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TITLE PAGE
DUAL SOLUTION IN MAGNETOHYDRODYNAMIC (MHD) FLOW ON A NON LINEAR POROUS SHRINKLING SHEET IN A VISCIOUS FLUID

BY

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EE/H2007/01430
DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING
SCHOOL OF ENGINEERING
INSTITUTE OF ---

DECEMBER,2018

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APPROVAL PAGE

This is to certify that the research work,"dual solution in magnetohydrodynamic (mhd) flow on a non linear porous shrinkling sheet in a viscious fluid" by ---, Reg. No. EE/H2007/01430 submitted in partial fulfillment of the requirement award of a Higher National Diploma on Electrical and Electronics Engineering has been approved.

By
Engr. ---                                                     Engr. ---
Supervisor                                                  Head of Department.
Signature……………….                           Signature……………….        

……………………………….
 Engr. ---
External Invigilator

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DEDICATION
This project is dedicated to Almighty God for his protection, kindness, strength over my life throughout the period and also to my --- for his financial support and moral care towards me.Also to my mentor --- for her academic advice she often gives to me. May Almighty God shield them from the peril of this world and bless their entire endeavour Amen.

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ACKNOWLEDGEMENT

The successful completion of this project work could not have been a reality without the encouragement of my --- and other people. My immensely appreciation goes to my humble and able supervisor mr. --- for his kindness in supervising this project.
My warmest gratitude goes to my parents for their moral, spiritual and financial support throughout my study in this institution.
My appreciation goes to some of my lecturers among whom are Mr. ---, and Dr. ---. I also recognize the support of some of the staff of --- among whom are: The General Manager, Deputy General manager, the internal Auditor Mr. --- and the ---. Finally, my appreciation goes to my elder sister ---, my lovely friends mercy ---, ---, --- and many others who were quite helpful.

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In this work, the problem of magnetohydrodynamic (MHD) flow of a viscous fluid on a nonlinear porous shrinking sheet is studied. The boundary layer partial differential equations are first transformed into an ordinary differential equation, which is then solved numerically by the shooting method. The features of the flow for various governing parameters are presented and discussed in detail. It is found that dual solutions only exist for positive values of the controlling parameter.

 

CHAPTER ONE
1.0                                   INTRODUCTION
Boundary layer flow over a stretching sheet has been studied in various aspects since the pioneering work done by Sakiadis [1]. Lately, many researchers have studied the shrinking sheet boundary layer flow problem due to its important applications in industries which involve packaging process, for example, shrink wrapping. The study on shrinking sheet was first initiated by Wang [2] by considering the stretching deceleration surface. Miklavcic and Wang [3] proved the existence and uniqueness for viscous flow due to a shrinking sheet, and dual solutions were reported for certain range of the suction parameter. Later, Wang [4] also studied the stagnation flow towards a shrinking sheet by considering the two-dimensional and axisymmetric stagnation flows, and dual solutions were also reported only for the two-dimensional case. Further, Sajid et al.[5] and Hayat et al.[6] considered the rotating flow over a shrinking surface. The steady boundary layer flow problems induced by a shrinking sheet can be found in [7, 8, 9, 10, 11] in different aspects. On the other hand, the unsteady case is described in papers by Fang et al.[12] and Ali et al.[13, 14]. It is worth mentioning that dual solutions are also found in the papers by Fang [11], Fang et al.[12] and Ali et al.[14].
Recently, Nadeem and Hussain [15] solved analytically the problem of magnetohydrodynamic (MHD) flow of a viscous fluid on a nonlinear porous shrinking sheet using the homotopy analysis method and dual solutions were not reported. Hence, the present paper aims to obtain the dual solutions numerically for the problem considered in Nadeem and Hussain [15] for various controlling parameters and magnetic parameters.

