I= SOME FLOW PROBLEMS OF AN OLDROYD FLUID: HOMOTOPY ANALYSIS METHOD AND NUMERICAL COMPUTATION
Pakistan Research Repository Home
 

Title of Thesis
SOME FLOW PROBLEMS OF AN OLDROYD FLUID: HOMOTOPY ANALYSIS METHOD AND NUMERICAL COMPUTATION

Author(s)
Masood Khan
Institute/University/Department Details
Department of Mathematics/ Quaid-i-Azam University
Session
2004
Subject
Number of Pages
206
Keywords (Extracted from title, table of contents and abstract of thesis)
oldroyd fluid, homotopy, non-newtonian fluid, newtonian fluid, fluid dynamics, rate type fluids, equation of continuity, plane couette flow,plane poiseuille flow, generalized couette flow

Abstract
In recent years the study of non-Newtonian fluids has become of increasing importance. This is mainly due to their wide use in chemical process industries, food industry, construction engineering, petroleum production, power engineering, commercial and technological applications. The rheological concept for such fluids is of special importance owing to its application to many engineering problems. Rheological properties of fluids are specified, in general by their so-called constitutive equations. Because of the non-linear relationship between stress and the rate of strain, the analysis of the behavior of the fluid motion of the nonNewtonian fluids tends to be really complicated and subtle in comparison with that of the Newtonian fluids.

The governing equation that describes the flow of a Newtonian fluid is the Navier-Stokes equation. For the flow of non-Newtonian fluids there is not a single governing equation which describes all of their properties and thus these fluids cannot be described as Newtonian fluids. Therefore, many non-Newtonian models or constitutive equations have been proposed, most of them are empirical and semi-empirical. The most important models for non-Newtonian fluids are the differential type and the rate type. However, the differential type fluids are incapable of predicting stress relaxation and therefore, have not been so successful for describing the flows of some polymeric suspensions. Of the numerous models that abound in non-Newtonian mechanics one that captures stress relaxation aspects of polymer suspensions, albeit only in a qualitative manner, and hence been accorded much scrutiny is rate type model which includes Oldroyd fluids. The most popular being an Oldroyd-B fluid (3-constant model). But unfortunately this fluid does not exhibit the non-Newtonian characteristics for steady unidirectional flow. Because of this shortfall, some steady flows may be well described by Oldroyd 6- and 8-constant fluids, which do show the non-Newtonian characteristics.

Another important aspect in the study of fluid dynamics is the consideration of slip boundary condition. The no-slip condition is one of the central tenets of Navier-Stokes theory and the modern fluid dynamics. In the literature much of the analysis relies on no-slip condition. This is convenient, and somewhat surprising, but hides the issue of whether the adherence boundary condition is infect a valid model of the fluid-boundary interaction, and if not, how to determine and apply more realistic boundary condition. The adherence condition is inadequate when one considers non-Newtonian fluids such as polymer melts which exhibit macroscopic wall slip and that in general is governed by a nonlinear non-monotone relation between the slip velocity and traction. This may be important factor in sharkskin, spurt and hysteresis effects. The existing theory for non-Newtonian fluids with wall slippage is scant.

Motivated by these facts, the entire work in this thesis is divided into seven chapters. Chapter 0 is introductory in nature. The basic preliminaries regarding rate type fluids, governing laws and homotopy analysis method (HAM) are presented in chapter 1. In chapter 2, the governing non-linear differential equation for an Oldroyd 6-constant fluid has been modeled. Also, the analytical solutions of non-linear equation for three fundamental flows (namely Couette, Poiseuille and generalized Couette) have been accomplished using HAM. It is noted that not only the non-Newtonian parameters appear in the solution even for steady flow which is distinct from the case of Oldroyd 3-constant fluid but the solution is strongly dependent on these emerging non-Newtonian parameters. Chapter 3 is prepared to see the effects of slip on the flows considered in chapter 2. The analysis of this chapter treats the solvability of non-linear equation subject to non-linear conditions. The slip effects have been shown and analytical solutions are obtained. It is found that with the slip boundary condition, the reduced shearing force from the boundaries causes the velocity to become flatter than for no-slip boundary condition.

In chapters 4 and 5, the flows of magnetohydrodynamic (MHD) Oldroyd 8-constant fluid are considered. These analyzes are important because the interaction of electromagnetic field with fluids gained importance in view of its promising applications in areas like nuclear fusion, chemical engineering, high speed noiseless printing, transformer cooling oil, etc. Chapter 4 describes the MHD flow analysis for three shearing flows which involve Oldroyd 8-constant fluid. Both analytical and numerical solutions have been given and compared. It is observed that with an increase of the magnitude of the magnetic field the flow velocity decreases monotonically due to the effect of the magnetic force. In chapter 5, the MHD flow of an Oldroyd 8-constant fluid between two coaxial cylinders is discussed analytically as well as numerically. It is noted that HAM and numerical solutions are in good agrement. Finally, the conclusions are presented in chapter 6.

Download Full Thesis
10430.26 KB
S. No. Chapter Title of the Chapters Page Size (KB)
1 0 Contents
349.25 KB
2 1 Introduction 4
659.34 KB
3 2 Preliminaries 13
938.34 KB
  2.1 Non-Newtonian Fluids 13
  2.2 Rate Type Fluids 14
  2.3 Equation of Continuity 20
  2.4 Equation of Motion for a Magnetohydrodynamic Fluid 21
  2.5 Homotopy 23
  2.6 Methods of Solution 23
4 3 Some Fundamental Shearing Flows of an Oldroyd 6-Constant Fluid 28
1241.73 KB
  3.1 Mathematical Modelling 28
  3.2 Simple Plane Shearing Flows 32
  3.3 Plane Couette Flow 33
  3.4 Plane Poiseuille Flow 40
  3.5 Generalized Couette Flow 44
  3.6 Results and Discussion 48
5 4 Effects of Slip Condition on Non-linear Flows 57
754.23 KB
  5.1 Governing Flows 57
  5.2 Plane Couette Flow 58
  5.3 Plane Poiseuille Flow 59
  5.4 Generalized Couette Flow 59
  5.5 Solutions of Shearing Flows 60
  5.6 Results and Discussion 66
6 6 MHD Flows of an Oldroyd 8-Constant Fluid 77
4588.3 KB
  6.1 Description of the Governing Equation 78
  6.2 Plane Couette Flow 81
  6.3 Plane Poiseuille Flow 86
  6.4 Generalized Couette Flow 91
  6.5 Numerical Method 91
  6.6 Numerical Results and Discussion 94
  6.7 Flow of an Oldroyd 8-Constant Fluid between Coaxial Cylinders 106
  6.8 The Constitutive Model 106
  6.9 Physical Model 108
  6.10 Explicit Analytic Solution 110
  6.11 Numerical Method 115
  6.12 Numerical Results and Discussion 116
  6.13 Conclusions 123
  6.14 Bibliography 127