VOL. 5, NO. 5, May 2015 ISSN ARPN Journal of Science and Technology All rights reserved.

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1 ARPN Journal o Science and Technology All rights reserved. Impact o Heat Transer on MHD Boundary Layer o Copper Nanoluid at a Stagnation Point Flo Past a Porous Stretching and Shrinking Surace ith Variable Stream Conditions 1 E N. Ashin Kumar, Norasikin Binti Mat Isa, 3 R. Kandasamy 1, Faculty o Mechanical Engineering, 3 Faculty o Science, Technology and Human Development, Research Centre or Computational Mathematics, University Tun Hussein Onn Malaysia, Johor, Malaysia uture990@gmail.com ABSTRACT In this paper, the eects o MHD boundary layer on a stagnation point lo and heat transer over a porous stretching/shrinking surace in nanoluids are analyzed. The resulting system is then solved numerically by Maple 18 sotare. It is concluded that the magnetic ield can be used as a means o controlling the heat transer characteristics in the presence o copper nanoluid. Keyords: Copper nanoluid, porous medium, Stagnation-point lo, Thermal radiation, Magnetic ield, Heat source /sink. NOMENCLATURE speciic heat at constant pressure C p E c K K N Pr q r Eckert number permeability o the porous medium Roseland mean spectral absorption coeicient thermal radiation parameter Prandtl number thermal radiative heat lux Q 0 dimensional heat generation/absorption coeicient Re x local Reynolds number suction/injection parameter S T temperature o the luid T ree stream temperature T temperature at the all u velocity component in x-direction u stretching/shrinking sheet velocity U ree stream velocity o the nanoluid V velocity component in y-direction x, y direction along and perpendicular to the plate, respectively Greek symbols n eective thermal diusivity o the nanoluid luid thermal diusivity n solid volume raction o the nanoparticles similarity variable heat generation/absorption parameter eective dynamic viscosity o the nanoluid dynamic viscosity o the luid kinematic viscosity o the luid n eective density o the nanoluid Stean Boltzmann constant dimensionless temperature o the luid all temperature excess ratio parameter stream unction eective thermal conductivity o the nanoluid n thermal conductivity o the luid 1. INTRODUCTION An innovative ay to increase conductivity coeicient o the luid is to suspend solid nanoparticles in it and make a mixture called nanoluid, having larger thermal conductivity coeicient than that o the base luid. This higher thermal conductivity enhances the rate o heat transer in industrial applications. Many researchers have investigated dierent aspects o nanoluids and have paid much attention to viscous luid motion near the stagnation region o a solid body, here body corresponds to either ixed or moving suraces in a luid. This multidisciplinary lo has requent applications in high speed los, thrust bearings, and thermal oil recovery. All studies mentioned above reer to the stagnation-point lo toards a stretching/shrinking sheet in a viscous and Netonian luid. Most conventional heat transer luids, such as ater, ethylene glycol, and engine oil, have limited capabilities in terms o thermal properties, hich, in turn, may impose serve restrictions in many thermal applications. On the other hand, most solids, in particular, metals, have thermal conductivities much higher, say by one to three orders o magnitude, compared ith that o liquids. Stagnation point lo is continuing to be an interesting area o research among scientists and investigators due to its importance in a ide variety o applications both in industrial and scientiic applications. Some o the 19

