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MHD FLOW AND HEAT TRANSFER THROUGH A POROUS MEDIUM OVER A STRETCHING / SHRINKING SURFACE WITH SUCTION By Ali Mansoor Alanbri & Rakan Saud Alharbi Mathematics Department, College of Science, Majmaah University, Alzul, KSA Rajab 14, 1436

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MHD FLOW AND HEAT TRANSFER THROUGH A POROUS MEDIUM OVER A STRETCHING / SHRINKING SURFACE WITH SUCTION

By Ali Mansoor Alanbri & Rakan Saud Alharbi Mathematics Department, College of Science, Majmaah University, Alzul…, KSA Rajab 14, 1436

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In The Name of ALLAH

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Supervised By Supervised By

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Supervised By Supervised By

  • Dr. Farooq Ahmad Gujar

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Acknowledgments Acknowledgments

Praise and glory is to Allah, the Cherisher and Sustainer of the worlds, without His will, faith and support, we would not have been able to accomplish this laborious task. We bow to Him in reverence and gratitude.

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Acknowledgments Acknowledgments

Praise and glory is to Allah, the Cherisher and Sustainer of the worlds, without His will, faith and support, we would not have been able to accomplish this laborious task. We bow to Him in reverence and gratitude. Peace and blessing to Hazrat Muhammad (Peace Be Upon Him) who is the cause of conceiving the universe.

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Acknowledgments

We are highly obliged and very grateful to our respected teacher and supervisor, Dr. Farooq Ahmad Gujar, whom we always found helpful and courteous. We owe a great debt of gratitude to him for his valuable suggestions in the selection of a suitable topic and the research problems. We are very much obliged for his day to day guidance, their keen supervision and friendly a¤ection. They have always been the most cordial and cooperative throughout the research work and during the compilation of this project. We are highly

  • bliged. We are also very grateful to our respected teacher Dr. Sajjad

Hussain, whom we always found helpful.

Ali & Rakan (College of Science Alzul…) Fluids Over Stretcing / Shrinking Sheet 14/07/1436 5 / 48

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Acknowledgments

We are highly obliged and very grateful to our respected teacher and supervisor, Dr. Farooq Ahmad Gujar, whom we always found helpful and courteous. We owe a great debt of gratitude to him for his valuable suggestions in the selection of a suitable topic and the research problems. We are very much obliged for his day to day guidance, their keen supervision and friendly a¤ection. They have always been the most cordial and cooperative throughout the research work and during the compilation of this project. We are highly

  • bliged. We are also very grateful to our respected teacher Dr. Sajjad

Hussain, whom we always found helpful. We are obliged and very grateful to my respected teacher, Prof. Dr. Adel Zaki, Head of the Department of Mathematics, Majmaah University, College of Science, Alzul… whom we always found helpful. We am very much obliged for his day to day guidance.

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Acknowledgments

We are also highly obliged and very grateful to respected Dr. Mohammad Saleh Aloboudi, Dean of College of Science, Majmaah University, Alzul…. It has been a privilege and great honor for us to learn with a great man of such prominent characteristics. We love and respect him the most for his high morality and honorable sincerity, his guidance is a ray of light in our research.

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Acknowledgments

We are also highly obliged and very grateful to respected Dr. Mohammad Saleh Aloboudi, Dean of College of Science, Majmaah University, Alzul…. It has been a privilege and great honor for us to learn with a great man of such prominent characteristics. We love and respect him the most for his high morality and honorable sincerity, his guidance is a ray of light in our research. We express heartiest thanks to Dr. Thamir Shelih Al-Harbi, Dean of Scienti…c research, Majmaah University, KSA who also helped us in publishing the research work in ISI listed journal.

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Abstract

Abstract

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Abstract

Abstract

This article examines MHD ‡ow and heat transfer through a porous medium over a shrinking surface with suction. The governing partial di¤erential equations of ‡uid motion are converted in to ordinary di¤erential form by using similarity functions. The resulting equations have been solved numerically to observe the e¤ects of physical parameters of this study namely magnetic parameter M2, permeability parameter K, suction parameter λ and Prandtl number Pr. Results are presented in graphical form.

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AMS Subject Classi…cation

AMS Subject Classi…cation:

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AMS Subject Classi…cation

AMS Subject Classi…cation:

[2000] Primary 05C38, 15A15; Secondary 05A15, 15A18

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Key Words Classi…cation

Key Words:

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Key Words Classi…cation

Key Words:

MHD ‡ow, Heat Transfer, Porous medium, Prandtl number, Suction, Shrinking / Stretching surface.

