Radiative Effects on BoundaryLayer Flow of a Nanofluid on a Continuously Moving or Fixed Permeable Surface


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1 176 Recent Patents on Mechanical Engineering 01, 5, Radiative Effects on BoundaryLayer Flo of a Nanofluid on a Continuously Moving or Fixed Permeable Surface Ali J. Chamkha a, Mohamed Modather b,c, Saber M.M. ELKabeir b,c and Ahmed M. Rashad*,b a Manufacturing Engineering Department, The Public Authority for Applied, Education and Training, Shueikh 70654, Kuait; b Department of Mathematics, Asan University, Faculty of science, Egypt; c Department of Mathematics, Salman Bin Abdul Aziz University, College of Science and Humanity Studies, AlKharj, KSA Received: May 16, 01; Accepted: July 16, 01; Revised: July 0, 01 Abstract: In recent years, significant patents have been devoted to developing nanotechnology in industry since materials ith sizes of nanometers possess unique physical and chemical properties. Due to recent advances in nanotechnology, a ne class of heat transfer fluids called nanofluids as discovered. A nanofluid is a mixture produced by dispersing metallic nanoparticles in lo thermal conductivity liquids such as ater, oils, lubricants and others to enhance the heat transfer performance. Many patents have been devoted to the methods of preparation of ne nanofluids heat transfer. Nanofluids are no being developed for medical applications, including cancer therapy and safer surgery by cooling. In this paper, a study of a boundary layer flo and heat transfer characteristics of an incompressible nanofluid floing over a permeable uniform heat flux surface moving continuously in the presence of radiation is reported. The Rosseland diffusion approximation is used to describe the radiative heat flux in the energy equation. The resulting system of nonlinear ordinary differential equations are solved numerically using a finite difference method. Numerical results are obtained for the velocity, temperature and nanoparticles volume fraction profiles, as ell as the friction factor, local Nusselt number and local Sherood number for several values of the parameters, namely the velocity ratio parameter, suction/injection parameter, Bronian motion parameter, thermophoresis parameter and radiation parameter. It is found that an increase in either of the Bronian motion parameter or the thermophoresis parameter leads to reductions in the local Nusselt number and the local Sherood number. In addition, increasing either of the suction/injection parameter or the radiation parameter causes enhancements in both the local Nusselt number and the local Sherood number. Keyords: Moving surface, nanofluid, radiation effect, suction/injection. 1. INTRODUCTION The study of convective heat transfer of nanofluids is gaining a lot of attention. Nanofluids have many applications in the industries since materials of nanometer size have unique physical and chemical properties. Nanofluids are solidliquid composite materials consisting of solid nanoparticles or nanofibers ith sizes typically of 1100nm suspended in liquid. Recently, nanofluids have attracted a great interest because of reports of significant enhancement in their thermal properties. For example, a small amount (< 1% volume fraction of Cu nanoparticles or carbon nanotubes dispersed in ethylene glycol or oil is reported to increase the inherently poor thermal conductivity of the liquid by 40% and 150%, respectively [1, ]. Conventional particleliquid suspensions require high concentration (>10% of particles to achieve such enhancement. Hoever, problems of rheology and stability are amplified at high concentration, precluding the idespread use of conventional slurries as heat transfer fluids. In some cases, the observed enhancement in thermal conductivity of nanofluids is orders of magnitude larger than predicted by ellestablished theories. Other perplexing results in this rapidly evolving field include a *Address correspondence to this author at the Department of Mathematics, Asan University, Faculty of Science, Egypt; Tel: ; Fax: ; surprisingly strong temperature dependence of the thermal conductivity [3] and a threefold higher critical heat flux compared ith the base fluids [4, 5]. These enhanced thermal properties are not merely of academic interest. If confirmed and found consistent, they ould make nanofluids promising for application in thermal management. Furthermore, suspensions of metal nanoparticles are also being developed for other purposes, such as medical applications including cancer therapy. The interdisciplinary nature of nanofluid research presents a great opportunity for exploration and discovery at the frontiers of nanotechnology. Kuznetsov and Nield [6] have also studied the classical problem of free convection boundary layer flo of a viscous and incompressible fluid (Netonian fluid past a vertical flat plate to the case of nanofluids. Khan and Pop [7] analyzed the development of the steady boundary layer flo, heat transfer and nanoparticle fraction over a stretching surface in a nanofluid. Bachok et al. [8] have investigated the steady boundarylayer flo of a nanofluid past a moving semiinfinite flat plate in a uniform free Stream. Syakila and Pop [9] studied the steady mixed convection boundary layer flo past a vertical flat plate embedded in a porous medium filled ith nanofluids using different types of nanoparticles. Application of nanofluids for heat transfer enhancement of separated flos encountered in a backard facing step X/1 $ Bentham Science Publishers