2 ARPN Journal o Science and Technology All rights reserved. applications are cooling o electronic devices by ans, cooling o nuclear reactors during emergency shutdon, solar central receivers exposed to ind currents, and many hydrodynamic processes in engineering applications. Magneto hydrodynamic (MHD) boundary layer lo is o considerable interest in the technical ield due to its requent occurrence in industrial technology and geothermal application, high-temperature plasmas applicable to nuclear usion energy conversion, liquid metal luids and (MHD) poer generation systems. The eect o thermal radiation on lo and heat transer processes is o major importance in the design o many advanced energy conversion systems operating at high temperature. Thermal radiations ithin such systems occur because o the emission by the hot alls and orking luid. The process o using o metals in an electrical urnace by applying a magnetic ield and the process o cooling o the irst all inside a nuclear reactor containment vessel here the hot plasma is isolated rom the all by applying a magnetic ield are some examples o such ields. The main subject o the present study is to study the to-dimensional Magneto hydrodynamic (MHD) boundary layer o stagnation-point lo in a nanoluid in the presence o thermal radiation. Using a similarity transorm the Navier-Stokes equations have been reduced to a set o nonlinear ordinary dierential equations. The resulting nonlinear system has been solved numerically using the Maple 18 sotare technique. Finally, the results are reported or to dierent types o nanoparticles namely alumina and copper ith ater as the base luid. Nanoluids could be used in major process industries, including materials and chemicals, ood and drink, oil and gas, paper and printing, and textiles. Nanoluids, hen used as coolants can provide dramatic improvements in the thermal conductivity o host luids compared to that o the traditional luids. By using nanoluids highest possible thermal properties at the smallest possible concentrations by uniorm dispersion and stable suspension o nanoparticles in base luids can be achieved. Xuan and Li [1] used pure copper particles in the study o convective heat transer and lo eatures o nanoluids. A comprehensive survey o convective transport in nanoluids as made by Buongiorno [], ho gave an explanation or the abnormal increase o the thermal conductivity o nanoluids. Thereore, by mixing the nanoparticles in the luid, thermal conductivity o the luids improve the heat transer capability. Pantzali et al. [3] presented an experimental study that shoed the role o CuO ater nanoluids as coolant in the plate heat exchangers. Recently, an analysis has been carried out by Vajravelu et al. [4] to study the convective heat transer in a nanoluid lo over a stretching surace. In particular, they have ocused on Ag ater and Cu ater nanoluids to investigate the eects o the nano particle volume raction on the lo and heat transer characteristics in the presence o thermal buoyancy and temperature-dependent internal heat generation/absorption. The thermal radiation eects become intensiied at high absolute temperature levels due to basic dierence beteen radiation and the convection and conduction energy-exchange mechanisms. A comprehensive revie o the literature about stagnation point lo o nanoluids is given by reerences rom [5 14]. Despite several orks have been reported on lo and heat transer o nanoluids, there seems to be no attempts in literature to consider the combined eects o buoyancy orce and convective heating on hydro magnetic stagnation point lo and heat transer o nanoluid toards a stretching / shrinking surace.. MATHEMATICAL ANALYSIS Consider the steady to-dimensional stagnationpoint laminar lo o a viscous nanoluid past a stretching / shrinking plate ith linear velocity u ( x) c x (or stretching sheet) / u ( x) c x (or shrinking sheet) and the velocity o ree stream lo U ( x) a x in the presence o magnetic ield and the thermal radiation, here a and c are constants. The uniorm magnetic ield o strength B 0 is applied in the positive direction o x- axis. Fig 1: Physical lo model over a porous edge sheet The ambient uniorm temperature o nanoluid is T here the body surace is kept at a constant temperature (Fig. 1). It is assumed that the base luid T and nanoparticles are in thermal equilibrium and no slip occurs beteen them. Under these assumptions, the governing equations or the continuity, momentum and energy in boundary layer lo can be ritten as u v 0 x y (1) 0

3 ARPN Journal o Science and Technology All rights reserved. u u u v U x y du dx n n u B n y K n n 0 ( U u) g( T T ) () T T u v x y T Q0 ( T T ) 1 q r n u ( ) ( ) ( ) (3) n y c p c y c y n p n p n ith boundary conditions (or stretching / shrinking sheet) x u u( x), v v0, T T T b l at y = 0; u U ( x) ax, v 0, T T as y (4) here + stands or stretching sheet and or shrinking sheet in (4) or momentum ield, u and v are the velocity components along the x and y directions, respectively. U ( x ) stands or the stagnation-point velocity in the in viscid ree stream. T- temperature o the nanoluid, K- permeability o the porous media, Q 0 - heat generation or absorption coeicient, a, b and c are positive constants and v 0 - all mass lux ith v 0 0 or suctions and v 0 0 or injection, respectively. Further ρ is the luid density, n is the coeicient o viscosity o the nanoluid, k n is the thermal conductivity o the nanoluid, n is the thermal diusivity o the nanoluid, n is the eective density o the nanoluid, ( c ) is the heat capacitance o the nanoluid, hich is deined as ollos: p n k n k ( ks k ( ks k ) ( k ) ( k ks ) ks ) here is the solid volume raction o the nanoluid, is the reerence density o the luid raction, s is the reerence density o the solid raction, is the viscosity o the luid raction, k is the thermal conductivity o the luid, and k s is the thermal conductivity o the solid raction. We no look or a similarity solution o the Eqs. (1 )- (3) ith boundary conditions (4) in the olloing orm: c (5) T T c ( ) x, ( ), y (6) T T here is the kinematic viscosity o the luid. The radiation heat lux is given as q r 4 3K 4 T y here is Stean Boltzmann constant and K is the Roseland mean spectral absorption coeicient. n ( 1 ), s 4 T ( 1 ) 4 4 T 1 (7) n, ( ).5 n (1 )( ) ( ) s ( 1 ) p n n k n ( c ),, p n ( c ) (1 )( c ) ( c ), p p s T here is the all temperature excess ratio T parameter. Deining the stream unction ψ in the usual ay such that u,, v hich identically y x satisies the Eq. (1) and the equations (1)-(3) take the non-dimensional orm.5 a a M K Gr 0 1 c s (8) c 1

