# Stagnation-point flow over a permeable stretching/shrinking sheet in a copper-water nanofluid

- Norfifah Bachok
^{1}, - Anuar Ishak
^{2}Email author, - Roslinda Nazar
^{2}and - Norazak Senu
^{1}

**2013**:39

**DOI: **10.1186/1687-2770-2013-39

© Bachok et al.; licensee Springer. 2013

**Received: **30 August 2012

**Accepted: **8 February 2013

**Published: **26 February 2013

## Abstract

An analysis is carried out to study the heat transfer characteristics of steady two-dimensional stagnation-point 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 stagnation-point. 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 fluid-solid interface.

**MSC:** 34B15, 76D10.

### Keywords

nanofluid stagnation-point stretching/shrinking sheet suction/injection heat transfer dual solutions## 1 Introduction

Nanofluids are the suspension of metallic, nonmetallic or polymeric nano-sized 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 non-Newtonian 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, micro-manufacturing, thermal therapy for cancer treatment, chemical and metallurgical sectors, as well as heating, cooling, ventilation and air-conditioning. 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 Cu-water nanofluid using the Tiwari and Das model. It is worth mentioning that this model was recently employed in Refs. [18–33], and the flow over a shrinking sheet was considered in Refs. [34–44]. The velocity distribution of the two-dimensional stagnation flow was first analyzed by Hiemenz (see White [45]) who discovered that this flow can be analyzed exactly by the Navier-Stokes 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 (Al-Sanea [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

*i.e.*, ${U}_{w}(x)=ax$ and ${U}_{\mathrm{\infty}}(x)=bx$, where

*a*and

*b*are constant with $b>0$. We note that $a>0$ and $a<0$ correspond to stretching and shrinking sheets, respectively. The simplified two-dimensional equations governing the flow in the boundary layer of a steady, laminar, and incompressible nanofluid are (see Ahmad

*et al.*[26])

*u*and

*v*are the velocity components along the

*x*- and

*y*-axes, respectively, ${V}_{w}$ is the mass transfer velocity,

*T*is the temperature of the nanofluid, ${T}_{w}$ is the surface temperature, ${T}_{\mathrm{\infty}}$ is the ambient temperature, ${\mu}_{nf}$ is the viscosity of the nanofluid, ${\alpha}_{nf}$ is the thermal diffusivity of the nanofluid and ${\rho}_{nf}$ is the density of the nanofluid, which are given by (Oztop and Abu-Nada [28])

Here, *φ* is the nanoparticle volume fraction, ${(\rho {C}_{p})}_{nf}$ 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 ${\rho}_{f}$ and ${\rho}_{s}$ 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 (Abu-Nada [18]). Also, the viscosity of the nanofluid ${\mu}_{nf}$ has been approximated by Brinkman [56] as viscosity of a base fluid ${\mu}_{f}$ containing dilute suspension of fine spherical particles.

*η*is the similarity variable and

*ψ*is the stream function defined as $u=\partial \psi /\partial y$ and $v=-\partial \psi /\partial x$, which identically satisfies Eq. (1). Employing the similarity variables (6), Eqs. (2) and (3) reduce to the following ordinary differential equations:

*η*, Pr is the Prandtl number,

*S*is the suction/injection parameter and

*ε*is the stretching/shrinking parameter defined respectively as

with $\epsilon >0$ for stretching and $\epsilon <0$ for shrinking.

where ${\mathrm{Re}}_{x}={U}_{\mathrm{\infty}}x/{\nu}_{f}$ is the local Reynolds number.

## 3 Numerical scheme

*et al.*[58]. In this method, we choose suitable finite values of

*η*, say ${\eta}_{\mathrm{\infty}}$, which depend on the values of the parameters considered. First, the system of equations (7) and (8) is reduced to a first-order system (by introducing new variables) as follows:

As we notice, we do not have the values of $q(0)$ and $r(0)$. To solve Eqs. (15) and (16) as an IVP, we need the values of $q(0)$ and $r(0)$, *i.e.*, ${f}^{\u2033}(0)$ and ${\theta}^{\prime}(0)$. We guess these values and apply the Runge-Kutta-Fehlberg method, then see if this guess matches the boundary conditions at the very end. Varying the initial slopes gives rise to a set of profiles which suggest the trajectory of a projectile ‘shot’ from the initial point. That initial slope is sought which results in the trajectory ‘hitting’ the target, that is, the final value (Bailey *et al.* [59]).