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Publication in ISI listed Journal Research work has been got published in ISI listed Journal: [2]

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Publication in ISI listed Journal Research work has been got published in ISI listed Journal: [2] Farooq Ahmad, Sajjad Hussain, Ali Mansoor Alanbari and Rakan Saud Alharbi, MHD Flow And Heat Transfer Through A Porous Medium Over A Stretching/Shrinking Surface With Suction, Sci.Int.(Lahore), 27(2), 931-935, 2015. [2]

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Publication in ISI listed Journal Research work has been got published in ISI listed Journal: [2] Farooq Ahmad, Sajjad Hussain, Ali Mansoor Alanbari and Rakan Saud Alharbi, MHD Flow And Heat Transfer Through A Porous Medium Over A Stretching/Shrinking Surface With Suction, Sci.Int.(Lahore), 27(2), 931-935, 2015. [2]

http://sci-int.com/pdf/749916686931-935-Farooq.%20Ahmad- paper%202-MATH%20-%20KSA-.pdf

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Publication Paper in ISI listed Journal

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Detail of the Project

The project comprises 5 chapters. Chapter 1. This chapter consists of seven sections. The section 1.1 gives the detail of the project and the section 1.2 covers the nomenclature of the project. The section 1.3 covers general terms and concepts, the section 1.4 covers types of the ‡uids and the section 1.5 covers some important de…nitions. The section 1.6 consists objects of the project, and the last section 1.7 shows research publications from the project / research work.

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Detail of the Project

The project comprises 5 chapters. Chapter 1. This chapter consists of seven sections. The section 1.1 gives the detail of the project and the section 1.2 covers the nomenclature of the project. The section 1.3 covers general terms and concepts, the section 1.4 covers types of the ‡uids and the section 1.5 covers some important de…nitions. The section 1.6 consists objects of the project, and the last section 1.7 shows research publications from the project / research work. Chapter 2. This chapter covers literature review, it contains some detail research work already done about the MHD ‡ow and heat transfer through a porous medium over a shrinking surface with suction.

Ali & Rakan (College of Science Alzul…) Fluids Over Stretcing / Shrinking Sheet 14/07/1436 12 / 48

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Detail of the Project

The project comprises 5 chapters. Chapter 1. This chapter consists of seven sections. The section 1.1 gives the detail of the project and the section 1.2 covers the nomenclature of the project. The section 1.3 covers general terms and concepts, the section 1.4 covers types of the ‡uids and the section 1.5 covers some important de…nitions. The section 1.6 consists objects of the project, and the last section 1.7 shows research publications from the project / research work. Chapter 2. This chapter covers literature review, it contains some detail research work already done about the MHD ‡ow and heat transfer through a porous medium over a shrinking surface with suction. Chapter 3. This chapter has only two sections, the section 3.1 represents formation of the problem and the section 3.2 deals with governing equations of the problem and mathematical analysis.

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Detail of the Project

Chapter 4. This chapter consists of 2 sections. The section 4.1 covers the discussion on graphical results. The section 4.2 covers graphical results from graph numbered 4.1-4.11.

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Detail of the Project

Chapter 4. This chapter consists of 2 sections. The section 4.1 covers the discussion on graphical results. The section 4.2 covers graphical results from graph numbered 4.1-4.11. Chapter 5. In this chapter, the section 5.1 gives conclusion of the

  • study. The section 5.2 represents an overview of the research. The

section 5.3 consists of some future extensions.

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Detail of the Project

Chapter 4. This chapter consists of 2 sections. The section 4.1 covers the discussion on graphical results. The section 4.2 covers graphical results from graph numbered 4.1-4.11. Chapter 5. In this chapter, the section 5.1 gives conclusion of the

  • study. The section 5.2 represents an overview of the research. The

section 5.3 consists of some future extensions.

In the last references related to the research, have been mentioned.

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Introduction

Introduction

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Introduction

Introduction

The ‡ow and heat transfer over a stretching surface bears important research interest due to its various applications in industries such as hot rolling, wire drawing, glass …ber production, manufacturing plastic …lms and extrusion of a polymer in a melt spinning process. Sakiadis [24, 25] was the …rst to propose and analyze the surface stretching problem based on the boundary layer approximation.

