JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 10.1155/2014/517273 517273 Research Article Three-Dimensional Flow and Heat Transfer Past a Permeable Exponentially Stretching/Shrinking Sheet in a Nanofluid http://orcid.org/0000-0003-0057-9035 Mansur Syahira 1 Ishak Anuar 2 Pop Ioan 3 Valencia Alvaro 1 Department of Mathematics and Statistics Faculty of Science, Technology and Human Development Universiti Tun Hussein Onn Malaysia Parit Raja, 86400 Batu Pahat, Johor Malaysia uthm.edu.my 2 Centre for Modelling and Data Analysis School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia (UKM) 43600 Bangi, Selangor Malaysia ukm.my 3 Department of Mathematics Babeș-Bolyai University 400084 Cluj-Napoca Romania ubbcluj.ro 2014 2182014 2014 14 04 2014 11 08 2014 24 8 2014 2014 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.

The three-dimensional 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 x -direction, suction increases the skin friction coefficient in the same direction while decreasing the skin friction coefficient in the y -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 nanometer-sized particles in a base fluid such as water, ethylene glycol, and propylene glycol, to increase their thermal conductivities. Choi and Eastman  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  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 , Khan and Pop , Bachok et al. , Mansur and Ishak [10, 11], and Zaimi et al.  among others.

The boundary layer flow over a stretching sheet is significant in applications such as extrusion, wire drawing, metal spinning, and hot rolling . Wang [14, 15], Mandal and Mukhopadhyay , P. S. Gupta and A. S. Gupta , Andersson , Ishak et al. , and Makinde and Aziz  are among various names who published their papers on a stretching sheet. Miklavčič and Wang  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 , Fang et al. , Bachok et al. , Bhattacharyya et al. , Zaimi et al. , and Roşca and Pop  among others.

All the above-mentioned 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  and has attracted many researchers to delve into this study such as Bachok et al. , Bhattacharyya and Vajravelu , and Rohni et al. . However, most studies revolved around two-dimensional flows. Motivated by this, the objective of this paper is to solve the problem of three-dimensional 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 three-dimensional boundary layer flow of a viscous nanofluid past a permeable stretching/shrinking flat surface in a quiescent fluid. A locally orthogonal set of coordinates ( x , y , z ) is chosen with the origin O in the plane of the stretching/shrinking sheet. The x - and y -coordinates are in the plane of the sheet, while the coordinate z 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 x - and y -directions with the velocities u ( x ) =    u w ( x ) and v ( y ) = v w ( y ) , respectively. It is also assumed that the mass flux velocity is w w ( x , y ) , where w w ( x , y ) < 0 is for suction and w w ( x , y ) > 0 is for injection or withdrawal of the fluid. Further, we assume that the constant surface temperature and the constant surface volume fraction are T w and C w , while the constant temperature and the constant surface volume fraction of the ambient (inviscid) fluid are T and C , respectively. Under these conditions, the boundary layer equations are (1) u x + v y + w z = 0 , (2) u u x + v u y + w u z = ν 2 u z 2 , (3) u v x + v v y + w v z = ν 2 v z 2 , (4)       u w x + v w y + w w z = ν 2 w z 2 , (5) u T x + v T y + w T z = α 2 T z 2 + τ [ D B C z T z + D T T ( T z ) 2 ] , (6) u C x + v C y + w C z = D B 2 C z 2 + D T T 2 T z 2 , along with the boundary conditions (7) u = u w ( x ) = λ 1 U w ( x ) , v = v w ( y ) = λ 2 V w ( y ) , w = w w = w 0 , T = T w , D B C z + D T T T z = 0 at z = 0 u 0 , w 0 , T T , C C as z . Here u ,    v , and w are the velocity components along x - ,    y -, and z -axes, respectively; ν is the kinematic viscosity of the fluid, λ 1 is the constant stretching ( λ 1 > 0 ) or shrinking ( λ 1 < 0 ) parameter in the x -direction, and λ 2 is the constant stretching ( λ 2 > 0 ) or shrinking ( λ 2 < 0 ) parameter in the y - direction, respectively. Further, we assume that U w ( x , y ) and V w ( x , y ) are of the following form: (8) U w ( x , y ) = V w ( x , y ) = U 0    e ( x + y ) / L , where L is the characteristic length and U 0 is the characteristic velocity of the stretching/shrinking sheet.

Geometry of the problem.

