On Boundary Layer Flow and Heat Transfer to...
154
On Boundary Layer Flow and Heat Transfer to Burgers Fluid By Waqar Azeem Khan
Transcript of On Boundary Layer Flow and Heat Transfer to...
On Boundary Layer Flow and Heat Transfer to
Burgers Fluid
By
Waqar Azeem Khan
Department of Mathematics
Quaid-i-Azam University, Islamabad
PAKISTAN
2017
On Boundary Layer Flow and Heat Transfer to
Burgers Fluid
By
Waqar Azeem Khan
Supervised by
Prof. Dr. Masood Khan
Department of Mathematics
Quaid-i-Azam University, Islamabad
PAKISTAN
2017
5
On Boundary Layer Flow and Heat Transfer
to Burgers Fluid
By
Waqar Azeem Khan
6
A Thesis
Submitted in the Partial Fulfillment of the
Requirements for the Degree of
DOCTOR OF PHILOSOPHY
IN
MATHEMATICS
Supervised by
Prof. Dr. Masood Khan
7
Department of Mathematics
Quaid-i-Azam University, Islamabad
PAKISTAN
2017
8
Dedicated to
My Parents
&
Wife, Brother and Sisters
9
Acknowledgement
All praise for Allah the creator and the most merciful, who guides me in
darkness and enables me to view stumbling blocks as stepping stones to the
stars to reach the ultimate with courage. I am nothing without Allah but I can
achieve everything with His assistance. All of my veneration and devotion
goes to our beloved Holy Prophet Muahmmad (peace be upon him) the
source of knowledge and guidance to humanity.
I express deepest gratitude to my respected, affectionate and devoted
supervisor Prof. Dr. Masood Khan for his intellectual guidance, constant
encouragement, suggestions and inexhaustible inspiration throughout my
research work.
I am extremely thankful to Prof. Dr. Muhammad Yousaf Malik (Chairman,
department of mathematics) for providing me opportunity to learn and seek
knowledge in educated environment.
10
I am very grateful to the honorable Distinguished National Prof. Dr. Tasawar
Hayat for his Excellency throughout in my research and other educational
circumstances.
I am very thankful to my loving parents for their guidance, support and
encouragement. I owe my heartiest gratitude for their assistance and never
ending prayers for my success. I highly commend the cooperative behavior
of my brother Mazhar Hussain, my wife, sisters who endeavored for my
edification and betterment. Also thanks to my little sister Tahira for her
support.
I am especially thankful to all my friends whose presence around me made
my life unforgettable and joyful. I would love to mention the name of
Muhammad Waqas, Zeeshan Asghar, Muhammad Irfan, Dr. Sabir Ali
Shehzad, Dr. Rizwan Ul Haq, Dr. Fahad Munir Abbasi, Dr. Muhammad Bilal
Ashraf, Dr. Salman Saleem, Dr. Muhammad Farooq, Dr. Majid Khan, Ata Bhai,
M. Ijaz Khan, Muhammad Azam, Fiaz-Ur-Rehman, Asif Jaffar, Shahid Farooq,
Hashim, Latif Ahmad, Kaleem Iqbal, Aamir Hamid, Jawad Ahmed, Masood Ur
Rehman, Taimoor Salahuddin, Taseer Muhammad, Zakir Hussain, Sajid
Qayyum, M. Waleed Ahmed Khan, Faisal Shah, Khaleel Ur Rehman, Sardar
11
Bilal, Ikram Ullah and Atif Khan. There company made my time beautiful and
full of joys with everlasting memories.
In the end, I am really grateful to all those who have true love for me and
whose moral support and useful suggestions encouraged me at every step.
Waqar Azeem Khan
12
Table of Contents
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiv
1 Introduction
1
1.1 Motivation of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 1
1.2 Aims and Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 8
2 Mathematical Modeling
12
2.1 Fundamental Laws . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 12
2.1.1 Relation for Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . .. . 12
2.1.2 Law of Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.3 Law of Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 13
2.1.4 Law of Conservation of Concentration . . . . . . . . . . . . . . . . . . . . . . . .
14
13
2.1.5 Law of Conservation of Energy for Nanofluid . . . . . . . . . . . . . . . . .
14
2.1.6 Law of Conservation of Concentration for Nanofluid . . . . . . . . . . . . .
15
2.1.7 Modified Fourier's and Fick's Relations . . . . . . . . . . . . . . . . . . . . . . .
15
2.2 Homogeneous-Heterogeneous Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 17
2.3 Boundary Layer Equations of Burgers Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.4 Mathematical Modeling of Generalized Burgers Fluid . . . . . . . . . . . . . . . . . . . .
26
2.5 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 33
3 Forced Convective Heat transfer to Burgers Nanofluid with Heat Genera-
tion/Absorption
34
3.1 Formulation of Problem . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . 35
3.2 Solution by HAM . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .
37
3.2.1 The zeroth order deformation problems . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.2 The mth order deformation problems . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 Convergence of the Homotopy Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 41
14
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
4 3D Flow and Heat Transfer Mechanisms to Burgers Fluid 51
4.1 Mathematical Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 The Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
4.3 Graphical Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5 Heterogeneous-Homogeneous Processes in 3D Flow of Burgers Fluid
62
5.1 Mathematical Description of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 Homotopic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3 Graphical Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6 3D Convectively Nonlinear Radiative Flow of Burgers Fluid
74
6.1 Mathematical Description of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.2 Homotopic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.3 Analysis of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7 Features of Brownian Motion and Thermophoresis for 3D Burgers Fluid Flow 87
7.1 Mathematical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.2 Homotopic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
8 Characteristics of Thermophoresis Particle Deposition on 3D Flow of
15
Burgers Fluid 99
8.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.2 The Analytic Series Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.3 Graphical Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
9 Melting Heat/Mass Transfer in Generalized Burgers Fluid Flow
109
9.1 Problem Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
9.2 The Analytic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
9.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
10 2D Analysis for Generalized Burgers Fluid Flow in presence of Nanoparticles
121
10.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
10.2 Homotopic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
10.3 Interpretation of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
11 Conclusions, Summary and Future Work
132
11.1 Contributions of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
11.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Bibliography 136
16
Abstract
The aim of this thesis is to furnish some theoretical results in the field of non-Newtonian fluid
mechanics. The research presented in the thesis is concerned with the mathematical modeling
and development of analytical solutions for non-Newtonian fluids. Particularly, this thesis
focuses on the boundary layer flows of Burgers and generalized Burgers fluids induced by the
stretching surface. Thus the theme of thesis is twofold. First, the development of the boundary
layer equations for steady two- and three-dimensional flows of Burgers and generalized
Burgers fluids. Second, to give a better understanding of their behaviors, the development of
the analytical results for them in diverse circumstances. The problems considered here
involve, the forced convective heat transfer over linear stretching surfaces by assuming
different situations like nanofluid, Cattaneo-Christov heat and mass flux models,
homogeneous-heterogeneous and melting processes. The modeled PDEs are transformed into
ODEs by utilizing suitable transformations which are then solved by employing HAM. In the
limiting cases, our solutions are in excellent agreement with previously reported results in the
literature. To assess and demonstrate the physical aspects of our results, some of the velocity,
17
temperature and concentration profiles are presented graphically for emerging parameters and
discussed in detail. Moreover, the local Nusselt and Sherwood numbers are presented in tabular
form for a set of values of the non-dimensional parameters. A profound observation is that the
velocity and associated momentum boundary layer thickness diminish with augmented values
of the materials parameter 𝛽2 of Burgers fluid; however, quite the opposite is true in case of
the material parameter 𝛽4 of generalized Burgers fluids. In addition, it is noticed that
temperature and concentration profiles enhance as the material parameter 𝛽2 is incremented.
It is further observed that the temperature and concentration distribution possess a reverse
behavior for the material parameter 𝛽4 when compared with 𝛽2. Indeed this thesis leads us to
suggest that the results owing from the Burgers and generalized Burgers models do provide a
much improved understanding of their rheological characteristics.
18
Chapter 1
Introduction
The motivation behind the heat transfer mechanisms of nonlinear materials and related literature
survey are demonstrated through this chapter. It further highlights the importance of studying such
uids, speci cally the Burgers uids. The description of all chapters is also presented in this chapter.
1.1 Motivation of this Work
The inspiration for this investigation has been originated from the widespread applications of
nonlinear materials in industry and engineering. The research work related to nonlinear materials has
become relatively ubiquitous in industry and engineering now a days. Particularly, such liquids are
encountered in certain oils, exotic lubricants, polymer solutions, paints, suspension and colloidal
solutions, cosmetic and clay coating products. The basic Navier-Stokes equations are not appropriate
to describe the diverse physical structures of nonlinear materials. As a consequence, di⁄erent types of
non-Newtonian relations have been anticipated according to numerous features of nonlinear liquids.
Amongst these nonlinear materials, rate type uids exhibit the properties memory and elastic impacts.
