Sains Malaysiana 39(2)(2010): 285–290

Aliran Titik Genangan Terhadap Permukaan Meregang dalam Bendalir Mikropolar dengan Fluks Haba Permukaan Boleh Ubah

(Stagnation-point Flow towards a Strecthing Surface Immersed in a  Micropolar Fluid with Prescribed Surface Heat Flux)

Nor Azizah M. Yacob

Fakulti Sains Komputer dan Matematik, Universiti TeknologiMara, Pahang

Lintasan Semarak, 26400, Bandar Jengka, Pahang, D.M., Malaysia

Anuar Mohd Ishak*

Pusat Pengajian Sains Matematik, Fakulti Sains dan Teknologi

Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor D.E., Malaysia

Received: 15 May 2009 / Accepted: 26 August 2009

ABSTRAK

Aliran lapisan sempadan mantap dua matra bersebelahan titik genangan pada permukaan meregang dalam bendalir mikropolar dengan fluks haba permukaan boleh ubah dikaji. Halaju regangan dan halaju aliran bebas diandaikan berubah secara linear dari titik genangan. Persamaan dalam bentuk persamaan pembezaan separa dijelmakan kepada persamaan pembezaan biasa menggunakan penjelmaan keserupaan dan diselesaikan secara berangka menggunakan skim beza-terhingga yang dikenali sebagai kaedah kotak Keller. Kedua-dua aliran membantu dan aliran menentang dipertimbangkan. Lapisan sempadan terbentuk apabila halaju aliran bebas melebihi halaju regangan, sebaliknya, lapisan sempadan yang terbalik terbentuk apabila halaju aliran bebas kurang daripada halaju regangan. Keputusan berangka menunjukkan bahawa daya seretan berkurangan bagi bendalir mikropolar berbanding dengan bendalir Newtonan, dan ini seterusnya mengurangkan kadar pemindahan haba pada permukaan.

Kata kunci: Bendalir mikropolar; lapisan sempadan; regangan permukaan; titik genangan

                                                              ABSTRACT          

The steady two dimensional boundary layer flow adjacent to the stagnation point on a stretching surface immersed in a micropolar fluid was investigated. The stretching and the free stream velocities were assumed to vary linearly from the stagnation point. The governing partial differential equations were transformed into ordinary differential equations before being solved numerically by a finite-difference scheme known as the Keller box method. Both assisting and opposing flows were considered. The boundary layer was formed when the free stream velocity exceeds the stretching velocity, whereas the inverted boundary layer was formed when the free stream velocity is less than the stretching velocity. The numerical results showed that the shear force decreased for micropolar fluid compared to Newtonian fluid, and in consequence decreased the heat transfer rate at the surface.

Keywords: Boundary layer; micropolar fluid; stagnation point; stretching surface

REFERENCES

Ahmadi, G. 1976. Self-similar solution of incompressible micropolar boundary layer flow over a semi-infinite plate. Int. J. Eng. Sci. 14: 639-646.

Cebeci, T. & Bradshaw, P. 1988. Physical and Computational Aspects of Convective Heat Transfer. New York: Springer-Verlag.

Chen, C.H. 1998. Laminar mixed convection adjacent to vertical, continuously stretching sheets. Heat Mass Transfer 33: 471-476.

Cheng, L.C. & Zong, Y.L. 2008. Free convection on a vertical plate with uniform and constant heat flux in a thermally stratified micropolar fluid. Mech. Res. Commun. 35: 421-427.

Chiam, T.C. 1994. Stagnation-point flow towards a stretching plate. J. Phys. Soc. Japan 63: 2443-2444.

Crane, L.J. 1970. Flow past a stretching plate. J. Appl. Math. Phys. (ZAMP) 21: 645-647.

Elbashbeshy, E.M.A. 1998. Heat transfer over a stretching surface with variable surface heat flux. J. Phys. D: Appl. Phys. 31:1951-1954.

Eringen, A.C. 1966. Theory of micropolar fluids. J. Math. Mech. 16: 1-18.

Eringen, A.C. 1972. Theory of thermomicropolar fluids. J. Math. Anal. Appl. 38: 480-496.

Hayat, T., Abbas, Z. & Javed, T. 2008. Mixed convection flow of a micropolar fluid over a non-linearly stretching sheet. Phys. Lett. A. 372: 637-647.

Ishak, A., Nazar, R. & Pop, I. 2008a. Heat transfer over a stretching surface with variable heat flux in micropolar fluids. Phys Lett. A. 372: 559-561.

Ishak, A., Nazar, R. & Pop, I. 2006b. Mixed convection boundary layers in the stagnation-point flow towards a stretching vertical sheet. Meccanica 41: 509-518.

Ishak, A, Nazar, R & Pop, I. 2008b. Magnetohydrodynamic (MHD) flow of a micropolar fluid towards a stagnation point on a vertical surface. Comput. Math. Appl. 56: 3188-3194.

Kumaran, V., Banerjee, A.K., Vanav Kumar, A. & Vajravelu, K. 2009. MHD flow past a stretching permeable surface. Appl. Math Comput. 210: 26-32.

Lok, Y.Y., Amin, N. & Pop, I. 2007. Mixed convection flow near a non-orthogonal stagnation point towards a stretching vertical plate. Int. J. Heat Mass Transfer 50: 4855-4863.

Mahapatra, T.R. & Gupta, A.S. 2002. Heat transfer in stagnation-point flow towards a stretching sheet. Heat Mass Transfer 38: 517-521.

Nazar, R., Amin, N., Filip, D. & Pop, I. 2004. Stagnation flow of a micropolar fluid towards a stretching sheet. Int. J. Non-Linear Mech. 39: 1227-1235.

Peddieson, J. 1972. An application of the micropolar fluid model to the calculation of turbulent shear flow. Int. J. Eng. Sci. 10: 23-32.

Takhar, H.S., Agarwal, R.S., Bhargava, R. & Jain, S. 1998. Mixed convection flow of a micropolar fluid over a stretching sheet. Heat Mass Transfer 34: 213-219.

*Corresponding author; email: anuar_mi@ukm.my