Research Article  Open Access
Syahira Mansur, Anuar Ishak, Ioan Pop, "ThreeDimensional Flow and Heat Transfer Past a Permeable Exponentially Stretching/Shrinking Sheet in a Nanofluid", Journal of Applied Mathematics, vol. 2014, Article ID 517273, 6 pages, 2014. https://doi.org/10.1155/2014/517273
ThreeDimensional Flow and Heat Transfer Past a Permeable Exponentially Stretching/Shrinking Sheet in a Nanofluid
Abstract
The threedimensional flow and heat transfer of a nanofluid over a stretching/shrinking sheet is investigated. Numerical results are obtained using bvp4c in MATLAB. The results show nonunique solutions for the shrinking case. The effects of the stretching/shrinking parameter, suction parameter, Brownian motion parameter, thermophoresis parameter, and Lewis number on the local skin friction coefficient and the local Nusselt number are studied. Suction increases the solution domain. Furthermore, as the sheet is shrunk in the direction, suction increases the skin friction coefficient in the same direction while decreasing the skin friction coefficient in the direction. The local Nusselt number is consistently lower for higher values of thermophoresis parameter and Lewis number. On the other hand, the local Nusselt number increases as the Brownian motion parameter increases.
1. Introduction
Nanofluids are dispersions of nanometersized particles in a base fluid such as water, ethylene glycol, and propylene glycol, to increase their thermal conductivities. Choi and Eastman [1] showed that the addition of a small amount (less than 1% by volume) of nanoparticles to conventional heat transfer liquids increased the thermal conductivity of the fluid up to approximately two times. In his paper, Buongiorno [2] developed a model for convective transport in nanofluids which takes into account the Brownian diffusion and thermophoresis effects. Buongiorno’s nanofluid model was used in many recent papers, for example, Nield and Kuznetsov [3–5], Khan and Pop [6], Bachok et al. [7–9], Mansur and Ishak [10, 11], and Zaimi et al. [12] among others.
The boundary layer flow over a stretching sheet is significant in applications such as extrusion, wire drawing, metal spinning, and hot rolling [13]. Wang [14, 15], Mandal and Mukhopadhyay [16], P. S. Gupta and A. S. Gupta [17], Andersson [18], Ishak et al. [19], and Makinde and Aziz [20] are among various names who published their papers on a stretching sheet. Miklavčič and Wang [21] studied flow over a shrinking sheet in which they observed that the vorticity is not confined within a boundary layer and the steady flow cannot exist without exerting adequate suction at the boundary. As the studies of shrinking sheet garner considerable attention, this finding proves to be crucial to these researches. In response to Miklavčič and Wang, numerous studies on these problems have been conducted by researches, namely, Wang [22], Fang et al. [23], Bachok et al. [24], Bhattacharyya et al. [25], Zaimi et al. [26], and Roşca and Pop [27] among others.
All the abovementioned studies dealt with problems involving linear stretching/shrinking sheet. The boundary layer flow induced by a stretching/shrinking sheet is very important in engineering processes [28] and has attracted many researchers to delve into this study such as Bachok et al. [29], Bhattacharyya and Vajravelu [30], and Rohni et al. [31]. However, most studies revolved around twodimensional flows. Motivated by this, the objective of this paper is to solve the problem of threedimensional flow and heat transfer past a permeable exponentially stretching/shrinking sheet in a nanofluid. The dependency of the local skin friction coefficient and the local Nusselt number on several parameters, namely, the stretching/shrinking, Brownian motion, and thermophoresis parameters is the main focus of the present investigation. Numerical solutions are presented graphically and in tabular forms to show the effects of these parameters on the local skin friction coefficient and the local Nusselt number.
2. Problem Formulation
We consider the steady threedimensional boundary layer flow of a viscous nanofluid past a permeable stretching/shrinking flat surface in a quiescent fluid. A locally orthogonal set of coordinates is chosen with the origin in the plane of the stretching/shrinking sheet. The  and coordinates are in the plane of the sheet, while the coordinate is measured in the perpendicular direction to the stretching/shrinking surface as shown in Figure 1. It is assumed that the flat surface is stretched/shrunk continuously in the both  and directions with the velocities and , respectively. It is also assumed that the mass flux velocity is , where is for suction and is for injection or withdrawal of the fluid. Further, we assume that the constant surface temperature and the constant surface volume fraction are and , while the constant temperature and the constant surface volume fraction of the ambient (inviscid) fluid are and , respectively. Under these conditions, the boundary layer equations are along with the boundary conditions Here , and are the velocity components along , and axes, respectively; is the kinematic viscosity of the fluid, is the constant stretching or shrinking parameter in the direction, and is the constant stretching or shrinking parameter in the  direction, respectively. Further, we assume that and are of the following form: where is the characteristic length and is the characteristic velocity of the stretching/shrinking sheet.