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Introduction

Introduction

The ‡ow and heat transfer over a stretching surface bears important research interest due to its various applications in industries such as hot rolling, wire drawing, glass …ber production, manufacturing plastic …lms and extrusion of a polymer in a melt spinning process. Sakiadis [24, 25] was the …rst to propose and analyze the surface stretching problem based on the boundary layer approximation. Crane [9] gave a closed-form solution for steady two-dimensional ‡ow

  • f an incompressible viscous ‡uid caused by the stretching of an

elastic sheet, which moves in its own plane with a velocity that varies linearly with distance from a …xed point. Gupta and Gupta [13] extended the work of Crane [9] by investigating the e¤ect of mass transfer on a stretching sheet with suction or blowing for linear surface velocity subject to uniform temperature.

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Introduction

Magyari and Keller [17] introduced a new type of the stretching sheet problem by considering the ‡ow due to a sheet stretched exponentially in its own plane, and they investigated also the heat transfer characteristics for the ‡ow taking the exponentially varying wall temperature. Barik et al. [7] studied heat and mass transfer on MHD ‡ow through a porous medium over a stretching surface with heat source.

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Introduction

Miklavcic and Wang [18] obtained the solution for steady two-dimensional, as well as axisymmetric viscous, ‡ow over a shrinking sheet. Wang [26] studied two dimensional stagnation ‡ow towards a shrinking sheet. Muhaimin et al. [19] observed the e¤ects

  • f heat and mass transfer on MHD boundary layer ‡ow past a

shrinking sheet subject to suction. Cortell [10] discussed MHD viscous ‡ow caused by a shrinking sheet with suction for two-dimensional and axisymmetric cases.

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Introduction

Miklavcic and Wang [18] obtained the solution for steady two-dimensional, as well as axisymmetric viscous, ‡ow over a shrinking sheet. Wang [26] studied two dimensional stagnation ‡ow towards a shrinking sheet. Muhaimin et al. [19] observed the e¤ects

  • f heat and mass transfer on MHD boundary layer ‡ow past a

shrinking sheet subject to suction. Cortell [10] discussed MHD viscous ‡ow caused by a shrinking sheet with suction for two-dimensional and axisymmetric cases. Fang and Zhang [12] obtained the closed form solution for steady MHD ‡ow over a shrinking surface subject to applied suction. Fang [11] investigated the ‡ow over a shrinking sheet to power law surface velocity and obtained the multiple solutions for certain mass transfer with controlling parameters.

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Introduction

Noor et al. [20] analyzed the magnetohydrodynamic viscous ‡ow due to a shrinking sheet analytically and found that the result obtained by a domain decomposition and homotopy analysis methods are well

  • agreed. Sajid and Hayat [23] investigated the e¤ect of the MHD for

two-dimensional and axisymmetric shrinking sheet. Bhattacharyya and Pop [8] investigated MHD boundary layer ‡ow due to an exponentially shrinking sheet.

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Introduction

Noor et al. [20] analyzed the magnetohydrodynamic viscous ‡ow due to a shrinking sheet analytically and found that the result obtained by a domain decomposition and homotopy analysis methods are well

  • agreed. Sajid and Hayat [23] investigated the e¤ect of the MHD for

two-dimensional and axisymmetric shrinking sheet. Bhattacharyya and Pop [8] investigated MHD boundary layer ‡ow due to an exponentially shrinking sheet. Asghar et al. [5] presented an exact analytical solution for the boundary layer ‡ow of a viscous ‡uid over an impermeable shrinking

  • sheet. Hayat et al. [14, 15] reported an analytic HAM solution for the

MHD ‡ow of a second grade ‡uid over a shrinking sheet without and with rotation e¤ects.

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Introduction

In recent years, the study of magnetohydrodynamic (MHD) ‡ow problems has gained considerable interest of researchers because of its huge applications in many engineering problems. Pavlov [22] discussed the MHD boundary layer ‡ow of an electrically conducting ‡uid due to the stretching sheet. Whereas, Al-Odat et al. [21] explained the e¤ect of magnetic …eld on the thermal boundary layer

  • ver an exponentially stretching continuous surface taking exponential

temperature distribution and Ishak [16] studied the MHD boundary layer ‡ow over an exponentially stretching sheet in the presence of thermal radiation.

