Problems on Ferrofluid Flow Due to Rotating Disk Thesis

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PROBLEMS ON FERROFLUID FLOW DUE TO ROTATING DISK

THESIS

Submitted to Department of Mathematics National Institute of Technology, Kurukshetra For the award of the degree of

DOCTOR OF PHILOSOPHY

MATHEMATICS

Under the Supervision of Submitted by Dr. PARAS RAM KUSHAL SHARMA (Reg. No. 2K 08-NITK-Ph.D. -1196)

DEPARTMENT OF MATHEMATICS National Institute of Technology Kurukshetra-136119 Haryana, INDIA (February, 2012)

DEPARTMENT OF MATHEMATICS National Institute of Technology Kurukshetra-136119, INDIA

CERTIFICATE

This is to certify that the thesis entitled “Problems on Ferrofluid Flow due to Rotating Disk” submitted by Mr. Kushal Sharma in fulfillment of the requirement for the award of degree of Doctor of Philosophy in Mathematics of National Institute of Technology, Kurukshetra is a bonafide record of the research work carried out by him under my supervision. Further, this thesis work as a whole or a part thereof has not been submitted elsewhere for the award of any other degree, associateship, fellowship or other similar title.

Date:

(Dr. Paras Ram) Associate Professor & Head Department of Mathematics National Institute of Technology Kurukshetra-136119 (INDIA)

ACKNOWLEDGEMENTS

Apart from the grace of almighty, numbers of persons have been extremely generous and helpful to me during the course of this study. I cannot afford to miss at least few of them who have been constant source of inspiration and moral support towards the completion of my thesis.

I express my deep sense of gratitude to Dr. Paras Ram, Associate Professor and Head, Department of Mathematics, National Institute of Technology, Kurukshetra, for his valuable and untiring guidance, constant encouragement and constructive criticism which have enabled me to successfully complete this study and to learn lot many things. My thankfulness to him goes beyond this formal acknowledgement and cannot be fully expressed in my words.

A special thanks to the faculty and supporting staff of the Department of Mathematics and the Librarian, N.I.T. Kurukshetra for their cooperation during my study stay of around three and half years in the Department.

I avail the opportunity to express my love and reverence to my mother and grandmother whose blessings has been source of strength to this accomplishment. My special thanks to my elder brother Shri Bhagwan Bhardwaj, who kept me worriless during whole the period of my research. My warm thanks to my wife Sarita and my younger brother Dinesh for their encouraging moral support.

It is beyond me to express my appreciation for the unforgettable help extended by my colleagues Mr. Vikas Kumar, Mr. Anupam Bhandari, Mr. Hawa Singh. I also wish to express my gratefulness to my friends Harish, Bajinder.

(Kushal Sharma)

AUTHOR’S PUBLICATIONS

The following research papers based on the work in the thesis have been published / accepted for publication in different journals.

Research Papers in National / International Journals

1. On the revolving ferrofluid flow due to rotating disk. International Journal of Nonlinear Sciences (IJNS) (accepted) (2012).

2. Revolving ferrofluid flow under the influence of MFD viscosity and porosity with rotating disk. Journal of Electromagnetic Analysis and Applications (JEMAA), 3 (9), pp. 378-386 (2011).

3. Effect of porosity on revolving ferrofluid flow with rotating disk. International Journal of Fluids Engineering (IJFE), 3 (3), pp. 261-271 (2011).

4. Effect of magnetic field-dependent viscosity on revolving ferrofluid. Journal of Magnetism and Magnetic Materials (JMMM), 322 (21), pp. 3476-3480 (2010).

5. Effect of porosity on ferrofluid flow with rotating disk. International Journal of Applied Mathematics and Mechanics (IJAMM), 6 (16), pp. 65-76 (2010).

6. Axi-symmetric ferrofluid flow with rotating disk in a porous medium, International Journal of Fluid Mechanics (IJFM), 2 (2), pp. 151-161 (2010).

7. Effect of rotation and MFD viscosity on ferrofluid flow with rotating disk, Indian Journal of Pure and Applied Physics (under review since Oct. 2011).

Research Papers in Conferences / Symposia / International Meet

1. Revolving ferrofluid flow due to a rotating disk. International Conference of Mechanical Engineering (ICME-2011) organized by World Congress on Engineering (WCE-2011) held from 6-8 July, 2011 at Imperial College London, UK.

2. Effect of MFD viscosity and porosity on revolving Axi-symmetric ferrofluid with rotating disk. International Conference of Mechanical Engineering (ICME-2011) organized by World Congress on Engineering (WCE-2011) held from 6-8 July, 2011 at Imperial College London, UK.

3. Study of ferrofluid flow by recurrence relation method, Proceedings of 55th Congress of ISTAM, held from 18-21 Dec. 2010, at National Institute of Technology, Hamirpur, Himachal Pradesh, INDIA.

4. Axi-symmetric ferrofluid flow with variable viscosity, Proceedings of 55th Congress of ISTAM, held from 18-21 Dec. 2010, at National Institute of Technology, Hamirpur, Himachal Pradesh, INDIA.

CONTENTS

OVERVIEW OF THE THESIS 1 NOMENCLATURE (LIST OF SYMBOLS) 5 LIST OF TABLES 7 LIST OF FIGURES 9

Chapter Description Page No.

CHAPTER 1 INTRODUCTION 11

CHAPTER 2 EFFECT OF ROTATION ON FERROFLUID FLOW 27 WITH ROTATING DISK

CHAPTER 3 FERROFLUID FLOW BEHAVIOUR DUE TO ROTATING 39 DISK IN A POROUS MEDIUM

CHAPTER 4 EFFECT OF POROSITY ON FERROFLUID FLOW 47 WITH ROTATING DISK

CHAPTER 5 REVOLVING FERROFLUID FLOW INFLUENCED BY 57 POROSITY WITH ROTATING DISK

CHAPTER 6 FERROFLUID FLOW BEHAVIOUR INFLUENCED 67 BY THE EFFECT OF ROTATION AND MAGNETIC FIELD-DEPENDENT VISCOSITY WITH ROTATING DISK

CHAPTER 7 FERROFLUID FLOW BEHAVIOUR INFLUENCED BY 77 THE EFFECT OF ROTATION, POROSITY AND MAGNETIC FIELD-DEPENDENT VISCOSITY WITH ROTATING DISK

CONCLUSIONS 91

REFERENCES 97

OVERVIEW OF THE THESIS

The purpose of the present study is to characterize the flow behaviour of non- conducting ferrofluid under the influence of uniform magnetic field in the laminar boundary layer with a rotating disk. Power series approximation technique is employed for analytical solution and visualization research problems on ferrofluid flow with a rotating disk. In the numerical investigations, the flow is characterized by using the dimensionless numbers: porosity value  , Darcy number (    ), MFD viscosity parameter k . Using similarity transformations, and introducing the rheological behaviour of the fluid into conservation equations, the corresponding nonlinear boundary value problems are formulated. The solutions of the problems under investigation are obtained by power series approximation method. The influence of these numbers: porosity value, Darcy Number, MFD viscosity parameter; on the flow behaviour has been observed. As for all research problems, fluid is assumed to be flowing though a disk, so cylindrical coordinate system is under consideration.

In all cases, numerical computations are made and graphical illustrations are provided. Comparison between the numerical results for all three components of velocity, pressure profile and displacement thickness for ferrofluid has been found to be interesting for rotating disk problems. Significant discrepancies are noticed between numerical results for ferrofluid with rotating disk especially at low porosity values. At the end of each chapter, the discussion of results obtained is given in detail and an overall comparison of results for various parameters has been reported in the last section of the thesis heading by “CONCLUSIONS”.

A brief account of research work, which is presented in the thesis, is given here.

Chapter 1 is of introductory nature. In this chapter, a brief overview of ferrofluid is given; some concepts related to rotating disk, porosity, porous medium, effect of rotation, magnetic-field dependent viscosity, boundary layer displacement thickness, angle of rotation and work done by different researchers on these aspects has been presented in short.

Chapter 2 deals with the theoretical investigation of effect of rotation on ferrofluid flow due to rotating disk within the framework of Neuringer-Rosensweig model. Power series approximations are used to solve the non-linear coupled partial differential equations for characterizing the profiles for radial velocity, tangential velocity, axial

1 velocity and pressure. In this chapter, the boundary layer displacement thickness is calculated and compared with the case of ordinary viscous-incompressible fluid.

Chapter 3 concerns with the Axi-symmetric ferrofluid with rotating disk in a porous medium. Here, some interesting results of fluid flow characteristics (radial velocity, tangential velocity, axial velocity and pressure profile) have been shown due to effect of porosity parameter (Darcy parameter).

Chapter 4 describes the effect of porosity on ferrofluid flow behaviour due to rotating disk and calculated the displacement thickness of boundary layer for different values of porosity resulting in the fact that the displacement thickness increases with increase in the porosity. Here, results are compared with the case of ordinary viscous fluid flow reported by Benton (1966).

Chapter 5 presents the effect of rotation and porosity under steady state conditions assuming the ferrofluid as viscous, incompressible, electrically non- conducting, Axi-symmetric in isotropic medium considering the z-axis as axis of rotation excluding the thermal effects.

Chapter 6 deals with the combined effect of magnetic-field dependent (MFD) viscosity and rotation on ferrofluid flow due to a rotating disk. Here, the velocity components (radial, tangential and axial) and pressure profiles are computed numerically and presented graphically for various values of magnetic-field dependent viscosity parameter along with the variation of Karman‟s dimensionless parameter.

Chapter 7 deals with the combined effect of rotation, porosity and MFD viscosity on ferrofluid flow due to a rotating disk. The velocity components and pressure profile are illustrated graphically for various values of MFD viscosity parameter and porosity value along with the variation of Karman‟s dimensionless parameter.

Finally, under the heading “Conclusions” of thesis, a comparative study of results obtained in all research problems is carried out. Firstly, the results obtained in chapters 2, 3 and 4 for rotation, porous medium, porosity, respectively, are compared with each other and it is found that these parameters affects the velocity profile, pressure profile, displacement thickness significantly. Further, the results of research problems discussed in chapters 5 and 6 for effect of “rotation & porosity” and “rotation & MFD viscosity” respectively, are compared with each other. Also, these results are compared with the results obtained in chapters 2, 3 and 4 for the parameters: rotation, porous medium, porosity, respectively. In the last, an overall comparison is made out one by one,

2 between the hypothesis of combined effect of rotation, MFD viscosity and porosity with the results reported in chapters 2, 3, 4, 5, 6 and 7. The objective of this study is to better understand the behaviour of ferrofluid flow due to rotating disk influenced by rotation, porous medium, porosity, MFD viscosity. In nut shell, our work is a theoretical motivation explaining physical effects of rotation, porous medium, porosity and variable field dependent viscosity on various flow characteristics of ferrofluid and projects certain practical applications in different areas such as rotating machinery, lubrication, oceanography, computer storage devices, crystal growth processes etc. The research for this doctoral thesis focuses on the effects of magnetic force, rotation, porous medium, porosity value, MFD viscosity on behaviour of ferrofluid flow.

NOMENCLATURE (LIST OF SYMBOLS)

Latin Symbols

V Velocity of ferrofluid (m/ s)

2 p Fluid pressure (kg/ ms )

p Reduced pressure

M Magnetization (A/ m)

H Magnetic field intensity (A/ m)

B Magnetic induction

r Radial direction (m)

z Axial direction (m)

vr Radial velocity (m/ s)

v Tangential velocity (rad / s)

vz Axial velocity (m/ s)

E Dimensionless component of radial velocity

F Dimensionless component of tangential velocity

G Dimensionless component of axial velocity

P Karman‟s dimensionless pressure

P0 Initial pressure (absolute value)

d Displacement thickness of the ferrofluid layer (m)

k MFD viscosity parameter

Q Total volume flowing outward the z-axis (m3 )

Greek Symbols

 Angular velocity of the disk (rad / s)

Ω Angular velocity of whole system (rad / s)

 Magnetic susceptibility

 Gradient operator (m1)

 Fluid density (kg/ m3 )

0 Magnetic permeability of free space (H / m)

 f Reference viscosity of fluid (kg/ ms)

 Kinematic reference viscosity (m2 / s)

 1 MFD kinematic variable viscosity

 Karman‟s parameter (dimensionless)

 Darcy number (porosity parameter)

 Variation coefficient of magnetic field-dependent (MFD) viscosity

 Porosity value

 Tangential direction (rad)

0 Angle of rotation (deg)

 Porous permeability (Darcy permeability)

LIST OF TABLES

Table 2.1: The steady state velocity field and pressure as functions of  for the effect of rotation

Table 3.1: Various coefficients in equations (3.9a) - (3.9d)

Table 3.2: The steady state velocity field and pressure as functions of for the effect of Darcy number 

Table 4.1: The coefficients involved in the equations (4.9a) - (4.9d)

Table 4.2: The steady state velocity field and pressure as functions of for the effect of porosity value 

Table 5.1: The coefficients involved in the equations (5.9a) – (5.9d)

Table 5.2: The steady state velocity field and pressure as functions of for the effect of rotation and porosity value

Table 6.1: First five coefficients in algebraic equations for A 's, B 's, C 's and D 's i i i i

Table 6.2: The steady state velocity field and pressure as functions of for the effect of rotation and MFD viscosity parameter k

Table 7.1: The coefficients A , A , A , A in equation (7.9a) for the effect of rotation, 1 2 3 4 MFD viscosity parameter k  1.1, 1.2 and 1.3 and porosity value   0.01, 0.02 and 0.03

Table 7.2: The coefficients C , C , C , C in equation (7.9c) for the effect of 1 2 3 4 rotation, MFD viscosity parameter and and porosity value and

Table 7.3: The coefficients D , D , D , D in equation (7.9d) for the effect of 1 2 3 4 rotation, MFD viscosity parameter and and porosity value and

Table 7.4: The coefficients B , B , B , B in equation (7.9b) for porosity value 1 2 3 4 and

Table 7.5: The steady state radial velocity E as a function of for the effect of rotation, porosity, MFD viscosity parameter

Table 7.6: The steady state axial velocity G as a function of  for the effect of rotation, porosity, MFD viscosity parameter k

Table 7.7: The steady state pressure P as a function of for the effect of rotation, porosity, MFD viscosity parameter

Table 7.8: The steady state tangential velocity F as a function of for the effect of rotation, porosity, MFD viscosity parameter

LIST OF FIGURES

Fig. 2: Coordinate system for rotating disk system Fig. 2.1: Effect of rotation on radial velocity profile

Fig. 2.2: Effect of rotation on tangential velocity profile

Fig. 2.3: Effect of rotation on axial velocity profile

Fig. 2.4: Effect of rotation on pressure profile

Fig. 2.5: Effect of rotation on derivative of radial velocity

Fig. 2.6: Effect of rotation on derivative of tangential velocity

Fig. 3.1: Effect of Darcy number (porosity parameter)  on radial velocity profile

Fig. 3.2: Effect of Darcy number (porosity parameter) on tangential velocity profile

Fig. 3.3: Effect of Darcy number (porosity parameter) on axial velocity profile

Fig. 3.4: Effect of Darcy number (porosity parameter) on pressure profile

Fig. 3.5: Effect of Darcy number (porosity parameter) on derivative of radial velocity profile

Fig. 3.6: Effect of Darcy number (porosity parameter) on derivative of tangential velocity profile

Fig. 4.1: Effect of porosity  on radial velocity profile

Fig. 4.2: Effect of porosity on tangential velocity profile

Fig. 4.3: Effect of porosity on axial velocity profile

Fig. 4.4: Effect of porosity on pressure profile

Fig. 4.5: Effect of porosity on derivative of radial velocity profile

Fig. 4.6: Effect of porosity on derivative of tangential velocity profile

Fig. 5.1: Effect of porosity and rotation on radial velocity

Fig. 5.2: Effect of porosity and rotation on tangential velocity

Fig. 5.3: Effect of porosity and rotation on axial velocity

Fig. 5.4: Effect of porosity  and rotation on pressure profile

Fig. 5.5: Effect of porosity on derivative of radial velocity profile

Fig. 5.6: Effect of porosity and rotation on derivative of tangential velocity profile

Fig. 6.1: Effect of MFD viscosity and rotation on radial velocity profile

Fig. 6.2: Effect of MFD viscosity and rotation on tangential velocity profile

Fig. 6.3: Effect of MFD viscosity and rotation on axial velocity profile

Fig. 6.4: Effect of MFD viscosity and rotation on pressure profile

Fig. 6.5: Effect of MFD viscosity and rotation on derivative of radial velocity

Fig. 6.6: Effect of MFD viscosity and rotation on derivative of tangential velocity

Fig. 7.1 (a): Effect of rotation and porosity   0.01 along with variation of MFD viscosity parameter k on radial velocity

Fig. 7.1 (b): Effect of rotation and porosity   0.02 along with variation of MFD viscosity parameter on radial velocity

Fig. 7.1 (c): Effect of rotation and porosity   0.03 along with variation of MFD viscosity parameter on radial velocity

Fig. 7.2: Effect of porosity  on tangential velocity for the case of rotation, MFD viscosity and porosity

Fig. 7.3 (a): Effect of rotation and porosity along with variation of MFD viscosity parameter on axial velocity

Fig. 7.3 (b): Effect of rotation and porosity along with variation of MFD viscosity parameter on axial velocity

Fig. 7.3 (c): Effect of rotation and porosity along with variation of MFD viscosity parameter on axial velocity

Fig. 7.4 (a): Effect of rotation and porosity along with variation of MFD viscosity parameter on pressure profile

Fig. 7.4 (b): Effect of rotation and porosity along with variation of MFD viscosity parameter on pressure profile

Fig. 7.4 (c): Effect of rotation and porosity along with variation of MFD viscosity parameter on pressure profile

Chapter 1

INTRODUCTION

1.1 CONCEPTS IN FLUID MECHANICS

Fluid (liquid or gas) is the substance; that is capable to flow under the action of forces acting on it and fluid dynamics is the branch of science which deals with fluid behaviour in motion under the influence of body and surface forces acting on it. The subject dealing with the motion of incompressible fluids i.e. liquids is termed as hydrodynamics. The origin of hydrodynamics is found in the middle of 18th century when Euler discovered the equation of motion of an inviscid fluid. Later on, many scientists gave their significant contributions towards the development of this subject. Navier and Stokes gave the equation of motion of viscous fluids. The equation of turbulent motion was discovered by Reynolds and the well known boundary layer theory was introduced by Prandtl. Besides, many other scientists / mathematicians including Benard, Kelvin, Taylor, Karman etc. gave their excellent contributions to this subject. On the basis of various fields and fluid interactions, hydrodynamics may be classified as

(i) Electro hydrodynamics (EHD) (ii) Magneto hydrodynamics (MHD) (iii) Ferrohydrodynamics (FHD)

Electro hydrodynamics (EHD) is the branch of science dealing with the motion of fluids with electric force effects and magneto hydrodynamics (MHD) deals with the motion of electrically conducting fluids in the presence of magnetic field. In MHD, when electric current flows at an angle to the direction of an impressed magnetic field, the Lorentz body force arises, however in FHD; there is no electric current flowing in the fluid. During mid sixties, just after the formation procedure of ferrofluids, the importance of Ferrohydrodynamics (FHD) was realized, because of large potential application in various fields. Ferrohydrodynamics (FHD) is the mechanics of fluid motion influenced by strong forces of magnetic polarization and in this branch; we study the interaction of magnetic fields with non-conducting ferromagnetic fluids. Many physicists and engineers gave great contributions (theoretical and experimental) to Ferrohydrodynamics and its applications.

