksþ

kf2�

u2ðkf�ksÞ

ksþkf2þuðkf�ks Þ

lnf

lf¼ 1ð1�uÞ2:5

4 Cylindrical (nanotubes) knf

kf¼ ksþ

kf2�

u2ðkf�ksÞ

ksþkf2þuðkf�ks Þ

lnflf¼ 1þ 7:3uþ 123u2

Table 2Thermophysical properties of water and nanoparticles [2].

q (kgm�3) Cp (Jkg�1K�1) k (Wm�1K�1) b � 10�5(K�1)

H2O 997.1 4179 0.613 21Cu 8933 385 401 1.67CuO 6320 531.8 76.5 1.8Ag 10500 235 429 1.89TiO2 4250 686.2 8.9538 0.9

S.E. Ahmed et al. / Applied Mathematics and Computation 238 (2014) 149–162 153

The similar solution of these equations can be obtained by introducing the following non-dimensional transformations:

g ¼ ra

� �2; u ¼ � caffiffiffigp f ðgÞ; w ¼ 2zcf 0ðgÞ; h ¼ T � T1

T � Tsð7Þ

where prime denotes differentiation with respect to g. By inserting Eq. (7) into Eqs. (2) and (4), the following ordinary dif-ferential equations are obtained:

11�uþ uqs

qf

lnf

lfgf 000 þ f 00½ �

( )þ Re ff 00 � f 02

� �¼ 0 ð8Þ

1

1�uþu ðqcpÞsðqcpÞf

knf

kfgh00 þ h0½ � þ Qh

� �þ RePr fh0 ¼ 0 ð9Þ

These dimensionless governing equations have been obtained by employing the following non-dimensional variables aslisted below:

Q ¼ Qoa2

4kf; Re ¼ ca2

2mfand Pr ¼ mf

afð10Þ

By the same way, the dimensionless forms of the boundary conditions are listed below:

1- At the surface of the stretching tube

at g ¼ 1; f 0 ¼ 1; f ¼ c; h ¼ 1 ð11Þ

2- Far away from the surface of the stretching tube

at g!1 f 0 ! 0; h! 0 ð12Þ

The most important physical quantities in the present problem are the skin friction coefficient (Cf) and the local Nusseltnumber (Nu) which are defined as [12]:

Cf ¼2sw

qf w2w

and Nu ¼ aqw

kf ðTw � T1Þð13Þ

where sw, qw represent the shear stress and the heat transfer from the stretching tube wall, respectively. They are definedby [21]:

sw ¼ lnf@w@r

r¼a

and qw ¼ �knf@T@r

r¼a

ð14Þ

154 S.E. Ahmed et al. / Applied Mathematics and Computation 238 (2014) 149–162

By using the non-dimensional transformations given in Eq. (7), Eq. (13) becomes:

Table 3Compar

Re

0.5125

10

Table 4Compar

Pr

0.727

10

CfRe z

a

� �¼

lnf

lff 00ð1Þ and Nu ¼ �2

knf

kfh0ð1Þ ð15Þ

3. Numerical procedure and validation

Eqs. (8) and (9) are nonlinear equations and it is difficult to get a closed form solution for these systems of equations.Therefore, these equations subject to the boundary conditions (11) are solved numerically by fourth order Runge–Kuttamethod with shooting technique. In order to check the accuracy of the computational method, the skin friction coefficientin terms of (f0(1)) and the local Nusselt number of the fluid in terms of (�h0(1)) for pure water and Pr = 0.7 and pure water andRe = 10 respectively are compared with those reported earlier by Wang [10] and Ishak et al. [12]. As shown in Tables 3 and 4,the present results are found to be in excellent agreement with the results of Wang [10] and Ishak et al. [12].

