MHD flow of a variable viscosity nanofluid over a radially stretching convective surface with radiative heat

Journal of Molecular Liquids 219 (2016) 624–630

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MHD flow of a variable viscosity nanofluid over a radially stretching convective surface with radiative heat O.D. Makinde a,⁎, F. Mabood b, W.A. Khan c, M.S. Tshehla a a b c

Faculty of Military Science, Stellenbosch University Private Bag X2, Saldanha 7395, South Africa Department of Mathematics, University of Peshawar, Pakistan Department of Mechanical and Industrial Engineering, College of Engineering, Majmaah University, Majmaah 11952, Saudi Arabia

a r t i c l e

i n f o

Article history: Received 25 September 2015 Received in revised form 27 December 2015 Accepted 6 March 2016 Available online xxxx Keywords: MHD nanofluid Radially stretching sheet Thermal radiation Convective heating Brownian motion Thermophoresis

a b s t r a c t This study investigates the combined effects of thermal radiation, thermophoresis, Brownian motion, magnetic field and variable viscosity on boundary layer flow, heat and mass transfer of an electrically conducting nanofluid over a radially stretching convectively heated surface. The stretching velocity is assumed to vary linearly with the radial distance. Using similarity transformation, the governing nonlinear partial differential equations are reduced to a set of nonlinear ordinary differential equations which are solved numerically by employing shooting method coupled with Runge-Kutta Fehlberg integration technique. Graphical results showing the effects of various pertinent parameters on the dimensionless velocity, temperature, nanoparticle concentration, local skin friction, local Nusselt and local Sherwood numbers are presented and discussed quantitatively. Comparisons with the earlier results have been made and good agreements are found. The present results reveal that the heat transfer rate is reduced with viscosity and nanofluid parameters whereas the mass transfer rates is enhanced with Brownian motion parameter and Lewis number. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Many engineering and industrial processes involve heat transfer by means of a flowing fluid in either laminar or turbulent regimes. A decrease in thermal resistance of heat transfer in the fluids would significantly benefit many of these applications/processes. It is well known that conventional heat transfer fluids, such as oil, water and ethylene glycol mixture are poor heat transfer fluids due to lower thermal conductivity. Nanofluids have the potential to reduce thermal resistances in many applications such as electronics, medical, food and manufacturing. Nanofluids are dilute suspensions of functionalized nanoparticles with higher thermal conductivity and smaller than 100 nm in diameter. They have been utilized in diverse technologies including turbulent flows [1], bio-fluids and polymer solutions [2,3], drug delivery and food biophysics [4], propellant combustion [5], crystal growth [6] and automotive engine cooling [7]. Makinde and Aziz [8] reported the similarity solutions for the thermal boundary layer of a nanofluid past a stretching sheet with a convective boundary condition, Aziz et al. [9] considered free convection boundary layer flow past a horizontal flat plate embedded in porous medium filled by nanofluid containing gyrotactic microorganisms. Motsumi and Makinde [10] examined the effects of thermal radiation and viscous dissipation on nanofluids flow over a permeable moving flat plate. Alsaedi et al. [11] provided an ⁎ Corresponding author. E-mail address: [email protected] (O.D. Makinde).

http://dx.doi.org/10.1016/j.molliq.2016.03.078 0167-7322/© 2016 Elsevier B.V. All rights reserved.

analysis to discuss the stagnation point flow of nanofluid near a permeable stretched surface with thermal convective condition. Butt and Ali [12] performed entropy analysis of magnetohydrodynamic flow and heat transfer over a convectively heated radially stretching surface. The sub-branch of nanofluids termed magnetic nanofluids has also shown significant promise in numerous engineering fields. These fluids respond to applied magnetic fields and allow further manipulation of heat transfer and hydrodynamic characteristics. Significant experimental analyses of magnetic nanofluids have been conducted by Parekh and Lee [13]. The tribological performance of magnetic nanofluids has been recently elucidated by Andablo-Reyes et al. [14]. Ellahi [15] studied magnetohydrodynamic (MHD) flow of non-Newtonian nanofluid in a pipe and observed that the MHD parameter decreases the fluid motion and the velocity profile is larger than that of the temperature profile even in the presence of variable viscosities, Rashidi et al. [16] considered the analysis of the second law of thermodynamics applied to an electrically conducting incompressible nanofluid fluid flowing over a porous rotating disc. Sheikholeslami et al. [17] studied the magnetic field effects on CuO–water nanofluid flow and heat transfer in an enclosure which is heated from below. Effects of nanofluid on heat transfer enhancement have been considered by several researchers including [18–25] among others. In most of the studies, the viscosity of the fluid was assumed to be constant. When the effects of variable viscosity is taken into account, the flow characteristics are significantly changed compared to the constant property case. Hence, main goal of the present work is to conduct a

