Numerical investigation of stagnation point flow over a stretching sheet with convective boundary conditions
© Mohamed et al.; licensee Springer. 2013
Received: 31 August 2012
Accepted: 26 December 2012
Published: 16 January 2013
In this study, the mathematical modeling for stagnation point flow over a stretching surface with convective boundary conditions is considered. The transformed boundary layer equations are solved numerically using the shooting method. Numerical solutions are obtained for the skin friction coefficient, the surface temperature as well as the velocity profiles. The features of the flow and heat transfer characteristics for various values of the Prandtl number, stretching parameter and conjugate parameter are analyzed and discussed.
Keywordsconvective boundary conditions mathematical modeling stagnation point flow stretching sheet
Problems related to convection boundary layer flows are important in engineering and industrial activities. Such flows are applied to manage thermal effects in many industrial outputs, for example, in electronic devices, computer power supply and also in an engine cooling system such as a heatsink in a car radiator. Sakiadis  was the first to study the boundary layer flow on a continuous solid surface moving at a constant speed. Due to entrainment of the ambient fluid, this boundary layer flow is quite different from the Blasius flow past a flat plate. Sakiadis’s theoretical predictions for Newtonian fluids were later corroborated experimentally by Tsou et al. . Flow of a viscous fluid past a stretching sheet is a classical problem in fluid dynamics. Crane  was the first to study the convection boundary layer flow over a stretching sheet. The heat and mass transfer on a stretching sheet with suction or blowing was investigated by Gupta and Gupta . They considered an isothermal moving plate and obtained the temperature and concentration distributions. Chen and Char  studied the laminar boundary layer flow and heat transfer from a linearly stretching, continuous sheet subjected to suction or blowing with prescribed wall temperature and heat flux. Stagnation flow towards a shrinking sheet was then investigated by Wang  who considered the prescribed wall temperature case. Ishak et al. [7–9] studied the MHD stagnation point flow towards a stretching sheet, mixed convection towards a vertical and continuosly stretching sheet and post stagnation-point towards a vertical and linearly stretching sheet. This type of problem was then extended to viscous fluids, viscoelastic fluids or micropolar fluids by many investigators by considering the usually applied boundary conditions, either prescribed wall temperature or prescribed wall heat flux. Recently, Mohamed et al.  studied the stagnation point flow over a stretching sheet and Hayat et al.  investigated the flow of a second grade fluid over a stretching surface with Newtonian heating.
On the other hand, Merkin  has shown that in general, there are four common heating processes specifying the wall-to-ambient temperature distributions, namely (i) constant or prescribed wall temperature; (ii) constant or prescribed surface heat flux; (iii) Newtonian heating (NH); and (iv) convective/conjugate boundary conditions (CBC), where heat is supplied through a bounding surface of finite thickness and finite heat capacity. The interface temperature is not known a priori but depends on the intrinsic properties of the system, namely the thermal conductivity of the fluid or solid. Recent demands in heat transfer engineering have requested researchers to develop various new types of heat transfer equipments with superior performance, especially compact and light-weight ones. With the increasing need for small-size units, focus has been cast on the effects of the interaction between developments of thermal boundary layers in both fluid streams and of axial wall conduction, which usually affects the heat exchanges performance. Since the early paper by Luikov et al. , many contributions to the topic of conjugate heat transfer have been made. The conjugate/convective boundary condition has been used only quite recently by Aziz  who studied the laminar thermal boundary layer over a flat plate. This Blasius flow with the conjugate boundary condition then has been revisited by Rashidi and Erfani  and Magyari . Makinde and Aziz  considered the hydromagnetic heat and mass transfer over a vertical plate. Ishak et al. [18, 19] have studied the thermal boundary layer flow on a moving plate (Sakiadis flow) with radiation effects. Recently, Merkin and Pop , Yao et al. , Yacob et al.  and Yacob and Ishak  investigated the boundary layer flow past a shrinking/stretching sheet with convective boundary conditions in a viscous fluid, nanofluid or micropolar fluid, respectively. Excellent reviews of the topics of convective heat transfer problems can be found in the books by Kimura et al.  and Martynenko and Khramtsov .
Motivated by the works of Wang  and Yacob and Ishak , we aim in this study to investigate the problem of stagnation point flow over a stretching sheet with convective boundary conditions. The governing nonlinear partial differential equations are first transformed into a system of ordinary differential equations by a similarity transformation before being solved numerically using the shooting method (see Salleh et al.  for more details about this method).
2 Mathematical formulation
where is the local Reynolds number and is the local Nusselt number.
