Boundary layer flow past a stretching/shrinking surface beneath an external uniform shear flow with a convective surface boundary condition in a nanofluid
 Nor Azizah Yacob^{1},
 Anuar Ishak^{2},
 Ioan Pop^{3}Email author and
 Kuppalapalle Vajravelu^{4}
https://doi.org/10.1186/1556276X6314
© Yacob et al; licensee Springer. 2011
Received: 19 November 2010
Accepted: 7 April 2011
Published: 7 April 2011
Abstract
The problem of a steady boundary layer shear flow over a stretching/shrinking sheet in a nanofluid is studied numerically. The governing partial differential equations are transformed into ordinary differential equations using a similarity transformation, before being solved numerically by a RungeKuttaFehlberg method with shooting technique. Two types of nanofluids, namely, Cuwater and Agwater are used. The effects of nanoparticle volume fraction, the type of nanoparticles, the convective parameter, and the thermal conductivity on the heat transfer characteristics are discussed. It is found that the heat transfer rate at the surface increases with increasing nanoparticle volume fraction while it decreases with the convective parameter. Moreover, the heat transfer rate at the surface of Cuwater nanofluid is higher than that at the surface of Agwater nanofluid even though the thermal conductivity of Ag is higher than that of Cu.
Introduction
Blasius [1] was the first who studied the steady boundary layer flow over a fixed flat plate with uniform free stream. Howarth [2] solved the Blasius problem numerically. Since then, many researchers have investigated the similar problem with various physical aspects [3–6]. In contrast to the Blasius problem, Sakiadis [7] introduced the boundary layer flow induced by a moving plate in a quiescent ambient fluid. Tsou et al. [8] studied the flow and temperature fields in the boundary layer on a continuous moving surface, both analytically and experimentally and verified the results obtained in [7]. Crane [9] extended this concept to a stretching plate in a quiescent fluid with a stretching velocity that varies with the distance from a fixed point and presented an exact analytic solution. Different from the above studies, Miklavčič and Wang [10] examined the flow due to a shrinking sheet where the velocity moves toward a fixed point. Fang [11] studied the boundary layer flow over a shrinking sheet with a powerlaw velocity, and obtained exact solutions for some values of the parameters.
It is well known that Choi [12] was the first to introduce the term "nanofluid" that represents the fluid in which nanoscale particles are suspended in the base fluid with low thermal conductivity such as water, ethylene glycol, oils, etc. [13]. In recent years, the concept of nanofluid has been proposed as a route for surpassing the performance of heat transfer rate in liquids currently available. The materials with sizes of nanometers possess unique physical and chemical properties [14]. They can flow smoothly through microchannels without clogging them because they are small enough to behave similar to liquid molecules [15]. This fact has attracted many researchers such as [16–27] to investigate the heat transfer characteristics in nanofluids, and they found that in the presence of the nanoparticles in the fluids, the effective thermal conductivity of the fluid increases appreciably and consequently enhances the heat transfer characteristics. An excellent collection of articles on this topic can be found in [28–33], and in the book by Das et al. [14].
It is worth mentioning that while modeling the boundary layer flow and heat transfer of stretching/shrinking surfaces, the boundary conditions that are usually applied are either a specified surface temperature or a specified surface heat flux. However, there are boundary layer flow and heat transfer problems in which the surface heat transfer depends on the surface temperature. Perhaps the simplest case of this is when there is a linear relation between the surface heat transfer and surface temperature. This situation arises in conjugate heat transfer problems (see, for example, [34]), and when there is Newtonian heating of the convective fluid from the surface; the latter case was discussed in detail by Merkin [35]. The situation with Newtonian heating arises in what is usually termed as conjugate convective flow, where the heat is supplied to the convective fluid through a bounding surface with a finite heat capacity. This results in the heat transfer rate through the surface being proportional to the local difference in the temperature with the ambient conditions. This configuration of Newtonian heating occurs in many important engineering devices, for example, in heat exchangers, where the conduction in a solid tube wall is greatly influenced by the convection in the fluid flowing over it. On the other hand, most recently, heat transfer problems for boundary layer flow concerning with a convective boundary condition were investigated by Aziz [36], Makinde and Aziz [37], Ishak [38], and Magyari [39] for the Blasius flow. Similar analysis was applied to the Blasius and Sakiadis flows with radiation effects by Bataller [4]. Yao et al. [40] have very recently investigated the heat transfer of a viscous fluid flow over a permeable stretching/shrinking sheet with a convective boundary condition. Magyari and Weidman [41] investigated the heat transfer characteristics on a semiinfinite flat plate due to a uniform shear flow, both for the prescribed surface temperature and prescribed surface heat flux. It is worth pointing out that a uniform shear flow is driven by a viscous outer flow of rotational velocity whereas the classical Blasius flow is driven over the plate by an inviscid outer flow of irrotational velocity.
The objective of this study is to extend the study of Magyari and Weidman [41] to a stretching/shrinking surface with a convective boundary condition immersed in a nanofluid, that is, to study the steady boundary layer shear flow over a stretching/shrinking surface beneath an external uniform shear flow with a convective surface boundary condition in a nanofluid. This problem is relevant to several practical applications in the field of metallurgy, chemical engineering, etc. A number of technical processes concerning polymers involve the cooling of continuous strips or filaments by drawing them through a quiescent fluid. In these cases, the properties of the final product depend to a great extent on the rate of cooling, which is governed by the structure of the boundary layer near the stretching/shrinking surface. The governing partial differential equations are transformed into ordinary differential equations using a similarity transformation, before being solved numerically by the RungeKuttaFehlberg method with shooting technique.
Mathematical formulation
with γ defined by Equation 12, the solutions of Equations 79 yield the similarity solutions. However, with γ defined by Equation 10, the generated solutions are local similarity solutions. We notice that the solution of Equations 7 and 8 approaches the solution for the constant surface temperature as γ → ∞. This can be seen from the boundary conditions (9), which gives θ(0) = 1 as γ → ∞. Further, it is worth mentioning that Equations 7 and 8 reduce to those of Magyari and Weidman [41] when φ = 0 (regular fluid) and λ = 0 (fixed surface).
Results and discussion
Thermophysical properties of water and the elements Cu and Ag
Physical Properties  Fluid Phase (Water)  Cu  Ag 