4 ARPN Journal o Science and Technology All rights reserved. 1 Pr K K n N 1 ( 1) 3 ( c ) p 1 ( c p ) ith boundary conditions (or stretching / shrinking sheet) S, 1, 1 at 0 ( or stretching sheet) S, 1, 1 at 0 ( or shrinking sheet) at y = 0; a, 0, (10) c s Ec 1 Prandtl number, (9) 16 T N be the radiation parameter 3 k K Gr and is the buoyancy parameter, here Re g ( T T Gr ) u be the Grash o number, x Re be c u the Reynolds number. Here prime denotes the dierentiation ith respect to η.for practical purposes, the unctions ( ) and ( ) allo us to determine the skin riction coeicient o here Q ( c ) is the heat source ( λ 0) or p sink ( λ < 0 ) parameter, K1 is the porous c K vo parameter, S is the mass lux parameter c (s 0 corresponds to the suction and s < 0 corresponds to B0 injection), M is the magnetic parameter, c u Ec K n T is the Eckert number, Pr is the v C u n u y at y0 1 (1 ).5 1 Re x (0) (11) and the Nusselt number Nu x k xk n ( T T T ) y at y0 1 k 4 3 n 3 Re x (0) 1 N( C (0)) k T (1) respectively. Here, number. Re x u x is the local Reynolds 3. RESULTS AND DISCUSSION The set o equations (8) and (9) is highly nonlinear. Hence coupled and cannot be solved analytically and numerical solutions subject to the boundary conditions (10) are obtained using the very robust computer algebra sotare Maple 18. This sotare uses a ourth-ith order Runge Kutta Fehlberg method as deault to solve boundary value problems numerically using the dsolve command. For the beneit o the readers the Maple orksheet is listed in Appendix A. The transormed system o coupled nonlinear ordinary dierential Equations (8) and (9) including boundary conditions (10) depend on the various parameters. The numerical results are represented in the orm o the dimensionless velocity and temperature. During computation e choose parameter such that Pr = 6., correspond physically to air (nanoluid). In Table, the present results o (0) our compared ith those o Mahapatra& Gupta [17] and Hamad & Pop [16]. The results sho a very good agreement ith their results as seen rom the calculated percentage o errors since the errors are ound to be very less. This may be due to the act that e have used Runge Kutta Fehlberg method hich has ith-order accuracy. Thus the present results are more accurate than their results.

5 ARPN Journal o Science and Technology All rights reserved. Table 1: Thermo physical properties o luid and nanoparticles (Oztop and Abu-nada [15]) Physical properties Fluid phase (ater) Cu A l O 3 TiO C p (J/kg K ρ (kg/m 3 ) κ (W/m K) Table : Comparison o result or - θ (0) in isothermal case hen λ = ϕ =K 1 = N r = E c = b = 0 or stretching sheet Pr a / c Mahapatra and Gupta [17] Hamad and Pop [16] Present results Fig (a): Dulal Pal et al. [18] Fig (b): Present Result Fig (c): Dulal Pal et al. [18] Fig (d): present result Figs (a)-(d): Nano particle volume raction on the velocity and temperature ield-comparison ith Dulal Pal et al. [18] 3

