Radiation effect on viscous flow of a nanofluid and heat transfer over a nonlinearly stretching sheet
- Fekry M Hady^{1},
- Fouad S Ibrahim^{1, 3},
- Sahar M Abdel-Gaied^{2} and
- Mohamed R Eid^{2}Email author
DOI: 10.1186/1556-276X-7-229
© Hady et al; licensee Springer. 2012
Received: 14 December 2011
Accepted: 22 April 2012
Published: 22 April 2012
Abstract
In this work, we study the flow and heat transfer characteristics of a viscous nanofluid over a nonlinearly stretching sheet in the presence of thermal radiation, included in the energy equation, and variable wall temperature. A similarity transformation was used to transform the governing partial differential equations to a system of nonlinear ordinary differential equations. An efficient numerical shooting technique with a fourth-order Runge-Kutta scheme was used to obtain the solution of the boundary value problem. The variations of dimensionless surface temperature, as well as flow and heat-transfer characteristics with the governing dimensionless parameters of the problem, which include the nanoparticle volume fraction ϕ, the nonlinearly stretching sheet parameter n, the thermal radiation parameter N_{ R }, and the viscous dissipation parameter Ec, were graphed and tabulated. Excellent validation of the present numerical results has been achieved with the earlier nonlinearly stretching sheet problem of Cortell for local Nusselt number without taking the effect of nanoparticles.
Keywords
nanofluid nonlinearly stretching surface viscous dissipation thermal radiation.Background
The problem of viscous flow and heat transfer over a stretching sheet has important industrial applications, for example, in metallurgical processes, such as drawing of continuous filaments through quiescent fluids, annealing and tinning of copper wires, glass blowing, manufacturing of plastic and rubber sheets, crystal growing, and continuous cooling and fiber spinning, in addition to wide-ranging applications in many engineering processes, such as polymer extrusion, wire drawing, continuous casting, manufacturing of foods and paper, glass fiber production, stretching of plastic films, and many others. During the manufacture of these sheets, the melt issues from a slit and is subsequently stretched to achieve the desired thickness. The final product with the desired characteristics strictly depends upon the stretching rate, the rate of cooling in the process, and the process of stretching. In view of these applications, Sakiadis [1, 2] investigated the boundary-layer flow of a viscous fluid past a moving solid surface; various aspects of the problem have been explored by many authors in the past decades.
However, all these studies are restricted to linear stretching of the sheet. It is worth mentioning that the stretching is not necessarily linear. In view of this, Kumaran and Ramanaih [3] studied flow over a quadratic stretching sheet, but only a few recent studies focused on exponentially and nonlinearly stretching sheet are cited here. Magyari and Keller [4], Elbashbeshy [5], Khan and Sanjayanand [6], Sanjayanand and Khan [7], Sajid and Hayat [8], and Partha et al. [9] studied the heat transfer characteristics of viscous and viscoelastic fluid flows over an exponentially stretching sheet. Vajravelu [10], Vajravelu and Cannon [11], Cortell [12–15], Prasad et al. [16], Afzal [17], and Nandeppanavar et al. [18] studied the effects of various parameters governing the flow of a viscous fluid over a nonlinearly stretching sheet.
A nanofluid is a new class of heat transfer fluids that contain a base fluid and nanoparticles. The use of additives is a technique applied to enhance the heat transfer performance of base fluids. The thermal conductivity of ordinary heat transfer fluids is not adequate to meet today's cooling rate requirements. Nanofluids have been shown to increase the thermal conductivity and convective heat transfer performance of the base liquids. Nanofluids are suspensions of submicronic solid particles (nanoparticles) in common fluids. The term was coined by Choi [19]. The characteristic feature of nanofluids is thermal conductivity enhancement, a phenomenon observed by Masuda et al. [20]. This phenomenon suggests the possibility of using nanofluids in advanced nuclear systems [21]. A comprehensive survey of convective transport in nanofluids was made by Buongiorno [22], who says that a satisfactory explanation for the abnormal increase of the thermal conductivity and viscosity is yet to be found. He focused on further heat transfer enhancement observed in convective situations. Very recently, Kuznetsov and Nield [23] have examined the influence of nanoparticles on natural convection boundary-layer flow past a vertical plate using a model in which Brownian motion and thermophoresis are accounted for. The authors have assumed the simplest possible boundary conditions, namely those in which both the temperature and the nanoparticle fraction are constant along the wall. Furthermore, Nield and Kuznetsov [24, 25] have studied the Cheng and Minkowycz [26] problem of natural convection past a vertical plate in a porous medium saturated by a nanofluid. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis for the porous medium. The Darcy model has been employed.
