Stagnation-point flow over a stretching/shrinking sheet in a nanofluid
© Bachok et al; licensee Springer. 2011
Received: 14 August 2011
Accepted: 8 December 2011
Published: 8 December 2011
An analysis is carried out to study the steady two-dimensional stagnation-point flow of a nanofluid over a stretching/shrinking sheet in its own plane. The stretching/shrinking velocity and the ambient fluid velocity are assumed to vary linearly with the distance from the stagnation point. The similarity equations are solved numerically for three types of nanoparticles, namely copper, alumina, and titania in the water-based fluid with Prandtl number Pr = 6.2. The skin friction coefficient, Nusselt number, and the velocity and temperature profiles are presented graphically and discussed. Effects of the solid volume fraction φ on the fluid flow and heat transfer characteristics are thoroughly examined. Different from a stretching sheet, it is found that the solutions for a shrinking sheet are non-unique.
Stagnation-point flow, describing the fluid motion near the stagnation region of a solid surface exists in both cases of a fixed or moving body in a fluid. The two-dimensional stagnation-point flow towards a stationary semi-infinite wall was first studied by Hiemenz , who used a similarity transformation to reduce the Navier-Stokes equations to nonlinear ordinary differential equations. This problem has been extended by Homann  to the case of axisymmetric stagnation-point flow. The combination of both stagnation-point flows past a stretching surface was considered by Mahapatra and Gupta [3, 4]. There are two conditions that the flow towards a shrinking sheet is likely to exist, whether an adequate suction on the boundary is imposed  or a stagnation flow is considered . Wang  investigated both two-dimensional and axisymmetric stagnation flow towards a shrinking sheet in a viscous fluid. He found that solutions do not exist for larger shrinking rates and non-unique in the two-dimensional case. After this pioneering work, the flow field over a stagnation point towards a stretching/shrinking sheet has drawn considerable attention and a good amount of literature has been generated on this problem [7–10].
All studies mentioned above refer to the stagnation-point flow towards a stretching/shrinking sheet in a viscous and Newtonian fluid. The present paper deals with the problem of a steady boundary-layer flow, heat transfer, and nanoparticle fraction over a stagnation point towards a stretching/shrinking sheet in a nanofluid, with water as the based fluid. Most conventional heat transfer fluids, such as water, ethylene glycol, and engine oil, have limited capabilities in terms of thermal properties, which, 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 of magnitude, compared with that of liquids. Hence, one can then expect that fluid-containing solid particles may significantly increase its conductivity. The flow over a continuously stretching surface is an important problem in many engineering processes with applications in industries such as the hot rolling, wire drawing, paper production, glass blowing, plastic films drawing, and glass-fiber production. The quality of the final product depends on the rate of heat transfer at the stretching surface. On the other hand, the new type of shrinking sheet flow is essentially a backward flow as discussed by Goldstein  and it shows physical phenomena quite distinct from the forward stretching flow . The enhanced thermal behavior of nanofluids could provide a basis for an enormous innovation for heat transfer intensification for the processes and applications mentioned above.
Many of the publications on nanofluids are about understanding of their behaviors so that they can be utilized where straight heat transfer enhancement is paramount as in many industrial applications, nuclear reactors, transportation, electronics as well as biomedicine and food. The broad range of current and future applications involving nanofluids have been given by Wong and Leon . Nanofluid as a smart fluid, where heat transfer can be reduced or enhanced at will, has also been reported. These fluids enhance thermal conductivity of the base fluid enormously, which is beyond the explanation of any existing theory. They are also very stable and have no additional problems, such as sedimentation, erosion, additional pressure drop and non-Newtonian behavior, due to the tiny size of nanoelements and the low volume fraction of nanoelements required for conductivity enhancement. These suspended nanoparticles can change the transport and thermal properties of the base fluid. The comprehensive references on nanofluids can be found in the recent book by Das et al.  and in the review papers by Buongiorno , Daungthongsuk and Wongwises , Trisaksri and Wongwises , Ding et al. , Wang and Mujumdar [19–21], Murshed et al. , and Kakaç and Pramuanjaroenkij .
The nanofluid model proposed by Buongiorno  was very recently used by Nield and Kuznetsov [24, 25], Kuznetsov and Neild [26, 27], Khan and Pop , and Bachok et al.  in their papers. The paper by Khan and Pop  is the first which considered the problem on stretching sheet in nanofluids. Different from the above model, the present paper considers a problem using the nanofluid model proposed by Tiwari and Das , which was also used by several authors (cf. Abu-Nada , Muthtamilselvan et al. , Abu-Nada and Oztop , Talebi et al. , Ahmad et al. , Bachok et al. [36, 37], Yacob et al. ). The model proposed by Buongiorno  studies the Brownian motion and the thermophoresis on the heat transfer characteristics, while the model by Tiwari and Das  analyzes the behavior of nanofluids taking into account the solid volume fraction. In the present paper, we analyze the effects of the solid volume fraction and the type of the nanoparticles on the fluid flow and heat transfer characteristics of a nanofluid over a stretching/shrinking sheet.
Here, φ is the nanoparticle volume fraction, (ρ C p )nf is the heat capacity of the nanofluid, k nf is the thermal conductivity of the nanofluid, k f and k s are the thermal conductivities of the fluid and of the solid fractions, respectively, and ρ f and ρ s are the densities of the fluid and of the solid fractions, respectively. It should be mentioned that the use of the above expression for k nf is restricted to spherical nanoparticles where it does not account for other shapes of nanoparticles . Also, the viscosity of the nanofluid μ nf has been approximated by Brinkman  as viscosity of a base fluid μ f containing dilute suspension of fine spherical particles.
where ε > 0 for stretching and ε < 0 for shrinking.
where Rex = U ∞ x /ν f is the local Reynolds number.
Results and discussion
Thermophysical properties of fluid and nanoparticles 
Fluid phase (water)
C p (J/kg K)
Values of f″(0) for some values of ε and φ for Cu-water working fluid
φ = 0
φ = 0
φ = 0.1
φ = 0.2
Values of for some values of ε and φ
Yacob et al.
Al 2 O 3 -water
TiO 2 -water
Al 2 O 3 -water
TiO 2 -water
Values of for some values of ε and φ
Yacob et al.
Al 2 O 3 -water
TiO 2 -water
Al 2 O 3 -water
TiO 2 -water
We have presented an analysis for the flow and heat transfer characteristics of a nanofluid over a stretching/shrinking sheet in its own plane. The stretching/shrinking velocity and the ambient fluid velocity are assumed to vary linearly with the distance from the stagnation point. The resulting system of nonlinear ordinary differential equations is solved numerically for three types of nanoparticles, namely copper (Cu), alumina (Al2O3), and titania (TiO2) in the water-based fluid with Prandtl number Pr = 6.2, to investigate the effect of the solid volume fraction parameter φ on the fluid and heat transfer characteristics. Different from a stretching sheet, it is found that the solutions for a shrinking sheet are non-unique. The inclusion of nanoparticles into the base water fluid has produced an increase in the skin friction and heat transfer coefficients, which increases appreciably with an increase of the nanoparticle volume fraction. Nanofluids are capable to change the velocity and temperature profile in the boundary layer. The type of nanofluids is a key factor for heat transfer enhancement. The highest values of the skin friction coefficient and the local Nusselt number were obtained for the Cu nanoparticles compared with the others.
The authors are indebted to the anonymous reviewers for their constructive comments and suggestions which led to the improvement of this paper. This work was supported by a Research Grant (Project Code: UKM-GGPM-NBT- 080-2010) from the Universiti Kebangsaan Malaysia.
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