Abstract
An analysis is carried out to study the heat transfer characteristics of steady twodimensional stagnationpoint flow of a copper (Cu)water nanofluid over a permeable stretching/shrinking sheet. The stretching/shrinking velocity and the ambient fluid velocity are assumed to vary linearly with the distance from the stagnationpoint. Results for the skin friction coefficient, local Nusselt number, velocity as well as the temperature profiles are presented for different values of the governing parameters. It is found that dual solutions exist for the shrinking case, while for the stretching case, the solution is unique. The results indicate that the inclusion of nanoparticles into the base fluid produces an increase in the skin friction coefficient and the heat transfer rate at the surface. Moreover, suction increases the surface shear stress and in consequence increases the heat transfer rate at the fluidsolid interface.
MSC: 34B15, 76D10.
Keywords:
nanofluid; stagnationpoint; stretching/shrinking sheet; suction/injection; heat transfer; dual solutions1 Introduction
Nanofluids are the suspension of metallic, nonmetallic or polymeric nanosized powders in base liquid which are employed to increase the heat transfer rate in various applications. The term nanofluid, first introduced by Choi [1], refers to the fluids with suspended nanoparticles. Most of the convectional heat transfer fluids such as water, ethylene glycol and mineral oils have low thermal conductivity and thus are inadequate to meet the requirements of today’s cooling rate. An innovative way of improving the thermal conductivities of such fluids is to suspend small solid particles in the base fluids to form slurries. An industrial application test was carried out by Liu et al.[2] and Ahuja [3], in which the effect of particle volumetric loading, size and flow rate on the slurry pressure drop and heat transfer behavior was investigated (Xuan and Li [4]). Experimental results by Eastman et al.[5] showed that an increase in thermal conductivity of approximately 60% is obtained for the nanofluid consisting of water and 5% volume fraction of CuO nanoparticles. The procedure for preparing a nanofluid is given in the paper by Xuan and Li [4].
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 (see Ding et al.[6]). Nanofluid is a smart fluid, where the heat transfer capabilities can be reduced or enhanced at will. 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 nonNewtonian behavior, due to the tiny size of nanoelements and the low volume fraction of nanoelements required for conductivity enhancement. Much attention has been paid in the past to this new type of composite material because of its enhanced properties and behavior associated with heat transfer, mass transfer, wetting and spreading as well as antimicrobial activities, and the number of publications related to nanofluids increases in an exponential manner. The enhanced thermal behavior of nanofluids could provide a basis for an enormous innovation for heat transfer intensification, which is of major importance to a number of industrial sectors including transportation, power generation, micromanufacturing, thermal therapy for cancer treatment, chemical and metallurgical sectors, as well as heating, cooling, ventilation and airconditioning. Nanofluids are also important for the production of nanostructured materials, for the engineering of complex fluids, as well as for cleaning oil from surfaces due to their excellent wetting and spreading behavior (Ding et al.[6]).
There are some nanofluid models available in the literature. Among the popular models are the model proposed by Buongiorno [7] and Tiwari and Das [8]. Buongiorno [7] noted that the nanoparticle absolute velocity can be viewed as the sum of the base fluid velocity and a relative velocity (that he calls the slip velocity). He considered in turn seven slip mechanisms: inertia, Brownian diffusion, thermophoresis, diffusiophoresis, Magnus effect, fluid drainage and gravity settling (Nield and Kuznetsov [9]). The nanofluid mathematical model proposed by Buongiorno [7] was very recently used by several researchers such as, among others, Nield and Kuznetsov [9,10], Kuznetsov and Nield [11,12], Khan and Pop [13], Khan and Aziz [14], Makinde and Aziz [15], Bachok et al.[16,17], etc. On the other hand, the Tiwari and Das model analyzes the behavior of nanofluids taking into account the solid volume fraction of the nanofluid. In the present paper, we study the flow and heat transfer characteristics near a stagnation region of a permeable stretching/shrinking sheet immersed in a Cuwater nanofluid using the Tiwari and Das model. It is worth mentioning that this model was recently employed in Refs. [1833], and the flow over a shrinking sheet was considered in Refs. [3444]. The velocity distribution of the twodimensional stagnation flow was first analyzed by Hiemenz (see White [45]) who discovered that this flow can be analyzed exactly by the NavierStokes equations. Homann (see White [45]) extended this problem to the axisymmetric stagnation flow and found that the solution differs a little from the plane flow, where the displacement and boundary layer thicknesses are slightly smaller and the wall shear stress is slightly larger. On the other hand, the temperature distributions of the Hiemenz and Homann flows were given by Goldstein [46] and Sibulkin [47], respectively. The governing partial differential equations are first transformed into a system of ordinary differential equations before being solved numerically. We study the effects of suction and injection at the boundary. Suction or injection of a fluid through the bounding surface, as, for example, in mass transfer cooling, can significantly change the flow field and, as a consequence, affect the heat transfer rate at the surface. In general, suction tends to increase the skin friction and heat transfer coefficients, whereas injection acts in the opposite manner (AlSanea [48]). Injection of fluid through a porous bounding heated or cooled wall is of general interest in practical problems involving film cooling, control of boundary layer, etc. This can lead to enhance heating (or cooling) of the system and can help to delay the transition from laminar flow (see Chaudhary and Merkin [49]). We mention to this end that studies of the boundary layer flows of a Newtonian (or regular) fluid past a permeable static or moving flat plate have been done by Merkin [50], Weidman et al.[51], Ishak et al.[52], Zheng et al.[53] and Zhu et al.[54,55], while Bachok et al.[32] have considered the boundary layers over a permeable moving surface in a nanofluid.