1.1                             BACKGROUND OF THE STUDY

Boundary layer flow and heat transfer over a stretching sheet is significant due to its many appli- cations in engineering processes such as in the extraction of polymer sheets, paper production, wire drawing and glass-fibre production. During the manufacturing process, a stretching sheet interacts with the ambient fluid both thermally and mechanically. The study of boundary layer flow caused by a stretching surface was initiated by Crane (1970) who gave an exact similar- ity solution in closed form. Mahapatra & Gupta (2001) reconsidered the steady stagnation point flow towards a stretching sheet taking different stretching and stagnation point velocities and observed two different kinds of boundary layer structure near the sheet. Recently, several papers on the dynamics of the boundary layer flow over a stretching surface have appeared in the liter- ature (Dutta et al 1985; Kameswaran et al 2012, 2013). Mukhopadhyay (2013) has studied the effects of Casson fluid flow and heat transfer over a nonlinearly stretching surface. She found that temperature increases with an increase in nonlinear stretching parameter and the momentum boundary layer thickness decreases with an increase in Casson parameter.
When heat and mass transfer occur simultaneously in a moving fluid, the relationship between the fluxes and the driving potentials is complex. It has been observed that an energy flux can  be generated not only by temperature gradients but also by concentration gradients. The energy flux caused by a concentration gradient is termed as the diffusion-thermo (Dufour) effect. On the other hand, mass fluxes can also  be created  by temperature gradients and this  is  termed as the thermal-diffusion (Soret) effect. Soret and Dufour effects have been utilized for isotope separation, in areas of hydrology, petrology, geosciences, etc. Srinivasacharya & RamReddy (2011) have investigated the Soret and Dufour effects on heat and mass transfer along a semi- infinite vertical plate embedded in a non-Darcy porous medium. Soret and Dufour effects on the magnetohydrodynamic (MHD) flow of a Casson fluid over a stretched surface were also studied by Hayat et al (2012a, b) and Nawaz et al (2012). Chamkha & Aly (2010) presented an analysis on heat and mass transfer in stagnation point flow of a polar fluid towards a stretching surface in porous media in the presence of Soret, Dufour and chemical reaction effects. Their study reveals that the velocity of fluid increased as the Soret number increased and the Dufour number decreased.
Casson fluids have a yield stress below which no flow occurs and a zero viscosity at an infinite rate of shear. The nonlinear Casson constitutive equation was derived by Casson (1959) and describes the properties of many polymers. At low shear rates when blood flows through small vessels, the blood flow is described by the Casson fluid model (McDonald 1974; Shaw et al 2009).
Non-Newtonian fluid flow generated by a stretching or shrinking sheet has many applica- tions in industry. The flow of various non-Newtonian fluids over stretching or shrinking sheets was analysed by Liao (2003), Hayat et al (2008) and Ishak et al (2012). Stagnation point flow and heat transfer in a Casson fluid flow from a stretching sheet was studied by Mustafa et al (2012). An exact solution of the steady boundary layer flow of Casson fluid over a porous stretching or shrinking sheet was studied by Bhattacharyya et al (2011, 2013a, b). The effect of mass transfer on the magnetohydrodynamic flow of a Casson fluid flow over a porous stretch- ing sheet was studied by Shehzad et al (2013). Hayat et al (2012a, b) studied the effects of mixed convection stagnation point flow of Casson fluid with convective boundary condi- tions. In their model, they showed that heat transfer characteristics depending on the embedded parameters.

1.2                                  OBJECTIVE OF THE STUDY
The main objective is to investigate the magnetohydrodynamic (MHD) flow of a viscous fluid towards a nonlinear porous shrinking sheet. The governing equations are simplified by similarity transformations. The reduced problem is then solved by the homotopy analysis method.
1.3                                      SCOPE OF THE STUDY
The steady laminar boundary layer flow of two classes of incompressible visco-elastic and electrically conducting fluids over a nonlinearly shrinking sheet with appropriate wall transpiration under the influence of a magnetic field is analyzed. It is shown that the problem permits an analytical solution of two classes of visco-elastic fluids, namely the second grade fluid and fluid obeying Walters’ liquid B model. The effects of visco-elasticity, suction/blowing parameter and magnetic parameter on both the skin friction parameter and velocity profiles are studied. It is shown that dual solution exists for certain range of physical parameters.

1.4                           OUTCOME OF THE STUDY
The pertinent parameters appearing in the problem are discussed graphically and presented in tables. It is found that the shrinking solutions exist in the presence of MHD. It is also observed from the tables that the solutions for f″(0) with different values of parameters are convergent.

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