6 ARPN Journal o Science and Technology All rights reserved. It is also observed rom the Figs. (a) (d) that the agreement ith the theoretical solution o velocity proile Fig.(b) and temperature proile Fig. (d) or dierent values o is excellent compared ith Fig.(a) and Fig.(c) o Dulal Pal et al. [18] respectively. Fig 3(a) Fig 3(b) Fig 3(c) Fig 3(d) Figs 3(a)-3(d): present typical proiles or velocity and temperature or dierent values o magnetic strength In the presence o pure ater(figs. 3(a) and 3(b)),it is clearly shon that the velocity o the luid decreases or stretching sheet and increases or shrinking sheet hereas the temperature o the luid increases or stretching sheet and decreases or shrinking sheet ith increase o magnetic strength, hich implies that the applied magnetic ield tends to heat the luid and enhances the heat transer rom the all or stretching sheet hile the opposite reaction are shon or shrinking sheet. In the presence o copper nanoluid (Figs. 3(c) and 3(d)), the velocity o the luid is uniorm or stretching sheet and increases or shrinking sheet hereas the temperature o the luid is uniorm or stretching sheet and decreases or shrinking sheet ith increase o magnetic strength. As it move aay rom the plate, the eect o M becomes less pronounced. The eects o a transverse 4

7 ARPN Journal o Science and Technology All rights reserved. magnetic ield to an electrically conducting luid gives rise to a resistive-type orce called the Lorentz orce. The Lorentz orce clearly indicates that the transverse magnetic ield opposes the transport phenomena and it has the tendency to slo don the motion o the luid and to accelerate its temperature proiles or stretching sheet and the opposite eects are observed or shrinking sheet in the presence o pure ater. Due to the copper nanoluid, it is interesting to note that velocity and temperature o the copper nanoluid or stretching sheet are uniorm hile the velocity increases and temperature o the nanoluid decreases or shrinking sheet. It implies that the strength o the magnetic ield on copper nanoluid play a dominant role on shrinking sheet as ell as on stretching sheet. In all cases, the temperature vanishes at some large distance rom the surace o the vertical plate. This result qualitatively agrees ith the expectations, since magnetic ield exerts retarding orce on the natural convection lo. Physically, it is interesting to note that the changes on velocity and temperature o the copper nanoluid (Cu-Water) and base luid signiicantly or shrinking sheet as compared to that o the stretching sheet because o the thermal conductivity o the nanoluids. Copper nano particle is a unique material that has both the liquid and magnetic properties. Many o the physical properties o these luids can be tuned by varying magnetic ield. These results clearly demonstrate that the magnetic ield can be used as a means o controlling the lo and heat transer characteristics. Fig 4(a) Fig 4(b) Figs 4(a)-4(b): present the temperature proiles or dierent values o nano particle volume raction Figs. 4(a) and 4(b) predict the characteristic temperature proiles or dierent values o the nano particle volume raction or stretching and shrinking sheet in the presence o base luid (pure ater) and nanoluid (Cu-ater). In both the cases (pure ater and copper nanoluid), it is note that the temperature o the luid accelerates or stretching surace ith increase o and there is a signiicance changes beteen them. For shrinking sheet, it is interesting to note that the temperature o the luid decreases ithin the thermal boundary layer region hile outside , the temperature proiles gradually increases ith increase o or pure ater hereas the temperature o the luid decreases ithin the thermal boundary layer region hile outside , the temperature proiles gradually increases ith increase o or copper nanoluid because o high thermal conductivity o the copper nanoluid compared to that o base luid. Further, it is noticed that values o the temperature proiles o and tends to zero hen 1. 1 and or stretching and shrinking sheets, respectively or pure ater hereas the temperature o coincide and tends to zero hen and or stretching and shrinking sheets, respectively or Cu-ater. It is important to note that the values o temperature proiles or shrinking sheet is more pronounced than the temperature proiles or the stretching sheet or pure and copper nanoluid. This is due to the act that thermal boundary layer thickness changes ith change in nano particle volume raction strength present in the Cuater. Thus it is an evident that the ormation o peak in the temperature proile or shrinking sheet is due to the act the thermal boundary layer increases in this case hereas no peak ormation is observed in the case o stretching sheet as thermal boundary layer is less than the 5