4. Results and discussion

The boundary layer flow and heat transfer due to uncertainties of thermal conductivity and dynamic viscosity of water-based nanofluids in a permeable stretching tube with the presence of heat source/sink has been investigated numerically inthis study. In the present work, four models of thermal conductivity and dynamic viscosity depending on the shape of nano-particles which are named Model 1, Model 2, Model 3 and Model 4 are considered. The range of Reynolds number is taken as(0.5 6 Re 6 10), the range of suction / injection parameter (c) is varied as �0.5 (injection case), 0 (impermeable surface) and0.5 (suction case) respectively while the solid volume fraction range is taken as (0% 6 u 6 5%) with an increment of 1%. Therange of the heat generation/absorption parameter (Q) is varied as �0.5, 0 and 0.5. The results of the present study are shownin Figs. 2–13 and Tables 1–5 to give a clear description of the present problem. The fluid outside the stretching tube is awater-based nanofluid containing four various types of solid spherical nanoparticles: Cu, Ag, CuO and TiO2. Table 1 explainsdifferent models of nanofluid based on different formulas for thermal conductivity and dynamic viscosity to explore theeffects of uncertainties associated with various formulas for effective thermal conductivity and dynamic viscosity on the heattransfer characteristics in a permeable stretching tube. Table 2 shows thermophysical properties of water and nanoparticles.

4.1. The effects of solid volume fraction on skin friction coefficient

Fig. 2 displays the effects of solid volume fraction on skin friction coefficient for different models of thermal conductivityand dynamic viscosity related with Cu–water nanofluid at Re = 5, Q = 0.5 and c = 0.5. It is found from this figure that as thesolid volume fraction of nanoparticles (u) increases, the skin friction coefficient decreases for various models of thermal con-ductivity and dynamic viscosity. This is indicates that a low skin friction coefficient can be obtained for higher solid volumeconcentration of nanoparticles.

ison of f0(1) with Wang [10] and Ishak et al. [12] for pure water and Pr = 0.7.

Impermeable surface c = 0 Permeable surface c = � 0.5 Permeable surface c = 0.5

Wang [10] Ishak et al. [12] Present Ishak et al. [12] Present Ishak et al. [12] Present

�0.8822 �0.8827 �0.88234 �0.7719 �0.78011 �1.0084 �1.00798�1.1777 �1.1781 �1.17801 �0.9623 �0.97417 �1.4400 �1.45495�1.5939 �1.5941 �1.59444 �1.1810 �1.18487 �2.1468 �2.15195�2.4174 �2.4175 �2.41798 �1.4811 �1.48164 �3.9308 �3.93119�3.3444 �3.3444 �3.34511 �1.6776 �1.67822 �6.6222 �6.62227

ison of �h0(1) with Wang [10] and Ishak et al. [12] for pure water and Re = 10.

Impermeable surface c = 0 (flat plate case) Permeable surface c = �0.5 (injection case) Permeable surface c = 0.5 (suction case)

Wang [10] Ishak et al. [12] Present Ishak et al. [12] Present Ishak et al. [12] Present

1.568 1.5683 1.58679 0.2573 0.30719 4.1961 4.196843.035 3.0360 3.03553 0.0631 0.06363 11.1517 11.151676.160 6.1592 6.15776 0 7E�005 36.612 36.61154

10.77 7.4668 7.46419 0 0 51.7048 51.70387

0.00 0.01 0.02 0.03 0.04 0.05

-3.8-3.7-3.6-3.5-3.4-3.3-3.2-3.1-3.0-2.9-2.8-2.7-2.6-2.5-2.4-2.3

Cu-waterRe=5Q=0.5γ=0.5

Model 1 Model 2 Model 3 Model 4C

f(Rez

/a)

ϕ

Fig. 2. Effects of solid volume fraction on skin friction coefficient for different models for thermal conductivity and dynamic viscosity.

0.00 0.01 0.02 0.03 0.04 0.058.05

8.10

8.15

8.20

8.25

8.30

8.35

8.40

8.45

8.50

8.55

8.60

8.65

8.70

8.75

8.80

Cu-waterRe=5Q=0.5γ=0.5

Nu

ϕ

Model 1 Model 2 Model 3 Model 4

Fig. 3. Effects of solid volume fraction on local Nusselt number for different models of thermal conductivity and dynamic viscosity.