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 $\eta ={\eta}_{\mathrm{\infty}}$ asymptotically. This procedure is repeated for other guessing values of $q(0)$ and $r(0)$ for the same values of parameters. If a different solution is obtained and the profiles satisfy the far field boundary conditions asymptotically but with different boundary layer thickness, then this solution is also a solution to the boundary-value problem (second solution).

## 4 Results and discussion

*φ*, the stretching/shrinking parameter

*ε*and the suction/injection parameter

*S*are analyzed. Following Oztop and Abu-Nada [28], Abu-Nada 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 ($0\le \phi \le 0.2$) in which $\phi =0$ corresponds to the regular (Newtonian) fluid. The thermophysical properties of the base fluid (water) and the nanoparticles are given in Table 1. The numerical results are presented in Figures 1-6.

**Thermophysical properties of fluid and nanoparticles (Oztop and Abu-Nada** [28]**)**

Physical properties | Fluid phase (water) | Cu |
---|---|---|

${C}_{p}$ (J/kg K) | 4,179 | 385 |

| 997.1 | 8,933 |

| 0.613 | 400 |

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 ${f}^{\u2033}(0)$ and ${\theta}^{\prime}(0)$, where all profiles satisfy the far fields boundary conditions (9) asymptotically but with different shapes. The variations of ${f}^{\u2033}(0)$ and $-{\theta}^{\prime}(0)$ with *ε* are shown in Figures 1 and 2 for some values of the suction/injection parameter *S*. These figures show that there are regions of unique solutions for $\epsilon \ge -1$, dual (upper and lower branches) solutions for ${\epsilon}_{c}\le \epsilon <-1$ and no solutions for $\epsilon <{\epsilon}_{c}<0$, where ${\epsilon}_{c}$ is the critical value of *ε* ($<{\epsilon}_{c}<0$) beyond which Eqs. (7) and (8) have no solutions. Based on our computation, the critical values of *ε*, say ${\epsilon}_{c}$, are −1.1098, −1.2465 and −1.4311 for $S=-0.3$, $S=0$ and $S=0.3$, respectively. Thus, from this observation, the value of $|{\epsilon}_{c}|$ increases as *S* increases. Hence, suction widens the range of *ε* for which the solution exists, while injection acts in the opposite manner. Further, it should be mentioned that similar to other studies where dual solutions exist, we postulate that upper branch (the first) solutions of Eqs. (7) and (8) are stable and physically realizable, while the lower branch (the second) solutions are not. The procedure for showing this has been described by Merkin [50], Weidman *et al.* [51] and very recently by Postelnicu and Pop [61], so that we will not repeat it here.

Figures 3 and 4 illustrate the variations of the skin friction coefficient ${C}_{f}{\mathrm{Re}}_{x}^{1/2}$ and the local Nusselt number ${\mathrm{Nu}}_{x}{\mathrm{Re}}_{x}^{-1/2}$ given by Eqs. (13) and (14) with the nanoparticle volume fraction parameter *φ* for three different values of *S* with $\epsilon =0.5$ (stretching sheet). These figures show that these quantities increase almost linearly with *φ*. The presence of the nanoparticles in the fluids increases appreciably the effective thermal conductivity of the fluid and consequently enhances the heat transfer characteristics, as seen in Figure 4. Nanofluids have a distinctive characteristics, which is quite different from those of traditional solid-liquid mixtures in which millimeter and/or micrometer-sized particles are involved. Such particles can clot equipment and can increase pressure drop due to settling effects. Moreover, they settle rapidly, creating substantial additional pressure (Khanafer *et al.* [60]). Figure 3 also shows that the effect of suction ($S>0$) is to increase the skin friction coefficient, and in consequence it increases the local Nusselt number, as presented in Figure 4.

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 ($\phi =0$). The terms first solution and second solution refer to the curves shown in Figures 1 and 2, where the first solution has larger values of ${f}^{\u2033}(0)$ and $-{\theta}^{\prime}(0)$ compared to the second solution. Figures 5 and 6 show that the far field boundary conditions (9) are satisfied asymptotically, thus support the validity of the numerical results, besides supporting the existence of the dual solutions presented in Figures 1 and 2.

