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Radiation effects on MHD stagnation point flow of nano fluid towards a stretching surface with convective boundary condition
Chinese Journal of Aeronautics, (2013),26(6): 1389–1397
Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
Radiation effects on MHD stagnation point flow of nano fluid towards a stretching surface with convective boundary condition Noreen Sher Akbar a, S. Nadeem b, Rizwan Ul Haq a b c
b,*
, Z.H. Khan
c
DBS&H, CEME, National University of Sciences and Technology, Islamabad 44000, Pakistan Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan School of Mathematical Sciences, Peking University, Beijing 100871, China
Received 13 November 2012; revised 30 December 2012; accepted 14 March 2013 Available online 5 November 2013
KEYWORDS Convective boundary condition; Nanoparticles; Numerical solutions; Radiation effects; Stagnation point; Stretching surface
Abstract The aim of the present paper is to study the numerical solutions of the steady MHD two dimensional stagnation point flow of an incompressible nano fluid towards a stretching cylinder. The effects of radiation and convective boundary condition are also taken into account. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The resulting nonlinear momentum, energy and nano particle equations are simplified using similarity transformations. Numerical solutions have been obtained for the velocity, temperature and nanoparticle fraction profiles. The influence of physical parameters on the velocity, temperature, nanoparticle fraction, rates of heat transfer and nanoparticle fraction are shown graphically. ª 2013 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA.
1. Introduction The impingement of flow on the cylinder creates a stagnation point on the approaching surface. The departure of the flow away from the cylinder creates another stagnation point on
* Corresponding author. Address: Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan. Tel.: +92 51 90642182. E-mail address: [email protected] (R. Ul Haq). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
the trailing surface. Flow and heat transfer of an incompressible viscous fluid over a stretching sheet have been discussed in several mechanized processes from industry such as the extrusion of polymers, the cooling of metallic plates, the aerodynamic extrusion of plastic sheets, etc. In the glass industry, blowing, floating or spinning of fibers are processes, which involve the flow due to a stretching surface. When scientific processes take place at high temperatures, i.e., cooling of a metal or glass sheet, thermal radiation effects start to play an important role and cannot be neglected.1–4 Pop et al.5 study the radiation effects (Rosseland Model) on the flow of an incompressible viscous fluid over a flat sheet near the stagnation point. Two-dimensional stagnation-point flow of viscoelastic fluids has been studied theoretically by Sadeghy et al.6 They assume that the fluid obeys the upper-convected Maxwell (UCM) model. Boundary-layer theory is used to simplify the
1000-9361 ª 2013 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA. http://dx.doi.org/10.1016/j.cja.2013.10.008
1390 equations of motion which are further reduced to a single nonlinear third-order ODE using the concept of stream function coupled with the technique of the similarity solution. The steady hydromagnetic laminar stagnation point flow of an incompressible viscous fluid impinging on a permeable stretching surface with heat generation or absorption has been analyzed by Attia.7 The steady MHD mixed convection flow of a viscoelastic fluid in the vicinity of two-dimensional stagnation point with magnetic field has been investigated by Kumari and Nath under the assumption that the fluid obeys the upper-convected Maxwell (UCM) model.8 Prandtl introduced the concept of boundary layer in 1904. The aim of his study is to concentrate the application of boundary layer theory primarily to aerodynamics, and secondary to fluid mechanics. Prandtl’s important contribution to 20th century aerodynamics ranges beyond his boundary-layer concept. He develops a theory for calculating the lift and pitch related moment coefficients for thin, cambered airfoil, and thin airfoil theory. During the same period, Prandtl developed his lifting-line theory for wings. Later on many authors modify the concept of Prandtl related to aerodynamics approach.9–12 Nanofluid is a fluid containing small particles, called nanoparticles. These fluids are engineered colloidal suspensions of nanoparticles in a base fluid.13 The nanoparticles used in nanofluids are typically made of metals, oxides, carbides, or carbon nanotubes. The base fluids used are usually water, ethylene glycol and oil.14,15 Recently, boundary-layer flow of a nanofluid past a stretching sheet was presented by Khan and Pop.16 The natural convective boundary-layer flow of a nanofluid past a vertical plate is presented analytically by Kuznetsov and Nield.17 Vajravelu et al.18 presented the detailed analysis of convective heat transfer in the flow of viscous Ag–water and Cu–water nanofluids over a stretching surface. Very recently the boundary layers of an unsteady stagnationpoint flow in a nanofluid is investigated by Bachok et al.19 They analyzed that fluid containing solid particles may significantly increase its conductivity. Recently, Bachok et al. coated the boundary layers of an unsteady stagnation-point flow in a nanofluid. Convective boundary condition is mostly used to define a linear convective heat exchange condition for one or more algebraic entities in thermal. Heat transfer analysis with convective boundary conditions is evoked in processes such as thermal energy storage, gas turbines, nuclear plants, etc. In view of the above analysis, Aziz presented a similarity solution for Blasius flow of viscous fluid with convective boundary conditions.20 Makinde investigated MHD mixed convection flow in a vertical plate in a porous medium. They take convective boundary condition into account.21 Makinde and Aziz examined the hydromagnetic boundary layer flow with heat and mass transfer over a vertical plate in the presence of magnetic field and a convective heat exchange at the surface.22 In another article, Makinde and Aziz visualized boundary layer flow of a nano fluid past a stretching sheet with a convective boundary condition.23 Free convection boundary layer flow past a horizontal flat plate embedded in porous medium filled by nanofluid containing gyrotactic microorganisms have been discussed by Aziz et al.24 Makinde et al.25 analyzed the combined effects of buoyancy force, magnetic field, thermophoresis, Brownian motion, and convective heating on stagnation-point flow and heat transfer
N.S. Akbar et al. due to nanofluid flow towards a stretching sheet. A novel analytical solution of the non-uniform convective boundary conditions problem for heat conduction in cylinders has been addressed by Tonini and Cossali.26 The present article gives the axisymmetric stagnation point flow of nano fluid towards a stretching surface with convective boundary condition. Radiation effects are also taken into account. The resulting nonlinear momentum, energy and nano particle equations are simplified using similarity transformations. Numerical solutions have been developed for the velocity, temperature and nanoparticle fraction. The influence of physical parameters on the velocity, temperature, nanoparticle fraction, rates of heat transfer and nanoparticle fraction are shown graphically. The present numerical results are compared with the literature.