3 Numerical method
The Runge-Kutta-Fehlberg method will be adopted to solve the applicable initial value problem. In order to integrate Equations (14) and (15) as an IVP, we require a value for and , i.e., and respectively. Since these values are not given in the boundary conditions (16), suitable guess values for and are made and integration is carried out. Then, we compare the calculated values for and at with the given boundary conditions and respectively and adjust the estimated values of , and to give a better approximation for the solution. This computation is done with the aid of shootlib file in Maple software. In this study, the boundary layer thickness between 2 and 8 was used in the computation, depending on the values of the parameters considered so that the boundary condition at ‘infinity’ is achieved.
4 Results and discussion
In this paper, we have theoretically and numerically studied the problem of stagnation point flow over a stretching sheet with the convective boundary condition. It is shown how the Prandtl number Pr, stretching parameter ε and conjugate parameter γ affect the values of the surface temperature and skin friction coefficient .
We can conclude that the thermal boundary layer thickness depends strongly on these three parameters. Further, it is seen that an increase in the Prandtl number Pr and stretching parameter ε results in a decrease of the temperature. The reason is that smaller values of Pr are equivalent to increasing thermal conductivity and therefore, heat is capable of diffusing away from the heated wall more rapidly than at higher values of Pr. However, the increase of conjugate parameter γ leads to an increase of the surface temperature .
The authors wish to thank the anonymous reviewers for their valuable comments and suggestions. The financial support received from the Universiti Malaysia Pahang (Project Codes: RDU110108 and RDU110390) and the Universiti Kebangsaan Malaysia (Project Code: DIP-2012-31) is gratefully acknowledged.
- Sakiadis BC: Boundary-layer behavior on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow. AIChE J. 1961, 7(1):26-28. 10.1002/aic.690070108View ArticleGoogle Scholar
- Tsou FK, Sparrow EM, Goldstein RJ: Flow and heat transfer in the boundary layer on a continuous moving surface. Int. J. Heat Mass Transf. 1967, 10(2):219-235. 10.1016/0017-9310(67)90100-7View ArticleGoogle Scholar
- Crane LJ: Flow past a stretching plate. Z. Angew. Math. Phys. 1970, 21: 645-647. 10.1007/BF01587695View ArticleGoogle Scholar
- Gupta PS, Gupta AS: Heat and mass transfer on a stretching sheet with suction or blowing. Can. J. Chem. Eng. 1977, 55(6):744-746. 10.1002/cjce.5450550619View ArticleGoogle Scholar
- Chen CK, Char M: Heat transfer on a continuous, stretching surface with suction and blowing. J. Math. Anal. Appl. 1988, 135: 568-580. 10.1016/0022-247X(88)90172-2MATHMathSciNetView ArticleGoogle Scholar
- Wang CY: Stagnation flow towards a shrinking sheet. Int. J. Non-Linear Mech. 2008, 43(5):377-382. 10.1016/j.ijnonlinmec.2007.12.021View ArticleGoogle Scholar
- Ishak A, Nazar R, Pop I: Mixed convection on the stagnation point flow toward a vertical, continuously stretching sheet. J. Heat Transf. 2007, 129(8):1087-1090. 10.1115/1.2737482View ArticleGoogle Scholar
- Ishak A, Nazar R, Pop I: Post stagnation point boundary layer flow and mixed convection heat transfer over a vertical, linearly stretching sheet. Arch. Mech. 2008, 60: 303-322.MATHMathSciNetGoogle Scholar
- Ishak A, Jafar K, Nazar R, Pop I: MHD stagnation point flow towards a stretching sheet. Phys. A, Stat. Mech. Appl. 2009, 388(17):3377-3383. 10.1016/j.physa.2009.05.026View ArticleGoogle Scholar
- Mohamed MKA, Salleh MZ, Nazar R, Ishak A: Stagnation point flow over a stretching sheet with Newtonian heating. Sains Malays. 2012, 41(11):1467-1473.Google Scholar
- Hayat T, Iqbal Z, Mustafa M: Flow of a second grade fluid over a stretching surface with Newtonian heating. J. Mech. 2012, 28(1):209-216. 10.1017/jmech.2012.21View ArticleGoogle Scholar
- Merkin JH: Natural-convection boundary-layer flow on a vertical surface with Newtonian heating. Int. J. Heat Fluid Flow 1994, 15(5):392-398. 10.1016/0142-727X(94)90053-1View ArticleGoogle Scholar