2 Mathematical formulation
Consider a stagnation flow of an incompressible nanofluid over a stretching/shrinking
surface located at
subject to the boundary conditions
where u and v are the velocity components along the x and yaxes, respectively,
Here, φ is the nanoparticle volume fraction,
The governing Eqs. (1)(3) subject to the boundary conditions (4) can be expressed in a simpler form by introducing the following transformation:
where η is the similarity variable and ψ is the stream function defined as
subjected to the boundary conditions (4) which become
In the above equations, primes denote differentiation with respect to η, Pr is the Prandtl number, S is the suction/injection parameter and ε is the stretching/shrinking parameter defined respectively as
with
The physical quantities of interest are the skin friction coefficient
where the surface shear stress
with
where
3 Numerical scheme
The nonlinear differential equations (7) and (8) along with the boundary conditions
(9) form a twopoint boundary value problem (BVP) and are solved using a shooting
method, by converting them into an initial value problem (IVP). This method is very
well described in the recent papers by Bhattacharyya and Layek [57] and Bhattacharyya et al.[58]. In this method, we choose suitable finite values of η, say
with the boundary conditions
Now we have a set of ‘partial’ initial conditions
As we notice, we do not have the values of
To determine either the solution obtained is valid or not, it is necessary to check
the velocity and the temperature profiles. The correct profiles must satisfy the boundary
conditions at
4 Results and discussion
We have considered one type of nanoparticle, namely, copper (Cu), with water as the
base fluid. The effects of the solid volume fraction of nanoparticles φ, the stretching/shrinking parameter ε and the suction/injection parameter S are analyzed. Following Oztop and AbuNada [28], AbuNada and Oztop [29] and Khanafer et al.[60], the value of the Prandtl number Pr is taken as 6.2 (for water) and the volume fraction
of nanoparticles is from 0 to 0.2 (
Figure 1
. Variation of
Figure 2
. Variation of
Figure 3
. Variation of the skin friction coefficient withφfor different values ofSwhen
Figure 4
. Variation of the local Nusselt number withφfor different values ofSwhen
Figure 5
. Velocity profiles for different values ofϕwhen
Figure 6
. Temperature profiles for different values ofϕwhen
Solving Eqs. (7) and (8) subject to the boundary conditions (9), it was found that
dual solutions exist, which were obtained by setting different initial guesses for
the missing values of
Figures 3 and 4 illustrate the variations of the skin friction coefficient
The samples of velocity and temperature profiles for some values of parameters are
presented in Figures 5 and 6. These profiles have essentially the same form as in the case of regular fluid (
5 Conclusions
We have numerically studied the existence of dual similarity solutions in boundary layer flow over a stretching/shrinking sheet immersed in a nanofluid with suction and injection effects. Discussions were carried out for the effects of suction/injection parameter S, the nanoparticle volume fraction φ and the stretching/shrinking parameter ε on the skin friction coefficient and the local Nusselt number. It was found that dual solutions exist for the shrinking case, while for the stretching case, the solution is unique. The results indicate that the inclusion of nanoparticles into the base fluid produced an increase in the skin friction coefficient and the local Nusselt number. Moreover, these quantities increase with suction but decrease with injection.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The paper is the result of joint work of all authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the anonymous reviewers for their comments and suggestions which led to the improvement of this paper. The financial supports received from the Ministry of Higher Education, Malaysia (Project Code: FRGS/1/2012/SG04/UKM/01/1) and the Universiti Kebangsaan Malaysia (Project Code: DIP201231) are gratefully acknowledged.
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