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Title of Thesis

New Results in the Theory of Ordinary and Generalized Non-Newtonian Fluids

Author(s)

Muhammad Kamran

Institute/University/Department Details
Abdus Salam School of Mathematical Sciences / GC University, Lahore
Session
2011
Subject
Mathematics
Number of Pages
101
Keywords (Extracted from title, table of contents and abstract of thesis)
Asymptotic, Fractional, Equations, Results, Fluids, Finite, Motion, Generalized, Newtonian, Cases, General, Ordinary, New, Theory

Abstract
This thesis concerns with the results regarding the flow behavior of some non- Newtonian fluids under different circumstances.First of all, some concepts regarding Newtonian and non-Newtonian fluids, constitutive equations, equations of motion, and integral transforms have been discussed.Then the exact solutions for the velocity field and the shear stress corresponding to some flows with technical relevance have been established for second grade, Maxwell, and Oldroyd-B fluids with fractional derivatives model.
In Chapter 2, the velocity field and the adequate shear stress, corresponding to the flow of a second grade fluid with fractional derivatives in an annular region, due to a constant/time-dependent shear stress, are determined by means of the Laplace and the finite Hankel transforms.The corresponding solutions for a second grade and Newtonian fluids, performing the same motion, are obtained from our general solutions.
Chapter 3 deals with the motion of a Maxwell fluid with fractional derivatives, and we studied the flow starting from rest due to the sliding of the cylinder along its axis with a constant acceleration.The velocity and the adequate shear stress, obtained by means of the finite Hankel and Laplace transforms, are presented under series form in terms of the generalized G functions.The similar solutions for the ordinary Maxwell fluid, performing the same motion, are obtained as special cases of our general solution.
Chapter 4 concerns with the unsteady flow of an incompressible Oldroyd-B fluid with fractional derivatives, induced by a constantly accelerating plate between two side walls perpendicular to the plate.The solutions have been studied using Fourier sine and Laplace transforms. The expressions for the velocity field and the shear stresses, written in terms of the generalized G and R functions, are presented as sum of the similar Newtonian solutions and the corresponding non-Newtonian contributions. Furthermore, the solutions for Maxwell fluid with fractional derivatives, ordinary Oldroyd-B, Maxwell and Newtonian fluids, performing the same motion, are also obtained as limiting cases of our general solutions. In the absence of the side walls, namely when the distance between the two walls tends to infinity, the solutions corresponding to the motion over an infinite constantly accelerating plate are recovered. Finally, the effect of the material parameters on the velocity profile is spotlighted by means of the graphical illustrations.
Chapter 5 intends to establish exact and approximative expressions for dissipation, the power due to the shear stress at the wall and the boundary layer thickness corresponding to the unsteady motion of a second grade fluid, induced by an infinite plate subject to a shear stress.As a limiting case of our general solutions, the similar results for Newtonian fluids performing the same motion, are obtained. The results that have been here obtained are different of those corresponding to the Rayleigh- Stokes problem.A series solution for the velocity field is also determined. Its form, as it was to be expected, is identical to that resulting from the general solution by asymptotic approximations.

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S. No. Chapter Title of the Chapters Page Size (KB)
1 0 CONTENTS

 

 
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2

1

PRELIMINARIES

1.1 Newtonian and non-Newtonian Fluids
1.2 Constitutive Equations
1.3 Continuity Equation
1.4 Energetic Balance
1.5 Equation of Motion
1.6 Some Integral Transforms

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3 2 AXIAL COUETTE FLOW OF A SECOND GRADE FLUID WITH FRACTIONAL DERIVATIVES DUE TO A LONGITUDINAL SHEAR STRESS

2.1 Introduction
2.2 Governing Equations
2.3 Starting flow due to a longitudinal constant shear
2.4 Starting flow due to a longitudinal timedependent shear
2.5 Conclusions

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4 3 UNSTEADY FLOW OF A MAXWELL FLUID WITH FRACTIONAL DERIVATIVES THROUGH A CIRCULAR CYLINDER

3.1 Introduction
3.2 Governing Equations
3.3 Starting flow through an infinite circular cylinder
3.4 The special case β → 1 (Maxwell fluid)
3.5 Conclusions

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5 4 EXACT SOLUTIONS FOR THE FLOW OF AN OLDROYD-B FLUID WITH FRACTIONAL DERIVATIVES INDUCED BY A SUDDENLY MOVED PLATE BETWEEN TWO SIDE WALLS PERPENDICULAR TO THE PLATE

4.1 Introduction
4.2 Governing Equations
4.3 Statement of the problem and its solution
4.4 Special cases
4.5 Limiting case h → ∞ (flow over an infinite plate)
4.6 Conclusions and numerical results

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6 5 ENERGETIC BALANCE FOR THE FLOW OF A SECOND GRADE FLUID DUE TO A PLATE SUBJECT TO A SHEAR STRESS

5.1 Introduction
5.2 Statement of the problem
5.3 Exact expressions for L, and δ
5.4 Asymptotic approximations for t ≪ 1
5.5 Analogy to the Teipel series expansion
5.6 Conclusions

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7 6 BIBLIOGRAPHY

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