8 ARPN Journal o Science and Technology All rights reserved. shrinking sheet or all the values o nano particle volume raction. It is important to note that the increase o or shrinking sheet plays a dominant role on temperature proiles compared to that o stretching sheet. This is in agreement ith the physical act that the thermal boundary layer thickness decreases ith increasing nano particle volume raction,. Fig 5(a) Fig 5(b) Figs 5(a)-5(b): Present the temperature proiles or dierent values o thermal radiation Figs. 5(a) and 5(b) display the eects o thermal radiation on the temperature distribution ith or ithout magnetic ield. In the presence o magnetic and nonmagnetic ield, it is note that the temperature o the luid enhances ith increase o N or stretching sheet hereas it decreases ith increase o N or shrinking sheet. The sensitivity o thermal boundary layer thickness ith N is related to the increased thermal conductivity o the nanoluid. Both the cases (ith or ithout magnetic ield), it is observed that the temperature o the nanoluid decreases ith increase o N or shrinking sheet. In act, higher values o thermal conductivity are accompanied by higher values o thermal diusivity. It is important to note that the values o temperature proiles or shrinking sheet is more pronounced than the temperature proiles values or the stretching sheet. This is due to the act that thermal boundary layer thickness changes ith change in the type o nanoparticles present in the base luid (ater). The high value o thermal diusivity causes a drop in the temperature gradients and accordingly increases the boundary thickness as demonstrated in Fig. 5(b). This agrees ith the physical behavior that hen the volume raction o copper nanoluids in the presence o magnetic ield raises the thermal conductivity and then the thermal boundary layer thickness increases. Enhancement in thermal conductivity can lead to eiciency improvements, although small, via more eective luid heat transer. In general nanoluids sho many excellent properties promising or heat transer applications. Despite many interesting phenomena described and understood there are still several important issues that need to be solved or practical application o nanoluids ith magnetic ield. For stretching sheet, temperature proile gradually decreases ith η hereas or shrinking sheet there is a ormation o a peak near η = 0. 5 hich gradually increases linearly thereater till the value o temperature goes to zero as η 1.5 or stretching sheet and η 1.75 or shrinking sheet (matching the boundary condition θ 0 as η ). Thus it is an evident that the ormation o peak in the temperature proile or shrinking sheet is due to the act the thermal boundary layer increases in this case hereas no peak ormation is observed in the case o stretching sheet as thermal boundary layer is less than the shrinking sheet or all the values o N. It is important to note that the temperature proile or shrinking sheet is higher compared to that o stretching sheet. 6

9 ARPN Journal o Science and Technology All rights reserved. S S S S S S Fig 6(a) Fig 6(b) Figs 6(a)-6(b): present the temperature proiles or dierent values o suction parameter Figs. 6(a) 6(b) depict the inluence o the suction S on the temperature proiles in the boundary layer hen the magnetic strength is uniorm, i.e. M 0.5, in the case o pure and Cu-ater or stretching and shrinking sheet respectively. It is observed that the temperature in the boundary layer decreases ith the increase o suction parameter (S 0). The explanation or such behavior is that the luid is brought closer to the surace and reduces the thermal boundary layer thickness in case o suction. As such then the presence o all suction decreases the thermal boundary layer thickness, i.e. thins out the thermal boundary layer in the nanoluid Cu-ater as ell as pure ater. It is interesting to note that the temperature o the luid gradually changes rom higher value to loer value only hen the strength o suction parameter higher than the magnetic strength in the presence o Cu-ater or shrinking sheet. For heat transer characteristics mechanism or shrinking sheet, interesting result is the large distortion o the temperature ield caused or S 3 because o high thermal conductivity o the copper nanoluid. Negative value o the temperature proile or shrinking sheet is seen in the outer boundary region or S 3. The decrease in thickness o the temperature layer as caused in to ays: (i) the direct action o suction, and (ii) the indirect action o suction causing a thicker thermal boundary layer hich corresponded to a loer temperature gradient, a consequent increase in the buoyancy orce and a higher temperature gradient. Hoever, the exact opposite behavior is produced by imposition o all luid bloing or injection. It is note that, or particular value o M and at each position, the corresponding value o the temperature in the presence o nanoparticles is smaller than the value o the temperature or basic luid. All these physical behavior are due to the combined eects o the strength o magnetic eect in the presence volume raction o the nanoparticles. 7