## 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.

## Declarations

### 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: DIP-2012-31) are gratefully acknowledged.

## Authors’ Affiliations

## References

- Choi US:
**Enhancing thermal conductivity of fluids with nanoparticles.***ASME FED*1995, 231: 99-103.Google Scholar - Liu, KV, Choi, US, Kasza, KE: Measurements of pressure drop and heat transfer in turbulent pipe flows of particulate slurries. Argonne National Laboratory Report, ANL-88-15, Argonne (1988)Google Scholar
- Ahuja AS:
**Augmentation of heat transfer in laminar flow of polystyrene suspensions.***J. Appl. Phys.*1975, 46: 3408-3425. 10.1063/1.322107View ArticleGoogle Scholar - Xuan Y, Li Q:
**Heat transfer enhancement of nanofluids.***Int. J. Heat Fluid Flow*2000, 21: 58-64. 10.1016/S0142-727X(99)00067-3View ArticleGoogle Scholar - Eastman JA, Choi US, Li S, Thompson LJ, Lee S:
**Enhanced thermal conductivity through the development of nanofluids.**In*Nanophase and Nanocomposite Materials II*. Edited by: Komarneni S, Parker JC, Wollenberger HJ. MRS, Pittsburgh; 1997:3-11.Google Scholar - Ding Y, Chen H, Wang L, Yang C-Y, He Y, Yang W, Lee WP, Zhang L, Huo R:
**Heat transfer intensification using nanofluids.***Kona*2007, 25: 23-38.View ArticleGoogle Scholar - Buongiorno J:
**Convective transport in nanofluids.***ASME J. Heat Transf.*2006, 128: 240-250. 10.1115/1.2150834View ArticleGoogle Scholar - Tiwari RK, Das MK:
**Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids.***Int. J. Heat Mass Transf.*2007, 50: 2002-2018. 10.1016/j.ijheatmasstransfer.2006.09.034MATHView 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.2009.07.024MATHView 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.2010.09.034MATHView ArticleGoogle Scholar - Kuznetsov AV, Nield DA:
**Natural convective boundary-layer 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 - Kuznetsov AV, Nield DA:
**Double-diffusive natural convective boundary-layer flow of a nanofluid past a vertical plate.***Int. J. Therm. Sci.*2011, 50: 712-717. 10.1016/j.ijthermalsci.2011.01.003View ArticleGoogle Scholar - Khan AV, 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.2010.01.032MATHView ArticleGoogle Scholar - Khan WA, Aziz A:
**Natural convection flow of a nanofluid over a vertical plate with uniform surface heat flux.***Int. J. Therm. Sci.*2011, 50: 1207-1214. 10.1016/j.ijthermalsci.2011.02.015View 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 - Bachok N, Ishak A, Pop I:
**Boundary layer 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, Pop I:
**Unsteady boundary layer flow of a nanofluid over a permeable stretching/shrinking sheet.***Int. J. Heat Mass Transf.*2012, 55: 2102-2109. 10.1016/j.ijheatmasstransfer.2011.12.013View ArticleGoogle Scholar - Abu-Nada E:
**Application of nanofluids for heat transfer enhancement of separated flow encountered in a backward facing step.***Int. J. Heat Fluid Flow*2008, 29: 242-249. 10.1016/j.ijheatfluidflow.2007.07.001View ArticleGoogle Scholar - Muthtamilselvan M, Kandaswamy P, Lee J:
**Heat transfer enhancement of copper-water nanofluids in a lid-driven enclosure.***Commun. Nonlinear Sci. Numer. Simul.*2010, 15: 1501-1510. 10.1016/j.cnsns.2009.06.015MATHMathSciNetView ArticleGoogle Scholar - Abu-Nada E, Oztop HF:
**Effect of inclination angle on natural convection in enclosures filled with cu-water nanofluid.***Int. J. Heat Fluid Flow*2009, 30: 669-678. 10.1016/j.ijheatfluidflow.2009.02.001View ArticleGoogle Scholar - Talebi F, Houshang A, Shahi M:
**Numerical study of mixed convection flows in a square lid-driven cavity utilizing nanofluid.***Int. Commun. Heat Mass Transf.*2010, 37: 79-90. 10.1016/j.icheatmasstransfer.2009.08.013View ArticleGoogle Scholar - Bachok N, Ishak A, Pop I:
**Flow and heat transfer over a rotating porous disk in a nanofluid.***Physica B*2011, 406: 1767-1772. 