2. Mathematical formulation We discussed the steady two-dimensional stagnation point flow of a nano fluid over a stretching cylinder with convective boundary conditions and radiation effects. For the formulation of the problem we have used the cylindrical coordinates r, u, z and assume that the wall is at z = 0, the stagnation point is at the origin and that the flow is in the direction of the negative z-axis. u = u(r, z) and w = w(r, z) are the velocity components along r and z directions, where the velocity along u direction vanishes due to symmetry. A uniform magnetic field B0 is applied normal to the cylinder where the induced magnetic field is neglected by assuming very small magnetic Reynolds number. Moreover, it is assumed that the surface has a temperature of Tw and concentration of nano particles Cw and fluid has uniform ambient temperature T1 and C1 (Tw > T1 and Cw > C1). The velocity distribution in the frictionless flow in the neighborhood of the stagnation point is given by
U ¼ ar W ¼ 2az
where a > 0 is a constant characterizing the velocity of the mainstream flow. The continuity, momentum, heat and nano particle fraction equations for the two-dimensional steady state flows under radiation effects, using the usual boundary layer approximations,7,15 reduce to @u u @w þ þ ¼0 @r r @z
ð1Þ
2 @u @u @p @U @ u 1 @u u @ 2 u ¼ þU q u þw þl þ þ @r @z @r @r @r2 r @r r2 @z2 rB20 ðu UÞ
ð2Þ
2 @w @w @p @ w 1 @w @ 2 w ¼ þl q u þw þ þ @r @z @z @r2 r @r @z2
ð3Þ
2 @T @T @ T 1 @T @ 2 T @C @T @C @T u þw þ þ 2 þ s DB þ ¼a 2 @r @z @r r @r @z @r @r @z @z " 2 #) 2 DT @T @T @qr þ þ ð4Þ @r @z @z T1
Radiation effects on MHD stagnation point flow of nano fluid towards a stretching surface 2 @C @C @ C 1 @C @ 2 C ¼ DB u þw þ þ @r @z @r2 r @r @z2 2 D @ T 1 @T @ 2 T þ þ þ T T1 @r2 r @r @z2
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8 pffiffiffiffiffi u ¼ crf0 ðgÞ; w ¼ 2 cmfðgÞ > > p ffi > > > g ¼ z ct; m ¼ lq > > > > > 1 1 > U ¼ CCC ; h ¼ TTT > > w C1 w T1 > > > Pr ¼ m=a ; Le ¼ DaB > > <
ð5Þ
ðqcÞ
1Þ Nt ¼ TD1T ðqcÞp ðTw T m f > > > > ðqcÞp ðUw U1 Þ > > > Nb ¼ DB ðqcÞf m > > > pffi > > h f m > c ¼ kf > > c > > > : 4r T31 Nr ¼ k k ; s ¼ ac
The corresponding boundary conditions are stated as 8 u ¼ cr; w ¼ 0; C ¼ Cw ; > > < @T ¼ hf ðTf TÞ kf > @z > : u ! ax; T ! T1 ; C ! C1
ð6Þ
z¼0 z!1
And using non-dimensional variables presented in Eq. (9) and Eqs.(7) and (8) in Eqs. (1)-(5) we have
In the above expressions u and w are the velocity components along the r and z axes, respectively, q is the fluid density, v the kinematic viscosity, T the temperature, and C the concentration of fluid; the ambient values of T and C as z tends to infinity are denoted by T1 and C1, cp is the specific heat of the material DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient, a = k/(qc)f is the thermal diffusivity of the fluid, s = (qc)p/(qc)f is the ratio between the effective heat capacity of the nano particle material and heat capacity of the fluid with q being the density, c the volumetric volume expansion coefficient hf the convective heat transfer coefficient, Tf the convective fluid temperature and qp the density of the particle. Using Rosseland approximation of radiation 4r @T4 qr ¼ 3k @z
2
4 2 1 þ Nr h00 þ Prð2fh0 þ NbU0 h0 þ Nbðh0 Þ Þ ¼ 0 3
ð11Þ
0
ð12Þ
8 > < fð0Þ ¼ 0 f0 ð0Þ ¼ 1 > : 0 f ð1Þ ¼ s
where k is the mean absorption coefficient and r is the Stefan–Boltzmann constant and T4 is a linear temperature function which is expanded by using Taylor series expansion about T1 as
ð13aÞ
h0 ð0Þ ¼ Bið1 hð0ÞÞ hð1Þ ¼ 0
ð13bÞ
Uð0Þ ¼ 1
ð13cÞ
Uð1Þ ¼ 0
M, s, Pr, Nb, Nt, Le, Bi and Nr are the Hartmann number, stagnation parameter, Prandtl number, the Brownian motion parameter, the thermophoresis parameter, Lewis number, Biot number and local concentration thermal radiation effects, respectively. Expressions for the local Nusselt number Nu and the local Sherwood number Sh are
ð8Þ
Introducing the following quantities
Table 1
ð10Þ
Nt 00 h ¼0 U þ 2LePrfU þ Nb
*
T4 ¼ 4T31 T 3T41
f000 þ 2ff00 ðf0 Þ þ M2 ðs f0 Þ þ s2 ¼ 0
00
ð7Þ
*
ð9Þ
Comparison of numerical results for local Nusselt number in the absence of nanofluid for various values of M, s and Pr.
(a) When Pr = 0.7, Nr = 0 and Bi = 1 Result
By Attia7
M
0
1
2
0
1
2
s = 0.1 s = 1.0 s = 1.5
0.6454 0.9109 1.0263
0.5974 0.9109 1.0309
0.5112 0.9109 1.0412
0.6454 0.9109 1.0263
0.5974 0.9109 1.0309
0.5112 0.9109 1.0412
Present
(b) When M = 1, Nr = 0 and Bi = 1 By Attia7
Present
Pr !
0.1
0.5
1.0
0.1
0.5
1.0
s = 0.1 s = 1.0 s = 1.5
0.1467 0.2533 0.2989
0.3471 0.5674 0.6498
0.5539 0.8043 0.9057
0.1467 0.2533 0.2989
0.3471 0.5674 0.6498
0.5539 0.8043 0.9057
1392 (
N.S. Akbar et al.
w Nu ¼ kðTrq f T1 Þ
ð14Þ
Sh ¼ DB ðCrqwmC1 Þ
8 h i < q ¼ 16rT3 þ k @T ; w 3k @z z¼0 : q ¼ D @C ; m
where qw and qm are the heat flux and mass flux, respectively.