This thesis presents the theoretical investigation about hydrodynamics of ferrofluid flow over a rotating disk. Before discussing about Ferrohydrodynamics (FHD), a brief introduction to ferrofluid (a colloidal suspension) is given here.

1.2 FERROFLUID

There are many fascinating materials which have been attracting scientists and researchers for their extraordinary physical properties and technical usage. Ferrofluid is one of such smart materials not available in nature freely, but is to be synthesized by different processes. Ferrofluid is a liquid which becomes strongly magnetized in the presence of magnetic field. There are at least three components required to prepare ferrofluid i.e. magnetic particles of colloidal size, carrier liquid and stabilizer (surfactant). They are stable suspensions of colloidal single domain ferromagnetic particles of the order of 10nm in suitable non-magnetic carrier liquid (Bibik and Lavrov 1965; Rosensweig et al. 1965; Rosensweig 1985; Berkovsky et al. 1993, Odenbach 2009). If the size of permanently magnetized nano-particles will be less than 1-2 nm, the magnetic properties will disappear and colloidal motion increases with decreasing the size of the particle. The colloidal particles, typically made from magnetite (Fe3O4), are coated with surfactants to avoid their agglomeration under Vander Waals attraction forces and dipole-dipole interaction among them. The presence of surfactant helps to maintain proper spacing between the particles to provide colloidal stability.

A rich set of flow patterns and instabilities in the presence of DC, AC and rotating magnetic fields is exhibited by ferrofluids which are opaque to visible light (Cowley and Rosensweig 1967, Rosensweig 1997). Ferrofluids were first discovered at National Aeronautics and Space Administration (NASA) Research Center in mid 1960‟s. The scientists at NASA found that they could make to flow this amazing ferrofluid by varying the external magnetic field. After the discovery of ferrofluid, not only original publications in journals and conferences have been released, but some textbooks like “Ferrohydrodymics” by Rosensweig (1985), “Magnetic Fluids: Engineering Applications” by Berkovsky et al. (1993), “Magnetic Fluids and Applications Handbook” by Berkovsky and Bastovoy (1996), “Magnetic Fluids” by Blums et al. (1997), “Magnetoviscous Effects in Ferrofluids” by Odenbach (2002) etc. also have been published in this area to supplement the basis for its engineering applications.

These fluids have variety of applications in the field of sciences and engineering like instrumentation, electrical and electronics engineering etc. which are being commercialized. Ferrofluids are widely used in sealing of computer hard disk drives,

12 rotating X-ray tubes, rotating shafts, rods and sink-float systems for separation of materials. These are used as lubricants in bearing and dumpers. They are also used as heat controller in electric motors and hi-fi speaker systems without the need of change in their geometrical shape (Hathaway 1979). Ferrofluids are being greatly used in many magnetic fluid based scientific devices like sensors, densimeters, accelerometer, pressure transducers etc. and are also used in actuating machines like electromechanical converters, energy converters etc.(Raj and Moskowitz 1990). One special application of ferrofluids is their use as magnetic ink for high-speed, inexpensive and silent printers (Maruno et al. 1983).

They are also found to be very useful in the field of biomedicine due to magnetically targeted drug delivery (anti-cancer agents such as radio-nuclides, cancer specific antibodies, genes etc.) (Ruuge and Rusetski 1993) to a certain area of human body, targeted destruction of tumors, in-vivo monitoring of chemical activity in the brain and toxin removal from the body for cancer treatment (Goodwin et al. 1999, Pulfer and Gallo 2000, Kim et al. 2001a, 2001b). Due to its viscous action, in a non-uniform magnetic field, a drop of magnetic fluid can move as a whole fluid body. Magnetic nanoparticles can reach even the smallest capillaries of the body, which are 5 6  m in diameter. Magnetic fluids are also used in the contrast medium in X-ray examinations (Papisov et al. 1993) and for positioning tamponade for retinal detachment repair in eye surgery (Dailey et al. 1999). A potential application of ferrofluids is found in the subsurface environmental engineering, in which externally applied magnetic fields are used to direct and control the flow of ferrofluids under the ground (Moridis et al. 1998, Oldenburg and Moridis 1998).

1.3 SIGNIFICANCE OF FLOW IN ROTATING DISK SYSTEMS

The rotating disk is a popular geometry for studying different flows, because of its simplicity and the fact that it represents a classical fluid dynamics problem. Rotating disk geometry is of widespread practical interest in connection with steam turbines, gas turbines, pumps and other rotating fluid machines (Owen and Rogers 1989) covered under the subject of classical fluid dynamics. Possible types of disk used in fluid flows related to rotating disks can be classified into two main categories:

1. “Free disk is a disk that rotates within an infinite medium.

2. “Enclosed disk” is a disk that rotates within a chamber of finite dimensions.

The ferrofluid flow behaviour can be characterized by considering a rotating disk system. The rotating disk has proven to be a successful system which allows a complete analytical solution to the equation of motion in the three dimensional flow. There are several reasons for considering various problems of ferrofluid flow due to the rotating disk system as a prototype, where a boundary layer type of flow occurs leading to engineering applications of rotating flows.

Here, we choose to study three-dimensional boundary layers with a theoretical investigation in the rotating disk boundary layer for a number of reasons. Firstly, the boundary layer is three dimensional throughout its entire development from laminar to turbulence transition. This helps us to describe three-dimensional effects driven by pressure gradients. The disk flow has an analytical solution for laminar flow which provides a genuine motivation and useful checks on experimental techniques. The rotating boundary layer is Axi-symmetric in the mean, eliminating one independent variable and simplifying the analysis. Finally, it shares similarities with other rotating flows. The frame of reference for the disk flow is rotating right handed cylindrical coordinate system with the rotation rate Ω , allowing us to use the symmetry of flow to simplify the average boundary layer equations.

The pioneering study of ordinary viscous fluid flow due to the infinite rotating disk was carried out by Karman (1921). Karman swirling viscous flow (1921) is a famous classical problem in fluid mechanics. He introduced the famous transformation, which reduces the governing partial differential equations into ordinary differential equations. Karman rotating disk problem is extended to the case of flow impulsively started from rest, and also the steady state is solved to a higher degree of accuracy than previously done by a simple analytical method which neglects the resembling difficulties in Cochran (1934) well known solution. Cochran obtained asymptotic solutions for the steady hydrodynamic problem formulated by Karman. However, the solution obtained is of the boundary-layer type. This problem was further investigated both theoretically and experimentally by Goldstein (1935) and Gregory et al. (1955). Stuart (1954) studied the effects of uniform suction on the flow due to a rotating disk. The most accurate solution so far seems to have been reported by Ackroyd (1978).The boundary layer has a constant thickness for the laminar case (Karman 1921) and explained experimentally by Gregory et al. (1955).

The steady flow of ordinary viscous fluid due to the rotating disk with uniform suction was studied by Mithal (1961). Benton (1966) improved Cochran‟s solutions and also solved the unsteady case. Attia (1998) studied the unsteady state in the presence of

14 an applied uniform magnetic field. Karman‟s problem of a rotating disk in an infinite viscous fluid is extended to the case where the disk surface admits partial slip (Miklavcic and Wang 2004). Attia and Aboul-Hasan (2004) discussed about the flow due to an infinite disk rotating in the presence of a uniform axial magnetic field by taking Hall effect into consideration. Turkyilmazoglu (2010a) obtained analytical expressions for the solution of the boundary layer flow due to a rotating disk in the presence of a uniform suction or injection and homotopy analysis method was employed to obtain the exact solutions. Turkyilmazoglu (2010b) discussed the solution of the steady laminar flow of an incompressible viscous electrically conducting fluid over a rotating disk in the presence of a uniform transverse magnetic field and obtained purely explicit analytical expressions for solution of magneto-hydrodynamic equations via homotopy analysis method.

As the motion of an object in an inertial frame of reference is governed by Newton‟s laws of motion, it becomes essential to take the pseudo-forces (Centrifugal and Coriolis) into account. A body that is stationary relative to a non-rotating inertial frame seems to be rotating when viewed from a frame rotating with an angular rate  . For example, take the clear revolution of a stationary object (such as a distant star or planet), which is seen to be rotating when viewed from the rotating frame. Therefore, by application of Newton‟s laws, an inward centripetal force is required for circulatory motion in the frame of rotation at a distance r from the rotational axis.

In the rotating frame, this centripetal force is given as a net force i.e. the total of the radially inward Coriolis force and the radially outward centrifugal force (Rudraiah et al. 1986, Lee et al. 2004, Sleiti and Kapat 2008, Ghosh et al. 2010, Javed et al. 2011, Prasad and Kumar 2011, Sunil and Mahajan 2011). When a fluid rotates, viscous forces may be balanced by Coriolis force, rather than inertial forces. Coriolis force is defined as an apparent deflection of moving objects when they are viewed from a rotating reference frame. The Coriolis force acts in a direction normal to the rotational axis and to the velocity of the body and proportional to the object‟s speed in the rotating frame. Centrifugal force points directly away from the rotational axis and proportional to the distance of the body from the axis of the rotating frame.

Mathematically,

 2 1. Centrifugal force  grad Ω r 2

2. Coriolis force 2(ΩV)

15 where , Ω, r and V are density of fluid, angular velocity, radius vector and velocity of the fluid, respectively. Gupta and Gupta (1979) examined the onset convection in a horizontal layer of ferromagnetic fluid heated from below and rotating about a vertical axis in the presence of a uniform magnetic field. Venkatasubramanian and Kaloni (1994) investigated the effect of rotation on the thermo-convective instability of a horizontal layer of ferrofluid heated from below in the presence of a uniform vertical magnetic field. Sunil et al. (2005a) studied the effect of rotation on thermosolutal convection in a ferromagnetic fluid considering a horizontal layer of an incompressible ferromagnetic fluid. Javed et al. discussed the rotating flow of a viscous incompressible fluid over an exponentially stretching continuous surface and modeled equations has been transformed to a system of coupled non-linear differential equations; solved numerically for a non-similar solution.

Porous medium is also under consideration in one of our research problems. Porous medium is defined as a solid body, which contains „pores (void spaces). Void spaces must be distributed more or less frequently through the material. In a porous medium, the pores may be interconnected or non-interconnected. Examples of porous materials are rocks, human bones, soils, beach sand, glass beads, catalyst pellets, soil, gravel, sandstone, limestone, concrete, cement, bricks, paper, cloth, rye bread, wood etc. It (flow through porous media) is of great interest in chemical engineering (for filtration, adsorption etc.), geophysics, biophysics. The four macroscopic properties of non-ideal porous media used to describe flow of the fluid are porosity, permeability, tortuosity and connectivity. ‘Porosity’ is the ratio of the void spaces to the total volume of the medium. This is denoted by  , expressed either in percent or as a fraction of one i.e. it lies between 0 and 1. In homogeneous isotropic materials,  is a pure constant but in non-homogeneous materials, may depend upon position. Porosity value may approach the value one for man-made materials (such as cements and ceramics), however, for natural media, does not normally exceed 0.6. The ‘medium permeability’ is measured in Darcy and „Darcy‟ is defined as the permeability that permeates a velocity of 1 cm/sec of a fluid with viscosity 1 centipoises under a pressure gradient of 1 atm/cm. That is, permeability is a measure of the flow conductivity in a porous medium. The porous medium of very low permeability allows us to use Darcy‟s model and of moderately large permeability allows us to use the Brinkman‟s model.

An important characteristic for the combination of the fluid and the porous medium is the ‘tortuosity’ which represents the hindrance to the flow diffusion imposed by local viscosity or local boundaries. It was noticed that the tortuosity is also a function

16 of the porosity and can be represented by square root of porosity (  ) (Liu and Masliyah 1999). Finally, ‘connectivity’ is the number of pore-connections; but when the pores are of small size, connectivity is the average numbers of pores per junction. Some complexities due to the interactions between fluids and porous material are to be taken when flow through porous medium is considered. When a fluid passes through a porous material, the real path of the individual particles cannot be traced analytically.

The effect of magnetic field along the vertical axis on thermo-convective instability in a ferromagnetic fluid saturating a rotating porous medium along with stress free boundaries has been studied by Sekar et al. (1993) by using the Darcy model. Darcy‟s law relates linearly the flow velocity to the pressure gradient across the porous medium. The flow in the porous media deals with the analysis in which the differential equations governing the fluid motion is based on the Darcy‟s law which accounts for the drag exerted by the porous medium (Kaloni and Qiao 2001, Khaled and Vafai 2003).

Finlayson (1970), Gupta and Gupta (1979), Sekar and Vaidyanathan (1993) discussed the effect of temperature, rotation and porous medium on ferromagnetic fluids as a single-component fluid. Sunil et al. (2004) have reported the effect of rotation on ferromagnetic fluid heated and soluted from below saturating a porous medium.

Frusteri and Osalusi (2007) examined the laminar convective and slip flow of an electrically conducting Newtonian fluid with variable properties over a rotating porous disk. Attia (2008) investigated the steady laminar flow of an incompressible viscous fluid due to the uniform rotation of a porous disk of infinite extent with heat transfer by applying a uniform injection or suction through the surface of the disk. Attia (2009) studied the steady laminar flow of an incompressible viscous fluid due to the uniform rotation of an infinite rotating disk in a porous medium with heat transfer and also discussed the effect of porosity of the medium on the velocity and temperature distribution. Chauhan and Agrawal (2010) studied the MHD flow in a parallel-disk channel partially filled with a porous medium in a rotating system including Hall current.

Also, in one of our research problem, the fluid is assumed to be incompressible having variable viscosity, given by  f (1 δ.B) , where δ is the variation coefficient of magnetic field-dependent (MFD) viscosity and is considered to be isotropic (Vaidyanathan et al. 2002, Ramanathan and Suresh 2004, Sunil et al. 2005b, 2008a,

2008b, Nanjundappa et al. 2009, 2010a, 2010b, Shivakumara et al. 2011),  is taken f as viscosity of the fluid when the applied magnetic field is absent. Viscosity is also one of

17 astounding rheological property of ferrofluid influencing convection flow problems. A detailed account of magneto-viscous effects in ferrofluid has been given in a monograph by Odenbach (2002). Sunil et al. (2005b) discussed the influence of rotation on medium permeability and how MFD viscosity affects the magnetization in ferromagnetic fluid heated from below in the presence of dust particles saturating a porous medium of very low permeability using Darcy model. Sunil and Mahajan (2009) studied the nonlinear stability analysis of magnetized ferrofluid heated and soluted from below with MFD viscosity via generalized energy method. Ram et al. (2010) characterized the effect of MFD viscosity on ferrofluid fluid behaviour describing the velocity components and pressure profile by solving the non-linear coupled differential equations using power series approximations.

1.4 FORCE MECHANISM IN FERROFLUID FLOW

In the presence of an external magnetic field, the ferromagnetic colloidal particles suspended in the carrier liquid of a ferrofluid become magnetized and produce attractive forces on each particle that produce a body force on the liquid. The magnetic (Kelvin) force F on ferrofluid per unit volume is given by F   (M .)H , where  is magnetic 0 0 permeability of free space, M is magnetization, H is magnetic field strength of the external magnetic field. The body force in FHD is due to polarization force.

The pioneering study of ferrofluids has been started by Rosensweig and his co- workers described the first principles of Ferrohydrodynamics i.e. mechanics of fluid motion influenced by strong force of magnetic polarization. Rosensweig (1985) has given an authoritative introduction to the research on magnetic liquids in his monograph and discussed the effect of magnetization resulting in interesting information. Ferrofluid particles rotate to align their magnetic axis with the instantaneous applied field, when subjected to rotating or time varying magnetic-fields. One of the many fascinating features of the ferrofluids is the prospect of influencing the flow of fluids by the magnetic field and vice-versa (Feynman et al. 1963, Shliomis 2004).

1.5 BASIC DEFINITIONS

1.5.1 Magnetic Field Intensity (H) : Magnetic field intensity at a point is the number

of magnetic lines of force crossing per unit area around that point, the area being held perpendicular to the direction of lines of force. In SI system, the unit of magnetic intensity is Ampere/metre.