The simplest subclasses of these liquids whose exhibit the properties of memory and elastic e⁄ects are
Maxwell and Oldroyd-B liquids.
19
However, these nonlinear materials do not portray rheological features of numerous real liquids such
as cheese in food products and asphalt in geomechanics. Consequently, the Burgers in 1935 presented
a modi ed 1D relation which interpret the features of these real liquids. Rajagopal and Srinivasa [2]
extended the Burgers relation for 3D ow to interpret the characteristics of memory and elastics e⁄ects.
This liquid relation is su¢ ciently utilized to interpret behavior of earth s mantle. This relation is a
favored relation to represent the response of asphalt concrete. This relation is the most appropriate
one to symbolize the response of asphalts. The Burgers relation is utilized to develop the other
structures, such as Olivine rocks [3]. Additionally, the Burgers liquid relation has been broadly utilized
to portray diverse viscoelastic materials: food products such as cheese [4,5]. The elastic and memory
e⁄ect for the Burgers liquid are examined by Lee and Markwick [6]. Saal and Labout [7] have observed
that with the minor modi cation of Burgers liquid the behavior of asphalts from the mechanical point
of view can approximated to Burgers relation [8,9]. These asphalt and asphalt mixes have sveral
applications such as binder for aggregate materials in the construction of highways and runways.
Furthermore, the Burgers liquid relation has been extensively used for calculating the properties of
earth,s mantle and speci cally related to post-glacial uplift [10-13]. The Burgers liquid relation has
been utilized in modeling of high temperature viscoelasticity of ne-grained polycrystalline olivine [14].
Extensive applications of Burgers uids can be found in the studies [15-23]. Further, this model has
been inspected by a few recent researchers [24-30].
In spite of the fact that new evolutions in electronics enhance the performance of electronic
devices. These advancements often mean the downsizing of these devices due to the power density
concomitant with these components and it increases drastically which increases the heat ux spawned
by the electronic components. The dissipation of this ux inside the device can lead to thermal
20
problems such as over heating, which minimizes the devices enactment levels and their lifespan. As a
result, it is a prodigious technological challenge to design and develop e⁄ective cooling systems. Such
systems, will able to evacuate the signi cant heat produced to conserve the temperature of electronic
instruments below a certain value. New cooling methods are mandatory to be identi ed in order to
encounter such challenges. In this milieu, the heat transfer rate of heat transfer devices can be
elevated amongst other cooling technologies by adding additives to their working uids. This alters the
uid transport properties and ow topographies. Choi [31] in 1995 rst time provide the idea of addition
of nanoparticles to the base liquid. Afterward, the investigation of this research work study of
nanoliquid had been started. Additionally, heat transfer mechanisms utilizing nanoliquids has
considered as one of the most e⁄ective procedures. This mechanisms possess numerous applications
in precincts like compact heat exchangers, heat pipes etc. Keeping in mind, Oztop and Abu-Nada [32]
scrutinized numerically the natural convection ow of nano uids in partially heated rectangular
enclosures. The features of viscous liquid by utilizing nanoparticles towards a convectively heated
surface was numerically addressed by Makinde and Aziz [33]. Kuznetsou and Nield [34] provided the
modi ed relation for natural convective ow by utilizing nanoparticles past a vertical plate subject to
the newly suggested conditions. Pal and Mondal [35] scrutinized the features of magneto-nanoliquid
over stretched surface. The impact of mixed convective ow for nanoliquids over a stretched with
internal heat source/sink was pondered by Pal and Mondal [36]. The characteristics of heat ux and
mass di⁄usion e⁄ects in hydromagnetic ow for viscous liquid by utilizing slip conditions and various
types of nanoparticles were inspected by Turkyilmazoglu [37]. Khan etal. [38] analyzed 3D forced
convective ow of Oldroyd-B nano uid towards a stretched sheet with heat source/sink. They concluded
from their graphical observation that with the augmented values of Nb and Nt temperature of liquid
21
rises. Further, Khan etal. [39] inspected the 2D forced convective ow of Sisko-nanoliquid over a
stretched sheet. They estimated from their graphical observation that the temperature of the
Siskonanoliquid rises with the boosted values of thermophoresis and random motion parameters.
Hayat etal. [40] inspected characteristics of magneto-nanoparticles for Burgers uid by utilizing
convectively heated surface. The impact of magneto-nanoparticles for Williamson liquid over melted
surface was explored by Hayat etal. [41].
An investigation on the radiative ow of forced convection problems has been incessantly attracting
more attention. This is due to its intriguing industrial applications for instance, glass production,
furnace design, nuclear power plants, comical ight aerodynamics rocket, propulsion systems, and
space craft reentry aerodynamics that operate at high temperatures. Moreover, the intriguing
e⁄ectiveness of thermal radiation is indispensable on the ow and heat transfer processes in the design
of developed energy conversion systems. Besides, solar energy plays a substantial part in the heat
transfer characteristic of absorbing-emitting uids when convection heat transfer is small,
predominantly in free convection problems. Likewise, the amount of thermal radiation manifestation
within such systems is on account of emanation by hot walls and functioning uids. It is well known
that the blood ow regulates the temperature of the human body and controls it according to the
environment. Nowadays, the thermal regulation in human blood ow by means of thermal radiation is
very signi cant in several medical treatments formuscle spasm, myalgia, chronicwide-spread pain and
permanent shortening of muscle. Cortell [42] deliberated the characteristics of radiative heat
transport for stretched sheet. Features of Sakiadis ow by utilizing nonlinear radiation in the energy
equation has been considered by Pantokratoras and Fang [43]. Numerical study of forced convective
radiative ow for nanoliquid was explored Mushtaq etal. [44]. Hussain etal. [45] focused on the impact
22
of mixed and nonlinear connective ow convection over stretched surface. Atlas et al. [46] securitized
the impacts of active and zero ux of nanoparticles on squeezing channels. Hayat etal. [47] scrutinized
the characteristics of Maxwell liquid by utilizing nanoparticles and thermal radiation. Features of heat
transport and mass di⁄usion mechanisms in the presence of solar energy and variable viscosity over
an unsteady stretched surface are explored by Pal and Saha [48]. Impact of binary chemical processes
on Casson uid over a stretched sheet with solar energy aspects have been studied by Abbas etal. [49].
Narayana and Babu [50] explored the characteristics of magneto-Je⁄rey uid over a stretched sheet
with chemical processes and solar enery.
Nowadays, the researchers from all over the world have shown a great interest in analyzing the
phenomenon of heat/mass transport because of its enormous applications in manufacturing
processes and industry. Features of heat/mass transport mechanisms have been inspected by Fourier
s relation for the heat transfer mechanism and Fick s relation for di⁄usion in last two centuries instead
of considering general anomalous thermal and mass di⁄usion. Moreover, it is obvious that by changing
the combination of relaxation times for velocity pro les should a⁄ect both the temperature and
concentration distributions. One of the major de ciency in Fourier s relation is that this relation gives
energy equation in the form of parabola according which when we disturb the temperature of the
substance then this disturbance is felt suddenly throughout the whole substance. By keeping in mind
the obstacle in this relation, Cattaneo [51] recommended an improved Fourier s relation by utilizing
the relaxation of time term in the energy equation term. One can observed that amendment yields
hyperbolic energy equation and facilitate the heat transfer mechanism in the form of propagation of
waves. Christov [52] further revised the Cattaneo relation by utilizing the thermal relaxation time with
Oldroyd s derivatives. Ciarletta and Straughan [53] deliberated the characteristics of developed heat
23
relation and also provide the stability of the solutions. Hayat etal. [54] inspected the features of
developed heat ux relation by utilizing the variable thicked surface. Additionally, they determined
from their graphical illustrations that the temperature of the liquid reduces for augmented values of
relaxation time term. Recently, Sui etal. [55] analyzed the usefulness of developed heat ux relation
and more convincing realistic conditions for Maxwell liquid by utilizing nanoparticles over a stretched
sheet. The features of wave transport mechanisms and convection di⁄usion by employing developed
heat ux relation were explored by Liu etal. [56]. Nadeem and Muhammad [57] investigated the
characteristics of strati cation and revised heat ux relation by utilizing the porous stretched surface.
Malik etal. [58] inspected the features of revised heat ux relation on Sisko uid ow past a nonlinear
stretched cylinder. Salahuddin et al. [59] pondered the impact of magneto-Williamson uid by utilizing
revised heat ux relation MHD over stretched surface. Khan etal. [60] inspected features of advanced
heat ux and mass di⁄usion mechanisms for Sisko liquid.