10.1016/j.physb.2011.02.024View ArticleGoogle Scholar - Bachok N, Ishak A, Nazar R, Pop I:
**Flow and heat transfer at a general three dimensional stagnation point flow in a nanofluid.***Physica B*2010, 405: 4914-4918. 10.1016/j.physb.2010.09.031View ArticleGoogle Scholar - Bachok N, Ishak A, Pop I:
**Stagnation-point flow over a stretching/shrinking sheet in a nanofluid.***Nanoscale Res. Lett.*2011, 6: 623. 10.1186/1556-276X-6-623View ArticleGoogle Scholar - Bachok N, Ishak A, Pop I:
**Flow and heat transfer characteristics on a moving plate in a nanofluid.***Int. J. Heat Mass Transf.*2012, 55: 642-648. 10.1016/j.ijheatmasstransfer.2011.10.047MATHView ArticleGoogle Scholar - Ahmad S, Rohni AM, Pop I:
**Blasius and Sakiadis problems in nanofluids.***Acta Mech.*2011, 218: 195-204. 10.1007/s00707-010-0414-6MATHView ArticleGoogle Scholar - Yacob NA, Ishak A, Pop I, Vajravelu K:
**Boundary layer flow past a stretching/shrinking surface beneath an external uniform shear flow with a convective surface boundary condition in a nanofluid.***Nanoscale Res. Lett.*2011., 6: Article ID 314Google 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 - Abu-Nada E, Oztop HF:
**Effect of inclination angle on natural convection in enclosures filled with Cu-water nanofluid.***Int. J. Heat Fluid Flow*2009, 30: 669-678. 10.1016/j.ijheatfluidflow.2009.02.001View ArticleGoogle Scholar - Talebi F, Houshang A, Shahi M:
**Numerical study of mixed convection flows in a square lid-driven cavity utilizing nanofluid.***Int. Commun. Heat Mass Transf.*2010, 37: 79-90. 10.1016/j.icheatmasstransfer.2009.08.013View ArticleGoogle Scholar - Rohni AM, Ahmad S, Pop I:
**Boundary layer flow over a moving surface in a nanofluid beneath a uniform free stream.***Int. J. Numer. Methods Heat Fluid Flow*2011, 21: 828-846. 10.1108/09615531111162819View ArticleGoogle Scholar - Bachok N, Ishak A, Pop I:
**Boundary layer flow over a moving surface in nanofluid with suction or injection.***Acta Mech. Sin.*2012, 28: 34-40. 10.1007/s10409-012-0014-xMATHMathSciNetView ArticleGoogle Scholar - Yacob NA, Ishak A, Pop I:
**Falkner-Skan 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 - Bhattacharyya K:
**Dual solutions in boundary layer stagnation-point flow and mass transfer with chemical reaction past a stretching/shrinking sheet.***Int. Commun. Heat Mass Transf.*2011, 38: 917-922. 10.1016/j.icheatmasstransfer.2011.04.020View ArticleGoogle Scholar - Rosali H, Ishak A, Pop I:
**Stagnation point flow and heat transfer over a stretching/shrinking sheet in a porous medium.***Int. Commun. Heat Mass Transf.*2011, 38: 1029-1032. 10.1016/j.icheatmasstransfer.2011.04.031View ArticleGoogle Scholar - Bhattacharyya K:
**Boundary layer flow and heat transfer over an exponentially shrinking sheet.***Chin. Phys. Lett.*2011., 28: Article ID 074701Google Scholar - Bhattacharyya K:
**Dual solutions in unsteady stagnation-point flow over a shrinking sheet.***Chin. Phys. Lett.*2011., 28: Article ID 084702Google Scholar - Bhattacharyya K:
**Effects of radiation and heat source/sink on unsteady MHD boundary layer flow and heat transfer over a shrinking sheet with suction/injection.***Front. Chem. Sci. Eng.*2011, 5: 376-384. 10.1007/s11705-011-1121-0View ArticleGoogle Scholar - Bhattacharyya K, Pop I:
**MHD boundary layer flow due to an exponentially shrinking sheet.***Magnetohydrodynamics*2011, 47: 337-344.Google Scholar - Bhattacharyya K, Arif MG, Pramanik WA:
**MHD boundary layer stagnation-point flow and mass transfer over a permeable shrinking sheet with suction/blowing and chemical reaction.***Acta Tech.*2012, 57: 1-15.MathSciNetGoogle Scholar - Bhattacharyya K, Mukhopadhyay S, Layek GC, Pop I:
**Effects of thermal radiation on micropolar fluid flow and heat transfer over a porous shrinking sheet.***Int. J. Heat Mass Transf.*2012, 55: 2945-2952. 10.1016/j.ijheatmasstransfer.2012.01.051View ArticleGoogle Scholar - Bhattacharyya K, Vajravelu K:
**Stagnation-point flow and heat transfer over an exponentially shrinking sheet.