1.5.2 Magnetic Induction (B) : When a magnetic material is placed in a uniform

magnetising field (H) , it acquires magnetism and develops its own magnetic field due to induction. As a result of this induction, the original magnetic field is modified both inside as well as outside the magnetic material. This modified or resultant field is called magnetic induction and is measured as the number of lines of induction passing normally through unit area of the material and is denoted by B . It is expressed in Tesla in SI units. Thus total number of magnetic lines crossing per unit area normally through a magnetic substance is called magnetic induction.

1.5.3 Intensity of Magnetization (M) : It is a measure of the extent to which a substance gets magnetized. Intensity of magnetization M of a magnetic substance is defined as its magnetic moment per unit volume, the specimen being so small that its magnetization can be supposed to be uniform. In SI system, the unit of intensity of magnetization is Weber/metre2.

1.5.4 Magnetic Susceptibility () : It measures the ease with which a specimen takes magnetism. Magnetic susceptibility of a magnetic substance is defined as the ratio of the intensity of magnetisation induced in the substance to the strength of magnetizing field H in which the substance is placed. M Mathematically,   H Susceptibility is zero for air, is positive in case of paramagnetism, ferromagnetism and negative in case of diamagnetism. As it is the ratio of same quantities, so it has no units.

1.5.5 Magnetic Permeability (0 ) : It measures the degree to which the specimen can be penetrated. The magnetic permeability of a material is defined as the ratio of magnetic induction to the strength of magnetization . B Mathematically,   0 H

Units: In SI system, the unit of 0 is henry/metre. For free space, permeability is 4 107 henry/metre.

Relation between B, H and : B  H  4M

1.6 EQUATION OF MOTION FOR A MAGNETIC FLUID

Formulating momentum equation for a magnetic fluid is decisive to the study of Ferrohydrodynamics. A momentum equation for magnetic fluid flow was first proposed by Neuringer and Rosensweig (1964). The equation continues to serve as the point of departure for most of the problems in which viscous friction or inertia along with magnetic forces play an important role.

Consider the dynamic equilibrium per unit volume of an infinitesimal element of magnetic fluid. The element is assumed to be large enough to contain a number of colloidal magnetic particles yet small in size compared to the dimension of the flow field. The time rate of change of momentum for the constant mass contained in a deformable element having volume dx dy dz is

D DV D( dx dy dz) (V dx dy dz)   dx dy dz V Dt Dt Dt

Because the mass  dx dy dz is constant, the last in the above equation vanishes and Newton‟s law normalized to unit volume can be written as DV   f  f  f Dt p q m pressure force viscous force magnetic force where D / Dt is the substantial derivative (convective or material). To be precise, the D  convention  V. is adopted. Dt t

The right hand side of this equation is the sum of the body forces normalized to a unit volume. The viscous effects are not dominant but inertial effects are the dominant one over a region at a small distance from the disk, if Reynolds number is large, which gives rise to a boundary layer over the surface of the disk.

A fairly simple model describing the steady or quasi steady flow of magnetic fluids and convective heat transfer in slowly as slightly changing external magnetic fields is Neuringer-Rosensweig (NR) model (1964). The model considers the liquid particle in magnetic fluid as a mathematical point with only three degrees of freedom. As a complete set of independent variables, the following functions are chosen: (liquid density, pressure and temperature) and three vectors (velocity, magnetization and magnetic field). This model considers the magnetization M as being always parallel to

20 the applied magnetic field, thus resulting in no interaction of magnetic fluid with external magnetic field through magnetic body couples and kinetic processes.

This model leads to governing equations which are considered from Navier- Stokes equations of magnetization. The system of equations consists of the following:

1.6.1 The Equation of Continuity

.V  0 … (1.1)

1.6.2 The Equation of Motion (Momentum Equation)

V   2 … (1.2)   (V.)V   p  0 (M.)H   f  V  t 

1.6.3 Maxwell Equations

  H  0, .B  0  … (1.3) where B  H  4 M 

1.6.4 Model Assumptions

 f M  H,   1, M  H  0 … (1.4) 0

Using the prescribed NR model, several boundary value problems have been solved. Verma and Vedan (1978, 1979), Verma and Singh (1981) have solved the research problems on paramagnetic Couette flow, helical flow with heat and flow through a porous annulus, respectively by using this model.

1.7 MODELED MOMENTUM EQUATIONS (For effects of various parameters)

To characterize the flow behaviour of ferrofluid over a rotating disk, we need the equations governing the motion of an incompressible electrically non-conducting ferrofluid. In this section, we are writing the following basic equations.

For effect of rotation, equation (1.2) is modelled as

V   2  2   (V.)V   p  0 (M.)H   f  V  2(ΩV )   Ω r … (1.5)  t  2

Here, the additional forces influencing the flow are: (i) Coriolis force (ii) Centrifugal force.

For porous medium, equation (1.2) is replaced by

V    2 … (1.6)   (V.)V   p  0 (M.)H   f  V  V  t  

For effect of porosity  , equation (1.2) is modelled as

 V 1   f  2 … (1.7)   (V.)V   p  0 (M.)H   V   t   

For effect of porosity, equation (1.5) is modified as

 V 1   f 2  2  2 … (1.8)   (V.)V   p  0 (M.)H   V  (ΩV )   Ω r   t     2

For the effect of MFD viscosity, equation (1.5) is modelled as

V   2  2   (V.)V   p  0 (M.)H   f (1 δ.B) V  2(ΩV )   Ω r  t  2 … (1.9)

For the effect of MFD viscosity and porosity, equation (1.5) is modelled as

 V 1   f (1 δ.B) 2  2  2   (V.)V   p  0 (M.)H   V  (ΩV )   Ω r   t     2

… (1.10)

1.8 METHOD DESCRIBING THE SOLUTION

Generally, the flow characterization is discussed after finding the solution of continuity equation and momentum equation for obtaining velocity and pressure profiles. Now a days, there are various methods for the solution of deformed equations which are strongly coupled non-linear differential equations after application of similarity transformations (Karman 1921, Cochran 1934, Benton 1966, Ackroyd 1978, Kumar et al. 1988, Yang and Liao 2006, Attia 2008, Sahoo 2009, Turkyilmazoglu 2009a, Abdou 2010, Makukula et al. 2010, Turkyilmazoglu 2010a, 2010b, 2010c), and power series approximation method is employed.

By using appropriate dimensionless variables and boundary layer approximations, one arrives at a set of ordinary differential equations that can be solved

22 numerically by various methods (Balaram and Luthra 1973, Wichterle and Mitschka 1998, Anderrson et al. 2001, Attia 2008, Turkyilmazoglu 2009b, Makukula et al. 2010, Turkyilmazoglu 2011).

In our research work, a classical problem considered by Karman is extended to the steady, Axi-symmetric, laminar flow of an incompressible electrically non-conducting ferrofluid over a rotating disk excluding thermal effects for various types of effects.

1.9 GENERAL ASSUMPTIONS

In the research work, the whole system is rotating with angular velocity Ω  (0,0,Ω) along the vertical axis, which is taken as z-axis. In some of our research problems, the fluid layer is assumed to be flowing through an isotropic medium of porosity  . Basic Assumptions are as follows:

(a) The flow is steady and Axi-symmetric. (b) The fluid and the ferrous particles have the same velocity. (c) The fluid and disk are electrically non-conducting. (d) The magnetic field affects only viscosity and not other properties. (e) The thermal effects are excluded.

1.10 BOUNDARY LAYER APPROXIMATIONS

Since the motion of the fluid is caused by the rotation of the disk, at sufficiently high Reynolds number the viscous effects will be confined within a thin layer near the disk. Therefore, further simplification can be obtained by considering the usual boundary- layer approximations (Schlichting 1960, Owen and Rogers 1989).

For our research work, we considered here boundary layer approximation as 1 p     0 M H  r 2 and very less variation of magnetic-field along z-direction  r  r and the component of velocity vz is very much smaller in magnitude than the other two components. The pressure depends only on the axial distance from the axis of rotation.

1.11 BOUNDARY CONDITIONS

The boundary conditions for the velocity components at the surface and far away from the disk are respectively, given by:

at z  0; vr  0, v  r, vz  0  … (1.11) at z  ; vr  0, v  0 

The value of vz vanishes near the surface of the disk, since there is no

penetration. However, the value of vz as z   is not specified; it adjusts to a negative value, which provides sufficient fluid, necessary to maintain the pumping effect. In contrast to the axial velocity, both the radial and tangential velocities go to zero at large axial distances from the disk. These boundary conditions are the part of the solution for research problems considered.

1.12 SIMILARITY TRANSFORMATIONS

The similarity transformation is the classical approach for finding exact solutions of linear and non-linear partial differential equations. A system of partial differential equations with m-independent variables can be converted to a system with m-1 independent variables with the help of similarity transformations.

Using similarity transformations, we can solve the Axi-symmetric momentum equations associated with rotating disk flow, which allows the governing partial differential equation set to be transformed into a set of ordinary differential equations. In the similarity solution, analytical relationships will be used and dimensionless parameters will be substituted so that the number of variables to be solved is reduced.

1.13 BOUNDARY LAYER DISPLACEMENT THICKNESS

The boundary layer displacement thickness (Benton 1966); defined in the monograph by Schlichting (1960), is calculated as:

1   d  v dz  F()d … (1.12) r z0  0

1.14 ANGLE OF ROTATION

The fluid is taken to rotate at a large distance from the wall (Schlichting 1960), the angle becomes E(0) tan   … (1.13) 0 F(0)

Since by the assumptions for E and F at the surface of the disk as reported in chapter 2, we have calculated angle between the wall and the revolving ferrofluid, which is approximately 410 .

1.15 VOLUME FLOWING OUTWARD THE Z-AXIS

Total volume flowing outward the z-axis (as description given on page 229 in the monograph written by Schlichting, 1960):

  2 Q  2R vr dz  2R E()   d z0   0 … (1.14)   R 2  G()  2.786094R 2   2.786094R 2 z

The research work carried out for this doctoral thesis focuses on the effects of magnetic force, rotation, porous medium, porosity value and MFD viscosity on the behaviour of ferrofluid flow with a rotating disk.

Chapter 2

EFFECT OF ROTATION ON FERROFLUID FLOW WITH ROTATING DISK

This research work deals with the theoretical investigation of the effect of rotation on ferrofluid flow due to rotating disk by solving boundary layer equations. Here, we have solved the coupled non-linear differential equations by power series approximations. Expressions for the components of velocity and pressure profile are obtained in cylindrical co-ordinate system by considering the z-axis as the axis of rotation. It is observed that there is an increment in the thickness of the boundary layer over rotating disk in comparison to the ordinary case of viscous fluid flow without rotation. The results for all above variables are obtained numerically and discussed graphically.

2.1 FORMULATION OF THE PROBLEM

The laminar flow produced by a disk rotating in an infinite fluid, where the effects of flow confinement do not exist, is a classical fluid mechanics problem. In such systems, it is often convenient to use a stationary frame of reference.

The flow in an isotropic medium is considered to be steady ( t  0) and Axi- symmetric (   0) excluding the thermal effects. The fluid and the ferrous particles have the same velocity. The fluid and disk are assumed to be electrically non- conducting. The whole system is rotating with angular velocity Ω  (0,0,Ω) along the vertical axis, which is taken as z-axis and the origin, 0, is taken as the point where the axis of rotation intersects the rotating disk as shown in figure 2.

A cylindrical co-ordinate system (r, , z) is taken such that  is oriented in the direction of rotation with vr , v and vz representing the radial, tangential and axial components of the velocity vector, respectively.

Additional assumptions are: (i) the flow is completely described by the continuity and conservation of momentum equations; and (ii) the fluid density  , is constant.

The work presented in this chapter has been accepted in the form of research paper entitled “On the revolving ferrofluid flow due to rotating disk”, International Journal of Nonlinear Sciences (IJNS), (2012).

Fig. 2: Coordinate system for rotating disk system

2.1.1 Equations of Motion

Applying the general assumptions, the transport equations (1.1 and 1.5) for conservation of mass and conservation of momentum, in cylindrical co-ordinates for the effect of rotation, can be written as follows:

Continuity equation

v v v r  r  z  0 … (2.1) r r z Momentum Equations

in the radial-direction:

2 2 2  vr vr v  1 p 0   vr   vr   vr  vr  vz      M H         2v r z r  r  r r 2 r r z 2      

… (2.2)

in the tangential-direction:

2 2  v v vr v   v   v   v  vr  vz         2vr … (2.3)  r z r  r 2 r r z 2       in the axial-direction:

2 2  vz vz  1 p 0   vz 1 vz  vz  vr  vz    M H      … (2.4)  r z   z  z r 2 r r z 2    

The terms 2v and 2vr are the projections of the Coriolis force onto the axes r and  , respectively, while the projection of centrifugal force onto the axis adjusts with the pressure term. The solution of the problem is based on the appropriate non- dimensional transformation variable i.e.    z … (2.5)  along with the associated set of dimensionless velocity components and pressure, i.e. v E()  r … (2.6a) r v F()   … (2.6b) r v G()  z … (2.6c)  

p P()   … (2.6d) 

This similarity transformation implies that all three dimensionless velocity components and pressure depend only on the distance from the disk,  . The boundary conditions (1.11) are transformed into the  coordinate as follows:

E(0)  F(0) 1  G(0)  0, P(0)  P0  0 … (2.7a)

E()  F()  0 … (2.7b)

It is possible to reduce the continuity and momentum equations to a set of ordinary differential equations by using similarity transformations (2.6a) - (2.6d) into equations (2.1) - (2.4) for velocity and pressure equations. The resultant equations are presented below.

Continuity Equation

G  2E  0 … (2.8)

Momentum Equations in the radial-direction:

2 2 E  GE  E  F  2F 1  0 … (2.9) in the tangential-direction:

F  GF  2EF  2E  0 … (2.10)

in the axial-direction:

P G  GG  0 … (2.11) where prime denotes differentiation with respect to  .

2.2 SOLUTION OF THE PROBLEM

Here, we have used power series approximation method to solve this system of coupled non-linear differential equations for all non-dimensional velocity parameter

(E, F, G) and pressure (P) . For large , the formal asymptotic expansions of the system of equations (2.8) - (2.11) are the power series in exp(c), i.e.

 E()   Ai exp(ic) … (2.12a) i1

 F()   Bi exp(ic) … (2.12b) i1

 G()  G()  Ci exp(ic) … (2.12c) i1

 P()  P0   Di exp(ic) … (2.12d) i1

Here, G must tend to a finite limit, say  c as    , i.e. G()  c, (c  0) .

Assuming the missing boundary conditions E(0)  a and F(0)  b in equations (2.8) - (2.11) along with the boundary conditions (2.7a), we get the following additional boundary conditions for the approximate solution for first four coefficients of series solution for the variables E, F, G and P, respectively:

E(0)  2, E(0)  4b … (2.13a)

F(0)  0, F(0)  4a … (2.13b)

G(0)  0, G(0)  2a, G(0)  4 … (2.13c)

P(0)  2a, P(0)  4, P(0)  8b … (2.13d)

With the help of assumed missing boundary condition E(0)  a and additional boundary conditions (2.13a) along with the boundary conditions (2.7a); up to 4th term in equation (2.12a) and in equations obtained after 1st, 2nd and 3rd differentiation of equation

(2.12a) at   0 gives the following set of algebraic equations:

A1  A2  A3  A4  0 … (2.14a)

a A  2A  3A  4A   … (2.14b) 1 2 3 4 c 2 A  4A  9A 16A   … (2.14c) 1 2 3 4 c 2 4b A  8A  27A  64A  … (2.14d) 1 2 3 4 c3 On solving these algebraic equations by matrix method, we get the values of variables A , A , A and A involved in the set of algebraic equations (2.14a) - (2.14d). 1 2 3 4 Similarly, we can find the values of other coefficients involved in equations (2.12b, 2.12c, 2.12d). First four coefficients in each equation (2.12), calculated with the help of (2.7a) and (2.13), are as follows:

 2b 3 13a  A1      … (2.15a)  3c3 c 2 3c 

 2b 8 19a  A2      … (2.15b)  c3 c 2 2c   2b 7 7a  A3   3  2   … (2.15c)  c c c   2b 2 11a  A4      … (2.15d)  3c3 c 2 6c   2a 13b  B1     4 … (2.15e)  3c3 3c   2a 19b  B2     6 … (2.15f)  c3 2c   2a 7b  B3     4 … (2.15g)  c3 c   2a 11b  B4    1 … (2.15h)  3c3 6c   2 3a  C1   3  2  4c … (2.15i)  3c c   2 8a  C2     6c … (2.15j)  c3 c 2   2 7a  C3     4c … (2.15k)  c3 c 2 

 2 2a  C4     c … (2.15l)  3c3 c 2   4b 6 26a  D1      … (2.15m)  3c3 c 2 3c   4b 16 19a  D2   3  2   … (2.15n)  c c c   4b 14 14a  D3      … (2.15o)  c3 c 2 c   4b 4 11a  D4      … (2.15p)  3c3 c 2 3c 

Precise values of the two missing boundary conditions E (derivative of the dimensionless radial velocity) and F (derivative of the dimensionless tangential velocity) were required at the surface of the disk i.e. at   0; due to sensitivity of the research problem. Using the values E(0)  a  0.54, , F(0)  b  0.62 and c  0.886

(Cochran 1934), we calculate the numerical values of the coefficients A1 , A2 , A3 , A4 ;

B1 , B2 , B3 , B4 ; C1 , C2 , C3 , C4 ; D1 , D2 , D3 , and D4 ; and draw the graphs of velocity components and asymptotic pressure with the dimensionless parameter  .