C _{p} (J/kgK)  4179  385  235 
ρ (KG/m^{3})  997.1  8933  10500 
k (W/mK)  0.613  400  429 
α × 10^{7} (m^{2}/s)  1.47  1163.1  1738.6 
Values of λ_{c} for Cuwater and Agwater nanofluids
φ  λ_{c}  

Cu  Ag  
0  0.62228  0.62228 
0.1  0.55512  0.53870 
0.2  0.53929  0.51800 
It is observed from Figures 2, 3, 6, and 7 that the skin friction coefficient and the local Nusselt number are more influenced by the nanoparticle volume fraction than the types of nanoparticles. This observation is in agreement with those obtained by Oztop and AbuNada [20] and AbuNada and Oztop [43]. In addition, water has the lowest skin friction coefficient and local Nusselt number compared with Cuwater and Agwater nanofluids. The range of λ for which the solution exists is wider for water compared with the others.
Conclusions
The problem of a steady boundary layer shear flow over a stretching/shrinking sheet in a nanofluid was studied numerically. The governing partial differential equations were transformed into ordinary differential equations by a similarity transformation, before being solved numerically using the RungeKuttaFehlberg method with shooting technique. We considered two types of nanofluids, namely, Cuwater and Agwater. It was found that the heat transfer rate at the surface increases with increasing nanoparticle volume fraction while it decreases with the convective parameter. The variations of the skin friction coefficient and the heat transfer rate at the surface are more influenced by the nanoparticle volume fraction than the types of the nanofluids. Moreover, the heat transfer rate at the surface of Cuwater nanofluid is higher than that of the Agwater nanofluid even though Ag has higher thermal conductivity than that of Cu.
Abbreviations
List of symbols
 c :