Heterogeneous-homogeneous reactions characteristic phenomenon has received much attention
by its applications in industrial processes, combustion, biochemical systems and catalysis. There are
certain chemical reaction which have the ability to proceed slowly. Chemical reactions are classi ed
into two types namely, heterogeneous or homogeneous processes depending on whether they occur
in bulk of the uid (homogeneous) or occur on some catalytic surfaces (heterogeneous). Moreover,
there are numerous chemical processes contain both heterogeneous and homogeneous process. The
association between heterogeneous and homogeneous processes correlated with consumption and
production of reactant species have di⁄erent rates together within liquid and on catalyst surface is
usually complex. Particularly, chemical process impacts are quite signi cant in industry of
hydrometallurgical, polymer production and manufacturing of ceramics, fog dispersion and formation
24
and many others. Merkin [61] presented the characteristics of heterogeneous-homogeneous
processes for viscous liquid. He investigated the homogeneous process by utilizing cubic autocatalysis.
It was estimated from their observation that foremost mechanisms adjacent leading edge is surface
processes. Chaudhary and Merkin [62] analyzed e⁄ects of the heterogeneous-homogenous process
for viscous liquid. They have computed the numerical solutions near leading edge for at plate. Khan
and Pop [63] examined the impacts of chemical process in the ow of viscoelastic uid. Kameswaran
etal. [64] presented the characteristic of nano uid ow due to stretching surface by utilizing the
chemical processes. Hayat etal. [65] addressed the features of developed heat ux relation and
chemical processes. Abbas etal. [66] deliberated the in uence of chemical process on hydromagnetic
viscous liquid past a stretched sheet by utilizing generalized slip relation. Ramzan et al. [67] explored
the features of chemical processes and revised heat ux relation for the third grade liquid. Yasmeen
etal. [68] observed the impact of chemical processes for the magneto-ferro uid ow over a stretched
surface. The characteristics of chemical processes in the presence of MHD ow due to an unsteady
stretched surface have been inspected by Imtiaz etal. [69].
The phenomenon of melting mechanisms have fascinated the attention of investigators in view of
its relevance to technological and industrial processes. Particularly, melting mechanisms are very signi
cant in numerous processes including freezing of soil and melting of permafrost. Epstein and Cho [70]
explored melting impacts for heat transfer mechanisms to submerged bodies. Cheng and Lin [71]
deliberate features of melting phenomenon for mixed convective ow transport over stretched
surface. Ishak etal. [72] analyzed the melting mechanisms for steady ow over stretched surface. The
features of melting mechanisms for stagnation-point ow toward a stretched sheet has been explored
25
by Bachok et al. [73]. Hayat et al. [74] scrutinized features of melting mechanisms for Powell-Eyring
liquid.
1.2 Aims and Structure of the Thesis
The foremost theme of this description is to improve the current research work of nonlinear liquids
particularly the Burgers liquids. Consequently, within this thesis we will focus upon the analytic
solutions of the boundary layer ows of Burgers uids and their heat transfer characteristics. There are
ELEVEN chapters in all covering various aspects of Burgers uids.
We have systematized this dissertation as follows:
Chapter 1 of this dissertation provides the inspiration, background and structure.
In Chapter 2, we provided the detailed mathematical modeling of Burgers and generalized Burgers
uids and brie y introduces the solution methodology.
Consecration of Chapter 3 is to visualize the impact of heat source/sink parameter and
nanoparticles on the Burgers uid. The current research work contain very complicated nonlinear PDEs
which are simpli ed through boundary layer approximations. These appropriate transformations
reduces the number of independent variable and the resultant equation take the form of ODEs. The
reduced transformed equations of Burgers nanoliquid relation are solved analytically by employing
HAM. The analytical outcomes are obtained for the temperature and concentration of Burgers
nanoliquid through sketches and these diagrams are also elucidated in detail. The contents of this
chapter has been published in J. Brazilian Society Mech.
Sci. Eng., (2016) 38:2359 2367, doi: 10.1007/s40430-014-0290-4.
26
Chapter 4 focuses on features of developed heat ux relation involving Burgers uid. The revised
heat ux relation is the modi cation of Fourier s relation of heat conduction that ponders fascinating
features of relaxation time in the energy equation. The subsequent equations of motion and energy
are reduced to a set of ODEs by implementation of appropriate transformations which are solved by
HAM. This research work is recently published in J. Molecular Liquids, 221 (2016) 651 657 .
Chapter 5 is an extension of the chapter 4 by considering the features of revised heat ux and mass
di⁄usion relation on steady 3D ow of Burgers liquid over a bidirectional stretched surface. The revised
heat ux and mass di⁄usion relations are the modi cations of Fourier s relation for heat mechanisms
and Fick s relation for mass phenomenon. Moreover, characteristics of heterogeneous and
homogeneous reactions are investigated. The analytical outcomes of the coupled ODEs is determined
by employing the HAM. This research work have been published in J. Molecular Liquids, 223 (2016)
1039 1047 .
Chapter 6 focusses on the chemical processes for the steady three-dimensional ow of Burgers uid
over a bidirectional convectively heated surface. Additionally, we assume that the size of the chemical
are comparable and heat released during chemical reactions is insigni cant. The governing PDEs in this
research work are render into a set of ODEs by utilizing appropriate similarity transformations. The
resultant ODEs appear in this research work are solved analytically by employing HAM. Results
achieved in this research work are compared with the previously published work and we establish an
outstanding agreement of present result with published data. The results of present work are
published in Results in Physics, 6 (2016) 772 779 .
We analyzed in chapter 7 the heat/mass transport characteristics of 3D ow of Burgersnano uid over
a bidirectional convectively heated surface by utilizing the revised conditions for the nanoliquid.
27
Moreover, the impacts of nonlinear radiation and heat source/sink are considered in the energy
relation. The resultant equations in this research work are appeared to be PDEs. We have applied
similarity approach to these equations and convert these equations into ODEs which are solved by
employing HAM. The features of pertinent parameters involving in the problem are deliberated on
temperature of the Burgers-nano uid. The aforementioned work has been published inInt. J. Heat
Mass Transfer, 101 (2016) 570 576 .
Chapter 8 deals with the combined e⁄ects of the heat generation/absorption and thermophoretic
on the heat transfer and mass di⁄usion mechanisms past a bidirectional stretched sheet for the
Burgers uid. Additionally, the heat transfer mechanisms is examined by utilizing the thermal radiation.
The PDEs are transformed into set of ODEs by utilizing appropriate similarity approach. The resultant
coupled set of equations are solved by employing HAM.
The outcomes for the temperature of the Burgers liquid are plotted sketchily and - 0 (0) and
- 0 (0) are provided in form of tables. These observations have been published in Results in Physics, 6
(2016) 829 836 .
The analytical analysis of melting mechanisms and mass di⁄usion features for the generalized
Burgers uid over a stretched sheet is presented in chapter 9. Moreover, the characteristics of heat
transfer mechanism are scrutinized by utilizing non-linear thermal radiation. The resulting nonlinear
problem is computed for the series solution. In uence of numerous physical parameters on
temperature of the generalized Burgers uid are scrutinized graphically and we also discussed these
graphs in detail. Additionally, - 0 (0) and - 0 (0) are introduced in the form of table. The outcomes of
this research work has been submitted inChinese J. Aeronautics .
28
Chapter 10 is actually an extension of the chapter 9 by considering the impact of nanosized material
particles on the generalized Burgers uid over a stretched sheet. The features of heat transfer and
mass di⁄usion mechanisms are analyzed by considering the Brownain motion and thermophoresis
e⁄ects in the in the energy and concentration equations. A set of coupled nonlinear ODEs is obtained
by utilizing appropriate similarity approach. Convergent series solution are derived by utilizing the
homotopy analysis method. Analysis of the obtained results show that the Brownian motion
parameter has reverse behavior on the temperature and concentrations elds. Moreover, it is
observed that the incremented values of the random motion of the particles lead to a quite
opposite e⁄ect on the rates of heat transfer mechanism and the concentration of nanoliquid at the
wall. The outcomes of this chapter are published in AIP Advances, 5, 107138 (2015); doi:
10.1063/1.4935043 .
We have concluded and summarized our main ndings in chapter 11 and also provided the
extensions of this research work.
29
Chapter 2
Mathematical Modeling
This chapter is devoted to the background of theoretical analysis of the current investigation.
In this portion of the thesis, the fundamental laws and development of governing equations of Burgers
and generalized Burgers uids are included.
2.1 Fundamental Laws
2.1.1 Relation for Conservation of Mass
In absence of sources or sinks relation for conservation of mass commonly known as continuity
equation in vector form can be stated as
r fV = 0; (2.1)
in which f is the liquid density, t the time and V liquid velocity.
The relation (2.1) for incompressible liquid is conveyed as
r V = 0: (2.2)
2.1.2 Law of Conservation of Momentum
For the steady incompressible liquids, it is of the form
fai = rp + divS+ fB; (2.3)
30
where ai is the acceleration vector, p the pressure, S the extra stress tensor and B the body
forces.
2.1.3 Law of Conservation of Energy
The energy equation is based on rst law of thermodynamics and for incompressible uids, it is of the
form
divq; (2.4)
where cf is speci c heat of uid, T the temperature and q the energy ux.
