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# Dual Solutions for the Magnetohydrodynamic Stagnation-Point Flow of a Power-Law Fluid Over a Shrinking Sheet

[+] Author and Article Information
Tapas Ray Mahapatra

Department of Mathematics,Visva-Bharati, Santiniketan-731 235, India

Samir Kumar Nandy

Department of Mathematics,A. K. P. C. Mahavidyalaya,Bengali, Hooghly-712 611, India

Kuppalapalle Vajravelu

Department of Mathematics,University of Central Florida, Orlando, FL 32816

Robert A. Van Gorder1

Department of Mathematics,University of Central Florida, Orlando, FL 32816rav@knights.ucf.edu

1

Corresponding author.

J. Appl. Mech 79(2), 024503 (Feb 24, 2012) (6 pages) doi:10.1115/1.4005584 History: Received March 09, 2011; Revised September 05, 2011; Posted February 01, 2012; Published February 13, 2012; Online February 24, 2012

## Abstract

We show that there exist bounded self-similar solutions to the steady state problem of the MHD stagnation point flow of a power-law fluid over a shrinking sheet. We then discuss the stability of the unsteady solutions about each steady solution, showing that one steady state solution corresponds to a stable solution whereas the other corresponds to an unstable solution. The stable solution corresponds to the physically relevant solution. Further, we obtain numerical results for each solution, which enable us to discuss the features of the respective solutions. Our method of finding dual solutions and analyzing stability is of practical application to those interested in engineering analysis, as it provides one with a way to determine whether a given steady state solution is physically meaningful. Hence, our study is useful not only as a discussion of the problem of the MHD stagnation point flow of a power-law fluid over a stretching or shrinking sheet but as a demonstration of the treatment of fluid flow problems with multiple solutions.

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Copyright © 2012 by American Society of Mechanical Engineers
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## Figures

Figure 2

Variation of F′(η) with η for several values of (a) the magnetic parameter M with n = 0.4 and α  = −1.4, (b) the power-law index n with M = 0.5 and α  = −1.4, and (c) the shrinking rate parameter α with M = 0.3 and n = 0.4

Figure 1

Wall shear stress F″(0) versus α for different values of (a) the magnetic parameter M with n = 0.4 and (b) the power-law index n with M = 0.3

Figure 3

Variation of V(η) with η for several values of (a) the magnetic parameter M with n = 0.4 and α  = −1.4, (b) the power-law index n with α  = −1.4 and M = 0.5, and (c) the shrinking rate parameter α with n = 0.4 and M = 0.3

Figure 4

Plot of lowest eigenvalues γ 1 as a function of α for (a) different values of M with n = 0.4 and (b) different values of n with M = 0.3

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