Unsteady Magnetohydrodynamic (MHD)
Non-Newtonian Fluid Flows in Porous Medium
BY
Kashif Ali
Thesis submitted for the
Degree of Doctorate of Philosophy
Department of Mathematics
NED University of Engineering & Technology
University Road, Karachi-75270, Pakistan
2018
2
Unsteady Magnetohydrodynamic (MHD)
Non-Newtonian Fluid Flows in Porous Medium
PhD Thesis
By
Kashif Ali
Batch: 2015-2016
Supervisor
Professor Dr. Mirza Mahmood Baig
Co-Supervisor
Dr. Mukkaram Hussain
2018
Department of Mathematics
NED University of Engineering & Technology
University Road, Karachi-75270, Pakistan
3
Statement of Copyright
© 2018 NED University of Engineering &
Technology
This copy of thesis has been supplied under the condition that anyone who consults it, is
understood to recognize that the copyright rests with NED university of Engineering &
Technology and that no quotations from the Thesis and no information derived from it
may be used/published without the permission of the university.
4
Author’s Declaration
I Kashif Ali hereby states that my PhD thesis titled Unsteady Magnetohydrodynamic
(MHD) non-Newtonian Fluid in Porous Medium is my own work and has not been
submitted previously by me for taking any degree from this university.
NED University of Engineering & Technology, University Road, Karachi-75270, Karachi
or anywhere else in the country/world.
At any time if my statement is found to be incorrect even after my Graduate the university
has the right to withdraw my PhD degree.
Kashif Ali
Date: 12-10-2018
5
Plagiarism Undertaking
I solemnly declare that research work present in the thesis titled “Unsteady
Magnetohydrodynamic (MHD) non-Newtonian Fluid in Porous Medium” is solely my
research work with no significant contribution from any other person. Small
contribution/help wherever taken has been duly acknowledge and that complete thesis
written by me.
I understand the zero tolerance policy of the HEC and the university NED University of
Engineering and Technology, University Road, Karachi-75270, Karachi towards
plagiarism. Therefore I as an Author of an above titled thesis declare that no portion of
my thesis has been plagiarized and any material used as reference is properly referred /
cited.
I undertake that if I am found guilty of any formal plagiarism in the above titled thesis
even after award of my PhD degree, the university reserves the rights to withdraw/revoke
my PhD degree and that HEC and the university has the right to publish my name on the
HEC/university website on which names of students are placed who submitted
plagiarized thesis.
Student/Author Signature: _______________
Name: Kashif Ali
6
Certificate of Approval
This is to certify that the research work presented in this thesis entitled “Unsteady
Magnetohydrodynamic (MHD) non-Newtonian Fluid in Porous Medium” was conducted
by Mr. Kashif Ali under the supervision of Professor Dr. Mirza Mahmood Baig.
No part of this thesis has been submitted anywhere else for any other degree. This thesis
is submitted to the NED University of Engineering and Technology, University Road,
Karachi-75270, Karachi in partial fulfillment of the requirements for the degree of Doctor
of Philosophy in the field Applied Mathematics Department of Mathematics NED
University of Engineering and Technology, University Road, Karachi-75270, Karachi.
Student Name: Kashif Ali Signature: _________________
Examination Committee:
a) External Examiner: Dr. Abdul Wasim Shaikh Signature: _________________
Professor & University of Sindh, Jamshoro,
Pakistan.
b) Internal Examiner: Dr. Faheem Raees Signature: _________________
Assistant Professor & NED University of
Engineering and Technology, Karachi
Pakistan.
Supervisor Name: Dr. Mirza Mahmood Baig Signature: _________________
Professor & NED University of Engineering
and Technology, Karachi, Pakistan.
Co-Supervisor Name: Dr. Mukkaram Hussain Signature: _________________
Institute of Space Technology, Karachi, Pakistan.
Name of Dean/HOD: Professor Dr. Noman Ahmed Signature: _________________
Faculty of Information Sciences and Humanities,
& NED University of Engineering and Technology,
Karachi, Pakistan
7
Office of the Controller of Examinations
Notification
No _____________________ Date: ________________
It is notified for the information of all concerned that Mr/Ms: Kashif Ali PhD Scholar of
Department of Mathematics of NED University of Engineering and Technology,
University Road, Karachi-75270, Karachi has completed all the requirements for the
award of the PhD degree in the discipline Applied Mathematics as per detail given
hereunder:
PhD in Education Cumulative Result
Registration
No.
Scholar’s
Name
Father’s
Name
Gcredit Hours Cumulative
Grade Point
Average CGPA Course
Work
Research
Work Total
NED/2683/
2015-2016
Kashif Ali Ali Sher 18 36 54 3.68
Research Topic: Unsteady Magnetohydrodynamic (MHD) non-Newtonian
Fluid in Porous Medium
Local Supervisor-I Name: Professor Dr. Mirza Mahmood Baig
Local Supervisor-II Name: Dr. Mukkaram Hussain
Foreign/External Examiners:
a) Name: Professor Dr. Abdon Atangana
University: University of the Free State, 9300, Bloemfontein, South Africa
Address: Faculty of Natural and Agriculture Sciences, University of the Free
State, South Africa.
b) Name: Assoc. Professor Dr. Sharidan Shafie
University: Univesiti Teknologi Malaysia
Address: Faculty of Science, Univesiti Teknologi Malaysia.
Detail of Research Articles Published on the basis of thesis research work:
8
[1] Kashif Ali Abro, Mukarrum Hussain, Mirza Mahmood Baig, An Analytic Study
of Molybdenum Disulfide Nanofluids Using Modern Approach of Atangana-
Baleanu Fractional Derivatives, European Physical Journal Plus, Eur. Phys. J. Plus
(2017) 132: 439, DOI 10.1140/epjp/i2017-11689-y (2017)
[2] Kashif Ali Abro, Mukarrum Hussain, Mirza Mahmood Baig, A Mathematical
Analysis of Magnetohydrodynamic Generalized Burger Fluid for Permeable
Oscillating Plate, Punjab University Journal of Mathematics, 50(2) 97-111 (2018).
Note: This result is declaration as notice only. Errors and omissions, if any, are subject to
subsequent rectification.
Signed by
Controller of Examinations
9
Dedication
I dedicate this thesis to my loving parents, sisters, brothers, wife, and sons.
i
Acknowledgements
In the name of almighty ALLAH, the most Benevolent, the most Merciful, the creator of
the universe and the master of life and death, who inculcated his countless blessings upon
me to fulfill the requirements of this Ph.D. thesis at NED University of Engineering and
Technology, Karachi, Pakistan. I offer my extremely humblest, sincerest Darood-O-
Salam to our beloved Prophet Hazrat Muhammad (peace be upon him) who is forever a
symbol of complete guidance in every walk of life for humanity.
I would like to express my supreme and the sincerest gratitude and warm thanks to my
supervisor Professor Dr. Mirza Mahmood Baig (Chairman at Department of Mathematics,
NED University of Engineering and Technology, Karachi, Pakistan) for the continuous
support, patience, motivation, enthusiasm, and immense knowledge in completing my
doctorate degree in Applied Mathematics. His guidance helped me throughout my
research process and writing of this thesis. I could not have imagined having a better
advisor, overseer and mentor for my Ph.D. study. His kind support and guidance have
been of great value in this study. My special sincere thanks also goes to my co-supervisor
Dr. Mukkarum Hussain (General Manager at Institute of Space Technology, SUPARCO,
Karachi, Pakistan) for his sympathetic help, care, concern, and continuous contribution in
this achievement. His extensive discussions around my work and interesting explorations
in difficult concepts have been very helpful for this study.
I would also like to record my gratitude to Dr. Mushtaque Hussain (Assistant Professor at
Department of Mathematics, NED University of Engineering and Technology, Karachi,
Pakistan) and Dr. Ilyas Khan (Assistant Professor at Basic Engineering Sciences
Department, College of Engineering Majmaah University, Saudi Arabia) for their
advices, invaluable and invigorating encouragement and support in various ways. Their
elegant personality will always remain a source of inspiration to me. I further
categorically acknowledge all of my honorable teachers (Dr. Muhammad Jamil and Dr.
Azam Khan); without them I would not be able to reach this stage of academic zenith. My
heartiest regards to my colleagues for their continually increasing encouragement,
appreciations, invaluable suggestions, cooperation and to provide me the wonderful
environment. I also want to pay my thanks to all the official staff of Department of
Mathematics particularly Arman Hussain Siddiqui and Furqan Ali for their help in office
work and procedures.
ii
I am failing in my duties and it will be skimpy on my part not to acknowledge the
benevolence of my Mehran University of Engineering and Technology, Jamshoro,
Pakistan for providing me an excellent opportunity to complete my Ph.D. at NED
University of Engineering and Technology, Karachi, Pakistan. It is not possible for me to
name all those who have contributed, directly or indirectly, towards the completion of my
research work. I am grateful to all my well-wishers for their sincere support. I express my
apology to those entire not mentioned personally by me individually. Words wane in
expressing my veneration to my loving, grateful, graceful, delicate and simple parents and
all my family members. I owe my heartiest gratitude for their assistance and never-ending
prayers for my success. I would never have been able to stand today without their
continuous support and generous help.
Department of Mathematics, NEDUET Kashif Ali
12th
October, 2018
iii
Table of Contents
Acknowledgement i
Table of contents iii
List of Figures vi
List of Tables x
Nomenclatures xi
Abbreviations xiv
List of Publications xv
Additional List of Publications xvii
Draft of Thesis xix
Abstract xxi
1. CHAPTER 1 Preliminaries of Fluid Flows and Mathematical Techniques 1
1.1. Introduction 2
1.2. Review of literature 6
1.3. Newtonian and non-Newtonian fluids 13
1.4. Equation of continuity 15
1.5. Magnetohydrodynamics 15
1.6. Porous medium 17
1.7. Nanofluid and nanoparticles 18
1.8. Heat transfer and dimensionless numbers 18
1.9. Fractional derivatives 19
1.10. Special functions 21
1.11. Constitutive equations of fluids 22
1.12. Integral transforms 24
2. CHAPTER 2 Analytic Solutions of MHD Generalized Burger’s Fluid with
Porous Flow 29
2.1. Introduction 30
2.2. Modeling of the governing equations 30
2.3. Accelerated plate with electrically conducting Burger fluid in porous flow 32
2.4. Solution of the problem 32
2.4.1 Calculation of the velocity field 32
2.4.2 Calculation of the shear stress 35
2.5. Limiting cases 36
iv
2.5.1 Solution of Burger fluid 36
2.5.2 Solution of Oldroyd-B fluid 37
2.5.3 Solution of Maxwell fluid 37
2.5.4 Solution of second grade fluid 38
2.6. Results and concluding remarks 38
2.7. Validation of the results 40
3. CHAPTER 3 A Mathematical Analysis of Fractional Generalized Burger’s
Fluid for the Oscillations of Plate with Magnetic Field 47
3.1. Introduction 48
3.2. Modeling of the governing equations 48
3.3. Porous flow of fractional Burger fluid on oscillating plate with magnetic field 50
3.4. Solution of the problem 51
3.4.1 Mathematical analysis of the velocity field 51
3.4.2 Mathematical analysis of the shear stress 52
3.5. Results and concluding remarks 54
3.6. Validation of the results 56
4. CHAPTER 4 Helices of Generalized Burger’s Fluid in Circular Cylinder: A
Caputo Fractional Derivative Approach 62
4.1. Introduction 63
4.2. Modeling of the governing equations 63
4.3. Oscillations of cylinder due to helicity of fluid 65
4.4. Solution of the problem 66
4.4.1 Investigation of the velocity field 66
4.4.2 Investigation of the shear stress 70
4.5. Results and concluding remarks 72
5. CHAPTER 5 An Analytic Study of MolyBdenum Disulfide Nanofluids: An
Atangana-Baleanu Fractional Derivative Approach 78
5.1. Introduction 79
5.2. Formulations of flow equations 80
5.3. Analytical solution of the problem 83
5.3.1 Temperature distribution via Atangana-Baleanu fractional derivatives 83
5.3.2 Velocity field via Atangana-Baleanu fractional derivatives 84
v
5.4. Results and concluding remarks 86
6. CHAPTER 6 Applications of This Research and Future Recommendations 93
6.1. Applications of non-Newtonian fluid 94
6.2. Applications of magnetohydrodynamics (MHD) 95
6.3. Applications of nanotechnology 96
6.4. Future recommendations 98
Appendix 99
References 100
Publication Snaps 110
vi
List of Figures
CHAPTER 1
Figure 1.1 Natural flows and weather 2
Figure 1.2 Power plants 2
Figure 1.3 Piping system 3
Figure 1.4 Industrial application 3
Figure 1.5 Process of magnetic separation of magnetite iron ore 4
Figure 1.6 Electromagnetic stirring 5
Figure 1.7 Electromagnetic pump 5
Figure 1.8 Geometry of fluid describing the relation between shear rate with shear
strain 14
Figure 1.9 Structure with magnetic 16
Figure 1.10 Structure without magnetic 16
Figure 1.11 Plate with porous medium 18
Figure 1.12 Plate without porous medium 18
CHAPTER 2
Figure 2.1 Geometrical configuration of accelerated plate 32
Figure 2.2 Plot of velocity field and shear stress for at 41
Figure 2.3 Plot of velocity field and shear stress for at 41
Figure 2.4 Plot of velocity field and shear stress for at 42
Figure 2.5 Plot of velocity field and shear stress for at 42
Figure 2.6 Plot of velocity field and shear stress for at 43
Figure 2.7 Plot of velocity field and shear stress for at 43
Figure 2.8 Plot of velocity field and shear stress for at 44
vii
Figure 2.9 Plot of velocity field and shear stress for at 44
Figure 2.10 Comparison of velocity field and shear stress at 45
Figure 2.11 Comparison of velocity field and shear stress at 45
Figure 2.12 Comparison of velocity field and shear stress at 46
Figure 2.13 Validation of present solutions with obtained solutions by Jamil [41] for the
velocity field when and remaining parameters are at
46
CHAPTER 3
Figure 3.1 Plot of velocity field and shear stress for with
different values of 57
Figure 3.2 Plot of velocity field and shear stress for with
different values of 57
Figure 3.3 Plot of velocity field for with different values of and 58
Figure 3.4 Plot of velocity field for with different values of and
58
Figure 3.5 Plot of velocity field and shear stress for with
different values of 59
Figure 3.6 Plot of velocity field and shear stress for with different
values of 59
Figure 3.7 Comparison of velocity fields for four models for 600
Figure 3.8 Comparison of velocity fields for four models with and without magnetic
field and porous medium for 60
Figure 3.9 Comparison of present solution with the solution obtained by Ilyas et al.
[96] 61
viii
CHAPTER 4
Figure 4.1 Geometrical configuration of helical cylinder 65
Figure 4.2 Plot of velocity fields for and distinct values of 74
Figure 4.3 Plot of velocity fields for and distinct values of 74
Figure 4.4 Plot of velocity fields for and distinct values of 75
Figure 4.5 Plot of velocity fields for and distinct values of
75
Figure 4.6 Plot of velocity fields for and distinct values of
76
Figure 4.7 Plot of velocity fields for and distinct
values of 76
Figure 4.8 Plot of velocity fields for fractionalized Newtonian, fractionalized
Maxwell, fractionalized Oldroyd-B and fractionalized Burger, for 77
Figure 4.9 Plot of velocity fields for ordinary Newtonian, ordinary Maxwell, ordinary
Oldroyd-B, ordinary Burger for 77
CHAPTER 5
Figure 5.1 A principle ore of 79
Figure 5.2 Van der walls interaction of 79
Figure 5.3 Plot of velocity profile for four types of nanoparticles in an ethylene glycol
based nanofluid when . 88
Figure 5.4 Plot of velocity profile for molybdenum disulfide in ethylene Glycol based
nanofluid when with different values of 88
Figure 5.5 Plot of velocity profile for four shapes of molybdenum disulfide in
ethylene glycol based nanofluid when 89
Figure 5.6 Plot of velocity profile for molybdenum disulfide in ethylene glycol based
nanofluid when with different values of 89
ix
Figure 5.7 Plot of velocity profile and temperature distribution for molybdenum
disulfide in ethylene glycol based nanofluid when with
different values of 90
Figure 5.8 Plot of velocity profile for molybdenum disulfide in ethylene glycol based nanofluid
with different values of 90
Figure 5.9 Plot of velocity profile for molybdenum disulfide in ethylene glycol based nanofluid
with different values of 91
Figure 5.10 Comparison of velocity profile for four types of molybdenum disulfide in
ethylene glycol based nanofluids for smaller and larger times when
91
Figure 5.11 Comparison of velocity profile for Atangana-Baleanu fractional derivative
verses ordinary derivative when 92
CHAPTER 6
Figure 6.1 Role of non-Newtonian fluids in industries for unidirectional and
oscillating flows 94
Figure 6.2 The process of magnetic separation of magnetite iron ore 95
Figure 6.3 Plasma confinement 96
Figure 6.4 Coolant tower of power plant 97
x
List of Tables
Table 3.1 Rheological parameters for limiting solutions 54
Table 3.2 Rheological parameters for validation of results 56
Table 4.1 Rheological parameters for particular solutions 63
Table 5.1 The Sphericity for different nanoparticle shapes with constants a and b 81
Table 5.2 Thermo-physical properties of ethylene glycol and nanoparticles 81
xi
Nomenclatures
Electrical conductivity
Magnetic field
Velocity field
Newtonian velocity field
Non-Newtonian velocity field
Relaxation time
Material parameters
Retardation time
Shear stress
Dynamic viscosity of fluid
Shear rate of deformation
Density of fluid
Cylindrical coordinates
Cartesian coordinates
, Physical components
Total magnetic field
Magnetic field strength
Electric field
Permeability of the free space
Current density
Conductivity
Characteristic dimension
Thermal diffusivity of the fluid
Kinematic viscosity
Grashof number
Fractional parameter
Caputo fractional operator
Caputo-Fabrizio fractional operator
Atangana-Baleanu fractional operator
Mittag-Leffler function
Fox-H function
xii
Generalized M-function
Cauchy stress tensor
Indeterminate spherical stress
Dynamic viscosity
First Rivlin Ericksen tensor
Dell operator
T Transpose operation
Extra-stress tensor
, Normal stress moduli
Second Rivlin Ericksen tensor
Velocity gradient
Material time derivative
Upper convected derivative
Integral transform operator
Kernel of the transform
Image of transform
Function of time
Transform parameter
Laplace transform operator
Inverse Laplace transform operator
Fourier Sine transform operator
Inverse Fourier Sine transform operator
Finite Hankel transform operator
Inverse finite Hankel transform operator
Convolution product
Applied magnetic field’s magnitude
Darcy’s resistance
Permeability of the porous medium
Porosity
Non-zero constant
Unit step function
Fourier sine transform parameter
Letting parameters for cylinder
Angular velocity
xiii
Oscillating velocity
Radius of cylinder
Molybdenum disulfide
Dynamic viscosity of nanofluids
Electrical conductivity of nanofluids
Thermal expansion coefficient of nanofluids
Density of nanofluids
Thermal conductivity of nanofluids
( )
Heat capacitance of nanofluids
Temperature distribution
Empirical shape factor
Sphericity
Constants of particle shape
Copper
Alumina
Silver
Ethylene glycol
Volume fraction of the nanoparticles
Peclet number
Reynold’s number
Radiation parameter
Letting parameters of governing equations
Letting parameters of calculations
xiv
Abbreviation
Magnetohydrodynamics
HAM Homotopy analysis method
Caputo
Caputo-Fabrizio
Atangana-Baleanu
Molybdenum disulfide
Copper
Alumina
Silver
Ethylene glycol
xv
List of Publications
[1] Kashif Ali Abro, Mukarrum Hussain, Mirza Mahmood Baig, Analytical solution
of MHD generalized Burger’s fluid embedded with porosity, International Journal
of Advanced and Applied Sciences, 4(7) 80-89, (2017).