Energy ux is given by
q = krT; (2.5)
in which k is the thermal conductivity. Utilizing Eq. (2.5) in Eq. (2.4), the energy equation can be written
as
(2.6)
2.1.4 Law of Conservation of Concentration
The concentration equation is based on the Fick s relation and it is of the form
V rC = r J; (2.7)
where J represents the normal mass ux and C the concentration of the uid. The normal ux mass ux is
given by
J = DrC: (2.8)
31
In view of Eq. (2.8), Eq. (2.7) can be written as
(2.9)
2.1.5 Law of Conservation of Energy for Nano uids
The energy equation for incompressible uids in the presence of nanoparticles, has the form
divq + hpr Jp; (2.10)
where hp the speci c enthalpy for nanoparticles, q the energy ux for the nano uid and Jp the
nanoparticles di⁄usion mass ux. Energy ux q and nanoparticles di⁄usion mass ux Jp are given by
q = krT + hpJp; (2.11)
(2.12)
in which p the nanoparticle mass density, DB the Brownian motion parameter, DT the thermophoretic
di⁄usion coe¢ cient and C the nanoparticles volume fraction.
Thus, in view of Eqs. (2.11) and (2.12), the energy equations for nano uids can be written as
T T
: (2.13)
2.1.6 Law of Conservation of Concentration for Nano uid
The concentration equation for nano uids is given as
1
V rC = r Jp: (2.14)
32
p
In view of Eq. (2.12), it can be written as
: (2.15)
2.1.7 Modi ed Fourier s and Fick s Relations
The energy and concentration equations in absence of nanoparticles can be written as
r q; (2.16)
r J; (2.17)
where (q;J) represent the normal heat and ux mass uxes, respectively. The modi ed Fourier s relation
for heat mechanisms and Fick s relation for mass phenomenon, namely Cattaneo Christov anomalous
di⁄usion models, in vectorial form, can be stated as
where ( E; C)
characterize the
relaxation times for heat mechanisms and mass di⁄usion phenomena, respectively.
We consider the velocity components (u;v;w) in Cartesian coordinates (x;y;z) and eliminate q
from Eqs. (2.16) and (2.18) and J from Eqs. (2.17) and (2.19). So the energy and concentration
q+ @q
V rq+ @t + (r V)q q rV
E = krT;
(2.18)
J+
@J
V rJ+ @t +(r V)J J rV C = DrC; (2.19)
33
equations, in the absence of nano uids, corresponding to Cattaneo Christov heat and mass ux models
are presented in the form
It is pertained to mentioned that the aftermention Eqs. (2.20) and (2.21) are reduced to classical Fourier
s and Fick s laws when E = C = 0.
2.2 Homogeneous-Heterogeneous Processes
The chemical processes for a cubic autocatalysis in which two chemical reactants are associated in a
boundary layer ow can be expressed as follows:
C + 2D ! 3D ; rate = kccd 2; (2.22)
C !D ; rate = ksc: (2.23)
34
In the above expressions (c;d ) denote the concentrations of chemical species (C ;D ), respectively,
and ki (i = c;s) are rate constants. . Furthermore, we assume both processes are isothermal and c0 is
concentration of reactant C as z tends to in nity and the reactant D has no auto catalyst. By
implementing overhead assumptions, the governing equations for homogeneous-heterogeneous
processes based on the law of conservation of concentrations of two chemical species for steady ow
are expressed as
(2.24)
: (2.25)
2.3 Boundary Layer Equations of Burgers Fluid
Let us consider the three-dimensional ow of an incompressible Burgers uid in Cartesian coordinates.
The momentum and the constitutive equations governing the steady ow of an incompressible Burgers
uid [25] in the absence of body forces are expressed as
fai = rp + divS; (2.26)
S A1: (2.27)
In the above equations V is the velocity vector, p pressure, S extra stress tensor, A1 = (rV)+ (rV)> rst
Rivlin-Ericksen tensor, dynamic viscosity, 1 and 3 ( 1) relaxation and retardation times, respectively, 2
material parameter of the Burgers uid and denotes the upper convected derivative de ned by
35
: (2.28)
For 3D ow, we ponder the velocity of the Burgers liquid and stress eld of the form
V = [u(x;y;z);v (x;y;z);w(x;y;z)]; S = S(x;y;z): (2.29) By applying the operator
to Eq. (2.26), we obtain
D
rp + 1 + 3 divA1: (2.30)
Dt
For i = 1, Eqs. (2.28) and (2.30) for steady ow become
D
rp + 1 + 3 divA1; (2.31)
Dt
(2.32)
where the values of a1;a2 and a3 are given by
(2.33)
(2.34)
: (2.35)
Equation (2.31) can be re-written as
36
divA1: (2.36)
We need to compute all the quantities in Eq. (2.36). These are given by
There are six quantities in Eq. (2.38) and are found to be
(2.39)
= 2
(2.40)
= 2
37
(2.42)
(2.43)
Finally, the last term on right hand side of Eq. (2.36) is found to be
divA
Substitution of Eqs. (2.37) to (2.44) into Eq. (2.36) gives the x component of the momentum
equation as
38
=
and the y and z components of the momentum are, respectively
39
=
40
=
In the standard boundary layer approximations for three-dimensional ow z and w of order and x, y, u,
v, p; 1; 2 and 3 are of order 1. Therefore, by utilizing the boundary layer approximations the overhead
equations reduces to
41
=
42
=
: (2.51)
2.4 Mathematical Modeling of Generalized Burgers Fluid
Here we derive the governing equations of generalized Burgers uid for two-dimensional ow. The extra-
stress tensor for generalized Burgers liquid [22-30] is related to liquid motion satis es the following
constitutive equation:
S A1; (2.52)
where 4 is the material parameter of the generalized Burgers uid.
For 2D ow, we have utilized the velocity of the generalized Burgers liquid and stress elds of form
43
V = [u(x;y);v (x;y);0]; S = S(x;y): (2.53)
As before, we apply the operator to Eq. (2.26), we obtain
rp
divA1: (2.54)
For i = 1, Eqs. (2.28) and (2.54) for steady ow become
rp
divA1; (2.55)
(2.56)
where the values of a1 and a2 are found to be
(2.57)
: (2.58)
Equation (2.55) can be re-written as
rp
divA1: (2.59)
Thus, similar to the previous case, we are now to determine all the quantities in Eq. (2.59) and are given
by
44
(2.60)
(2.61)
(2.62)
and
(2.63)
(2.64)
: (2.65)
The last term on right hans side of Eq. (2.59) is thus given by
45
divA
Using Eqs. (2.60) to (2.66) in Eq. (2.59) we determine the x component of the momentum equation for
the generalized Burgers uid as
46
while y component of momentum equation is
=
Note that by utilizing the standard boundary layer approximation for two-dimensional ow
47
x, u, p and 4 are of order 1 while y and v are of order . Consequently, by utilizing the boundary layer
approximation the above equations give
=
: (2.70)
Having derived system (2.67) and (2.70) which governs steady 2D ow of generalized Burgers uid it
remains focus to reformulate the derivation for 2D Burgers uid ow. Thus by xing 4 = 0 in the above
system we determine an analogous set of governing equations for steady two-dimensional ow of
Burgers uid and is given as follow;
48
= (2.71)
(2.72)
2.5 Solution Procedure
Flow equations occuring in the eld of science and engineering are highly nonlinear in general.
Consequently, it is exceptionally di¢ cult to nd exact solution of such equations. Usually perturbation,
Adomian decomposion and homotopy perturbation methods are used to nd the solution of nonlinear
equations. But these methods have some drawback through involvement of large/small parameters
in the equations and convergence. An e¢ cient analytical method namely homotopy analysis method
(HAM) [75 78] is one which is independent of small/large parameters. HAM also gives us a way to
adjust and control the convergence region (i.e. by plotting h-curve). It also provides exemption to
choose di⁄erent sets of base functions. We have utilized this technique in the subsequent chapters to
develop the series solutions. The details of this method is provided in chapter 3.
Chapter 3
49
Forced Convective Heat transfer to Burgers
Nano uid with Heat Generation/Absorption
This chapter addresses the 2D forced convection ow of Burgers nanoliquid over stretched surface. The
features of heat source/sink are also incorporated. Appropriate transformations reduce the nonlinear
PDEs to ODEs. The reduced ODEs are then solved by utilizing HAM. The analytical results obtained for
the temperature and concentration of the Burgers nanoliquid are portrayed through several plots and
deliberated in detail. Additionally, the relations Nux and Shx for Burgers nanoliquid are tabulated for
numerous values of the pertinent parameters. One can also detect that the formulated relation in this
chapter can be successfully used to predict the relaxation and retardation times. Additionally, the
results indicate that an enhancement in the material parameters of the Burgers liquid (i.e, Deborah
numbers 1 and 2) correspond to enhancement in the temperature of the Burgers liquid. However, the
impacts of Deborah number 2 on the temperature and concentration elds are quite the opposite to
those of 1 and 2.