We introduce now the following similarity variables: (9) u = U 0 e ( x + y ) / L f ( η ) , v = U 0 e ( x + y ) / L g ( η ) , w = - ( ν U 0 2 L ) 1 / 2 e ( x + y ) / 2 L [ f ( η ) + η f ( η ) + g ( η ) + η g ( η ) ] , θ ( η ) = ( T - T ) ( T w - T ) , ϕ ( η ) = ( C - C ) ( C w - C ) , η = ( U 0 2 ν L ) 1 / 2 e ( x + y ) / 2 L z , where primes denote differentiation with respect to η . Next we take (10) w w ( x , y ) = - ( ν U 0 2 L ) 1 / 2 e ( x + y ) / 2 L S , where S is the surface mass transfer parameter with S > 0 for suction and S < 0 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: (11) f ′′′ + ( f + g ) f ′′ - 2 ( f + g ) f = 0 , g ′′′ + ( f + g ) g ′′ - 2 ( f + g ) g = 0 , 1 Pr θ ′′ + ( f + g ) θ + Nb ϕ θ + Nt θ 2 = 0 , ϕ ′′ + Le ( f + g ) ϕ + Nt Nb    θ ′′ = 0 subject to the boundary conditions (12) f ( 0 ) = S , g ( 0 ) = 0 , f ( 0 ) = λ 1 , g ( 0 ) =    λ 2 , θ ( 0 ) = 1 , Nt ϕ ( 0 ) + Nb θ ( 0 ) = 0 f ( η ) 0 , g ( η ) 0 , θ ( η ) 0 , ϕ ( η ) 0 as η , where Pr is the Prandtl number, Le is the Lewis number, Nb is the Brownian motion parameter, and Nt is the thermophoresis parameter, which are defined as follows: (13) Pr = ν α , Le = ν D B , Nb = β D B ( C w - C ) ν , Nt = β D T ( T w - T ) T ν . The physical quantities of practical interest are the local skin friction coefficients, C f x and C f y , and the local Nusselt number Nu x , which are defined as follows: (14) C f x = 2 τ w x ρ U w 2 , C f y = 2 τ w y ρ U w 2 , Nu x =    q w k ( T w - T ) , where τ w x and τ w y are the shear stresses in the x - and y -directions of the stretching/shrinking sheet and q w is the heat flux from the surface of the sheet, which are given by (15) τ w x = μ ( u z ) z = 0 , τ w y = μ ( v z ) z = 0 , q w = -    k ( T z ) z = 0 . Substituting (9) into (14) and using (15), we obtain (16) R e x 1 / 2 C f x = 2 f ′′ ( 0 ) , R e y 1 / 2 C f y = 2 g ′′ ( 0 ) , R e x -    1 / 2 Nu x = -    θ ( 0 ) , where R e x = U w L / ν and R e y = V w L / ν 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 x -direction λ 1 , suction S , 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 y -direction λ 2 to be 1 ( λ 2 = 1 ) throughout the paper. The relative tolerance is set to 10−10 and the boundary conditions (12) at η = are replaced by η = 10 . 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 three-dimensional 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 S and the stretching/shrinking parameter λ 1 . Figures 2 and 3 show that the skin friction coefficient in the x -direction and the y -direction, respectively, decreases as λ 1 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 λ 1 - 1 and λ 1 = λ c , where λ c is the critical values of λ 1 . Furthermore, it is seen that the range of λ 1 , where solutions exist, increases as S increases, as shown in Table 1. In addition, in Figure 2, it is shown that when the sheet is shrunk in the x -direction, the skin friction coefficient parallel to the direction increases as S increases. However, the skin friction coefficient in the y -direction decreases with increasing S as illustrated in Figure 3. Moreover, it is interesting to note that the shear stress in the x -direction is prominently higher than the shear stress in the y -direction.

Values of λ 1 c .

S λ c
2.0 −1.7785
2.5 −2.2164
3.0 −2.7157

Variation of the skin friction coefficient in x -direction with λ 1 for different values of S when λ 2 = 1 .

Variation of the skin friction coefficient in y -direction with λ 1 for different values of S when λ 2 = 1 .

Figure 4 shows that the local Nusselt number increases with λ 1 . 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.

The local Nusselt number for different Nb and Le when λ 1 = 1, Pr = 6.8, and Nt = 0.1.

Nb Le Nu x Re x - 1 / 2
0.10 5 11.689893
0.12 14.032429
0.14 19.445658

0.10 5 11.689893
8 11.260672
10 11.105908

Variation of the local Nusselt number with λ 1 for different values of Nt when λ 2 = 1 , Pr = 6.8, Nb = 0.1, and Le = 5.