[2] Kashif Ali Abro, Mukarrum Hussain, Mirza Mahmood Baig, Khalil-ur-Rehman
Channa, Analysis of generalized Burger’s fluid in Rayleigh stokes problem,
Journal of Applied Environmental and Biological Sciences (JAEBS), 7(5) 55-63
(2017).
[3] Kashif Ali Abro, Mukarrum Hussain, Mirza Mahmood Baig, A mathematical
analysis of magnetohydrodynamic generalized Burger fluid for permeable
oscillating plate, Punjab University Journal of Mathematics, 50(2) 97-111 (2018).
[4] Kashif Ali Abro, Mukarrum Hussain, Mirza Mahmood Baig, Helices of
generalized Burger fluid in circular cylinder: An analytic analysis, Applied
Sciences (under review).
[5] Kashif Ali Abro, Mukarrum Hussain, Mirza Mahmood Baig, An analytic study of
molybdenum disulfide nanofluids using modern approach of Atangana-Baleanu
fractional derivatives, European Physical Journal Plus, 132: 439, DOI
10.1140/epjp/i2017-11689-y (2017).
[6] Kashif Ali Abro, Mukarrum Hussain, Mirza Mahmood Baig, Impacts of
magnetic field on fractionalized viscoelastic fluid, Journal of Applied
Environmental and Biological Sciences (JAEBS), 6(9) 84-93 (2016).
[7] Kashif Ali Abro, Mukarrum Hussain, Mirza Mahmood Baig, Slippage of
fractionalized Oldroyd-B fluid with magnetic field in porous medium, Progress in
Fractional Differentiation and Applications: An international Journal, 3(1) 69-80
(2017).
[8] Kashif Ali Abro, Mukarrum Hussain, Mirza Mahmood Baig, Influences of
magnetic field in viscoelastic fluid, International Journal of Nonlinear Analysis
and Applications, 9(1) 99-109 (2018).
xvi
[9] Kashif Ali Abro, Sumera Dero, Mirza Mahmood Baig, Effects of Transverse
Magnetic Field on Oscillating Plate of Second Grade Fluid, Sindh University
Research Journal (Science Series), 48(3) 605-610 (2016).
[10] Kashif Ali Abro, Zubair Ahmed Kalhoro, Mukarrum Hussain, Accelerating Flow
of Oldroyd-B over the Boundary with No Slip Assumption, Science International
(Lahore), 28(4), 4163-4169, (2016).
[11] Kashif Ali Abro, Zubair Ahmed Kalhoro, Mirza Mahmood Baig, Rajab Ali
Malookani, Impacts of Permeability on Oldroyd-B Fluid in the Absence of
Slippage, Science International (Lahore), 28(4) 4171-4176 (2016).
xvii
Additional List of Publications
[1] Kashif Ali Abro, Ilyas Khan, Asifa Tassaddiq, Application of Atangana-Baleabu
fractional derivative to convective flow of MHD Maxwell fluid in a porous
medium over a vertical plate, Mathematical Modeling of Natural Phenomena, 13
(2018) 1, https://doi.org/10.1051/mmnp/2018007
[2] Kashif Ali Abro, Ilyas Khan, J.F. G´omez-Aguilar, A mathematical analysis of a
circular pipe in rate type fluid via Hankel transform, Eur. Phys. J. Plus (2018) 133:
397, DOI 10.1140/epjp/i2018-12186-7
[3] Kashif Ali Abro, Ilyas Khan, Effects of CNTs on magnetohydrodynamic flow of
methanol based nanofluids via Atangana-Baleanu and Caputo-Fabrizio fractional
derivatives, Thermal Science, (2018), DOI: 10.2298/TSCI180116165A
[4] Kashif Ali Abro, Ali Dad Chandio, Irfan Ali Abro, Ilyas Khan, Dual thermal
analysis of magnetohydrodynamic flow of nanofluids via modern approaches of
Caputo-Fabrizio and Atanagana-Baleanu fractional derivatives, Journal of
Thermal Analysis and Calorimetry, (2018), DOI: 10.1007/s10973-018-7302-z.
[5] Qasem Al-Mdallal, Kashif Ali Abro, Ilyas Khan, Analytical solutions of
fractional Walter's-B fluid with applications, Complexity, (2018), Volume 2018,
Article ID 8131329, 10 pages, https://doi.org/10.1155/2018/8131329
[6] Kashif Ali Abro, Anwar Ahmed Memon, Muhammad Aslam Uqaili, A
comparative mathematical analysis of RL and RC electrical circuits via Atangana-
Baleanu and Caputo-Fabrizio fractional derivatives, European Physical Journal
Plus, (2018) (2018) 133: 113, DOI 10.1140/epjp/i2018-11953-8.
[7] Ilyas Khan, Kashif Ali Abro, Thermal analysis in Stokes’ second problem of
nanofluid: Applications in thermal engineering, Case Studies in Thermal
Engineering, (2018), Available online 10 April 2018, DOI:
https://doi.org/10.1016/j.csite.2018.04.005
[8] Kashif Ali Abro, Ilyas Khan, Analysis of Heat and mass transfer in MHD flow of
generalized Casson fluid in a porous space via non-integer order derivative
without singular kernel, Chinese Journal of Physics, 55(4) 1583-1595 (2017).
xviii
[9] Arshad Khan, Kashif Ali Abro, Asifa Tassaddiq, Ilyas Khan, Atangana-Baleanu
and Caputo Fabrizio analysis of fractional derivatives for heat and mass transfer of
second grade fluids over a vertical plate: A comparative study, Entropy, 19(8) 1-
12 (2017).
[10] Muzaffar Hussain Laghari, Kashif Ali Abro, Asif Ali Shaikh, Helical flows of
fractional viscoelastic fluid in a circular pipe, International Journal of Advanced
and Applied Sciences, 4(10) 97-105 (2017).
[11] Shakeel Ahmed Kamboh, Zubair Ahmed Kalhoro, Kashif Ali Abro, Jane
Labadin, Simulating electrohydrodynamic ion-drag pumping on distributed
parallel computing systems, Indian Journal of Science and Technology, 10(24) 1-5
(2017).
[12] Kashif Ali Abro, Porous effects on second grade fluid in oscillating plate, Journal
of Applied Environmental and Biological Sciences (JAEBS), 6(5) 71-82 (2016).
[13] Muhammad Jamil, Kashif Ali Abro, Najeeb Alam Khan, Helices of
fractionalized Maxwell fluid, Nonlinear Engineering, 4(4) 191-201 (2015).
xix
Draft of Thesis
The thesis is categorized as the following order:
Chapter 1 elucidates fundamentals of fluid flows, few types of fractional derivatives,
different special functions, constitutive equations of fluids, mathematical techniques
(some integral transforms).
Chapter 2, pertains to study magnetohydrodynamic (MHD) generalized Burger fluid in
porous medium as a sum of Newtonian and non-Newtonian forms. The integral
transforms (Laplace and Fourier Sine transforms) are invoked on governing partial
differential equations for investigating the analytical solutions of velocity field
corresponding to shear stress. The general solutions have been established to retrieve
several special cases of the problem. The effects of different pertinent parameters, for
instance, time, viscosity, magnetic field, material parameters, porosity are underlined by
graphical illustrations. The contents of this chapter have been published in “International
Journal of Advanced and Applied Sciences, (2017) 4(7) 80-89”.
Chapter 3 is particularized for an electrically conducting flow of fractionalized Burger
fluid over an oscillating plate with permeable plate. The governing partial differential
equations have been converted into non-integer order derivative (i-e Caputo fractional
operator). The analytical solutions are investigated for velocity field and shear stress
satisfying initial and boundary conditions as well. The general solutions are focused for
four models namely (i) the solutions in the absence of magnetic field, (ii) the solutions in
the absence of permeability, (iii) the solutions for ordinary differential operator, (iv) the
solutions for fractional Burger, Oldroyd-B, Maxwell and Newtonian fluids. The effects of
these models have also been checked graphically by imposing rheological parameters.
The contents of this chapter have been published in “Punjab University Journal of
Mathematics, 50(2) 97-111 (2018)”.
In Chapter 4, the effects of generalized Burger fluid flow for infinite helically moved
cylinder are analyzed. The analytical solutions are investigated by invoking Hankel and
Laplace transforms on the governing partial differential equations. The expressions of
velocity field and shear stress are expressed in the layout of Fox-H function satisfying
initial and boundary conditions. The helical flows of four models as Burger, Oldroyd-B,
Maxwell and Newtonian fluids are presented for comparisons with existing published
xx
findings which exhibit good agreement and reveal the accuracy and validity of this
analysis. This work is under review process in “Applied Sciences”.
Chapter 5 concerns with an analytic study of molybdenum disulfide nanofluids using
modern approach of Atangana-Baleanu fractional derivatives. Here, the core object is to
check the significance of different shapes of Molybdenum disulfide nanoparticles
containing ethylene glycol in mixed convection flow with magnetic field and porous
medium. Non-spherically shaped molybdenum disulfide nanoparticles namely platelet,
blade, cylinder and brick are utilized in this analysis because ethylene glycol is chosen as
a base fluid in which molybdenum disulfide nanoparticles are suspended. A modern
approach of Atangana-Baleanu fractional derivatives is applied for the modeling of the
problem. The analytical solutions are investigated by Laplace transforms with inversion
and expressed in terms of compact form of M-function Ta
bM . The ordinary derivative
models have been compared with the Atangana-Baleanu fractional derivatives models
graphically by setting various rheological parameters. This investigation has been
published in “The European Physical Journal Plus (2017) 132: 439”.
Chapter 6 is devoted for the applications of this thesis in real world problems; for
instance, role of non-Newtonian fluids in industries, magnetohydrodynamics (MHD), heat
transfer of nanofluids and coolant tower of power plant. Meanwhile, future
recommendations of this research with different geometries of non-Newtonian fluid have
been highlighted as well.
xxi
Abstract
The objectives of this thesis is to explore analytical solutions of velocity field, shear stress
and temperature distribution subject to the electrically conducting flows of fractionalized
non-Newtonian fluids embedded with porous medium. The mathematical modeling of
governing equations for fluid flow has been established in terms of fractional derivatives
and solved by employing discrete Laplace, Fourier Sine and Hankel transforms. The
newly defined fractional derivatives namely Atangana-Baleanu and Caputo fractional
derivatives have been implemented on the problems of fluid flows. The general solutions
have been investigated under the influence of fractional and non-fractional (ordinary)
parameters, magnetohydrodynamics (MHD), porous medium, heat and mass transfer and
nanoparticles suspended in base fluids. The obtained solutions satisfy initial, boundary
and natural conditions, expressed in terms of special functions and have been reduced for
special and limiting cases as well. Moreover, influence of magnetic field, porosity,
fractional parameter, heat and mass transfer, nanoparticles and different rheological
parameters of practical interest have been investigated. At the end, in order to highlight
the differences and similarities among various rheological parameters, the graphical
illustration has been depicted for fluid flows.
1
CHAPTER 1
Preliminaries of Fluid Flows and Mathematical
Techniques
2
1.1 Introduction
The mechanics of fluid is dealt with the deformation of Substances under the
influence of shearing force. The velocity of the deformation will be correspondingly
small because a small shearing force can deform a fluid body. In simple words, fluid
mechanics is the study of fluids either in motion or at rest. It has been utilized in several
technological advancements; for instance, piping systems in chemical plants, design of
canal and dam systems, lubrication systems, aerodynamics of automobiles and supersonic
airplanes, ducting and piping utilized in the water and air conditioning systems of
businesses and homes and many others [1]. The classification of fluid mechanics is
divided into various categories: for instance, the study of movement of fluids with
incompressibility is called hydrodynamics; the study of fluid flows in open channels and
pipes is termed as hydraulics; the study of fluid flows that undergo changes of density is
known as gas dynamics; the study of fluid flows of gases over bodies is categorized as
aerodynamics [4]. In brevity, due to naturally occurring flows, few subcategories are
hydrology, oceanography and meteorology. However, fluid mechanics plays an adhesive
and vital role in the development of science and technology. It is extremely useful in the
design and analysis of wind turbines, aircraft, jet engines, boats, rockets, submarines,
natural gas, crude oil, the transportation of water, the cooling of electronic components
and biomedical devices etc., some examples are shown from figure 1.1 to figure 1.4. It is
also useful for the design of bridges and buildings. Various naturally occurring
phenomena are also ruled by ideologies of fluid mechanics such as rain cycle, the rise of
ground water to the top of trees, ocean waves, weather patterns and winds.
Figure 1.1 Natural flows and weather Figure 1.2 Power plants
3
Figure 1.3 Piping system Figure 1.4 Industrial application
The analytical solutions of governing equations of non-Newtonian fluid flows
perform an adhesive role due to various reasons: for instance, the accuracy and
correctness of numerous approximate methods can be checked for standard validity by
analytical solutions; numerical and empirical results can also be recognized through the
comparison with analytical solutions; the numerical solutions/schemes for various
complex problems of unsteady flows are also verified by analytical solutions. However,
analytical solutions are not only important in non-Newtonian fluid flows but also play a
significant role in certain physical circumstances. Hence the diversity of analytical
solutions lies in various disciplines of science like electromagnetic theory, chemical
kinematics, optical fibers, meteorology, hydrodynamics and several others. At the current
scenario, the governing partial differential equations of fluid mechanics problems are
usually solved by various attractive numerical schemes. This is due to the fact of
availability of computer codes and programs. Such approximate solutions may be
insignificant if they are not compared with analytical solutions or experimental data.
Hence, analytical solutions are fundamental tool for serving the accuracy/correctness of
experimental as well as asymptotic methods. In brevity, analytical solutions arise in
several scientific modeling for instance, fluid mechanics, beam theory, the propagation of
shallow water waves, nonlinear optics, earthquake stress, astrophysics, elastic waves in
soil and optimization and many others [7].
The study of the interaction between magnetic fields and conducting fluids is
termed as magnetohydrodynamics. In other words, study of flows in which the fluid is
electrically conducting is known as magnetohydrodynamics. Magneto means magnetic
field; hydro means fluids; and dynamics mean forces and the laws of motion [8-9]. Some
familiar examples are liquid metals (molten magnesium, liquid sodium, mercury, molten
antimony, gallium, etc.) and plasmas (electrically conducting gases or ionized gases).
Fundamentally, the relative movement of a conducting fluid and a magnetic field causes
an electromotive force and that relative movement generates electrical currents. The
4
concept behind magnetohydrodynamics depends upon the order of density ;
here are electrical conductivity, velocity field, magnetic field respectively. Due to
this fact, the fluid generates flow with magnetic field lines because current give rise to
another induced magnetic field. In short, one can observe that the magnetic fields can pull
on the conducting fluids while fluid can drag magnetic field lines. The applications of
magnetohydrodynamics can be categorized in numerous engineering processes. Processes
are described briefly as follows;
For removal of non-metallic inclusions
It is well established fact that all metallic materials are magnetic in nature. The
removal of non-metallic inclusions is extracted from iron ore (magnetite) by
various methods such as magnetic separation. For instance, one can consider the
magnetic roller for removal of magnetic and non-magnetic particles form finely
ground ore as shown in the figure 1.5
Figure 1.5 Process of magnetic separation of magnetite iron ore
However, several researchers use certain amount of aluminum in molten steel by
which they discuss the removal of non-metallic inclusions by utilizing high frequency
magnetic field.
Solidification Processing of Materials in Magnetic Fields
Solidification describes the phenomenon of liquids transforming into solids as a
result of a decrease in liquid temperature. Researchers have diverted their attention for
improved process performance and better-quality products because use of external
magnetic fields controls the behavior of the melts during solidification. Furthermore, the
5
purpose of magnetic field during the solidification process is to eliminate the impurities
(bridging and macrosegregation) from the products.
The metallurgical industries where magnetic fields are routinely used to heat,
stir, pump and levitate liquid metals
One of the most important applications of magnetic effect is pumping of materials
that are hard to pump using conventional pumps. The main advantage of this lies in the
MHD molten salt pump that is used for nuclear reactor coolants due to its no-moving-
parts feature and propulsion.
Consequently, the study of magnetohydrodynamic can be discussed further;
however it is justified to the end here by adding few supportive applications, for instance,
dampen the motion of liquid metal, Astrophysics (planetary magnetic field),
electromagnetic stirring (see figure 1.6), electrolysis cells where it is used to reduce
aluminum oxide to aluminum, electromagnetic pump (see figure 1.7) and MHD
generators.