The current solution to the problem determines the values E(0)  0.54886 and F(0)  0.62000 (up to five digits) those are very much comparable to the assumed values. The values of E, F, G and P are compared numerically and graphically with their corresponding values in classical case.

E F  α E F G P() P0 0 0 1 0 0 0.54886 -0.62000 0.4 0.08894 0.76572 -0.05778 -0.17909 -0.00241 -0.53308 0.8 0.05266 0.57609 -0.17309 -0.10562 -0.13267 -0.41783 1.2 0.00351 0.42861 -0.30498 -0.00710 -0.10197 -0.32301 1.6 -0.02627 0.31524 -0.43171 0.05253 -0.04825 -0.24658 2.0 -0.03740 0.22935 -0.54162 0.07480 -0.01078 -0.18521 4.0 -0.01482 0.04218 -0.81773 0.02965 0.01132 -0.03673 6.0 -0.00282 0.00727 -0.87408 0.00563 0.00244 -0.00643 8.0 -0.00049 0.00124 -0.88397 0.00098 0.00043 -0.00110 12.0 -1.4E-05 3.6E-05 -0.88594 2.8E-05 1.3E-05 -3.2E-05  0 0 -0.886 0 0 0 Table 2.1: The steady state velocity field and pressure as functions of  for the effect of rotation

The present numerical results give very good approximate solution of the above system of non-linear coupled differential equations.

2.3 BOUNDARY LAYER DISPLACEMENT THICKNESS

The boundary layer displacement thickness is calculated as:

1   d  v dz  F() d 1.3456145 … (2.16) r z0  0

2.4 RESULTS AND DISCUSSION

1 p   If we remove boundary layer approximation   0 M H  r 2 and  r  r

 2 the effects of both, the centrifugal force   Ω r and Coriolis acceleration 2 2(ΩV) ; the problem reduces to Cochran‟s case (1934) of ordinary viscous fluid flow without rotation. Here, a wide range of numerical results have been derived. Of these results, a small section is presented here for brevity. The numerical results for the velocity profiles, for (r, , z) components of the velocity, commonly known as radial, tangential, vertical (axial) velocities, are shown in figures 2.1, 2.2, 2.3 respectively and for pressure profile in figure 2.4.

2.4.1 Radial Velocity Profile

In figure 2.1, the curve E2 represents the effects of rotation on the radial velocity in case of ferrofluid flow due to rotating disk. Whereas, E1 indicates the radial velocity profile of the Cochran's study of ordinary viscous fluid flow. Due to rotation, the radial velocity reaches its maximum value near the surface of the disk with magnitude 0.08894 at   0.4, whereas, in Cochran's case, the maximum value 0.181 of radial velocity is attained at comparatively distant point   0.9. Here, it is noticed that the radial velocity

E2 has very less peak value in comparison to because of thickening of the fluid layer due to the rotation of the whole system. Thus, the effect of rotation is more pronounced than the force of magnetization in the sense of fluid thickening. Also it is quite interesting to see that in case of rotation before converging to zero, the radial velocity once becomes negative.

2.4.2 Tangential Velocity Profile

Figure 2.2, is the graphical comparison of the tangential velocity of ferrofluid flow with rotation to that of Cochran's ordinary viscous fluid flow . In our case, if we (F2 ) (F1 ) increase the value of  , the tangential velocity F2 decreases continuously and tends to zero for large value of . It is observed from the table 1, the value of tangential velocity is 0.49762 at  1, whereas for the ordinary viscous fluid, the tangential velocity is 0.46800 for the same value of . Therefore, at   1, there is an approximate increment of 6.7% in the value of tangential velocity in comparison to that of Cochran's value. Also from the figure, it is clear that F1 converges to zero little faster than F2 , however, both the curves have similar trends.

2.4.3 Axial (Vertical) Velocity Profile

Figure 2.3 shows the axial velocity profile, which is zero in the beginning and tends to a finite value  0.886 for  14.8 onwards. When we increase the value of  , it decreases continuously in the negative region. Our axial velocity value is  0.23858 at

  1, whereas for Cochran, the axial velocity is  0.266 for the same value of  . Meaning thereby, for the same value of , the axial velocity component acquires larger value than the value in the ordinary case.

2.4.4 Pressure Profile

The pressure profile P()  P0 with the initial pressure, P0 at   0 is shown in figure 2.4. Here, pressure goes to negative region near the surface of the disk, and at   0.4, it goes to maximum negative value  0.17909 . Onwards to , pressure starts increasing with increasing value of  and at  1.3, it enters in the positive region and attains maximum value 0.07667 at   2.2 . Finally converges to zero i.e. P() converges to .

Comparing figures 2.1 and 2.4, we conclude that when radial velocity increases, the pressure of the ferrofluid decreases and when radial velocity decreases, ferrofluid pressure increases i.e. they are converse in convergence behaviour. Also, the tangential velocity diminishes slower than axial velocity component. The change in the curve of radial velocity is faster due to the effect of external magnetic field resulting in reducing the time required for velocity profile to reach their convergence level.

The derivatives E and F , along the axial direction are shown in figures 2.5 and 2.6, respectively. In our problem, the disk is rotating along with revolution of the

34 ferrofluid, due to which thickness is increasing. In nut shell, the rotation of the disk along with revolution of the ferrofluid results in an increased displacement thickness 1.34562 more than that of 1.27144 (Benton 1966). Therefore, we conclude that if we rotate the disk with rotational effect on ferrofluid, there is an increment in the thickness of boundary layer.

From these results, we conclude that magnetization force i.e., 0 (M.)H reduces the pressure. Also, it has been observed that magnetic field intensity increases the radial velocity; whereas, the fluid rotation has reverse effect. The effect of rotation is more pronounced than the force of magnetization due to which the radial velocity takes very less peak value in comparison to ordinary viscous fluid flow case. Due to the rotation, the retardation of the radial velocity increases the thickness of the magnetic fluid layer.

FIGURES

E(α) E1 = Cochran’s case 0.2 E2 = Case of rotation

0.15

0.1

E E₁ 0.05 E₂

0 0 1 2 3 4 5 -0.05

Fig. 2.1: Effect of rotation on radial velocity profile

F(α) F = Cochran’s case 1.2 1 F2 = Case of rotation 1

0.8

F 0.6 F₁ 0.4 F₂

0.2

0 0 1 2 3 4 5 6

Fig. 2.2: Effect of rotation on tangential velocity profile

G(α) 0 2 4 6 8 10 0 -0.1

-0.2 G1 = Cochran’s case -0.3 G2 = Case of rotation -0.4

G -0.5 -0.6 G₁ -0.7 G₂ -0.8 -0.9 -1

Fig. 2.3: Effect of rotation on axial velocity profile

0.1 P()-P₀

0.05

₀ 0 2 4 6 8 10

P -

) -0.05

α P( -0.1

-0.15

-0.2 Fig. 2.4: Effect of rotation on pressure profile

0.6 E'(α) 0.5

0.4

0.3

0.2 E'

0.1

0 0 2 4 6 8 10 12 -0.1

-0.2 Fig. 2.5: Effect of rotation on derivative of radial velocity

F'(α) 0 5 10 15 0 -0.1 -0.2

-0.3 F' -0.4 -0.5 -0.6

-0.7 Fig. 2.6: Effect of rotation on derivative of tangential velocity

Chapter 3

FERROFLUID FLOW BEHAVIOUR DUE TO ROTATING DISK IN A POROUS MEDIUM

Here, ferrofluid flow with rotating disk through a porous medium is investigated within the framework of NR model. All the three velocity components (radial, tangential and axial), pressure profile and the displacement thickness of the boundary layer are calculated numerically. The non-linear coupled differential equations have been solved by power series approximation. Here, the effect of Darcy number  (porosity parameter) has shown some interesting results on the ferrofluid flow characteristics.

3.1 FORMULATION OF THE PROBLEM 3.1.1 Equations of Motion

Applying the assumptions given under the heading (1.9) “General Assumption”, the constitutive equations (1.2 and 1.6) for conservation of mass and conservation of momentum for porous medium, in cylindrical co-ordinates, can be written as follows:

Continuity equation

v v v r  r  z  0 … (3.1) r r z

Momentum Equations

in the radial-direction:

2 2 2  vr vr v  1 p 0   vr   vr   vr   vr  vz      M H         vr r z r  r  r r 2 r r z 2       

… (3.2)

in the tangential-direction:

2 2  v v vr v   v   v   v   vr  vz          v … (3.3)  r z r  r 2 r r z 2       

The work presented in this chapter has been published in the form of research paper entitled “Axi-symmetric ferrofluid flow with rotating disk in a porous medium”, International Journal of Fluid Mechanics (IJFM), 2 (2), pp. 151-161 (2010).

in the axial-direction:

2 2  vz vz  1 p 0   vz 1 vz  vz   vr  vz    M H       vz … (3.4)  r z   z  z r 2 r r z 2      A system of non-linear coupled differential equations in E, F, G and P from continuity equation (3.1) and momentum equations (3.2 - 3.4) on applying similarity transformations from equations (2.6a) - (2.6d) (discussed on page no. 29 in second chapter), is obtained as follows:

G  2E  0 … (3.5)   2 2 E  GE  E  F   E 1  0 … (3.6) F  GF  2EF  F  0 … (3.7)

P  G  GG   G  0 … (3.8)

 where   is the porosity parameter (Darcy Number). 

3.2 SOLUTION OF THE PROBLEM

The formal asymptotic expansion (for large  ) for the system of equations (3.5) - (3.8) in the form of power series is given as:

 E()  A exp(ic) A ec  A e2c  A e3c  A e4c  ...... … (3.9a)  i 1 2 3 4 i1

 c 2c 3c 4c F()  Bi exp(ic)  B1e  B2e  B3e  B4e  ...... … (3.9b) i1

 c 2c 3c G()  G()  Ci exp(ic)  c  C1e  C2 e  C3e  ...... … (3.9c) i1

 c 2c 3c 4c P()  P0  Di exp(ic)  D1e  D2 e  D3e  D4 e  ...... … (3.9d) i1 Here, G must tend to a finite limit, say  c as    , where c is a positive quantity. Using the precised values of E(0)  a and F(0)  b (as reported in chapter 2) along with the equation (2.7a) and the set of equations (3.5) - (3.8), we get the following additional boundary conditions for the approximate solution (3.9a) - (3.9d) for first four coefficients of series solution for the variables and respectively:

E(0)  0, E(0)   a  2b … (3.10a)

F(0)  , F(0)   b  2a … (3.10b)

G(0)  0, G(0)  2a, G(0)  0 … (3.10c)

P(0)  2a, P(0)  0, P(0)  2 a  8b … (3.10d)

By using these additional boundary conditions and taking the precised values E(0)  a  0.54 and F0)  b  0.62 and c  0.886 in MATLAB environment, we calculate the first four coefficients involved in each equation of the system (3.9) for different values of Darcy number  (porosity parameter) which are as follows:

 A1 A2 A3 A4 0 2.9383 -6.6815 5.1578 -1.4146 1 3.0677 -7.0697 5.5460 -1.5440 2 3.1971 -7.4579 5.9342 -1.6734 3 3.3265 -7.8461 6.3224 -1.8028

 B1 B2 B3 B4 0 1.2264 -0.1285 -0.1220 0.0241 1 2.9886 -4.7784 3.8909 -1.1012 2 4.7508 -9.4283 7.9038 -2.2265 3 6.5130 -14.078 11.9167 -3.3518

D1 D2 D3 D4 0 10.5947 -1.9195 -12.0985 3.4234 1 10.3359 -1.1431 -12.8749 3.6822 2 10.0771 -0.3667 -13.6513 3.9410 3 9.8183 0.4097 -14.4277 4.1998

C1 C 2 C 3 C 4 1.4803 0.1872 -1.2713 0.4898

Table 3.1: Various coefficients in equations (3.9a) - (3.9d)

Here values of constants C , does not depend the porosity C1 , C2 , 3 C4 parameter (Darcy number), because there is no involvement of Darcy number in the boundary conditions as well as in the algebraic equations formed by assumed series solution (3.9c) for the non-dimensional axial velocity parameter G .

  E F G E F  P(  )  P 0

0 0 0 1 0 0.54019 -0.62002 0 2 0.33057 0.20419 -0.63479 -0.16357 -0.17661 -1.68898 4 0.07945 0.03533 -0.84310 -0.06567 -0.03120 -0.30427 6 0.01427 0.00602 -0.87872 -0.01250 -0.00533 -0.05199 8 0.00245 0.00102 -0.88476 -0.00217 -0.00091 -0.00885 1 0 0 1 0.54019 -0.61993 0 2 0.34314 0.38815 -0.16787 -0.25298 -1.66382 4 0.08288 0.08247 -0.06843 -0.06969 -0.29742 6 0.01490 0.01457 -0.01305 -0.0128 -0.05075 8 0.00256 0.00249 -0.00226 -0.00221 -0.00863 2 0 0 1 0.54019 -0.61985 0 2 0.35572 0.57211 -0.17217 -0.32936 -1.63867 4 0.08630 0.12960 -0.07120 -0.10819 -0.29057 6 0.01553 0.02311 -0.01360 -0.02028 -0.04949 8 0.00267 0.00396 -0.00236 -0.00350 -0.00842 3 0 0 1 0.54019 -0.61976 0 2 0.36830 0.75608 -0.17646 -0.40573 -1.61351 4 0.08973 0.17674 -0.07396 -0.14668 -0.28372 6 0.01615 0.03166 -0.01414 -0.02775 -0.04824 8 0.00277 0.00543 -0.00245 -0.00480 -0.00820

Table 3.2: The steady state velocity field and pressure as functions of  for the effect of Darcy number 

3.3 BOUNDARY LAYER DISPLACEMENT THICKNESS

 The displacement thickness d  F()d  (as stated in introductory part of   0  the thesis) of the boundary layer, for   0, 1, 2 and 3, is 1.2725828, 1.8296464, 2.3867099 and 2.9497735, respectively. However, the boundary layer displacement thickness for   0 i.e. an ordinary case without porous medium (1.2725828) as reported in Benton (1966). Here, the fluid layer is thickening with increasing the porosity parameter .

3.4 RESULTS AND DISCUSSION

3.4.1 Radial Velocity Profile

Figure 3.1 shows the radial velocity profile. The radial velocity for E1  0.41083, an ordinary viscous incompressible fluid without porous medium, is maximum at  1.2, whereas the radial velocity components E2 , E3 and E4 , for ferrofluid flow, have the maximum values 0.423051, 0.436136 and 0.449200 at  1.3 for Darcy number

  1, 2 and 3, respectively indicating that increase in the values of Darcy number , increases the radial velocity.

3.4.2 Tangential Velocity Profile

Figure 3.2 shows the tangential velocity profile. Here, the tangential velocity F1, without porous medium, converges to zero faster than F , F and F , respectively. For 2 3 4

  3, the tangential velocity is 1.004077 at   0.7 ; however, from   0.7 onwards to  1.2, the tangential velocity fluctuate around the maximum value 1 and an increase in the Darcy number also increases the tangential velocity.

3.4.3 Axial Velocity Profile

Figure 3.3 shows the axial velocity profile. The axial velocity in the present case is free from the variation in the porosity parameter (Darcy number).

3.4.4 Pressure Profile

Figure 3.4 shows the pressure profile for different values of Darcy number. Here, for   0.7, is maximum at   0, however, and have the P1  3.547151 P2 , P3 P4 maximum values 3.533411, 3.519670 and 3.505930 for  1, 2 and 3, respectively. It is clear from the figures 3.1 and 3.4, increase in the Darcy number  increases the peak value of the radial velocity, however, decreases the peak value of the fluid pressure means these two are converse in behaviour.

Figures 3.5 and 3.6 are the derivatives of the radial and tangential velocity, respectively. In figure 3.6, the derivative of the tangential velocity, i.e. F takes positive values for   3. The effect of porosity parameter is much pronounced in the derivative of the tangential velocity profile F in comparison to the derivative of the radial velocity profile E .

FIGURES

E(α) E1 for   0 0.5 E2 for  1

0.4 E3 for   2

E4 for   3 0.3

E E₁ 0.2 E₂

0.1 E₃ E₄ 0 0 2 4 6 8

Fig. 3.1: Effect of Darcy number (porosity parameter) 훽 on radial velocity profile

F(α) F1 for   0 1.2 F2 for  1

1 F3 for   2

0.8 F4 for   3

F 0.6 F₁

0.4 F₂ F₃ 0.2 F₄ 0 0 2 4 6 8 10

Fig. 3.2: Effect of Darcy number (porosity parameter) 훽 on tangential velocity profile

G(α) 0 2 4 6 8 10 0

-0.2

-0.4

G G -0.6

-0.8

-1

Fig. 3.3: Effect of Darcy number (porosity parameter) 훽 on axial velocity profile

P(α)-Po 0 1 2 3 4 5 0 -0.5 -1 -1.5 P for   0 P -2 1 P for  1 -2.5 2 P₁ P for   2 -3 3 P₂ P for   3 -3.5 4 P₃

-4 P₄

Fig. 3.4: Effect of Darcy number (porosity parameter) 훽 on pressure profile

E'(α) E for   0 0.6 1 E for  1 0.5 2  0.4 E3 for   2 E'₁ E for   3 0.3 E'₂ 4

0.2 E'₃ E' 0.1 E'₄ 0 -0.1 0 2 4 6 8 -0.2 -0.3

Fig. 3.5: Effect of Darcy number (porosity parameter) 훽 on derivative of radial velocity profile

F'(α) F1 for   0

F2 for  1 0.3 F for   2 0.2 3  0.1 F4 for   3 0 -0.1 0 2 4 6 8 10

-0.2 F' -0.3 F'₁ -0.4 F'₂ -0.5 F'₃ -0.6 -0.7 F'₄

Fig. 3.6: Effect of Darcy number (porosity parameter) 훽 on derivative of tangential velocity profile

Chapter 4

EFFECT OF POROSITY ON FERROFLUID FLOW WITH ROTATING DISK

A ferrofluid flow over a rotating disk influenced by porosity is theoretically investigated by solving the boundary layer equations with boundary conditions using NR model. Besides, numerically calculating the velocity components and pressure profile for different porosity values ( ) with the variation of dimensionless parameter () , we have also calculated the thickness of the boundary layer for different porosity values. Here, the solution of strongly coupled non-linear differential equations is obtained by power series approximation.

