Sains Malaysiana 41(10)(2012): 1263–1270

Aliran Genangan bagi Bendalir Mikrokutub terhadap Permukaan Mencancang yang

Telap dengan Fluks Haba Boleh Ubah

(Stagnation Flow of a Micropolar Fluid towards a Vertical Permeable Surface with

Prescribed Heat Flux)

W.M.K.A Wan Zaimi

Institut Matematik Kejuruteraan, Kampus Pauh Putra, Universiti Malaysia Perlis, 02000 Arau, Perlis, Malaysia

Anuar Ishak*

Pusat Pengajian Sains Matematik, Fakulti Sains dan Teknologi

Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia

Received: 27 March 2012 / Accepted: 21 May 2012

ABSTRAK

Dalam makalah ini, masalah aliran genangan dalam bendalir mikrokutub terhadap permukaan mencancang yang telap dengan fluks haba boleh ubah dipertimbangkan. Dengan menggunakan penjelmaan keserupaan, persamaan asas yang menakluk aliran bendalir dan pemindahan haba dijelmakan kepada satu set persamaan perbezaan biasa. Persamaan yang dijelmakan tertakluk kepada syarat-syarat sempadan yang berkaitan kemudiannya diselesaikan secara berangka menggunakan kaedah tembakan. Kesan parameter sedutan/semburan fw ke atas profil halaju dan suhu serta pekali geseran kulit dan nombor Nusselt setempat diperoleh dan dibincangkan. Keputusan berangka menunjukkan bahawa penyelesaian dual wujud dalam kedua-dua aliran membantu dan aliran menentang.

Kata kunci: Aliran genangan; bendalir mikrokutub; mekanik bendalir; penyelesaian dual; sedutan/semburan

ABSTRACT

This paper deals with the stagnation flow of a micropolar fluid towards a vertical permeable surface with prescribed heat flux. By using a similarity transformation, the basic equations governing the fluid flow and heat transfer are transformed into a set of ordinary differential equations. The transformed equations subjected to the associated boundary conditions are then solved numerically using the shooting method. The effects of suction/injection parameter fw on the velocity and the temperature profiles as well as the skin friction coefficient and the local Nusselt number are obtained and discussed. The numerical results indicate that dual solutions exist in both assisting and opposing flows.

Keywords: Dual solutions; fluid mechanics; micropolar fluid; stagnation flow; suction/injection

REFERENCES

Ahmadi, G. 1976. Self-similar solution of incompressible micropolar boundary layer flow over a semi-infinite plate. International Journal of Engineering Science 14: 639–646.

Asgharian, A., Domairry Ganji, D., Soleimani, S. & Asgharian, S. 2010. Analytical solution of stagnation flow of a micropolar fluid towards a vertical permeable surface. Thermal Science 14: 383-392.

Devi, C.D.S., Takhar, H.S. & Nath, G. 1991. Unsteady mixed convection flow in stagnation region adjacent to a vertical surface. Heat and Mass Transfer 26: 71-79.

Eringen, A.C. 1966. Theory of micropolar fluids. Journal of Mathematics and Mechanics 16: 1-18.

Gorla, R.S.R. 1988. Combined forced and free convection in micropolar boundary layer flow on a vertical flat plate. International Journal of Engineering Science 26: 385–391.

Hassanien, I. & Gorla, R.S.R. 1990. Combined forced and free convection in stagnation flows of micropolar fluids over vertical non-isothermal surfaces. International Journal of Engineering Science 28: 783-792.

He, X. & Fan, J. 2012. A regularity criterion for 3D micropolar fluid flows. Applied Mathematics Letters 25: 47-51.

Ishak, A. 2010. Thermal boundary layer flow over a stretching sheet in a micropolar fluid with radiation effect. Meccanica 45: 367-373.

Ishak, A., Nazar, R., Arifin, N.M. & Pop, I. 2008. Dual solutions in mixed convection flow near the stagnation point on a vertical porous plate. International Journal of Thermal Sciences 47: 417-422.

Kline, K.A. 1977. A spin-vorticity relation for unidirectional plane flows of micropolar fluids. International Journal of Engineering Science 15: 131-134.

Lok, Y.Y., Amin, N. & Pop, I. 2006. Unsteady mixed convection flow of a micropolar fluid near the stagnation point on a vertical surface. International Journal of Thermal Sciences 11: 49-57.

Lok, Y.Y. & Pop, I. 2011. Wang’s shrinking cylinder problem with suction near a stagnation point. Physics of Fluids 23: 083102.

Merrill, K., Beauchesne, M., Previte, J., Paullet, J. & Weidman P. 2006. Final steady flow near a stagnation point on a vertical surface in a porous medium. International Journal of Heat and Mass Transfer 49: 4681-4686.

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

Ramachandran, N., Chen, T.S., & Armaly, B.F. 1988. Mixed convection in stagnation flows adjacent to vertical surfaces. ASME Journal of Heat Transfer 110: 373-377.

Ridha, A. 1996. Aiding flows non-unique similarity solutions of mixed-convection boundary-layer equations. Journal of Applied Mathematics and Physics (ZAMP) 47: 341-352.

Schlichting, H. & Gersten, K. 2003. Boundary Layer Theory. Berlin: Springer.

Sherief, H.H., Faltas M.S. & Ashmawy, E.A. 2011. Slow motion of a sphere moving normal to two infinite parallel plane walls in a micropolar fluid. Mathematical and Computer Modelling 53: 376-386.

Wang, Y.Z, Yuan, H. 2012. A logarithmically improved blow-up criterion for smooth solutions to the 3D micropolar fluid equations. Nonlinear Analysis: Real World Applications 13: 1904-1912

Yacob, N.A. & Ishak, A. 2010. Aliran titik genangan terhadap permukaan meregang dalam bendalir mikropolar dengan fluks haba permukaan boleh ubah. Sains Malaysiana 39(2): 285-290.

Yacob, N.A. & Ishak, A. 2012. Micropolar fluid flow over a shrinking sheet. Meccanica 47: 293-299.

*Corresponding author; email: anuar_mi@ukm.my