Constant
 C _{f} :

Skin friction coefficient
 C _{p} :

Specific heat at constant pressure
 f :

Dimensionless stream function
 h _{f} :

Heat transfer coefficient
 k :

Thermal conductivity
 L :

Reference length
 Nu _{ x } :

Local Nusselt number
 Pr:

Prandtl number
 q _{w} :

Surface heat flux
 T :

Fluid temperature
 T _{f} :

Temperature of the hot fluid
 T _{w} :

Surface temperature
 T _{ ∞ } :

Ambient temperature
 u :

v Velocity components along the x :and ydirections, respectively
 u _{ e }(y):

Free stream velocity
 u _{w}(x):

Stretching/shrinking velocity
 U _{w} :

Reference stretching/shrinking velocity
 x :

y: Cartesian coordinates along the surface and normal to it, respectively
Greek symbols
 α:

Thermal diffusivity
 β:

Constant strain rate
 γ:

Convective parameter
 η:

Similarity variable
 θ:

Dimensionless temperature
 λ:

Stretching/shrinking parameter
 μ:

Dynamic viscosity
 ν:

Kinematic viscosity
 ρ:

Fluid density
 φ:

Nanoparticle volume fraction
 ψ:

Stream function
 τ_{ w } :

Wall shear stress
Subscripts
 f:

Fluid
 nf:

Nanofluid
 s:

Solid.
Declarations
Acknowledgements
The authors express their sincere thanks to the anonymous reviewers for their valuable comments and suggestions for the improvement of the article. This study was supported by research grants from the Ministry of Science, Technology and Innovation, Malaysia (Project Code: 060102SF0610) and the Universiti Kebangsaan Malaysia (Project Code: UKMGGPMNBT0802010).
Authors’ Affiliations
References
 Blasius H: Grenzschichten in Flussigkeiten mit Kleiner Reibung. Zeitschrift Fur Angewandte Mathematik Und Physik 1908, 56: 1–37.Google Scholar
 Howarth L: On the solution of the laminar boundary layer equations. Proc R Soc Lond A 1938, 164: 547–579. 10.1098/rspa.1938.0037View ArticleGoogle Scholar
 Merkin JH: The effect of buoyancy forces on the boundarylayer flow over a semiinfinite vertical flat plate in a uniform free stream. J Fluid Mech 1969, 35: 439–450. 10.1017/S0022112069001212View ArticleGoogle Scholar
 Bataller RC: Radiation effects for the Blasius and Sakiadis flows with a convective surface boundary condition. Appl Math Comput 2008, 206: 832–840. 10.1016/j.amc.2008.10.001View ArticleGoogle Scholar
 Pantokratoras A: Asymptotic suction profiles for the Blasius and Sakiadis flow with constant and variable fluid properties. Arch Appl Mech 2009, 79: 469–478. 10.1007/s0041900802434View ArticleGoogle Scholar
 Fang T, Liang W, Lee CF: A new solution branch for the Blasius equation  a shrinking sheet problem. Comput Math Appl 2008, 56: 3088–3095. 10.1016/j.camwa.2008.07.027View ArticleGoogle Scholar
 Sakiadis BC: Boundarylayer behaviour on continuous solid surfaces. I. Boundarylayer equations for twodimensional and axisymmetric flow. AIChE J 1961, 7: 26–28. 10.1002/aic.690070108View ArticleGoogle Scholar
 Tsou FK, Sparrow EM, Goldstein RJ: Flow and heat transfer in the boundary layer on a continuous moving surface. Int J Heat Mass Transfer 1967, 10: 219–235. 10.1016/00179310(67)901007View ArticleGoogle Scholar
 Crane LJ: Flow past a stretching plate. Zeitschrift Für Angewandte Mathematik Und Physik 1970, 21: 645–647. 10.1007/BF01587695View ArticleGoogle Scholar
 Miklavčič M, Wang CY: Viscous flow due to a shrinking sheet. Q Appl Math 2006, 64: 283–290.View ArticleGoogle Scholar
 Fang T: Boundary layer flow over a shrinking sheet with powerlaw velocity. Int J Heat Mass Transfer 2008, 51: 5838–5843. 10.1016/j.ijheatmasstransfer.2008.04.067View ArticleGoogle Scholar
 Choi SUS: Enhancing thermal conductivity of fluids with nanoparticles. The Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition, San Francisco, USA. ASME, FED 231/MD 66 1995, 99–105.Google Scholar
 Wang XQ, Mujumdar AS: Heat transfer characteristics of nanofluids: a review. Int J Therm Sci 2007, 46: 1–19. 10.1016/j.ijthermalsci.2006.06.010View ArticleGoogle Scholar
 Das SK, Choi SUS, Yu W, Pradeep T: Nanofluids: Science and Technology. NJ: Wiley; 2007.View ArticleGoogle Scholar
 Khanafer K, Vafai K, Lightstone M: Buoyancydriven heat transfer enhancement in a twodimensional enclosure utilizing nanofluids. Int J Heat Mass Transfer 2003, 46: 3639–3653. 10.1016/S00179310(03)00156XView ArticleGoogle Scholar
 AbuNada E: Application of nanofluids for heat transfer enhancement of separated flows encountered in a backward facing step. Int J Heat Fluid Flow 2008, 29: 242–249. 10.1016/j.ijheatfluidflow.2007.07.001View ArticleGoogle Scholar
 Tiwari RJ, Das MK: Heat transfer augmentation in a twosided liddriven differentially heated square cavity utilizing nanofluids. Int J Heat Mass Transfer 2007, 50: 2002–2018. 10.1016/j.ijheatmasstransfer.2006.09.034View ArticleGoogle Scholar
 Maïga 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: 530–546.View ArticleGoogle Scholar
 Polidori G, Fohanno S, Nguyen CT: A note on heat transfer modelling of newtonian nanofluids in laminar free convection. Int J Therm Sci 2007, 46: 739–744. 10.1016/j.ijthermalsci.2006.11.009View ArticleGoogle Scholar
 Oztop HF, AbuNada 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
 Nield DA, Kuznetsov AV: The ChengMinkowycz problem for natural convective boundarylayer flow in a porous medium saturated by a nanofluid. Int J Heat Mass Transfer 2009, 52: 5792–5795. 10.1016/j.ijheatmasstransfer.2009.07.024View ArticleGoogle Scholar