10 ARPN Journal o Science and Technology All rights reserved. Fig 7(a) Fig 7(b) Figs 7(a)-7(b): present the temperature proiles or dierent values o heat source parameter Eect o heat source ( 0 ) on temperature distribution or stretching and shrinking sheet in the presence o base luid (pure ater) and nanoluid (Cu ater) are shon in Figs. 7(a) and 7(b). Both stretching and shrinking sheet, it is observed that the temperature o the copper nanoluid increases ith increase o heat source parameter. It is interesting to note that the heat source generates energy hich causes the temperature o the luid to increase in the Cu-ater boundary layer or stretching and shrinking sheet. For pure ater, it is predict that the temperature o the luid initially increases and then decreases ith increase o heat source or stretching sheet hereas the opposite eects are shon or shrinking sheet hen but beyond this value o ( ) the temperature increases ith increase o heat source. Hence, the source plays an important role on controlling the thermal boundary layers quite eectively. It is important to note that the temperature proiles or stretching sheet is loer than shrinking sheet ith no peak ormation or stretching sheet hereas the inluence o internal heat generation on temperature distribution is more pronounced on the copper nanoluid than that o the base luid or shrinking sheet. Increasing the heat source parameter 0 has the tendency to increase the thermal state o the copper nanoluid. On the other hand, the presence o heat sink in the boundary layer absorbs energy hich causes the temperature o the luid to decrease. All these physical behavior are due to the combined eects o the strength o convective radiation and the size and shape o the nanoparticles in the nanoluid. 4. CONCLUSION In this investigation, the eect o MHD laminar lo and heat transer o copper nanoluid on a stagnation point lo and heat transer over a porous stretching / shrinking sheet ith variable stream conditions are analyzed. The governing partial dierential equations are transormed into a set o nonlinear ordinary dierential equations using similarity transormation and these equations are solved numerically using Maple 18 sotare. Folloing conclusions are dran rom the present study: In the presence o copper nanoluid, the temperature o the luid is uniorm or stretching sheet and decreases or shrinking sheet ith increase o magnetic strength. These results clearly demonstrate that the magnetic ield can be used as a means o controlling the lo and heat transer characteristics. It is important to note that the temperature proile or shrinking sheet is higher to stretching sheet or copper nanoluids because the volume raction o copper nanoluids in the presence o magnetic ield raises the thermal conductivity and then the thermal boundary layer thickness increases. It is interesting to note that the increasing o thermal radiation on shrinking sheet plays a dominant role on temperature proiles compared to that o stretching sheet. This is because o improved thermal conductivity or speciic volume concentration o copper nanoparticles. 8