3.1 Formulation of Problem
Ponder 2D ow of Burgers liquid over a stretched sheet by utilizing nanoparticles. The Burgers
nanoliquid is restricted above the y axis and ow is produced due to the elongating of the sheet along
x axis with velocity ax, where a is a positive constant. In this research work we also assumed that the
temperature and concentration of the Burgers liquid at the stretched surface are Tw and Cw,
respectively, which are greater than the ambient liquid temperature T1 and concentration C1;
respectively. Employing overhead revealed assumptions the continuity, momentum, energy and
50
concentration equations (2.2), (2.13), (2.15), (2.71) and (2.72) (cf. Chapter 2) in the presence of heat
source/sink take the form:
(3.1)
(3.2)
(3.3)
(3.4)
The conditions
for the problem under consideration are
u = ax; v = 0; T = Tw; C = Cw at y = 0; (3.5)
as y!1: (3.6)
Where is the thermal di⁄usivity, Q0 the heat generation/absorption parameter and
ratio of heat capacities of nanoparticle to base liquid.
51
The overhead research problem can be expressed in a simpler form by utilizing the following
transformations
(3.7)
By employing the similarity variables (3.7) and after mathematical simpli cation; we obtain the following
problem:
f000 f02 + ff00 1 f2f000 2ff0f00 2 3f2f002 + 2ff02f00 f3fiv
3 ffiv f002 = 0; (3.8)
00 + Pr + PrNb 0 0 + Prf 0 + PrNt 02 = 0; (3.9)
(3.10)
f = 0; = 1; = 1; f0 = 1; at = 0; (3.11)
f0 !
0; f00 ! 0; ! 0; ! 0
as !1;
(3.12)
where 1;3 (= 1;3a) and 2 = 2a2 are the Deborah numbers, heat source
(> 0) and heat sink (< 0) parameter, Pr = 1 Prandtl number, Nb = DB(Cw C1)
Brownian motion parameter, thermophoresis parameter,
Lewis number and Nb = DB(Cw C1) Brownian motion parameter, respectively.
52
The quantities of interest from the industrial point of view are Nux and Shx, which are de ned by
at y = 0: (3.13)
Theses quantities of industrial interest dimensionless form are
Nux = 0 Re Re at y = 0; (3.14)
in which Re=ax2/ characterize the Reynolds number
3.2 Solution by HAM
Appropriate approximations (f0; 0; 0) and operators (Lf;L ;L ) are required in order to determine the
approximate series solutions by virtue of HAM. In the present problem, these are
f0( ) = 1 e ; 0( ) = e ; 0( ) = e ; (3.15)
: (3.16)
The overhead operators have the characteristics given as under
(3.17)
where elucidate the constants.
53
3.2.1 The zeroth order deformation problems
Deformation of the present research work at zeroth-order are
(c-1)Lf hf0( ) f~( ;c)i = Nf[f~( ;c)]c~f;
(3.18)
(c-1)L h 0( ) ~( ;c)i = N [f~( ;c);~( ;c); ~( ;c)]c~ ; (3.19)
(c-1)L h 0( ) ~( ;c)i = N [f~( ;c);~(;c); ~( ;c)]c~ :
f~(0;c) = 0; ~(0;c) = 1; ~(0;c) = 1; f~0(0;c) = 1; f~0(1;c) =
0; ~(1;c) = 0; ~(1;c) = 0;
(3.20)
(3.21)
!
(3.22)
~
2
2 : (3.24) @
Here c 2 [0;1] indicates the embedding parameter.
When changes from 0 to 1 then we obtain the nal solution
1 !
at c = 0; (3.25) m~! m
@~ +PrNt( )2 + Pr
@
~; (3.23)
54
~
! 1 ; at c = 0;; (3.26) m~! m~
~! 1 ; at c = 0: (3.27) m~!
The convergence regarding Eqs. (3:25) (3:27) is strongly based on (~f;~ ;~ ): By selecting appropiate
values of (~f;~ ;~ ) so that Eqs. (3:25) (3:27) converge at = 1 then
(3.28)
(3.29)
: (3.30)
3.2.2 The m~th order deformation problems The m~th-
ordeR problems are de ned as follows:
Lf [fm~ ( ) m~ fm~ 1( )] = ~fR^mf~ ( ); (3.31)
L [ m~ ( ) m~ m~ 1( )] = ~R^m~ ( ); (3.32)
L m~ ( ) m~ m~ 1( ) = ~R^m~ ( ); (3.33)
(3.34)
m~ 1 m~ 1 m~ 1 k
^mf~ ( ) = fm000~ 1 + Xfm~ 1 kfk00 Xfm0~ 1 kfk0 + 2 1 Xfm~ 1 k Xfk0 lfl00 R
55
k=0 k=0 k=0 l=0 m~ 1 k m~ 1 k l
1 Xfm~ 1 k Xfk lfl000 + 2 Xfm~ 1 k Xfk l Xfl ifi0000
k=0 l=0 k=0 l=0 i=0 m~ 1 k l m~ 1 k l
2 2 Xfm~ 1 k Xfk0 l Xfl0 ifi00 3 2 Xfm~ 1 k Xfk l Xfl00 ifi00
k=0 l=0 i=0 k=0 l=0 i=0 m~ 1 m~ 1
+ 3 Xfm00~ 1 kfk00 3 Xfm~ 1 kfk0000; (3.35)
k=0 k=0
m~ 1 m~ 1
R^m~ ( ) = 00m~ 1 + PrNb X m0~ 1 k k0 + PrNt X 0m~ 1 k k0
k=0 k=0
+Pr m~ 1; (3.36)
(3.37)
and
m~ =(3.38)
terms of solutions General expressions (fm~ ; m~ ; m~ ) for Eqs. in
are presented by the following expressions:
(3.39)
(3.40)
56
(3.41)
in which the constants G?j (j = 1 7) through the boundary conditions (3:37) are given by
(3.42)
: (3.43)
3.3 Convergence of the Homotopy Solutions
The solutions of nonlinear coupled ordinary di⁄erential equations (3:8) (3:10) subject to boundary
conditions (3:11) and (3:12) are obtained with the help of well-known homotopy analysis technique
(HAM). The homotopy analysis technique (HAM) provides a way to check and adjust the convergence
of the obtained solution with the help of the auxiliary parameters
}f, } and } and the base functions. These parameters (}f;} ;} ) have vital role for the series solutions. The
appropriate values of (}f;} ;} ) are determined by considering minimum square which is de ned as
: (3.47)
Table 3.1 is plotted to ensure the convergence of this research work. This table portrays about the
convergence of given problem it is anticipated from the table that convergent solutions are achieved
for velocity of Burgers naonoliquid at 15th-order of approximation while the convergent solutions for
the temperature and concentration of Burgers nanoliquids are gained at 30th-order of approximation.
57
Table 3.1 : The convergent homotopic solutions for Burgers nanoliquids are achieved when 1 = 3 = 2
= 0:2; = 0:2;Le = 1:0;Pr = 1:4;Nt = 0:1 and Nb = 0:1.
Approximations -f 00(0) - 0(0) - 0(0)
1
5
10
15
20
26
30
35
0:972461
0:973283
0:973324
0:973323
0:973323
0:973323
0:973323
0:973323
0:377585
0:386163
0:383310
0:382695
0:382631
0:382623
0:382622
0:382622
0:0456303
0:497787
0:520282
0:520219
0:520261
0:520267
0:520267
0:520267
3.4 Discussion
This segment of research work is focused to explore the features of numerous material parameters
on the temperature and concentration of the Burgers nanoliquid. The coupled set of Eqs. (3.8)-(3.10)
along with conditions (3.11) and (3.12) are utilized to solve the resultant problem by means of HAM.
Graphs are presented to envisage physical behavior of di⁄erent parameters on the Burgers nanoliquid.
Additionally, results of Nux and Shx relations for Burger nanoliquid in 2D ow are presented in tabular
form and discussed in detail.