Figure 1.6 Electromagnetic stirring Figure 1.7 Electromagnetic pump
A material that contains pores (voids) is typically filled with a fluid (liquid or
gas). The main characteristics of porous medium are porosity and permeability; here the
porosity refers to the relative fractional volume of the void space and permeability refers
to capacity of the medium to transmit fluid. The porous media encompasses several major
applications for instance, hydrogeology (groundwater modeling, nuclear waste disposal,
soil drainage, tracking the distribution of pollutants, etc.), flows in heavy oil reservoirs
and petroleum for field tests and filtrations processes, hydraulics and drilling fractures
due to complex rheological behavior in pore materials, subsurface flow models, geo-
6
mechanical engineering and several others. On the other hand, there is no denying fact
that nanofluids have become more popular among the researchers due to scientific
development and technological advancement. Nanofluids are formulated by the
suspension of the nanoparticles (carbides, oxides, carbon nanotubes or metals) in
conventional base fluids (water, engine oil, kerosene or certain liquids). Due to significant
role of nanofluids in industries, Several researchers have diverted their attention towards
the analytical study of nanofluids for improving the heat transfer of the conventional
fluids and enhancement of thermal conductivities.
1.2 Review of Literature
The analysis of the Newtonian and non-Newtonian fluid flows has great
importance and significance because of distinct engineering and industrial applications,
for instance pharmaceuticals, chemical, oil and gas, polymeric liquids, polymer solutions,
cosmetics, filtration and ceramic processes, biomechanics and enhanced oil recovery
processes. Due to this reason the industrial fluids are mostly referred as the non-
Newtonian fluids because of their complex flow characteristics, the rate of deformation
and structures. This diversity of applications in different fields of science and technology
has created an opportunity for scientists, mathematicians, engineers, and numerical
analysts to exhibit the behavior of typical characteristic of non-Newtonian fluids. The
typical characteristic of non-Newtonian fluids includes shear thinning and thickening,
relaxation and retardation, yield stress and non-zero normal stress differences in shear
flow, asphalts in geomechanics and asphalt concrete and many others. In continuation,
there is no denying fact that the non-Newtonian fluids are not easy to handle in
comparison with Newtonian fluids and it is also not an easy to explain all the
characteristics of non-Newtonian fluids. Due to non-linear relation between the shear
stress and shear rate, there does not exist single governing equation that possesses the
potential to predict all kinds of rheological characteristics of non-Newtonian fluids.
Therefore, various models have been proposed by scientific community to predict the
complete properties and characteristics of non-Newtonian fluids [1-7].
It is well known fact that non-Newtonian fluids are enumerated into following
classes as (i) the class of differential types, (ii) the class of rate type and (iii) the class of
integral type models. The Maxwell, Oldroyd-B and Burger models lie in the category of
rate type fluids because they take into account the elastic and memory effects and they
also describe slight memory. The main advantage of rate type fluid is to characterize the
viscoelastic fluids (i-e viscous and elastic memory effects). The first viscoelastic model
7
was originated by James Clerk Maxwell that could describe the stress relaxation. He
recognized that a body has the means to dissipate energy for the description of its viscous
nature and to store energy for characterizing the fluids elastic response [8]. Choi et al. [9]
studied incompressible and steady suction flow of Maxwell fluid for investigating the
combined effects of inertia and viscoelasticity. Fetecau and Corina [10] analyzed the
flow of a Maxwell fluid over a suddenly moved an infinite flat plate by using integral
transforms. They recovered the well-known solution for Navier–Stokes equations by
neglecting relaxation time parameter. Corina et al. [11] studied the second problem of
stokes for the flow of incompressible Maxwell fluid over an oscillating plate at small and
large times. They investigated the steady-state and transient solutions by invoking integral
transforms and analyzed several rheological effects of the material parameters. Friederich
[12] applied the science of fractional derivative on the Maxwell model for retardation and
relaxation functions. He applied Riemann-Liouville definition of fractional operator and
found the relaxation function in the time domain by employing power law series. Hayat et
al. [13] observed the effects of viscoelastic fluid via fractional Maxwell model for the
unidirectional flow periodically. They invoked integral transform and found analytical
solutions for the flows induced by periodic oscillations of lower plate when the upper
plate is at rest or being free. In a similar vein, a fractional Maxwell model to the unsteady
flow was investigated by Corina et al. [14] for constantly accelerated plate. They
presented the solutions for the contribution as a sum of non-Newtonian as well as
Newtonian fluids and also retrieved the ordinary solutions from fractional solutions by
making fractional parameter equal to one. Vieru and Rauf [15] analyzed the Maxwell
fluid under the wall slip and non-slip assumptions for stokes flows. They found the
solutions for velocity field and corresponding shear stress along with two particular cases
namely translation with a constant velocity and sinusoidal oscillations at the wall. In
exaggeration, several researchers extended Maxwell model with and without porosity,
magnetohydrodynamics, heat and mass transfer, stretching sheet and modern fractional
operators. Ilyas et al. [16] investigated the impact of magnetohydrodynamic Maxwell
fluid for oscillatory flow embedded in porous medium. They established the steady and
transient solutions for the cosine and sine oscillations of a plate satisfying the initial and
boundary conditions. Nadeem et al. [17] worked on numerical analysis for heat transfer of
a viscoelastic fluid in the presence of nanoparticles on a stretching sheet. They replaced
their flow equations into coupled ordinary differential equations and presented the
numerical solutions by invoking similarity transformations. Ilyas et al. [18] employed a
modern definition of fractional Caputo-Fabrizio operator on the unsteady flow of a
8
generalized Maxwell fluid over the oscillating plate with constant temperature. The
mathematical formulation of the problem was modeled via Caputo-Fabrizio fractional
derivatives; they found exact solutions and presented them in terms of special function
namely modified Bessel functions and complementary error. Imran et al. [19] examined
heat and mass transfer for natural convection flow of Maxwell fluid with non-integer
order derivative. They applied Laplace transform techniques on the non-dimensional
governing differential equations of temperature distribution, mass concentration, and
velocity field and presented the general solutions in terms of special functions namely
Wright's function, Robotnov-Hartley function Mittage-Leffler function and G-function.
Ilyas et al. [20] presented an interesting scientific report on mixed convection flow of
Maxwell fluid with constant wall temperature. They formulated the problem on
oscillating plate with coupled partial differential equations and introduced some non-
dimensional variables for converting the governing problem into dimensionless form. In
this work, the impacts of Grashof number and Prandtl number on different times for
velocity field and temperature distributions and a comparative graphical illustration was
presented for Maxwell and Newtonian fluid as well.
After that there is another model namely Oldroyd-B model commonly known as
rate type model which was proposed by Oldroyd [21]. Although, an Oldroyd-B model is
not sufficient for the rheological descriptions of shear-thinning/thickening yet it can
describe the normal stress differences, the stress relaxation, and retardation in a shear
flow. This is analyzed deeply by several researchers with different geometries. Few
contributions in this regard are discussed as, Rajagopal and Bhatnagar [22] observed the
flow effects of an Oldroyd-B fluid via analytic solutions under the consideration of two
cases; one is the case of longitudinal and torsional oscillations and the second is the of an
infinite porous plate. The findings of this problem admit asymptotically decaying solution
in terms of Bessel functions. Fetecau and Corina [23] presented interesting solutions on
stokes’ first problem for an Oldroyd-B liquid. This study was extended by Wenchang [24]
a modified Darcy’s law for a viscoelastic liquid. He extended an Oldroyd-B liquid in a
porous by invoking the mathematical methodology of Fourier Sine transform and
established exact solution. The main investigation of this extension was to highlight the
effects of viscoelasticity in porous medium. On the other hand, some previous solutions
were retrieved from [23] in the absence of porous medium. Hayat et al. [25] presented
analytical solutions for the steady flow of an Oldroyd 6-constant with magnetic field by
employing Homotopy analysis method (HAM), namely (i) Poiseuille flow (ii) generalized
Couette flow and (iii) Couette flow and presented this analysis by the comparison of
9
Homotopy analysis method (HAM) with numerical solutions as well. In another study,
this model was analyzed in the presence of an external magnetic field. Ghosh and Sana
[26] investigated exact solutions for the fluid velocity and the shear stress on the plate by
invoking operational methods. The main focus was to study the rheological effects of the
fluid elasticity and the magnetic field simultaneously on the wall shear stress and on the
flow for different periods. Haitao and Jin [27] worked on the unsteady helical flow within
an infinite cylinder and between two infinite coaxial cylinders for fractionalized Oldroyd-
B fluid. They investigated the exact analytical solutions of governing partial differential
equations by applying Weber transform, Laplace transform and finite Hankel transform
satisfying imposed initial and boundary conditions. Ellahi et al. [28] observed the impacts
of slippage on an Oldroyd 8-constant liquid by the consideration of non-linear boundary
conditions with Couette, Poiseuille and generalized Couette flows. Zheng et al. [29]
analyzed the impacts of magnetic field on fractional Oldroyd-B fluid induced by an
accelerated plate with no slip assumptions. They investigated the closed form solutions
and expressed them in terms of Fox-H function.
In this continuation, the Burger model is also lying among the category of rate
type fluids that enables the behavior of fluids and characterize the typical rheology of
fluid for instance, asphalts in geomechanics, response of a variety of geological materials,
simultaneous effects of relaxation and retardation phenomenon, response of asphalt and
asphalt mixtures, propagation of seismic waves in the interior of the earth, geological
structures like Olivine rocks, motion of the earths’ mantle, the post-glacial uplift and
several other geological structures [30-37]. The Burger model is highly non-linear model
which contains four special cases depending on relaxation time , retardation time
and material parameters and ; for instance (i) if then the
Burger model is reduced for the Newtonian fluid, (ii) if then the
Burger model is reduced for the Maxwell fluid, (iii) if then the Burger
model is reduced for the Oldroyd-B fluid and if then the Burger model is reduced
for the simple Burger fluid. In brevity, Krishnan and Rajagopal [37] investigated the
constitutive modeling of asphalt concrete with thermodynamic framework. They
generalized upper convected Burger’s model for describing the nonlinear behavior of
materials such as asphalt concrete. They emphasized their investigations for tensile creep
of asphalt concrete using numerical scheme on the initial value problem and compared
via experimental data. Even the diverse application of Burger’s model, the Burgers’
model is still not contributed deeply because of complexities in terms of mathematical
modeling. Limited studies have been carried out on the flows of the Burgers fluids.
10
Ravindran et al. [38] investigated the solutions of Burgers’ fluid through orthogonal
rheometer. They solved the governing couple differential equations under the
assumptions that the fluid adheres to the top and bottom plates. Hayat et al. [39] analyzed
the Burger fluid for the unidirectional flow based on imposed conditions on geometry of
the problem. They investigated exact analytical solutions using Fourier Sine transform
and retrieved the solutions of Oldroyd-B, Maxwell, Second grade, Newtonian fluid from
open literature as the limiting cases. Chen et al. [40] observed the effects of unsteady flow
of Burger fluid by considering four types of flow situations namely (i) constant
acceleration piston motion, (ii) suddenly started flow, (iii) trapezoidal piston motion, and
(iv) oscillatory piston motion. They concluded that Burger’s fluid parameters have
significant effects on pressure gradient and velocity fields. Jamil [41] presented
generalized Burger fluid with first problem of stokes’; here sum of steady and transient
solutions for velocity field and adequate shear stress were generated by invoking integral
transforms on governing partial differential equations govern the generalized Burger fluid
flow. Of course the study on Burger fluid is continued but the study on Burger fluid is
categorically discussed in terms of magnetohydrodynamic, porous medium, heat transfer,
cylindrical geometry, fractional operators and few others in following sections.
Burger fluid with magnetohydrodynamic
Siddiqui et al. [42] investigated exact solutions from the governing momentum
and energy equations and they analyzed the effects of Hall current and heat transfer on
the magnetohydrodynamic flow. In exaggeration, they checked various pertinent
parameters on fluid flow, such as, the fixed Hartmann number, the hall parameter, the
vorticity, the heat transfer, Prandtl number etc. Ilyas et al. [43] studied rotating flow of an
incompressible generalized Burgers fluid in presence of magnetic field and porous
medium. They applied Laplace transform techniques to solve the modeled governing
equations and investigated closed form solutions using modified Darcy’s law. They
concluded their analysis that the real and the imaginary part of velocity decreases and
increases respectively for large hall parameter.
Burger fluid with porous medium
Masood et al. [44] observed the impacts of flow of the fractional generalized
Burgers’ fluid in a porous space by implementing modified Darcy's law. They established
the solutions for velocity field for three types of problems: (i) flow due to rigid plate, (ii)
periodic flow between two plates and (iii) periodic Poiseuille flow. Their major findings
11
were to compare the present analytic solutions using fractional calculus approach with
previously published results which were found in excellent agreement. Aziz et al. [45]
investigated exact steady-state solutions of magnetohydrodynamic for rotating flow of
generalized Burgers fluid. They invoked Fourier Sine transform techniques on the
imposed governing partial differential equations over variable accelerated plate. They
particularized the obtained solutions for various cases; for instance, (i) MHD Burger fluid
in porous medium, (ii) MHD Oldroyd-B fluid in porous medium, (iii) MHD Maxwell
fluid in porous medium, (iv) MHD second grade fluid in porous medium, (v) MHD
viscous fluid in porous medium and few others as well.
Burger fluid with cylindrical geometry
Tong and Shan [46] analyzed annular pipe for unsteady unidirectional transient
flows of generalized Burgers fluid. They utilized Hankel and Laplace transforms for
investigating the two types of flow problems namely axial Couette flow in annulus and
Poiseuille flow with a constant pressure gradient. Fetecau et al. [47] worked on
generalized Burgers fluid due to longitudinal oscillations of circular cylinder with
pressure gradient. They presented starting solutions for generalized Burgers fluid as the
sum of steady-state and transient solutions. Jamil and Najeeb [48] studied flows of
Burgers’ fluid with fractional derivatives model through a circular cylinder by utilizing
integral transforms on the governing partial differential equations. They focused the
effects of linear and angular velocities. They presented the general solutions in terms of
special function commonly known as G-function, while all the solutions were satisfying
imposed initial and boundary conditions. Moreover, a recent study by, Masood [49]
considered the generalized Burgers fluid constitutive model with the electrostatic body
force in the electric double layer in cylindrical domain. They applied the temporal Fourier
and finite Hankel transforms on the Cauchy momentum equation and presented analytical
solutions from non-dimensional governing differential equations of the generalized
Burgers fluid.
Burger fluid with fractional operator
Xue et al. [50] investigated the exact solutions for the fractional generalized
Burgers’ fluid under the influence of modified Darcy’s law. The general solutions were
established by invoking the Fourier sine transform and the fractional Laplace transform
on fractional partial differential equations. They also focused on the first problem of
stokes for Burger, Oldroyd-B, Maxwell, Second grade and viscous fluids as well. Hyder
12
[51] analyzed the fractional Burgers’ fluid model between two parallel plates with some
unidirectional flows. In this study, the exact solutions were obtained by invoking the
finite Fourier sine and Laplace transforms for two types of flow namely Plane Couette
flow and Plane Poiseuille flow. More recently, Shihao et al [52] analyzed slippage of
generalized Burgers’ fluid via Caputo fractional derivative. The flow is induced between
two side walls caused by a constant pressure gradient and an exponential accelerating
plate. The fractional Calculus approach was used on governing partial differential
equations to investigate exact analytical solutions via integral transformation techniques.
They also depicted the 2 and 3-dimensional graphs for the rheological effects of pertinent
parameters on velocity field and shear stress.
Burger fluid with heat transfer
Chang feng and Jun xiang [53] analyzed the Stokes’ first problem of a heated
Burgers’ fluid via calculus of fractional differentiation in presence of porous medium.
They traced out the solutions of temperature and velocity distributions by invoking
Laplace and Fourier Sine transforms. Yaqing et al. [54] considered incompressible
generalized Burgers’ fluid with heat transfer for exponentially accelerating plate under
the influences of magnetic field and radiation. They investigated exact analytical
solutions for temperature and velocity distributions by using fractional calculus approach
via integral transforms and expressed their mathematical results in terms of special
function.
Burger nanofluid
Masood and Khan [55] studied free convection boundary layer flow of a Burgers’
nanofluid suspended with nanoparticles under influences of heat generation/absorption.
They invoked similarity transformations on the differential equations of fluid flow and
investigated analytic results via homotopy analysis method. Their main interest was to
analyze the effects of few non-dimensional numbers and some rheological parameter for
instance, Lewis number, Deborah numbers, Prandtl number and the thermophoresis
parameter and the Brownian motion parameter. In this connection, the same authors
Masood and Waqar [56] considered two-dimensional forced convective flow of a
generalized Burgers with nanoparticles over a linearly stretched sheet. They investigated
analytic results through the homotopy analysis method (HAM) on the set of coupled
nonlinear ordinary differential equations. They also presented the graphical illustrations
of velocity, temperature and concentration fields and discussed them in detail. In another
13
study, the same authors [57] observed steady three-dimensional steady flow of Burgers
nanofluid and emphasized on the heat and mass transfer characteristics over a
bidirectional stretching surface. They sketched the graphical and numerical computations
and ensured the convergence of series solutions as well. On the other hand, a very
interesting study was presented by Rashidi et al. [58] in which they achieved the
mathematical modeling for hydro-magnetics. They analyzed the effects on the flow of a
Burgers' nanofluid and constructed set of coupled nonlinear ordinary differential system
by employing the suitable transformations. The significant work in this research paper
was the comparative study of present results with an already published data. Off course,
there is a list of research related to Burger fluids but few recently published studies are
discussed here. [59-64].
The objectives of this thesis is to explore analytical solutions of velocity field,
shear stress and temperature distribution subject to the electrically conducting flows of
fractionalized non-Newtonian fluids embedded with porous medium. The mathematical
modeling of governing equations for fluid flow have been established in terms of
fractional derivatives and solved by employing discrete Laplace, Fourier Sine and Hankel
transforms. The newly defined fractional derivatives namely Atangana-Baleanu and
Caputo fractional derivatives have been implemented on the problems of fluid flows. The
general solutions have been investigated under the influence of fractional and non-
fractional (ordinary) parameters, magnetohydrodynamics (MHD), porous medium, heat
and mass transfer and nanoparticles suspended in base fluids. The obtained solutions
satisfy initial, boundary and natural conditions, expressed in terms of special functions
and have been reduced for special and limiting cases as well. Moreover, influence of
magnetic field, porosity, fractional parameter, heat and mass transfer, nanoparticles and
different rheological parameters of practical interest have been investigated. At the end, in
order to highlight the differences and similarities among various rheological parameters,
the graphical illustration is depicted for fluid flows.
