Sains Malaysiana 41(11)(2012): 1475–1482

Free Convection Boundary Layer Flow of a Nanofluid from a Convectively Heated Vertical Plate with Linear Momentum Slip Boundary Condition

(Aliran Lapisan Sempadan Olakan Bebas Nanobendalir terhadap Plat Menegak yang dipanaskan

Secara Berolak dengan Syarat Sempadan Gelinciran Momentum Linear)

Md. Jashim Uddin & A.I. Md. Ismail

School of Mathematical Sciences, University of Sains Malaysia, 1800 USM, Penang, Malaysia

I. Pop*

Faculty of Mathematics, University of Cluj, CP 253, R-3400 Cluj, Romania

Diserahkan: 2 Februari 2012 / Diterima: 9 April 2012

ABSTRACT

Two dimensional steady laminar boundary layer flow of a nanofluid over a convectively heated vertical flat plate with linear momentum slip boundary condition has been studied numerically. The governing boundary layer equations are non-dimensionalized and transformed into a two point boundary value problem of coupled nonlinear ordinary differential equations in similarity variable before being solved numerically. The resulting equations with corresponding boundary conditions have been solved numerically by Maple 13 which uses Runge-Kutta-Fehlberg fourth- fifth order numerical algorithm for solving nonlinear ordinary boundary value problems. Our analysis reveals that the similarity solution is possible if the convective heat transfer coefficient is directly proportional to x–1/4, where x is the axial distance from the leading edge of the plate. Solutions depend on the seven parameters: Prandtl number, buoyancy ratio, Brownian motion, thermophoresis, Lewis number, momentum slip and convective heat transfer. The effects of the governing parameters on the flow and heat transfer characteristics have been shown graphically and discussed. Comparisons of the present numerical solution with the existing results in the literature are made and our results are in very good agreement. Results for the skin friction factor, the reduced Nusselt and the Sherwood numbers are provided in tabular form for various values of the convective heat transfer parameter. It is found that the skin friction coefficint reduces with the momentum slip and the buoyancy ratio parameters whilst it enhances with the convective heat transfer parameter. It is also found that mass transfer rate enhances with the Lewis number and the convective heat transfer parameter whilst it falls with the thermophoresis parameter.

Keywords: Free convection; momentum slip boundary condition; nanofluids; thermal convective boundary condition

ABSTRAK

Aliran lapisan sempadan berlamina dua dimensi yang mantap bagi nanobendalir ke atas plat menegak yang dipanaskan secara berolak dengan syarat sempadan gelinciran momentum linear dikaji secara berangka. Persamaan menakluk lapisan sempadan dijadikan tanpa dimensi dan dijelmakan kepada masalah nilai sempadan dua titik yang terdiri daripada persamaan perbezaan biasa tidak linear terganding dalam bentuk pemboleh ubah keserupaan sebelum diselesaikan secara berangka. Persamaan yang terhasil bersama-sama syarat sempadan yang sepadan telah diselesaikan secara berangka melalui Maple 13 dengan menggunakan algoritma berangka Runge-Kutta-Fehlberg peringkat keempat-kelima untuk menyelesaikan masalah nilai sempadan tak linear. Analisis ini mendedahkan bahawa penyelesaian keserupaan adalah mungkin sekiranya pekali pemindahan haba berolak berkadar langsung dengan x–1/4, dengan x ialah jarak paksi dari pinggir depan plat. Penyelesaian bergantung kepada parameter berikut: pergerakan Brownian, perpindahan haba, nombor Lewis, nombor Prandtl, gelinciran momentum dan pemindahan haba berolak. Kesan parameter menakluk ke atas aliran dan ciri-ciri pemindahan haba telah dipamerkan secara bergraf dan dibincangkan. Perbandingan keputusan berangka kajian ini dilakukan dan didapati menepati keputusan yang sedia ada dalam kajian lepas. Keputusan bagi faktor geseran kulit, nombor Nusselt terturun dan nombor Sherwood diberikan dalam bentuk berjadual untuk pelbagai nilai parameter pemindahan haba berolak. Didapati bahawa faktor geseran kulit mengurang dengan slip momentum dan parameter nisbah keapungan dan pada masa yang sama ia meningkat dengan parameter pemindahan haba perolakan. Kadar pemindahan jisim juga didapati meningkat dengan nombor Lewis dan parameter pemindahan haba berolak dan pada masa yang sama ia jatuh dengan parameter thermophoresis.

Kata kunci: Nanobendalir; olakan bebas; syarat sempadan gelinciran momentum; syarat sempadan haba berolak

RUJUKAN

Arifin, N.M., Nazar, R. & Pop, I. 2011. Viscous flow due to a permeable stretching/shrinking sheet in a nanofluid. Sains Malaysiana40(12): 1359-1367.

Aziz, A., Uddin, M.J., Hamad, M.A.A. & Ismail, AI.M. 2012. MHD flow over an inclined radiating plate with the temperature-dependent thermal conductivity, variable reactive index, and heat generation. Heat Transfer Asian Research (Online).

Aziz, A. 2009. A similarity solution for laminar thermal boundary over a flat plate with a convective boundary condition. Communications in Nonlinear Science and Numerical Simulations 15: 1064-1068.

Aziz, A. 2010. Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition. Communications in Nonlinear Science and Numerical Simulations 15: 573-580.

Bachok, N., Ishak, A. & Pop, I. 2010. Boundary-layer flow of nanofluids over a moving surface in a flowing fluid. International Journal of Thermal Sciences 49: 1663-1668.