Hamad and Bashir [27] numerically investigated the problem of forced convection heat transfer to the power law non-Newtonian nanofluid from the stretching surface. Khan and Pop [28] focused on the problem of laminar fluid flow, which results from the stretching of a flat surface in a nanofluid. A similarity solution of the steady boundary layer flow near the stagnation-point flow on a permeable stretching sheet in a porous medium saturated with a nanofluid and in the presence of internal heat generation/absorption was theoretically studied by Hamad and Pop [29]. Hamad and Ferdows [30] investigated the heat and mass transfer analysis for boundary layer stagnation-point flow over a stretching sheet in a porous medium saturated by a nanofluid with internal heat generation/absorption and suction/blowing. The problem of laminar fluid flow, which results from the stretching of a vertical surface with variable stream conditions in a nanofluid, was investigated numerically by Kandasamy et al. [31]. Makinde and Aziz [32] studied numerically the boundary layer flow induced in a nanofluid due to a linearly stretching sheet. Hamad [33] examined the convective flow and heat transfer of an incompressible viscous nanofluid past a semi-infinite vertical stretching sheet in the presence of a magnetic field. All these researchers studied the linear stretching sheet in the nanofluid, but only the numerical investigation by Rana and Bhargava [34] studied the steady laminar boundary fluid flow, which results from the non-linear stretching of a flat surface in a nanofluid, and incorporated the effects of Brownian motion and thermophoresis. Also, more recently, Nadeem and Lee [35] investigated analytically the problem of steady boundary layer flow of nanofluid over an exponential stretching surface including the effects of Brownian motion parameter and thermophoresis parameter.
Presentation of the hypothesis
To the authors' knowledge, no studies have thus far been communicated with regard to the boundary layer viscous flow and heat transfer of a nanofluid past a nonlinearly stretching sheet in the presence of the radiation effect in a one-phase model. The aim of the present paper is therefore to extend the work of Cortell [13] by taking the steady thermal boundary-layer flow with nonlinearly stretching sheet in a nanofluid. The present study is of immediate interest to all those processes which are highly affected with heat enhancement concept, e.g., cooling of metallic sheets or electronic chips, etc. An efficient numerical shooting technique with a fourth-order Runge-Kutta scheme was used to solve the normalized boundary layer equations, and the effects of nanoparticle volume fraction ϕ, nonlinearly stretching sheet parameter n, thermal radiation parameter N_{ R }, and viscous dissipation parameter Ec are described in details and are further presented in tabular form.
Testing the hypothesis
Problem formulation
Thermo-physical properties of fluid and nanoparticles (Oztop and Abu-Nada [36]).
Physical properties | Fluid phase (water) | Cu | Al_{2}O_{3} | TiO_{2} |
---|---|---|---|---|
C_{ p } (J/kgK) | 4179 | 385 | 765 | 686.2 |
ρ (kg/m^{3}) | 997.1 | 8933 | 3970 | 4250 |
k(W/mK) | 0.613 | 401 | 40 | 8.9538 |
β ×10^{5} (K^{-1}) | 21 | 1.67 | 0.85 | 0.9 |
where ${k}_{0}=\frac{3{N}_{R}}{3{N}_{R}+4}$. It is worth citing here that the classical solution for energy equation, Equation 8, without thermal radiation influence can be obtained from the above equation, which reduces to $u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}={\alpha}_{nf}\frac{{\partial}^{2}T}{\partial {y}^{2}}$as N_{ R } →∞ (i.e., k_{0} → 1) and eliminates viscous dissipation.
where Pr = υ_{ f }/α_{ f } is the Prandtl number, and Ec = u_{ w }^{2}/[(c_{ p })_{ f }(T_{w} - T_{∞})] is the Eckert number. In the above equations, primes denote differentiation with respect to η.
Results and discussion
Comparison of - θ' (0) with ϕ = 0 and N_{ R } → ∞ (i.e., k_{0} = 1).