1.3 Newtonian and Non-Newtonian Fluids
The fluid that satisfies Newton’s constitutive equation of viscosity is termed as
Newtonian fluid. Newton’s law of a viscosity is given by
14
Here,
are shear stress, dynamic viscosity, shear rate of deformation
respectively. In simple words, the rate of shear and applied shear stress possess linear
relationship. There are several Newtonian fluids which include such as air, water, gases,
glycerin, silicone oils, ethyl alcohol, benzene, hexane and other solutions of the lowest
weightage of molecules. In exaggeration, the experimental characteristics of Newtonian
fluid are enumerated below depending upon constant temperature and pressure, (i) when
viscosities are measured during various types of deformations for Newtonian fluids they
turn to be in simple proportions to each other. (ii) In simple shear flows of Newtonian
fluids the only non-zero stress is the shear stress, whereas the two normal stresses give
zero difference. (iii) The viscosity remains constant with respect to the shear applied to
the fluid, over a wide range. As long as the shear applied to the fluid is stopped the
resulting shear in the fluid tends to zero. (iv) In a Newtonian fluid the shear viscosity
remains independent of its shear rate. On the other hand, the fluid that does not satisfy
Newton’s constitutive equations of viscosity is termed as non-Newtonian fluid. The
rheological characteristics of non-Newtonian fluid are exhibited by
(
)
Where, characterizes the behavior of fluid flows. Equation (1.2) can be particularized
for equation (1.1) by employing . The non-Newtonian fluids include for instance,
suspensions, elastomers, lubricants, clay coatings, paints, cosmetics, extrusion of polymer
liquids, certain oils, toothpaste, blood, ketchup, jellies and few others.
Figure 1.8 Geometry of fluid describing the relation between shear rate with shear strain
15
Basically, the classifications of non-Newtonian fluids are usually divided as (i) the
rate type fluid, (ii) the differential type fluid and (iii) the integral type fluid. These three
types of fluids are discussed in detail in the open literature as well.
1.4 Equation of Continuity
Assume that a control volume is bound with surface along with assumption of
the fact that fluid is not leaving or entering the surface. Meanwhile, having in mind the
law of conservation of mass, the expression can be written for the total mass as:
∫
Here, describes the density of fluid. The control volume is considered arbitrary;
hence the differential form of the continuity equation for an unsteady flow of
compressible fluid as
Where, and are density and velocity of fluid respectively. The reduced form
of equation (4) for an unsteady flow of incompressible flow is described as
The equation (1.5) represents the condition of incompressibility can also be
expressed in terms of cylindrical coordinates and cartesian coordinates
respectively as
(
)
Here in equations (1.6) and (1.7), and are the physical
components of the velocity field .
1.5 Magnetohydrodynamics (MHD)
Magnetohydrodynamics (MHD) refers to the electrically conducting fluid where
the magnetic field and velocity field are coupled. It is also meant as hydromagnetics or
magneto fluid dynamics. The electrolytes or salt water, plasmas and liquid metals,
16
mercury, sodium or molten iron, gallium and ionized gases are well known examples of
electrically conducting fluid. The concept of magnetohydrodynamics was suggested by
Hannes Alfven. He described the operational meaning of magnetohydrodynamics in three
words; “magneto” refers to magnetic field, "hydro" refers to liquid and "dynamics" refers
to movement. Magnetohydrodynamics plays an adhesive and significant role in the
analysis of magnetics. The main theme of magnetohydrodynamics is that a magnetic field
can generate current in conductive fluids. It also affects the magnetic field itself and also
responsible for imposing forces on the fluid. Magnetohydrodynamics (MHD) has
uncountable applications in various areas, for instance; the description of ionosphere is
analyzed by magnetohydrodynamics, generation of Earth's magnetic field,
electromagnetic forces can be used to pump liquid metals. Induction furnace and casting
is usually analyzed by magnetohydrodynamics and several others [65]. In short, structures
with and without magnetic are sketched in figures 1.9 to 1.10.
Figure 1.9 Structure with magnetic Figure 1.10 Structure without magnetic
The description of magnetohydrodynamics can be characterized by the
combination of Maxwell’s equations of electromagnetism and Navier-Stokes equations of
fluid dynamics. The momentum equation with magnetohydrodynamics is
(
)
Here, the Lorentz force is represented by , defined as
⁄ ⁄
The following equations represent the different formats of Maxwell’s equations
with the nature of magnetic field , as
17
Here, equations (1.10-1.15) represent solenoidal law, ampere equation, Faraday’s
law, charge conversation, Ohm’s law and Lorentz force respectively. Whereas, for above
equation, and are the total magnetic field, the magnetic field strength, the
electric field, the permeability of the free space, the current density and conductivity
respectively.
1.6 Porous Medium
The study of porous medium has diverted the interest of researchers due to its
applications in ceramic and filtration processes, chromatography, biomechanics,
insulation system and several others as well. The porous medium is a term stated as a
substance that encompasses spaces between solid areas through which liquid or gas can
be transmitted. The complexities of pore structure usually occur in context of physical
aspects to characterize the permeability and porosity of porous medium. Where,
permeability refers capacity of the medium to transmit fluid and porosity describes the
fractional volume of the void space to the total volume. In this continuation, the
mathematical description of the flow in porous medium is extremely challenging and
complex task for the analytical solution of fluid mechanics problem. Such problems of
fluid flows through porous medium are rarely found in literature. In order to have an
analytic study of porous flow problem, several researchers are usually employing the
Carman-Kozeny equation commonly known as Darcy’s law which can be found in
literature [66-67]. In short, structures describing the vivid characteristics with and without
porous medium are sketched in figures 1.11 to 1.12.
18
Fig. 1.11. Plate with porous medium Fig. 1.12. Plate without porous medium
1.7 Nanofluids and Nanoparticles
Nanofluids are generally stated as the suspension of nanometer sized particles
which can increase few properties of base fluid. In simple word, the colloidal suspension
of nanoparticles (1-100 mm) in the base fluid is termed as nanofluids. In order to enhance
the thermal conductivities, various researchers utilize the concept of nanofluids.
Typically, nanoparticles are made up of metal oxides, stable metals and carbon in various
forms. The size of nanoparticles varies in some unique properties to the base fluids,
erosion of the containing surface, reduced tendency for sedimentations, greatly enhanced
mass transfer and energy momentum. They have also unique characteristics varying from
conventional solid liquid mixtures of (mu) m and mm sized particles dispersed in non-
metals and metals. However, nanofluids are widely applicable for the enhancement of
heat transfer due to their excellent characteristics. The term nanofluid was suggested by
Choi in 1995 [68]. He described that fluids with good heat transfer characteristics have
superior thermal properties as compare to conventional fluids. Nanofluids are stable
colloidal suspension of nanoparticles, nanocomposites, nanofibers in base fluids includes
polymer solution, ethylene glycol, oil, water and several others. Dimensionally,
nanoparticles are usually less than 100 mm. The goal of nanofluids is to achieve the
highest possible thermal properties at the smallest possible concentrations by stable
suspension of nanoparticles and uniform dispersion in base fluids.
1.8 Heat Transfer and Dimensionless Numbers
It is well established fact that heat transfer ceases when thermal equilibrium is
reached. The heat is transferred from hot to cold medium. Fundamentally, heat transfer is
classified into three categories namely (i) conduction, (ii) convection and (iii) radiation.
19
Whereas, Conduction is the way through which energy is transferred by the movement of
electrons or ions. Convection is the transfer of heat energy into or out of the body by
actual movement of fluids particles that transfer energy with its mass. Thermal radiation
or radiation is the process of heat transfer due to emission of electromagnetic waves [69].
In brevity, there is a variety of dimensionless numbers but a few dimensionless numbers
used in this thesis are described below:
Peclet number
The Peclet number is a dimensionless number that is applied to perform the
convective heat transfer. It is denoted by . In other words, the ratio of thermal energy
convected to the fluid to the thermal energy conducted within the fluid is called Peclet
number. Its mathematical form is
Here is fluid velocity, is a characteristic
dimension, and is thermal diffusivity of the fluid.
Reynold number
The Reynold number is a dimensionless number that is applied to find flow
behaviors such as laminar, turbulent or transitional flows. It is denoted by . In other
words, the ratio of inertial force to the viscous force is called Reynold number. Its
mathematical expression is
Here represents the free stream velocity,
denotes the characteristics length and stands for kinematic viscosity.
Grashof number
The Grashof number is a dimensionless number that is used in natural convection.
It is denoted by . It approximates the ratio between buoyancy force and viscous force
acting on fluid. Its mathematical expression is ( )
.
1.9 Fractional Derivatives
The description of the complex dynamics is explained by the fractional derivatives
in theoretical and practical areas. Due to increasing attention of fractional derivatives, the
fractional derivatives ascertained to be an intensive tool in the study of fluid mechanics.
In general, a fractional model is generated from ordinary model via interchanging the
derivatives of integer order into derivatives of fractional order. Furthermore, in order to
describe the characteristics of polymer solution and melts, the rheological constitutive
equations are modeled via certain fractional operator. Nowadays, it has been rectified that
20
fractional derivatives can be utilized for the description of certain physical problems, and
also for processes where memory effects are important [70-71]. In brevity, few types of
fractional derivatives are elucidated as below
The Caputo fractional derivative
It is an established fact that the Riemann–Liouville fractional derivative and the
Caputo fractional derivatives are the most useful fractional derivatives in literature. The
Riemann–Liouville fractional derivative exhibits some difficulties in the applications in
comparison with the Caputo fractional derivative. For instance, Laplace transform of the
Riemann–Liouville derivative contains terms without physical significance and the
Riemann–Liouville derivative of a constant is not zero. Due to these facts, the Caputo
fractional derivative has eliminated both difficulties because this operator contains a
singular kernel. The Caputo fractional derivative of order is stated in [72-73]
∫
Where, the fractional operator is so called Caputo fractional operator.
While, { } of Caputo fractional operator can be obtained from equation (1.16).
On the other hand, it is noted that Caputo fractional operator can be extended
significantly by letting in equation (1.16) as well.
The Caputo-Fabrizio fractional derivative
Although Riemann–Liouville and Caputo fractional derivatives are the most
useful fractional derivatives yet a new definition of the fractional derivatives is presented
namely Caputo-Fabrizio fractional derivatives in literature. The main significance of
Caputo-Fabrizio fractional derivatives is that its kernel is based on exponential function
having no singularities. In a nut shell, the Caputo-Fabrizio fractional derivative is
commonly known as CF fractional operator of order is stated in [74-76]
∫ (
)
where, the fractional operator is so called Caputo-Fabrizio fractional
operator. While, { } of Caputo-Fabrizio fractional operator can be obtained
21
from equation (1.17). I contrast, it is essential to note that Caputo-Fabrizio fractional
operator can also be extended significantly by letting in equation (1.17) as well.
The Atangana-Baleanu fractional derivative
In relation to be with above definitions of fractional derivatives Riemann–
Liouville to Caputo and Caputo to Caputo-Fabrizio fractional derivatives, the diversity of
definitions is due to the fact that fractional operators take different kernel. Meanwhile,
Atangana-Baleanu presented newly proposed fractional derivatives based on the
generalized Mittag-Leffler function. The powerful significance of Atangana and Baleanu
fractional derivatives is the non-singularity and non-locality of the kernel. To fix the
shortcoming of the non-singularity and non-locality of the kernel, Atangana-Baleanu
fractional derivative is commonly known as AB fractional operator of order is
stated in [76-78]
∫ (
)
where, the fractional operator is so called Atangana-Baleanu fractional
operator. While, { } of Atangana-Baleanu fractional operator can be obtained
from equation (1.18). On the other hand, it is essential to note that Atangana-Baleanu
fractional operator can also be extended significantly by letting in equation (1.18)
as well.
1.10 Special Functions
The special functions have significant applications in a large variety of problems
encompassing the fractional differential equations, integral equations, diffusion, reaction,
reaction–diffusion and several other areas of theoretical physics and mathematics. The
special functions cover various problems of engineering, physical, biological and earth
sciences. For instance, rheology of fluid flow, kinematics in viscoelastic media, diffusion
processes in complex systems, diffusion in porous media, propagation of seismic waves,
anomalous diffusion, relaxation, turbulence and several others. However, to have the
usefulness of the Caputo, the Caputo-Fabrizio and the Atangana-Baleanu fractional
derivatives in this research work, first discusses some useful mathematical definitions of
special functions that will commonly be encountered and are inherently tied to fractional
derivatives. This thesis includes here some well-known special functions namely Mittag-
22
Leffler function, Fox-H function and Generalized M-function which are used in this
thesis for writing the lengthy and cumbersome calculation. The mathematical expressions
of Mittag-Leffler, Fox-H and Generalized M-functions are defined [79-84] as
respectively
∑
∑ ∏ ( )
∏ ( )
[ |
( )
( )]
∑ ∏ ( )
∏ ( )
[ |
( )
( )]
1.11 Constitutive Equations of Fluids
The relation between rate of deformation and stress is termed as constitutive
equation. In simple words, the properties of rheological materials are generally specified
by their, so called constitutive equation. Constitutive equations do not lie in the truth of
universality but offer some specific and particular properties for certain class of
substances. In continuation, some general principles such as objectivity principles and
symmetry principles are also verified and satisfied by constitutive equations. In Short, the
dimensional simplest constitutive equations are listed below
Newtonian fluid
The Newtonian fluid characterizes the relationship between the shear stress and
the rate of deformation, such equation are usually called Constitutive equation of
Newtonian fluid which are defined as
In this continuation, the following described equations are the constitutive equations for
non-Newtonian fluids
Second grade fluid
The second grade fluid model is preferred due to its relatively simple structure
which forms the simplest subclass of differential type fluids. The constitutive equations of
this model are
23
Where
denotes the material time derivative. From this model a classical
Newtonian fluid model can be reduced when and .
Maxwell fluid
The Maxwell fluid model is considered as a viscoelastic model which
characterizes the relaxation phenomenon of certain fluids. This model recognizes that the
body has a means to dissipate energy and to store energy. Where, the dissipation of the
energy describes its viscous nature and the storing energy characterizes the elastic
response of the fluids. In short, the constitutive equations of Maxwell model are
Here, represents relaxation time and
elucidates upper convected derivative.
A special case namely Newtonian fluid model when is substituted in the
constitutive equations (1.25).
Oldroyd-B fluid
The Oldroyd-B fluid model is also called Oldroyd 3-constant model and
categorized as a rate type fluid which enables the retardation time, stress-relaxation and
normal stress differences. These characteristics occur frequently in the motion of fluids.
Furthermore, the Oldroyd-B fluid model is inadequate to characterize the shear-
thinning/thickening phenomenon of fluid. The constitutive equations of Oldroyd-B fluid
are
[
]
Here, represents retardation time. A special case namely Maxwell fluid model
can be reduced when and are substituted in the constitutive equations (1.27).
24
Burger fluid
The Burger fluid model is also a rate type fluid model developed by Burger. This
model enables the most significant characteristics for the motion of fluid includes, the
transient creep property of earth mantle, the response of a variety of geological materials,
the post-glacial uplift and the response of asphalt and asphalt mixes. The constitutive
equations for Burger fluid model are
[
]
(
)
Generalized Burger fluid
This model is an extension of Burger fluid model which performs similar
characteristics as described in Burger fluid models. The generalized Burger fluid includes
some special cases, such as if then Burger model is achieved, if then
Oldroyd-B model is achieved, if then Maxwell model is achieved and
if then Newtonian model is achieved [85-86]. The constitutive
equations for generalized Burger fluid model are
[
]
1.12 Integral Transforms
Generally, origination of an integral transform was developed via Fourier integral
formula. Integral transform has been employed for the solutions of several problems in
engineering science and applied mathematics. An integral transform simply refers a
unique mathematical operation for real or complex-valued function which is transformed
into new function. The main significance of the integral transform is to converts a
difficult mathematical problem to relatively easy problem, which can be solved easily
without any cumbersome and lengthy calculations. Just like, for the solution of initial-
boundary value problems and initial value problems of integral and linear differential
equations, the integral transforms have been proved to be powerful operational methods.
Such problems mostly arise in the modeling of fluid mechanics problems. The basic
definition and concepts of integral transform can be described in the following standard
definition. Consider the function defined in as [87-89]
25
{ } ∫
Where, integral transform operator is represented by , the function is so
called kernel of the transform, the transform function is the image of function ,
and is the transform parameter. An integral transform of the function is stated in
then the improper integral is stated as below:
{ } ∫
∫
Here, the interval of integration is the unbounded interval . Meanwhile, an
equation (1.32) is said to be convergent if limit exists and if limit does not exist one can
say that an equation (1.32) is divergent. In continuation, one can retrieve number of
important integral transforms which include Laplace, Hankel, Fourier and several other
transforms by interchanging different kernel function into equation (1.32) with
different values of and . Few major transforms are discussed below which are utilized
in this thesis for investigating the general solutions of governing partial differential
equations of fluid flows.
Laplace and inverse Laplace transform
Suppose be a real or complex-valued function of time variable and is
real or complex parameter then the Lebesgue integral (1.31) is defined for Laplace
transform as [87-89]
∫
Here, the function is called original function, is a kernel function
of Laplace transform and the function defined by (1.34) is called Laplace image of
the function . The formal definition of inverse Laplace transform is stated as
∫
26
Applying mathematical definitions say equations (1.33) and (1.34), one can
investigate Laplace and inverse Laplace transform of various elementary and simple
functions as well which can be found in any standard of transforms [87-89].
Fourier sine and inverse Fourier sine transforms
There is no denying fact that linear initial value and boundary value problems
arising from fluid mechanics phenomenon can effectively be solved by invoking Fourier
sine and inverse Fourier sine transform. This transform is very significant for the general
solution of integral and differential equations for the below reasons:
The integral and differential equations are converted into elementary algebraic
equations, which capable us to investigate solution of transformed function.
Solutions of boundary value problems are then converted in the format of
original variables by inversions.
Solutions provided by Fourier sine and inverse Fourier sine transform are
interconnected with the convolution theorem generates an elegant compact form
of the solutions.