We introduce now the following similarity variables: where primes denote differentiation with respect to . Next we take where is the surface mass transfer parameter with for suction and for injection. Substituting the similarity variables (9) into (1) to (6), it is found that the continuity equation (1) is automatically satisfied, and (2) to (6) are reduced to the following ordinary (similarity) differential equations: subject to the boundary conditions where is the Prandtl number, is the Lewis number, is the Brownian motion parameter, and is the thermophoresis parameter, which are defined as follows: The physical quantities of practical interest are the local skin friction coefficients, and , and the local Nusselt number , which are defined as follows: where and are the shear stresses in the  and directions of the stretching/shrinking sheet and is the heat flux from the surface of the sheet, which are given by Substituting (9) into (14) and using (15), we obtain where and are the local Reynolds numbers.
3. Results and Discussions
The system of ordinary differential equations (11) subject to the boundary conditions (12) was solved numerically using the package bvp4c in MATLAB for different values of parameters: the stretching/shrinking parameter in direction , suction , Brownian motion parameter Nb, thermophoresis parameter Nt, and Lewis number Le. We fixed the Prandtl number to be equal to 6.8 and the stretching/shrinking parameter in the direction to be 1 () throughout the paper. The relative tolerance is set to 10^{−10} and the boundary conditions (12) at are replaced by . This choice is sufficient for the velocity and the temperature profiles to reach the far field boundary conditions asymptotically.
In this paper, we intend to study the threedimensional flow and heat transfer of a nanofluid over a stretching/shrinking sheet. The analysis shows that the existence of solution depends on the suction parameter and the stretching/shrinking parameter . Figures 2 and 3 show that the skin friction coefficient in the direction and the direction, respectively, decreases as increases. From these figures, we can see that dual solutions exist for the problem. However, based on the previous studies [27, 32, 33], only the first solution is physically realizable and thus relevant to that studies. It is portrayed in Figures 2 and 3 that unique solution exists for and , where is the critical values of . Furthermore, it is seen that the range of , where solutions exist, increases as increases, as shown in Table 1. In addition, in Figure 2, it is shown that when the sheet is shrunk in the direction, the skin friction coefficient parallel to the direction increases as increases. However, the skin friction coefficient in the direction decreases with increasing as illustrated in Figure 3. Moreover, it is interesting to note that the shear stress in the direction is prominently higher than the shear stress in the direction.

Figure 4 shows that the local Nusselt number increases with . However, the local Nusselt number decreases as thermophoresis parameter increases. This phenomenon may be caused by the thermal boundary layer that thickens as the thermophoresis parameter is increased. As opposed to this occurrence, the thermal boundary layer becomes thinner as the Brownian motion parameter increases. This leads to the increase of the local Nusselt number as Brownian motion parameter increases as shown in Table 2. The table also shows that the Lewis number lowers the local Nusselt number.

Figures 5–7 show the velocity profiles for the flow in the  and directions for different values and . These profiles show that the far field boundary conditions are satisfied which validates the numerical result. Furthermore, these profiles also support the existence of dual solutions. The effect of on both and is shown in Figures 5 and 6, respectively. From the two figures, it is noted that while increases the velocity , it decreases the velocity . Figure 7 then shows the effect of on the velocity profiles and . Increasing the stretching parameter in the direction causes to increase. On the other hand, the velocity is consistently lower for higher although it is seen that the changes are minuscule.
4. Conclusions
The threedimensional flow and heat transfer of a nanofluid over a stretching/shrinking sheet was investigated numerically. The effects of various parameters on the skin friction coefficient and the local Nusselt number were discussed. The results showed that suction parameter increases the solution domain. Furthermore, as the sheet is shrunk in the direction, suction increases the skin friction coefficient in the same direction while decreasing the skin friction coefficient in the direction. As thermophoresis parameter and Lewis number increase, the local Nusselt number decreases. On the other hand, the local Nusselt number increases as Brownian motion parameter increases.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
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.
