I= ROTATING DISK IN NON-NEWTONIAN FLUIDS
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Title of Thesis
ROTATING DISK IN NON-NEWTONIAN FLUIDS

Author(s)
Zahid Hussain
Institute/University/Department Details
Department of Mathematics/ Quaid-i-Azam University Islamabad
Session
2006
Subject
Mathematics
Number of Pages
115
Keywords (Extracted from title, table of contents and abstract of thesis)
rotating disk, non newtonian fluids, navier-stokes equations, biological fluids, polymeric liquids, suspensions, liquid crystals, newtonian model, magnetohydrodynamic, third grade fluids, second grade fluids, porous disk

Abstract
The Navier-Stokes equations are a good model for the flow of a wide class of fluids. The underlying constitutive model is a linear Newtonian relationship between shear stress and shear rate governed by the fluidā€™s (constant) viscosity. Known exact solutions of Navier-Stokes equation are few in number. This is in general due to the non-linearities which occur in the inertial part of these equations. However, many flow situations of interest are such that a number of terms in the equations of motion either disappear automatically or may be neglected, and the resulting equations reduce to a form that can be readily solved. There are many fluids with complex microstructure such as biological fluids, as well as polymeric liquids, suspensions, liquid crystals which are used in current industrial processes and show non-linear viscoelastic behavior that can not be described by the classical linearly Newtonian model. For such fluids the non-linearities occur not only in the inertial part but also in viscosity part of the governing equations. As a result, the number of exact solutions become rare as compared to the analytical solutions of Navier Stokes equations. But the inadequacy of the Navier-Stokes theory to describe rheological complex fluids has led to the development of several theories of non-Newtonian fluids. Amongst these, the fluids of second and third-grade have acquired a special status.

Another important aspect in the study of fluid dynamics is the consideration of magnetohydrodynamic (MHD) situation. Magnetic fields are used in many engineering applications that involve electrically conducting fluids. They are employed, for example, to drive flows, induce stirring, levitation or to suppress turbulence. Keeping the above aspects into account, this thesis consists of five chapters. In chapter zero there is a development of history and introduction of the work given in this thesis. Chapter one consists of some basic ideas and equations which will be used subsequently. The modeling of the equations which govern the MHD flow of third grade is also presented.

Exact solutions of two flow problems, namely unsteady MHD flow due to eccentric rotations of porous disk and an oscillating fluid at infinity and unsteady MHD flow due to eccentric rotations of porous disk and a fluid at infinity oscillating with different frequencies are obtained in chapter 2. It is observed that the boundary layer thickness decreases by increasing magnetic and suction parameters.

The viscous flow analysis of chapter 2 has been extended to the case of second grade fluid in chapter 3. The exact solutions of both problems are obtained using Laplace transform and perturbation techniques. It is found that boundary layer thickness increases with the increase in material parameter of the second grade fluid. The values where the second grade parameter starts influencing the velocity are determined.

Chapter 4 is devoted to the solutions of the problems presented in chapter 3 for the third grade fluid. The governing partial differential equations are non-linear for which closed form solution is impossible. Thus numerical solutions are developed. It is noted that boundary layer thickness increases with the increase in material parameter of third grade fluid.

The unsteady MHD flows due to non-coaxial rotations of a porous disk with variable axes of rotation and a fluid at infinity for the viscous and second grade fluid are presented in chapter 5. It is found that increase in suction and magnetic field causes reduction in the boundary layer thickness and blowing enhances the layer thickness in comparison to that of suction.

Download Full Thesis
2246 KB
S. No. Chapter Title of the Chapters Page Size (KB)
1 0 Contents
148.34 KB
2 1 Introduction 3
114.69 KB
3 2 Preliminaries And Governing Equations 7
202.43 KB
  2.1 Newtonian And Non-Newtonian Fluids 7
  2.2 Thermodynamic Second And Third Grade Fluids 8
  2.3 Equation Of Continuity 10
  2.4 The Equation Of Motion For MHD Fluid 10
  2.5 Equations Of Motion For MHD Third Grade Fluid 12
  2.6 Governing Equation For MHD Flow Due To Non-Coaxial Rotations Of Porous Boundary And Thermodynamic Third Grade Fluid 18
4 3 Unsteady MHD Flow Due To Eccentric Rotations Of A Porous Disk And A Viscous Fluid At Infinity 20
584.37 KB
  3.1 Unsteady MHD Flow Due To Eccentric Rotations Of A Porous Disk And An Oscillating Fluid At Infinity 21
  3.2 Unsteady MHD Flow Due To Eccentric Rotations Of Porous Disk And A Fluid At Infinity Oscillating With Different Frequencies 34
5 4 Unsteady MHD Flow Induced By Eccentric Rotations Of A Porous Disk And A Second Grade Fluid 47
601.55 KB
  4.1 MHD Flow Due To Eccentric Rotations Of A Porous Disk And An Oscillating Second Grade Fluid At Infinity 48
  4.2 MHD Flow Due To Eccentric Rotations Of A Porous Disk And A Second Grade Fluid At Infinity Oscillating With Different Frequencies 63
6 5 Unsteady MHD Flow Induced By Non-Coaxial Rotations Of A Porous Disk And A Third Grade Fluid 76
436.66 KB
  5.1 MHD Flow Caused By Non-Coaxial Rotations Of A Porous Disk And An Oscillating Third Grade Fluid At Infinity 76
  5.2 MHD Flow Caused By Non-Coaxial Rotations Of A Porous Disk And A Third Grade Fluid At Infinity Oscillating With Different Frequencies 90
7 6 MHD Flow With Variable Axis Of Rotation 101
304.96 KB
  6.1 Unsteady MHD Flow Due To Non-Coaxial Rotations Of A Porous Disk With Variable Axes Of Rotation And A Fluid At Infinity 101
  6.2 Unsteady MHD Second Grade Flow Due To Non Coaxial Rotations Of A Porous Disk With Variable Axes Of Rotation And A Fluid At Infinity 106