(a) f ' (η )
(b) θ (η )
ð15Þ
B @z z¼0
Dimensionless form of Eq. (15) takes the form
(a) f ' (η )
(b) θ (η )
(c) Φ (η )
Fig. 1 Effects of M on velocity, temperature and nanoparticle fraction with Nr = Nb = Nt = Bi = s = 0.1, Le = 1, Pr = 0.8.
Fig. 2 Effects of s on velocity, temperature and nanoparticle fraction with Nr = Nb = Nt = Bi = 0.1, Le = M = 1, Pr = 0.8.
Radiation effects on MHD stagnation point flow of nano fluid towards a stretching surface (
Re1=2 Nu ¼ ð1 þ 43 NrÞh0 ð0Þ r Sh Re1=2 r
0
¼ u ð0Þ
( ð16Þ
3. Numerical technique The system of coupled nonlinear coupled differential Eqs. (10)(12) along with the boundary conditions Eq. (13a) to Eq. (13c) are solved numerically using fourth-order Runge–Kutta– Fehlberg method with a shooting technique. The step size and the convergence criteria were taken as Dg = 0.001 and 106 respectively. The asymptotic boundary conditions were replaced by using a value of 20 for the similarity variable g1 as follows
(a) θ (η )
(b) Φ (η )
Fig. 3 Effects of Pr on temperature and nanoparticle fraction with Nr = s = Bi = 0.5, Le = M = 1, Nb = 0.3, Nt = 0.1.
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g1 ¼ 20 f0 ð20Þ ¼ hð20Þ ¼ Uð20Þ ¼ 0
The choice of g1 = 20 ensures that all numerical solutions approached the asymptotic values correctly. The present results of dimensionless heat transfer rate for pure fluid are compared with those reported by Attia7 and are shown in Table 1. We notice that the comparison shows good agreement for each value considered. We are, therefore, confident that the present results are correct and accurate. 4. Results and discussion The numerical solutions are presented through graphs (see Figs. 1–11) for the physical interpretation of the proposed
(a) θ (η )
(b) Φ (η )
Fig. 4 Effects of Bi on temperature and nanoparticle fraction with Nr = Nt = 0.5, Le = M = 1, Nb = 0.3, s = 0.1, Pr = 6.2.
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N.S. Akbar et al.
(a) θ (η )
(b) Φ (η )
Fig. 5 Effects of Le on temperature and nanoparticle fraction with Nr = s = Bi = 0.1, Nt = 0.5, M = 1, Nb = 0.3, Pr = 6.2.
study. These figures describes the influence of magnetic field M, Prandtl number Pr, stretching parameter s, Brownian motion parameter Nb, thermophoresis parameter Nt, Lewis number Le and the Biot number Bi on the dimensionless velocity, temperature, nanoparticle fraction, heat transfer rate, skin friction coefficient and nanoparticle fraction rate. Effect of magnetic field on the velocity, temperature and nanoparticle fraction is shown in Fig. 1. It is shown that velocity profile f0 (g) decreases with the increase of M and the boundary layer thickness also decreases. It can also be concluded that the temperature profile h(g) and nanoparticle fraction U(g) increases with an increase in M. Furthermore, with an increase in M, the thermal boundary layer as well as concentration boundary layer decreases. Fig. 2 illustrates the effect of stagnation parameter s on f0 (g), h(g) and U(g). It is found that the velocity profile increases while tempera-
(a) θ (η )
(b) Φ (η )
Fig. 6 Effects of Nb on temperature and nanoparticle fraction with s = 0.1, Nt = 0.4, Nr = Le = M = 1, Bi = 0.5, Pr = 6.2.
ture profile and nanoparticle fraction decrease with the increasing value of s. We notice from Fig. 3 that Prandtl number Pr has the same sort influence on temperature profile as well as on nanoparticle fraction, as Pr has decreasing behavior for both h(g) and U(g). The effects of Bi on temperature profile and nanoparticle fraction have exactly the same increasing behavior (see Fig. 4). As expected, it is found that temperature profile is an increasing function in view of Le. While, decrease in Le leads to increase in nanoparticle fraction (see Fig. 5). Fig. 6 shows the effect of Nb on the h(g) and U(g). It is seen that temperature profile increase with the increasing value of Nb, whereas nanoparticle fraction decreases. Fig. 7 shows that temperature profile and nanoparticle fraction show the same behavior against the value of Nt Temperature profile as well as the nanoparticle fraction in-
Radiation effects on MHD stagnation point flow of nano fluid towards a stretching surface
(a) θ (η )
(b) Φ (η )
Fig. 7 Effects of Nt on temperature and nanoparticle fraction with s = 0.1, Nr = Nb = 0.3, Le = M = 1, Bi = 0.7, Pr = 6.2.
creases with an increase in Nt. Effects of Nr on temperature and nanoparticle fraction are presented in Fig. 8. Here we say that h(g) and U(g) shows the same increasing behavior with an increasing values of Nr. Figs. 9–11, show the influence of governing parameters on dimensionless heat transfer rates, skin friction coefficient and dimensionless nanoparticle fraction rates. In Fig. 9, it depicts that for higher values of M, skin friction coefficient presents the increasing behavior corresponding to the increasing values of s. This shows that fluid motion on the wall of the sheet accelerated, when we strengthen the effects of parameter and stagnation parameter. From Fig. 10(a), it is noticed that local Nusselt number decreases for increasing values of Nb and Nt whereas for Nr and Bi, the heat transfer rate increases (see Fig. 10(b)). Fig. 11 shows the variation in dimensionless nanoparticle fraction rates against Nt and Le, respectively. As Nt increase the concentration rate decreases whereas Pr increase the mass rate increase.
1395
(a) θ (η )
(b) Φ (η )
Fig. 8 Effects of Nr on temperature and nanoparticle fraction with s = Nb = 0.1, Nt = 0.7, Le = M = Pr = 1, Bi = 0.5.
Fig. 9 Effects of M and s on Skin friction coefficient with Nb = 0.3, Nt = 0.5, Nr = Bi = 0.1, Le = 1, Pr = 6.2.
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N.S. Akbar et al.
(a) Nt
(a) Nb
(b) Pr
(b) Nr
Fig. 10 Effects of Nb and Nr on reduced Nusselt number with s = 0.1, Le = 10, Pr = 6.2.
Fig. 11 Effects of Nt and Pr on reduced Sherwood number with s = 0.1, M = 1, Nr = Bi = 0.5.