11 ARPN Journal o Science and Technology All rights reserved. The temperature o the luid gradually changes rom higher value to loer value only hen the strength o suction parameter higher than the magnetic strength in the presence o Cu-ater or shrinking sheet because the copper nanoluid has high thermal conductivity hereas the temperature proiles or stretching sheet is loer than shrinking sheet and there is no peak ormation or stretching sheet. Inluence o internal heat generation on temperature distribution is more pronounced on the nanoluid than that o the base luid or shrinking sheet. Increasing the heat source parameter 0 has the tendency to increase the thermal state o the copper nanoluid. It is important to note that the increase o nano particle volume raction or shrinking sheet plays a dominant role on temperature ield compared to that o stretching sheet. Hence the luid containing solid particles may signiicantly increase its conductivity. The lo over a continuously stretching / shrinking surace is an important problem in many engineering processes ith applications in industries such as the hot rolling, ire draing, paper production, glass bloing, plastic ilms draing and glass-iber production. ACKNOWLEDGEMENT The ork as partly supported by University Tun Hussein Onn Malaysia, Johor, Malaysia, under the Fundamental Research Grant Scheme No. 108/013. REFERENCES [1] Y. Xuan, Q. Li,Investigation on convective heat transer and lo eatures o nanoluids,j. Heat Transer, 15 (003), pp [] J. Buongiorno, Convective transport in nanoluids, J. Heat Transer, 18 (006), pp convective boundary condition, Chin. J. Aeronaut., 6 (6) (013), pp [6] S. Nadeem, R. Ul Haq, Z.H. Khan, Heat transer analysis o ater-based nanoluid over an exponentially stretching sheet, Alexandria Eng. J., 53 (014), pp [7] [X. Wang, A.S. Mujumdar, A revie on nanoluids. Part I: Theoretical and numerical investigation, Braz. J. Chem. Eng. 5 (04) (008) [8] A.V. Kuznetsov, D.A. Nield, Natural convective boundary layer lo o a nanoluid past a vertical plate, Int. J. Therm. Sci. 49 (010) [9] J.A. Eastman, S.U.S. Choi, S. Li, W. Yu, L.J. Thompson, Anomalously increased eective thermal conductivity o ethylene glycol based nanoluids containing copper nanoparticles, Appl. Phys. Lett. 78 (6) (001) [10] M. Mostaa, T. Hayat, I. Pop, S. Asghar, S. Obaidat, Stagnation point lo o a nanoluid toards a stretching sheet, Int. J. Heat Mass Transer 54 (011) [11] W.A. Khan, I. Pop, Boundary-layer lo o a nanoluid past a stretching sheet, Int. J. Heat Mass Transer 53 (010) [1] W.A. Khan, A. Aziz, Natural convection lo o a nanoluid over a vertical plate ith uniorm surace heat lux, Int. J. Therm. Sci. 50 (011) [13] R. Nazar, M. Jaradat, M. Ariin, I. Pop, Stagnationpoint lo past a shrinking sheet in a nanoluid, Cent. Eur. J. Phys. 9 (5) (011) [14] W. Ibrahim, B. Shankar, M.M. Nandeppanavar, MHD stagnation point lo and heat transer due to nanoluid toards a stretching sheet, Int. J. Heat Mass Transer 56 (013) 1 9 [3] M.N. Pantzali, A.A. Mouza, S.V. Paras, Investigating the eicacy o nanoluids as coolants in plate heat exchangers (PHE), Chem. Eng. Sci., 64 (009), pp [4] W. Duangthongsuk, S. Wongises, Heat transer enhancement and pressure drop characteristics o T i O ater nanoluid in a double-tube counter lo heat exchanger, Int. J. Heat Mass Transer, 5 (009), pp [5] N.S. Akbar, S. Nadeem, R. Ul Haq, Z.H. Khan, Radiation eects on MHD stagnation point lo o nano luid toards a stretching surace ith [15] Oztrop and Abu-nada, Numerical study o natural convection in partially heated rectangular enclosures illed ith nanoluids, Int. J. Heat Fluid Flo, 9 (008) [16] M.A.A. Hamad, I. Pop, Scaling transormations or boundary layer lo near the stagnation-point on a heated permeable stretching surace in a porous medium saturated ith a nanoluid and heat generation/absorption eects, Trans. Porous Medium 87 (010)

12 ARPN Journal o Science and Technology All rights reserved. [17] T.R. Mahapatra, A.S. Gupta, Heat transer in stagnation-point lo toards a stretching sheet, Heat Mass Transer 38 (00) [18] Dulal Pal, Gopinath Mandal, K. Vajravelu, Flo and heat transer o nanoluids at a stagnation point APPENDIX A lo over a stretching/shrinking surace in a porous medium ith thermal radiation, Applied Mathematics and Computation, 38(014)

13 ARPN Journal o Science and Technology All rights reserved. shoot: Step # 1 shoot: Parameter values : alpha = beta = shoot: Step # shoot: Parameter values : alpha = HFloat( ) beta = HFloat( ) shoot: Step # 3 shoot: Parameter values : alpha = HFloat( ) beta = HFloat( ) shoot: Step # 4 shoot: Parameter values : alpha = HFloat( ) beta = HFloat( ) shoot: Step # 5 shoot: Parameter values : alpha = HFloat( ) beta = HFloat( ) shoot: Step # 6 shoot: Parameter values : alpha = HFloat( ) beta = HFloat( ) 31