2 BoundaryLayer Flo of Nanofluid on Continuously Moving Permeable Surface Recent Patents on Mechanical Engineering 01, Vol. 5, No presented by AbuNada [10]. Duangthongsuk and Wongises [11] analyzed the effect of thermophysical properties models on the predicting of the convective heat transfer coefficient for lo concentration nanofluid. A similarity analysis for the problem of a steady boundarylayer flo of a nanofluid on an isothermal stretching circular cylindrical surface is presented by Gorla et al. [1]. Gorla et al. [13] have also studied mixed convective boundary layer flo over a vertical edge embedded in a porous medium saturated ith a nanofluid. Noghrehabadi et al. [14] studied effect of partial slip boundary condition on the flo and heat transfer of nanofluids past a stretching sheet ith prescribed constant all temperature. The characteristics of flo and heat transfer of a viscous and incompressible fluid over flexed or continuously moving flat surfaces in a moving or a quiescent fluid are ell understood. These flos occur in many manufacturing processes in modern industry, such as hot rolling, hot extrusion, ire draing and continuous casting. For example, in many metallurgical processes such as draing of continuous filaments through quiescent fluids and annealing and tinning of copper ires, the properties of the end product depends greatly on the rare of cooling involved in these processes. Sakiadis [15] as the first one to analyze the boundary layer flo on continuous surfaces. Crane [16] obtained an exact solution the boundary layer flo of Netonian fluid caused by the stretching of an elastic sheet moving in its on plane linearly. ElKabeir [17] studied the interaction of forced convection and thermal radiation during the flo of a surface moving continuously in a floing stream of micropolar fluid ith variable viscosity. ElKabeir et al. [18] discussed a similarity analysis for the problem of the steady boundarylayer flo of a micropolar fluid on a continuous moving permeable surface ith uniform heat flux. One of the first related patents regarding the development of nanofluids as that by Choi and Eastman [19] ho invented an apparatus and method for enhancing the heat transfer in liquids namely, ater, ethylene glycol and oil. The method involves dispersing nanoparticles of copper, copper oxide and aluminum oxide in these liquids. Li et al. [0] invented a onestep method for preparing nanofluids consisting of metal oxide nanoparticles dispersed in oily fluids such as lubricant oils. A more recent related patent is that of Zhang and Lockood [1] on enhancing the thermal conductivity of fluids using high thermal conductivity graphite nanoparticles and carbon nanotubes. Lockood et al. [] invented a method for gear oil composition containing nanoparticles. Hajikata et al. [3] invented a heat transport nanofluid important for heat exchanger thermal energy transfer. This nanofluid includes a solvent, surfacecoated particles together ith organic components dispersed in the solvent. Hong and Wensel [4] invented a nanofluid that contains carbon nanoparticles, metal oxide nanoparticles and a surfactant in a thermal transfer fluid. Their invention relates to processes for producing such a nanofluid ith enhanced thermal conductive properties. Hayes and McCants [5] recently invented a nanofluid for thermal management systems in hich they used a base liquid ith zinc oxide nanoparticles at a concentration of of about 1% to about 5% by volume. Cardenas et al. [6] invented special nanofluids and methods of use for drilling and completion fluids. These fluids containing nanomaterials, such as carbon nanotubes, meet the required rheological and filtration properties for application in challenging HPHT drilling and completions operations. We present here a similarity analysis for the problem of steady boundarylayer flo of a nanofluid on a continuous moving permeable uniform heat flux surface in the presence of the radiation effect. The development of the velocity, temperature and nanoparticles volume fraction profiles distributions have been illustrated for several values of nanofluid parameters, velocity ratio, suction/injection parameters and radiation parameter.. ANALYSIS Consider a flat surface moving at a constant velocity u in a parallel direction to a free stream of a nanofluid of uniform velocity u. Either the surface velocity or the freestream velocity may be zero but not both at the same time. The physical properties of the fluid are assumed to be constant. Under such condition, the governing equations of the steady, laminar boundarylayer