4.1 FORMULATION OF THE PROBLEM

4.1.1 Equations of Motion

Applying the assumptions given under the heading (1.9) “General Assumption”, the constitutive equations (1.2 and 1.7) for conservation of mass and conservation of momentum for porosity value, in cylindrical co-ordinates, can be written as follows:

Continuity equation

v v v r  r  z  0 … (4.1) r r z

Momentum Equations

in the radial-direction:

2 2 2 1  vr vr v  1 p 0    vr   vr   vr  vr  vz      M H        … (4.2)  2 r z r  r  r  r 2 r r z 2       in the tangential-direction:

2 2 1  v v vr v    v   v   v  vr  vz         … (4.3)  2  r z r   r 2 r r z 2      

The work presented in this chapter has been published in the form of research paper entitled “Effect of porosity on ferrofluid flow with rotating disk”, International Journal of Applied Mathematics and Mechanics (IJAMM), 6 (16), pp. 65-76 (2010).

in the axial-direction:

2 2 1  vz vz  1 p 0    vz 1 vz  vz  vr  vz    M H      … (4.4)  2  r z   z  z  r 2 r r z 2     Applying the similarity transformations (2.6a) - (2.6d) (discussed on page no. 29 in second chapter), in the system of equations (4.1) - (4.4), a system of strongly coupled non-linear differential equations in E, F, G and P ; is obtained as follows:

G  2E  0 … (4.5)

  2 2 2  E  GE  E  F   0 … (4.6)  F  GF  2EF  0 … (4.7)

 2 P  G  GG  0 … (4.8)

4.2 SOLUTION OF THE PROBLEM

The formal asymptotic expansion for the system of equations (4.5) - (4.8), is a  c  power series in exp  , i.e.      c  E()  A exp  i … (4.9a)  i   i1      c  F()   Bi exp i  … (4.9b) i1      c  G()  G()  Ci exp i  … (4.9c) i1      c  P()  P0   Di exp i  … (4.9d) i1   

Using the supposition E(0)  a and F(0)  b (as in chapter 2) along with the equations (2.7a), (4.5) - (4.8), we get the following additional boundary conditions for the approximate solution (4.9a) - (4.9d) for first four coefficients of series solution for the variables E, F, G and P, respectively:

 2 1 2b E(0)  , E(0)   … (4.10a)   2a F(0)  0, F(0)  … (4.10b) 

  2 1 G(0)  0, G(0)  2a, G(0)  2  … (4.10c)   

a 1 2  b P(0)  2 , P(0)  2 , P(0)  4 … (4.10d)  2  2     

By using these additional boundary conditions and taking the values a  0.54 , b  0.62 and c  0.886 in MATLAB environment, we calculate the values of first four coefficients A , A , A , A ; B , B , B , B ; C , C , C , C ; D , D , D and D ; 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 involved in each equation of the system (4.9a) - (4.9d), for porosity values   0.01,

0.02 and 0.03, which are as follows:

Coefficients   0.01   0.02   0.03 0.00733 0.01474 0.02223 A1 A2 -0.00704 -0.01429 -0.02177

A3 -0.00183 -0.00345 -0.00484

A4 0.00153 0.00300 0.00439 3.96970 3.93946 3.90926 B1 B2 -5.93360 -5.86735 -5.80126

B3 3.95109 3.90234 3.85375

B4 -0.98720 -0.97445 -0.96175 3.54384 3.54337 3.54257 C1 C2 -5.31559 -5.31437 -5.31234

C3 3.54366 3.54265 3.54096

C4 -0.88591 -0.88564 -0.88519 -1.46682 -1.47391 -1.48176 D1 D2 1.40785 1.42873 1.45166 0.36579 0.34529 0.32300

-0.30682 -0.30012 -0.29290 Table 4.1: The coefficients involved in the equations (4.9a) - (4.9d)

The present results, if rounded to the five decimal places, we get the good approximate solutions of the above system of non-linear coupled differential equations.

  E F G E F P(α)  P0 0.01 0 0 1 0 0.54046 -0.61852 0 0.03 0.00048 0.25042 -0.66253 -0.03926 -19.8392 -0.09577 0.06 3.6E-05 0.01936 -0.86872 -0.00316 -1.70251 -0.00717 0.09 2.5E-06 0.00137 -0.88478 -0.00022 -0.12097 -0.00050 0.12 1.8E-07 9.6E-05 -0.88591 -1.6E-05 -0.00849 -3.5E-05 0.15 1.2E-08 6.7E-06 -0.88599 -1.1E-06 -0.00059 -2.5E-06 0.02 0 0 1 0 0.53988 -0.61940 0 0.03 0.00285 0.69933 -0.25901 -0.07823 -18.5420 -0.28514 0.06 0.00096 0.24861 -0.66256 -0.03940 -9.85237 -0.09617 0.09 0.00027 0.07110 -0.82206 -0.01168 -3.06254 -0.02685 0.12 7.2E-05 0.01921 -0.86872 -0.00318 -0.84481 -0.00721 0.15 1.9E-05 0.00511 -0.88140 -0.00085 -0.22608 -0.00191 0.18 5.1E-06 0.00136 -0.88478 -0.00023 -0.06002 -0.00051 0.21 1.3E-06 0.00036 -0.88568 -6.0E-05 -0.01590 -0.00013 0.03 0 0 1 0 0.53934 -0.61932 0 0.03 0.00525 0.86794 -0.10585 -0.03699 -9.99972 -0.34999 0.06 0.00313 0.51503 -0.42064 -0.07275 -11.3066 -0.20860 0.09 0.00145 0.24680 -0.66260 -0.03955 -6.52350 -0.09662 0.12 0.00062 0.10822 -0.78798 -0.01788 -3.05835 -0.04160 0.15 0.00026 0.04576 -0.84454 -0.00764 -1.32751 -0.01745 0.18 0.00011 0.01906 -0.86873 -0.00319 -0.55892 -0.00724 0.21 4.5E-05 0.00789 -0.87885 -0.00132 -0.23244 -0.00300 0.24 1.9E-05 0.00326 -0.88305 -0.00055 -0.09617 -0.00124 0.27 7.7E-06 0.00135 -0.88478 -0.00023 -0.03971 -0.00051 0.30 3.2E-06 0.00056 -0.88550 -9.3E-05 -0.01638 -0.00021

Table 4.2: The steady state velocity field and pressure as functions of  for the effect of porosity value

4.3 BOUNDARY LAYER DISPLACEMENT THICKNESS

Here, the boundary layer displacement thickness becomes 0.02340, 0.04657 and 0.06951 for different porosity values   0.01, 0.02 and 0.03, respectively. It is clear from these results that the boundary layer displacement thickness increases with increasing porosity. For Benton‟s case (1966), the boundary layer displacement

50 thickness is 1.27261 which is much larger than our case. If we substitute  1 in equations (4.5) - (4.8), we get the boundary layer thickness 1.27262 , which is approximately equal to Benton‟s case.

4.4 RESULTS AND DISCUSSION

4.4.1 Radial Velocity Profile

Figure 4.1 shows the radial velocity profile with the variation of dimensionless parameter  for different porosity values. The radial component of velocity

E1  0.001743 is maximum at   0.01 for   0.01. However, E2 and E3 have the maximum values 0.003493 and 0.005251 at   0.02 and 0.03 for   0.02 and 0.03, respectively.

4.4.2 Tangential Velocity Profile

The tangential velocity profile is shown in figure 4.2. Here, F1 converges to zero faster than and similarly, converges to zero faster than . Further, and F2 , F2 F3 F1, F2 for and become approximately zero at and F3   0.01, 0.02   0.11, 0.24 0.36, respectively.

4.4.3 Axial Velocity Profile

Figure 4.3 represents the axial velocity profile which is zero in the beginning as shown by for different porosity values. From this figure, it is clear that G1, G2 , G3 G1 converges faster than and G . takes constant value at for G2 3 G1  0.886   0.16   0.01, whereas and become constant at   0.31 and 0.45 for   0.02 and

0.03, respectively.

4.4.4 Pressure Profile

Figure 4.4 exhibits the pressure profile with the initial pressure P at   0 . Here, 0 reaches the maximum negative value at for . However, P1 0.34868   0.01   0.01

P2 and P3 take the maximum negative values 0.34929 and 0.34999 at   0.02 and 0.03 for and 0.03, respectively.

Thus we see that increasing porosity values increases all three velocity components and pressure; but at the same time, diminishes convergence of all the four (radial, tangential, axial components of velocity and pressure).

On comparing these results with Benton (1966), we get the following inferences:

1. In Benton‟s case radial velocity is 0.0462 at   0.1, whereas in present case, it

is 0.00000104 for the same value of , for   0.01. This much difference looks due to the effect of porosity. So, we can say that porosity in ferrofluid flow decreases the radial velocity.

2. Tangential velocity in Benton‟s case is 0.9386 at whereas in the

present case, it is 0.0000563 for the same value of for . Hence, the tangential velocity is also decreasing due to the effect of porosity. 3. In Benton‟s case the axial velocity is 0.0048 at whereas in our case at

same value of  is 0.8855 for . Hence, due to the effect of porosity, fluid velocity is going into more negative region.

FIGURES

E(α) E1 for   0.01 0.006 E2 for   0.02

0.005 E3 for   0.03

0.004

E 0.003 E₁ 0.002 E₂

0.001 E₃

0 0 0.05 0.1 0.15 -0.001

Fig. 4.1: Effect of porosity ε on radial velocity profile

F for   0.01 1.2 F(α) 1 F2 for   0.02 1 F3 for   0.03 0.8

F 0.6

0.4 F₁ F₂ 0.2 F₃ 0 0 0.05 0.1 0.15 0.2

Fig. 4.2: Effect of porosity ε on tangential velocity profile

G(α) 0 0.05 0.1 0.15 0.2 0.1 0 -0.1 G₁ -0.2 G1 for   0.01 -0.3 G₂ G2 for   0.02 -0.4 G₃ G -0.5 G3 for   0.03 -0.6 -0.7 -0.8 -0.9 -1 Fig. 4.3: Effect of porosity ε on axial velocity profile

P(α)-P₀ 0 0.05 0.1 0.15 0.2 0 -0.05 P for   0.01 -0.1 1 P for   0.02 -0.15 2 P3 for   0.03

P -0.2

-0.25 -0.3 P₁ -0.35 P₂ -0.4 P₃

Fig. 4.4: Effect of porosity ε on pressure profile

E'(α) E for   0.01 0.6 1 E2 for   0.02 0.5 E3 for   0.03 0.4 0.3 E'₁

E 0.2 E'₂ 0.1 E'₃ 0 0 0.05 0.1 0.15 0.2 -0.1

-0.2 Fig. 4.5: Effect of porosity ε on derivative of radial velocity profile

F'(α) 0 0.1 0.2 0.3 0 -5 F1 for   0.01 -10 F2 for   0.02 -15 F3 for   0.03 -20 F' -25 F'₁ -30 F'₂ -35 F'₃ -40

Fig. 4.6: Effect of porosity ε on derivative of tangential velocity profile

Chapter 5

REVOLVING FERROFLUID FLOW INFLUENCED BY POROSITY WITH ROTATING DISK

This research work deals with the theoretical investigation of effect of porosity and rotation taken together on ferrofluid flow over a rotating disk. In the present case, the effects of porosity value  and rotation under steady state conditions have been discussed for velocity components, pressure profile and thickness of the boundary layer, considering the z-axis as axis of rotation. Highly non-linear coupled differential equations with boundary conditions involved in this problem are solved numerically by using power series approximation method. Boundary layer displacement thickness is calculated numerically and compared with ferrofluid flow for the case of rotation only i.e. a case without porosity.

5.1 FORMULATION OF THE PROBLEM

5.1.1 Equations of Motion

Applying the general assumptions, given under the heading (1.9) “General Assumption”, the transport equations (1.1 and 1.8) for conservation of mass and conservation of momentum, in cylindrical co-ordinates for the effect of rotation and porosity, can be written as follows:

Continuity equation

v v v r  r  z  0 … (5.1) r r z

Momentum Equations in the radial-direction:

2 2 2 1  vr vr v  1 p 0    vr   vr   vr  2v vr  vz      M H          2 r z r  r  r  r 2 r r z 2       

… (5.2)

The work presented in this chapter has been published in the form of research paper entitled “Effect of porosity on revolving ferrofluid flow with rotating disk”, International Journal of Fluids Engineering (IJFE), 3 (3), pp. 261-271 (2011).

in the tangential-direction:

2 2 1  v v vr v    v   v   v  2vr vr  vz          … (5.3)  2  r z r   r 2 r r z 2        in the axial-direction:

1  v v  1 p     2v 1 v  2v  v z  v z    0 M H  z  z  z … (5.4) 2  r z   2 2    r z   z  z   r r r z 

Applying the similarity transformations (2.6a - 2.6d), the continuity equation (5.1) and momentum equations (5.2 - 5.4), for the combined effect of rotation and porosity, are transformed into a system of strongly coupled non-linear differential equations in E, F,

G and P ; can be written as follows:

G  2E  0 … (5.5)   2 2 2  E GE  E  F  2 F   0 … (5.6)  F GF  2EF  2 E  0 … (5.7)

 2 P  G  GG  0 … (5.8)

5.2 SOLUTION OF THE PROBLEM

The formal asymptotic expansion for the system of equations (5.5) - (5.8), is a  c  power series in exp , i.e.      c  E()   Ai exp i  … (5.9a) i1      c  F()   Bi exp i  … (5.9b) i1      c  G()  G()  Ci exp i  … (5.9c) i1      c  P()  P0   Di exp i  … (5.9d) i1    Here, G must tend to a finite limit, say  c as    , i.e. G()  c, (c  0) .

By using the supposition E(0)  a and F(0)  b (precise value of the two missing boundary conditions as used in previous chapters) along with the equations (2.7a), (5.5) - (5.8), we get the following additional boundary conditions for the

58 approximate solution (5.9a) - (5.9d) for first four coefficients of series solution for the variables E, F, G and P respectively:

  2  2 1 1   E(0)   , E(0)   2b … (5.10a)      

1   F(0)  0, F(0)   2a … (5.10b)   

  2  2 1 G(0)  0, G(0)  2a, G(0)  2  … (5.10c)   

a   2  2 1 1   P(0)  2 , P(0)  2 , P(0)   4b … (5.10d)    2   2     Using the values a  0.54, b  0.62 and c  0.886 , we calculate the values of the first four coefficients A1 , A2 , A3 , A4 ; B1, B2 , B3 , B4 ; C1, C2 , C3 , C4 ; D1, D2 , D3 and involved in equations (5.9a) - (5.9d). D4

Coefficients   0.01   0.02   0.03 0.006952 0.013213 0.018795 A1 -0.006021 -0.010218 -0.012627 A2

A3 -0.002720 -0.007013 -0.012847 0.001789 0.004018 0.006679 A4 3.969703 3.939458 3.542599 B1 -5.933600 -5.867360 -5.801284 B2

B3 3.951094 3.902348 3.853767 -0.987197 -0.974447 -0.961752 B4 3.543843 3.543374 3.542599 C1 -5.315596 -5.314397 -5.312418 C2

C3 3.543665 3.542672 3.541037 -0.885911 -0.885649 -0.885219 C4 -1.390442 -1.321280 -1.252994 D1 1.204205 1.021803 0.841799 D2 0.543958 0.701264 0.856430 -0.357720 -0.401790 -0.445231

Table 5.1: The coefficients involved in the equations (5.9a) – (5.9d)

The following numerical values for E, F, G, P, E and F, if rounded to five decimal places, we get the good approximate solutions of the above system of strongly coupled non-linear differential equations.

  E F G E F P(α)  P0 0.01 0 0 1 0 0.53975 -0.61887 0 0.03 0.00046 0.25042 -0.66253 -0.03770 -19.8393 -0.09136 0.06 3.4E-05 0.01936 -0.86872 -0.00300 -1.70251 -0.00680 0.09 2.4E-06 0.00137 -0.88478 -0.00021 -0.12097 -0.00048 0.12 1.7E-07 9.9E-05 -0.88591 -1.5E-05 -0.00849 -3.7E-05 0.15 1.2E-08 6.7E-06 -0.88599 -1.0E-06 -0.00059 -2.4E-06 0.02 0 0 1 0 0.53980 -0.61940 0 0.03 0.00267 0.69933 -0.25901 -0.07771 -18.5419 -0.26714 0.06 0.00087 0.24861 -0.66256 -0.03627 -9.85237 -0.08736 0.09 0.00024 0.07110 -0.82206 -0.01050 -3.06254 -0.02416 0.12 6.8E-05 0.01921 -0.86872 -0.00285 -0.84481 -0.00647 0.15 1.7E-05 0.00511 -0.88140 -0.00076 -0.22608 -0.00172 0.18 4.6E-06 0.00136 -0.88478 -0.00020 -0.06002 -0.00046 0.21 1.2E-06 0.00036 -0.88568 -5.3E-05 -0.01590 -0.00012 0.03 0 0 1 0 0.54055 -0.62020 0 0.03 0.00490 0.86794 -0.10585 -0.04504 -9.99973 -0.32635 0.06 0.00277 0.515034 -0.42064 -0.06787 -11.3067 -0.18484 0.09 0.00125 0.246800 -0.66260 -0.03487 -6.52351 -0.08340 0.12 0.00053 0.10822 -0.78798 -0.01539 -3.05835 -0.03549 0.15 0.00022 0.04576 -0.84454 -0.00651 -1.32751 -0.01481 0.18 9.2E-05 0.01906 -0.86873 -0.00271 -0.55893 -0.00614 0.21 3.8E-05 0.00789 -0.87885 -0.00112 -0.23244 -0.00253 0.24 1.6E-05 0.00326 -0.88305 -0.00046 -0.09617 -0.00105 0.27 6.5E-06 0.00135 -0.88478 -0.00019 -0.03971 -0.00043 0.30 2.7E-06 0.00056 -0.88550 -7.9E-05 -0.01638 -0.00018

Table 5.2: The steady state velocity field and pressure as functions of  for the effect of rotation and porosity value

5.3 BOUNDARY LAYER DISPLACEMENT THICKNESS

Here, the boundary layer displacement thickness becomes 0.023399, 0.046568 and 0.069507 for   0.01,   0.02 and   0.03, respectively. Hence, it is clear from these results that the boundary layer displacement thickness increases with increasing the porosity value. The boundary layer displacement thickness, 1.34562, for the case of rotation only, is much larger than the present case of rotation and porosity taken together.

