Sains Malaysiana 46(2)(2017): 349–358

http://dx.doi.org/10.17576/jsm-2017-4602-20

Dual Solutions of Forced Convection Flow along a Stretching Sheet with Variable Thickness in Presence of Free Stream and Magnetic Field

(Dual Penyelesaian Aliran Perolakan Dipaksa di Sepanjang Lembaran Regangan dengan Ketebalan Pemboleh Ubah dalam Kehadiran Aliran Bebas dan Medan Magnet)

UPENDRA MISHRA¹* & GURMINDER SINGH²

¹Department of Mathematics, Amity University Rajasthan, Jaipur, India

²Department of Applied Mathematics, Birla Institute of Technology (Ranchi), Ext. Center Jaipur, 27 Malviya Industrial Area, Jaipur-302017, India

Received: 9 November 2015/Accepted: 18 June 2016

ABSTRACT

The paper aims at studying forced convection in a viscous incompressible electrically conducting fluid along stretching sheet with variable thickness in the presence of variable free stream and magnetic field. The governing equations of flow and heat transfer are subjected to similarity transformation using boundary layer assumption and are then solved numerically. The system of equations possesses dual solutions for negative value of velocity power index (m). The impact of velocity parameter (λ) and other parameters on velocity and temperature distributions, skin friction and heat transfer are studied when system possesses dual solutions and is presented through graphs and discussed suitably. It is found that the first solution is in tune with natural physical phenomena. The second solution possesses very large skin-friction and fluid velocity as compared to the first solution. The second solution is stable and is a mere outcome of non-linearity and does not follow natural phenomena.

Keywords: Free stream; magnetic field; similarity solution; stretching surface; variable thickness

ABSTRAK

Kertas ini bertujuan mengkaji perolakan dipaksa dalam pengaliran bendalir likat tak termampat elektrik di sepanjang lembaran regangan dengan ketebalan pemboleh ubah dengan kehadiran pemboleh ubah aliran bebas dan medan magnet. Persamaan pemindahan haba dan aliran tertakluk kepada transformasi persamaan menggunakan andaian lapisan sempadan dan kemudian diselesaikan secara berangka. Sistem persamaan mempunyai dua penyelesaian untuk nilai negatif indeks tenaga halaju (m). Kesan parameter halaju (λ) dan lain-lain parameter ke atas halaju dan taburan suhu, geseran kulit dan pemindahan haba dikaji apabila sistem memiliki dual penyelesaian dan ditunjukkan melalui graf dan dibincangkan penyesuaiannya. Didapati bahawa penyelesaian yang pertama adalah sealiran dengan fenomena fizikal semula jadi. Penyelesaian yang kedua memiliki geseran kulit yang sangat besar dan halaju bendalir berbanding dengan penyelesaian yang pertama. Penyelesaian kedua adalah stabil dan hasil daripada kelinearan dan tidak mengikut fenomena semula jadi.

Kata kunci: Aliran bebas; medan magnet; penyelesaian persamaan; permukaan regangan; ketebalan pemboleh ubah

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*Corresponding author; email: dr_umishra@yahoo.com