I= NUMERICAL SOLUTIONS OF THE PROBLEMS RELATED TO MICROPOLAR FLUIDS FLOW
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Title of Thesis
NUMERICAL SOLUTIONS OF THE PROBLEMS RELATED TO MICROPOLAR FLUIDS FLOW

Author(s)
Muhammad Ashraf
Institute/University/Department Details
Centre for Advanced Studies in Pure and Applied Mathematics/ Bahauddin Zakariya University Multan
Session
2005
Subject
Mathematics
Number of Pages
239
Keywords (Extracted from title, table of contents and abstract of thesis)
micropolar fluids flow, numerical solutions, newtonian fluids, iterative methods, discritisation, nested iterations, laminar flow, viscous flow, impinging stagnation flow, stagnation point flow

Abstract
The numerical solutions of some problems of micropolar fluids have been undertaken. The flow is assumed to be steady laminar and incompressible. The body force and body couples are neglected. The similarity transformations are used to reduce partial differential equations to ordinary differential equations. The resulting boundary value problems have been solved using appropriate numerical techniques. The results have been calculated on three different grid sizes. The central differences are applied to these differential equations. The difference equations thus obtained are solved by Successive Over Relaxation parameter SOR method. The calculated results are further enhanced for more accuracy using Richardson's extrapolation method. The corresponding equations for Newtonian fluids are also solved for comparison purposes.

The chapter 2 contains some numerical procedures which we used to solve the fluid flow problems. The SOR method is explained by taking the elliptic problem subject to mixed boundary conditions. For the fast rate of convergence, the relaxation parameter ω is optimized. The extrapolation schemes are explained for higher order accuracy. In order to check whether the required accuracy tolerance is obtained at a given extrapolation step, we have described a relation for the local discritisation error.

Typical sets of solutions showing the effects of the cross flow parameter R and the permeability parameter α in case of flow between two porous walls are presented in chapter 3. The behavior of the normal and streamwise velocity profiles is discussed. The position of the viscous layer is determined. The profiles of the microrotation are also presented. The fluid flow problems between two porous disks are studied in chapters 4 & 5. The flow was driven by suction or injection at the two disks. The micropolar model due to Eringen [34] was used to explain the fluid flow phenomenon. The flow due to moving boundaries for micropolar fluids is described in chapters 6-8. The shear stresses at the two walls are calculated for various R. The results of the normal and streamwise velocities are discussed. The two-dimensional flow phenomenon of the micropolar fluid due to accelerating disks is explained in chapter 7 & 8. In chapter 7 the flow is driven by accelerating one or both the disks where as in chapter 8 the flow is driven by asymmetrically accelerating disks. The flow of a micropolar fluid due to rotating disks is described in chapter 9. The stagnation point flows of micro polar fluids for orthogonal and non-orthogonal cases are examined in chapter 10-12. The corresponding results for the Newtonian fluids are presented for each problem considered for comparison purposes. A brief description of the abstract for each problem is given in the beginning of the related chapter.

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S. No. Chapter Title of the Chapters Page Size (KB)
1 0 Contents
86.14 KB
2 1 General Introduction 1
87.27 KB
3 2 Description On The Iterative Methods 10
117.17 KB
  2.1 Sor Method
  2.2 Discritisation
  2.3 Extrapolation Schemes
  2.4 Error Estimation
  2.5 Use Of Nested Iterations
  2.6 Higher Order Solutions At The Fine Grids
  2.7 Summary Of The Algorithm
4 3 Laminar Flow Of A Micripolar Fluid With Large Injection Through Parallel And Uniformly Porous Walls Of Different Permeability 24
301.65 KB
  3.1 Introduction
  3.2 Basic Analysis
  3.3 Finite Difference Equations
  3.4 Computational Procedure
  3.5 Results And Discussions
5 4 Steady Viscous Flow Of A Micripolar Fluid Driven By Injection Between Two Porous Disks 43
285.22 KB
  4.1 Introduction
  4.2 Basic Analysis
  4.3 Finite Difference Equations
  4.4 Computational Procedure
  4.5 Results And Discussions
6 5 On The Flow Of A Micripolar Fluid Between Two Porous Disks Of Different Permeability 61
391 KB
  5.1 Introduction
  5.2 Basic Analysis
  5.3 Finite Difference Equations
  5.4 Computational Procedure
  5.5 Results And Discussions
7 6 Laminar Channel Flow Of A Micripolar Fluid Driven By Accelerating Walls 85
396.82 KB
  6.1 Introduction
  6.2 Basic Analysis
  6.3 Finite Difference Equations
  6.4 Computational Procedure
  6.5 Results And Discussions
8 7 Laminar Flow Of A Micripolar Fluid Driven By One Or Both Accelerating Disks 106
441.57 KB
  7.1 Introduction
  7.2 Basic Analysis
  7.3 Finite Difference Equations
  7.4 Computational Procedure
  7.5 Results And Discussions
9 8 Laminar Flow Of A Micripolar Fluid Asymmetrically Driven By Accelerating Disks 130
564.45 KB
  8.1 Introduction
  8.2 Basic Analysis
  8.3 Finite Difference Equations
  8.4 Computational Procedure
  8.5 Results And Discussions
10 9 Micripolar Asymmetric Flow Due To Rotating Disks 153
402.81 KB
  9.1 Introduction
  9.2 Basic Analysis
  9.3 Finite Difference Equations
  9.4 Computational Procedure
  9.5 Results And Discussions
11 10 Impinging Stagnation Flows Of Micropolar Fluids 175
357.38 KB
  10.1 Introduction
  10.2 Basic Analysis
  10.3 Finite Difference Equations
  10.4 Computational Procedure
  10.5 Results And Discussions
12 11 Stagnation Point Flow And Heat Transfer Of Micropolar Fluids 198
368.46 KB
  11.1 Introduction
  11.2 Basic Analysis
  11.3 Finite Difference Equations
  11.4 Computational Procedure
  11.5 Results And Discussions
13 12 Non-Orthogonal Stagnation Point Flow Of A Micropolar Fluid In Two Dimensions 217
298.53 KB
  12.1 Introduction
  12.2 Basic Analysis
  12.3 Finite Difference Equations
  12.4 Computational Procedure
  12.5 Results And Discussions
  12.6 References 232