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Introduction

In recent years, the study of magnetohydrodynamic (MHD) ‡ow problems has gained considerable interest of researchers because of its huge applications in many engineering problems. Pavlov [22] discussed the MHD boundary layer ‡ow of an electrically conducting ‡uid due to the stretching sheet. Whereas, Al-Odat et al. [21] explained the e¤ect of magnetic …eld on the thermal boundary layer

  • ver an exponentially stretching continuous surface taking exponential

temperature distribution and Ishak [16] studied the MHD boundary layer ‡ow over an exponentially stretching sheet in the presence of thermal radiation. Baag et al. [6] analyzed MHD ‡ow on a stretching sheet embedded in a porous Medium. The magnetohydrodynamic ‡ow over a stretching surface has been studied by [10] & [18] for both permeable and impermeable surfaces.

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Introduction

Flow of an electrically conducting non- Newtonian ‡uid past a stretching surface was studied by Able et al. [1] when a uniform magnetic …eld acts transverse to the surface. Ahmad and Sajjad [3]

  • btained exact solution for a viscous, incompressible, MHD ‡ow over

a porous stretching sheet. Ali et al. [4] presented numerical solution

  • f MHD ‡ow of ‡uid and heat transfer over porous stretching sheet.

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Mathematical Analysis Mathematical Analysis

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Mathematical Analysis Mathematical Analysis

Consider steady, two dimensional and incompressible laminar ‡ow of viscous ‡uid through porous medium over a stretching/ shrinking

  • surface. The ‡uid is electrically conducting. Magnetic …eld of

strength B0 is applied in normal direction to the sheet. A convective heat source with heat ‡ux boundary conditions provides temperature Tw at the surface. The Cartesian coordinates are used. The x axis is along the sheet and y axis is perpendicular to it.

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Mathematical Analysis Mathematical Analysis

Consider steady, two dimensional and incompressible laminar ‡ow of viscous ‡uid through porous medium over a stretching/ shrinking

  • surface. The ‡uid is electrically conducting. Magnetic …eld of

strength B0 is applied in normal direction to the sheet. A convective heat source with heat ‡ux boundary conditions provides temperature Tw at the surface. The Cartesian coordinates are used. The x axis is along the sheet and y axis is perpendicular to it. The origin is …xed. Here u, v are velocity components along horizontal and vertical directions. The induced magnetic …eld is

  • neglected. The permeability of medium is K0. The governing

equations of the motion are:

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Nomenclature

M2 = Magnetic Parameter λ = Suction Parameter Re = Reynolds’s Number Pr = Prandtl Number B0 =Strength of Transverse Magnetic Field σ =Electrical Conductivity µ = The Coe¢cient of Viscosity c = Positive Constant ρ = Fluid Density V0 = Injection Velocity η = Dimensionless Variable

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Mathematical Analysis

∂u/∂x + ∂v/∂y = 0, (3.1) u∂u/∂x + v∂u/∂y = υ∂2u/∂y2 σB2

0 u

ρ υ K0 u, (3.2) u∂T/∂x + v∂T/∂y = K 0 ρCp ∂2T/∂y2 (3.3) where µ is dynamic viscosity, ρ is ‡uid density, υ kinematic viscosity, Cp is the speci…c heat at constant pressure and K 0 is thermal conductivity.

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Mathematical Analysis

The associated boundary conditions are: u = ax, υ = v0, K 0 ∂T ∂y = qw = E0x2 at y = 0 (3.4) u ! 0, T ! Tw as y ! ∞ where u > 0 corresponds with ‡ow over stretching surface and u < 0 for ‡ow over shrinking surface.

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Mathematical Analysis

The similarity transformations are as follows: u = cxf 0(η), v = p υcf (η), (3.5) T T∞ = E0x2 K 0 r υ a θ(η), where η = yp a

υis dimensionless variable, E0 is positive constant, T∞

is temperature far away from the surface, qw is ratio of heat transfer.

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Mathematical Analysis

The similarity transformations are as follows: u = cxf 0(η), v = p υcf (η), (3.5) T T∞ = E0x2 K 0 r υ a θ(η), where η = yp a

υis dimensionless variable, E0 is positive constant, T∞

is temperature far away from the surface, qw is ratio of heat transfer. ∂u/∂x = c f 00, ∂u/∂y = c f 00(∂η/∂y) = c f 00 rc υ = c2/3 υ1/2 f 00,

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Mathematical Analysis

and ∂2u/∂y2 = c2/3 υ1/2 rc υf 000 = c2 υ f 000,

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Mathematical Analysis

and ∂2u/∂y2 = c2/3 υ1/2 rc υf 000 = c2 υ f 000, also ∂v/∂x = and ∂v/∂y = p υcf 0 rc υ = cf 0,