Table 2 Numerical values of the local Nusselt number and local Sherwood number for various values of Nt and Nb when Pr = 6.2, Le = 10, M = 1 and s = 0.1. Nr = 0 and Bi = 1 Nt #
0.1 0.2 0.3 0.4 0.5
Nr = 0.5 and Bi = 0.5
Nb = 0.1
Nb = 0.3
Nb = 0.1
Nb = 0.3
Nb = 0.1
Nb = 0.3
Rer1=2 Nu
Re1=2 Sh r
Rer1=2 Nu
Re1=2 Nu r
Rer1=2 Sh
Re1=2 Sh r
0.37263 0.28899 0.22993 0.18742 0.15619
8.77962 8.94717 9.07920 9.18033 9.25679
0.59963 0.59070 0.58100 0.57045 0.55898
0.46726 0.45020 0.43196 0.41262 0.39232
8.56609 8.56793 8.58369 8.61380 8.65851
8.63009 8.68759 8.74943 8.81457 8.88166
Finally, Table 2 gives the numerical values of skin friction coefficient, local Nusselt number and the local Sherwood number. With an increase in Pr and Nt, local Nusselt number
and skin friction coefficient decrease while local Sherwood number increases.
Radiation effects on MHD stagnation point flow of nano fluid towards a stretching surface 5. Conclusions Numerical simulation of steady two-dimensional stagnation point flow of a nano fluid is investigated with convective boundary conditions. The behavior of the flow parameters is observed. Summary of the present work is listed as follows. (1) It is seen that velocity profile f0 (g) decreases with the increasing values of M and the boundary layer thickness also decreases. (2) It is also depicted that the temperature profile h(g) and nanoparticle fraction U(g) increase with an increase in M. (3) The effects of Bi on temperature profile h(g) and nanoparticle fraction U(g) have exactly the same behavior. (4) It is found that the velocity profile f0 (g) and boundary layer thickness increase while temperature profile h(g) and nanoparticle fraction U(g) decrease with the increasing value of s. (5) It is found that temperature profile h(g) is increasing function in view of Le. An increase in Le leads to decrease in nanoparticle fraction U(g). (6) It is seen that temperature profile and nanoparticle fraction increase with the increasing value of Nt. (7) Temperature profile increases while nanoparticle fraction decreases with an increase in Nt. (8) It is observed that the influence of Pr for temperature profile and nanoparticle fraction has exactly the same behavior, as Pr has decreasing behavior for h(g) and U(g). (9) The heat transfer rate and skin friction coefficient decrease with the increasing values of Nb, Nt and Pr.
References 1. Siegel R, Howell JR. Thermal radiation heat transfer. 3rd ed. New York: Hemisphere; 1992. 2. Chiam TC. Stagnation point flow towards a stretching plate. J Phys Soc Jpn 1994;63:2443–4. 3. Mahapatra TR, Gupta AS. Heat transfer in stagnation point flow towards a stretching sheet. Heat Mass transfer 2002;38(6): 517–21. 4. Modest MF. Radiative heat transfer. 2nd ed. New York: Acad. Press; 2003. 5. Pop SR, Grosan T, Pop I. Radiation effects on the flow near the stagnation point of a stretching sheet. Tech Mech 2004;25(2): 100–6. 6. Sadeghy K, Hajibeygi H, Taghavi SM. Stagnation-point flow of upper-convected Maxwell fluids. Int J Non-Linear Mech 2006;41(10):1242–7. 7. Attia HA. Stagnation point flow towards a stretching surface through a porous medium with heat generation. Turkish J Eng Environ Sci 2006;30(5):299–306.
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8. Kumari M, Nath G. Steady mixed convection stagnation-point flow of upper convected Maxwell fluids with magnetic field. Int J Non-Linear Mech 2009;44(10):1048–55. 9. Liu JX, Tang DB, Yang YZ. On nonlinear evolution of C-type instability in nonparallel boundary layers. Chin J Aeronaut 2007;20(4):313–9. 10. Wang Q, Guo ZY, Zhou C, Feng GT, Wang ZQ. Coupled heat transfer simulation of a high-pressure turbine nozzle guide vane. Chin J Aeronaut 2009;22(3):230–6. 11. Guo X, Tang DB. Nonlinear stability of supersonic nonparallel boundary layer flows. Chin J Aeronaut 2010;23(3):283–9. 12. Guo SA, Chen SW, Song YP, Song YF, Chen F. Effects of boundary layer suction on aerodynamic performance in a highload compressor cascade. Chin J Aeronaut 2010;23(2):179–86. 13. Buongiorno J. Convective transport in nanofluids. J Heat Transfer 2010;128(3):240–50. 14. Kakac S, Pramuanjaroenkij A. Review of convective heat transfer enhancement with nanofluids. Int J Heat Mass Transfer 2009;52(13):3187–96. 15. Marga SEB, Palm SJ, Nguyen CT, Roy G, Galanis N. Heat transfer enhancement by using nanofluids in forced convection flows. Int J Heat Fluid Flow 2005;26(4):530–46. 16. Khan WA, Pop I. Boundary-layer flow of a nanofluid past a stretching sheet. Int J Heat Mass Transfer 2010;53(11):2477–83. 17. Kuznetsov AV, Nield DA. Natural convective boundary-layer flow of a nanofluid past a vertical plate. Int J Therm Sci 2010;49(2):243–7. 18. Vajravelu K, Prasad KV, Lee J, Lee C, Pop I, Gorder RAV. Convective heat transfer in the flow of viscous Ag-water and Cu– water nanofluids over a stretching surface. Int J Therm Sci 2011;50(5):843–51. 19. Bachok N, Ishak A, Pop L. The boundary layers of an unsteady stagnation-point flow in a nanofluid. Int J Heat Mass Transfer 2012;55(22–23):6499–505. 20. Aziz A. A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition. Commun Non-Linear Sci Numer Simul 2009;14(4):1064–8. 21. Makinde OD. Similarity solution of hydromagnetic heat and mass transfer over a vertical plate with a convective surface boundary condition. Int J Phys Sci 2010;5(6):700–10. 22. Makinde OD, Aziz A. MHD mixed convection from a vertical plate embedded in a porous medium with a convective boundary condition. Int J Therm Sci 2010;49(9):1813–20. 23. Makinde OD, Aziz A. Boundary layer flow of a nano fluid past a stretching sheet with a convective boundary condition. Int J Therm Sci 2011;53(11):2477–83. 24. Aziz A, Khan WA, Pop I. Free convection boundary layer flow past a horizontal flat plate embedded in porous medium filled by nanofluid containing gyrotactic microorganisms. Int J Therm Sci 2012;56:48–57. 25. Makinde OD, Khan WA, Khan ZH. Buoyancy effects on MHD stagnation point flow and heat transfer of a nanofluid past a convectively heated stretching/shrinking sheet. Int J Heat Mass Transfer 2013;62:526–33. 26. Tonini S, Cossali GE. A novel analytical solution of the nonuniform convective boundary conditions problem for heat conduction in cylinders. Int Commun Heat Mass Transfer 2012;39(8): 1059–65.
Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
Radiation effects on MHD stagnation point flow of nano fluid towards a stretching surface with convective boundary condition Noreen Sher Akbar a, S. Nadeem b, Rizwan Ul Haq a b c
b,*
, Z.H. Khan
c
DBS&H, CEME, National University of Sciences and Technology, Islamabad 44000, Pakistan Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan School of Mathematical Sciences, Peking University, Beijing 100871, China
Received 13 November 2012; revised 30 December 2012; accepted 14 March 2013 Available online 5 November 2013
KEYWORDS Convective boundary condition; Nanoparticles; Numerical solutions; Radiation effects; Stagnation point; Stretching surface
Abstract The aim of the present paper is to study the numerical solutions of the steady MHD two dimensional stagnation point flow of an incompressible nano fluid towards a stretching cylinder. The effects of radiation and convective boundary condition are also taken into account. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The resulting nonlinear momentum, energy and nano particle equations are simplified using similarity transformations. Numerical solutions have been obtained for the velocity, temperature and nanoparticle fraction profiles. The influence of physical parameters on the velocity, temperature, nanoparticle fraction, rates of heat transfer and nanoparticle fraction are shown graphically. ª 2013 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA.
1. Introduction The impingement of flow on the cylinder creates a stagnation point on the approaching surface. The departure of the flow away from the cylinder creates another stagnation point on
* Corresponding author. Address: Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan. Tel.: +92 51 90642182. E-mail address: [email protected] (R. Ul Haq). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
the trailing surface. Flow and heat transfer of an incompressible viscous fluid over a stretching sheet have been discussed in several mechanized processes from industry such as the extrusion of polymers, the cooling of metallic plates, the aerodynamic extrusion of plastic sheets, etc. In the glass industry, blowing, floating or spinning of fibers are processes, which involve the flow due to a stretching surface. When scientific processes take place at high temperatures, i.e., cooling of a metal or glass sheet, thermal radiation effects start to play an important role and cannot be neglected.1–4 Pop et al.5 study the radiation effects (Rosseland Model) on the flow of an incompressible viscous fluid over a flat sheet near the stagnation point. Two-dimensional stagnation-point flow of viscoelastic fluids has been studied theoretically by Sadeghy et al.6 They assume that the fluid obeys the upper-convected Maxwell (UCM) model. Boundary-layer theory is used to simplify the
1000-9361 ª 2013 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA. http://dx.doi.org/10.1016/j.cja.2013.10.008
1390 equations of motion which are further reduced to a single nonlinear third-order ODE using the concept of stream function coupled with the technique of the similarity solution. The steady hydromagnetic laminar stagnation point flow of an incompressible viscous fluid impinging on a permeable stretching surface with heat generation or absorption has been analyzed by Attia.7 The steady MHD mixed convection flow of a viscoelastic fluid in the vicinity of two-dimensional stagnation point with magnetic field has been investigated by Kumari and Nath under the assumption that the fluid obeys the upper-convected Maxwell (UCM) model.8 Prandtl introduced the concept of boundary layer in 1904. The aim of his study is to concentrate the application of boundary layer theory primarily to aerodynamics, and secondary to fluid mechanics. Prandtl’s important contribution to 20th century aerodynamics ranges beyond his boundary-layer concept. He develops a theory for calculating the lift and pitch related moment coefficients for thin, cambered airfoil, and thin airfoil theory. During the same period, Prandtl developed his lifting-line theory for wings. Later on many authors modify the concept of Prandtl related to aerodynamics approach.9–12 Nanofluid is a fluid containing small particles, called nanoparticles. These fluids are engineered colloidal suspensions of nanoparticles in a base fluid.13 The nanoparticles used in nanofluids are typically made of metals, oxides, carbides, or carbon nanotubes. The base fluids used are usually water, ethylene glycol and oil.14,15 Recently, boundary-layer flow of a nanofluid past a stretching sheet was presented by Khan and Pop.16 The natural convective boundary-layer flow of a nanofluid past a vertical plate is presented analytically by Kuznetsov and Nield.17 Vajravelu et al.18 presented the detailed analysis of convective heat transfer in the flow of viscous Ag–water and Cu–water nanofluids over a stretching surface. Very recently the boundary layers of an unsteady stagnationpoint flow in a nanofluid is investigated by Bachok et al.19 They analyzed that fluid containing solid particles may significantly increase its conductivity. Recently, Bachok et al. coated the boundary layers of an unsteady stagnation-point flow in a nanofluid. Convective boundary condition is mostly used to define a linear convective heat exchange condition for one or more algebraic entities in thermal. Heat transfer analysis with convective boundary conditions is evoked in processes such as thermal energy storage, gas turbines, nuclear plants, etc. In view of the above analysis, Aziz presented a similarity solution for Blasius flow of viscous fluid with convective boundary conditions.20 Makinde investigated MHD mixed convection flow in a vertical plate in a porous medium. They take convective boundary condition into account.21 Makinde and Aziz examined the hydromagnetic boundary layer flow with heat and mass transfer over a vertical plate in the presence of magnetic field and a convective heat exchange at the surface.22 In another article, Makinde and Aziz visualized boundary layer flow of a nano fluid past a stretching sheet with a convective boundary condition.23 Free convection boundary layer flow past a horizontal flat plate embedded in porous medium filled by nanofluid containing gyrotactic microorganisms have been discussed by Aziz et al.24 Makinde et al.25 analyzed the combined effects of buoyancy force, magnetic field, thermophoresis, Brownian motion, and convective heating on stagnation-point flow and heat transfer
N.S. Akbar et al. due to nanofluid flow towards a stretching sheet. A novel analytical solution of the non-uniform convective boundary conditions problem for heat conduction in cylinders has been addressed by Tonini and Cossali.26 The present article gives the axisymmetric stagnation point flow of nano fluid towards a stretching surface with convective boundary condition. Radiation effects are also taken into account. The resulting nonlinear momentum, energy and nano particle equations are simplified using similarity transformations. Numerical solutions have been developed for the velocity, temperature and nanoparticle fraction. The influence of physical parameters on the velocity, temperature, nanoparticle fraction, rates of heat transfer and nanoparticle fraction are shown graphically. The present numerical results are compared with the literature.