References
 S. U. S. Choi and J. A. Eastman, “Enhancing thermal conductivities of fluids with nanoparticles,” in Proceedings of the ASME International Mechanical Engineering Congress and Exposition, vol. 66, pp. 99–105, San Francisco, Calif, USA, 1995. View at: Google Scholar
 J. Buongiorno, “Convective transport in nanofluids,” Journal of Heat Transfer, vol. 128, no. 3, pp. 240–250, 2006. View at: Publisher Site  Google Scholar
 D. A. Nield and A. V. Kuznetsov, “The ChengMinkowycz problem for natural convective boundarylayer flow in a porous medium saturated by a nanofluid,” International Journal of Heat and Mass Transfer, vol. 52, no. 2526, pp. 5792–5795, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 D. A. Nield and A. V. Kuznetsov, “Thermal instability in a porous medium layer saturated by a nanofluid,” International Journal of Heat and Mass Transfer, vol. 52, no. 2526, pp. 5796–5801, 2009. View at: Publisher Site  Google Scholar
 D. A. Nield and A. V. Kuznetsov, “The ChengMinkowycz problem for the doublediffusive natural convective boundary layer flow in a porous medium saturated by a nanofluid,” International Journal of Heat and Mass Transfer, vol. 54, no. 1–3, pp. 374–378, 2011. View at: Publisher Site  Google Scholar
 W. A. Khan and I. Pop, “Boundarylayer flow of a nanofluid past a stretching sheet,” International Journal of Heat and Mass Transfer, vol. 53, no. 1112, pp. 2477–2483, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 N. Bachok, A. Ishak, and I. Pop, “Boundarylayer flow of nanofluids over a moving surface in a flowing fluid,” International Journal of Thermal Sciences, vol. 49, no. 9, pp. 1663–1668, 2010. View at: Publisher Site  Google Scholar
 N. Bachok, A. Ishak, and I. Pop, “Unsteady boundarylayer flow and heat transfer of a nanofluid over a permeable stretching/shrinking sheet,” International Journal of Heat and Mass Transfer, vol. 55, no. 78, pp. 2102–2109, 2012. View at: Publisher Site  Google Scholar
 N. Bachok, A. Ishak, and I. Pop, “Boundary layer stagnationpoint flow toward a stretching/shrinking sheet in a nanofluid,” ASME Journal of Heat Transfer, vol. 135, no. 5, Article ID 054501, 5 pages, 2013. View at: Publisher Site  Google Scholar
 S. Mansur and A. Ishak, “The flow and heat transfer of a nanofluid past a stretching/shrinking sheet with a convective boundary condition,” Abstract and Applied Analysis, vol. 2013, Article ID 350647, 9 pages, 2013. View at: Publisher Site  Google Scholar  MathSciNet
 S. Mansur and A. Ishak, “The magnetohydrodynamic boundary layer flow of a nanofluid past a stretching/shrinking sheet with slip boundary conditions,” Journal of Applied Mathematics, vol. 2014, Article ID 907152, 7 pages, 2014. View at: Publisher Site  Google Scholar
 K. Zaimi, A. Ishak, and I. Pop, “Unsteady flow due to a contracting cylinder in a nanofluid using Buongiorno's model,” International Journal of Heat and Mass Transfer, vol. 68, pp. 509–513, 2014. View at: Google Scholar
