Sains Malaysiana 43(1)(2014): 151-159

 

Combined Similarity-numerical Solutions of MHD Boundary Layer Slip Flow of

Non-Newtonian Power-law Nanofluids over a Radiating Moving Plate

(Gabungan Penyelesaian Keserupaan Persamaan-berangka Aliran Slip bagi Lapisan Sempadan MHD dengan Nano Bendalir Hukum Kuasa yang Bukan-Newtonian atas Permukaan Beradiasi yang Bergerak)

 

NUR HUSNA MD. YUSOFF, MD. JASHIM UDDIN* & AHMAD IZANI MD. ISMAIL

School of Mathematical Sciences, Universiti Sains Malaysia, Penang 11800, Malaysia

 

Received: 24 August 2012/Accepted: 26 March 2013

 

ABSTRACT

A combined similarity-numerical solution of the magnetohydrodynamic boundary layer slip flow of an electrically conducting non-Newtonian power-law nanofluid along a heated radiating moving vertical plate is explored. Our nanofluid model incorporates the influences of the thermophoresis and the Brownian motion. The basic transport equations are made dimensionless first and then suitable similarity transformations are applied to reduce them into a set of nonlinear ordinary differential equations with the associated boundary conditions. The reduced equations are then solved numerically. Graphical results for the non-dimensional flow velocity, the temperature and the nanoparticles volume fraction profiles as well as for the friction factor, the local Nusselt and the Sherwood numbers are exhibited and examined for various values of the controlling parameters to display the interesting aspects of the solutions. It was found that the friction factor increases with the increase of the magnetic field (M), whilst it is decreased with the linear momentum slip parameter (a). The linear momentum slip parameter (a) reduces the heat transfer rates and the nanoparticles volume fraction rates. Our results are compatible with the existing results for a special case.

 

Keywords: Magnetic field; momentum slip boundary condition; non-Newtonian power–law nanofluids; radiation

 

ABSTRAK

Gabungan penyelesaian keserupaan-berangka aliran tergelincir bagi lapisan sempadan magnetohidrodinamik nano bendalir hukum kuasa yang bukan-Newtonian yang boleh mengalirkan elektrik atas permukaan bergerak serta beradiasi diterokai. Model nanobendalir kami menggabungkan pengaruh termoforesis dan gerakan Brownian. Persamaan pengangkutan asas dijadikan tidak berdimensi dahulu dan transformasi persamaan yang sesuai digunakan untuk mengurangkan ke satu set persamaan pembezaan biasa tak linear dengan syarat sempadan yang berkaitan. Persamaan yang dikurangkan kemudian diselesaikan secara berangka. Keputusan grafik untuk halaju tidak berdimensi, suhu, profil pecahan isi padu zarah-zarah nano, nombor Nusselt serta Sherwood dipamerkan dan dikaji bagi pelbagai parameter kawalan untuk memaparkan aspek-aspek yang menarik daripada penyelesaian. Kajian mendapati bahawa regangan dinding ricih meningkatkan peningkatan medan magnet M, manakala menurun dengan momentum lelurus slip parameter (a). Momentum lelurus slip parameter a mengurangkan kadar pemindahan haba dan pecahan isi padu zarah nano. Keputusan kami adalah selaras dengan keputusan yang sedia ada bagi kes khas.

 

Kata kunci: Keadaan sempadan momentum slip; medan magnet; nanobendalir yang bukan Newtonian; radiasi

 

 

 

REFERENCES

Andersson, H.I. & Bech, K.H. 1992. Magnetohydrodynamic flow of a power-law fluid over a stretching sheet. International Journal Non-Linear Mechanics 27: 929-936.

Boutros, Y.Z., Abd-el-Malek, M.B., Badran, N.A. & Hassan, H.S. 2006. Lie-group method of solution for steady two-dimensional boundary-layer stagnation-point flow towards a heated stretching sheet placed in a porous medium. Meccanica 41: 681-691.

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

Cheng, P. & Minkowycz, W.J. 1977. Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike. J. Geophysics Research 82: 2040-2044.

Cortell, R. 2008. Radiation effects in the Blasius flow. Applied Mathematics and Computations 198: 333-338.

Cortell, R. 2011. Suction, viscous dissipation and thermal radiation effects on the flow and heat transfer of a power law fluid past an infinite porous plate. Chemical Engineering Research and Design 89: 85-93.

Crane, L.J. 1970. Flow past a stretching plate. Journal of Applied Mathematics and Physics 21: 645-647.

Elhajjar, B., Bachir, G., Mojtabi, A., Fakih, C. & Charrier-Mojtabi M.C. 2010. Modeling of Rayleigh-Benard natural convection heat transfer in nanofluid. Comptes Rendus Mecaniquue 338: 350-354.