S.E. Ahmed et al. / Applied Mathematics and Computation 238 (2014) 149–162 155

4.2. The effect of solid volume fraction on local Nusselt number

Fig. 3 depicts the effects of solid volume fraction on local Nusselt number for different models of thermal conductivity anddynamic viscosity related with Cu–water nanofluid at Re = 5, Q = 0.5 and c = 0.5. It can be shown from this figure that as thesolid volume fraction of nanoparticles (u) increases, the local Nusselt number increases for different models of thermal con-ductivity and dynamic viscosity. This is refers that a high heat transfer rate can be obtained for higher solid volume concen-tration of nanoparticles. From the other side, the addition of nanofluid to the base fluid leads to increase the temperaturedistribution and this effect increases as the solid volume fraction range increases. Since as the solid volume fractionincreases, more heat is transferred into the system and thus the local Nusselt number increases. The figure shows also that

-1.0 -0.5 0.0 0.5 1.0

-4.0

-3.8

-3.6

-3.4

-3.2

-3.0

-2.8 Model 2Re=5ϕ=0.04γ=0

Cf(R

ez/a

)

Q

Cu-water Ag-water CuO-water TiO

2-water

Fig. 4. Effects of heat generation/absorption parameter on skin friction coefficient for different nanoparticles.

-1.0 -0.5 0.0 0.5 1.08.3

8.4

8.5

8.6

8.7

8.8

8.9

9.0

9.1Model 2Re=5ϕ=0.04γ=0

Cu-water Ag-water CuO-water TiO

2-water

Nu

Q

Fig. 5. Effects of heat generation/absorption parameter on local Nusselt number for different nanoparticles.

156 S.E. Ahmed et al. / Applied Mathematics and Computation 238 (2014) 149–162

the Model 2 gives a high value of the local Nusselt number. This behavior can be go back to the higher dynamic viscosity forthis model which causes to increase the Prandtl number and enhances the local Nusselt number.

4.3. The effect of heat generation/absorption parameter on skin friction coefficient

Fig. 4 displays effects of heat generation/absorption parameter (Q) on skin friction coefficient for different nanoparticles atRe = 5, u = 0.04, c = 0 (impermeable surface) related with Model 2 (i.e., spherical shape of nanoparticles). From this figure, itcan be noticed that all values of the skin friction coefficient have a negative sign. In this case, the negative sign means thatthe stretching tube achieves a drag force of the water–nanofluid mixture. The same observation can be noticed for all valuesof f0(1) which is given in Table 3 where the f0(1) values represents the skin friction coefficient. It can be shown in Fig. 4 amonotonic variation between the skin friction coefficient with the heat generation/absorption parameter (Q) for different

0 2 4 6 8 10-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

Model 2Cu-waterQ=0.5ϕ=0.04

Cf(R

ez/a

)

Re

γ=−0.5γ=0γ=0.5

Fig. 6. Effects of suction/injection parameter on skin friction coefficient for different values of Reynolds number.

0 2 4 6 8 10

0

10

20

30

40

50

60

70

Model 2Cu-waterQ=0.5ϕ=0.04

γ=−0.5γ=0γ=0.5N

u

Re

Fig. 7. Effects of suction/injection parameter on local Nusselt number for different values of Reynolds number.

S.E. Ahmed et al. / Applied Mathematics and Computation 238 (2014) 149–162 157

nanoparticles. Also, it can be observed that TiO2–water nanofluid has a high value of skin friction while the Ag–water nano-fluid has a lower value of it. The reason of this behavior is due to the lower value of thermal conductivity of TiO2 (see Table 2)which causes to increase the heat generation/absorption parameter and leads to increase the skin friction coefficient. Fromthe other hand, the high value of thermal conductivity of Ag (see Table 2) causes to decrease the heat generation/absorptionparameter and leads to decrease the skin friction coefficient.

4.4. The effect of heat generation/absorption parameter on local Nusselt number

Fig. 5 explains effects of heat generation/absorption parameter (Q) on local Nusselt number for different nanoparticles atRe = 5, u = 0.04, c = 0 (impermeable surface) related with Model 2 (i.e., spherical shape of nanoparticles). It can be observedthat, CuO–water nanofluid gives higher value of local Nusselt number while the Ag–water nanofluid gives a lower value of it.Therefore, it can be deduced that CuO–water nanofluid performs better heat transfer performance for the present problem.