O.D. Makinde et al. / Journal of Molecular Liquids 219 (2016) 624–630

numerical investigation of the effects of varying viscosity on the MHD flow of nanofluid past a radially stretching sheet in the presence of radiative heat. The numerical investigation is carried out for different governing parameters. Consequently, this paper completes a theoretical analysis of convective heat transfer of variable viscosity nanofluid over a stretching surface in the presence of magnetic field and thermal radiation. The results with respect to dimensionless velocity, temperature, concentration, friction and heat and mass transfer rates can be used as a reference for any future study on micro/nanoscale thermal-fluid transport phenomena.

2. Model formulation Consider the steady flow of a variable viscosity electrically conducting optically thick nanofluid over an infinite radially stretching and convectively heated disc surface. The radially stretching surface is placed at z = 0 and the nanofluid is confined above the stretching surface as shown in the Fig. 1. The ambient fluid temperature is taken as T∞ while the lower side of the disc surface is convectively heated by a hot fluid at temperature Tf with heat transfer coefficient hf. A uniform magnetic field of strength B0 is applied normal to the disc surface and the induced magnetic field due to the motion of a conducting nanofluid is neglected. The heat transfer analysis is carried out in the presence of Brownian motion, thermophoresis and thermal radiation. Following the Buongiorno [26] model and considering the above assumptions, the governing equations for momentum, energy balance and nanoparticle concentration are given as; ∂u u ∂w þ þ ¼ 0; ∂r r ∂z

ð1Þ



 ∂u ∂u 1 ∂ ∂u σB2 u ¼ − 0 ; μf u þw ρ f ∂z ρf ∂r ∂z ∂z

u

! 2 ! ∂T ∂T 16σ  T 3∞ ∂ T þw ¼ αf 1 þ  ∂r ∂z ∂z2 3kk (
"
2 #) ∂C ∂T DT ∂T þ τ DB ; þ T∞ ∂z ∂z ∂z

2

∂C ∂C ∂ C u þw ¼ DB ∂r ∂z ∂z2

!


 2 ! DT ∂ T þ ; T∞ ∂z2

625

with the boundary conditions u ¼ uw ðrÞ ¼ ar; u→0;

C→C ∞ ;

w ¼ 0; T→T ∞ ;

  ∂T ¼ h f T f −T ; C ¼ C w ∂z as z →∞: −k

at z ¼ 0; ð5Þ

where u and w are the velocity components in r and z directions respectively, T is the temperature, C is the nanoparticles concentration, Cw is the surface concentration, C∞ is the free stream concentration, ρf is the fluid density, cpf is the specific heat at constant pressure, DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient, αf is the thermal diffusion coefficient, k is the thermal conductivity coefficient, σ is the electrical conductivity, σ* is the Stefan–Boltzman constant, k* is the mean absorption coefficient, uw(r) (=ar) is the prescribed stretching surface radial velocity such that a is positive constant with dimension of inverse time. The nanofluid dynamic viscosity is assumed to be an exponential decreasing function of temperature given by μ f ðT Þ ¼ μ ∞ e−βðT−T ∞ Þ ;

ð6Þ

where μ∞ is the free stream dynamic viscosity at temperature T∞. We introduce the following parameters and similarity transformation; rffiffiffi pffiffiffiffiffiffiffiffiffi a μ T−T ∞ uw r 0 ; υ ¼ ∞ ; u ¼ ar f ðηÞ; w ¼ −2 ðaυÞ f ðηÞ; θðηÞ ¼ ; ; Rer ¼ υ υ ρf T f −T ∞ 2   C−C ∞ σ B0 τDB ðC w −C ∞ Þ ϕðηÞ ¼ ;M ¼ ; ; δ ¼ β T f −T ∞ ; Nb ¼ υ C w −C ∞ ρf a   rffiffiffi 3  τDT T f −T ∞ αf 16σ T ∞ υ h υ ; Nr ¼ ; Nc ¼ : Nt ¼ ; Le ¼ ; Pr ¼ T∞υ αf k a DB 3kk