flo on the moving surface are given by (see, ElKabeir et al. [18]: u v + = 0 x y u u u u + v = ( x y y T T T 1 q r u + v = x y y C p y (3 T C DT T + DB + ( y y T y C C C DT T u + v = DB + (4 x y y T y In the above equations, u, v, T and C are the x and y components of fluid velocity, temperature and nanoparticles volume fraction, respectively.,,, q r, D B and D T are the thermal diffusivity, density of ambient fluid, nanofluid heat capacity ratio, heat flux in the ydirection, Bronian diffusion coefficient, and the thermophoretic diffusion coefficient, respectively. The corresponding initial and boundary conditions for this problem can be ritten as: T q C m u =± u, v = v, =, =, at y = 0, y k y DB u u, T T, C C as y (5 The boundary condition of u = u in (5 represents the case of a plane surface moving in parallel to the free stream and q and m represent the all heat and mass fluxes, respectively. To analyze the effect of both the moving and the free stream on the boundarylayer flo, e propose a ne similarly coordinate and a dimensionless stream function y = (Re, x (Re f = (6 (1
3 178 Recent Patents on Mechanical Engineering 01, Vol. 5, No. 3 Chamkha et al. hich are the combinations of the traditional ones:, y Re f B = B = (7 Re x for the Blasius problem (stationary all and uniform free stream velocity and f, y = = Re (8 s s Re x from the Sakiadis (uniformly moving all ith stagnant free stream problem. The Reynolds numbers are defined as: u x ux Re =, Re = (9 A velocity ratio parameter is defined as u u 1 Re 1 = = (1 + = (1 + (10 u + u u Re Note that form the Blasius problem u = 0, therefore = 0. On the other hand, for the Sakiadis problem u = 0 and thus = 1. In addition, e also define dimensionless temperature and concentration function as T T C C =, = (11 qx / (Re mx / (Re and the radiation * 4 4 T q r = * 3k y T = T + 4 T( T T + 6 T ( T T +... ( T + 4TT 3 * qr 16T T = * y 3k y Using the transformation variables defined in equations (61, the governing transformed equations may be ritten as: 1 f + ff = 0 ( (1 + Rd + f + Nb + Nt = 0 Pr 3 ( Nt f = 0 Le Le Nb (15 The transformed initial and boundary conditions become: f (0 = f, f (0 = ±, (0 = 1, (0 = 1, (16 f ( = 1, ( = ( = 0 here Eq. (1 is identically satisfied. In Eqs. (1316, a prime indicates differentiation ith respect to and the parameters Dq T (Re Nt = T( u + u * 3 4 T R d = *, Pr =, Le k k DBm (Re, Nb =, ( u + u (Re f = v. ( u + u (17 =, D Here, Pr, Rd, f, Le, Nb and Nt denote the Prandtl number, the radiation parameter, suction/injection parameter, the Leis number, the Bronian motion parameter and the thermophoresis parameter respectively. It is important to note that this boundary value problem reduces to the classical problem of flo and heat and mass transfer due to a the Blasius problem hen Nb and Nt are zero. Most nanofluids examined to date have large values for the Leis number Le > 1 (see Nield and Kuznetsov [7]. For ater nanofluids at room temperature ith nanoparticles of nm diameters, the Bronian diffusion coefficient D B ranges from to m  /s. Furthermore, the ratio of the Bronian diffusivity coefficient to the thermophoresis coefficient for particles ith diameters of nm can be varied in the ranges of  for alumina, and from to 0 for copper nanoparticles (see Buongiorno [8] for details. Hence, the variation of the nondimensional parameters of nanofluids in the present study is considered to vary in the mentioned range. Of special significance for this type of flo and heat transfer situation are the local skinfriction coefficient, local Nusselt number and the local Sherood number. These physical parameters can be defined in dimensionless form as: 3/ Cf Re = (1 f (0, (18 3/ Cf Re = f (0 4 (1 + Rd Nu /Re 3 x = (19 (0 Sh /Re = x (0 ( RESULTS AND DISCUSSIONS In order to get a clear insight of the physical problem, numerical results are displayed ith the help of graphical and tabulated illustrations. The system of equations (1315 ith the boundary conditions (16 ere solved numerically by means of an efficient, iterative, tridiagonal implicit finitedifference method discussed previously by Blottner [9]. Since the changes in the dependent variables are large in the immediate vicinity of the plate hile these changes decrease greatly as the distance aay from the plate increases, variable step sizes in the direction ere used. The used initial step size in the direction as 1 = 01 and the groth factor as K = 375 such that n = K n1. This indicated that the edge of the boundary layer = 35. The convergence criterion used as based on the relative difference beteen the current and the previous iterations hich as set to 105 in the present ork. It is possible to compare the results obtained by this numerical method ith the previously published ork of ElKabeir et al. [18]. Table 1 shos that excellent agreement beteen the results exists. This lends confidence in the numerical results to be reported subsequently. Computations ere carried out for various values of parameters at Pr = 0.7 and Le = 10. The results of this parametric study are shon in Figs. (15.