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Mathematical Analysis

T T0 = E0x2 K 0 r υ c θ(η), ∂T/∂x = E0x2 K 0 r υ c θ(η)

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Mathematical Analysis

T T0 = E0x2 K 0 r υ c θ(η), ∂T/∂x = E0x2 K 0 r υ c θ(η) and ∂T/∂y = E0x2 K 0 r υ c θ0(η) rc υ = E0x2 K 0 θ0(η), ∂2T/∂y2 = E0x2 K 0 rc υθ00(η).

Ali & Rakan (College of Science Alzul…) Fluids Over Stretcing / Shrinking Sheet 14/07/1436 26 / 48

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Mathematical Analysis

The equation (3.1) is readily satis…ed and the equations (3.2) and (3.3) become: f 000 + ¤ 00 f 02 (M2 + 1 K )f 0 = 0, (3.6) θ00 + Prf θ0 2Prf 0θ = 0, (3.7) where prime denotes the di¤erentiation with respect to η, M2 = σB 2

ρc

is magnetic parameter, K = K0c

υ

is permeability parameter and Pr = µCp

K 0 .

Ali & Rakan (College of Science Alzul…) Fluids Over Stretcing / Shrinking Sheet 14/07/1436 27 / 48

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Mathematical Analysis

The boundary conditions (3.4) then become: f (0) = λ, f 0(0) = 1, θ0(0) = 1, f 0(∞) = 0, θ(∞) = 0.

  • (3.8)

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Results and Discussion

Discussion on Graphical Results

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Results and Discussion

Discussion on Graphical Results

The equations (3.6) and (3.7) are solved subject to the boundary conditions (3.8). The numerical results have been computed by using Mathematica 5.5 software. The e¤ects of the physical parameters namely magnetic parameter M2, suction parameter λ, permeability parameter K and Prandtl number Pr have been noticed on velocity and temperature distributions. The results have been presented graphically.

Ali & Rakan (College of Science Alzul…) Fluids Over Stretcing / Shrinking Sheet 14/07/1436 29 / 48

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Results and Discussion

Discussion on Graphical Results

The equations (3.6) and (3.7) are solved subject to the boundary conditions (3.8). The numerical results have been computed by using Mathematica 5.5 software. The e¤ects of the physical parameters namely magnetic parameter M2, suction parameter λ, permeability parameter K and Prandtl number Pr have been noticed on velocity and temperature distributions. The results have been presented graphically. The ‡uid ‡ow due to stretching surface is described in …g-4.1 to …g-4.3.

Ali & Rakan (College of Science Alzul…) Fluids Over Stretcing / Shrinking Sheet 14/07/1436 29 / 48

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Results and Discussion

Discussion on Graphical Results

The equations (3.6) and (3.7) are solved subject to the boundary conditions (3.8). The numerical results have been computed by using Mathematica 5.5 software. The e¤ects of the physical parameters namely magnetic parameter M2, suction parameter λ, permeability parameter K and Prandtl number Pr have been noticed on velocity and temperature distributions. The results have been presented graphically. The ‡uid ‡ow due to stretching surface is described in …g-4.1 to …g-4.3. Fig-4.1 and …g-4.2 respectively show that the velocity component f 0decreases with increasing values of M2 and λ.

Ali & Rakan (College of Science Alzul…) Fluids Over Stretcing / Shrinking Sheet 14/07/1436 29 / 48

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Results and Discussion

Discussion on Graphical Results

The equations (3.6) and (3.7) are solved subject to the boundary conditions (3.8). The numerical results have been computed by using Mathematica 5.5 software. The e¤ects of the physical parameters namely magnetic parameter M2, suction parameter λ, permeability parameter K and Prandtl number Pr have been noticed on velocity and temperature distributions. The results have been presented graphically. The ‡uid ‡ow due to stretching surface is described in …g-4.1 to …g-4.3. Fig-4.1 and …g-4.2 respectively show that the velocity component f 0decreases with increasing values of M2 and λ. Fig- 4.3 demonstrates that velocity f 0 increases with increase in the value of K.

Ali & Rakan (College of Science Alzul…) Fluids Over Stretcing / Shrinking Sheet 14/07/1436 29 / 48

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Results and Discussion

Discussion

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Results and Discussion

Discussion The ‡uid ‡ow over a shrinking surface is described in …g-4.4 to …g-4.6.