1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

Cu-water (ϕ=0.04)

Pure water

Model 2Q=0.5γ=0

Re=2,4,6,8,10

f'

η

Fig. 8. Effects of Reynolds number on velocity profiles.

3210.0

0.2

0.4

0.6

0.8

1.0

Model 2Q=0.5γ=0

Cu-water (ϕ=0.04)Pure water

Re=2,4,6,8,10

θ

η

Fig. 9. Effects of Reynolds number on temperature distributions.

158 S.E. Ahmed et al. / Applied Mathematics and Computation 238 (2014) 149–162

This is due to the fact that for impermeable surface (c = 0), the rate of heat transfer increases with low thermal conductivityor high viscosity. The reduction in the thermal conductivity and the increasing in the viscosity cause to reduce the thicknessof thermal boundary layer and as a result enhance the rate of the heat transfer or the local Nusselt number. Also, it is inter-esting to see that although CuO nanoparticles has a low thermal conductivity (see Table 2), however, it still shows a com-parable enhancement on heat transfer compared to the high thermal conductivity of usual metals such as Cu and Ag. This isdue to the low thermal diffusivity of CuO nanoparticles which leads to increase the temperature gradients and as a resultincreases the local Nusselt number. While, the high thermal diffusivity of Cu and Ag plays a reverse role by reducing the tem-perature gradients and as a result decreases the local Nusselt number.

4.5. The effects of suction/injection parameter on skin friction coefficient

Fig. 6 depicts effects of suction/injection parameter (c) on skin friction coefficient for different values of Reynolds numberat Q = 0.5, u = 0.04 related with Model 2 (i.e., spherical shape of nanoparticles) and Cu–water nanofluid. The suction/injec-

1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

Model 2Re=5Q=0.5ϕ=0.04γ=0

f'

η

Ag-water Cuo-water TiO

2-water

Cu-water Pure water

Fig. 10. Velocity profiles for different nanoparticles.

1 20.0

0.2

0.4

0.6

0.8

1.0

Model 2Re=5Q=0.5ϕ=0.04γ=0

θ

η

Ag-water CuO-water TiO2-water Cu-water Pure water

Fig. 11. Temperature distributions for different nanoparticles.

S.E. Ahmed et al. / Applied Mathematics and Computation 238 (2014) 149–162 159

tion parameter (c) is varied as �0.5 (injection case), 0 (impermeable surface) and 0.5 (suction case) respectively. It is worthmentioning that the Reynolds number represents the relative significance of the inertia effect compared to the viscous effect.This figure shows that the skin friction coefficient decreases as the Reynolds number increases and this result is consistentwith those presented in Table 3. This is because that as the Reynolds number increases, the velocity gradient decreases andleading to decrease the skin friction coefficient. Also, it is obvious that the skin friction coefficient decreases as the suction/injection parameter (c) increases. Again this result is consistent with those presented in Table 3. Furthermore, Fig. 6 showsthat the suction decreases the skin friction, whereas injection increases it. From this observation, the suction process can beused to reduce the friction effect at the surface. In Table 4, it can be shown for impermeable surface (c = 0), that as the Prandtlnumber increases, the local Nusselt number in terms of (�h0(1)) increases. This is due to increase the effect of viscosity anddecrease the effect of thermal conductivity when the Prandtl number increases. Moreover, the reduction in the thermal con-ductivity leads to decrease the thermal boundary layer thickness and as result increases the local Nusselt number. In Table 5,

20.0

0.2

0.4

0.6

0.8

1.0

Model IIRe=5γ=0

Cu-water (ϕ=0.04)Pure water

Q=-5, 0, 5

θ

η

Fig. 12. Effects of heat generation/absorption parameter on temperature distributions.

1 2 3 4 5 6 70.0

0.2

0.4

0.6

0.8

1.0

Model 2Re=5Q=0.5

Pure waterCu-water (ϕ=0.04)

γ=−0.5,0,0.5

f'

η

Fig. 13. Effects of suction/injection parameter on velocity profiles.