4 BoundaryLayer Flo of Nanofluid on Continuously Moving Permeable Surface Recent Patents on Mechanical Engineering 01, Vol. 5, No Table 1. Comparison of Skin Friction f (0 for Various Values of Velocity Ratio. ELKabeir et al. [18] Present Results Figures 4 display the effect of velocity ratio on the velocity, temperature and volume fraction profiles, respectively. It can be seen that, the velocity of the fluid increases as velocity ratio increases in cases < 0.5 and decreases in case > 0.5, on the other hand the temperature of the fluid increases as velocity ratio increases, hile nanoparticle volume fraction decreases as velocity ratio increases. Figures 5 & 6 present the effect of thermophoresis parameter N t and Bronian motion parameter N b on the temperature and nanoparticle volume fraction, respectively. It can be seen that, temperature and nanoparticle volume fraction of the fluid increase as thermophoresis parameter N t and Bronian motion parameter N b increase. 6 Fig. (1. Physical model and coordinates system. f'( Nb=0.3,Nt=0.3,,,, = =0. = = = = Fig. (. Velocity profiles ith various values of velocity ratio parameter. θ( ,0.,,,, Nb=0.3 Nt=0.3 = Fig. (3. Temperature profiles ith various values of velocity ratio parameter. Figures 7 & 8 display the effect of thermophoresis parameter N t and Bronian motion parameter N b on local Nusselt number and local Sherood number respectively. It can be seen that the thermophoresis parameter Nt appears in the
5 180 Recent Patents on Mechanical Engineering 01, Vol. 5, No. 3 Chamkha et al ,0.,,,, =,,, Nb,Nt=0.1,0.3,0.5,0.7 φ( Nb=0.3 Nt= = (1+4 /3/θ( Fig. (4. Volume fraction profiles ith various values of velocity ratio parameter. Fig. (7. Local Nusselt number ith various values of Nb and Nt =,,, Nb,Nt=0.1,0.3,0.5, θ( 3 1/φ(0 0.9 =0.5 1 Nb,Nt=0.1,0.3,0.5,0.7,0.9 = Fig. (5. Temperature profiles ith various values of Nb and Nt. φ( Nb,Nt=0.1,0.3,0.5,0.7,0.9 =0.5 = Fig. (6. Volume fraction profiles ith various values of Nb and Nt. thermal and concentration boundary layer equations. As e note, it is coupled ith the temperature function and plays a strong role in determining the diffusion of heat and nanoparticles concentration in the boundary layer. Moreover, In nanofluid systems, oing to the size of the nanoparticles, Fig. (8. Local Sherood number ith various values of Nb and Nt. θ( Fig. (9. Temperature profiles ith various values of and. Bronian motion takes place, and this can enhance the heat transfer properties. This is due to the fact that the Bronian diffusion promotes heat conduction. The nanoparticles increase the surface area for heat transfer. A nanofluid is a to phase fluid here the nanoparticles move randomly and increase the energy exchange rates. Hoever, the Bronian motion reduces nanoparticles diffusion. The increase in the =0.,0.1,,0.1,0. =.0 =0.5 Nb=0.3 Nt=0.3
6 BoundaryLayer Flo of Nanofluid on Continuously Moving Permeable Surface Recent Patents on Mechanical Engineering 01, Vol. 5, No local Sherood number as Nb changes is relatively small. Therefore, as indicated before, increasing the Bronian motion parameter N b and the thermophoresis parameter N t cause increasing the temperature and volume fraction profiles. This yields reduction in the local Nusselt number and local Sherood number. Figures 9 & 10 sho the representative of temperature and nanoparticles volume fraction profiles for different values of suction/injection and radiation R parameters, respectively. It can be observed that the increasing the value of suction/injection parameter causes decreasing in both temperature and volume fraction profiles, hile radiation R leads to a significant change in temperature and slight change in volume fraction, as radiation parameter increases the temperature of the fluid increases and volume fraction