- θ'(0) | |||||
---|---|---|---|---|---|
Ec | n | Pr = 1 | Pr = 5 | ||
Cortell [12] | Present study | Cortell [12] | Present study | ||
0.0 | 0.75 | 1.252672 | 1.253454 | 3.124975 | 3.123518 |
1.5 | 1.439393 | 1.439378 | 3.567737 | 3.566532 | |
7 | 1.699298 | 1.698781 | 4.185373 | 4.184386 | |
10 | 1.728934 | 1.728383 | 4.255972 | 4.254939 | |
0.75 | 1.219985 | 1.220285 | 3.016983 | 3.013524 | |
0.1 | 1.5 | 1.405078 | 1.404805 | 3.455721 | 3.453154 |
7 | 1.662506 | 1.661742 | 4.065722 | 4.063757 | |
10 | 1.691822 | 1.691031 | 4.135296 | 4.133338 |
Comparison of - θ'(0) for various values of thermal radiation parameter N_{ R } with ϕ = 0 (regular fluid).
N _{ R } | Pr | Ec | n | -θ'(0) | |
---|---|---|---|---|---|
Cortell [13] | Present study | ||||
1.5 | - | 0.832709 | |||
0.05 | 3 | - | 0.923306 | ||
1 | 1 | 10 | - | 1.011487 | |
1.5 | 0.823356 | 0.824127 | |||
0.1 | 3 | 0.913773 | 0.914364 | ||
10 | 1.001573 | 1.002161 | |||
1.5 | - | 0.755467 | |||
0.5 | 3 | - | 0.842838 | ||
10 | - | 0.927554 | |||
1.5 | 1.295677 | 1.295790 | |||
0.05 | 3 | - | 1.429987 | ||
10 | - | 1.560471 | |||
1.5 | 1.280575 | 1.280680 | |||
2 | 0.1 | 3 | - | 1.414247 | |
10 | - | 1.544069 | |||
1.5 | 1.159542 | 1.159609 | |||
0.5 | 3 | - | 1.288335 | ||
10 | - | 1.412856 | |||
0.05 | - | 2.209436 | |||
5 | 0.1 | 1.5 | 2.178778 | 2.178846 | |
0.5 | - | 1.934126 | |||
0.05 | - | 1.584762 | |||
2 | 2 | 0.1 | 1.5 | 1.564987 | 1.565049 |
0.5 | - | 1.407369 | |||
0.05 | - | 1.925487 | |||
5 | 2 | 0.1 | 1.5 | 1.833888 | 1.834037 |
0.5 | - | 1.639374 |
Values related to the skin friction for different values of n.
n | -f"(0) | ||
---|---|---|---|
Cu | Al _{ 2 } O _{ 3 } | TiO _{ 2 } | |
0 | 0.737218 | 0.626792 | 0.633534 |
1 | 1.174748 | 0.998779 | 1.009523 |
2 | 1.293408 | 1.099665 | 1.111494 |
3 | 1.349309 | 1.147192 | 1.159532 |
4 | 1.381883 | 1.174886 | 1.187525 |
5 | 1.403223 | 1.193030 | 1.205863 |
10 | 1.450669 | 1.233367 | 1.246635 |
20 | 1.477159 | 1.255889 | 1.269399 |
50 | 1.494071 | 1.270267 | 1.283932 |
100 | 1.499890 | 1.275215 | 1.288933 |
Figures 2 and 3 illustrate the effect of nanoparticle volume fraction ϕ on the nanofluid velocity and temperature profile, respectively, in the case of Cu nanoparticles and water base fluid (Pr = 6.2) when ϕ = 0, 0.05, 0.1, and 0.2, with Ec = 0.1, n = 10, and NR = 1. It is clear that, as the nanoparticles volume fraction increases, the nanofluid velocity decreases, and the temperature increases. These figures illustrate this agreement with the physical behavior. When the volume of nanoparticles increases, the thermal conductivity increases, and then the thermal boundary layer thickness increases. Figures 4 and 5 depict the effect of nonlinearly stretching sheet parameter n on velocity distribution f'(η) and temperature profile θ(η), respectively. Figure 4 illustrates that an increase of nonlinear stretching sheet parameter n tends to decrease the nanofluid velocity in the case of Cu-water when n = 0.75, 1.5, 3, 7, and 10, with Ec = 0.1, N_{ R } = 1, and ϕ = 0.1. Furthermore, Figure 5 shows that increasing the nonlinear stretching sheet parameter n tends to decrease the temperature distribution the same values, thus leading to higher heat transfer rate between the nanofluid and the surface. The effect of the viscous dissipation parameter Ec on the temperature profile in the case of Cu-water when the Eckert number Ec = 0, 0.5, 1, 1.5, 2, and 2.5 with n = 10, N_{ R } = 1, and ϕ = 0.1 is shown in Figure 6. It is clear that the temperature distribution increases with an increase in the viscous dissipation parameter Ec. Figure 7 shows the influence of thermal radiation parameter N_{ R } on the temperature profile in the case of Cu-water. It is clear that the temperature decreases with an increase in the thermal radiation parameter N_{ R }; this leads to an increase in the heat transfer rate. Moreover, Figure 8 shows this effect of the thermal radiation parameter on the temperature distribution but for the different types of nanoparticles with water as the base fluid. It can be seen from Figure 8 that θ(η) decreases with an increase in the thermal radiation parameter as shown in Figure 7, and the Cu nanoparticles have the highest value of temperature distribution than the nanoparticles Al_{2}O_{3} and TiO_{2}. The influence of Ec and n on the temperature profiles for all types of nanoparticles is shown in Figures 9 and 10, respectively. It is found that the temperature decreases with n and increases with Ec as shown in Figures 5 and 6, respectively, and the TiO_{2} nanoparticles proved to have the highest cooling performance for this problem.