***Commun. Nonlinear Sci. Numer. Simul.*2012, 17: 2728-2734. 10.1016/j.cnsns.2011.11.011View ArticleGoogle Scholar - Fang T, Zhong Y:
**Viscous flow over a shrinking sheet with an arbitrary surface velocity.***Commun. Nonlinear Sci. Numer. Simul.*2010, 15: 3768-3776. 10.1016/j.cnsns.2010.01.034MATHMathSciNetView ArticleGoogle Scholar - Fang T, Yao S, Zhang J, Aziz A:
**Viscous flow over a shrinking sheet with a second order slip flow model.***Commun. Nonlinear Sci. Numer. Simul.*2010, 15: 1831-1842. 10.1016/j.cnsns.2009.07.017MATHMathSciNetView ArticleGoogle Scholar - White FM:
*Viscous Fluid Flow*. McGraw-Hill, Boston; 2006.Google Scholar - Goldstein S:
*Modern Development in Fluid Dynamics*. Oxford University Press, London; 1938.Google Scholar - Sibulkin M:
**Heat transfer near the forward stagnation point of a body of revolution.***J. Aeronaut. Sci.*1952, 19: 570-571.Google Scholar - Al-Sanea SA:
**Mixed convection heat transfer along a continuously moving heated vertical plate with suction or injection.***Int. J. Heat Mass Transf.*2004, 47: 1445-1465. 10.1016/j.ijheatmasstransfer.2003.09.016MATHView ArticleGoogle Scholar - Chaudhary MA, Merkin JH:
**The effect of blowing and suction on free convection boundary layers on vertical surface with prescribed heat flux.***J. Eng. Math.*1993, 27: 265-292. 10.1007/BF00128967MATHMathSciNetView ArticleGoogle Scholar - Merkin JH:
**A note on the similarity equations arising in free convection boundary layers with blowing and suction.***Z. Angew. Math. Phys.*1994, 45: 258-274. 10.1007/BF00943504MATHMathSciNetView ArticleGoogle Scholar - Weidman PD, Kubitschek DG, Davis AMJ:
**The effect of transpiration on self-similar boundary layer flow over moving surfaces.***Int. J. Eng. Sci.*2006, 44: 730-737. 10.1016/j.ijengsci.2006.04.005MATHView ArticleGoogle Scholar - Ishak A, Nazar R, Pop I:
**Boundary layer on a moving wall with suction and injection.***Chin. Phys. Lett.*2007, 24: 2274-2276. 10.1088/0256-307X/24/8/033View ArticleGoogle Scholar - Zheng L, Wang L, Zhang X:
**Analytic solutions of unsteady boundary flow and heat transfer on a permeable stretching sheet with non-uniform heat source/sink.***Commun. Nonlinear Sci. Numer. Simul.*2011, 16: 731-740. 10.1016/j.cnsns.2010.05.022MATHView ArticleGoogle Scholar - Zhu J, Zheng L, Zhang X:
**Homotophy analysis method for hydromagnetic plane and axisymmetric stagnation-point flow with velocity slip.***World Acad. Sci., Eng. Technol.*2010, 63: 151-155.Google Scholar - Zhu J, Zheng L, Zhang X:
**The effect of the slip condition on the MHD stagnation-point over a power-law stretching sheet.***Appl. Math. Mech.*2010, 31: 439-448. 10.1007/s10483-010-0404-zMATHMathSciNetView ArticleGoogle Scholar - Brinkman HC:
**The viscosity of concentrated suspensions and solutions.***J. Chem. Phys.*1952, 20: 571-581. 10.1063/1.1700493View ArticleGoogle Scholar - Bhattacharyya K, Layek GC:
**Effects of suction/blowing on steady boundary layer stagnation-point flow and heat transfer towards a shrinking sheet with thermal radiation.***Int. J. Heat Mass Transf.*2011, 54: 302-307. 10.1016/j.ijheatmasstransfer.2010.09.043MATHView ArticleGoogle Scholar - Bhattacharyya K, Mukhopadhyay S, Layek GC:
**Slip effects on boundary layer stagnation-point flow and heat transfer towards a shrinking sheet.***Int. J. Heat Mass Transf.*2011, 54: 308-313. 10.1016/j.ijheatmasstransfer.2010.09.041MATHView ArticleGoogle Scholar - Bailey PB, Shampine LF, Waltman PE:
*Nonlinear Two Point Boundary Value Problems*. Academic Press, New York; 1968.MATHGoogle 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-XMATHView ArticleGoogle Scholar - Postelnicu A, Pop I:
**Falkner-Skan boundary layer flow of a power-law fluid past a stretching wedge.***Appl. Math. Comput.*2011, 217: 4359-4368. 10.1016/j.amc.2010.09.037MATHMathSciNetView ArticleGoogle Scholar

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