η¼z

ð7Þ Substituting Eq. (7) into Eqs. (1)–(6), we obtain:

ð2Þ

!
2 3 2 2 d f dθ d f d f df df δθ −δ þe 2f 2 − −M ¼ 0; dη dη2 dη dη dη3 dη

ð8Þ


2 2 ð1 þ Nr Þ d θ dθ dθ dϕ dθ þ 2f ¼ 0; þ Nb þ Nt 2 Pr dη dη dη dη dη ð3Þ

2

ð9Þ

2

d ϕ dϕ Nt d θ þ 2PrLef ¼ 0; þ dη Nb dη2 dη2

ð10Þ

with ð4Þ f ð0Þ ¼ 0; df ð∞Þ ¼ 0; dη

df ¼ 1; dη θð∞Þ ¼ 0;

dθ ð0Þ ¼ −Ncð1−θð0ÞÞ; dη

ϕð0Þ ¼

1;

ϕð∞Þ ¼ 0:

ð11Þ

where Pr is the Prandtl number, Le is the Lewis number, Nc is the convective heat transfer parameter, Nb is the Brownian motion parameter, Nt is the thermophoresis parameter, δ is the viscosity variation parameter, M is the magnetic field parameter and Nr is the radiation parameter. It is important to note that when Nc → ∞, the convective

Table 1 Numerical Values of f″(0) when δ=0.

Fig. 1. Physical model and coordinate system.

M

HAM [12]

Present results

0 0.5 1 2 3

−1.17372 −1.36581 −1.53571 −1.83049 −2.08484

−1.17372 −1.36581 −1.53571 −1.83049 −2.08484

626

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Table 2 Variation of skin friction with different parameters at Pr=6.2,Nb=Nt=0.1,Le=1. δ

M=0

M=1

Nc = 0.5

0 0.5 1 1.5 2

Nc = 1

Nc = 0.5

Nr= 1

Nr = 0.5

Nr = 1

Nr = 0.5

Nr= 1

Nr = 0.5

Nr = 1

−1.1737 −1.1296 −1.0846 −1.0388 −0.9923

−1.1737 −1.1229 −1.0712 −1.0187 −0.9656

−1.1737 −1.1011 −1.0268 −0.9512 −0.8747

−1.1737 −1.0920 −1.0089 −0.9249 −0.8406

−1.5357 −1.4725 −1.4080 −1.3421 −1.2750

−1.5357 −1.4629 −1.3886 −1.3129 −1.2359

−1.5357 −1.4325 −1.3268 −1.2188 −1.1092

−1.5357 −1.4197 −1.3014 −1.1813 −1.0602

boundary condition reduces to a uniform surface temperature boundary condition and when Nc → 0, the surface shows the adiabatic condition. Other quantities of interest are the skin friction coefficient (Cf), Nusselt number (Nu) and the Sherwood number (Sh) which are given as Cf ¼

pffiffiffi pffiffiffi τw q υ qw υ pffiffiffi ; ; Nu ¼  m pffiffiffi ; Sh ¼ 2 ρ f uw k T f −T ∞ a DB ðC w −C ∞ Þ a

ð12Þ

where τw ¼ μ f

∂u ; ∂z

qm ¼ −k 1 þ

16σ  T 3∞ 3kk

!

∂T ; ∂z

qw ¼ −DB

∂C ; ∂z

ð13Þ

are the stretching disc surface shear stress, heat flux and mass flux respectively. Substituting Eq. (13) into (12), we obtain 

Re1=2 r Cf

¼

2 d f  e−δθ 2  dη 

¼−

 dφ ; dη η¼0

η¼0

 dθ  ; Re−1=2 Nu ¼ −ð1 þ NrÞ  ; r dη η¼0

Re−1=2 Sh r ð14Þ

Following Kuznestov and Nield [28], the reduced skin friction, reduced Nusselt and Sherwood numbers can be defined as C f r ¼ Re1=2 r Cf ;