Figures 3:1(a;b) are sketches to predict the impact of 1 and 2 on the temperature of the Burgers
nanoliquid. We detected from the graphical observation that with the incremented values of Deborah
numbers the temperature of the Burgers nanoliquid boosts up. Figure 3:2(a) is designed to visualize
the features of Deborah number 3 for the temperature of the Burgers nanoliquid in 2D ow. It is
58
observed from the graphical illustrations that the temperature of the Burgers nanoliquid decline for
the higher value of 3. Figure 3:2(b) demonstrate the characteristics of Pr for the temperature of the
Burgers nanoliquid. It is perceived that a diminution in the temperature of the Burgers nanoliquid is
detected as enriches Pr. From the mathematical point of view Pr is ratio of momentum di⁄usivity of
Burgers nanoliquid to thermal di⁄usivity of Burgers nanoliquid. By keeping this de nition of Prandtl
number in mind, it is clear from this relation that as we augmented the Pr the thermal di⁄usivity of
the Burger nanoliquid drops due to which temperature of the Burgers nanoliquid decays. Figure 3:3(a)
has been sketched to visualize the impacts of thermophoresis parameter Nt on the temperature of the
Burgers nanoliquid. The temperature of the Burgers nanoliquid is enlarged with the incremented
values of thermophoresis parameter. As we boosts up the values of thermophoresis parameter the
temperature of Burgers nanoliquid enriches this is because of the fact that for the augmented values
of thermophoresis parameter di⁄erence between the temperature of the liquid at the and far away
from plate enhances. Characteristics of Nb for temperature of the Burgers nanoliquid can be
understood through gures 3:3(b). It is detected from the displays illustrations that as we boosts up
the values of Nb the temperature of the Burgers nanoliquid enriches. It mechanisms appear because
with the incremented values of Nb the random motion of the nanoparticles rises due to which
temperature of the Burgers nanoliquid enhances. Figure 3:4(a) is designed to deduce impact of heat
source parameter (> 0) on the temperature of the Burgers nanoliquid. It is detected from the graphs
that the temperature of the Burgers nanoliquid enriches for higher values of heat generation
mechanism because heat generation phenomenon provides more heat to the liquid of the Burgers
nanoliquid. Figure 3:4(b) displays the temperature of the Burgers nanoliquid for heat absorption
59
mechanisms (< 0). These gures reveal that temperature of the Burgers nanoliquid declines for heat
absorption mechanisms.
Figures 3:5(a;b) are delineated to visualize the in uence of Deborah numbers 1 and 2 for
concentration of the Burgers nanoliquid. We observed from these graphics that the concentration of
the nanoliquid is growing function 1 and 2. Figure 3:6(a) interprets that escalation in 3 corresponds to
the decline the concentration of the Burgers nanoliquid. Figure 3.6 (b) is prepared to explored the
features of Le for concentration of the nanoliquid. We observed from the mathematical point of that
Le is the raton of 1 to DB. As we augmented the Le then Brownian di⁄usion decline due to which the
concentration of the Burgers nanoliquid drops. The features of Nt for concentration of Burgers liquid
is designated through sketches 3:7(a). One can from the graphical illustrations that concentration of
the Burgers liquid boosts up as we rises Nt. Figure 3:7(b) interpret the features of Nb for the
concentration of Burgers nanoliquid. One can perceive from these plots that increase in Nb cause the
concentration of the Burgers nanoliquid to diminish.
Table 3.2 is presented for 0(0) and 0(0) for distinct values of Pr; ;Nb;Nt and Le.
We can noticeable from table that heat/mass transport rates are boosted with the rise in the
Prandtl number. Furthermore, it is detected that augmentation in the heat source parameter,
Nb;Nt and Le decline rate of heat transport mechanisms
60
0 2 4 6 8 10 12 0 2 4 6 8 10
12 η η
Figs. 3:1: Variation of ( ) via 1 (panel-a) and 2 (panel-b).
0 2 4 6 8 10 12 0 2 4 6 8 10
12 η η
Figs. 3:2: Variation of ( ) via 3 (panel-a) and Pr (panel-b).
61
0 2 4 6 8 10 12 0 2 4 6 8 10
12 η η
Figs. 3:3: Variation of ( ) via Nt (panel-a) and Nb (panel-b).
0 2 4 6 8 10 12 0 2 4 6 8 10
12 η η
Figs. 3:4: Variation of ( ) via > 0 (panel-a) and < 0) (panel-b).
62
0 2 4 6 8 10 12 0 2 4 6 8 10
12 η η
Figs. 3:5: Variation of ( ) via 1 (panel-a) and 2 (panel-b).
0 2 4 6 8 10 12 0 2 4 6
8 10 12 η η
Figs. 3:6: Variation of ( ) via 3 (panel-a) and Le (panel-b).
63
0 2 4 6 8 10 12 0 2 4 6
8 10 12 η η
Figs. 3:7: Variation of ( ) via Nt (panel-a) and Nb (panel-b).
64
Table 3.2: The rate of heat and mas transfer mechanisms for distinct values of Pr; ;Nb;Nt
and Le when 3 = 2 = 1 = 0:2.
Pr Nt Nb Le 0(0) 0(0)
1.4 0.2 0.1 0.1 1.0 0.382622 0.520267
1.6 0.430023 0.567626
1.7
0.450711 0.591845
1.5 0.0 0.668390 0.335028
0.1
0.553222 0.428434
0.3
0.172336 0.721386
0.2
0.374922 0.380974
0.3
0.343763 0.262409
0.4
0.313862 0.184418
0.2
0.358295 0.668851
0.3
0.312513 0.709515
0.4
0.270043 0.729081
1.1 0.405471 0.601278
1.2 0.403769 0.655788
1.3 0.402293 0.707634
65
Chapter 4
3D Flow and Heat Transfer Mechanisms to
Burgers Fluid
Features of developed heat ux relation on steady 3D ow of Burgers liquid are explored in this
chapter. The developed heat ux relation is modi cation of Fourier s relation of heat conduction that
ponders fascinating characteristic of relaxation time in energy equation. The governing PDEs reduced
to a set of ODEs by implementation of appropriate transformations. These transformed nonlinear
ODEs are solved by employing HAM. Characteristics of thermal relaxation time and ratio of stretching
rates parameter on the temperature eld is analyzed and presented graphically. It is found that the
temperature is signi cantly a⁄ected with varying values of the thermal relaxation time. More speci
cally, we detected from graphical illustrations that temperature of the Burgers liquid for 3D diminish
for greater values of E. Thus, temperature of Burgers liquid is higher in case of Fourier s relation when
compared to developed heat ux relation.
4.1 Mathematical Formulation of the Problem
Ponder the 3D forced convection ow of Burgers uid over a bidirectional stretched surface.
The sheet coincides with z = 0 and ow takes place in domain z> 0. The developed heat ux
relation is utilized in the energy equation to visualize impact of relaxation time term on the heat transfer
mechanisms. The ambient temperature of the Burgers liquid is taken T1; while the temperature of the
66
Burgers liquid at the stretched surface is preserved at a certain value Tw such that Tw >T1. By utilizing
overhead approximations and simplifying governing equations
(2.2), (2.20) and (2.49)-(2.50) for Burgers uid we can write (cf. Chapter 2)
Fig. 4.1. Physical model of the problem.
(4.1)
67
68
The appropriate conditions of present problem for the velocity and temperature of Burgers liquid are
as follows:
as z!1; (4.5)
We consider the following transformations:
(4.7) In view of Eq. (4:7),
Eqs. (4:1) to (4:6) are reduced to the following boundary value problems:
f000 + (f + g)f00 f02 + 1 2(f + g)f0f00 (f + g)2f000 + 2[(f + g)3fiv
(f + g)2f00(f00 + g00) 2(f + g)2 f00 2 2(f + g)(f0 + g0)f0f00
2g0(f + g)3f000] + 3[(f00 + g00)f00 (f + g)fiv] = 0; (4.8)
69
(4.9)
00 + Pr(f + g) 0 Pr E 2(f + g)(f0 + g0) 0 + (f + g)2 00 = 0; (4.10)
f(0) = 0; f0(0) = 1; g(0) = 0; g0(0) = ;(0) = 1; (4.11)
f0 ! 0;f00 ! 0;g0 ! 0;g00 ! 0; ! 0; as !1: (4.12)
In the above equations, is the ratio of stretching rates parameter, E = a E the nondimensional
thermal relaxation time and Pr the Prandtl number de ned in chapter 3.
Note that = 0 yields the two-dimensional case (g = 0), that is, f000 + ff00 (f0)2 + 1 2ff0f00
f2f000 + 2 f3fiv 2f(f0)2f00 3f2f002
+ 3 (f00)2 ffiv = 0: (4.13)
4.2 The Analytical Solution
We have utilized the HAM to obtain the analytical solutions corresponding to the governing nonlinear
coupled ordinary di⁄erential equations (4:8) (4:10) and the related boundary conditions. The guesses
(f0;g0; 0) and operators ($f;$g;$ ) are taken as follow:
70
(4.14)
$
$ (4.15)
subject to the properties
$f[N1 + N2e + N3e ] = 0; $g[N4 + N5e + N6e ] = 0;
$ [N7e + N8e ] = 0; (4.16)
where Ni (i = 1 8) are characterize the constants.
4.3 Graphical Results and Discussion
Since the transformed equations (4:8) (4:10) together with the boundary conditions (4:11) and
(4:12) are highly nonlinear and hence the homotopy analysis treatment can be more appropriate. The
outcomes obtained by the implementation of HAM are sketched graphically.
Figures 4:2(a) and 4:2(b) explore the e⁄ect of the 3 and on the velocity component f0( ),
respectively. These gures demonstrate an increasing behavior of velocity for larger values of the 3 and
. Physically, Deborah number dependent on the 3. Therefore, with increase in
3 the retardation time also enhances. Consequently, the uid ow is accelerated. Moreover, this
mechanism spectacles that the velocity of the Burgers liquid drop for augmented value . Clearly an
escalation in the means that the velocity of the Burgers liquid in the y-direction, consequently f0(
)drops.