decreases. Cf Re =,Nb=0.3,Nt=0.3,, =0.,0.1,,0.1, Fig. (11. Local skinfriction coefficient ith various values of. C Re φ( =0.,0.1,,0.1,0. =.0 =0.5 Nb=0.3 Nt= (1+4 /3/θ( =,Nb=0.3,Nt=0.3,, =0.,0.1,,0.1,0. Fig. (10. Volume fraction profiles ith various values of and. Figure 11 displays the effect of velocity ratio and suction/injection parameters on the values of local skin friction coefficient, It can be seen that, local skin friction decreases as velocity ratio increases in cases < 0.5 and increases in case > 0.5, hile local skin friction coefficient increases as suction/injection parameter increases. Finally, Figs. (115 sho the effect of suction/injection and radiation parameters on local Nusselt number and local Sherood number, respectively. It can be seen that increasing the value of suction/injection and radiation parameters led to increasing in both local Nusselt number and local Sherood number. 4. CURRENT & FUTURE DEVELOPMENTS The objective of the present ork as to study the effect of thermal radiation on boundary layer flo of an incompressible nanofluid floing over a permeable uniform heat flux surface moving continuously. Some related patents for the preparation of heat transfer nanofluids are highlighted. The model used for the nanofluid incorporated the effects of Bronian motion and thermophoresis. The governing boundary layer equations ere solved numerically using an efficient, implicit finitedifference method. The local skinfriction coefficient, local Nusselt number and the local Fig. (1. Local Nusselt number ith various values of. 1/φ( =,Nb=0.3,Nt=0.3,, =0.,0.1,,0.1, Fig. (13. Local Sherood number ith various values of. Sherood number as ell as the temperature, concentration and velocity distributions for various values of the velocity ratio, suction/injection parameter, radiation parameter and nanofluid parameters ere illustrated graphically and discussed. Based on the obtained results of the above numerical investigation, the folloing conclusions could be dran:
7 18 Recent Patents on Mechanical Engineering 01, Vol. 5, No. 3 Chamkha et al. (1+4 /3/θ(0 Fig. (14. Local Nusselt number ith various values of. 1/φ( ,,.0, Nb=0.3,Nt=0.3,,, Nb=0.3,Nt=0.3,,, 0. Fig. (15. Local Sherood number ith various values of. 1 It as found that both of the temperature of the fluid and the nanoparticles volume fraction increase as either of the thermophoresis parameter or the Bronian motion parameter increased.  Increasing the suction/injection parameter caused reductionds in both the temperature and nanoparticles volume fraction profiles. On the other hand, increasing the thermal radiation parameter led to a significant change in temperature and a slight change in the nanoparticles volume fraction. 3 The fluid temperature increased hile the nanoparticles volume fraction decreased as the velocity ratio increased. 4 Increases in either of the Bronian motion parameter or the thermophoresis parameter led to reductions in the local Nusselt and Sherood numbers. 5 As either of the suction/injection parameter or the radiation parameter increased, both the local Nusselt number and the local Sherood number enhanced. CONFLICT OF INTEREST The authors confirm that this article content has no conflicts of interest.,,.0,3.0 ACKNOWLEDGEMENTS The authors are thankful to the revieers for their positive comments and their valuable suggestions hich led to definite improvements in the paper. REFERENCES [1] Eastman JA, Choi SUS, Li S, Yu W, Thompson LJ. Anomalously increased effective thermal conductivities containing copper nanoparticles. Appl Phys Lett 001; 78: [] Choi SUS, Zhang ZG, Yu W, Lockood FE, Grulke EA. Anomalous thermal conductivity enhancement on nanotube suspensions. Appl Phys Lett 001; 79: 54. [3] Patel HE, Das SK, Sundararajan T, Sreekumaran A, George B, Pradeep T. Thermal