Consider a function which is piecewise continuous and absolutely integrable
over the Fourier sine and inverse Fourier sine transforms are stated as respectively
[87-89]
{ } √
∫
{ } √
∫
Here, the function is called original function of Fourier sine transform,
is a kernel function of Fourier sine transform and { } and { } are
the Fourier sine and inverse Fourier sine operators respectively.
Finite Hankel and inverse finite Hankel transforms
A German mathematician namely Hermann Hankel (1839-1873) is a real founder
of finite Hankel transformation which is commonly known as finite Fourier-Bessel
transform. He worked on Hankel functions so called Bessel functions of the third kind.
27
Owing to his scientific contribution towards finite Hankel transformation, he
suggested the usefulness of finite Hankel transformation when the axisymmetric problems
are formulated in cylindrical polar coordinates and spherically symmetric coverage
problems. The finite Hankel transformation is an integral transform whose kernel is the
Bessel function. The main significance of finite Hankel transform is that the differential
equations involving partial derivatives in cylindrical coordinates can be converted into the
differential equation involving ordinary derivatives by using the finite Hankel
transformation. In brevity, let a function be a function which is defined on ]
then finite Hankel and inverse finite Hankel transforms are stated as respectively [87-89]
{ } ∫
{ }
∑
Here, the function is called original function of finite Hankel transform,
is a kernel function of finite Hankel transform and { } and { }
are the finite Hankel and inverse finite Hankel operators respectively, the summation is
taken over all positive roots of . Meanwhile, few important relations which
have been highly used in this thesis are described as [90]
∫ (
)
∫ (
)
Convolution integral
Convolution integral is a mathematical relation that is extremely important in the
study of fractional calculus. This mathematical relation provides an elegant representation
to the solution in terms of integral form. Consider and be two functions of time
variable then the convolution of a product of two transformed functions is denoted by
. The inverse Laplace transform of convolution is defined as
28
{ }
alternatively, one can define as
∫
{ }
29
CHAPTER 2
Analytical Solutions of MHD Generalized Burger Fluid
with Porous Flow
30
2.1 Introduction
In this chapter the MHD generalized Burger’s fluid embedded with porous
medium is studied. The investigations of analytical solutions are presented as a sum of
Newtonian part and non-Newtonian part. The solutions are derived for the velocity field
and the shear stress while governing partial differential equations have been solved via
the integral transforms; and solutions are expressed into the compact form of infinite
series format. The general solutions also satisfy initial and boundary condition and
particularized for special cases along with sum of Newtonian and non-Newtonian forms.
The impacts on six models namely (i) Generalized Burger model, (ii) Burger model, (iii)
Oldroyd-B model, (iv) Maxwell model, (v) Second Grade model and (vi) Newtonian
model are investigated. These models are also discussed with and without porous medium
and magnetohydrodynamics effects on fluid flow. At the end, the impacts of permeability
(porosity), magnetism and several rheological parameters have been analyzed for fluid
flows by portraying graphical illustrations. Finally, the graphical results show that
Newtonian model moves slower than other models in presence and absence of magnetic
field and porous medium.
2.2 Modeling of the Governing Equations
The constitutive equations for an incompressible generalized Burgers' fluid are
[41, 86, 91, 92]
(
)
Where,
denotes the upper convected time derivative defined as
(
)
represents material time derivative. The unsteady flow of an incompressible fluid is
governed by:
31
For this problem, the equation (2.5) is employed for velocity field with an extra-stress
tensor as
Introducing equation (2.5) in equation (2.1) and considering the initial conditions as
yields and
(
)
(
)
in which tangential stress is . With reference [93], the generalized Burger’s fluid has
relation for is
(
) (
)
Assuming that there is no pressure gradient in the flow direction and introducing
equation (2.5) into equation (2.4) and using equations (2.7- 2.8), the following governing
equations are obtained as
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
Where,
,
,
are kinematic viscosity, applied magnetic field,
porous medium of the fluid, respectively.
32
2.3 Accelerated Plate with Electrically Conducting Burger Fluid in Porous Flow
Here, consider electrically conducting generalized Burgers fluid embedded in
porous medium lying above a plate perpendicular to the y-axis. The plate is permeated
under an influence of magnetic field normal to the flow in porous medium. For
the plate begins to accelerate in its own plane with velocity . Due to
the shear, the fluid above the plate is gradually moved as sketched in figure 2.1, while the
governing equations (2.9) and (2.10) and imposed conditions are:
Fluid
Figure 2.1 Geometrical configuration of accelerated plate
2.4 Solution of the Problem
2.4.1 Calculation of the Velocity Field
For finding the solutions of governing equation (2.9), the Fourier sine transform is
applied on equation (2.9), the expression is
{(
)
} { (
)
𝑥
𝑧
𝑦
33
(
) (
) }
simplifying equation (2.14) and employing the imposed conditions (2.11-2.13) gives
(
)
(
)
{ √
} (
)
(
)
Where, is the image of Fourier sine transform of as defined in
equation (A1) and the Fourier sine transform has to justify imposed condition equation
(2.11),
applying Laplace transform on equation (2.15) and keeping imposed conditions equations
(2.11) and (2.12), we obtain
√
[
{ }
{ } ]]
The sum of non-Newtonian and Newtonian part can be
obtained by making in equation (2.17). The Newtonian part of
equation (2.17) is
√
in order to balance the equation (2.18) for satisfying imposed conditions, equation (2.18)
is employed into equation (2.17), the balanced equation by using appendix (A1) is
√
√
34
√
[
{ }
]
applying inverse Fourier sine transform on equation (2.19), the suitable expression of
(2.19) is
∫
∫
∫
[
{ }
]
invoking inverse Laplace transform on equation (2.20) and using appendix (A2), the
obtained expression of velocity field in the format of convolution product is
( √
)
∫
∫
[
{ }
{ }
{ }
{ } ( )
]
finally, the velocity field is expressed in the compact form in terms of series, as
∫ ∫
[
{ } ( )
∑
{ }
( ) ( )
]
where,
35
( √
)
∫ ∫
is the Newtonian part of velocity field.
2.4.2 Calculation of the Shear Stress
For finding shear stress, the Laplace transform is applied to equation (2.10) as
(
)
The sum of non-Newtonian and Newtonian part can be
obtained by making in equation (2.24). The Newtonian part of
equation (2.24) is
employing the expression of equation (2.18) and (2.25) into equation (2.24), the balanced
equation for shear stress is found by using the appendix (A1) as
∫
∫
[
]
under the simplification, equation (2.26) is written for more suitable equivalent form as
∫
∫
[
{ }
{ }
{ }
{ }
]
36
invoking inverse Laplace transform on equation (2.27) and the shear stress is expressed in
the compact form in terms of series as
∫ ∫
[
{ } ( )
∑
{ }
( ) ( )
]
where,
∫ ∫
is the Newtonian part of shear stress.
2.5 Limiting Cases
2.5.1 Solution of Burger Fluid
Letting in equation (2.23) and (2.28) and using appendix (A3), the
solutions are obtained as
∫ ∫
[
{ }
( )
]
and corresponding shear stress
∫ ∫
37
[
( )
]
2.5.2 Solution of Oldroyd-B Fluid
Letting in equation (2.23) and (2.28) and using appendix (A4), the
solutions are obtained as
∫ ∫
[
( )
]
and corresponding shear stress
∫ ∫
[
( )
]
2.5.3 Solution of Maxwell Fluid
Letting in equation (2.23) and (2.28) and using appendix (A5),
the solutions are obtained as
∫ ∫
38
[
( )
]
and corresponding shear stress
∫ ∫
[
( )
]
2.5.4 Solution of Second Grade Fluid
Letting in equation (3.23) and (3.28), the solutions are
investigated as
∫ ∫
[
(
)
(
)
( )
(
)]
and corresponding shear stress
∫ ∫
[(
)
(
) ]
2.6 Results and Concluding Remarks
This portion is the analysis of the significance of permeability (porosity),
magnetism and several rheological parameters, material parameters on the fluid flow due
to accelerating plate for magnetohydrodynamic generalized Burger fluid embedded with
39
porous medium as a sum of Newtonian and non-Newtonian forms. In order to illustrate
the differences and similarities among various graphs for relevant physical aspects,
different numerical values are used for instance, time, permeability, porosity, magnetic
parameter, viscosity, non-zero constant, density, kinematic viscosity, relaxation time,
retardation phenomenon, material and rheological parameters few others. However, the
major findings/outcomes are listed below
(i). The general solutions for velocity field and shear stress have been expressed into
compact form i-e in terms of series form satisfying initial, boundary and natural
conditions as well. These solutions are obtained employing four translations of
integral transforms which are (i) Fourier Sine transform, (ii) Laplace transform,
(iii) Inverse Fourier Sine transform and (iv) Inverse Laplace transform. The
translation from (i) to (iv) are applied according to governing partial differential
equation and usual initial and boundary conditions.
(ii). The influence of time parameter is depicted in figure 2.2 in which both velocity
and shear stress profiles are absolute increasing function with increase in time.
(iii). Figure 2.3 is plotted for the rheology of viscosity of the fluid in which the elastic
behavior of fluid has a tendency to decline the profiles of velocity and shear stress
generated by motion of accelerating plate.
(iv). Figures 2.4,2.5,2.6 and 2.7 have been drawn for showing the influence on material
parameters ( and ), by considering nearer and smaller values of
and have similar and identical behavior of fluid flow as expected. It is worth
pointed out that and relaxation and retardation phenomenon respectively
have quiet contradictory effects on fluid flow for both profiles velocity as well as
shear stress.
(v). Figure 2.8 presents the profile of velocity and shear stress in which various
extreme small values are taken for magnetic field , it is clearly seen that it is a
Lorentz force which resist the fluid flow. This is due to the fact that magnetic field
B is applied in transverse direction.
(vi). Effects of permeability and porosity on the fluid motion are depicted in
figure 2.9 which has qualitatively scattering behavior on the fluid motion.
40
(vii). Figure 2.10 is drawn to give variations for the behavior on the fluid motion in
presence of magnetic field as well as porosity. In which the velocity field has
squeezed motion of fluid as compared with profile of shear stress.
(viii). Figures 2.11 and 2.12 display the variation in presence of porosity and magnetic
field respectively for six models namely (i) Generalized Burger model, (ii) Burger
model, (iii) Oldroyd-B model, (iv) Maxwell model, (v) Second Grade model and
(vi) Newtonian model, in which generalized Burger’s model is the efficient and
the Newtonian model has slowest behavior on motion either in the presence or
absence of porosity and magnetic field.
2.7 Validation of the Results
It is worth pointed out that this problem can also be considered for retrieving a
few solutions from the published literature. The obtained analytical solutions as equations
(2.33) and (2.34) can be particularized in the absence of magnetic field and porous
medium if and in equations (2.33) and (2.34) respectively, such solutions
can easily be retrieved in similar manners which can be recovered from literature as
investigated in [41]. The solutions obtained in [41] are in excellent agreement with the
present solutions in the absence of magnetic field and porous medium. The comparison is
shown in figure 2.13 for the graphical validation at three different times i-e (smaller time,
unit time and larger time), It is observed that for the smaller time both
velocities have identical behavior while for the unit time and larger time
velocity field obtained by present solution is increasing function. This graphical
illustration also confirms the accuracy of the present work.
41
Figure 2.2 Plot of velocity field and shear stress for at
Figure 2.3 Plot of velocity field and shear stress for at
42
Figure 2.4 Profile of velocity field and shear stress for at
Figure 2.5 Plot of velocity field and shear stress for at
43
Figure 2.6 Plot of velocity field and shear stress for at
Figure 2.7 Plot of velocity field and shear stress for at
44
Figure 2.8 Plot of velocity field and shear stress for at
Figure 2.9 Plot of velocity field and shear stress for at
45
Figure 2.10 Comparison of velocity field and shear stress at
Figure 2.11 Comparison of velocity field and shear stress at
46
Figure 2.12 Comparison of velocity field and shear stress at
Figure 2.13 Validation of the present solutions with obtained solutions by Jamil [41] for
the velocity field when and remaining parameters are at
47
CHAPTER 3
A Mathematical Analysis of Generalized Fractional
Burger Fluid for Permeable Oscillations of Plate with
Magnetic Field
48
3.1 Introduction
It is an established fact that mathematical analysis of fractional derivatives of
arbitrary order has several applications in various scientific fields. In this chapter, the
mathematical analysis for an electrically conducting flow of generalized fractional
Burgers' fluid with permeable oscillating plate is investigated. The governing partial
differential equations of fluid flow have been converted into fractional differential
equations by the Caputo fractional operator. For tracing out the analytical solutions of
velocity field and shear stress with and without magnetic field and porosity, the
techniques of Laplace transform are invoked. In order to get rid of the product of Gamma
functions, the analytical solutions are established in the format of Fox-H function which
satisfies imposed conditions. The final analytical solutions are particularized for limiting
cases, such as (i) the solutions are retrieved without magnetic field and permeability, (ii)
the solutions are converted into ordinary differential operator, and (iii) the solutions are
reduced for fractional Burger, fractional Oldroyd-B, fractional Maxwell fluid and
fractional Newtonian fluids. At the end, in order to highlight the similarities and
differences among various rheological parameters, the graphical illustration has been
depicted for fluid flows.
3.2 Modeling of the Governing Equations
The rheological equations for generalized Burgers' fluid can be characterized as
[94-96]
The upper convective time derivative is stated as
(
)
Here,
and
are material time derivative and upper convective derivative. The
well-known governing equations for an unsteady flow of incompressible fluid are
described as below
49
The equation (3.5) is considered for velocity field and extra-stress tensor as
Keeping initial conditions in mind, it is defined as
implementing equation (3.5) in equation (3.1), yields and the
suitable equation is
(
)
(
)
in which tangential stress is . As per previously published literature [93], the
generalized Burger fluid has relation for is
(
) (
)
Where are the permeability and porosity respectively. By considering that
flow direction is free from pressure gradient, balance of linear momentum, absence of
body forces and implementing equation (3.5) into equation (3.4) keeping with equations
(3.7-3.8), the governing equations for the generalized Burger fluid are obtained as
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
50
Where,
is applied magnetic field,
is porous medium and
is kinematic viscosity of the fluid. Using the concept of non-integer order derivative, so
called Caputo fractional operator, the governing equations for the generalized Burger
fluid in the form of non-integer order derivative or Caputo fractional derivative are
investigated as
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
where, the
is so called Caputo fractional operator described as [72-73]
{
∫
3.3 Porous Flow of Fractional Burger’s Fluid on Oscillating Plate with Magnetic Field
Here an electrically conducting incompressible generalized fractional Burgers
fluid is considered with porous medium occupying the space above an oscillating plate
which is placed perpendicular to the y-axis. The plate is saturated under an influence of
magnetic field and permeability. At the moment , the plate moves in its own
plane with oscillating velocity. When shear is functionalized, fluid and plate are gradually
moved, for this fractional equations which govern the fluid phenomenon are given by the
equations (3.11) and (3.12) and corresponding conditions are employed as
51
In order to have analytical solutions of fractional differential equations (3.11-3.12)
with above imposed conditions (3.14-3.16), integral transform methods are applied.
3.4 Solution of the Problem
Case-I: Cosine Oscillations
3.4.1 Mathematical Analysis of the Velocity Field
Apply the Laplace transform to equation (3.11) and the results (3.14-3.16), yields
( )
subject to the imposed conditions
and
and { }. The solution of equation (3.17) is obtained as
( √
)
In order to justify the initial and boundary condition, writing equation (3.18)
equivalently as by invoking appendix (A6) and (A7)
∑
( √ )
√
∑
∑
(
)
∑
∑
∑(
)
(
) (
) (
)
(
) (
) (
)
52
Inverting equation (3.19) by means of Laplace transform, and expressing final
expression of velocity field in terms of Fox-H function [79, 84] and convolution theorem,
the expression is
∑( √ )
√
∑
∑
(
)
∑
∑
[
(
)|
|
(
) (
) (
)
(
) (
) (
)
(
) ]
∫
3.4.2 Mathematical Analysis of the Shear Stress
Apply the Laplace transform to equation (3.12) and keeping the imposed
conditions (3.14-3.16), yields
(
)
differentiating equations (3.18) with respect to “ ” partially, then simplified expression as
√
(
)
writing equation (3.22) equivalently as by invoking appendix (A6) and (A7)
√ ∑
∑
∑
∑
∑
(
)
∑(
)
(
) (
) (
)
(
) (
) (
)
53
inverting equation (3.23) by means of Laplace transform, and expressing final expression
of shear stress in terms of Fox-H function and convolution theorem as
√ ∑
∑
∑
∑
∑
(
)
[
(
)
|
|(
) (
) (
)
(
) (
) (
)
(
)]
∫
Case-II: Sine Oscillations
Invoking the similar procedure, the calculation of velocity field and shear stress is
obtained as
∑( √ )
√
∑
∑
(
)
∑
∑
[
(
)|
|
(
) (
) (
)
(
) (
) (
)
(
) ]
∫
√ ∑
∑
∑
∑
∑
(
)
54
[
(
)
|
|(
) (
) (
)
(
) (
) (
)
(
)]
∫
3.5 Results and Concluding Remarks
In this section, the main features and concluding remarks for the present
mathematical analysis of magnetohydrodynamic and porosity are discussed. The
analytical solutions of velocity field and shear stress with and without
magnetohydrodynamics and porous medium have been established by employing discrete
Laplace transform with its inversion. The general analytic solutions are written in the
layout of the product of Gamma functions and Fox-H function. The equations (3.23-3.26)
are the analytical solutions of generalized fractional Burger flow with magnetic field and
permeability. In short, the generalized fractional Burger model has several special cases
which depend upon concerned rheological parameters. The important solutions can be
retrieved by setting certain rheological parameters as discussed in the table 3.1
Table 3.1- Rheological parameters for limiting solutions
Equation (s) Parameter (s) Type of solutions
(3.23-3.26) The solutions are in the absence of magnetic field
(3.23-3.26) The solutions are in the absence of porosity
(3.23-3.26) The solutions are in the ordinary differential operator
(3.23-3.26) The solutions are for fractional Burger fluid
(3.23-3.26)
The solutions are for fractional Oldroyd-B fluid
(3.23-3.26)
The solutions are for fractional Maxwell fluid
(3.23-3.26)
The solutions are for fractional Newtonian fluid
(3.23-3.26) The solutions are for first problem of stokes investigated
by Ilyas and Sharidan [96, see equation 20]
In short, the main features concerned with results are enumerated below
55
Figure 3.1 is presented for time parameter, it is observed that as time increases
then velocity field and shear stress are increasing function of time. It is noticed
that both velocity field and shear stress have qualitatively identical behavior of
fluid flows over the boundary.
