I= FLOWS OF THIRD GRADE FLUID IN A ROTATING FRAME
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Title of Thesis
FLOWS OF THIRD GRADE FLUID IN A ROTATING FRAME

Author(s)
Muhammad Mudassar Gulzar
Institute/University/Department Details
Department of Mathematics/ Quaid-i-Azam University Islamabad
Session
2006
Subject
Mathematics
Number of Pages
133
Keywords (Extracted from title, table of contents and abstract of thesis)
third grade fluid, rotating frame, fluid mechanics, newtonian fluid, navier-stokes equations, biological fluids, polymeric liquids, suspensions, liquid crystals, non-newtonian models, oscillating flows

Abstract
There are many models which are used to investigate different types of fluid mechanics problems. It is difficult to characterize in general way all necessary requirements since each problem is unique. However, we can broadly classify many of the problems on the basis of the general nature of the flow and the fluid and subsequently develop some general characteristics of model designs in each of these classifications. Amongst these models, the model of Newtonian fluid is the simplest one for which the Navier-Stokes equations can describe the flow problem. However, 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 behaviour that cannot be characterized by Navier-Stokes equations. Because of the fluids complexity, many constitutive equations have been proposed. The non-Newtonian models that have been developed to describe the other rheological characteristics can be classified under the following three categories: fluids of differential type, rate type and integral type. Amongst these types, the differential fluids have received the special attention from the recent researchers in order to describe the several non-standard features such as normal stress effects, rod climbing, shear thinning and shear thickening. The governing equations for such fluids are more non-linear and higher order than the Navier-Stokes equations.

In the literature much attention has been focused on the flows of second grade fluid which is simplest subclass of differential type fluids. The second grade fluid model is able to predict the normal stress differences but it does not take into account the shear thinning or shear thickening phenomena that many fluids show. The third grade fluid model represents a further, although inconclusive attempt toward a more comprehensive description of the behaviour of non-Newtonian fluids. Due to this fact in mind, the model in the present thesis is a third grade.

Another aspect in the study of non-Newtonian fluids is the slip boundary condition. Although there are rigorous mathematical researches on flows of Newtonian fluids with slip condition but due attention has not been given to flows of non-Newtonian fluids with slip condition. The non-Newtonian fluids such as polymer melts often exhibit macroscopic wall slip governed by a nonlinear and non-monotone relation between the slip velocity and the traction. The fluids that exhibit boundary slip are important from technological point of view for example, the polishing of artificial heart valves.

Keeping the above facts in view, the present thesis is organized as follows:

Chapter zero provides the introduction of the thesis. Basic preliminaries relevant to non-Newtonian fluids, governing laws and techniques are given in Chapter one. Equation which governs the rotating flow of a third grade fluid over a porous surface is also modeled here. Chapter two describes the steady flow of a third grade fluid in a rotating frame by using no-slip condition. The same problem has been solved employing another set of dimensionless variables for the influence of dynamic viscosity. Later, this problem is solved using partial slip boundary condition. Chapter three describes the oscillatory rotating flow of a third grade fluid passed a porous plate. An asymptotic solution has been obtained. Two cases of no-slip and partial slip have been considered. Homotopy analysis method is used to obtain the analytic solutions for the problems in chapters two and three. Convergence of the obtained solutions developed in these chapters is also ensured. Chapter four has been prepared for the numerical solutions of the two partial slip boundary value problems. A reasonable agreement between the HAM and numerical solutions is presented through graphs. The concluding remarks are made at the end of each Chapter. However, a brief summary of the important results from the thesis has been included in Chapter five.

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11925.02 KB
S. No. Chapter Title of the Chapters Page Size (KB)
1 0 Contents
996.13 KB
2 1 Introduction 6
857.85 KB
3 2 Preliminaries 13
2146.5 KB
  2.1 Non-Newtonian Fluids 13
  2.2 Equation Of Continuity 15
  2.3 The Momentum Equation 16
  2.4 Constitutive Equation For A Third Grade Fluid 17
  2.5 The Governing Equation For A Third Grade Fluid In A Rotating System 19
  2.6 Partial Slip Boundary Conditions 23
  2.7 Mathematical Techniques 26
4 3 Steady Flow Of A Third Grade Fluid In A Rotating Frame 37
2411.81 KB
  3.1 Mathematical Problem For Non-Slip Case 37
  3.2 Mathematical Problem For The Partial Slip Case 61
5 4 Oscillating Flows Of A Third Grade Fluid In A Rotating Frame 74
1745.09 KB
  4.1 Problem Formulation For The No-Slip Case 74
  4.2 Problem Formation For The Partial Slip Case 88
6 5 Numerical Solutions For Rotating Flows Of A Third Grade Fluid With Partial Slip 100
1430.58 KB
  5.1 Steady Flow Past A Porous Plate 101
  5.2 Oscillating Flow Past A Porous Plate 111
7 6 Conclusion 123
293.26 KB
8 7 References 126
2226.23 KB