Ali & Rakan (College of Science Alzul…) Fluids Over Stretcing / Shrinking Sheet 14/07/1436 30 / 48

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SLIDE 59

Results and Discussion

Discussion The ‡uid ‡ow over a shrinking surface is described in …g-4.4 to …g-4.6. The …g-4.4, …g-4.5 and …g-4.6 respectively show that the velocity f 0 increases with increasing values of M2, λ and K.

Ali & Rakan (College of Science Alzul…) Fluids Over Stretcing / Shrinking Sheet 14/07/1436 30 / 48

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SLIDE 60

Results and Discussion

Discussion The ‡uid ‡ow over a shrinking surface is described in …g-4.4 to …g-4.6. The …g-4.4, …g-4.5 and …g-4.6 respectively show that the velocity f 0 increases with increasing values of M2, λ and K. The temperature distributions for ‡ow due to stretching surface are presented in …g-4.7 and …g-4.8.

Ali & Rakan (College of Science Alzul…) Fluids Over Stretcing / Shrinking Sheet 14/07/1436 30 / 48

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SLIDE 61

Results and Discussion

Discussion The ‡uid ‡ow over a shrinking surface is described in …g-4.4 to …g-4.6. The …g-4.4, …g-4.5 and …g-4.6 respectively show that the velocity f 0 increases with increasing values of M2, λ and K. The temperature distributions for ‡ow due to stretching surface are presented in …g-4.7 and …g-4.8. The temperature function θ(η) decreases with increasing values of Pr and λ.

Ali & Rakan (College of Science Alzul…) Fluids Over Stretcing / Shrinking Sheet 14/07/1436 30 / 48

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SLIDE 62

Results and Discussion

Discussion The ‡uid ‡ow over a shrinking surface is described in …g-4.4 to …g-4.6. The …g-4.4, …g-4.5 and …g-4.6 respectively show that the velocity f 0 increases with increasing values of M2, λ and K. The temperature distributions for ‡ow due to stretching surface are presented in …g-4.7 and …g-4.8. The temperature function θ(η) decreases with increasing values of Pr and λ. Fig-4.9 and …g-4.10 respectively demonstrate the e¤ects of Pr and λ

  • n θ(η) for ‡ow over shrinking surface.

Ali & Rakan (College of Science Alzul…) Fluids Over Stretcing / Shrinking Sheet 14/07/1436 30 / 48

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SLIDE 63

Results and Discussion

Discussion The ‡uid ‡ow over a shrinking surface is described in …g-4.4 to …g-4.6. The …g-4.4, …g-4.5 and …g-4.6 respectively show that the velocity f 0 increases with increasing values of M2, λ and K. The temperature distributions for ‡ow due to stretching surface are presented in …g-4.7 and …g-4.8. The temperature function θ(η) decreases with increasing values of Pr and λ. Fig-4.9 and …g-4.10 respectively demonstrate the e¤ects of Pr and λ

  • n θ(η) for ‡ow over shrinking surface.

The temperature function θ(η) decreases for increase in the values of Pr and λ.

Ali & Rakan (College of Science Alzul…) Fluids Over Stretcing / Shrinking Sheet 14/07/1436 30 / 48

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SLIDE 64

Results and Discussion

Discussion The ‡uid ‡ow over a shrinking surface is described in …g-4.4 to …g-4.6. The …g-4.4, …g-4.5 and …g-4.6 respectively show that the velocity f 0 increases with increasing values of M2, λ and K. The temperature distributions for ‡ow due to stretching surface are presented in …g-4.7 and …g-4.8. The temperature function θ(η) decreases with increasing values of Pr and λ. Fig-4.9 and …g-4.10 respectively demonstrate the e¤ects of Pr and λ

  • n θ(η) for ‡ow over shrinking surface.

The temperature function θ(η) decreases for increase in the values of Pr and λ. Fig-4.11 shows that temperature function θ(η) is higher for ‡ow over shrinking surface than for stretching surface.

Ali & Rakan (College of Science Alzul…) Fluids Over Stretcing / Shrinking Sheet 14/07/1436 30 / 48

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SLIDE 65

Results and Discussion

Fig-4.1 shows that the velocity component f 0decreases with increasing values of M2.