 E. G. Fischer, Extrusion of Plastics, Wiley, New York, NY, USA, 1976.
 C. Y. Wang, “Flow due to a stretching boundary with partial slip—an exact solution of the NavierStokes equations,” Chemical Engineering Science, vol. 57, no. 17, pp. 3745–3747, 2002. View at: Publisher Site  Google Scholar
 C. Y. Wang, “Analysis of viscous flow due to a stretching sheet with surface slip and suction,” Nonlinear Analysis: Real World Applications, vol. 10, no. 1, pp. 375–380, 2009. View at: Publisher Site  Google Scholar  MathSciNet
 I. C. Mandal and S. Mukhopadhyay, “Heat transfer analysis for fluid flow over an exponentially stretching porous sheet with surface heat flux in porous medium,” Ain Shams Engineering Journal, vol. 4, no. 1, pp. 103–110, 2013. View at: Publisher Site  Google Scholar
 P. S. Gupta and A. S. Gupta, “Heat and mass transfer on a stretching sheet with suction or blowing,” The Canadian Journal of Chemical Engineering, vol. 55, pp. 744–746, 1977. View at: Google Scholar
 H. I. Andersson, “Slip flow past a stretching surface,” Acta Mechanica, vol. 158, no. 12, pp. 121–125, 2002. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 A. Ishak, R. Nazar, and I. Pop, “Heat transfer over an unsteady stretching permeable surface with prescribed wall temperature,” Nonlinear Analysis: Real World Applications, vol. 10, no. 5, pp. 2909–2913, 2009. View at: Publisher Site  Google Scholar  MathSciNet
 O. D. Makinde and A. Aziz, “Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition,” International Journal of Thermal Sciences, vol. 50, no. 7, pp. 1326–1332, 2011. View at: Publisher Site  Google Scholar
 M. Miklavčič and C. Y. Wang, “Viscous flow due to a shrinking sheet,” Quarterly of Applied Mathematics, vol. 64, no. 2, pp. 283–290, 2006. View at: Google Scholar  MathSciNet
 C. Y. Wang, “Stagnation flow towards a shrinking sheet,” International Journal of NonLinear Mechanics, vol. 43, no. 5, pp. 377–382, 2008. View at: Publisher Site  Google Scholar
 T. Fang, S. Yao, J. Zhang, and A. Aziz, “Viscous flow over a shrinking sheet with a second order slip flow model,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 7, pp. 1831–1842, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 N. Bachok, A. Ishak, and I. Pop, “Stagnationpoint flow over a stretching/shrinking sheet in a nanofluid,” Nanoscale Research Letters, vol. 6, article 623, 10 pages, 2011. View at: Publisher Site  Google Scholar
 K. Bhattacharyya, S. Mukhopadhyay, and G. C. Layek, “Slip effects on boundary layer stagnationpoint flow and heat transfer towards a shrinking sheet,” International Journal of Heat and Mass Transfer, vol. 54, no. 1–3, pp. 308–313, 2011. View at: Publisher Site  Google Scholar
 K. Zaimi, A. Ishak, and I. Pop, “Boundary layer flow and heat transfer past a permeable shrinking sheet in a nanofluid with radiation effect,” Advances in Mechanical Engineering, vol. 2012, Article ID 340354, 7 pages, 2012. View at: Publisher Site  Google Scholar
 A. V. Roşca and I. Pop, “Flow and heat transfer over a vertical permeable stretching/shrinking sheet with a second order slip,” International Journal of Heat and Mass Transfer, vol. 60, no. 1, pp. 355–364, 2013. View at: Publisher Site  Google Scholar
 K. Bhattacharyya, “Boundary layer flow and heat transfer over an exponentially shrinking sheet,” Chinese Physics Letters, vol. 28, no. 7, Article ID 074701, 2011. View at: Publisher Site  Google Scholar
 N. Bachok, A. Ishak, and I. Pop, “Boundary layer stagnationpoint flow and heat transfer over an exponentially stretching/shrinking sheet in a nanofluid,” International Journal of Heat and Mass Transfer, vol. 55, no. 2526, pp. 8122–8128, 2012. View at: Publisher Site  Google Scholar
 K. Bhattacharyya and K. Vajravelu, “Stagnationpoint flow and heat transfer over an exponentially shrinking sheet,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 7, pp. 2728–2734, 2012. View at: Publisher Site  Google Scholar
 A. M. Rohni, S. Ahmad, A. I. M. Ismail, and I. Pop, “Boundary layer flow and heat transfer over an exponentially shrinking vertical sheet with suction,” International Journal of Thermal Sciences, vol. 64, pp. 264–272, 2013. View at: Publisher Site  Google Scholar
 P. D. Weidman, D. G. Kubitschek, and A. M. J. Davis, “The effect of transpiration on selfsimilar boundary layer flow over moving surfaces,” International Journal of Engineering Science, vol. 44, no. 1112, pp. 730–737, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 A. Postelnicu and I. Pop, “FalknerSkan boundary layer flow of a powerlaw fluid past a stretching wedge,” Applied Mathematics and Computation, vol. 217, no. 9, pp. 4359–4368, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
Copyright
Copyright © 2014 Syahira Mansur et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.