Ellahi, R. 2009. Effects of the slip boundary condition on non- Newtonian flows in a channel. Communications in Nonlinear Sciences and Numerical Simulations 14: 1377-1384.

Ellahi, R., Raza, M. & Vafai, K. 2012. Series solutions of non-Newtonian nanofluids with Reynolds’ model and Vogels’ model by means of the Homotopy analysis method. Mathematical and Computer Modelling 55: 1876-1891.

Guo, S.Z., Jiang, Y.L.J. & Xie, H.Q. 2010, Nanofluids Containing γ-Fe2O3 nanoparticles and their heat transfer enhancements. Nanoscale Research Letters 5: 1222-1227.

Hady, F.M., Ibrahim, F.S., Abdel-Gaied, S.M. & Eid, M.R. 2012. Radiation effect on viscous flow of a nanofluid and heat transfer over a nonlinearly stretching sheet. Nanoscale Research Letters 7: 229.

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

Lioubashevski, O., Katz, E. & Willner, I. 2004. Magnetic force effects on electrochemical processes: A theoretical hydrodynamic model. Journal of Physical Chemistry B 108: 5778-5784.

Mahapatra, T.R., Dholey, S. & Gupta, A.S. 2007. Momentum and heat transfer in the magnetohydrodynamic stagnation point flow of a viscoelastic fluid toward a stretching surface. Meccanica 42: 263-272.

Mahmoud, M.A.A. 2011. Slip velocity effect on a non-Newtonian power law fluid over a moving permeable surface with heat generation. Mathematical and Computer Modelling 54: 1228-1237.

Makinde, O.D. & Aziz, A. 2011. Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition. International Journal of Thermal Sciences 50: 1326-1332.

Nadeem, S. & Lee, C. 2012. Boundary layer flow of nanofluid over an exponentially stretching surface. Nanoscale Research Letters 7: 94.

Nield, D.A. & Kuznetsov, A.V. 2011. 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 Mass Transfer 54: 374-378.

Noghrehabadi, A., Pourrajab, R. & Ghalambaz, M. 2012. Effect of partial slip boundary condition on the flow and heat transfer of nanofluids past stretching sheet prescribed constant wall temperature. International Journal of Thermal Sciences 54: 253-261.

Piazza, R. & Parola, A. 2008. Thermophoresis in colloidal suspensions. Journal of Physics: Condensed Matter 20: 153102.

Prasad, K.V., Vajravelu, K. & Datti, P.S. 2010. Mixed convection heat transfer over an on-linear stretching surface with variable fluid properties. International Journal of Non-Linear Mechanics 45: 320-330.

Prasad, R.V., Vasu, B., Bég, O.A. & Parshad, R.D. 2012. Thermal radiation effects on magnetohydrodynamic free convection heat and mass transfer from a sphere in a variable porosity regime. Communications in Nonlinear Science and Numerical Simulation 17: 654-671.

Putra, N., Roetzel, W. & Das, S.K. 2003. Natural convection of nanofluids. Heat Mass Transfer 39: 775-784.

Raptis, A., Perdikis, C. & Takhar, H.S. 2004. Effect of thermal radiation on MHD flow. Applied Mathematics and Computations 153: 645-649.

Rosenbaum, E.E. & Hatzikiriakos, S.G. 1997. Wall slip in the capillary flow of molten polymers subject to viscous heating. AI Chem E Journal 43: 598–608.

Roux, C.L. 2009. On flows of third-grade fluids with non-linear slip boundary conditions. International Journal of Non- Linear Mechanics 44: 31-41.

Sakiadis, B.C. 1961. Boundary layer behaviour on continuous solid surfaces: I. Boundary layer equations for two-dimensional and axis symmetric flow. American Institute Chemical Engineers Journal 7: 26-28.

Uddin, M.J., Pop, I. & Ismail, A.I.M. 2012. Free convection boundary layer flow of a nanofluid from a convectively heated vertical plate with linear momentum slip boundary condition. Sains Malaysiana 41(11): 1475-1482.

Uddin, M.J., Khan, W.A. & Ismail, A.I.M. 2012. MHD free convective boundary layer flow of a nanofluid past a flat vertical plate with Newtonian heating boundary condition. PLoS One 7(11): e49499.

Xue, H. & Liao, S.J. 2009. Laminar flow and heat transfer in the boundary-layer of non-Newtonian fluids over a stretching flat sheet. Computers and Mathematics with Applications 57: 1425-1431.

Yu, W. & Xie, H. 2012. A review on nanofluid: Preparation, stability mechanisms, and applications. Journal of Nanomaterials Articles ID 435873.

 

*Corresponding author; email: jashim_74@yahoo.com