Nur ¼ Re−1=2 Nu; r

Shr ¼ Re−1=2 Sh: r

ð15Þ

3. Numerical method In this study, an efficient numerical scheme Runge-Kutta Fehlberg method has been employed to investigate the flow model defined by Eqs. (8)–(10) with the boundary conditions (11) for different values of controlling parameters. The effects of the thermophysical parameters on the dimensionless velocity, temperature, concentration, skin friction coefficient and the rate of heat and mass transfer are investigated. The step size and convergence criteria are chosen to be 0.001 and 10−6 respectively. For the special case, in the absence of variable viscosity, our results agreed perfectly with those reported by Butt and Ali [12] as shown in Table 1 and this serves as a benchmark for the accuracy of our numerical procedure. Table 3 Variation of Nusselt numbers with different parameters at Pr=6.2, Le=1, Nb=Nt=0.1. δ

M=0 Nc = 1

4. Results and discussion The combined effects of viscosity variation, radiation, convective heating, and magnetic parameters on skin friction, and heat and mass transfer from a stretching sheet are investigated numerically and presented in Tables 2–4. Individual effects of these parameters on the dimensionless velocity, temperature, rescaled nanoparticle volume fraction, skin friction and heat and mass transfer rates are illustrated in Figs. 2–7. The effects magnetic and viscosity variation parameters on the dimensionless velocity f′(η)are presented in Fig. 2(a) for a water-based nanofluid. It is important to note that an increase in viscosity parameter represents a decrease in the nanofluid viscosity. Moreover, in the absence of magnetic field i.e. in a pure hydrodynamic case, the dimensionless velocity is higher within the hydrodynamic boundary layer. An increase in the magnetic field to higher values, i.e. in case of hydromagnetic flow, reduces the dimensionless velocity within the boundary layer due to a resisting force known as Lorentz force. This force decelerates the boundary layer flow and thickens the momentum boundary layer. From the same Fig. 2(a), we observed also that as δ increases the hydrodynamic boundary layer thickness reduces and the velocity become shallow. Physically, this is due to higher temperature difference between the surface and the ambient fluid. Fig. 2(b) shows the effects of radiation and convective parameters on the dimensionless velocity. It is observed that both dimensionless parameters tend to reduce the dimensionless velocity and hence reduce the velocity boundary layer thickness. Fig. 3(a), (b) and (c) illustrate the variation of the dimensionless temperature with the thermophysical parameters. As the value of these parameters increases, the dimensionless temperature increases, consequently, the thermal boundary thickness increases as well. This fact can be explained as follows: the Lorentz force is a resistive force which opposes the fluid motion, so heat is produced and as a result, the thermal boundary layer becomes thicker for stronger magnetic field. Meanwhile, it is also observed that the viscosity variation, thermal radiation, convective and nanofluid parameters lead to a rise in the dimensionless temperature. This is because an increase in Nr has a tendency to enhance the conduction effects and to increase temperature at each point away from the surface. Therefore higher value of radiation parameter implies higher surface heat flux. The convective parameter arises only in the wall temperature gradient boundary condition (11), [θ′(0) = − Nc(1 − θ(0))]. It is interesting to note that if Nc → ∞, the Table 4 Variation of Sherwood numbers with different parameters at Pr = 6.2, Le = 1, Nb=Nt=0.1. δ