160 S.E. Ahmed et al. / Applied Mathematics and Computation 238 (2014) 149–162

it can be shown the percentages for increasing of local Nusselt number for different models comparing to pure water at dif-ferent values of solid volume fraction at Cu–water, Re = 5, Q = 0.5 and c = 0. It is observed that as solid volume fractionincreases, the local Nusselt number increases. It is clearly seen from this table that addition small amounts of nanoparticlesimproves significantly the heat transfer performance. This result is consistent with those presented in Fig. 3. Furthermore, Itis found from this table that increasing the solid volume fraction up to 5%, can induce an increasing in the local Nusselt num-ber of 7.58% and 4.73% for Models 2 and 1, respectively.

4.6. The effect of suction/injection parameter on local Nusselt number

Fig. 7 displays effects of suction/injection parameter (c) on local Nusselt number for different values of Reynolds numberat Q = 0.5, u = 0.04 related with Model 2 (i.e., spherical shape of nanoparticles) and Cu–water nanofluid. It is found that thelocal Nusselt number increases as the Reynolds number increases. This is due to fact that the thermal boundary layer thick-

Table 5The percentages for increasing of local Nusselt number for different models comparing to pure water at different values of solid volume fraction at Cu–water,Re = 5, Q = 0.5 and c = 0.

u Model 1 (%) Model 2 (%) Model 3 (%) Model 4 (%)

0.01 0.93 1.30 0.25 0.620.02 1.86 2.73 0.53 1.380.03 2.81 4.28 0.84 2.240.04 3.76 5.90 1.16 3.180.05 4.73 7.58 1.51 4.17

S.E. Ahmed et al. / Applied Mathematics and Computation 238 (2014) 149–162 161

ness decreases as the Reynolds number increases. This reduction leads to increase the temperature gradient and leads toincrease the local Nusselt number. Also, it can be observed that the local Nusselt number increases as the suction/injectionparameter (c) increases. This is due to the thermal boundary layer thickness decreasing when the suction/injection param-eter (c) increases. Furthermore, this increasing causes to increase the temperature gradient at the surface of the stretchingtube and increases the local Nusselt number. In addition, it can be observed that the local Nusselt number is larger for suc-tion (c = 0.5) compared with injection (c = �0.5). Therefore, injection is a sufficient condition to reduce the local Nusseltnumber.

4.7. The effect of Reynolds number on velocity profiles

Fig. 8 displays effects of Reynolds number on velocity profiles for different values of Reynolds number at Q = 0.5, imper-meable surface (c = 0) related with Model 2 (i.e., spherical shape of nanoparticles) for pure water and Cu–water nanofluidwith u = 0.04. The figure shows that as the value of the Reynolds number increases the velocity profiles decrease. Therefore,the high values of the Reynolds number can be used to damping the fluid movement. Moreover, the figure shows that thevelocity profiles for pure water and Cu–water nanofluid begin from a high value and then decreases gradually far away fromthe surface of the stretching tube (i.e., at g = 1).

4.8. The effect of Reynolds number on temperature distributions

The effects of Reynolds number on temperature distributions are depicted in Fig. 9 for various values of Reynolds numberat Q = 0.5, impermeable surface (c = 0) related with Model 2 (i.e., spherical shape of nanoparticles) for pure water and Cu–water nanofluid with u = 0.04. The figure illustrates that the temperature distributions decrease as the value of the Reynoldsnumber increases. Therefore, the high values of the Reynolds number can be used to reduce the fluid temperature distribu-tions. Again, the figure explains that the temperature distributions for pure water and Cu–water nanofluid starts from a highvalue and then decreases gradually at some large distance from the surface of the stretching tube (i.e., at g = 1). This obser-vation is in agreement with the problem boundary conditions given in Eqs. (11) and (12). The velocity profiles and temper-ature distributions for different nanoparticles are exhibited in Figs. 10 and 11, respectively at Q = 0.5, impermeable surface(c = 0) related with Model 2 (i.e., spherical shape of nanoparticles) for Re = 5 and u = 0.04. It can be noticed that the velocityprofiles and temperature distributions for different nanoparticles decrease gradually far away from the surface of thestretching tube. Moreover, a slight increasing in the velocity profiles and temperature distributions can be detected by add-ing different nanoparticles to the base fluid (i.e., water). Therefore, both Figs. 10 and 11 exhibit that the addition of differenttypes of nanoparticles in water improves the velocity profiles and temperature distributions. Moreover, it can be observedthat the velocity profiles and the temperature distributions are not strongly affected by addition various nanoparticle withlow solid volume fraction concentrations (u = 0.04). Also, it can be noticed in Fig. 10 that the velocity profiles of TiO2–waternanofluid are the higher one and normally greater than the pure water.

