Jeffrey hamel.pdf - International Journal of Modern Mathematical Sciences 2013 7(3 236-247 International Journal of Modern Mathematical Sciences ISSN

Jeffrey hamel.pdf - International Journal of Modern...

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Copyright © 2012 by Modern Scientific Press Company, Florida, USA International Journal of Modern Mathematical Sciences, 2013, 7(3): 236-247 International Journal of Modern Mathematical Sciences Journal homepage: ISSN: 2166-286X Florida, USA Article On Jeffery-Hamel Flows Umar Khan 1 , Naveed Ahmed 1 , Z. A. Zaidi 1, 2 , S. U. Jan 2 , Syed Tauseef Mohyud-Din 1, * 1 Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan 2 COMSATS Institute of Information Technology, University Road, Abbottabad, Pakistan * Author to whom correspondence should be addressed; Email: [email protected] Article history : Received 1 May 2013, Received in revised form 7 July 2013, Accepted 10 July 2013, Published 15 July 2013. Abstract: Jeffery-Hamel flows describe flow inside nonparallel walls which is present in many practical situations. Due to its applications in industrial and biological sciences such flows have gained considerable attention. In this study we have consider flow of a viscous incompressible fluid in a converging and diverging channel; conservation laws along with suitable similarity transform are used to obtain a highly nonlinear ordinary differential equation (ODE) which governs the fluid flow. Due to abstractness of resulting ODE exact solution is unlikely therefore analytical solution is approximated using Variation of Parameters Method (VPM). Results are compared with already existing solutions in literature both analytical and numerical. It is observed that VPM is more efficient and requires less amount of computational to maintain a very high accuracy level. Influence of different parameters is also discussed and demonstrated graphically. Keywords: Jeffery-Hamel flows, Variation of Parameters Method (VPM), converging and diverging channels. Mathematics Subject Classification (2010): 76A05, 76W05, 76Z99 1. Introduction Flow between two nonparallel walls is common in practical situations. It may be considered one of the most important problems in fluid mechanics due to wide range of applications. These include aerospace, chemical, civil, environmental, mechanical and bio-mechanical engineering. These
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Int. J. Modern Math. Sci. 2013 , 7(3): 236-247 Copyright © 2012 by Modern Scientific Press Company, Florida, USA 237 mathematical models also serve as drastic simplified prototypes for flows in rivers and canals. Jeffery and Hamel were the pioneers to discuss these problems mathematically [1-2]. Jeffery-Hamel flows provide an exact similarity solution of Navier Stokes equations in special case of two-dimensional flow through a channel with inclined plane walls meeting at a vertex having a source or sink. After aforementioned founding work these types of flows have been studied extensively by several authors and discussed in many textbooks [3-8].
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