- Luikov AV, Aleksashenko VA, Aleksashenko AA: Analytical methods of solution of conjugated problems in convective heat transfer. Int. J. Heat Mass Transf. 1971, 14(8):1047-1056. 10.1016/0017-9310(71)90203-1MATHView ArticleGoogle Scholar
- Aziz A: A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition. Commun. Nonlinear Sci. Numer. Simul. 2009, 14(4):1064-1068. 10.1016/j.cnsns.2008.05.003View ArticleGoogle Scholar
- Rashidi MM, Erfani E: A novel analytical solution of the thermal boundary-layer over a flat plate with a convective surface boundary condition using DTM-Pade. 2009 International Conference on Signal Processing Systems 2009.Google Scholar
- Magyari E: Comment on ‘A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition’ by A. Aziz, Comm. Nonlinear Sci. Numer. Simul. 2009, 14:1064-1068. Commun. Nonlinear Sci. Numer. Simul. 2010, 16: 599-601.View ArticleGoogle Scholar
- Makinde OD, Aziz A: MHD mixed convection from a vertical plate embedded in a porous medium with a convective boundary condition. Int. J. Therm. Sci. 2010, 49(9):1813-1820. 10.1016/j.ijthermalsci.2010.05.015View ArticleGoogle Scholar
- Ishak A: Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition. Appl. Math. Comput. 2010, 217(2):837-842. 10.1016/j.amc.2010.06.026MATHMathSciNetView ArticleGoogle Scholar
- Ishak A, Yacob N, Bachok N: Radiation effects on the thermal boundary layer flow over a moving plate with convective boundary condition. Meccanica 2011, 46(4):795-801. 10.1007/s11012-010-9338-4MATHMathSciNetView ArticleGoogle Scholar
- Merkin JH, Pop I: The forced convection flow of a uniform stream over a flat surface with a convective surface boundary condition. Commun. Nonlinear Sci. Numer. Simul. 2011, 16(9):3602-3609. 10.1016/j.cnsns.2011.01.007MATHMathSciNetView ArticleGoogle Scholar
- Yao S, Fang T, Zhong Y: Heat transfer of a generalized stretching/shrinking wall problem with convective boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 2011, 16(2):752-760. 10.1016/j.cnsns.2010.05.028MATHView 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(1):1-7.View ArticleGoogle Scholar
- Yacob NA, Ishak A: Stagnation point flow towards a stretching/shrinking sheet in a micropolar fluid with a convective surface boundary condition. Can. J. Chem. Eng. 2012, 90(3):621-626. 10.1002/cjce.20517View ArticleGoogle Scholar
- Kimura S, Kiwata T, Okajima A, Pop I: Conjugate natural convection in porous media. Adv. Water Resour. 1997, 20: 111-126. 10.1016/S0309-1708(96)00025-5View ArticleGoogle Scholar
- Martynenko OG, Khramtsov PP: Free Convective Heat Transfer. Springer, Berlin; 2005.Google Scholar
- Salleh MZ, Mohamed N, Khairuddin R, Khasi’ie NS, Nazar R: Numerical study of free convection boundary layer flow on a vertical surface with prescribed wall temperature, heat flux and Newtonian heating using shooting method. Proceedings of the International Conference on Software Engineering and Computer Systems (ICSECS’09) UMP, Kuantan 19-21 October 2009, 94-98.Google Scholar
- Salleh MZ, Nazar R, Pop I: Boundary layer flow and heat transfer over a stretching sheet with Newtonian heating. J. Taiwan Inst. Chem. Eng. 2010, 41(6):651-655. 10.1016/j.jtice.2010.01.013View ArticleGoogle Scholar
- Bailey PB, Shampine LF, Waltman PE: Nonlinear Two Point Boundary Value Problems. Academic Press, New York; 1968.MATHGoogle Scholar
- Meade DB, Haran BS, White RE: The shooting technique for the solution of two-point boundary value problems. Maple Technol. 1996, 3: 85-93.Google 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
- Ishak A: Thermal boundary layer flow over a stretching sheet in a micropolar fluid with radiation effect. Meccanica 2010, 45: 367-373. 10.1007/s11012-009-9257-4MATHView ArticleGoogle Scholar
- Bachok N, Ishak A, Pop I: Melting heat transfer in boundary layer stagnation-point flow towards a stretching/shrinking sheet. Phys. Lett. A 2010, 374: 4075-4079. 10.1016/j.physleta.2010.08.032MATHView ArticleGoogle Scholar
- Hiemenz K: Die Grenzschicht an einem in den gleichformigen Flussigkeitsstrom eingetauchten geraden Kreiszylinder. Dinglers Polytech. J. 1911, 32: 321-410.Google Scholar