2. Mathematical formulation We discussed the steady two-dimensional stagnation point flow of a nano fluid over a stretching cylinder with convective boundary conditions and radiation effects. For the formulation of the problem we have used the cylindrical coordinates r, u, z and assume that the wall is at z = 0, the stagnation point is at the origin and that the flow is in the direction of the negative z-axis. u = u(r, z) and w = w(r, z) are the velocity components along r and z directions, where the velocity along u direction vanishes due to symmetry. A uniform magnetic field B0 is applied normal to the cylinder where the induced magnetic field is neglected by assuming very small magnetic Reynolds number. Moreover, it is assumed that the surface has a temperature of Tw and concentration of nano particles Cw and fluid has uniform ambient temperature T1 and C1 (Tw > T1 and Cw > C1). The velocity distribution in the frictionless flow in the neighborhood of the stagnation point is given by
U ¼ ar W ¼ 2az
where a > 0 is a constant characterizing the velocity of the mainstream flow. The continuity, momentum, heat and nano particle fraction equations for the two-dimensional steady state flows under radiation effects, using the usual boundary layer approximations,7,15 reduce to @u u @w þ þ ¼0 @r r @z
ð1Þ
2 @u @u @p @U @ u 1 @u u @ 2 u ¼ þU q u þw þl þ þ @r @z @r @r @r2 r @r r2 @z2 rB20 ðu UÞ
ð2Þ
2 @w @w @p @ w 1 @w @ 2 w ¼ þl q u þw þ þ @r @z @z @r2 r @r @z2
ð3Þ
2 @T @T @ T 1 @T @ 2 T @C @T @C @T u þw þ þ 2 þ s DB þ ¼a 2 @r @z @r r @r @z @r @r @z @z " 2 #) 2 DT @T @T @qr þ þ ð4Þ @r @z @z T1
Radiation effects on MHD stagnation point flow of nano fluid towards a stretching surface 2 @C @C @ C 1 @C @ 2 C ¼ DB u þw þ þ @r @z @r2 r @r @z2 2 D @ T 1 @T @ 2 T þ þ þ T T1 @r2 r @r @z2
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8 pffiffiffiffiffi u ¼ crf0 ðgÞ; w ¼ 2 cmfðgÞ > > p ffi > > > g ¼ z ct; m ¼ lq > > > > > 1 1 > U ¼ CCC ; h ¼ TTT > > w C1 w T1 > > > Pr ¼ m=a ; Le ¼ DaB > > <
ð5Þ
ðqcÞ
1Þ Nt ¼ TD1T ðqcÞp ðTw T m f > > > > ðqcÞp ðUw U1 Þ > > > Nb ¼ DB ðqcÞf m > > > pffi > > h f m > c ¼ kf > > c > > > : 4r T31 Nr ¼ k k ; s ¼ ac
The corresponding boundary conditions are stated as 8 u ¼ cr; w ¼ 0; C ¼ Cw ; > > < @T ¼ hf ðTf TÞ kf > @z > : u ! ax; T ! T1 ; C ! C1
ð6Þ
z¼0 z!1
And using non-dimensional variables presented in Eq. (9) and Eqs.(7) and (8) in Eqs. (1)-(5) we have
In the above expressions u and w are the velocity components along the r and z axes, respectively, q is the fluid density, v the kinematic viscosity, T the temperature, and C the concentration of fluid; the ambient values of T and C as z tends to infinity are denoted by T1 and C1, cp is the specific heat of the material DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient, a = k/(qc)f is the thermal diffusivity of the fluid, s = (qc)p/(qc)f is the ratio between the effective heat capacity of the nano particle material and heat capacity of the fluid with q being the density, c the volumetric volume expansion coefficient hf the convective heat transfer coefficient, Tf the convective fluid temperature and qp the density of the particle. Using Rosseland approximation of radiation 4r @T4 qr ¼ 3k @z
2
4 2 1 þ Nr h00 þ Prð2fh0 þ NbU0 h0 þ Nbðh0 Þ Þ ¼ 0 3
ð11Þ
0
ð12Þ
8 > < fð0Þ ¼ 0 f0 ð0Þ ¼ 1 > : 0 f ð1Þ ¼ s
where k is the mean absorption coefficient and r is the Stefan–Boltzmann constant and T4 is a linear temperature function which is expanded by using Taylor series expansion about T1 as
ð13aÞ
h0 ð0Þ ¼ Bið1 hð0ÞÞ hð1Þ ¼ 0
ð13bÞ
Uð0Þ ¼ 1
ð13cÞ
Uð1Þ ¼ 0
M, s, Pr, Nb, Nt, Le, Bi and Nr are the Hartmann number, stagnation parameter, Prandtl number, the Brownian motion parameter, the thermophoresis parameter, Lewis number, Biot number and local concentration thermal radiation effects, respectively. Expressions for the local Nusselt number Nu and the local Sherwood number Sh are
ð8Þ
Introducing the following quantities
Table 1
ð10Þ
Nt 00 h ¼0 U þ 2LePrfU þ Nb
*
T4 ¼ 4T31 T 3T41
f000 þ 2ff00 ðf0 Þ þ M2 ðs f0 Þ þ s2 ¼ 0
00
ð7Þ
*
ð9Þ
Comparison of numerical results for local Nusselt number in the absence of nanofluid for various values of M, s and Pr.
(a) When Pr = 0.7, Nr = 0 and Bi = 1 Result
By Attia7
M
0
1
2
0
1
2
s = 0.1 s = 1.0 s = 1.5
0.6454 0.9109 1.0263
0.5974 0.9109 1.0309
0.5112 0.9109 1.0412
0.6454 0.9109 1.0263
0.5974 0.9109 1.0309
0.5112 0.9109 1.0412
Present
(b) When M = 1, Nr = 0 and Bi = 1 By Attia7
Present
Pr !