The influence of nonlinear stretching sheet n on the skin friction at the surface -f'' (0) with N_{ R } = 5, Pr = 6.2, ϕ = 0.1, and Ec = 0.5 is shown in Figure 11. It can be noticed that, from Table 4 and Figure 11, the numerical values of |f'' (0)| for different kinds of nanofluids increase with an increase in the nonlinear stretching parameter n. This implies an increment of the skin friction at the surface where Cu nanoparticles have the highest skin friction than the other nanoparticles. Figures 12,13,14 display the behavior of the heat transfer rates under the effects of N_{ R }, Ec, and n, respectively, using different nanofluids for Pr = 6.2 and ϕ = 0.1. These figures show that, when using different kinds of nanofluids, the heat transfer rates change, which means that the nanofluids will be important in the cooling and heating processes. It can be noticed from the results above that, as expected, the heat transfer rate increases with an increase in the thermal radiation parameter N_{ R } and nonlinear stretching sheet parameter n, and decreases rapidly with an increase in the viscous dissipation parameter Ec.
Implications of the hypothesis
- 1.
The increase of the solid volume fraction ϕ and the nonlinear stretching sheet parameter n leads to the decrease of dimensionless surface velocity; this yields an increase in the skin friction at the surface.
- 2.
An increment in the solid volume fraction ϕ and the Eckert number Ec yields an increment in the nanofluid's temperature; this leads to a rapid reduction in the heat transfer rates.
- 3.
An increase in the thermal radiation parameter N_{ R } and the nonlinear stretching sheet parameter n yields a decrease in the nanofluid's temperature, which leads to an increase in the heat transfer rates.
- 4.
The TiO_{2} nanoparticles proved to have the highest cooling performance for this problem than the other two types of nanoparticles (cu and Al_{2}O_{3} nanoparticles).
Abbreviations
Nomenclature
b: constant; C: physical parameter related with stretched surface; C_{ f }: skin friction coefficient; c_{ p }: specific heat; Ec: Eckert number; f: dimensionless stream function; k: thermal conductivity; k*: mean absorption coefficient; m: surface temperature parameter; N_{ R }: radiation parameter; Nu_{ x }: Nusselt number; Pr: Prandtl number; q_{ r }: radiative heat flux; n: nonlinear stretching parameter; T: temperature; u and v: velocity components along x- and y-directions: respectively; x and y: Cartesian coordinates along the plate and normal to it: respectively.
Greek symbols
α: thermal diffusivity; η: similarity variable; θ: dimensionless temperature; μ: effective viscosity; υ: kinematic viscosity; ρ: density; σ*: Stefan-Boltzmann constant; (ρC_{ p })_{ nf, }heat capacitance of the nanofluid; (ρC_{ p })f: heat capacity of the fluid; (ρC_{ p })_{ s }: effective heat capacity of the nanoparticle material; ϕ: nanoparticle volume fraction.
Subscripts
f: fluid fraction; nf: nanofluid fraction; s: solid fraction; w: condition at the wall; ∞: stream function condition at infinity.
Authors' information
FMH and FSI are professors of applied mathematics. SMAG is a mathematics lecturer, and MR is a PhD student.
Endnotes
This is just a theoretical study; every experimentalist can check it experimentally with our consent.
Declarations
Acknowledgements
The authors are very thankful to the editor and the reviewers for their constructive comments and suggestions to improve the presentation of this paper.