M=1

Nc = 0.5

0 0.5 1 1.5 2

Nc = 1

Nr = 0.5

Nc = 0.5

M=0

M=1

Nc = 0.5

Nc = 1

Nr = 0.5

Nr= 1

Nr= 0.5

Nr = 1

Nr = 0.5

Nr = 1

Nr = 0.5

Nr= 1

0.5653 0.5642 0.5629 0.5615 0.5599

0.7322 0.7301 0.7278 0.7253 0.7225

0.8989 0.8938 0.8879 0.8813 0.8738

1.1457 1.1372 1.1275 1.1163 1.1035

0.5596 0.5579 0.5560 0.5540 0.5517

0.7226 0.7196 0.7162 0.7124 0.7081

0.8842 0.8769 0.8685 0.8588 0.8475

1.1221 1.1098 1.0956 1.0789 1.0593

0 0.5 1 1.5 2

Nc = 1

Nc = 0.5

Nc = 1

Nr = 0.5

Nr= 1

Nr= 0.5

Nr = 1

Nr = 0.5

Nr = 1

Nr = 0.5

Nr = 1

2.3904 2.3714 2.3510 2.3292 2.3058

2.3992 2.3783 2.3558 2.3314 2.3051

2.3028 2.2705 2.2347 2.1948 2.1503

2.3194 2.2847 2.2459 2.2023 2.1532

2.3037 2.2779 2.2501 2.2200 2.1874

2.3128 2.2843 2.2532 2.2193 2.1822

2.2122 2.1686 2.1196 2.0643 2.0016

2.2297 2.1827 2.1295 2.0687 1.9989

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627

Fig. 2. Variation of dimensionless velocity with (a) viscosity variation and magnetic parameters and with (b) radiation and convective parameters.

uniform wall temperature (UWT) boundary condition is achieved and when Nc = 0 an insulated boundary condition achieved. Convective boundary condition occurs in a variety of real situations such as fluid flow around micro-electromechanical system (MEMS) (Kiwan and AlNimr [27]). Brownian motion is the irregular motion exhibited by nanoparticles when suspended in a fluid. As a result of this motion, molecules of the base fluid and the nanoparticles move faster and generate more kinetic energy and the temperature rises. The thermophoretic force generated by the temperature gradient creates a fast flow away from the stretching surface. In this way more heated fluid is moved away from the surface, and consequently, as Nt increases, the temperature within the boundary layer increases. Fig. 4(a) and (b) shows the influence of Brownian motion parameterNb, thermophoresis parameterNt, Lewis number, and viscosity parameter on dimensionless nanoparticle concentration of a waterbased nanofluid. It is noticed that, within the concentration boundary layer, the dimensionless nanoparticle volume fraction decreases with increasing values of Brownian motion parameterNb, and increases with increasing thermophoretic parameter Nt. Physically, thermophoretic

force creates a fast flow away from the stretching surface, consequently, as Nt increases, the fast flow from the stretching sheet carries with it thermophoretic force leading to an increase in the concentration boundary layer thickness as shown in Fig. 4(a). On the other side, an increase in the Brownian motion parameter Nb increases the diffusion of nanoparticles due to the Brownian effect. An increase in the Brownian motion parameter Nb of the fluid leads to a decrease in the concentration inside the boundary layer. From Fig. 4(b), it is observed that the dimensionless concentration decreases and increases with Le andδ, respectively. In fact, Lewis number Le expresses the relative contribution of thermal diffusion rate to species diffusion rate in the boundary layer regime. A larger Lewis number has a relatively lower molecular diffusivity. Therefore, a gradual increase in Le corresponds to a thinner concentration boundary layer, as it is evident from Fig. 4(b). The variations in skin friction with different controlling parameters are shown in Fig. 5(a) and (b) for a water-based nanofluid. As the dimensionless velocity of the fluid is reduced by the Lorentz force produced by magnetic field, consequently, the skin friction is reduced for stronger magnetic field as shown in Fig. 5(a). It was also found that

Fig. 3. Variation of dimensionless temperature with (a) viscosity variation and magnetic parameters, with (b) radiation and convective parameters and with (c) nanofluids parameters.

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Fig. 4. Variation of dimensionless concentration with (a) nanofluids parameters and with (b) viscosity variation parameter and Lewis numbers.

the skin coefficient increases as the viscosity and radiation parameters are increased. The effects of convective and nanofluid parameters on skin friction are shown in Fig. 5(b). An increase in nanofluid parameters increases the skin friction. It is also noted that for large values of the convective parameter, the effects are found to be more prominent and thus, the dimensionless skin friction is found higher. The effects of nanofluids and convective parameters on the dimensionless Nusselt numbers are explored in Fig. 6(a). It is noticed that when Nc become too large (say Nc = 1000), we get the uniform surface temperature case. It is well known that the tendency of the nanofluid parameters is to reduce the Nusselt numbers. Physically, the surface temperature increases with an increase in Nt values. With an increase in Nt values, the thermal boundary layer thickens and hence, the dimensionless Nusselt number decreases. Similar trend can be observed for increasing values of Nb. Moreover, it is noted that for large values of the convective parameter Nc, the effects are found to be more prominent and thus, the Nusselt number is found higher. Therefore, the convective parameter helps in increasing both the temperature and heat transfer profiles over the heated

