I= ACCELERATED FLOWS OF VISCOELASTIC FLUIDS WITH NO-SLIP AND PARTIAL SLIP CONDITIONS
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Title of Thesis
ACCELERATED FLOWS OF VISCOELASTIC FLUIDS WITH NO-SLIP AND PARTIAL SLIP CONDITIONS

Author(s)
Khalid Hanif
Institute/University/Department Details
Department of Mathematics/ Quaid-i-Azam University
Session
2004
Subject
Mathematics
Number of Pages
150
Keywords (Extracted from title, table of contents and abstract of thesis)
accelerated flows, viscoelastic fluids, no-slip conditions, partial slip conditions, non-newtonian fluids, magnetohydrodynamic problems, magnetohydrodynamic, variable accelerated flows, maxwell's equations

Abstract
There has been a great deal of interest in understanding the behaviour of non-Newtonian fluids as they are used in various branches of science, engineering and technology: particularly in material processing, chemical industry, geophysics and bie-engineering. The study of non-Newtonian fluid flow is also of significant interest in oil reservoir engineering. Moreover, the non-Newtonian fluids such as mercury amalgams, liquid metals, biological fluids, plastic extrusions, paper coating, lubrication oils and greases have applications in many areas with or without magnetic field. Many magnetohydrodynamic problems of practical interest involving fluids as a working medium have attracted engineers, physicists and mathematicians alike. These problems are challenging because of non-linearity of the governing equations, field coupling, and complex boundary conditions. Further, using Newtonian fluid models to analyse, predict and simulate the behaviour of viscoelastic fluids has been widely adopted in industries. However, the flow characteristics of viscoelastic fluids are quite different from those of Newtonian fluids. This suggests that in practical applications the behaviour of viscoelastic fluids cannot be represented by that of Newtonian fluids. Hence, it is necessary to study the flow behaviour of viscoelastic fluids in order to obtain a thorough cognition and improve the utilization in various manufactures. Due to complexity of fluids in nature, non-Newtonian fluids are classified on the basis of their behaviour in shear. Amongst the many fluid models which have been used to describe the visco elastic behaviour exhibited by these fluids, the fluids of second and third grades have received a special attention. The major attraction of these fluid models is due to their popularity and the fact that they are derived from the first principle. Unlike many other phenomenological models, there are no curve-fittings or parameters to adjust for these models. Though, in both of these grade models, there are material properties that need to be measured. Also, the second grade fluid is a subclass of nonNewtonian fluids for which one can reasonably hope to obtain an analytical solution.

Another important aspect in fluid mechanics is the consideration of partial slip condition. One of the cornerstones on which the fluid mechanics is built is the no-slip condition. But, there are situations wherein this condition does not hold. In certain cases, partial slip between the fluid and the moving surface may occur. Mention may be made to the situations when the fluid is particulate such as emulsions, suspensions, foams and polymer solutions. However, literature for non-Newtonian fluids with wall slippage is scarce.

Keeping the above facts in mind, this thesis has been organized offering five chapters. Chapter zero is introductory. In chapter one, the basic equations and mathematical techniques are included for the succeeding chapters. The modeling of the general equation which govern the magnetohydrodynamic (MHD) flow of a third grade fluid is also given. Chapter two deals with the MHD flows due to non-coaxial rotations of a porous disk and a viscous fluid at infinity. Three types of unsteady flows namely, the flows induced by a constant accelerated disk with no-slip and partial slip and the flow due to variable accelerated disk with no-slip. Exact analytical solutions are constructed using Laplace transform technique. It is noted that in presence of partial slip, the reduced shearing force from the boundary causes the velocity to become flatter than that for no-slip case. Moreover, the velocity profiles in case of constant accelerated flow are greater than for the variable accelerated case for all values of time less than one. However, this situation is quite reverse for all times greater than one. Chapter three is devoted to the flows of a second grade fluid generated by a constant accelerated disk with no-slip and partial slip conditions. The influence of second grade parameters arises in the governing equation and the boundary conditions. Both analytical and numerical solutions are given and are compared for the no-slip case. But only the numerical solution is obtained for the partial slip case. It is worth noting that material parameter of the second grade fluid reduces the velocity profiles. In chapter four, the constant accelerated flows of a third grade fluid with no-slip and partial slip have been presented. The analysis of this chapter involves the solvability of a non-linear equation. Also, the boundary condition in partial slip situation is non-linear. Numerical solutions are given using the Crank-Nicolson scheme with modification. The objective of chapter five is to extend the contents of chapter four to the case of variable accelerated flows. The influence of acceleration against time in third grade fluid is found to be smaller than that of Newtonian fluid.

Download Full Thesis
4786.21 KB
S. No. Chapter Title of the Chapters Page Size (KB)
1 0 Contents
341.17 KB
2 1 Introduction 4
420.93 KB
3 2 Preliminaries 12
840.62 KB
  2.1 Non-Newtonian Fluid 14
  2.2 Equation Of Continuity 16
  2.3 Maxwell's Equations 17
  2.4 The Momentum Equation 18
  2.5 General Equations Of Motion For M H D Third Grade Fluid 19
  2.6 Partial Slip Boundary Condition 22
  2.7 Boundary Layer 25
  2.8 Mathematical Techniques 29
4 3 Unsteady Flows Of A Viscous Fluid Due To An Accelerated Disk 35
822.96 KB
  3.1 Constant Accelerated Flow With No-Slip Condition 36
  3.2 Constant Accelerated Flow With Partial Slip Condition. 39
  3.3 Variable Accelerated Flow With No-Slip Condition 46
5 4 Unsteady Flows Of A Second Grade Fluid Due To Constant Accelerated Disk 63
890.3 KB
  4.1 Flow With No-Slip Condition 64
  4.2 Flow With Partial Slip Condition 83
6 5 Transient Flows Of A Third Grade Fluid Due To Constant Accelerated Disk 94
829.59 KB
  5.1 Flow With No-Slip Condition 95
  5.2 Flow With Partial Slip Condition 109
7 6 Transient Flows Of A Third .Grade -Fluid Due To Variable Accelerated Disk 120
813.59 KB
  6.1 Flow With No-Slip Condition 120
  6.2 Flow With Partial Slip Condition 129
  6.3 Concluding Remarks 137