5.4 RESULTS AND DISCUSSION

5.4.1 Radial Velocity Profile

Figure 5.1 represents the dimensionless radial velocity profile with the variation of dimensionless parameter  for different porosity values. The dimensionless radial component of velocity E1  0.001704 is maximum at   0.01 for   0.01. However, at , and have the maximum values and for   0.02 E2 E3 0.003335 0.00498   0.02 and   0.03, respectively, whereas, in the case of porosity without rotation (chapter 4), the dimensionless radial component of velocity for   0.01, 0.02, 0.03 , gets the maximum values 0.001776, 0.003559, 0.005351 at   0.08, 0.017, 0.025, respectively. Meaning thereby, increasing the values of porosity  increases the peak value of the dimensionless radial velocity, but these values are obtained little nearer to the disk than in case of porosity only.

5.4.2 Tangential Velocity Profile

The dimensionless tangential velocity profile is shown in figure 5.2. Here, F1, F2 and F3 converge faster to zero in ascending order. Here, at   0.1, the dimensionless tangential velocities for   0.01, 0.02 and 0.03, are 0.000563, 0.04611 and 0.188683, respectively. For the case of porosity without rotation, the dimensionless tangential velocity is almost comparable for the same values of  . It is also clear from the figures, for increasing values of  , the dimensionless tangential velocity takes larger values of  for converging to zero. Also, the zero convergence of the dimensionless tangential velocity is rapid in our case.

5.4.3 Axial Velocity Profile

Figure 5.3 shows the dimensionless axial velocity profile for different porosity

61 values. It is clear from the figure; G , G and G converge rapidly to a finite negative 1 2 3 value (0.886) in their descending order. Here, for   0.01, the dimensionless axial velocity component converges to  0.886 at   0.16 onwards, indicating faster convergence rate than for the values for   0.02 and   0.03.

5.4.4 Pressure Profile

Figure 5.4 exhibits the dimensionless pressure profile with the initial pressure P 0 at the surface of the disk. The dimensionless pressure P1 reaches to the maximum negative value 0.34079 at   0.01for   0.01, whereas, the dimensionless pressure

P2 and P3 take the maximum negative values 0.33353 and 0.33212 at   0.02 for

  0.02 and 0.03, respectively. From graph 5.4, we observe that goes to more negative region than and similarly, goes to more negative region than . Also, it is noticed that figures 5.1 and 5.4 shows the converse behaviour of the radial velocity and pressure.

Figure 5.5 shows the derivative of the dimensionless radial velocity. Here,  E1 ,  and  enter into negative region at and for E2 E3   0.01, 0.02 0.03   0.01, 0.02 and respectively. However,   and for and take the 0.03, E1 , E2 0.03 maximum negative values  0.07086, 0.077709 and 0.07644 at   0.02, 0.03 and 0.05, respectively. Figure 5.6 represents the derivative of dimensionless tangential velocity profile, and shows the interesting results. Here, F1 converges faster than F2 and F3 , and it goes to larger negative region in comparison to F2 and F3 . The maximum negative value 34.272 is attained by at   0.02 for   0.01. However, and take the maximum negative values 18.5419 and 12.2532 at   0.03

and   0.05 for   0.02 and respectively.

FIGURES

0.006 E(α) E1 for   0.01 E for   0.02 0.005 2 E3 for   0.03 0.004 0.003

E E₁ 0.002 E₂ 0.001 E₃

0 0 0.05 0.1 0.15 0.2 -0.001

Fig. 5.1: Effect of porosity ε and rotation on radial velocity

F(α) F1 for   0.01 1.2 F2 for   0.02 1 F3 for   0.03

0.8

0.6 F

0.4 F₁ F₂ 0.2 F₃ 0 0 0.05 0.1 0.15 0.2 0.25

Fig. 5.2: Effect of porosity ε and rotation on tangential velocity

G(α) 0 0.1 0.2 0.3 0.1 0 -0.1 -0.2 G for   0.01 -0.3 1 -0.4 G2 for   0.02 G -0.5 G for   0.03 3 G₁ -0.6 -0.7 G₂ -0.8 G₃ -0.9 -1 Fig. 5.3: Effect of porosity ε and rotation on axial velocity

P(α)-P₀ 0 0.05 0.1 0.15 0.2 0.05 0 -0.05 P₁ -0.1 P₂ -0.15

P P₃ -0.2 -0.25 P for   0.01 -0.3 1 P for   0.02 -0.35 2 -0.4 P3 for   0.03

Fig. 5.4: Effect of porosity ε and rotation on pressure profile

E'(α) E1 for   0.01 0.6 E2 for   0.02

0.5 E3 for   0.03 0.4

0.3 E'₁ 0.2 E' E'₂ 0.1 E'₃ 0 0 0.05 0.1 0.15 0.2 -0.1

-0.2 Fig. 5.5: Effect of porosity ε on derivative of radial velocity profile

F'(α) 0 0.05 0.1 0.15 0.2 0.25 0 -5 F for   0.01 -10 1 F for   0.02 -15 2 F for   0.03 -20 3 F' -25 -30 F'₁ -35 F'₂ -40 F'₃

Fig. 5.6: Effect of porosity ε and rotation on derivative of tangential velocity profile

Chapter 6

FERROFLUID FLOW BEHAVIOUR INFLUENCED BY THE EFFECT OF ROTATION AND MAGNETIC FIELD-DEPENDENT VISCOSITY WITH ROTATING DISK

The present work deals with the theoretical investigation of the effect of magnetic field- dependent (MFD) viscosity on the revolving Axi-symmetric steady laminar flow of viscous incompressible electrically non-conducting ferrofluid with rotating disk. The non-linear coupled ordinary differential equations involved in the problem are solved by asymptotic approximations. Here, the velocity components and pressure profiles are computed numerically and discussed graphically for various values of MFD viscosity parameter (k) along with the variation of Karman‟s dimensionless parameter () . Also, we have calculated the displacement thickness of the boundary layer.

6.1 FORMULATION OF THE PROBLEM

Applying the general assumptions, given under the heading (1.9) “General Assumption”, the continuity equation (1.1) and conservation of momentum equation (1.9), for the effect of rotation and MFD viscosity in cylindrical co-ordinates; can be written as follows:

v v v r  r  z  0 … (6.1) r r z

2 2 2 1 p 0   vr   vr   vr  vr vr v   M H 1  2     2   2v  vr  vz  … (6.2)  r  r r r r z r z r    

2 2  v   v   v  v v vrv   2     2   2vr  vr  vz  … (6.3)  r r  r  z  r z r

2 2 1 p 0   vz 1 vz  vz  vz vz   M H 1  2   2   vr  vz … (6.4)  z  z  r r r z  r z

The work presented in this chapter has been communicated in the form of research paper entitled “Effect of rotation and MFD viscosity on ferrofluid flow with rotating disk”, Indian Journal of Pure and Applied Physics (IJPAP), (2011).

 (1 δ.B) where   f = Ratio of MFD viscosity and density of the fluid. 1  Applying the similarity transformations (2.6a - 2.6d), the continuity equation (6.1) and momentum equations (6.2 - 6.4), for the combined effect of rotation and MFD viscosity, are transformed into a system of strongly coupled non-linear differential

and ; equations in E, F, G P can be written as follows:

2 2 kE GE  E  F  2F 1 0 … (6.5) F  GF  2EF  2 E  0 … (6.6)

P  kG  GG  0 … (6.7)

G  2E  0 … (6.8)

Here, k 1   Ratio of MFD kinematic viscosity (variable) and kinematic reference viscosity i.e. MFD viscosity parameter.

6.2 SOLUTION OF THE PROBLEM

The formal asymptotic expansion of the system of equations (6.5), (6.7) and (6.8),  c  is a power series in exp  , and equation (6.6) is a power series in exp(c) :  k 

  c  E()   Ai exp i  … (6.9a) i1  k 

 F()   Bi exp(-ic) … (6.9b) i1   c  G()  G()  Ci exp i  … (6.9c) i1  k    c  P()  P0   Di exp i  … (6.9d) i1  k 

Here, G must tend to a finite limit, say  c as    , i.e. G()  c, (c  0) .

By using the supposition E(0)  a and F(0)  b (precised values of missing boundary conditions) along with the equations (2.7a), (6.5) - (6.8), we get the following additional boundary conditions for the approximate solution (6.9a) - (6.9d) for first five coefficients for series solution for the variables E, F, G and P, respectively:

 2 4b 2b2 E(0)  , E(0)   , E iv (0)   … (6.10a) k k k

8 F(0)  0, F(0)  4a, F iv (0)  2ab  … (6.10b) k 4 8b G(0)  0, G(0)  2a, G(0)  , Giv (0)  … (6.10c) k k

P(0)  2ak, P(0)  4, P(0)  8b, Piv (0)  4b2 12a2 … (6.10d)

With the help of these additional boundary conditions (6.10a) - (6.10d) along with the boundary conditions (2.7a) and by considering the precised values of a, b and c (discussed in previous chapters), we get the numerical values of A 's, B 's, C 's and i i i

Di 's for different values of k , up to four decimal places (Table 6.1): by using MATLAB programming. Here, the values of  are taken as 0.01, 0.02, 0.03 so that k 1.1, 1.2, 1.3; (Sunil et al. 2009).

Coefficients k 1.1 k 1.2 k 1.3

A1 -1.5414 -1.4462 -1.3132

A2 6.5311 6.2551 5.8396

A3 -8.7295 -8.3255 -7.7300

A4 4.7016 4.4019 3.9865

A5 -0.9618 -0.8853 -0.7828 B 1.7844 1.8254 1.8600 1 B 2 -2.1015 -2.2654 -2.4041 B 3 2.4492 2.6950 2.9031 B 4 -1.4312 -1.5951 -1.7338 B5 0.2991 0.3401 0.3748 3.1182 2.8206 2.4845 C1 C2 -5.7824 -5.0043 -4.1081 C3 6.6657 5.9588 5.1151 C4 -4.0269 -3.7571 -3.4136 C5 0.9115 0.8680 0.8082 3.0446 2.9803 2.7387 D1 D2 -12.9828 -13.0498 -12.4799 D 3 17.1264 17.0375 16.0435 D4 -8.9578 -8.6021 -7.6618 D5 1.7696 1.6341 1.3596

Table 6.1: First five coefficients in algebraic equations for Ai' s , Bi' s , Ci' s and Di' s

The following numerical values, for E, F, G, P, E and F, if rounded to five decimal places, we get the good approximate solution of the above system of strongly coupled non-linear differential equations.

E F k  E F G P(α)  P0 1.1 0 0 1 0 0.54056 -0.61993 0 1 0.00686 0.51234 -0.19688 -0.19893 -0.34430 -0.02957 2 -0.10966 0.25349 -0.44698 -0.02655 -0.18908 0.21288 3 -0.09143 0.11555 -0.64932 0.04104 -0.09464 0.17988 4 -0.05162 0.04987 -0.77045 0.03407 -0.04273 0.10182 5 -0.02544 0.02097 -0.83224 0.01890 -0.01832 0.05022 1.2 0 0 1 0 0.53992 -0.61993 0 1 0.03623 0.51436 -0.20484 -0.18310 -0.34107 -0.10439 2 -0.09182 0.25680 -0.44147 -0.05160 -0.18961 0.18064 3 -0.09357 0.11770 -0.63048 0.02871 -0.09597 0.19084 4 -0.05958 0.05092 -0.75164 0.03310 -0.04355 0.12234 5 -0.0323 0.02143 -0.81869 0.02116 -0.01871 0.06646 1.3 0 0 1 0 0.53952 -0.62002 0 1 0.06440 0.51604 -0.21501 -0.16147 -0.33833 -0.18786 2 -0.06701 0.25958 -0.44725 -0.07099 -0.19005 0.12119 4 -0.06306 0.05181 -0.73955 0.02873 -0.04425 0.13020 6 -0.02040 0.00908 -0.84550 0.01284 -0.00799 0.04246 10 -0.00143 0.00026 -0.88328 0.00097 -0.00023 0.00299

Table 6.2: The steady state velocity field and pressure as functions of  for the effect of rotation and MFD viscosity parameter k

6.3 BOUNDARY LAYER DISPLACEMENT THICKNESS

The displacement thickness of the boundary layer for ferrofluid flow is also calculated for the different values of MFD viscosity. In the present study, the displacement thickness is 1.413156, 1.422407 and 1.430234 for k  1.1, 1.2 and 1.3 , respectively. Hence, we conclude from this result that increase in the magnetic field intensity increases the boundary layer displacement thickness.

6.4 RESULTS AND DISCUSSION

6.4.1 Radial Velocity Profile

Figure 6.1 represents the radial velocity (dimensionless) profile with the variation of dimensionless parameter  at different values of MFD viscosity. The radial component of the velocity E1  0.95196 is the maximum at   0.4. However, E2

 0.104538 and E3  0.116384 are the maximum at   0.5 and   0.6 respectively.

However, for k  1, the problem reduces to Newtonian case of ordinary viscous fluid.

Identical cases for radial velocity, axial velocity and pressure profile are shown by E4 ,

G4 and P4 in figures 6.1, 6.3 and 6.4, respectively.

6.4.2 Tangential Velocity Profile

The dimensionless tangential velocity components, at different MFD viscosity parameters, shown in figure 6.2 are in proximity with each other and are difficult to identify separately. Precisely, the variation in magnetic field intensity has the negligible impact on the dimensionless tangential velocity. It is clear that the tangential velocity decreases continuously for different increasing values of dimensionless parameter  at different MFD viscosity.

6.4.3 Axial Velocity Profile

Figure 6.3 exhibits the dimensionless axial (vertical) velocity profile. It is clear from the figure that the effect of magnetic field in the z-direction is comparatively less than radial direction. Here, and tends to finite negative value at G1 , G2 G3  0.886  16.6, 18 and 19.3 for k  1.1, 1.2 and 1.3, respectively.

6.4.4 Pressure Profile

The pressure (dimensionless) profile is shown in the figure 6.4, reaches the maximum negative values P1  0.21085, P2  0.255398 and P3  0.30599 at

  0.4, 0.4 and 0.5 for 1.2 and 1.3, respectively. Here, we notice that the increase in the magnetic field increases the radial velocity and decreases the pressure on the fluid. On comparing figures 6.1 and 6.4, we conclude that the variations in radial velocity and pressure are converse in nature.

Figure 6.5 represents derivative of the dimensionless radial component of velocity. Here, E() decreases continuously as we move far from the disk and firstly, the curves for 1.2, 1.3 go to negative region and afterwards enter into positive 71 region. Further, E1, E2 and E3 take the maximum negative values for the same value

  1 at k  1.1, 1.2 and 1.3, respectively.

Derivative of the tangential velocity profile is exhibited in the figure 6.6. The effect of MFD viscosity on the derivative of tangential velocity is negligible, so the curves are in close proximity and it is difficult to see them distinctly, for different MFD viscosity parameters.

FIGURES

E(α) E1 for k 1.1

0.15 E2 for k 1.2 E for k 1.3 0.1 3 E4 for k 1.0 0.05

E 0 0 2 4 6 8 10 -0.05 E₁ -0.1 E₂

-0.15 E₃ E₄

Fig. 6.1: Effect of MFD viscosity and rotation on radial velocity profile

F(α) F1 for k 1.1

1.2 F2 for k 1.2 F for k 1.3 1 3

F 0.8

0.6 F₁ 0.4 F₂ 0.2 F₃ 0 0 2 4 6 8

Fig. 6.2: Effect of MFD viscosity and rotation on tangential velocity profile

G(α) 0 5 10 15 0 -0.1 -0.2 G1 for k 1.1 -0.3 G2 for k 1.2 -0.4 G₁ G3 for k 1.3

G -0.5 G₂ G for k 1.0 -0.6 4 G₃ -0.7 -0.8 G₄ -0.9 -1

Fig. 6.3: Effect of MFD viscosity and rotation on axial velocity profile

P(α)-P₀ P1 for k 1.1

0.3 P2 for k 1.2 P for k 1.3 0.2 3 P for k 1.0 0.1 4

0 P -0.1 0 2 4 6 8 10

-0.2 P₁ -0.3 P₂ -0.4 P₃ P₄

Fig. 6.4: Effect of MFD viscosity and rotation on pressure profile

 E'(α) E1 for k 1.1 0.6 E2 for k 1.2 0.5 E3 for k 1.3 0.4 E₁' 0.3 E₂' 0.2

E' E₃' 0.1 0 -0.1 0 2 4 6 8 -0.2 -0.3

Fig. 6.5: Effect of MFD viscosity and rotation on derivative of radial velocity

F'(α) 0 2 4 6 8 10 0

-0.1 F1 for k 1.1 F for k 1.2 -0.2 2 F3 for k 1.3 -0.3

F' -0.4

-0.5 F'₁

-0.6 F'₂ F'₃ -0.7

Fig. 6.6: Effect of MFD viscosity and rotation on derivative of tangential velocity

Chapter 7

FERROFLUID FLOW BEHAVIOUR INFLUENCED BY THE EFFECT OF ROTATION, POROSITY AND MAGNETIC FIELD-DEPENDENT VISCOSITY WITH ROTATING DISK

In the present case, we have studied the effect of rotation, magnetic field- dependent viscosity (MFD) along with porosity on the Axi-symmetric steady ferrofluid flow with rotating disk by solving the boundary layer equations. Here, we have calculated the velocity components and pressure for different values of MFD viscosity parameter k and porosity  with the variation of Karman‟s dimensionless parameter . Also, we have calculated the displacement thickness of the boundary layer. The numerical results which are obtained for various flow characteristics are shown graphically.