Buongiorno, J. 2006. Convective transport in nanofluids. ASME Journal of Heat Transfer 128: 240-250.

Godson, L.B., Raja, D., Mohan Lal, D. & Wongwisesc, S. 2010. Enhancement of heat transfer using nanofluids – An overview. Renewable and Sustainable Energy Reviews 14: 629-641.

Gorla, R.S.R. & Chamkha, A. 2011. Natural convective boundary layer flow over a horizontal plate embedded in a porous medium saturated with a nanofluid. Journal of Modern Physics 2: 62-71.

Hak. G.M. 2002. Flow Physics in the MEMS Handbook. Boca Raton, FL: CRC Press.

Incropera, F.P., Dewitt, D.P., Bergman, T.L. & Lavine, A.S. 2007. Fundamentals of Heat and Mass Transfer. 6th ed. New York: John Wiley.

Ishak A. 2010. Similarity solutions for flow and heat transfer over a permeable surface with convective boundary conditions. Applied Mathematics and Computation 217: 837-842.

Jaluria, Y. 1980. Natural Convection: Heat and Mass Transfer. Oxford: Pergamon Press.

Kandasamy, R., Loganathan, P. & Puvi Arasu, P. 2011. Scaling group transformation for MHD boundary-layer flow of a nanofluid past a vertical stretching surface in the presence of suction/injection, Nuclear Engineering and Design 241(6): 2053-2059.

Kakaç, S., Özerinç, S. & Yaz?c?oğlu, A.G. 2010. Enhanced thermal conductivity of nanofluids: A state-of-the-art review. Microfluid Nanofluid8: 145-170.

Kaufui V.W. & Omar, D.L. 2010. Applications of nanofluids: Current and future. Adv. In Mech. Eng. 2010. Article ID 519659, doi:10.1155/2010/519659.

Khair, K.R. & Bejan, A. 1985. Mass transfer to natural convection boundary layer flow driven by heat transfer. ASME Journal of Heat Transfer 107: 979-981.

Khan, W.A. & Aziz, A. 2011. Natural convection flow of nanofluid over a vertical plate with uniform surface heat flux. International Journal of Thermal Science 50: 207-1214.

Kuznetsov, A.V. & Nield, D.A. 2010a. Natural convective boundary-layer flow of a nanofluid past a vertical plate. International Journal of Thermal Science 49: 243-247.

Kuznetsov, A.V. & Nield, D.A. 2010b. Thermal instability in a porous medium layer saturated by a nanofluid: Brinkman model. Transport in Porous Media 81: 409-422.

Kuznetsov, A.V. & Nield, D.A. 2010c. Effect of local thermal non-equilibrium on the onset of convection in a porous medium layer saturated by a nanofluid. Transport in Porous Media 83: 425-436.

Magyari, E. 2011. Comment on ‘A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition’ by A. Aziz, Comm.Non. Sci. Numer. Simul. 2009;14:1064–8. Communications in Nonlinear Science and Numerical Simulations 16: 599-601.

Makinde, O.D. & Aziz A. 2011. Boundary layer flow of a nanofluid past a streaching sheet with convective boundary condition, Int. J. of Ther. Sci. 50: 1326-1332.

Martin, M.J. & Boyd, I.D. 2010. Falkner-Skan flow over a wedge with slip boundary conditions. AIAA Journal of Thermo physics and Heat Transfer 24(2): 263-270.

Mathews, M.T. & Hill, J.M. 2007. Micro/nano thermal boundary layer equations with slip-creep-jump boundary conditions. IMA Journal of Applied Mathematic 72: 894-911.

Mukhopadhyay, S. & Andersson, H.I. 2009. Effects of slip and heat transfer analysis of flow over an unsteady stretching surface. Heat Mass Transfer 45: 1447-1452.

Murshed, S.M.S., Leong, K.C. & Yang, C. 2008. Thermophysical and electrokinetic properties of nanofluids–a critical review. Applied Thermal Engineering 28: 2109-2125.

Nield, D.A. & Kuznetsov, A.V. 2009. The Cheng–Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid. International Journal of Heat Mass Transfer 52: 5792-5795.

Pantokratoras, A. 2009. A common error made in investigation of boundary layer flows. Appl. Math. Model.33: 413-422.

Patankar, S.V. & Spalding D.B. 1970. Heat and Mass Transfer in Boundary Layers. 2nd ed. London: Intertext Books.

Ram, P.C. 1991. Recent developments of heat and mass transfer in hydromagnetic flows. International Journal of Energy Research 15(9): 691-713.

Uddin, M.J., Khan, W.A. & Ismail, A.I.M. 2012. Free convection boundary layer flow from a heated upward facing horizontal flat plate embedded in a porous medium filled by a nanofluid and with a convective boundary condition. Transport in Porous Media 2012. DOI10.1007/s11242-01109938-z.

Wang, C.Y. 2009. Analysis of viscous flow due to stretching sheet with surface slip and suction. Nonlinear Analysis and Real World Applications 10: 375-80.

Yacob, N.A., Ishak, A. & Pop, I. 2011. Falkner-Skan problem for a static or moving wedge in nanofluids. International Journal of Thermal Science 50: 133-139.

Yao, S., Fang, T. & Zhong, Y. 2011. Heat transfer of a generalized stretching/shrinking wall problem with convective boundary conditions. Communications in Nonlinear Science and Numerical Simulations 16: 752-760.

*Pengarang untuk surat-menyurat; email: popm.ioan@yahoo.co.uk