Figures 57 show the velocity profiles for the flow in the x - and y -directions for different values S and λ 1 . 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 S on both f ( η ) and g ( η ) is shown in Figures 5 and 6, respectively. From the two figures, it is noted that while S increases the velocity f ( η ) , it decreases the velocity g ( η ) . Figure 7 then shows the effect of λ on the velocity profiles f ( η ) and g ( η ) . Increasing the stretching parameter in the x -direction causes f ( η ) to increase. On the other hand, the velocity g ( η ) is consistently lower for higher λ 1 although it is seen that the changes are minuscule.

Velocity profiles in x -direction for different values of S when λ 1 = - 1.7 and λ 2 = 1 .

Velocity profiles in y -direction for different values of S when λ 1 = - 1.7 and λ 2 = 1 .

Velocity profiles for different values of λ 1 when λ 2 = 1 .

4. Conclusions

The three-dimensional 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 x -direction, suction increases the skin friction coefficient in the same direction while decreasing the skin friction coefficient in the y -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: DIP-2012-31) are gratefully acknowledged.

Choi S. U. S. Eastman J. A. Enhancing thermal conductivities of fluids with nanoparticles 66 Proceedings of the ASME International Mechanical Engineering Congress and Exposition 1995 San Francisco, Calif, USA 99 105 Buongiorno J. Convective transport in nanofluids Journal of Heat Transfer 2006 128 3 240 250 10.1115/1.2150834 2-s2.0-33645634748 Nield D. A. Kuznetsov A. V. The Cheng-Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid International Journal of Heat and Mass Transfer 2009 52 25-26 5792 5795 10.1016/j.ijheatmasstransfer.2009.07.024 ZBL1177.80046 2-s2.0-77649338479 Nield D. A. Kuznetsov A. V. Thermal instability in a porous medium layer saturated by a nanofluid International Journal of Heat and Mass Transfer 2009 52 25-26 5796 5801 10.1016/j.ijheatmasstransfer.2009.07.023 2-s2.0-75849141594 Nield D. A. Kuznetsov A. V. The Cheng-Minkowycz problem for the double-diffusive natural convective boundary layer flow in a porous medium saturated by a nanofluid International Journal of Heat and Mass Transfer 2011 54 1–3 374 378 10.1016/j.ijheatmasstransfer.2010.09.034 2-s2.0-78449272982 Khan W. A. Pop I. Boundary-layer flow of a nanofluid past a stretching sheet International Journal of Heat and Mass Transfer 2010 53 11-12 2477 2483 10.1016/j.ijheatmasstransfer.2010.01.032 ZBL1190.80017 2-s2.0-77649341610 Bachok N. Ishak A. Pop I. Boundary-layer flow of nanofluids over a moving surface in a flowing fluid International Journal of Thermal Sciences 2010 49 9 1663 1668 10.1016/j.ijthermalsci.2010.01.026 2-s2.0-77955432252 Bachok N. Ishak A. Pop I. Unsteady boundary-layer flow and heat transfer of a nanofluid over a permeable stretching/shrinking sheet International Journal of Heat and Mass Transfer 2012 55 7-8 2102 2109 10.1016/j.ijheatmasstransfer.2011.12.013 2-s2.0-84856388565 Bachok N. Ishak A. Pop I. Boundary layer stagnation-point flow toward a stretching/shrinking sheet in a nanofluid ASME Journal of Heat Transfer 2013 135 5 5 054501 10.1115/1.4023303 2-s2.0-84876218991 Mansur S. Ishak A. The flow and heat transfer of a nanofluid past a stretching/shrinking sheet with a convective boundary condition Abstract and Applied Analysis 2013 2013 9 350647 MR3111810 10.1155/2013/350647 Mansur S. Ishak A. The magnetohydrodynamic boundary layer flow of a nanofluid past a stretching/shrinking sheet with slip boundary conditions Journal of Applied Mathematics 2014 2014 7 907152 10.1155/2014/907152 Zaimi K. Ishak A. Pop I. Unsteady flow due to a contracting cylinder in a nanofluid using Buongiorno's model International Journal of Heat and Mass Transfer 2014 68 509 513 Fischer E. G. Extrusion of Plastics 1976 New York, NY, USA Wiley Wang C. Y. Flow due to a stretching boundary with partial slip—an exact solution of the Navier-Stokes equations Chemical Engineering Science 2002 57 17 3745 3747 10.1016/S0009-2509(02)00267-1 2-s2.0-0037073060 Wang