7.1 FORMULATION OF THE PROBLEM

Applying the assumptions given under the heading (1.9) “General Assumption”, the constitutive equations (1.2 and 1.10) for conservation of mass and conservation of momentum for the effect of rotation, MFD viscosity and porosity, in cylindrical co- ordinates, can be written as follows:

Continuity equation

v v v r  r  z  0 … (7.1) r r z Momentum Equations

in the r-direction:

2 2 2 1  vr vr v  1 p 0   1  vr   vr   vr  2 2 vr  vz      M H   2     2   v   r z r   r  r   r r  r  z   … (7.2)

The work presented in this chapter has been published in the form of research paper entitled “Revolving ferrofluid flow under the influence of MFD viscosity and porosity with rotating disk”, Journal of Electromagnetic Analysis and Applications (JEMAA), 3 (9), pp. 378-386 (2011).

2 2 1  v v vr v    v   v   v  2 vr  vz          vr … (7.3)  2  r z r   r 2 r r z 2       

2 2 1  vz vz  1 p 0   1  vz 1 vz  vz  vr  vz    M H      … (7.4)  2  r z   z  z  r 2 r r z 2    

 (1 δ.B) where   f  Ratio of MFD viscosity and density of the fluid. On applying 1  similarity transformations (2.6a) - (2.6d), the system of non-linear strongly coupled differential equations in E, F, G and P from continuity equation (7.1) and momentum equations (7.2 - 7.4); is obtained as follows:

G  2E  0 … (7.5)

  1 E  GE  E 2  F 2  2 F   2  0 … (7.6) 

 F  GF  2EF  2 E  0 … (7.7)

  2 P  1 G  GG  0 … (7.8) 

7.2 SOLUTION OF THE PROBLEM

The formal asymptotic expansion for the system of equations (7.5), (7.7) and   c  (7.8) is a power series in exp   and equation (7.6) is a power series in   1   c  exp   , i.e.   

   c  E()  A exp i  … (7.9a)  i     i1  1 

  c  F()   Bi exp i  … (7.9b) i1   

   c  G()  G()  C exp i  … (7.9c)  i     i1  1 

   c  P()  P  D exp i  … (7.9d) 0  i     i1  1 

Here, G must tend to a finite limit, say  c as    , where c is a positive quantity. Using the precised values of missing boundary conditions E(0)  a and

F(0)  b (as reported in chapter 2) along with the equations (2.7a) and set of equations (7.5) - (7.8), we get the following additional boundary conditions for the approximate solution (7.9a) - (7.9d) for first four coefficients for series solution for the variables E,

F, G and P, respectively:

1 ( 2  2 1) 1 2b(1 ) E(0)  , E(0)  … (7.10a) k  k 

2a(1 ) F(0)  0, F(0)  … (7.10b) 

 2 ( 2  2 1) G(0)  0, G(0)  2a, G(0)  … (7.10c) k 

2a 2( 2  2 1) 4b(1 )    … (7.10d) P (0)  k , P (0)   2 , P (0)  2   

 where k  1  MFD viscosity parameter. By using these additional boundary conditions  and taking the precised values E(0)  a  0.54 and F(0)  b  0.62 and c  0.886 in

MATLAB environment, we calculate the first four coefficients A1 , A2 , A3 , A4 ; B1 , B2 ,

B3 , B4 ; C1 , C2 , C3 , C4 ; D1 , D2 , D3 , and D4 ; for the expansions (7.9a) - (7.9d) for different values of MFD viscosity parameter k which are as follows:

  0.01

k A1 A2 A3 A4 1.1 0.00765 -0.00663 -0.00298 0.00196 1.2 0.00835 -0.00725 -0.00324 0.00214 1.3 0.00905 -0.00786 -0.00350 0.00231   0.02

1.1 0.01455 -0.01128 -0.00767 0.00441 1.2 0.01588 -0.01235 -0.00833 0.00479 1.3 0.01722 -0.01343 -0.00897 0.00518   0.03

1.1 0.02070 -0.01398 -0.01404 0.00732 1.2 0.02262 -0.01535 -0.01522 0.00795 1.3 0.02454 -0.01674 -0.01638 0.00857

Table 7.1: The coefficients , , , in equation (7.9a) for the effect of rotation, MFD viscosity parameter k  1.1, 1.2 and 1.3 and porosity value   0.01, 0.02 and 0.03

C1 C 2 C 3 C 4 1.1 3.54369 -5.31516 3.54324 -0.88577 1.2 3.54363 -5.31500 3.54310 -0.88573 1.3 3.54357 -5.31482 3.54294 -0.88568   0.02

1.1 3.54276 -5.31261 3.54095 -0.88509 1.2 3.54252 -5.31197 3.54037 -0.88492 1.3 3.54227 -5.31127 3.53973 -0.88473   0.03

C 2 1.1 3.5412 -5.30835 3.53710 -0.88395 1.2 3.54067 -5.30689 3.53579 -0.88356 1.3 3.54009 -5.30531 3.53436 -0.88314

Table 7.2: The coefficients , , , in equation (7.9c) for the effect of rotation, MFD viscosity parameter and and porosity value and

  0.01

k D1 D2 D3 D4 1.1 -1.6832 1.45926 0.65601 -0.43211 1.2 -2.0040 1.73924 0.77811 -0.51339 1.3 -2.3529 2.04423 0.91016 -0.60150   0.02

1.1 -1.6002 1.24078 0.84413 -0.48470 1.2 -1.9061 1.48187 0.99935 -0.57509 1.3 -2.2391 1.74528 1.16670 -0.67288   0.03

1.1 -1.5183 1.02524 1.02961 -0.53651 1.2 -1.8096 1.22805 1.21739 -0.63584 1.3 -2.1269 1.45056 1.41944 -0.74313

Table 7.3: The coefficients , , , in equation (7.9d) for the effect of rotation, MFD viscosity parameter k  1.1, 1.2 and 1.3 and porosity value   0.01, 0.02 and 0.03

Coefficients

B1 3.96970 3.93946 3.90927

B2 -5.93360 -5.86736 -5.80128 3.95109 3.90235 3.85377 B3 -0.98720 -0.97445 -0.96175 B4

Table 7.4: The coefficients , , , in equation (7.9b) for the porosity value and

  0.01   0.02   0.03 k 1.1 k 1.2 k 1.3 

E1 E2 E3 E4 E5 E6 E7 E8 E9 0 0 0 0 0 0 0 0 0 0 0.004 0.00147 0.00152 0.00156 0.00179 0.00182 0.00185 0.00191 0.00193 0.00195 0.008 0.00191 0.00205 0.00218 0.00291 0.00302 0.00311 0.00333 0.00341 0.00348 0.012 0.00183 0.00205 0.00225 0.00352 0.00372 0.00389 0.00434 0.00450 0.00464 0.016 0.00155 0.00181 0.00205 0.00375 0.00405 0.00431 0.00500 0.00525 0.00547 0.020 0.00124 0.00150 0.00175 0.00374 0.00411 0.00445 0.00538 0.00573 0.00604 0.024 0.00096 0.00120 0.00144 0.00357 0.00400 0.00441 0.00554 0.00598 0.00638 0.028 0.00073 0.00093 0.00116 0.00332 0.00379 0.00424 0.00554 0.00607 0.00654 0.032 0.00054 0.00072 0.00092 0.00302 0.00351 0.00400 0.00543 0.00602 0.00657 0.036 0.00040 0.00055 0.00072 0.00271 0.00321 0.00371 0.00523 0.00588 0.00649 0.040 0.00029 0.00042 0.00056 0.00240 0.00290 0.00340 0.00498 0.00567 0.00633 0.044 0.00022 0.00031 0.00043 0.00211 0.00259 0.00309 0.00470 0.00541 0.00611 0.048 0.00016 0.00024 0.00033 0.00185 0.00231 0.00279 0.00439 0.00513 0.00584

Table 7.5: The steady state radial velocity E as a function of  for the effect of rotation, porosity and MFD viscosity parameter k

  0.01   0.02   0.03



G1 G2 G3 G4 G5 G6 G7 G8 G9 0 0 0 0 0 0 0 0 0 0 0.04 -0.75290 -0.71511 -0.67580 -0.36357 -0.31422 -0.27186 -0.16674 -0.13672 -0.11291 0.08 -0.88038 -0.87640 -0.87091 -0.75293 -0.71516 -0.67587 -0.53950 -0.48583 -0.43640 0.12 -0.88578 -0.88550 -0.88501 -0.85812 -0.84454 -0.82813 -0.75299 -0.71524 -0.67599 0.16 -0.88599 -0.88597 -0.88593 -0.88038 -0.87640 -0.87091 -0.83872 -0.81898 -0.79622 0.2 -0.886 -0.886 -0.886 -0.88488 -0.88380 -0.88212 -0.86963 -0.86049 -0.84895 0.24 -0.886 -0.886 -0.886 -0.88578 -0.88550 -0.88501 -0.88038 -0.87640 -0.87092 0.28 -0.886 -0.886 -0.886 -0.88596 -0.88589 -0.88575 -0.88408 -0.88241 -0.87990 0.32 -0.886 -0.886 -0.886 -0.88599 -0.88597 -0.88593 -0.88534 -0.88466 -0.88354 0.36 -0.886 -0.886 -0.886 -0.886 -0.88599 -0.88598 -0.88578 -0.88550 -0.88501 0.4 -0.886 -0.886 -0.886 -0.886 -0.886 -0.886 -0.88592 -0.88581 -0.88560

Table 7.6: The steady state axial velocity G as a function of for the effect of rotation, porosity and MFD viscosity parameter

  0.01   0.02   0.03 k 1.1 k 1.2 k 1.3 

P1 P2 P3 P4 P5 P6 P7 P8 P9 0 0 0 0 0 0 0 0 0 0 0.02 -0.27340 -0.35910 -0.45553 -0.41131 -0.49342 -0.5791 -0.39422 -0.45801 -0.52314 0.04 -0.06477 -0.09970 -0.14504 -0.26413 -0.34771 -0.44200 -0.36533 -0.45359 -0.54826 0.06 -0.01331 -0.02363 -0.03884 -0.13236 -0.18919 -0.25823 -0.25499 -0.33647 -0.42865 0.08 -0.00267 -0.00544 -0.01005 -0.06179 -0.09526 -0.13880 -0.16173 -0.22563 -0.30165 0.1 -0.00053 -0.00124 -0.00258 -0.02812 -0.04658 -0.07220 -0.09851 -0.14476 -0.20244 0.12 -0.00011 -0.00028 -0.00066 -0.01267 -0.0225 -0.03701 -0.05886 -0.09089 -0.13265 0.14 -2.1E-05 -6.5E-05 -0.00017 -0.00568 -0.01081 -0.01885 -0.03483 -0.05642 -0.08580 0.16 -4.3E-06 -1.5E-05 -4.3E-05 -0.00254 -0.00518 -0.00957 -0.02050 -0.03480 -0.05509 0.18 -8.5E-07 -3.4E-06 -1.1E-05 -0.00114 -0.00248 -0.00485 -0.01203 -0.02138 -0.03522 0.2 -1.7E-07 -7.7E-07 -2.8E-06 -0.00051 -0.00118 -0.00245 -0.00705 -0.01311 -0.02245 0.22 -3.4E-08 -1.8E-07 -7.2E-07 -0.00023 -0.00057 -0.00124 -0.00412 -0.00803 -0.01429 0.24 -6.8E-09 -4E-08 -1.9E-07 -0.0001 -0.00027 -0.00063 -0.00241 -0.00492 -0.00909 0.26 -1.4E-09 -9.2E-09 -4.7E-08 -4.5E-05 -0.00013 -0.00032 -0.00141 -0.00301 -0.00578 0.28 -2.7E-10 -2.1E-09 -1.2E-08 -2E-05 -6.2E-05 -0.00016 -0.00082 -0.00184 -0.00367 0.3 -5.4E-11 -4.8E-10 -3.1E-09 -9.1E-06 -3E-05 -8.1E-05 -0.00048 -0.00112 -0.00233

Table 7.7: The steady state pressure P as a function of  for the effect of rotation, porosity and MFD viscosity parameter k

  0.02   0.03

 F1 F2 F3 0 1 1 1 0.02 0.52194 0.87219 0.94988 0.04 0.10985 0.51848 0.75617 0.06 0.01936 0.24861 0.51503 0.08 0.00331 0.10903 0.31983 0.1 0.00056 0.04611 0.18868 0.14 1.6E-05 0.00795 0.06111 0.18 4.7E-07 0.00136 0.01906 0.2 8.0E-08 0.00056 0.01060

Table 7.8: The steady state tangential velocity F as a function of for the effect of rotation, porosity and MFD viscosity parameter

7.3 BOUNDARY LAYER DISPLACEMENT THICKNESS

7.4 RESULTS AND DISCUSSIONS

7.4.1 Radial Velocity Profile

Figures. 7.1a, 7.1b and 7.1c show the radial velocity profile with the variation of dimensionless parameter  (Karman‟s parameter) for different values of porosity  at MFD viscosity parameter k 1.1, 1.2 and 1.3, respectively. The radial velocity at porosity  1.0 for MFD viscosity parameter k  1, without rotation is the reduced case of viscous incompressible problem. For   0.01, the radial component of velocity is maximum at and and have the maximum values E1  0.00192   0.009, E2 E3

0.00209 and 0.00227 at   0.01 and 0.011, respectively. Whereas, in Ram et al.

(2010) case of MFD viscosity without porosity and without rotation; E1 , E2 and E3 have the maximum values 0.45465, 0.54455 and 0.59099 at  1.4, 1.7 and 1.8, respectively. Thus, the convergence rate for radial component of velocity is faster for revolving ferrofluid flow with MFD viscosity along with porosity than the case reported in Ram et al. (2010) for MFD viscosity only. However, from figures 7.1a - 7.1c; we observe that for different increasing values of porosity with same set values of MFD viscosity parameter k , the radial values of velocity lead to its slow convergence.

7.4.2 Tangential Velocity Profile

Figure 7.2 shows the tangential velocity profile for different values of porosity. There is no effect of MFD viscosity on tangential velocity component as we have not considered the effect of magnetic field in tangential direction. Here, for   0.01, the tangential velocity components are 0.00056, 0.04611 and 0.18868 for different values of porosity   0.01, 0.02 and 0.03, respectively. Also, it is observed that due to the effect of porosity, the tangential velocity component is increasing with the increase in Karman‟s parameter  . Whereas in Ram et al. (2010), the tangential velocity component is free from the effect of porosity. Here, the tangential velocity decreases smoothly and after

84 certain values of  , it converges to zero. But, as we increase the value of porosity, the convergence becomes slower.

7.4.3 Axial Velocity Profile

Figures. 7.3a, 7.3b and 7.3c, represent the axial velocity profiles which are zero in the beginning. It is clear that when we increase the magnetic-field, the axial velocity goes to more negative region and the component G3 tends to a finite negative value

 0.886 little faster than G1 and G2 . For   0.01, the axial velocity G1 , G2 and G3 converges to finite negative value  0.886 at   0.17, 0.19 and 0.20 for MFD viscosity parameter k 1.1, 1.2 and 1.3, respectively. Whereas, for   0.02 , the axial components of the velocity G4 , G5 and G6 for MFD viscosity parameter k 1.1, 1.2 and

1.3, converges to this finite negative value at   0.34, 0.37 and 0.40 , respectively. Similarly, we can conclude for   0.03, the late convergence is appeared for axial velocity components G7 , G8 and G9 for MFD viscosity parameter k 1.1, 1.2 and respectively. In Ram et al. (2010), the fluctuations in graph are negligible, but here fluctuations are prominent. There is a large variation in axial velocity components for different MFD viscosity parameter along with the effect of porosity and revolution of ferrofluid. Also from the graphs 7.3a, 7.3b and 7.3c, we can conclude that the axial velocity component is decreasing with increase in porosity and MFD viscosity, but for large values of , it takes finite negative value  0.886 .

7.4.4 Pressure Profile

Figures 7.4a, 7.4b and 7.4c show the pressure profile with initial pressure P0 , for different values of porosity  at MFD viscosity parameter k 1.1, 1.2 and 1.3, respectively. In figure 7.4b, the pressure P1 reaches maximum negative value  0.42214 at   0.009 for k 1.1 and   0.01, whereas, for the same value of , the pressures P2 and P3 take the maximum negative values  0.50232 and  0.58932 at

  0.01 and 0.011 for k 1.2 and respectively. We conclude from these graphs that due to the increment in magnetic-field and rotation of ferrofluid, the convergence rate is going slow and slow with the increment in porosity  along with the variation in Karman‟s parameter  . Also, in comparison to Ram et al. (2010), the pressure decreases due to the effect of porosity and rotation of ferrofluid.