0.1
0.5
1.0
0.1
0.5
1.0
s = 0.1 s = 1.0 s = 1.5
0.1467 0.2533 0.2989
0.3471 0.5674 0.6498
0.5539 0.8043 0.9057
0.1467 0.2533 0.2989
0.3471 0.5674 0.6498
0.5539 0.8043 0.9057
1392 (
N.S. Akbar et al.
w Nu ¼ kðTrq f T1 Þ
ð14Þ
Sh ¼ DB ðCrqwmC1 Þ
8 h i < q ¼ 16rT3 þ k @T ; w 3k @z z¼0 : q ¼ D @C ; m
where qw and qm are the heat flux and mass flux, respectively.
(a) f ' (η )
(b) θ (η )
ð15Þ
B @z z¼0
Dimensionless form of Eq. (15) takes the form
(a) f ' (η )
(b) θ (η )
(c) Φ (η )
Fig. 1 Effects of M on velocity, temperature and nanoparticle fraction with Nr = Nb = Nt = Bi = s = 0.1, Le = 1, Pr = 0.8.
Fig. 2 Effects of s on velocity, temperature and nanoparticle fraction with Nr = Nb = Nt = Bi = 0.1, Le = M = 1, Pr = 0.8.
Radiation effects on MHD stagnation point flow of nano fluid towards a stretching surface (
Re1=2 Nu ¼ ð1 þ 43 NrÞh0 ð0Þ r Sh Re1=2 r
0
¼ u ð0Þ
( ð16Þ
3. Numerical technique The system of coupled nonlinear coupled differential Eqs. (10)(12) along with the boundary conditions Eq. (13a) to Eq. (13c) are solved numerically using fourth-order Runge–Kutta– Fehlberg method with a shooting technique. The step size and the convergence criteria were taken as Dg = 0.001 and 106 respectively. The asymptotic boundary conditions were replaced by using a value of 20 for the similarity variable g1 as follows
(a) θ (η )
(b) Φ (η )
Fig. 3 Effects of Pr on temperature and nanoparticle fraction with Nr = s = Bi = 0.5, Le = M = 1, Nb = 0.3, Nt = 0.1.
1393
g1 ¼ 20 f0 ð20Þ ¼ hð20Þ ¼ Uð20Þ ¼ 0
The choice of g1 = 20 ensures that all numerical solutions approached the asymptotic values correctly. The present results of dimensionless heat transfer rate for pure fluid are compared with those reported by Attia7 and are shown in Table 1. We notice that the comparison shows good agreement for each value considered. We are, therefore, confident that the present results are correct and accurate. 4. Results and discussion The numerical solutions are presented through graphs (see Figs. 1–11) for the physical interpretation of the proposed
(a) θ (η )
(b) Φ (η )
Fig. 4 Effects of Bi on temperature and nanoparticle fraction with Nr = Nt = 0.5, Le = M = 1, Nb = 0.3, s = 0.1, Pr = 6.2.
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N.S. Akbar et al.
(a) θ (η )
(b) Φ (η )
Fig. 5 Effects of Le on temperature and nanoparticle fraction with Nr = s = Bi = 0.1, Nt = 0.5, M = 1, Nb = 0.3, Pr = 6.2.
study. These figures describes the influence of magnetic field M, Prandtl number Pr, stretching parameter s, Brownian motion parameter Nb, thermophoresis parameter Nt, Lewis number Le and the Biot number Bi on the dimensionless velocity, temperature, nanoparticle fraction, heat transfer rate, skin friction coefficient and nanoparticle fraction rate. Effect of magnetic field on the velocity, temperature and nanoparticle fraction is shown in Fig. 1. It is shown that velocity profile f0 (g) decreases with the increase of M and the boundary layer thickness also decreases. It can also be concluded that the temperature profile h(g) and nanoparticle fraction U(g) increases with an increase in M. Furthermore, with an increase in M, the thermal boundary layer as well as concentration boundary layer decreases. Fig. 2 illustrates the effect of stagnation parameter s on f0 (g), h(g) and U(g). It is found that the velocity profile increases while tempera-
(a) θ (η )
(b) Φ (η )
Fig. 6 Effects of Nb on temperature and nanoparticle fraction with s = 0.1, Nt = 0.4, Nr = Le = M = 1, Bi = 0.5, Pr = 6.2.
ture profile and nanoparticle fraction decrease with the increasing value of s. We notice from Fig. 3 that Prandtl number Pr has the same sort influence on temperature profile as well as on nanoparticle fraction, as Pr has decreasing behavior for both h(g) and U(g). The effects of Bi on temperature profile and nanoparticle fraction have exactly the same increasing behavior (see Fig. 4). As expected, it is found that temperature profile is an increasing function in view of Le. While, decrease in Le leads to increase in nanoparticle fraction (see Fig. 5). Fig. 6 shows the effect of Nb on the h(g) and U(g). It is seen that temperature profile increase with the increasing value of Nb, whereas nanoparticle fraction decreases. Fig. 7 shows that temperature profile and nanoparticle fraction show the same behavior against the value of Nt Temperature profile as well as the nanoparticle fraction in-
Radiation effects on MHD stagnation point flow of nano fluid towards a stretching surface
(a) θ (η )
(b) Φ (η )
Fig. 7 Effects of Nt on temperature and nanoparticle fraction with s = 0.1, Nr = Nb = 0.3, Le = M = 1, Bi = 0.7, Pr = 6.2.
creases with an increase in Nt. Effects of Nr on temperature and nanoparticle fraction are presented in Fig. 8. Here we say that h(g) and U(g) shows the same increasing behavior with an increasing values of Nr. Figs. 9–11, show the influence of governing parameters on dimensionless heat transfer rates, skin friction coefficient and dimensionless nanoparticle fraction rates. In Fig. 9, it depicts that for higher values of M, skin friction coefficient presents the increasing behavior corresponding to the increasing values of s. This shows that fluid motion on the wall of the sheet accelerated, when we strengthen the effects of parameter and stagnation parameter. From Fig. 10(a), it is noticed that local Nusselt number decreases for increasing values of Nb and Nt whereas for Nr and Bi, the heat transfer rate increases (see Fig. 10(b)). Fig. 11 shows the variation in dimensionless nanoparticle fraction rates against Nt and Le, respectively. As Nt increase the concentration rate decreases whereas Pr increase the mass rate increase.
1395
(a) θ (η )
(b) Φ (η )
Fig. 8 Effects of Nr on temperature and nanoparticle fraction with s = Nb = 0.1, Nt = 0.7, Le = M = Pr = 1, Bi = 0.5.
Fig. 9 Effects of M and s on Skin friction coefficient with Nb = 0.3, Nt = 0.5, Nr = Bi = 0.1, Le = 1, Pr = 6.2.