Figure 3.2 is presented to show the impact of viscosity on fluid flow. An
interesting fact is achieved that velocity field and shear stress have decreasing
behavior of fluid flow as lower value of viscosity is used. In general, this
phenomenon meets with true facts of shear thickening and shear thinning.
The influence of relaxation time, retardation time and material parameters are
shown in figures 3.3 and 3.4. Here, it is noted that velocity field has reciprocal
trend of fluid flow. Meanwhile, velocity field is increasing and decreasing with
respect to rheological parameter vice versa.
Figure 3.5 elaborates the effects of magnetic field on fluid flow. Increase in magnetic
field decreases the velocity field while increases the shear stress. This may be due to
the fact that resistive force is generated by applied magnetic field which is alike a
drag force. Consequently, velocity field and shear stress have opposite trend due to
applied magnetic field which slows down the motion of fluid flow.
Figure 3.6 is plotted for porosity; here velocity field and shear stress show
scattering behavior reciprocally.
Comparison of four ordinary and fractional models namely (i) Burger fluid model,
(ii) Oldroyd-B fluid model, (iii) Maxwell fluid model and (iv) Newtonian fluid
model is shown in figure 3.7. It is found that both ordinary as well as fractional
Newtonian fluid moves faster in comparison with remaining other models. It is
also noted that all ordinary models have sequestrating behavior and fractional
models have scattering one.
Comparison of only ordinary and fractional Burger model with and without
magnetic field and porosity is presented in figure 3.8. It is pointed out that both
ordinary as well as fractional Burger fluid model without magnetic field and
porosity moves faster in comparison with remaining other models with and
without magnetic field and porosity. On the other hand, models with magnetic
field have sequestrating behavior and models with porosity have scattering
behavior over the whole domain of plate.
56
3.6 Validation of Results
The analytical solutions of velocity field and shear stress with and without magnetic field
and porosity have been investigated via fractional calculus approach. The Caputo
fractional operator is applied on the governing fractional differential equations of fluid
flows. In order to have the validity of the obtained results, few rheological parameters are
to be set for fulfilling the gaps between present solutions and the solutions obtained by
[96]. The main features are listed in table 3.2 as
Table 3.2- Rheological parameters for validation of results
Present solutions Published solutions [96]
The Caputo fractional parameter . The rotating parameter .
Here the present and published analytical solutions of velocity field are validated
with published data in figure 3. 9. It is worth mentioning that comparison of the present
results with a previous published result is in good agreement.
57
Figure 3.1 Plot of velocity field and shear stress for
with different
values .
Figure 3.2 Plot of velocity field and shear stress for
with different
values .
58
Figure 3.3 Plot of velocity field for
with different values and .
Figure 3.4 Plot of velocity field for
with different values and .
59
Figure 3.5 Plot of velocity field and shear stress for
with different values .
Figure 3.6 Plot of velocity field and shear stress for
with different values .
60
Figure 3.7 Comparison of velocity fields for four models for
.
Figure 3.8 Comparison of velocity fields for four models with and without magnetic field
and porous medium for
.
61
Figure 3.9 Comparison of present solution with the solution obtained by Ilyas et al. [96].
62
CHAPTER 4
Helices of Generalized Burger Fluid in Circular
Cylinder: A Caputo Fractional Derivative Approach
63
4.1 Introduction
In this chapter, the analytical solutions are derived for the effects of generalized
Burger fluid flow for infinite helically moving cylinder. The analytical expressions are
traced out for velocity and shear stress profiles by utilizing mathematical Hankel and
Laplace transforms with their inversions. The expressions of analytical solutions have
been established in the layout of Fox-H function. The general solutions satisfy
initial and boundary conditions and reduced for limiting / particularized solutions of
Burger, Oldroyd-B, Maxwell, Second grade fluids. The helical flows of four models as
Burger, Oldroyd-B, Maxwell and Newtonian fluids are compared with existing published
research exhibit good agreement and reveal the accuracy and validity of present study.
With the help of graphs, the influence of rheological parameters such as dynamic
viscosity, time, fractional parameter, material parameters, oscillations, retardation and
relaxation periods are investigated for helicity of cylinder on fluid flow.
4.2 Modeling of the Governing Equations
The constitutive equations for an incompressible generalized Burgers' fluid are
(
)
The upper convective derivative respectively defined as
(
)
represents material time derivative. This incompressible generalized Burgers' model
has particular cases of fluid models as discussed in the Table. 4.1
Table 4.1- Rheological parameters for particular solutions
Parameter (s) Type of fluid model
Newtonian fluid model
Maxwell fluid model
Oldroyd-B fluid model
Burger fluid model
64
For this problem, the following assumptions can be considered for velocity field
and extra-stress tensor
Initially the fluid does not take motion because it is in rest when then due to
this flow constraint of incompressibility is consequently fulfilled,
Here, the term is then the corresponding result can be attained as[86, 97]
(
)
(
) (
)
where, the shear stresses and are considered vividly.
Neglecting body forces and balance of linear momentum without pressure gradient leads
to identical equation due to rotational symmetry [98]
(
)
(
)
cancelling and from equations (4.5) and (4.6), with the help of Caputo
derivatives operator fractionalized governing equations are
(
)
(
)
(
)
]
(
)
(
)
(
)
]
(
)(
)
(
)
65
(
)
(
)
where, Caputo-fractional operator of order is defined as [72, 73]
{
∫
4.3 Oscillations of Cylinder Due to Helicity of Fluid
Consider a flow of generalized fractional Burger fluid at rest in an infinite circular
cylinder subject to imposed conditions with radius . At the initial stage, the
cylinder starts to rotate around its axis with or
(angular velocity) and to oscillate along with the same axis
or (oscillating velocity). Fluid is
gradually moved due to shear stress; while the velocity is assumed of the form of
equation having the governing partial differential equations (4.7-4.10). Such type
of flow generates helices (flow in helicity), in general its streamlines yield helices. See
figure 4.1 for the detailed configuration of helical cylinder.
Figure 4.1 Geometrical configuration of helical cylinder
The corresponding boundary and initial conditions are
66
]
]
4.4 Solution of the Problem
4.4.1 Investigation of the Velocity Field
Case-I: and
Applying Laplace transform to equations (4.7-4.8) and considering the initial and
boundary conditions (4.12-4.14), the following set of partial differential equations are
obtained
(
)
(
)
(
)
]
(
)
(
)
(
)
]
where, and are the image functions of and respectively
and have to fulfill the imposed conditions defined as in equation (4.17)
Applying Hankel transform on equations (4.15-4.16) and taking into account
equation (4.17) along with the well-known results [ Debnath and Bhatta, 87 ]
( ) ( ) ∫ (
) ( )
( ) ( ) ∫ (
) ( )
we obtain that,
( )
67
(
)
(
)
( )
(
)
(
)
where,
( ) ∫ ( ) ( )
( ) ∫ ( ) ( )
are the Hankel transforms of and respectively. Inverting with Hankel
transform for perusing and , rewriting equations (4.20-4.21) in
equivalent form as
( )
[
(
)
{ (
)
(
)}
]
( )
[
(
)
{ (
) (
)}
]
inverting equations (4.24-4.25) by Hankel transform [87]
∑
( )
∑
( )
and using the fact of integrals as described below
68
∫ ( )
∫ ( )
The obtained simplified form is as below.
∑
( )
(
)
{ (
)
(
)}
∑
( )
(
)
{ (
) (
)}
writing equations (4.28-4.29) into series form by invoking appendix (A6) and (A7), the
equivalent expressions are
∑
( )
( )
∑( )
∑(
)
∑
(
)
∑(
)
∑
(
)
∑
∑
∑
( )
( )
∑( )
∑(
)
∑
(
)
∑(
)
∑
(
)
∑
∑
69
inverting equations (4.30-4.31) by Laplace transform and writing in terms Fox-H
function, the final expressions of velocities can be expressed as
∑ ( )
( )
∑( )
∑(
)
∑
(
)
∑(
)
∑
(
)
∑
[ |
]
∑ ( )
( )
∑( )
∑(
)
∑
(
)
∑(
)
∑
(
)
∑
[ |
]
equations (4.32-4.33) are general solutions for velocities satisfying initial and boundary
conditions. Where, the newly defined special Fox-H function is
∑ ∏
∏
[ |
]
70
4.4.2 Investigation of the Shear Stress
Applying Laplace transform on equation (4.9-4.10) with implementing equations (4.12-
4.14) as initial and boundary conditions, it is found that
(
)(
)
(
)
(
)
(
)
Where, using identities ( ) ( ) and ( )
( )
( ), it is obtained as
(
) ∑
( )
( )
(
)
{ (
)
(
)}
(
)
∑
( )
( )
(
)
{ (
) (
)}
employing equations (4.37-4.38) into equations (4.35-4.36) respectively and using
Laplace transform on both sides to equations (4.37-4.38), we obtain final expressions of
shear stress in terms of Fox-H function by invoking appendix (A6) and (A7) as
∑ ( )
( )
∑( )
∑(
)
∑
(
)
∑(
)
∑
(
)
71
[(
)
|
( )
]
∑
( )
( )
∑( )
∑(
)
∑
(
)
∑(
)
∑
(
)
[(
)
|
( )
]
Case-II: and
Employing identical procedure, solutions for velocities and shear stresses are
obtained for cosine oscillations.
∑ ( )
( )
∑( )
∑(
)
∑(
)
∑
(
)
∑
[ |
]
∑ ( )
( )
∑( )
∑(
)
∑
(
)
∑(
)
∑
(
)
∑
[ |
]
72
∑ ( )
( )
∑( )
∑(
)
∑
(
)
∑(
)
∑
(
)
[(
)
|
( )
]
∑
( )
( )
∑( )
∑(
)
∑
(
)
∑(
)
∑
(
)
[(
)
|
( )
]
4.5 Results and Conclusion
This portion is written for the final analytical expressions which are obtained for
velocity and shear stress profiles by utilizing mathematical Hankel and Laplace
transforms with their inversions are analyzed. The analytical expressions have been
established in the layout of Fox-H function and imposed conditions are fulfilled for the
oscillating velocity and angular velocity along with same axis. It is important to point out
that various solutions can be retrieved from literature under certain conditions. For the
sake of literature and the most elementary case, we set then the
fluid is termed as Newtonian fluid (the classical Navier-Stokes viscous fluid), for such
case the solutions have been investigated by [Jamil et al. 99 (see equations. 46-49)]. The
solutions and effects of helicity for second grade and viscoelastic fluid (Maxwell fluid)
can be retrieved by letting and [Jamil et al. 99 (see
equations, 32-33 and 40-41)] respectively. The similar solutions for Oldroyd-B fluid
have been deduced for longitudinal and torsional time-dependent shear stresses by
employing [Jamil et al. 100 (see equations, 25-26 and 32-33)]. In order to
reveal physical aspects from obtained results, the graphical illustrations are investigated
and discussed as below
73
The analytical solutions for two components of velocity and shear stress have
been established in terms of newly defined Fox-H function and the
correctness of these solutions have been determined by graphical illustrations with
comparison of four models for ordinary and fractionalized fluids, i-e (i) Ordinary
and fractionalized Burger model (ii) Ordinary and fractionalized Oldroyd-B model
(iii) Ordinary and fractionalized Maxwell model and (iv) Ordinary and
fractionalized Newtonian model.
Figure 4.2 is portrayed for velocity components, both velocity components are
increasing functions with variation of time. Additionally, it is obvious that due to
helical flow in circular cylinder the influence of the rigid boundary is larger over
the boundary.
The rheological effects of relaxation , retardation and material parameters
have been depicted in figures 4.3, 4.5, 4.4, 4.6 respectively. More precisely,
the relaxation parameter has reciprocal behavior of fluid flow in comparison
with retardation parameter. Qualitatively the same phenomenon is observed for
the material parameters on the whole domain. This phenomenon may be
due to the rotations and oscillations of cylinder because stream lines of rotations
and oscillations of cylinder are helices (Helical flow).
The impact of radius is shown in figure 4.7. It is noted that the fluid motion is
sequestrated about its own plane. This is due to the fact that as radius increases
the motion of fluid slows down and decelerates the helices in cylinder.
The viscoelastic behavior of real materials is modeled by fractional-order laws of
deformation. Figure 4.8 signifies the dynamics of fractionalized and ordinary fluid
models with comparison. The models are (i) ordinary and fractionalized Burger
model (ii) ordinary and fractionalized Oldroyd-B model (iii) ordinary and
fractionalized Maxwell model and (iv) ordinary and fractionalized Newtonian
model. It is pointed out that in figure 4.8 fractionalized models are scattered and
fractionalized Newtonian fluid moves faster in comparison with other
fractionalized models. On the contrary, in figure 4.9, ordinary models are
sequestrated and ordinary Newtonian fluid is slower in comparison with other
ordinary models. Hence, due to significance of fractional parameter helices of
cylinder for fractionalized and ordinary fluid have reciprocal behavior of fluid
flow over the boundary.
74
Figure 4.2 Plot of velocity fields from equations (32) and (33) for
and distinct values of
.
Figure 4.3 Plot of velocity fields from equations (32) and (33) for
and distinct
values of .
75
Figure 4.4 Plot of velocity fields from equations (32) and (33) for
and distinct values of .
Figure 4.5 Plot of velocity fields from equations (32) and (33) for
and distinct values
of .
76
Figure 4.6 Plot of velocity fields from equations (32) and (33) for
and distinct values
of .
Fig 4.7 Plot of velocity fields from equations (32) and (33) for
and distinct
values of .
77
Figure 4.8 Plot of velocity fields for fractionalized Newtonian, fractionalized Maxwell,
fractionalized Oldroyd-B and fractionalized Burger, for
.
Figure 4.9 Plot of velocity fields for ordinary Newtonian, ordinary Maxwell, ordinary
Oldroyd-B, ordinary Burger for
.
78
CHAPTER 5
Analytic Study of Molybdenum Disulfide Nanofluids:
An Atangana-Baleanu Fractional Derivatives Approach
79
5.1 Introduction
This chapter is devoted to the analysis of nanofluids because nanofluids have
attained the consideration of researchers for last two decades. Molybdenum disulfide is
usually considered as an inorganic compound consisting of alternative layers of sulfur and
Molybdenum atoms, having chemical formula MoS2. Molybdenum disulfide is a mineral
so called molybdenite containing silvery black solids. Molybdenum disulfide is largely
utilized as a lubricant in the form of solids; this is due to the fact that it has robustness and
low friction properties. MoS2 is unaffected through oxygen as well as dilute acids and is
quite stable. MoS2 is composed of lamellar crystal structure designed as sandwiched
between trigonal prismatic molecular structures of sulfur atoms in a hexagonally closed-
pack arrangement. In brevity, the principle ore and van der walls interaction of MoS2 are
shown in figure 5.1 and figure 5.2 respectively
Figure 5.1 A principle ore of MoS2 Figure 5.2 Van der walls interaction of MoS2
Here it is well established fact that, the significance of different shapes of
molybdenum disulfide nanoparticles containing in ethylene glycol have recently attracted
researchers, because of numerical or experimental analysis on shapes of molybdenum
disulfide and lack of fractionalized analytic approach. In order to analyze the shape
impacts of molybdenum disulfide nanofluids in mixed convection flow with magnetic
field and porous medium, Ethylene glycol is considered as a base fluid in which
Molybdenum disulfide nanoparticles are suspended. Non-spherically shaped
Molybdenum disulfide nanoparticles namely platelet, blade, cylinder and brick are
utilized in this analysis. The mathematical modeling of the problem is characterized by
employing modern approach of Atangana-Baleanu fractional derivatives and the
governing partial differential equations are solved via Laplace transforms with its
inverses. The general solutions are obtained for temperature distribution and velocity field
and are expressed in terms of compact form of M-function Ta
bM . A graphical
illustration is depicted to compare the non-spherically shaped Molybdenum disulfide
nanoparticles. The Atangana-Baleanu fractional derivatives model have also been
80
compared with ordinary derivative model and discussed graphically by choosing various
rheological parameters.
5.2 Formulation of Flow Equations
Assume unidirectional and incompressible convection flow of ethylene glycol
based nanofluids containing nanoparticles based on Molybdenum disulfide saturated
porous medium with effects of radiation. The convection flow in the absence of an
external pressure gradient is caused by a buoyancy force. The fluid is considered under
the effects of a transverse magnetic field 0B applied perpendicularly to the flow. The
impact of convinced magnetic field can be ignored when the magnetic Reynolds number
is assumed to be small. The electric field due to polarization is also ignored because
external electric field is considered to be zero, while flow is considered under the
assumption of no slip boundary condition. The x-axis and y-axis are taken along the flow
and normal to the flow direction respectively. The governing equations of momentum and
energy are described by implementing the assumptions of Boussinesq approximation as
(
2
0 ) ( )
( )
Here, dynamic viscosity and thermal conductivity are based on Crosser’s and
Hamilton (1962) model which gives validation for non-spherical as well as spherical
nanoparticles are utilized. In this model,
( )
( )
The empirical shape factor as
is appearing in equation (5.3), where, is
the sphericity. In this connection, sphericity is stated as the ratio between surface area of
the real particle and surface area of the sphere with equal volumes. While and are
constants which depend on the particle shape. In brevity, the particle shapes and constant
and are illustrated in the table 5.1 [101]
81
Table 5.1 Sphericity for different nanoparticle shapes with constants a and b .