4.1 Fig-4.1 Graph of f 0 for di¤erent values of M2 when K=100, λ =2, Pr=0.1

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SLIDE 66

Results and Discussion Fig-4.2 shows that the velocity component f 0decreases with increasing values of λ.

4.2 Fig-4.2 Graph of f 0 for di¤erent values of λ when K=100, M2 =2, Pr=0.1

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SLIDE 67

Results and Discussion

Fig-4.3 demonstrates that velocity f 0 increases with increase in the value of K.

4.3 Fig-4.3 Graph of f 0 for di¤erent values of K when M2=100, λ =2, Pr=0.1

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SLIDE 68

Results and Discussion

The ‡uid ‡ow over a shrinking surface is described in …g-4.4.

4.4 Fig-4.4 Graph of f 0 for di¤erent values of M2 when K=100, λ =2, Pr=0.1

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SLIDE 69

Results and Discussion

The ‡uid ‡ow over a shrinking surface is described in …g- 4.5.

4.5 Fig- 4.5 Graph of f 0 for di¤erent values of λ when K=100, M2 =2, Pr=0.1

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SLIDE 70

Results and Discussion

The ‡uid ‡ow over a shrinking surface is described in …g- 4.6.

4.6 Fig- 4.6 Graph of f 0 for di¤erent values of K when M2=100, λ =2, Pr=0.1

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SLIDE 71

Results and Discussion

The temperature distributions for ‡ow due to stretching surface are presented in …g-4.7.

4.7 Fig-4.7 Graph of θ for di¤erent values of Pr when K=100, λ =2, M2=2

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SLIDE 72

Results and Discussion

The temperature distributions for ‡ow due to stretching surface are presented in …g- 4.8.

4.8 Fig-4.8 Graph of θ for di¤erent values of λ when K=100, M2 =2, Pr=0.1

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SLIDE 73

Results and Discussion

Fig-4.9 demonstrate the e¤ects of Pr on θ(η) for ‡ow over shrinking surface.

4.9 Fig-4.9 Graph of f 0 for di¤erent values of Pr when K=100, λ =2, M2=2

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SLIDE 74

Results and Discussion

Fig-4.10 demonstrate the e¤ects of λ on θ(η) for ‡ow over shrinking surface.

4.10 Fig- 4.10 Graph of θ for di¤erent values of λ when K=100, M2 =2, Pr=0.1

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SLIDE 75

Results and Discussion

Fig-4.11 shows that temperature function θ(η) is higher for ‡ow over shrinking surface than for stretching surface.

4.11 Fig-4.11 Graph for the comparison of the values of θ for ‡ow over stretching/shrinking surface.

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SLIDE 76

Bibliography I

[1] S. Able, P. H. Veena, K. Rajagopal & V. K. Pravin, Non- Newtonian magnetohyderodynamic ‡ow over a stretching surface with heat and mass transfer. International Journal of Nonlinear Mechanics 39, 1067-1078, 2004. [2] Farooq Ahmad, Sajjad Hussain, Ali Mansoor Alanbari and Rakan Saud Alharbi, MHD Flow And Heat Transfer Through A Porous Medium Over A Stretching/Shrinking Surface With Suction, Sci.Int.(Lahore), 27(2), 931-935, 2015. [3] F. Ahmad, Sajjad Hussain and M. Ali, An Analytical Solution of MHD Flow over Porous Stretching Sheet, J. Basic. Appl. Sci. Res., 4(3),160-167, 2014. [4] M. Ali, F. Ahmad and S. Hussain, Numerical Solution of MHD Flow of Fluid and Heat Transfer over Porous Stretching Sheet, J.

  • Basic. Appl. Sci. Res., 4(5),146-152, 2014.

Ali & Rakan (College of Science Alzul…) Fluids Over Stretcing / Shrinking Sheet 14/07/1436 42 / 48

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Bibliography II

[5] S. Asghar, A. Ahmad and A. Alsaedi, Flow of a viscous ‡uid over an incompressible shrinking sheet, Applied Mathematics Letters, 26(12),1165-1168, 2013. [6] S. Baag, M. R. Acharya, G. C. Dash, MHD Flow Analysis Using DTM-Pade and Numerical Methods, American Journal of Fluid Dynamics, 4(1), 6-15, 2014. [7] R. N. Barik, G. C. Dashand & P. K. Rath, Heat and mass transfer

  • n MHD ‡ow through a porous Medium over a stretching surface with

heat source,Mathematical Theory and Modeling , 2(7), 49-60, 2012. [8] K. Bhattacharyya and I. Pop, MHD boundary layer ‡ow due to an exponentially shrinking sheet, MAGNETOHYDRODYNAMICS, 47(4), 337–344, 2011.