Concluding Remarks

(i) Under the influence of MFD viscosity, porosity and rotation, the radial and tangential components of velocity converge to zero faster than the case of MFD viscosity alone. The radial velocity increases with increase in porosity value and MFD viscosity parameter both, whereas the tangential component of velocity increases with the increase in porosity only and MFD viscosity has no effect on it. The effect of MFD viscosity is dominant in radial direction and is moderate in axial direction. (ii) Also, the axial velocity converges to finite negative value faster than the case of MFD viscosity alone. Numerical value of the axial component decreases with increase in porosity and MFD viscosity, both. (iii) Pressure profile shows smaller values in comparison to the case of MFD viscosity only. Here, we can also say that the radial velocity and pressure are converse in behaviour to each other. (iv) The boundary layer displacement thickness becomes very small in comparison to the case of MFD viscosity only as well as to the case of ordinary viscous flow reported in Ram et al. (2010) and Benton (1966), respectively.

FIGURES   0.01 0.0025 E(α) E1 for k 1.1

0.002 E2 for k 1.2 E for k 1.3 0.0015 3

E 0.001 E₁ E₂ 0.0005 E₃ -4E-18 0 0.02 0.04 0.06 0.08 -0.0005

Fig. 7.1 (a): Effect of rotation and porosity 휖 = 0.01 along with variation of MFD viscosity parameter 푘 on radial velocity

0.005 E(α)   0.02

0.004 E4 for k 1.1 E for k 1.2 0.003 5 E5 for k E1₄.3

E 0.002 E₅ 0.001 E₆ 0 0 0.05 0.1 0.15 0.2 -0.001

Fig. 7.1 (b): Effect of rotation and porosity 휖 = 0.02 along with variation of MFD viscosity parameter 푘 on radial velocity

  0.03 0.007 E(α) E7 for k 1.1 0.006 E8 for k 1.2 0.005 E9 for k 1.3 0.004

E 0.003 E₇ 0.002 E₈ 0.001 E₉ 0 -0.001 0 0.1 0.2 0.3

Fig. 7.1 (c): Effect of rotation and porosity 휖 = 0.03 along with variation of MFD viscosity parameter 푘 on radial velocity

F(α) F1 for   0.01 1.2 F2 for   0.02 1 F3 for   0.03 0.8

F 0.6

0.4 F₁ F₂ 0.2 F₃ 0 0 0.05 0.1 0.15 0.2 0.25 0.3

Fig. 7.2: Effect of porosity 휖 on tangential velocity for the case of rotation, MFD viscosity and porosity

G(α) 0 0.05 0.1 0.15 0

-0.2   0.01 G for k 1.1 -0.4 1 G for k 1.2 G 2 G₁ -0.6 G for k 1.3 3 G₂

-0.8 G₃

-1

Fig. 7.3 (a): Effect of rotation and porosity 휖 = 0.01 along with variation of MFD viscosity parameter 푘 on axial velocity

G(α) 0 0.1 0.2 0.3 0.4 0

-0.2   0.02

-0.4 G4 for k 1.1

G G for k 1.2 -0.6 5 G₄ G6 for k 1.3 G₅ -0.8 G₆

-1 Fig. 7.3 (b): Effect of rotation and porosity 휖 = 0.02 along with variation of MFD viscosity parameter k on axial velocity

G(α) 0 0.2 0.4 0.6 0

-0.2

  0.03 -0.4 G7 for k 1.1 G G₇ -0.6 G for k 1.2 8 G₈ G for k 1.3 -0.8 9 G₉

-1 Fig. 7.3 (c): Effect of rotation and porosity 휖 = 0.03 along with variation of MFD viscosity parameter 푘 on axial velocity

P(α)-P₀ 0 0.05 0.1 0.15 0.1 -6E-16 -0.1

-0.2   0.01 P -0.3 P1 for k 1.1

-0.4 P2 for k 1.2 P₁ -0.5 P3 for k 1.3 P₂ -0.6 P₃ Fig. 7.4 (a): Effect of rotation and porosity 휖 = 0.01 along with variation of MFD viscosity parameter 푘 on pressure profile

P(α)-P₀ 0 0.05 0.1 0.15 0.2 0.25 0.1 -6E-16 -0.1 -0.2   0.02

P P for k 1.1 -0.3 4 P for k 1.2 -0.4 5 -0.5 P6 for k 1.3 P₄ -0.6 P₅ P₆ Fig. 7.4 (b): Effect of rotation and porosity 휖 = 0.02 along with variation of MFD viscosity parameter 푘 on pressure profile

P(α)-P₀ 0 0.05 0.1 0.15 0.2 0.25 0.1 0 -0.1 -0.2

P   0.03 -0.3 P₇ P7 for k 1.1 -0.4 P₈ P8 for k 1.2 -0.5 P₉ P9 for k 1.3 -0.6

Fig. 7.4 (c): Effect of rotation and porosity 휖 = 0.03 along with variation of MFD viscosity parameter 푘 on pressure profile

CONCLUSION Here, a comparative study of results obtained in all research problems is carried out.

Firstly, the results obtained in chapters 2, 3 and 4 for the effect of rotation, porous medium, porosity, respectively, are compared with each other and it is found that these parameters diversely affects the velocity profile, pressure profile, displacement thickness significantly. Further, the results of research problems discussed in chapters 5 and 6 for effect of “rotation & porosity” and “rotation & MFD viscosity”, respectively, are compared with each other. Also, these results are compared with the results obtained in chapters 2, 3 and 4. In the last, an overall comparison is made out one by one, between the hypothesis of combined effect of rotation, MFD viscosity and porosity (chapter 7) with the results reported in chapters 2, 3, 4, 5 and 6.

The objective of this study is to better understand the behaviour of ferrofluid flow due to rotating disk influenced by rotation, porous medium, porosity and MFD viscosity together with rotation and porosity. In nut shell, our work is a theoretical motivation explaining physical effects of rotation, porous medium, porosity and variable field dependent viscosity on various flow characteristics of ferrofluid.

Dimensionless Radial Velocity Profile

The dimensionless radial velocity for the effect of rotation (chapter 2) is 0.04271 at   0.1 (near to the disk) whereas for the same value of  in the case of porous medium (chapter 3), it becomes 0.05412, 0.05419, 0.05426 and 0.05433 for Darcy number   0, 1, 2 and 3, respectively. Here   0.1 is the non-dimensional distance from the disk surface. For the effect of porosity value only (chapter 4), it is 0.00174,

0.00317 and 0.00384, for porosity values   0.01, 0.02 and 0.03, respectively at dimensionless distance   0.01. If we compare these three cases individually, the non- dimensional radial velocity has larger values in porous medium in comparison to the values due to rotational effect; whereas in the case of porosity, it attains very less values. However, increasing values of both Darcy number and porosity value, increases the radial velocity.

For a fixed distance   0.1 from the disk, the dimensionless radial velocity E , for the combined effect of rotation and porosity value (chapter 5), is 0.987 1006,

0.156103 and 0.944103 for different porosity values   0.01, 0.02 and 0.03, respectively. It is observed that these values are significantly less even to the case of porosity alone for corresponding values of  .

The non-dimensional radial velocity E , for the effect of rotation and MFD viscosity both (chapter 6), is 0.04533, 0.04600 and 0.04658 for viscosity parameter k  1.1, 1.2 and 1.3, respectively at non-dimensional distance   0.1 from the disk surface which are comparable to the results obtained for rotational effect only. Meaning thereby, MFD viscosity does not affect the radial velocity appreciably when taken along with the effect of rotation.

For the effect of rotation, MFD viscosity and porosity (chapter 7), at , the dimensionless radial velocity, for porosity value   0.01 and MFD viscosity parameter and 1.3, is 2.43106 , 5.19106 and 9.92106 , respectively. If we compare this case (effect of rotation, MFD viscosity and porosity) to all other cases individually, the dimensionless radial velocity gets very less values. If we increase to   0.02 , for the effect of rotation, MFD viscosity and porosity; becomes

0.256103 , 0.388103 and 0.555103 for MFD viscosity parameter and

respectively.

The maximum value (0.08894) for non-dimensional radial velocity for the effect of rotation is attained at   0.4 and the minimum value  0.03834 is attained at   2.19 in the negative region. In the case of porous medium; the peak value of dimensionless radial velocity is 0.41061, 0.42345, 0.43636 and 0.44932 for Darcy number   0, 1, 2 and 3, is attained at  1.25, 1.26, 1.27 and 1.28, respectively, whereas under the effect of porosity only, for porosity values and 0.03; the peak value of dimensionless radial velocity is 0.00178, 0.00356 and 0.00535 at

  0.008, 0.017 and 0.025, respectively (near the disk).

For the combined effect of rotation and porosity, the non-dimensional radial velocity attains the peak value 0.00174, 0.00343 and 0.00505, for porosity values

and 0.03, at   0.0082, 0.0159 and 0.0231, respectively, whereas for the combined effect of rotation and MFD viscosity, the peak value is 0.09520, 0.10552

92 and 0.11638, for different MFD viscosity parameter k 1.1, 1.2 and 1.3, at   0.4, 0.45 and 0.5, respectively.

For the effect of rotation, MFD viscosity and porosity, the maximum value for dimensionless radial velocity for porosity   0.01 is 0.00192, 0.00209 and 0.00227 at

  0.009, 0.01 and 0.011 for different MFD viscosity parameter 1.2 and respectively. If we increase the porosity value from to   0.02 ; the peak value becomes 0.00377, 0.00411 and 0.00446, at   0.018, 0.019 and 0.021,for MFD viscosity parameter 1.2 and respectively.

Hence on comparison, we see that maximum value of dimensionless radial velocity (0.05433) is achieved the porous medium for Darcy number   3 at a distance  1.27 from the disk surface. Meaning thereby, the convergence of radial velocity is slow in porous medium than in the case of rotation, MFD viscosity and porosity.

Dimensionless Tangential Velocity Profile

The dimensionless tangential velocity (chapter 2) at   0.1 (near to the disk) is

0.93831 for the effect of rotation, whereas at same value of  , for Darcy number   0,

1, 2 and 3, the dimensionless tangential velocity is 0.93817, 0.94289, 0.94762 and 0.95235, respectively. At the same distance from the disk surface, for porosity values

  0.01, 0.02 and 0.03, the dimensionless tangential velocity is 0.87644, 0.97790 and 0.98972, respectively. Therefore, at a fixed distance from the disk surface, if we compare these three cases to each other, the dimensionless tangential velocity is highest for the porosity value   0.03.

For the combined effect of rotation and porosity, for the porosity values

0.02 and the dimensionless tangential velocity is 0.00056, 0.04611 and 0.18868, respectively, whereas for the combined effect of rotation and MFD viscosity; F is

0.93833, 0.93834 and 0.93831, for MFD viscosity parameter 1.2 and respectively which is much comparable to the case of rotation only.

At   0.1, the dimensionless tangential velocity, for the effect of rotation taken together with MFD viscosity and porosity, is and for the porosity values 0.02 and respectively. These values are exactly same

93 as the values for the effect of rotation and porosity i.e. there is no effect of MFD effect viscosity on dimensionless tangential velocity when taken together with rotation and porosity. Meaning thereby, near to the disk, the effect of MFD viscosity are subdued when taken together with the effect of rotation and rotation.

Further the trend of the dimensionless tangential velocity is almost similar in all the cases except for the case of porous medium, where there is significant variation near the surface of the disk.

Dimensionless Axial Velocity Profile

The dimensionless axial velocity for the effect of rotation at   0.1, is  0.00481 whereas in case of porous medium, it becomes  0.00534 and for porosity values

  0.01, 0.02 and 0.03, it reaches to the value  0.10571,  0.01457 and  0.00383, respectively. If we compare these three cases to each other, we noticed that the dimensionless axial velocity is minimum (magnitude) for porosity value   0.03 at a fixed distance   0.1 from the disk surface. Also, here we see that G attains larger value (magnitude) in case of porous medium than the case of rotation only.

For the combined effect of rotation and porosity; at the dimensionless axial velocity is  0.88550,  0.84453 and  0.71516, for and respectively, whereas for the combined effect of rotation and MFD viscosity, it becomes  0.00486,  0.00486 and  0.00487, for MFD viscosity parameter k 1.1, 1.2 and

1.3, respectively. If we compare the case of rotation and porosity to the case of rotation and MFD viscosity, it is noticed that dimensionless axial velocity converges rapidly in case of rotation and porosity. If we compare the effect of rotation only to the effect of rotation and MFD viscosity, it is observed that there is negligible effect of MFD viscosity parameter on dimensionless axial velocity.

Whereas, for the combined effect of rotation, MFD viscosity and porosity; the non-dimensional axial velocity for porosity value   0.01, is  0.88488,  0.88380 and  0.88212, for MFD viscosity parameter 1.2 and respectively. On comparing the effect of rotation together with MFD viscosity and porosity to that of “rotation & porosity” and “rotation & MFD viscosity”, we conclude that the effect of porosity is dominant and the effect of MFD viscosity is dormant when porosity and MFD viscosity are taken with effect of rotation. Because of the values for the combined effect

94 of rotation, MFD viscosity and porosity is nearer to the combined effect of “rotation & porosity” and “rotation & MFD viscosity”.

Dimensionless Pressure Profile

At a distance   0.1 from the disk surface, for the effect of rotation, the dimensionless pressure P is  0.08892 , whereas in case of porous medium, for Darcy number   0, 1, 2 and 3 is 1.21582, 1.21567, 1.21553 and 1.21538, respectively.

For the porosity values   0.01, 0.02 and 0.03 ; at , becomes  0.34868,  0.31694 and  0.25540, respectively. Thus, dimensionless pressure decreases with increasing values of Darcy number while for porosity value, it behaves conversely.

For the combined effect of rotation and porosity, at , is  0.00200,

 0.01560 and  0.06295 for   0.01, and 0.03, respectively, whereas for the combined effect of rotation and MFD viscosity for the same value of  , the dimensionless pressure is  0.09972,  0.11041 and  0.12115 for MFD viscosity parameter k 1.1, 1.2 and 1.3, respectively.

Dimensionless pressure, for the effect of rotation, MFD viscosity and porosity, at

and , is  0.00054,  0.00125 and  0.00258, for MFD viscosity parameter and respectively, whereas for the same case, if we increase the porosity value from to 0.02 ; the dimensionless pressure becomes  0.02812,  0.04658 and  0.072198, for MFD viscosity parameter 1.2 and

respectively. Similarly, it is noticed that if we increase the porosity value from 0.02 to 0.03; it becomes  0.09851,  0.11477 and  0.20244, for MFD viscosity parameter

1.2 and respectively. From here we can conclude that for the same value of MFD viscosity parameter at a fixed distance from the disk surface, the dimensionless pressure increases (in magnitude).

If we compare all discussed cases with each other, at a fixed distance from the disk surface, the dimensionless pressure is maximum for the case of porous medium and minimum for the case of porosity value. Also the effect of porosity taken along with rotation increases the pressure while MFD viscosity along with effect of rotation decreases it.

Detailed account of Boundary layer displacement thickness for different cases (rotation, porous medium, porosity, MFD viscosity)

1. Effect of rotation d 1.345615 2. Effect of porous medium (Darcy Number  )

d1 1.272583 for   0

d2 1.829646 for  1

d3  2.386710 for   2 3. Effect of porosity value 

d1  0.023399 for   0.01

d2  0.046568 for   0.02

d3  0.069507 for   0.03 4. Effect of rotation with porosity for

d2  0.046568 for for

5. Effect of rotation with MFD viscosity

d1 1.413156 for k 1.1

d2 1.422407 for k 1.2

d3 1.430234 for k 1.3 6. Effect of rotation with MFD viscosity & porosity for

for

From above, we can conclude that the effect of rotation when taken along with MFD viscosity, increases the displacement thickness of boundary layer while boundary layer displacement thickness becomes very thin when effect of rotation and porosity are taken together i.e. porosity shows converse effect. Also when rotation, MFD viscosity and porosity, all three are taken together, the effect of porosity dominates over the effect of MFD viscosity.

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APPENDIX

Details of transformation of the system of PDE (page no. 28) to the system of ODE (page no. 29) under chapter 2:

Using similarity transformations (2.6a -2.6d) in equation (2.1), we have

E( ) E( ) G ( ) 0

G ( ) 2E( ) 0 … (2.1.1)

Using similarity transformations (2.6a -2.6d) in equation (2.2), we get

r 2 2 F 2 ( ) r E( ) E( ) G( ) r E ( ) r

1 p r 2 M H 0 0 E ( ) 2 r F( ) r 0 r … (2.2.1)

1 p Using boundary layer approximation M H r 2 ; equation (2.2.1) r 0 r becomes

r 2 2 F 2 ( ) r E( ) E( ) G( ) r E ( ) r

r 2 r 2 0 0 E ( ) 2 r F( )

r 2 E 2 ( ) r 2G( )E ( ) r 2 F 2 ( ) r 2 r 2 E ( ) 2r 2 F( )

2 2 E ( ) G( )E ( ) E ( ) F ( ) 2F( ) 1 0 … (2.2.2)

Using similarity transformations (2.6a -2.6d) in equation (2.3), we get

r 2 2 E( )F( ) r E( ) F( ) G( ) r F ( ) r

0 0 r F ( ) 2 r E( )

r 2 E( )F( ) r 2 E( )F( ) r 2G( )F ( ) r 2 F ( ) 2r 2 E( ) F ( ) G( )F ( ) 2E( )F( ) 2E( ) 0 … (2.3.1)

Using similarity transformations (2.6a -2.6d) in equation (2.4), along with H 0 as small z variation of magnetic field in z-direction is assumed, we get

1 0 G( ) G ( ) P ( ) 0 0 0 G ( )

1 3 1 3 1 3 2 2G( )G ( ) 2 2 P ( ) 2 2G ( )

P ( ) G( )G ( ) G ( ) 0 … (2.4.1)

**************