Sains Malaysiana 40(11)(2011): 1297–1300

Similarity Solutions for the Stagnation-Point Flow and Heat Transfer Over a Nonlinearly Stretching/Shrinking Sheet

(Penyelesaian Keserupaan bagi Aliran Titik Genangan dan Pemindahan

Haba terhadap Helaian Meregang/Mengecut)

Norfifah Bachok

Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia

43400 UPM Serdang Selangor D.E. Malaysia

Anuar Ishak*

Centre for Modelling & Data Analysis, School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor D.E. Malaysia

Received: 11 November 2010 / Accepted: 14 January 2011

ABSTRACT

This paper presents a numerical analysis of a stagnation-point flow towards a nonlinearly stretching/shrinking sheet immersed in a viscous fluid. The stretching/shrinking velocity and the external flow velocity impinges normal to the stretching/shrinking sheet are assumed to be in the form U ~ x^m, where m is a constant and x is the distance from the stagnation point. The governing partial differential equations are converted into ordinary ones by a similarity transformation, before being solved numerically. The variations of the skin friction coefficient and the heat transfer rate at the surface with the governing parameters are graphed and tabulated. Different from a stretching sheet, it is found that the solutions for a shrinking sheet are non-unique fo m > 1/3.

Keywords: Boundary layer; dual solutions; nonlinear stretching/shrinking; similarity solution

ABSTRAK

Makalah ini melaporkan analisis berangka aliran titik genangan terhadap helaian meregang/mengecut tak linear yang terendam dalam bendalir likat. Halaju regangan/pengecutan dan halaju aliran bebas normal ke arah helaian meregang/mengecut diandaikan dalam bentuk U ~ x^m, dengan m adalah pemalar dan x adalah jarak dari titik genangan. Persamaan-persamaan menakluk dalam bentuk persamaan pembezaan separa ditukar kepada persamaan pembezaan biasa menggunakan penjelmaan keserupaan, sebelum diselesaikan secara berangka. Perubahan-perubahan pekali geseran kulit dan kadar pemindahan haba pada permukaan terhadap parameter-parameter terlibat dilaporkan dalam bentuk graf dan jadual. Berbeza dengan kes regangan, didapati bahawa penyelesaian bagi kes pengecutan adalah tidak unik bagi m > 1/3.

Kata kunci: Lapisan sempadan; penyelesaian dual; penyelesaian keserupaan; regangan/pengecutan tak linear

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*Corresponding author; e-mail: anuarishak@yahoo.com