 Kuznetsov AV, Nield DA: Natural convective boundarylayer flow of a nanofluid past a vertical plate. Int J Therm Sci 2010, 49: 243–247. 10.1016/j.ijthermalsci.2009.07.015View ArticleGoogle Scholar
 Muthtamilselvan M, Kandaswamy P, Lee J: Heat transfer enhancement of cooperwater nanofluids in a liddriven enclosure. Commun Nonlinear Sci Numer Simul 2010, 15: 1501–1510. 10.1016/j.cnsns.2009.06.015View ArticleGoogle Scholar
 Bachok N, Ishak A, Pop I: Boundarylayer flow of nanofluids over a moving surface in a flowing fluid. Int J Therm Sci 2010, 49: 1663–1668. 10.1016/j.ijthermalsci.2010.01.026View ArticleGoogle Scholar
 Bachok N, Ishak A, Nazar R, Pop I: Flow and heat transfer at a general threedimensional stagnation point in a nanofluid. Physica B 2010, 405: 4914–4918. 10.1016/j.physb.2010.09.031View ArticleGoogle Scholar
 Yacob NA, Ishak A, Pop I: FalknerSkan problem for a static or moving wedge in nanofluids. Int J Therm Sci 2011, 50: 133–139. 10.1016/j.ijthermalsci.2010.10.008View ArticleGoogle Scholar
 Yacob NA, Ishak A, Nazar R, Pop I: FalknerSkan problem for a static and moving wedge with prescribed surface heat flux in a nanofluid. Int Commun Heat Mass Transfer 2011, 38: 149–153. 10.1016/j.icheatmasstransfer.2010.12.003View ArticleGoogle Scholar
 Buongiorno J: Convective transport in nanofluids. ASME J Heat Transfer 2006, 128: 240–250. 10.1115/1.2150834View ArticleGoogle Scholar
 Daungthongsuk W, Wongwises S: A critical review of convective heat transfer nanofluids. Renew Sustain Energy Rev 2007, 11: 797–817. 10.1016/j.rser.2005.06.005View ArticleGoogle Scholar
 Trisaksri V, Wongwises S: Critical review of heat transfer characteristics of nanofluids. Renew Sustain Energy Rev 2007, 11: 512–523. 10.1016/j.rser.2005.01.010View ArticleGoogle Scholar
 Wang XQ, Mujumdar AS: A review on nanofluids  Part I: theoretical and numerical investigations. Braz J Chem Eng 2008, 25: 613–630.Google Scholar
 Wang XQ, Mujumdar AS: A review on nanofluids  Part II: experiments and applications. Braz J Chem Eng 2008, 25: 631–648. 10.1590/S010466322008000400002View ArticleGoogle Scholar
 Kakaç S, Pramuanjaroenkij A: Review of convective heat transfer enhancement with nanofluids. Int J Heat Mass Transfer 2009, 52: 3187–3196.View ArticleGoogle Scholar
 Merkin JH, Pop I: Conjugate free convection on a vertical surface. Int J Heat Mass Transfer 1996, 39: 1527–1534. 10.1016/00179310(95)002383View ArticleGoogle Scholar
 Merkin JH: Naturalconvection boundarylayer flow on a vertical surface with Newtonian heating. Int J Heat Fluid Flow 1994, 15: 392–398. 10.1016/0142727X(94)900531View ArticleGoogle Scholar
 Aziz A: A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition. Commun Nonlinear Sci Numer Simul 2009, 14: 1064–1068. 10.1016/j.cnsns.2008.05.003View ArticleGoogle Scholar
 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: 1813–1820. 10.1016/j.ijthermalsci.2010.05.015View ArticleGoogle Scholar
 Ishak A: Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition. Appl Math Comput 2010, 217: 837–842. 10.1016/j.amc.2010.06.026View ArticleGoogle Scholar
 Magyari E: Comment on 'A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition' by A. Aziz. Commun Nonlinear Sci Numer Simul 2009, 14:1064–1068. Commun Nonlinear Sci Numer Simul 2011, 16: 599–601. 10.1016/j.cnsns.2010.03.020View ArticleGoogle Scholar
 Yao S, Fang T, Zhong Y: Heat transfer of a generalized stretching/shrinking wall problem with convective boundary conditions. Commun Nonlinear Sci Numer Simul 2011, 16: 752–760. 10.1016/j.cnsns.2010.05.028View ArticleGoogle Scholar
 Magyari E, Weidman PD: Heat transfer on a plate beneath an external uniform shear flow. Int J Therm Sci 2006, 45: 110–115. 10.1016/j.ijthermalsci.2005.05.006View ArticleGoogle Scholar
 Schlichting H, Gersten K: BoundaryLayer Theory. New York: Springer; 2000.View ArticleGoogle Scholar
 AbuNada E, Oztop HF: Effects of inclination angle on natural convection in enclosures filled with Cuwater nanofluid. Int J Heat Fluid Flow 2009, 30: 669–678. 10.1016/j.ijheatfluidflow.2009.02.001View ArticleGoogle Scholar
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