Model Platelet Blade Cylinder Brick
0.52 0.36 0.62 0.81
37.1 14.6 13.5 1.9
612.6 123.3 904.4 471.4
Table 5.2 Thermo-physical properties of ethylene glycol and nanoparticles
Nanoparticles /
Base fluid (m
-3 Kg) ρ (JKg
-1 K
-1) Cp (Wm
-1.K
-1) k
MoS2 5.06 × 103 397.21 85-110
Cu 8933 385 401
Al2O3 3970 765 40
Ag 10500 235 429
EG 1.115 0.58 0.1490
Solving equations (5.1-5.2) by using the relations investigated by [102, 103] as
( )
[
]
Here, subscripts and are denoted for base fluid and solid nanoparticle of the
fluid properties while is the volume fraction of the nanoparticles. It is also assumed that
temperature of both plates and are high and generates radiative heat transfer, which
is defined as
{ }
82
Employing equation (5.6) in equation (5.2), we arrive
( )
{ }
Implementing dimensional variables given below in equation (5.1) and equation
(5.7) and equation (5.3-5.5) as
The governing partial differential equations using equation (5.8) as
(
)
[
]
where,
( )
( )
( )
[ ( )
( )
]
Here, is Peclet number, is Reynold’s number, is thermal Grashof
number, is a magnetic field, is radiation parameter. The following conditions are
imposed as
Simplifying equations (5.9-5.10) and using newly defined Atangana-Baleanu
fractional operator of order as [77,78]
83
(
)
∫ (
)
For equation (5.15), ( ) ∑( )
is the Mittage-Leffler function, the
final expression of governing equations is
where,
[
]
5.3 Analytical Solution of Problem
5.3.1 Temperature Distribution Using Atangana-Baleanu Fractional Derivative
Employing discrete Laplace transform on equation (5.17) and imposing the
equations (5.12-5.14), taking
, yields
(
)
the solution equation (5.19) is traced out with imposed conditions as
{ √
}
expanding equation (5.20), into suitable format for satisfying the initial and boundary
conditions, an equivalent expression
84
∑
( √ )
∑
∑
(
) (
)
(
) (
)
Where, ,
and . Now inverting equation (5.21) by
means of discrete Laplace transform and expressing final solution of temperature
distribution in terms of M-Function as previously published papers [79, 82,84]
∑( √ )
∑
[ |
(
) (
)
(
) (
)
]
Here, the property of
M-Function is defined as [79, 82,84]
∑
∏
∏
[ |
( )
( )]
5.3.2 Velocity Field with Atangana-Baleanu Fractional Derivative
Employing discrete Laplace transform on equation (5.16) and imposing the
equation (5.12-5.14), taking
, it is obtained as
(
)
the solutions of differential equation (5.24) is traced out via initial and boundary
conditions, we get
{ √
} (
)
{ √
}
85
Where, the parameters are supposed as ,
, ,
and
⁄
Now expanding equation (5.25), into suitable format for satisfying the initial and
boundary conditions, an equivalent expression
∑
( √ )
∑
(
)
∑
(
)
(
) (
)
∑
( √ )
∑
(
)
∑
∑
(
) (
)
(
) (
)
∑
( √ )
∑
(
)
∑
∑
(
) (
)
(
) (
)
inverting equation (5.26) by means of discrete Laplace transform and expressing final
solution of velocity field in terms of M-Function as previously published papers
[79, 82,84]
∑( √ )
∑
(
)
[ |
(
)
(
) (
)
]
∑
( √ )
∑
(
)
∑ (
)
[ |
(
) (
)
(
) (
)
]
∑
( √ )
∑
(
)
∑ (
)
[ |
(
) (
)
(
) (
)
]
86
5.4 Results and Concluding Remarks
The different shapes of Molybdenum disulfide nanoparticles containing in
ethylene glycol have been analyzed using fractionalized analytic approach for mixed
convection flow with magnetic field and porous medium. Ethylene glycol is chosen as a
base fluid in which Molybdenum disulfide nanoparticles are suspended. The
nonspherically shaped Molybdenum disulfide nanoparticles namely platelet, blade,
cylinder and brick are used in this study as discussed in table 5.1 and table 5.2. Governing
partial differential equations have been modeled by employing modern approach of
Atangana-Baleanu fractional derivatives and then solved via Laplace transforms methods.
The final expression are investigated for velocity and temperature while these general
solutions are expressed in the layout of M-function . The Atangana-Baleanu
fractional derivatives models have been compared with ordinary derivatives models and
discussed in detail by varying various rheological parameters. However, the major
outcomes are enumerated below:
Figure 5.3 presents the comparison of different types of nanoparticles namely,
(silver), (copper), (molybdenum disulfide) and (alumina). The
results show that the moves faster and has highest velocity in comparison with
remaining types of nanoparticles, followed as (copper), (molybdenum
disulfide) and (alumina) in ethylene glycol based nanofluids. This may be
due to fact that thermal conductivity and viscosity of nanofluids increase for large
deferment/suspension with volume fraction of nanoparticles as stated in Hamilton
and Crosser’s (1962) model.
Figure 5.4 elucidates the impacts of Grashof number. It is observed that velocity
field is increasing function with respect to increase in Grashof number. Increase in
Grashof number indicates an increase in free convection and buoyancy force. This
is due to the fact that conduction is not dominant in comparison with convection
by increasing temperature.
The effects of different shapes of molybdenum disulfide nanoparticles namely
platelet, blade, cylinder and brick on the velocity field of ethylene glycol based
nanofluids have been plotted in figure 5.5. It is quite clear to mention that
nanoparticles namely platelet, cylinder and brick have least velocity in contrast
with blade-shaped nanoparticles. Brick and blade shaped nanoparticles have
lowest viscosities in discrepancy with platelet and cylinder nanoparticles. This is
87
due to the fact that particle shapes totally depend upon the strong behavior of
viscosity.
The influence of radiation parameter is emphasized in figure 5.6. It is observed
from physical point of view that an increase in thermal radiation generates an
amount of heat energy due to fluid molecules convection.
Volume fraction of molybdenum disulfide nanoparticles has interesting behavior
on temperature distribution and velocity field of ethylene glycol based nanofluids.
This behavior is depicted in figure 5.7. The velocity field has scattering behavior
by increasing volume fraction of molybdenum disulfide nanoparticles, while
temperature distribution shows sequestrating oscillations over whole boundary.
This may be due to the fact that an increase in volume fraction leads fluid more
viscous.
Figures 5.8 and 5.9 disclose the hidden phenomenon of transverse magnetic field
and permeability on velocity field as well as temperature distribution. It is noted
that due to Lorentz force, velocity field has resistive behavior of fluid. In simple
words, Lorentz force which is identical to drag force tends the fluid velocity to be
slowed down. On the other hand, in Fig.5.9 fluid friction is reduced due to
increase in permeability.
Figure 5.10 is presented for smaller and larger time for analyzing the effects of
velocity field on molybdenum disulfide nanoparticles with ethylene glycol namely
(silver), (copper), (molybdenum disulfide) and (alumina). For
smaller time , moves faster in
comparison with other nanoparticles, for larger time ,
moves slower in comparison with other nanoparticles. In
brevity, for a unit time , almost an identical behavior is observed for all
nanoparticles with resistivity.
Figure 5.11 explores the effects of Atangana-Baleanu fractional derivative verses
ordinary derivative on the velocity field. In this figure, an opposite direction of
fluid flow is observed. This may be due to the non-locality and non-singular
kernel of Atangana-Baleanu fractional derivative. It is noted that temperature
distribution was also analyzed for Atangana-Baleanu fractional derivative verses
ordinary derivative in which reciprocal behavior was observed. But for the sake of
simplicity, graphical illustration of temperature distribution is not included here.
88
Figure 5.3 Plot of velocity field for four types of nanoparticles in an ethylene glycol based
nanofluid when
.
Figure 5.4 Plot of velocity field for Molybdenum disulfide in ethylene Glycol based
nanofluid when
with different values of .
89
Figure 5.5 Plot of velocity field for four shapes of molybdenum disulfide in ethylene
glycol based nanofluid when
.
Figure 5.6 Plot of velocity field for molybdenum disulfide in ethylene glycol based
nanofluid when
with different values of .
90
Figure 5.7 Plot of velocity field and temperature distribution for molybdenum disulfide in
ethylene glycol based nanofluid when
with different values of .
Figure 5.8 Plot of velocity field for molybdenum disulfide in ethylene glycol based
nanofluid
with different values of .
91
Figure 5.9 Plot of velocity field for molybdenum disulfide in ethylene glycol based
nanofluid
with different values of .
Figure 5.10 Comparison of velocity field for four types of molybdenum disulfide in
ethylene glycol based nanofluids for smaller and larger times when
92
Figure 5.11 Comparison of velocity field for Atangana-Baleanu fractional derivative
verses ordinary derivative when
.
93
CHAPTER 6
Applications of This Research and Future
Recommendations
94
6.1 Applications of Non-Newtonian Fluid
The research in this thesis is based on the non-Newtonian liquids/fluids. Several
mathematicians, engineers, scientists and researchers have significant opinions that flows
of non-Newtonian liquids/fluids are extremely important in industrial and manufacturing
processes. Ultimately several industries will benefit as the result of the understanding of
the dynamics and physics of helical, oscillating, constantly rotating, and accelerating
processes of circular cylinder and plate as shown in Fig. 6.1. Meanwhile, the non-
Newtonian fluid mechanics provides the theoretical foundation for the turbomachines and
hydraulics. Both focus on the engineering uses of fluid properties. Any device that
extracts energy from a continuously moving stream of fluid is termed as a Turbomachine.
Turbomachines have direct applications in the following terms
Compressors, pumps and turbines in mechanical applications for example, irrigation,
sewage treatments plants and heating ventilation and air conditioning systems.
Power generation in thermal power plants, hydro power plants, wind turbine.
Jet propulsion, chemical and food processing industries.
On the other hand, hydraulics is the liquid version of pneumatics which is highly
used in applied science and engineering because it deals with mechanical properties of
fluids or liquids. From application point of view, a word Hydraulic covers most of
engineering modules,
Dam design, pipe flow, river channel behavior, pumps.
Turbines, hydropower, fluidics and fluid control circuitry, flow measurement and erosion.
Figure 6.1 Role of non-Newtonian fluids in industries for unidirectional and oscillating
flows.
95
In order to understand the unidirectional and oscillating flows in manufacturing
process, as shown in above figure. One has to follow the set of steps listed below:
(i). The flow of raw material, (ii). The flow of work-in-process, (iii). The flow of finished
goods (iv). The flow of operators, (v). The flow of machines, (vi). The flow of
information and (vii). The flow of engineering. It is very important not to skip these steps
while manufacturing processes.
6.2 Applications of Magnetohydrodynamics (MHD)
Magnetohydrodynamics (MHD) deals with the dynamics of a conducting fluid
which interacts with a magnetic field. The magnetohydrodynamic non-Newtonian fluids
have also significant role in science and engineering phenomenon. For instance
It is well established fact that all metallic materials are magnetic in nature. The
removal of non-metallic inclusions is extracted from iron ore (magnetite) by
various methods such as magnetic separation. For instance, one can consider the
magnetic roller for removal of magnetic and non-magnetic particles form finely
ground ore as shown in the following figure 6.2
Figure 6.2 Process of magnetic separation of magnetite iron ore
However, several researchers use certain amount of aluminum in molten steel by
which they discuss the removal of non-metallic inclusions by utilizing high frequency
magnetic field.
96
The efficient thermal power generating systems are usually influenced by heat,
this is due to the fact that the thermal energy is directly converted in to electrical
energy (MHD) in power plant. The magnetohydrodynamic processes can be
employed on many purposes, for instance hydroelectric power plants, nuclear
power plants, commercial power generation systems and few others.
Magnetohydrodynamics as a science of electrically conducting fluids has its
applications in astrophysics and geophysics. The magnetohydrodynamics is
interconnected to several engineering phenomenon such as liquid-metal cooling of
nuclear reactors, electromagnetic casting and power generations and plasma
confinement (see figure 6.3).
Figure 6.3 Plasma Confinement
6.3 Applications of Nanotechnology
Nanotechnology has diverted the attention of several scientists and researchers
because of its technological development and industrial applications. The nanotechnology
refers to the type of technology that scientists, engineers and researchers use for materials
at the nanometer size. Undoubtedly nanofluids would be a significant topic of the
upcoming era because of thermophysical properties and rich increase in heat transfer rate.
Nanofluids are considered better thermally effective in comparison with conventional
heat transfer fluids. It is also well established fact that the enhancement of thermal
conductivity is due to thermophysical properties which plays an adhesive role in
nanofluids for mass and heat transfer. It is well established fact that the suspension of
97
nanoparticles in nanofluids depends upon preparation and stability, shape and size,
volume fraction and temperature factors, because the suspension of nanoparticles
characterizes the perfect and effective heat transfer. In short, the cooling is one of the
most pressing needs of many industrial technologies because of their ever-increasing heat
generation rates. Conventional heat transfer fluids such as water, alcohol, air, engine oil
and kerosene shows very low thermal conductivity. In order to increase energy efficiency
and heat transfer, several industries have opted efficient coolants. For instance, nuclear
reactors coolants, car radiator coolants, industrial coolants, Smart Fluids, nanofluids in
automobile fuels, extraction of geothermal power and few others. A useful coolant is
sketched below in Fig. 6.4.
Figure 6.4 Coolant tower of power plant
98
6.4 Future Recommendations
It is very significant to highlight some extensions regarding this research work
with different geometries for non-Newtonian fluid as
The same problems can be extended with different fractional derivatives, for
instance the Caputo-Fabrizio, Atangana-Baleanu and Atangana-Koca fractional
derivatives and few others.
By invoking the natural and Sumudu transforms, the same problems can be
extended with and without magnetohydrodynamic and porous medium.
The problem discussed in chapter five can be analyzed via different base fluids,
for instance, engine oil, Kerosene, methanol excreta using few suitable
nanoparticles.
The same problems can be extended with different numerical schemes, like lattice
Boltzmann method, Adomian decomposition method, Homotopy perturbation
method and Keller Box method.
The problems of this thesis can be analyzed under the assumptions of no slip
conditions, first and second order slip conditions with Newtonian heating and
chemical reactions.
In order to enhance or modify the work of this thesis, the different numerical
techniques can be applied for three dimensional study of the problems.
99
Appendix
{ } { }
∫
{ }
{ }
∑
∑
100
References
[1] K. R. Rajagopal, “A Note on Unsteady Unidirectional Flows of a Non-Newtonian
Fluid”, International Journal of Non-Linear Mechanics, 17, 369-373 (1982).
[2] T. Sarpkaya and P.G. Rainey, “Stagnation Point Flow of a Second Order
Viscoelastic Fluid”, Acta Mechanica, 11, 237-246 (1971).
[3] M. J. Crochet and R. Keunings, “Die Swell of a Maxwell Fluid: Numerical
Prediction”, Journal of Non-Newtonian Fluid Mechanics, 7, 199-212 (1980).
[4] J. E. Dunn and K. R. Rajagopal, “Fluids of Differential Type: Critical Review and
Thermodynamic Analysis”, International Journal of Engineering and Science, 33,
689-729 (1995).
[5] M. E. Erdogan and C. E. Imrak, “On Some Unsteady Flows of a Non-Newtonian
Fluids”, Applied Mathematical Modeling, 31, 170-180 (2007).
[6] R. P. Chhabra and J. F. Richardson, “Non-Newtonian Flow and Applied
Rheology”, Engineering Applications, 2nd edition (Elsevier, New York) 2008.
[7] F. Irgens, “Rheology and Non-Newtonian Fluids”, (Springer, Berlin) 2014.
[8] J. C. Maxwell, “On the Dynamical Theory of Gases”, Philos. Transaction in Royal
Society of London, 157, 26-78 (1866).
[9] J. J. Choi, Z. Rusak, J. A. Tichy, “Maxwell Fluid Suction Flow in a Channel”,
Journal of Non-Newtonian Fluid Mechanics, 85, 165-187 (1999).
[10] C. Fetecau and F. Corina, “A New Exact Solution for the Flow of a Maxwell Fluid
Past an Infinite Plate”, International Journal of Non-Linear Mechanics, 38, 423-
427 (2003).
[11] F. Corina, M. Jamil, F, Constantin, S. A. Imran, “A Note on the Second Problem
of Stokes for Maxwell Fluids”, International Journal of Non-Linear Mechanics,
44,1085-1090 (2009).
[12] C, Friederich, “Relaxation and Retardation Functions of the Maxwell Model with
Fractional Derivatives”, Rheologica Acta, 30, 151-158 (1991).
101
[13] T. Hayat, S. Nadeem, S. Asghar, “Periodic Unidirectional Flows of a Viscoelastic
Fluid with the Fractional Maxwell Model”, Applied Mathematics Computation,
151, 153-161 (2004).
[14] C. Fetecau, M. Athar, C. Fetecau, “Unsteady Flow of a Generalized Maxwell
Fluid with Fractional Derivative due to a Constantly Accelerating Plate”,
Computers and Mathematics with Application, 57, 596-603 (2009).
[15] D. Vieru and A. Rauf, “Stokes Flows of a Maxwell Fluid with Wall Slip
Condition”, Canadian Journal of Physics, 89, 1061-1071 (2011).
[16] I. Khan, F. Ali, U. S. Haq, S. Shafie, “Exact Solutions for Unsteady MHD
Oscillatory Flow of a Maxwell Fluid in a Porous Medium”, Zeitschrift für
Naturforschung A: A Journal of Physical Sciences, 68, 635-645 (2013).
[17] S. Nadeem, R. U. Haq, Z. Khan, “Numerical Study of MHD Boundary Layer
Flow of a Maxwell Fluid Past a Stretching Sheet in the Presence of
Nanoparticles”, Journal of the Taiwan Institute of Chemical Engineers, 45, 121-
126 (2014).
[18] I. Khan, N. A. Shah, Y. Mahsud, D. Vieru, “Heat Transfer Analysis in a Maxwell
Fluid over an Oscillating Vertical Plate Using Fractional Caputo-Fabrizio
Derivatives”, European Physical Journal Plus, 132, (2017). DOI
10.1140/epjp/i2017-11456-2
[19] M. A. Imran, I. Khan, M. Ahmad, N. A. Shah, M. Nazar, “Heat and Mass
Transport of Differential Type Fluid with Non-Integer Order Time-Fractional
Caputo Derivatives”, Journal of Molecular Liquids, 229, 67-75 (2017).
[20] I. Khan, N. A. Shah, L. C. C. Dennis, “A Scientific Report on Heat Transfer
Analysis in Mixed Convection Flow of Maxwell Fluid over an Oscillating
Vertical Plate”, Scientific Reports, (2017). DOI: 10.1038/srep40147.