71
Figures 4:3(a) and 4:3(b) are depicted for the variation of the Deborah number 1 and Deborah
number 2 to velocity component g0( ). Apparently, these gures show a diminishing behavior of velocity
component g0( ) with the increasing values of the Deborah numbers 1 and 2. In fact 1 from the
mathematically point of is ratio of relaxation to observation times. So with augmentation in Deborah
number relaxation time also increases which provides more resistance to the uid motion. Therefore,
velocity pro le diminishes.
To interpret the features of velocity of the Burgers liquid for the boost up values of 3 and are
illustrated through gures 4:4(a) and 4:4(b), respectively. Figures 4(a) clearly shows that 3 has reverse
behaviors on the velocity component g0( ) when compared with 1 and 2. Furthermore, it is noted that
the velocity of the Burgers liquid enhance with the increasing value of . An increase in leads to fact
that velocity of Burgers liquid dominates in the x direction, consequently velocity component of the
Burgers liquid growths.
The impact of the Deborah numbers 1 and Deborah numbers 2 on the temperature of the Burgers
liquid ( ) is depicted through gures 4:5(a) and 4:5(b), respectively. We inferred that the impact of 1
and 2 on the velocity component g0( ) and temperature ( ) is reversed. It is anticipated by these gures
that the temperature distribution increases as the Deborah numbers 1 and 2 incremented. This is due
to fact that as we increase 1 and 2 collision between the uid particles increases and result in an
enhancement in temperature of the uid.
Figures 4:6(a) and 4:6(b) show temperature distribution variation for the Deborah numbers
3 and ratio of stretching rates parameter . These gures reveal that the temperature of the Burgers
liquid diminish for enlarged values of the Deborah number 3 and ratio of stretching rates parameter .
72
Furthermore, these gures lead to the conclusion that the temperature of the the Burgers liquid is
decreasing function of the Deborah number 3 and .
The behavior of Pr and E on the temperature distribution are sketched in gures 4:7(a) and 4:7(b),
respectively. One can detect from the graphical illustration that the temperature of the Burgers liquid
diminish with the augmentation in Pr. It is perceived that a diminution in the temperature of the
Burgers liquid is detected as the enriches the Prandtl number. From the mathematical point of view
the Prandtl number momentum di⁄usivity of Burgers nanoliquid to thermal di⁄usivity of Burgers liquid.
By keeping this de nition of Prandtl number in mind, it is clear from this relation that as we augmented
the Prandtl number the thermal di⁄usivity of the Burger liquid drops due to which temperature of the
Burgers liquid decays. Furthermore, one can detect that temperature of the Burgers liquid reduces for
higher values of E. Physically, as we boosts up the E, molecules of the Burgers liquid need extra time
to transport heat to
its neighboring particles.
0 2 4 6 8 0 2 4 6 8 η η
Figs. 4.2: Characteristics of f0( ) for augmented values of 3 (panel-a) and (panel-b).
73
0 2 4 6 8 0 2 4 6 8 η η
Figs. 4.3: Characteristics of g0( ) for augmented values of 1 (panel-a) and 2 (panel-b).
0 2 4 6 8 0 2 4 6 8 η η
Figs. 4.4: Characteristics of g0( ) for augmented valus of 3 (panel-a) and (panel-b).
74
0 2 4 6 8 0 2 4 6 8 η η
Figs. 4.5: Characteristics of ( ) for augmented values of 1 (panel-a) and 2 (panel-b).
0 2 4 6 8 0 2 4 6 8 η η
Figs. 4.6: Characteristics of ( ) for augmented values of 3 (panel-a) and (panel-b).
75
0 2 4 6 8 0 2 4 6 8 η η
Figs. 4.7: Characteristics of ( ) for augmented values of Pr (panel-a) and E (panel-b).
Chapter 5
Heterogeneous-Homogeneous Processes in
3D Flow of Burgers Fluid
This chapter communicates the features of chemical processes for 3D ow of Burgers liquid over a
bidirectional stretched surface. Additionally, developed heat transfer mechanism and mass di⁄usions
phenomena are utilized in the energy and concentration equation to visualize the characteristics of
these relations for the Burgers liquid. These are the advanced relation for the Fourier s through
relaxation time in the energy equation and Fick s through relaxation time in the concentration
equation. Similar to previous chapter, a transformation procedure is adopted to obtain the highly
coupled ODE. The developed nonlinear problem is then elucidated by utilizing the HAM. The features
of numerous industrial parameters involved in this problem are pondered and physical features of
these parameters from the engineering point of view are demonstrated through sketches and debated
76
through the reasonable judgment. Our investigation conveys that the temperature and concentration
pro les decay as the nondimensional thermal and concentration relaxation times parameters
incremented. On the other hand, it is perceived that the concentration of the Burgers liquid
moderates with the growth of the Deborah numbers 1 and 2. Additionally, it is fascinating to nd that
the concentration of the Burgers liquid declines as the Schmidt number escalates. To sum up, the
verdict of the present chapter is that the pro ciency of thermal and concentration systems can be
enriched by introducing the Cattaneo-Christov heat and mass ux models.
5.1 Mathematical Description of the Problem
We deliberate 3D ow of Burgers liquid over a bidirectional stretched surface with velocities u = ax and
v = by. We have utilized the relaxation of time mechanism in the energy and concentration equations
to visualize the features these mechanisms on the Burgers liquid. Additionally, the ow analysis in the
presence of heterogeneous-homogeneous process is investigated. The temperature of the Burgers
liquid at the stretched surface is Tw; similarly, concentration of the Burgers liquid at surface is Cw; at
in nite distance accomplished the values (T1;C1), respectively. Furthermore, we deliberate a simple
relation of chemical process in which two species associated for cubic autocatalysis. The overhead
norms lead us the continuity, momentum, energy and concentration equations (2.20), (2.21), (2.24),
(2.25) and (2.49) to (2.51) of the form as follows (cf. Chapter 2)
(5.1)
77
=
78
(5.4)
79
(5.5)
(5.6)
: (5.7)
The boundary conditions for the existing problem are
at z = 0;(5.8)
Introducing the following transformations (cf. Chapter 4):
By utilizing Eq. (5:10), Eqs. (5:1) to (5:9) are reduced to the following boundary value problems:
80
f000 + (f + g)f00 f02 + 1 2(f + g)f0f00 (f + g)2f000 + 2[(f + g)3fiv
(f + g)2f00(f00 + g00) 2(f + g)2 f00 2 2(f + g)(f0 + g0)f0f00
2g0(f + g)3f000] + 3[2(f00 + g00)f00 (f + g)fiv] = 0; (5.11)
g000 + (f + g)g00 g02 + 1 2(f + g)g0g00 (f + g)2g000 + 2[(f + g)3giv
(5.12)
00 + Pr(f + g) 0 Pr E 2(f + g)(f0 + g0) 0 + (f + g)2 00 = 0; (5.13)
00 + Scb(f + g) 0 Scb C 2(f + g)(f0 + g0) 0 + (f + g)2 00 = 0; (5.14)
(5.15)
ber, ratio of di⁄usion coe¢ cient, Sc = D C Schmidt number (for the
heterogeneoushomogeneous processes) and (k1;k2) measure of strength of homogeneous and
heterogeneous processes, respectively. Moreover, dimensionless parameters 1; 2; 3; and Pr are the
same as de ned in chapter 4.
h00 + (f + g)h0 + kh2
Sc 1 = 0;
f(0) = 0; f0(0) = 1; g(0) = 0; g0(0) = ;
(5.16)
(0) = 1; (0) = 1 r0(0) = k2r(0); h0(0) = k2r(0);
(5.17)
f0 ! 0;f00 ! 0;g0 ! 0;g00 ! 0; ! 0; ! 0;r! 0,h! 0 as !1;
(5.18)
where C = a C is the relaxation time parameter of concentration, Scb = DB Schmidt num-
81
For comparable in size by utilizing this assumption we can make a further assumption that is, i.e. =
1 and thus
h( ) + r( ) = 1;
and Eqs. (5.15) and (5.16) yield
(5.19)
(5.20)
with the boundary conditions
as !1: (5.21)
5.2 Homotopic Solutions
An e¢ cient technique namely HAM has been employed to construct the analytic solutions of the ow
model de ned by Eqs. (5.11)-(5.14) and (5.20) subject to conditions (5.17)-(5.18) and (5.21) for di⁄erent
various values of controlling parameters. For such homotopic solutions, the suitable initial
approximations (f0;g0; 0; 0;r0) and linear operators ($f;$g;$;$ ;$r) are de ned as follows:
f0( ) = 1 e ; g0( ) = (1 e ); 0( ) = e ;
0( ) = e ; r0( ) = (1 e ): (5.22)
$f [f( )] = f000 f; $g [g( )] = g000 g0; $ [ ( )] = 00 ;
$ [ ( )] = 00 ; $r [r( )] = r00 r: (5.23)
82
5.3 Graphical Results and Discussion
This section is organize to interpret features of chemical processes for Burgers liquid over a
bidirectional stretched surface by utilizing develop heat and mass ux relations. The analytical results
in the form of series solution are deliberated for numerous pertinent parameters. Furthermore, the
foremost intention of the subsequent debate is to interpret the impacts of newly parameters
announced in this chapter .