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N.S. Akbar et al.
(a) Nt
(a) Nb
(b) Pr
(b) Nr
Fig. 10 Effects of Nb and Nr on reduced Nusselt number with s = 0.1, Le = 10, Pr = 6.2.
Fig. 11 Effects of Nt and Pr on reduced Sherwood number with s = 0.1, M = 1, Nr = Bi = 0.5.
Table 2 Numerical values of the local Nusselt number and local Sherwood number for various values of Nt and Nb when Pr = 6.2, Le = 10, M = 1 and s = 0.1. Nr = 0 and Bi = 1 Nt #
0.1 0.2 0.3 0.4 0.5
Nr = 0.5 and Bi = 0.5
Nb = 0.1
Nb = 0.3
Nb = 0.1
Nb = 0.3
Nb = 0.1
Nb = 0.3
Rer1=2 Nu
Re1=2 Sh r
Rer1=2 Nu
Re1=2 Nu r
Rer1=2 Sh
Re1=2 Sh r
0.37263 0.28899 0.22993 0.18742 0.15619
8.77962 8.94717 9.07920 9.18033 9.25679
0.59963 0.59070 0.58100 0.57045 0.55898
0.46726 0.45020 0.43196 0.41262 0.39232
8.56609 8.56793 8.58369 8.61380 8.65851
8.63009 8.68759 8.74943 8.81457 8.88166
Finally, Table 2 gives the numerical values of skin friction coefficient, local Nusselt number and the local Sherwood number. With an increase in Pr and Nt, local Nusselt number
and skin friction coefficient decrease while local Sherwood number increases.
Radiation effects on MHD stagnation point flow of nano fluid towards a stretching surface 5. Conclusions Numerical simulation of steady two-dimensional stagnation point flow of a nano fluid is investigated with convective boundary conditions. The behavior of the flow parameters is observed. Summary of the present work is listed as follows. (1) It is seen that velocity profile f0 (g) decreases with the increasing values of M and the boundary layer thickness also decreases. (2) It is also depicted that the temperature profile h(g) and nanoparticle fraction U(g) increase with an increase in M. (3) The effects of Bi on temperature profile h(g) and nanoparticle fraction U(g) have exactly the same behavior. (4) It is found that the velocity profile f0 (g) and boundary layer thickness increase while temperature profile h(g) and nanoparticle fraction U(g) decrease with the increasing value of s. (5) It is found that temperature profile h(g) is increasing function in view of Le. An increase in Le leads to decrease in nanoparticle fraction U(g). (6) It is seen that temperature profile and nanoparticle fraction increase with the increasing value of Nt. (7) Temperature profile increases while nanoparticle fraction decreases with an increase in Nt. (8) It is observed that the influence of Pr for temperature profile and nanoparticle fraction has exactly the same behavior, as Pr has decreasing behavior for h(g) and U(g). (9) The heat transfer rate and skin friction coefficient decrease with the increasing values of Nb, Nt and Pr.
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8. Kumari M, Nath G. Steady mixed convection stagnation-point flow of upper convected Maxwell fluids with magnetic field. Int J Non-Linear Mech 2009;44(10):1048–55. 9. Liu JX, Tang DB, Yang YZ. On nonlinear evolution of C-type instability in nonparallel boundary layers. Chin J Aeronaut 2007;20(4):313–9. 10. Wang Q, Guo ZY, Zhou C, Feng GT, Wang ZQ. Coupled heat transfer simulation of a high-pressure turbine nozzle guide vane. Chin J Aeronaut 2009;22(3):230–6. 11. Guo X, Tang DB. Nonlinear stability of supersonic nonparallel boundary layer flows. Chin J Aeronaut 2010;23(3):283–9. 12. Guo SA, Chen SW, Song YP, Song YF, Chen F. Effects of boundary layer suction on aerodynamic performance in a highload compressor cascade. Chin J Aeronaut 2010;23(2):179–86. 13. Buongiorno J. Convective transport in nanofluids. J Heat Transfer 2010;128(3):240–50. 14. Kakac S, Pramuanjaroenkij A. Review of convective heat transfer enhancement with nanofluids. Int J Heat Mass Transfer 2009;52(13):3187–96. 15. Marga SEB, Palm SJ, Nguyen CT, Roy G, Galanis N. Heat transfer enhancement by using nanofluids in forced convection flows. Int J Heat Fluid Flow 2005;26(4):530–46. 16. Khan WA, Pop I. Boundary-layer flow of a nanofluid past a stretching sheet. Int J Heat Mass Transfer 2010;53(11):2477–83. 17. Kuznetsov AV, Nield DA. Natural convective boundary-layer flow of a nanofluid past a vertical plate. Int J Therm Sci 2010;49(2):243–7. 18. Vajravelu K, Prasad KV, Lee J, Lee C, Pop I, Gorder RAV. Convective heat transfer in the flow of viscous Ag-water and Cu– water nanofluids over a stretching surface. Int J Therm Sci 2011;50(5):843–51. 19. Bachok N, Ishak A, Pop L. The boundary layers of an unsteady stagnation-point flow in a nanofluid. Int J Heat Mass Transfer 2012;55(22–23):6499–505. 20. Aziz A. A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition. Commun Non-Linear Sci Numer Simul 2009;14(4):1064–8. 21. Makinde OD. Similarity solution of hydromagnetic heat and mass transfer over a vertical plate with a convective surface boundary condition. Int J Phys Sci 2010;5(6):700–10. 22. Makinde OD, Aziz A. MHD mixed convection from a vertical plate embedded in a porous medium with a convective boundary condition. Int J Therm Sci 2010;49(9):1813–20. 23. Makinde OD, Aziz A. Boundary layer flow of a nano fluid past a stretching sheet with a convective boundary condition. Int J Therm Sci 2011;53(11):2477–83. 24. Aziz A, Khan WA, Pop I. Free convection boundary layer flow past a horizontal flat plate embedded in porous medium filled by nanofluid containing gyrotactic microorganisms. Int J Therm Sci 2012;56:48–57. 25. Makinde OD, Khan WA, Khan ZH. Buoyancy effects on MHD stagnation point flow and heat transfer of a nanofluid past a convectively heated stretching/shrinking sheet. Int J Heat Mass Transfer 2013;62:526–33. 26. Tonini S, Cossali GE. A novel analytical solution of the nonuniform convective boundary conditions problem for heat conduction in cylinders. Int Commun Heat Mass Transfer 2012;39(8): 1059–65.