[21] J. G. Oldroyd, “On the Formulation of Rheological Equations of State”,
Proceeding of Royal Society of London: Series A, 38, 523-541 (1950).
[22] K. R. Rajagopal and R.K. Bhatnagar, “Exact Solutions for Some Flows of an
Oldroyd-B Fluid”, Acta Mechanica, 113, 233-239 (1995).
102
[23] C. Fetecau and Corina Fetecau, “The First Problem of Stokes’ for an Oldroyd-B
Fluid”, International Journal of Non-Linear Mechanics, 38, 1539-1544 (2003).
[24] T. Wenchang, “Stokes’ First Problem for an Oldroyd-B Fluid in a Porous Half
Space”, Physics of Fluids, 17, (2005). https://doi.org/10.1063/1.1850409
[25] T. Hayat, M. Khan, M. Ayub, “Couette and Poiseuille Flows of an Oldroyd 6-
Constant Fluid with Magnetic Field”, Journal of Mathematical Analysis and
Applications, 298, 225-244 (2004).
[26] A. K. Ghosh and P. Sana, On Hydromagnetic Flow of an Oldroyd-B Fluid near a
Pulsating Plate”, Acta Astronautica, 64, 272-280 (2009).
[27] T. Q. Haitao and H. Jin, “Unsteady Helical Flows of a Generalized Oldroyd-B
Fluid with the Fractional Derivative”, Non-Linear Analysis and Real World
Applications, 10, 2700-2708 (2009).
[28] R. Ellahi, T. Hayat, F.M. Mahomed, A. Zeeshan, “Exact Solutions for Flows of an
Oldroyd 8-Constant Fluid with Nonlinear Slip Conditions”, Communication in
Non-Linear Sciences and Numerical Simulation, 15, 322-330 (2010).
[29] L. Zheng, Y. Liu and X. Zhang, “Slip Effects on MHD Flow of a Generalized
Oldroyd-B Fluid with Fractional Derivative”, Non-Linear Analysis and Real
World Applications, 13, 513-523 (2012).
[30] J. M. Burgers, “Mechanical Considerations-Model Systems-Phenomenological
Theories of Relaxation of Viscosity in J. M. Burgers (Ed.)”, First Report on
Viscosity and Plasticity, Nordemann Publishing company, New York, 1935.
[31] A. R. Lee and A. H. D. Markwick, “The Mechanical Properties of Bituminous
Surfacing Materials under Constant Stress”, Journal of Society of Chemical
Industries, 56, 146-156 (1937).
[32] K. Majidzadeh and H.E. Schweyer, “Viscoelastic Response of Asphalts in the
Vicinity of the Glass Transition Point”, Association of Asphalt Paving
Technology Proceedings, 36, 80-105 (1967).
[33] M. C. Wang and K. Y. Lee, “Creep Behaviour of Cement Stabilized Soils”,
Highway Research Record, 442, 36-52 (1973).
103
[34] C. A. Tovar, C. A. Cerdeirina, L. Roman, B. Prieto, J. Carballo, “Viscoelastic
Behavior of Arzua-Ulloa Cheese”, Journal of Texture Studies, 34, 115-129
(2003).
[35] B. H. Tan, I. Jackson, J. D. F. Gerald, “High-Temperature Viscoelasticity of Fine
Grained Polycrystalline Olivine”, Physics and Chemistry of Minerals, 28, 641-664
(2001).
[36] W. R. Peltier, P. Wu D. A. Yuen, “The Viscosities of the Earth Mantle”, An
elasticity in the Earth, American Geophysical Union, Colorado, 1981.
[37] J. M. Krishnan and K. R. Rajagopal, “A Thermodynamic Framework for the
Constitutive Modelling of Asphalt Concrete: Theory and Application”, Journal of
Materials in Civil Engineering, 16, 155-166 (2004).
[38] P. Ravindran, J. M. Krishnan, K. R. Rajagopal, “A Note on the Flow of a Burgers
Fluid in an Orthogonal Rheometer”, International Journal of Engineering and
Science, 42, 1973-1985 (2004).
[39] T. Hayat, C. Fetecau, S. Asghar, “Some Simple Flows of a Burgers’ Fluid”,
International Journal of Engineering and Science, 44, 1423-1431 (2006).
[40] C. I. Chen, T. Hayat, J. L. Chen, “Exact Solutions for the Unsteady Flow of a
Burgers Fluid in Duct Induced by Time-Dependent Prescribed Volume Flow
Rate”, Heat and Mass Transfer, 43, 85-90 (2006).
[41] M. Jamil, “First problem of Stokes' for generalized Burgers' fluids”, Mathematical
Physics, Article ID 831063 (2012).
[42] A. M. Siddiqui, M. A. Rana N. Ahmed, “Effects of Hall Current and Heat
Transfer on MHD Flow of a Burgers Fluid due to a Pull of Eccentric Rotating
Disks”, Communication in Non-Linear Science and Numerical Simulation, 13,
1554-1570 (2008).
[43] I. Khan, K. Fakhar, S. Sharidan, “Magnetohydrodynamic Rotating Flow of a
Generalized Burgers Fluid in a Porous Medium with Hall Current”, Transport in
Porous Media, 91, 49-58 (2012).
104
[44] K. Masood and T. Hayat, “Some Exact Solutions for Fractional Generalized
Burgers’ Fluid in a Porous Space”, Nonlinear Analysis Real World Applications,
9, 1952-1965 (2008).
[45] Z. A. Aziz, S. Faisal, L. C. C. Dennis, “On Accelerated Flow for MHD
Generalized Burger Fluid in a Porous Medium and Rotating Frame”, International
Journal of Applied Mathematics, 41, 413-423 (2011).
[46] D. Tong, and L. Shan, “Exact Solutions for Generalized Burgers’ Fluid in an
Annular Pipe”, Meccanica, 44, 427-431 (2009).
[47] C. Fetecau, T. Hayat, M. Khan, F. Cornia, “A Note on Longitudinal Oscillations
of a Generalized Burgers Fluid in Cylindrical Domains”, Journal of Non-
Newtonian Fluid Mechanics, 165, 350-361 (2010).
[48] M. Jamil and A. K. Najeeb, “Helical Flows of Fractionalized Burgers Fluids”, AIP
Advances, 2, 1-15 (2012).
[49] K. Masood, F. Asma, A. K. Waqar, M. Hussain, “Exact Solution of an
Electroosmotic Flow for Generalized Burgers Fluid in Cylindrical Domain”,
Results in Physics, 6, 933-939 (2016).
[50] C. F. Xue, J. X. Nie, W. C. Tan, “An Exact Solution of Start-up Flow for the
Fractional Generalized Burgers’ Fluid in a Porous Half space”, Nonlinear
Analysis, 69, 2086-2094 (2008).
[51] M. S. Hyder, “Unsteady Flows of a Viscoelastic Fluid with the Fractional
Burgers’ Model”, Nonlinear Analysis and Real World Applications, 11, 1714-
1721 (2010).
[52] H. Shihao, Z. Liancun, Z. Xinxin, “Slip Effects on a Generalized Burgers’ Fluid
Flow Between Two Side Walls with Fractional Derivative”, Journal of the
Egyptian Mathematical Society, 24, 130-137 (2016).
[53] C. Xue and J. Nie, Exact Solutions of Stokes’ First Problem for Heated
Generalized Burgers’ Fluid in a Porous Half-Space”, Nonlinear Analysis and Real
World Applications, 9, 1628-1637 (2008).
[54] L. Yaqing, Z. Liancun, Z. Xinxin, “MHD Flow and Heat Transfer of a
Generalized Burgers’ Fluid due to an Exponential Accelerating Plate with the
105
Effect of Radiation” Computers and Mathematics with Applications, 62, 3123-
3131 (2011).
[55] K. Masood and W.A. Khan, “Steady Flow of Burgers Nanofluid over a Stretching
Surface with Heat Generation/Absorption”, Journal of Brazil Society of
Mechanical Science and Engineering, 29, (2014),
http://dx.doi.org/10.1007/s40430-014-0290-4
[56] K. Masood and W.A. Khan, “Forced Convection Analysis for Generalized
Burgers Nanofluid Flow over a Stretching Sheet”, AIP Advances, 5, (2015),
087178, http://dx.doi.org/10.1063/1.4929832
[57] K. Masood, W. A. Khan, A. S. Alshomrani, “Non-linear Radiative Flow of Three-
Dimensional Burgers Nanofluid with New Mass Flux Effect”, International of
Journal of Heat and Mass Transfer, 101, 570-576 (2016).
[58] M. M. Rashidi, Z. Yang, M. Awais ,N. Maria, T. Hayat, “Generalized Magnetic
Field Effects in Burgers' Nanofluid Model”, PlosOne, 3, (2017),
https://doi.org/10.1371/journal.pone.0168923
[59] Q. Sultan, M. Nazar, U. Ali, I. Ahmad, “On the Flow of Generalized Burgers'
Fluid Induced by Sawtooth Pulses”, Journal of Applied Fluid Mechanics, 8, 570-
576 (2015).
[60] M. Awais, T. Hayat, A. Alsaedi, “Investigation of Heat Transfer in Flow of
Burgers’ Fluid during a Melting Process”, Journal of the Egyptian Mathematical
Society, 23, 410-415 (2015)
[61] T. Hayat, M. Waqas, S. A. Shehzad, A. Alsaedi, “On Model of Burgers Fluid
Subject to Magneto Nano-particles and Convective Conditions”, Journal of
Molecular Liquids, 222, 181-187 (2016).
[62] A. K. Waqar, K.Masood, S. A. Ali, “Impact of Chemical Processes on 3D Burgers
Fluid Utilizing Cattaneo-Christov Double-Diffusion: Applications of non-
Fourier's Heat and Non-Fick's Mass Flux Models”, Journal of Molecular Liquids,
223, 1039-1047 (2016) .
106
[63] T. Hayat, M. Waqas, M. Ijaz Khan, A. Alsaedi, S. A. Shehzad,
“Magnetohydrodynamic Flow of Burgers Fluid With Heat Source and Power Law
Heat Flux”, Chinese Journal of Physics, 55(2), 318-330 (2017).
[64] T. Hayat, A. Arsalan, M. Taseer, A. Alsaedi, “On Model for Flow of Burgers
Nanofluid with Cattaneo-Christov Double Di¤usion”, Chinese Journal of Physics
(2017), doi 10.1016/j.cjph.2017.02.017
[65] R. H. Roberts, “An Introduction by Magnetohydrodynamics”, New Castle, (1967).
[66] C. U . Ikoku, and H. J. Jr Ramey, “Transient Flow of Non-Newtonian Power Law
Fluids in Porous Media”, Society of Petroleum Engineering, 164-174 (1979).
[67] P. Van, H. K, J. R Jargon, “Steady-State and Unsteady-State Flow of Non-
Newtonian Fluids through Porous Media”, Society of Petroleum Engineering,
Trans, AIME 246, 80-88 (1969).
[68] S. U. S. Choi, “Enhanced Thermal Conductivity of Nanofluids with
Nanoparticles, Development and Applications of Newtonian flows”, Fusion
Engineering and Design, 231, 99-105 (1995).
[69] J. Buongiorno. “Convective Transport in Nanofluids”, Journal of Heat Transfer,
128 (3), 240-250 (2006).
[70] R. Hilfer, “Threefold Introduction to Fractional Derivatives, in: Anomalous
Transport”, Wiley & Co. KGA, (2008)
[71] M. Du, Z. Wang, H. Hu, “Measuring Memory with the Order of Fractional
Derivative”, Scientific Reports, 3, (2013).
[72] I. Podlubny, “Fractional Differential Equations”, Academic press, San Diego,
California, USA, (1999).
[73] F. Mainardi, “Fractional Calculus and Waves in Viscoelasticity: An Introduction
to Mathematical Models”, Imperial College Press, London, (2010).
[74] M. Caputo, and M. Fabrizio, “A New Definition of Fractional Derivative without
Singular Kernel”, Progress in Fractional Differentiations and Applications, 1, 73-
85 (2015).
107
[75] A. A. Kashif, and K. Ilyas, “Analysis of Heat and Mass Transfer in MHD Flow of
Generalized Casson Fluid in a Porous Space Via Non-Integer Order Derivative
without Singular Kernel”, Chinese Journal of Physics, 55(4), 1583-1595,(2017).
[76] K. Arshad, A. A. Kashif, T. Asifa, K. Ilyas, “Atangana-Baleanu and Caputo
Fabrizio Analysis of Fractional Derivatives for Heat and Mass Transfer of Second
Grade Fluids over a Vertical Plate”, A Comparative study, Entropy, 19(8), 1-12
(2017).
[77] A. Abdon, and D. Baleanu, “New Fractional Derivatives with Nonlocal and Non-
Singular Kernel Theory and Application to Heat Transfer Model”, Thermal
Science, 18, (2016).
[78] A. S. Nadeem, A. Farhad, S. Muhammad, K. Ilyas. S. A. A. Jan, S. A. Ali, S. A.
Metib, “Comparison and Analysis of the Atangana-Baleanu and Caputo-Fabrizio
Fractional Derivatives for Generalized Casson Fluid Model with Heat Generation
and Chemical Reaction”, Results in Physics, 7, 789-800 (2017).
[79] C. Fox, “The G and H Functions as Symmetrical Fourier Kernels”, Transactions
of the American Mathematical Society, 98, 395-429 (1961).
[80] C. Stankovi, “On the Function of E. M. Wright”, Publication Institute of
Mathematical sciences, 10, 113-124 (1970).
[81] R. Gorenflo, Y. Luchko, F. Mainardi, “Analytical Properties and Applications of
Wright Function”, Fractional Calculus and Applied Analysis, 2, 383-414 (1999).
[82] C. F. Lorenzo,and T. T. Hartley, “Generalized Functions for the Fractional
Calculus”, Critical Reviews in Biomedical Engineering, 36, 1, 39-55 (2008).
[83] I. S. Gradshteyn, and I. M. Ryzhik, “Table of Integral, Series and Products”,
Eighth Edition, Academic Press, USA, (2014).
[84] A. A. Kashif, H. Mukarrum, M. B. Mirza, “Slippage of Fractionalized Oldroyd-B
Fluid with Magnetic Field in Porous Medium”, Progress in Fractional
Differentiation and Applications, 3(1), 69-80 (2017).
[85] M. Khan, A. Asia, C. Fetecau, H. T. Qi,” Exact Solutions for Some Oscillating
Motions of a Fractional Burgers' Fluid”, Mathematical and Computer Modeling,
51, 682-692 (2010).
108
[86] D. Tong, “Starting Solutions for Oscillating Motions of a Generalized Burgers
Fluid in Cylindrical Domains”, Acta Mechanica, 214, 395-407 (2010).
[87] L. Debnath, D. Bhatta, “Integral Transforms and Their Applications”, (Second
Edition) Chap-man & Hall/CRC, (2007).
[88] M. R. Spiegel, “Mathematical Handbook of Formulas and Tables”, (Third
Edition), McGraw Hill Schaum’s Outline Series, (1976).
[89] R. Hilfer, “Applications of Fractional Calculus in Physics”, World Scientific
Press, Singapore, (2000).
[90] J. Muhammad, A. A. Kashif, A. K. Najeeb, “Helices of Fractionalized Maxwell
fluid”, Nonlinear Engineering, 4(4), 191-201, (2015).
[91] M. Jamil, C. Fetecau, “Some Exact Solutions for Rotating Flows of a Generalized
Burgers' Fluid in Cylindrical Domains”, Journal of Non-Newtonian Fluid
Mechanics, 165, 1700-1712 (2010).
[92] M. Khan, T. Hayat, S. Asghar, “Exact solutions of MHD Flow of a Generalized
Oldroyd-B fluid with Modified Darcy’s law”, International Journal of Engineering
Science, 44, 333-339 (2006).
[93] W. C. Tana, and T. Masuoka, “Stability Analysis of a Maxwell Fluid in a Porous
Medium Heated From Below”, Physical Letters A, 360, 454-460 (2007).
[94] M. Jamil , A. A. Zafar , C. Fetecau , N. A.Khan, “Exact Analytic Solutions for the
Flow of a Generalized Burgers Fluid Induced by an Accelerated Shear Stress”,
Chemical Engineering Communications, 199, 17-39 (2012).
[95] K. Ilyas, A. Farhad, S. Shafie, “Stokes’ Second Problem for
Magnetohydrodynamics Flow in a Burgers’ Fluid: The Cases 4
2 and
4
2
”, PLoS ONE, 8, (2013).
[96] K. Ilyas, A. Farhad, M. Norzieha, S. Sharidan, “Transient Oscillatory Flows of a
Generalized Burgers’ Fluid in a Rotating Frame”, Zeitschrift für Naturforschung,
68, 305-309 (2013).
[97] C. Fetecau, F. Corina, D. Vieru, “On Some Helical Flows of Oldroyd-B fluids”,
Acta Mechanica., 189, 53-63 (2007).
109
[98] F. Corina. M. Imran, C. Fetecau, I. Burdujan, “Helical Flow of an Oldroyd-B
Fluid Due to a Circular Cylinder Subject to Time-Dependent Shear Stresses”, Z.
Angew. Mathematical Physics, DOI: 10.1007/s00033-009-0038-7.
[99] M. Jamil, A. Rauf, C. Fetecau, A. K. Najeeb, “Helical Flows of Second Grade
Fluid Due to Constantly Accelerated Shear Stresses”, Communication in Non-
linear Science and Numerical Simulation, 16, 1959-1969 (2011).
[100] M. Jamil, C. Fetecau, M. Imran, “Unsteady Helical Flows of Oldroyd-B Fluids”,
Communication in Non-linear Science and Numerical Simulation, 16, 1378-1386
(2011).
[101] E. V. Timofeeva, J. L. Routbort, D. Singh, “Particle Shape Effects on
Thermophysical Properties of Alumina Nanofluids”, Journal of Applied Physics,
106, (2009).
[102] K. Asma, I. Khan, S. Sharidan, “Exact Solutions for Free Convection Flow of
Nanofluids with Ramped Wall Temperature”, European Physical Journal Plus,
130, 57-71 (2015).
[103] S. Das and R. Jana, “Natural Convective Magneto-Nanofluid Flow and Radiative
Heat Transfer Past a Moving Vertical Plate”, Alexandria Engineering Journal, 54,
55-64 (2015).
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