stretching surface. Fig. 6(b) illustrates the influence of magnetic, viscosity and radiation parameter on the dimensionless Nusselt number. It is noticed that the powerful Lorentz force that arose in the flow field for larger magnetic field reduces the values of the Nusselt number. Also, the heat transfer rate decreases and increases with viscosity and radiation parameters, respectively. Finally, the combined effects of Lewis number, Brownian motion and thermophoresis parameters on the dimensionless Sherwood number are depicted in Fig. 7. The Sherwood number, which represents the dimensionless mass transfer rate at the surface decreases when Nt increases. This may be due to the fact that the thermophoresis parameter decreases the dimensionless mass transfer rate of nanofluids. For high rate of Brownian motion, that is, for higher values ofNb, the dimensionless Sherwood number increases. This is due to the fact that a higher value of the Brownian motion parameter leads to a decrease in the concentration inside the boundary layer. It is observed from Fig. 7 that an increase in Lewis number Le increases the dimensionless mass transfer rates. In fact, an increase in Lewis number reduces the diffusion rate and as a result Sherwood number increases.

Fig. 5. Variation of reduced skin friction with (a) magnetic, radiation and viscosity variation parameters and (b) convective and nanofluids parameters.

O.D. Makinde et al. / Journal of Molecular Liquids 219 (2016) 624–630

629

Fig. 6. Variation of reduced Nusselt numbers (a) convective and nanofluids parameters and with (b) magnetic, radiation and viscosity variation parameters.

5. Conclusions The combined effects of thermal radiation, thermophoresis, Brownian motion, magnetic field and variable viscosity on boundary layer flow, heat and mass transfer of an electrically conducting waterbased nanofluid are investigated over a radially stretching convectively heated surface. It is concluded that • The controlling parameters tend to reduce the dimensionless velocity and hence reduce the velocity boundary layer thickness. • The controlling parameters help in increasing the dimensionless temperature and hence increase the thermal boundary layer thickness. • The reduced skin friction decreases with an increase in magnetic field but increases with viscosity variation, radiation, convective and nanofluids parameters.

Fig. 7. Variation of reduced Sherwood numbers with Lewis number and nanofluids parameters.

• The reduced Nusselt number decreases with magnetic field, nanofluids, viscosity variation but increases with convective and radiation parameters. • The reduced Sherwood number increases with increasing Lewis number and Brownian motion but decreases with increasing thermophoresis parameter.

References [1] S.G. Etemad, B. Farajollahi, M. Hajipour, J. Thibault, Turbulent convective heat transfer of suspensions of Γ-AL2O3 and cuonanoparticles (nanofluids), J. Enhanc. Heat Transf. 19 (3) (2012) 191–197. [2] S.U.S. Choi, Nanofluids: from vision to reality through research, J. Heat Transf. 131 (3) (2009) 1–9. [3] K.V. Wong, O. Leon, Applications of nanofluids: current and future, Adv. Mech. Eng. 2010 (2010), 519659 (10 pages). [4] P. Esmaeilzadeh, Z. Fakhroueian, A.A.M. Beigi, Synthesis of biopolymeric αlactalbumin protein nanoparticles and nanospheres as green nanofluids using in drug delivery and food technology, J. Nano Res. 16 (2012) 89–96. [5] K. Jayaraman, K.V. Anand, S.R. Chakravarthy, R. Sarathi, Effect of nano-aluminium in plateau-burning and catalyzed composite solid propellant combustion, Combust. Flame 156 (8) (2009) 1662–1673. [6] M. Muthtamilselvan, R. Rakkiyappan, Mixed convection in a lid-driven square cavity filled with nanofluids, Int. J. Nanomech. Sci. Technol. 2 (4) (2011) 275–294. [7] K.Y. Leong, R. Saidur, S.N. Kazi, A.H. Mamun, Performance investigation of an automotive car radiator operated with nanofluid-based coolants (nanofluid as a coolant in a radiator), Appl. Therm. Eng. 30 (17–18) (2010) 2685–2692. [8] O.D. Makinde, A. Aziz, Boundary layer flow of a nano fluid past a stretching sheet with a convective boundary condition, Int. J. Therm. Sci. 53 (11) (2011) 2477–2483. [9] A. Aziz, W.A. Khan, I. Pop, Free convection boundary layer flow past a horizontal flat plate embedded in porous medium filled by nanofluid containing gyrotactic microorganisms, Int. J. Therm. Sci. 56 (2012) 48–57. [10] T.G. Motsumi, O.D. Makinde, Effects of thermal radiation and viscous dissipation on boundary layer flow of nanofluids over a permeable moving flat plate, Phys. Scr. 86 (2012), 045003 (8 pp.). [11] A. Alsaedi, M. Awais, T. Hayat, Effects of heat generation/absorption on stagnation point flow of nanofluid over a surface with convective boundary conditions, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 4210–4223. [12] A.S. Butt, A. Ali, Entropy analysis of magnetohydrodynamic flow and heat transfer over a convectively heated radially stretching surface, J. Taiwan Inst. Chem. Eng. 45 (4) (2014) 1197–1203. [13] K. Parekh, H.S. Lee, Experimental investigation of thermal conductivity of magnetic nanofluids, AIP Conf. Proc. 1447 (2011) 385–386. [14] E. Andablo-Reyes, R. Hidalgo-Alvarez, J. de Vicente, Controlling friction using magnetic nanofluids, Soft Matter 7 (3) (2011) 880–883. [15] R. Ellahi, The effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid in a pipe: analytical solutions, Appl. Math. Model. 37 (3) (2013) 1451–1467. [16] M. Sheikholeslami, M. Gorji-Bandpy, R. Ellahi, A. Zeeshan, Simulation of MHD CuO– water nanofluid flow and convective heat transfer considering Lorentz forces, J. Magn. Magn. Mater. 369 (2014) 69–80.