C. Y. Analysis of viscous flow due to a stretching sheet with surface slip and suction Nonlinear Analysis: Real World Applications 2009 10 1 375 380 10.1016/j.nonrwa.2007.09.013 MR2451717 2-s2.0-50349092405 Mandal I. C. Mukhopadhyay S. Heat transfer analysis for fluid flow over an exponentially stretching porous sheet with surface heat flux in porous medium Ain Shams Engineering Journal 2013 4 1 103 110 10.1016/j.asej.2012.06.004 2-s2.0-84875219840 Gupta P. S. Gupta A. S. Heat and mass transfer on a stretching sheet with suction or blowing The Canadian Journal of Chemical Engineering 1977 55 744 746 Andersson H. I. Slip flow past a stretching surface Acta Mechanica 2002 158 1-2 121 125 10.1007/BF01463174 ZBL1013.76020 2-s2.0-0036412278 Ishak A. Nazar R. Pop I. Heat transfer over an unsteady stretching permeable surface with prescribed wall temperature Nonlinear Analysis: Real World Applications 2009 10 5 2909 2913 10.1016/j.nonrwa.2008.09.010 MR2523255 2-s2.0-64449088714 Makinde O. D. Aziz A. Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition International Journal of Thermal Sciences 2011 50 7 1326 1332 10.1016/j.ijthermalsci.2011.02.019 2-s2.0-79955469700 Miklavčič M. Wang C. Y. Viscous flow due to a shrinking sheet Quarterly of Applied Mathematics 2006 64 2 283 290 MR2243864 2-s2.0-33748030110 Wang C. Y. Stagnation flow towards a shrinking sheet International Journal of Non-Linear Mechanics 2008 43 5 377 382 10.1016/j.ijnonlinmec.2007.12.021 2-s2.0-43249115375 Fang T. Yao S. Zhang J. Aziz A. Viscous flow over a shrinking sheet with a second order slip flow model Communications in Nonlinear Science and Numerical Simulation 2010 15 7 1831 1842 10.1016/j.cnsns.2009.07.017 MR2585050 ZBL1222.76028 2-s2.0-74449090532 Bachok N. Ishak A. Pop I. Stagnation-point flow over a stretching/shrinking sheet in a nanofluid Nanoscale Research Letters 2011 6, article 623 10 10.1186/1556-276X-6-623 2-s2.0-84856066520 Bhattacharyya K. Mukhopadhyay S. Layek G. C. Slip effects on boundary layer stagnation-point flow and heat transfer towards a shrinking sheet International Journal of Heat and Mass Transfer 2011 54 1–3 308 313 10.1016/j.ijheatmasstransfer.2010.09.041 2-s2.0-78449309241 Zaimi K. Ishak A. Pop I. Boundary layer flow and heat transfer past a permeable shrinking sheet in a nanofluid with radiation effect Advances in Mechanical Engineering 2012 2012 7 340354 10.1155/2012/340354 2-s2.0-84872972132 Roşca A. V. Pop I. Flow and heat transfer over a vertical permeable stretching/shrinking sheet with a second order slip International Journal of Heat and Mass Transfer 2013 60 1 355 364 10.1016/j.ijheatmasstransfer.2012.12.028 2-s2.0-84873346346 Bhattacharyya K. Boundary layer flow and heat transfer over an exponentially shrinking sheet Chinese Physics Letters 2011 28 7 074701 10.1088/0256-307X/28/7/074701 2-s2.0-80051662250 Bachok N. Ishak A. Pop I. Boundary layer stagnation-point flow and heat transfer over an exponentially stretching/shrinking sheet in a nanofluid International Journal of Heat and Mass Transfer 2012 55 25-26 8122 8128 10.1016/j.ijheatmasstransfer.2012.08.051 2-s2.0-84867528210 Bhattacharyya K. Vajravelu K. Stagnation-point flow and heat transfer over an exponentially shrinking sheet Communications in Nonlinear Science and Numerical Simulation 2012 17 7 2728 2734 10.1016/j.cnsns.2011.11.011 2-s2.0-84856428088 Rohni A. M. Ahmad S. Ismail A. I. M. Pop I. Boundary layer flow and heat transfer over an exponentially shrinking vertical sheet with suction International Journal of Thermal Sciences 2013 64 264 272 10.1016/j.ijthermalsci.2012.08.016 2-s2.0-84869079413 Weidman P. D. Kubitschek D. G. Davis A. M. J. The effect of transpiration on self-similar boundary layer flow over moving surfaces International Journal of Engineering Science 2006 44 11-12 730 737 10.1016/j.ijengsci.2006.04.005 ZBL1213.76064 2-s2.0-33747331317 Postelnicu A. Pop I. Falkner-Skan boundary layer flow of a power-law fluid past a stretching wedge Applied Mathematics and Computation 2011 217 9 4359 4368 10.1016/j.amc.2010.09.037 MR2745118 ZBL05843663 2-s2.0-78650169529