conductivities of naked and monolayer protected metal nanoparticle based nanofluids manifestation of anomalous enhancement and chemical effects. Appl Phys Lett 003; 83: [4] You SM, Kim JH, Kim KH. Effect of nanoparticles on critical heat flux oater in pool boiling heat transfer. Appl Phys Lett 003; 83: [5] Vassallo P, Kumar R, D'Amico S. Pool boiling heat transfer experiments in silica ater nonofluids. Int J Heat Mass Transfer 004; 47: [6] Kuznetsov AV, Nield DA. Natural convective boundarylayer flo of a nanofluid past a vertical plate. Int J Therm Sci 010; 49: [7] Khan WA, Pop I. Boundary layer flo of nanofluid past a stretching sheet. Int J Heat Mass Transfer 010; 53: [8] Bachok N, Ishak A, Pop I. Boundary layer flo of nanofluid over moving surface in a floing fluid. Int J Therm Sci 010; 49: [9] Syakila A, Pop I. Mixed convection boundary layer flo from a vertical flat plate embedded in a porous medium filled ith nanofluids. Int Comm Heat Mass Transfer 010; 37: [10] AbuNada E. Application of nanofluids for heat transfer enhancement of separated flos encountered in a backard facing step. Int J Heat Fluid Flo 008; 9: 49. [11] Duangthongsuk W, Wongises S. Effect of thermophysical properties models on the predicting of the convective heat transfer coefficient for lo concentration nanofluid. Int Comm Heat Mass Transfer 008; 35: [1] Gorla RSR, ELKabeir SMM, Rashad AM. Boundarylayer heat transfer from a stretching circular cylinder in a nanofluid. J Thermophys Heat Transfer 011; 5(1: [13] Gorla RSR, Chamkha AJ, Rashad AM. Mixed convective boundary layer flo over a vertical edge embedded in a porous medium saturated ith a nanofluid: Natural convection dominated regime. Nanoscale Res Lett 011; 6: 19. [14] Noghrehabadi A, Pourrajab R, Ghalambaz M. Effect of partial slip boundary condition on the flo and heat transfer of nanofluids past stretching sheet prescribed constant all temperature. Int J Therm Sciences 01; 54: [15] Sakiadis BC. Boundarylayer behavior on a continuous solid surface: IIThe boundary layer on a continuous flat surface. AIChE J 1961; 7: 15. [16] Crane LJ. Flo past a stretching plate. ZAMP 1970; 1: [17] ELKabeir SMM. Radiative effects on forced convection flos in micropolar fluids ith variable viscosity. Can J Phys 004; 8: [18] ELKabeir SMM, Rashad AM, Gorla RSR. Heat transfer in a micropolar fluid flo past a permeable continuous moving surface. ZAMM 011; 9(5: [19] Choi, S.U.S., Eastman, J.A. Enhanced heat transfer using nanofluids. US6175 (001. [0] Li C, Young M, Shih R, Wen M, Chang M. Process for preparing nanofluids ith rotating packed bed reactor. TW63675 (006. [1] Zhang, Z., Lockood, F.E. Enhancing thermal conductivity of fluids ith graphite nanoparticles and carbon nanotube. US (008. [] Lockood, F.E., Zhang, Z., Wu, G., Smith, T.R. Gear oil composition containing nanomaterial. US (008.
8 BoundaryLayer Flo of Nanofluid on Continuously Moving Permeable Surface Recent Patents on Mechanical Engineering 01, Vol. 5, No [3] Hijikata, Y., Torigo, E., Kaaguchi, T., Morishita, T. Heat transport fluid, heat transport structure, and heat transport method. US (008. [4] Hong, H., Wensel, J. Carbon nanoparticlecontaining hydrophilic nanofluid ith enhanced thermal conductivity. US (011. [5] Hayes, A.M., McCants, D.A. Nanofluids for thermal management systems. US (01. [6] Quintero, L., Cardenas, A.E., Clark, D.E. Nanofluids and methods of use for drilling and completion fluids. US (01. [7] Nield DA, Kuznetsov AV. Thermal instability in a porous medium layer saturated by a nanofluid. Int J Heat Mass Transfer 009; 5: [8] Buongiorno J. Convective transport in nanofluids. J Heat TransT, ASME 006;18: [9] Blottner FG. Finitedifference methods of solution of the boundarylayer equations. AIAA J 1970; 8:
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