Authors’ Affiliations
References
- Sakiadis BC: Boundary layer behavior on continuous solid surfaces: I. boundary layer equations for two dimensional and axisymmetric flows. AIChE J 1961, 7: 26–28. 10.1002/aic.690070108View ArticleGoogle Scholar
- Sakiadis BC: Boundary layer behavior on continuous solid surfaces: II. boundary layer on a continuous flat surface. AIChE J 1961, 7: 221–225. 10.1002/aic.690070211View ArticleGoogle Scholar
- Kumaran V, Ramanaih G: A note on the flow over stretching sheet. Arch Mech 1996, 116: 229–233.Google Scholar
- Magyari E, Keller B: Heat mass transfer in the boundary layers on an exponentially stretching sheet. J Phys D: Appl Phys 1999, 32: 577–585. 10.1088/0022-3727/32/5/012View ArticleGoogle Scholar
- Elbashbeshy EMA: Heat transfer over an exponentially stretching continuous surface with suction. Arch Mech 2001, 53: 643–651.Google Scholar
- Khan SK, Sanjayanand E: Viscoelastic boundary layer flow and heat transfer over an exponential stretching sheet. Int J Heat Mass Transf 2005, 48: 1534–1542. 10.1016/j.ijheatmasstransfer.2004.10.032View ArticleGoogle Scholar
- Sanjayanand E, Khan SK: On heat and mass transfer in viscoelastic boundary layer flow over an exponentially stretching sheet. Int J Therm Sci 2006, 45: 819–828. 10.1016/j.ijthermalsci.2005.11.002View ArticleGoogle Scholar
- Sajid M, Hayat T: Influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet. Int Comm Heat Mass Transf 2008, 35: 347–356. 10.1016/j.icheatmasstransfer.2007.08.006View ArticleGoogle Scholar
- Partha MK, Murthy PVSN, Rajasekhar GP: Effect of viscous dissipation on the mixed convection of heat transfer from an exponential stretching surface. Heat Mass Transf 2005, 41: 360–366. 10.1007/s00231-004-0552-2View ArticleGoogle Scholar
- Vajravelu K: Viscous flow over a nonlinearly stretching sheet. A ppl Math Comput 2001, 124: 281–288.View ArticleGoogle Scholar
- Vajravelu K, Cannon JR: Fluid flow over a nonlinearly stretching sheet. A ppl Math Comput 2006, 181: 609–618.View ArticleGoogle Scholar
- Cortell R: Viscous flow and heat transfer over a nonlinearly stretching sheet. A ppl Math Comput 2007, 184: 864–873.View ArticleGoogle Scholar
- Cortell R: Effects of viscous dissipation and radiation on the thermal boundary layer over a non-linearly stretching sheet. Phys Lett A 2008, 372: 631–636. 10.1016/j.physleta.2007.08.005View ArticleGoogle Scholar
- Cortell R: Similarity solutions for the flow and heat transfer of a quiescent fluid over a nonlinearly stretching sheet. J Mater Process Technol 2008, 203: 176–183. 10.1016/j.jmatprotec.2007.09.055View ArticleGoogle Scholar
- Cortell R: Heat and fluid flow due to non-linearly stretching surfaces. Appl Math Comput 2011, 217: 7564–7572. 10.1016/j.amc.2011.02.029View ArticleGoogle Scholar
- Prasad KV, Vajravelu K, Datti PS: Mixed convection heat transfer over a non-linear stretching surface with variable fluid properties. Int J Non-Linear Mech 2010, 45: 320–330. 10.1016/j.ijnonlinmec.2009.12.003View ArticleGoogle Scholar
- Afzal N: Momentum and thermal boundary layers over a two-dimensional or axisymmetric non-linear stretching surface in a stationary fluid. Int J Heat Mass Transf 2010, 53: 540–547. 10.1016/j.ijheatmasstransfer.2009.04.039View ArticleGoogle Scholar
- Nandeppanavar MM, Vajravelu K, Abel MS, Ng C: Heat transfer over a nonlinearly stretching sheet with non-uniform heat source and variable wall temperature. Int J Heat Mass Transf 2011, 54: 4960–4965. 10.1016/j.ijheatmasstransfer.2011.07.009View ArticleGoogle Scholar
- Choi SUS: Enhancing thermal conductivity of fluid with nanoparticles, developments and applications of non-Newtonian flow. ASME FED 1995, 231: 99–105.Google Scholar
- Masuda H, Ebata A, Teramae K, Hishinuma N: Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles. Netsu Bussei 1993, 7: 227–233. 10.2963/jjtp.7.227View ArticleGoogle Scholar