Ali & Rakan (College of Science Alzul…) Fluids Over Stretcing / Shrinking Sheet 14/07/1436 43 / 48

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Bibliography III

[9] L. J. Crane, Flow past a stretching plate, Zeitschrift f¨ ur Angewandte Mathematik und Physik, 21(4), 645–647, 1970. [10] R. Cortell. On a certain boundary value problem arising in shrinking sheet ‡ows. Appl. Math. Comput., 217, 4086–4093, 2010. [11] T. G. Fang, Boundary layer ‡ow over a shrinking sheet with power-law velocity, International Journal of Heat and Mass Transfer, 51(25-26), 5838–5843, 2008. [12] T. Fang and J. Zhang, Closed-form exact solutions of MHD viscous ‡ow over a shrinking sheet, Communications in Nonlinear Science and Numerical Simulation, 14(7), 2853–2857, 2009. [13] P. S. Gupta and A. S. Gupta, Heat and mass transfer on a stretching sheet with suction or blowing, The Canadian Journal of Chemical Engineering, 55 (6), 744–746, 1977.

Ali & Rakan (College of Science Alzul…) Fluids Over Stretcing / Shrinking Sheet 14/07/1436 44 / 48

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Bibliography IV

[14] T. Hayat, Z. Abbas, M. Sajid. On the analytic solution of magnetohydrodynamic ‡ow of a second grade ‡uid over a shrinking

  • sheet. ASME J. Appl. Mech., 74, 1165–1171, 2007.

[15] T. Hayat, T. Javed, M. Sajid. Analytic solution for MHD rotating ‡ow of a second grade ‡uid over a shrinking surface. Phys. Lett. A, 372, 3264–3273, 2008. [16] A. Ishak. MHD boundary layer ‡ow due to an exponentially stretching sheet with radiation e¤ect. Sains Malaysiana, 40, 391–395, 2011. [17] E. Magyari, B. Keller. Heat and mass transfer in the boundary layers on an exponentially stretching continuous surface. J. Phys. D.

  • Appl. Phys., 32, 577–585, 1999.

[18] M. Miklavcic and C. Y. Wang, Viscous ‡ow due to a shrinking sheet, Quarterly of Applied Mathematics, 64(2), 283–290, 2006.

Ali & Rakan (College of Science Alzul…) Fluids Over Stretcing / Shrinking Sheet 14/07/1436 45 / 48

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Bibliography V

[19] I. Muhaimin, R. Kandasamy and A.B. Khamis, E¤ects of heat and mass transferon non-linear MHD boundary layer ‡ow over a shrinking sheet in the presence of suction, Applied Mathematics and Mechanics, 29(10),1309-1317, 2008. [20] N. F. M. Noor, S. A. Kechil, I. Hashim. Simple non-perturbative solution for MHD viscous ‡ow due to a shrinking sheet. Commun. Nonlinear Sci. Numer. Simul., 15, 144–148, 2010. [21] M. Q. Al-Odat, R. A. Damseh, T. A. Al-Azab. Thermal boundary layer on an exponentially stretching continuous surface in the presence

  • f magnetic …eld e¤ect. Int. J. Appl. Mech. Eng., 11, 289–299, 2006.

[22] K. B. Pavlov. Magnetohydrodynamic ‡ow of an incompressible viscous ‡uid caused by the deformation of a plane surface. Magnetohydrodynamics, 10(4), 507–509, 1974.

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Bibliography VI

[23] M. Sajid and T. Hayat, The application of homotopy analysis method for MHD viscous ‡ow due to a shrinking sheet, Chaos, Solitons and Fractals, 39(3), 1317–1323, 2009. [24] B. C. Sakiadis, Boundary layer behavior on continuous solid surface: I. Boundary layer equations fortwo dimensional and axisymmetric ‡ow, AIChE Journal, 7(1), 26–28, 1961. [25] B. C. Sakiadis, Boundary layer behavior on continuous solid surface: II. Boundary layer equations for two dimensional and axisymmetric ‡ow, AIChE Journal, 7(1), 221–225, 1961. [26] C. Y. Wang, Stagnation ‡ow towards a shrinking sheet, Int. J. Non- Linear Mechanics, 43, 377-382, 2008.

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Thanks

THANKS

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