4.9. The effect of heat generation/absorption parameter on temperature distributions

Fig. 12 displays effects of heat generation/absorption parameter (Q) on temperature distributions for Re = 5, impermeablesurface (c = 0) related with Model 2 (i.e., spherical shape of nanoparticles) for pure water and Cu–water nanofluid withu = 0.04. The results of this figure show that as the heat generation/absorption parameter increases, the temperature distri-butions increase.

4.10. The effect of suction/injection parameter on velocity profiles

Fig. 13 explains effects of suction/injection parameter (c) on velocity profiles for Re = 5, Q = 0.5 related with Model 2 (i.e.,spherical shape of nanoparticles) for pure water and Cu–water nanofluid with u = 0.04. The results of this figure show that asthe suction/injection parameter increases, the velocity profiles decrease.

162 S.E. Ahmed et al. / Applied Mathematics and Computation 238 (2014) 149–162

5. Conclusions

The present study aims to identify effects due to uncertainties of thermal conductivity and dynamic viscosity of nanofluidon boundary layer flow and heat transfer characteristics due to permeable stretching tube in the presence of heat source/sink. It is found that the local Nusselt number increases and skin friction coefficient decreases with the increase in the solidvolume fraction of nanoparticles. The results show that the Ag–water nanofluid has a lower value of the skin friction coef-ficient, while the TiO2–water nanofluid has a high value of the skin friction coefficient. The skin friction coefficient decreaseswhile the local Nusselt number increases as the Reynolds number and the suction/injection parameter (c) increase. More-over, injection is a sufficient condition to reduce the local Nusselt number, while suction is a sufficient condition to reducethe skin friction. Finally, the uncertainties associated with different formulas adopted for the thermal conductivity anddynamic viscosity of the nanofluid have a strong effect on the boundary layer flow and heat transfer characteristics dueto permeable stretching tube.

References

[1] S. Kakaç, A. Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids, Int. J. Heat Mass Transfer 52 (2009) 3187–3196.[2] M. Mansour, R. Mohamed, M. Abd-Elaziz, S. Ahmed, Numerical simulation of mixed convection flows in a square lid-driven cavity partially heated from

below using nanofluid, Int. Commun. Heat Mass Transfer 37 (2010) 1504–1512.[3] Y. Xuan, O. Li, Heat transfer enhancement of nanofluids, Int. J. Heat Fluid Flow 21 (2000) 58–64.[4] D. Wen, Y. Ding, Experimental investigation into convective heat transfer of nanofluids at the entrance region under laminar flow conditions, Int. J.

Heat Mass Transfer 47 (2004) 5181–5188.[5] P. Bhattacharya, S. Saha, A. Yadav, P. Phelan, R. Prasher, Brownian dynamics simulation to determine the effect thermal conductivity of nanofluids, J.