- Buongiorno J, Hu W: Nanofluid coolants for advanced nuclear power plants. In Proceedings of ICAPP 05: May 2005. Seoul. Sydney: Curran Associates, Inc; 2005:15–19.Google Scholar
- Buongiorno J: Convective transport in nanofluids. ASME J Heat Transf 2006, 128: 240–250. 10.1115/1.2150834View ArticleGoogle Scholar
- Kuznetsov AV, Nield DA: Natural convective boundary-layer flow of a nanofluid past a vertical plate. Int J Thermal Sci 2010, 49: 243–247. 10.1016/j.ijthermalsci.2009.07.015View ArticleGoogle Scholar
- Nield DA, Kuznetsov AV: The Cheng-Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid. Int J Heat Mass Transf 2009, 52: 5792–5795. 10.1016/j.ijheatmasstransfer.2009.07.024View ArticleGoogle Scholar
- Nield DA, Kuznetsov AV: The Cheng-Minkowycz problem for the double-diffusive natural convective boundary-layer flow in a porous medium saturated by a nanofluid. Int J Heat Mass Transf 2011, 54: 374–378. 10.1016/j.ijheatmasstransfer.2010.09.034View ArticleGoogle Scholar
- Cheng P, Minkowycz WJ: Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike. J Geophysics Research 1977, 82(14):2040–2044. 10.1029/JB082i014p02040View ArticleGoogle Scholar
- Hamad MAA, Bashir MA: Boundary-layer flow and heat transfer of a power-law non-Newtonian nanofluid over a vertical stretching sheet. World Applied Sciences J (Special Issue for Applied Mathematics) 2009, 7: 172–178.Google Scholar
- Khan WA, Pop I: Boundary-layer flow of a nanofluid past a stretching sheet. Int J Heat Mass Transf 2010, 53: 2477–2483. 10.1016/j.ijheatmasstransfer.2010.01.032View ArticleGoogle Scholar
- Hamad MAA, Pop I: Scaling transformations for boundary layer flow near the stagnation-point on a heated permeable stretching surface in a porous medium saturated with a nanofluid and heat generation/absorption effects. Transp Porous Med 2011, 87: 25–39. 10.1007/s11242-010-9683-8View ArticleGoogle Scholar
- Hamad MAA, Ferdows M: Similarity solution of boundary layer stagnation-point flow towards a heated porous stretching sheet saturated with a nanofluid with heat absorption/generation and suction/blowing: a lie group analysis. Commun Nonlinear Sci Numer Simulat 2011, 17(1):132–140.View ArticleGoogle Scholar
- Kandasamy R, Loganathan P, Arasu PP: Scaling group transformation for MHD boundary-layer flow of a nanofluid past a vertical stretching surface in the presence of suction/injection. Nuclear Engineering Design 2011, 241: 2053–2059. 10.1016/j.nucengdes.2011.04.011View ArticleGoogle Scholar
- Makinde OD, Aziz A: Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition. Int J Therm Sci 2011, 50: 1326–1332. 10.1016/j.ijthermalsci.2011.02.019View ArticleGoogle Scholar
- Hamad MAA: Analytical solution of natural convection flow of a nanofluid over a linearly stretching sheet in the presence of magnetic field. Int Commun Heat Mass Transf 2011, 38: 487–492. 10.1016/j.icheatmasstransfer.2010.12.042View ArticleGoogle Scholar
- Rana P, Bhargava R: Flow and heat transfer of a nanofluid over a nonlinearly stretching sheet: a numerical study. Commun Nonlinear Sci Numer Simulat 2011, 17(1):212–226.View ArticleGoogle Scholar
- Nadeem S, Lee C: Boundary layer flow of nanofluid over an exponentially stretching surface. Nanoscale Res Lett 2012, 7: 94. 10.1186/1556-276X-7-94View ArticleGoogle Scholar
- Oztop HF, Abu-Nada E: Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids. Int J Heat Fluid Flow 2008, 29: 1326–1336. 10.1016/j.ijheatfluidflow.2008.04.009View ArticleGoogle Scholar
- Khanafer K, Vafai K, Lightstone M: Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. Int J Heat Mass Transf 2003, 46: 3639–3653. 10.1016/S0017-9310(03)00156-XView ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.