630

O.D. Makinde et al. / Journal of Molecular Liquids 219 (2016) 624–630

[17] M. Mustafa, S. Hina, T. Hayat, A. Alsaedi, Influence of wall properties on the peristaltic flow of a nanofluid: analytic and numerical solutions, Int. J. Heat Mass Transf. 55 (2012) 4871–4877. [18] S. Nadeem, R. Mehmood, N.S. Akbar, Optimized analytical solution for oblique flow of a Casson nanofluid with convective boundary conditions, Int. J. Therm. Sci. 78 (2014) 90–100. [19] O.D. Makinde, W.A. Khan, J.R. Culham, MHD variable viscosity reacting flow over a convectively heated plate in a porous medium with thermophoresis and radiative heat transfer, Int. J. Heat Mass Transf. 93 (2016) 595–604. [20] A.O. Ali, O.D. Makinde, Modelling the effect of variable viscosity on unsteady Couette flow of nanofluids with convective Cooling, J. Appl. Fluid Mech. 8 (4) (2015) 793–802. [21] S. Khamis, O.D. Makinde, Y. Nkansah-Gyekye, Buoyancy driven heat transfer of water based nanofluid in a permeable cylindrical pipe with Navier slip through a saturated porous medium, J. Porous Media 18 (12) (2015) 1169–1180. [22] S. Nadeem, R. Ul Haq, Effect of thermal radiation for magnetohydrodynamic boundary layer flow of a nanofluid past a stretching sheet with convective boundary conditions, J. Comput. Theor. Nanosci. 11 (2014) 32–40.

[23] S. Das, R.N. Jana, O.D. Makinde, MHD boundary layer slip flow and heat transfer of nanofluid past a vertical stretching sheet with non-uniform heat generation/absorption, Int. J. Nanosci. 13 (3) (2014), 1450019 (12 pages). [24] S. Das, R.N. Jana, O.D. Makinde, Mixed convective magnetohydrodynamic flow in a vertical channel filled with nanofluids, Eng. Sci. Technol. 18 (2) (2015) 244–255. [25] W.A. Khan, J.R. Culham, O.D. Makinde, Combined heat and mass transfer of thirdgrade nanofluids over a convectively-heated stretching permeable surface, Can. J. Chem. Eng. 93 (10) (2015) 1880–1888. [26] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Trans. 128 (2006) 240–250. [27] S. Kiwan, M.A. Al-Nimr, Investigation into the similarity solution for boundary layer flows in Microsystems, J. Heat Transf. 132 (2010) 041011. [28] A.V. Kuznetsov, D.A. Nield, Natural convective boundary-layer flow of a nanofluid past a vertical plate, Int. J. Therm. Sci. 49 (2) (2010) 243–247.