Appl. Phys. 95 (2004) 6492–6494.[6] A. Mokmeli, M. Saffar-Avval, Prediction of nanofluid convective heat transfer using the dispersion model, Int. J. Therm. Sci. 49 (2010) 471–478.[7] R. Lotfi, Y. Saboohi, A. Rashidi, Numerical study of forced convective heat transfer of nanofluids: comparison of different approaches, Int. Commun. Heat

Mass Transfer 37 (2010) 74–78.[8] S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, in: The Proceedings of the 1995 ASME International Mechanical Engineering

Congress and Exposition, ASME, San Francisco, USA, 1995, pp. 99–105. FED 231/MD 66.[9] S. Choi, Z. Zhang, W. Yu, F. Lockwood, E. Grulke, Anomalously thermal conductivity enhancement in nanotube suspensions, Appl. Phys. Lett. 79 (2001)

2252–2254.[10] C. Wang, Fluid flow due to a stretching cylinder, Phys. Fluids 31 (1988) 466–468.[11] Q. Li, Y. Xuan, Experimental investigation of transport properties of nanofluids, in: Buxuan Wang (Ed.), Heat Transfer Science & Technology, Higher

Education Press, 2000, pp. 757–784.[12] A. Ishak, R. Nazar, I. Pop, Uniform suction/blowing effect on flow and heat transfer due to a stretching cylinder, Appl. Math. Model. 32 (2008) 2059–

2066.[13] L. Sundar, K. Sharma, Numerical analysis of heat transfer and friction factor in a circular tube with Al2O3 nanofluid, Int. J. Dyn. Fluids 4 (2) (2008) 121–

129.[14] C. Ho, M. Chen, Z. Li, Numerical simulation of natural convection of nanofluid in a square enclosure: effects due to uncertainties of viscosity and

thermal conductivity, Int. J. Heat Mass Transfer 51 (2008) 4506–4516.[15] A. Ishak, R. Nazar, I. Pop, Magnetohydrodynamic (MHD) flow and heat transfer due to a stretching cylinder, Energy Convers. Manage. 49 (2008) 3265–

3269.[16] A. Ishak, R. Nazar, Laminar boundary layer flow along a stretching cylinder, Eur. J. Sci. Res. 36 (1) (2009) 22–29.[17] W. Khan, I. Pop, Boundary-layer flow of a nanofluid past a stretching sheet, Int. J. Heat Mass Transfer 53 (2010) 2477–2483.[18] L. Sundar, K. Sharma, Heat transfer enhancements of low volume concentration Al2O3 nanofluid and with longitudinal strip inserts in a circular tube,

Int. J. Heat Mass Transfer 53 (2010) 4280–4286.[19] M. Hojjat, S. Etemad, R. Bagheri, Laminar heat transfer of non-Newtonian nanofluids in a circular tube, Korean J. Chem. Eng. 27 (5) (2010) 1391–1396.[20] N. Bachok, A. Ishak, Flow and heat transfer over a stretching cylinder with prescribed surface heat flux, Malaysian J. Math. Sci. 4 (2) (2010) 159–169.[21] M. Mansour, R. Mohamed, M. Abd-Elaziz, S. Ahmed, Thermal stratification and suctioninjection effects on flow and heat transfer of micropolar fluid

due to stretching cylinder, Int. J. Numer. Methods Biomed. Eng. 8 (2011) 1951–1963.[22] N. Yacob, A. Ishak, I. Pop, K. Vajravelu, Boundary layer flow past a stretching/shrinking surface beneath an external uniform shear flow with a

convective surface boundary condition in a nanofluid, Nanoscale Res. Lett. 6 (2011) 314–320.[23] E. Abu-Nada, K. Ziyad, M. Saleh, Y. Ali, Heat Transfer enhancement in combined convection around a horizontal cylinder using nanofluids, J. Heat

Transfer – Trans. ASME 130 (2008) 1–4.[24] S. Anjali Devi, J. Andrews, Laminar boundary layer flow of nanofluid over a flat plate, Int. J. Appl. Math. Mech. 7 (6) (2011) 52–71.[25] R. Gorla, S. EL-Kabeir, A. Rashad, Heat transfer in the boundary layer on a stretching circular cylinder in a nanofluid, J. Thermophys. Heat Transfer

(2011) 183–186.[26] S. EL-Kabeir, R. Gorla, Variable viscosity effects on boundary layer heat transfer to a stretching sheet including viscous dissipation and internal heat

generation, Int. J. Fluid Mech. Res. 34 (1) (2007) 42–51.