Figure 5.1(a) has been sketched to visualize the impact of Pr on the temperature of the Burgers
liquid. One can detect from these sketches that enlargement of Pr leads to fallo⁄ the temperature of
the Burgers liquid. From the mathematical point of view Pr is ratio of momentum to thermal di⁄usivity,
as we enlarged Pr the thermal di⁄usivity of the Burgers liquid decline due to which temperature of the
Burgers liquid drops. Additionally, it can perceived from graphical illustrations that as we boosts up
Pr, temperature of the Burgers liquid decay. By utilizing the relaxation time term in the equation
means that particles of Burgers liquid need extra time to transfer heat.
The characteristics of the Deborah numbers 1 and 2 on the dimensionless concentration
distribution can be seen through gures 5:2(a;b). We can perceive from these gures that the
temperature of the Burgers liquid enriches with the bigger values of the 1 and 2. This mechanism
appear due to fact that thermal relaxation time associated with Deborah number enhances as we
increase it and produces more resistance to the motion of the Burgers uid
enhances.
The uctuation of concentration of the Burgers liquid for 3 and C is plotted through sketches 5:3(a;b).
One can detect from these diagrams that the concentration of the Burgers liquid drop as 3 and C
incremented. This mechanism occur because an enhancement in the
83
Deborah number causes an increase in retardation time and consequently concentration of the Burgers
liquid decay.
Figures 5:4(a;b) are presented to visualize the features of the concentration of the Burgers liquid
for distinct values of and Scb. It can be seen that and Scb diminish concentration of Burgers liquid. An
enhancement in the stretching rates parameter corresponds the velocity in y direction dominates then
the velocity in x direction due to which collision of the particles enhances which as a result diminishes
the concentration of the Burgers liquid. Furthermore, from the mathematically point of view Scb is the
ratio momentum to molecular di⁄usivity. As we augmented Scb the molecular di⁄usivity declines due
to which concentration of Burgers liquid reduces.
The variation of Sc (for the heterogeneous-homogeneous processes) and homogeneous reaction
parameter k1 on the concentration of the Burgers liquid is demonstrated through gures 5:5(a;b). It is
found through these gures that the concentration distribution diminishes for k1 while con icting
behavior is detected for Sc. This mechanism appear due to fact that the reactants are consumed
during homogeneous reaction which causes the concentration pro le to decrease. Moreover, as we
augmented the values of the Schmidt number the molecular diffusivity for the heterogeneous-
homogeneous processes diminishes due which the concentration distribution decays while the
associated concentration boundary layer thickness intensi es.
84
0 2 4 6 8 0 2 4 6 8 η η
Figs. 5.1: Features of ( ) for distinct values of Pr (panel-a) and E (panel-b).
0 2 4 6 8 0 2 4 6 8 η η
Figs. 5.2: Features of ( ) for distinct values of 1 (panel-a) and 2 (panel-b).
85
0 2 4 6 8 0 2 4 6 8 η η
Figs. 5.3: Features of ( ) for distinct values of 3 (panel-a) and C (panel-b).
0 2 4 6 8 0 2 4 6 8 η η
Figs. 5.4: Features of ( ) for distinct values of (panel-a) and Scb (panel-b).
86
0 2 4 6 8 10 12 14 0 2 4
6 8 10 12 14 η η
Figs. 5.5: Features of r( ) for distinct values of k1 (panel-a) and Sc (panel-b).
87
Chapter 6
3D Convectively Nonlinear Radiative Flow
of Burgers Fluid
In this chapter an analysis is performed to explore the 3D ow of Burgers uid over a bidirectional
stretched surface. Moreover, in this chapter we again utilized the advanced model of homogeneous-
heterogeneous reactions with equal di⁄usivities for reactant and autocatalysis. The basic governing
nonlinear problem is presented and reduced into self-similar form with the aid of suitable similarity
approach. The non-linear problem is then tackled by employing HAM. The e⁄ectiveness of relevant
physical from the industrial and engineering point of view is explored in depth. One of the interesting
observations is that higher estimations of the Biot number, thermal radiation and temperature ratio
parameters have the tendency to enhance the temperature of the Burgers liquid in 3D. The present
results are also validated in the absence of non-Newtonian e⁄ects by comparison with the previous
pertinent literature. Here we establish outstanding agreement of the existing results with the existing
ones.
6.1 Mathematical Description of the Problem
We ponder 3D nonlinear radiative ow of Burgers uid over a bidirectional stretched sheet.
The stretched velocity along x axis an y axis are u = ax and v = by respectively, in which a;b> 0 are
constants and ow takes place in domain z> 0. In this research work we made the Burgers liquid is in
contact with hot liquid. Additionally, we have explored the heat/mass transport mechanisms by utilizing
88
the chemical processes. Furthermore, the species as z tends to in nity have the concentration c0. By
utilizing the overhead assumptions, the governing of the Burgers liquid are written as (cf. Chapter 2)
89
(6.3)
(6.4)
: (6.5)
The appropriate conditions are
at z =
0; (6.6)
as z!1; (6.7)
90
where qr represents the radiative heat ux. By utilizing the Rosseland approximation [79], the qr is simpli
ed as
(6.8)
In view of Eq. (6:8), energy equation (6:3) becomes
: (6.9)
Introducing the dimensionless velocities, temperature, concentration and variable as :
(6.10) Inserting (6.10), in
(6.1)-(6.9) one can achieve
f000 + (f + g)f00 f02 + 1 2(f + g)f0f00 (f + g)2f000 + 2[(f + g)3fiv
(f + g)2f00(f00 + g00) 2(f + g)2 f00 2 2(f + g)(f0 + g0)f0f00
2g0(f + g)3f000] + 3[2(f00 + g00)f00 (f + g)fiv] = 0; (6.11)
g000 + (f + g)g00 g02 + 1 2(f + g)g0g00 (f + g)2g000 + 2[(f + g)3giv
(6.12)
91
hf1 + Rd (1 + ( f 1) )3g 0i0 + Pr(f + g) 0 = 0; (6.13)
(6.14)
h00 + (f + g)h0 + k1h2 = 0; (6.15)
Sc
f(0) = 0; g(0) = 0; f0(0) = 1; g0(0) = ;
0 (0) = [1 (0)]; r0(0) = k2r(0); h0(0) = k2r(0);; (6.16) f0 ! 0; f00 ! 0; g0 ! 0; g00 ! 0; ! 0; r!
0,h! 0 as !1; (6.17)
where radiation parameter and temperature ratio parameter.
For comparable size, thus
h( ) + r( ) = 1:
Consequently, Eqs. (6.14) and (6.15) result in
(6.18)
(6.19)
with
r0(0) = k2r(0); r! 1 as
Mathematical expression for heat transfer rate (Nux) is
!1: (6.20)
. (6.21)
In view of Eqs. (6.10), it reduces to
92
1
Re (6.22)
6.2 Homotopic Solutions
The set of coupled non-linear di⁄erential Eqs. (6.11) (6.13) , (6.19) subject to the boundary conditions
(6.16), (6.17) and (6.20) are solved analytically by utilizing the HAM. To emanate with such method, it
is indispensable to de ne the initial guesses and linear operators. For such an analytic solution, initial
guesses (f0;g0; 0;r0) and linear operators ($f;$g;$;$r) are chosen as follows:
$ $
6.3 Analysis of Results
The foremost prominence of this segment is to interpret the physical features of chemical process for
3D ow of Burgers liquid over a bidirectional stretched surface by utilizing non-linear thermal radiation
with convective conditions.
The impinging of the Rd and f on the temperature of the Burgers liquid is portrayed through gures
6:1(a;b). These gures elucidated that higher values of Rd and f have the tendency to enhance both the
temperature of the Burgers liquid. From the mathematical point of view as we augmented Rd the hf
increases. Hence Rd enhances the rate of heat transfer of Burgers liquid. Moreover, as we enriches f
the temperature of the Burgers liquid at the sheet is much higher that Burgers liquid away from the
stretched surface. Additionally, it is straightforward clear from energy equation (6.9) that the e⁄ective
93
thermal di⁄usivity is sum of two di⁄usivities. Thus we perceive that f, the coe¢ cient of the later term,
support to boosts up the temperature of the Burgers liquid.
Figures 6:2(a;b) demonstrate the impact of the Pr and on the temperature of the Burgers liquid.
From the mathematical point of view Pr is ratio of momentum to thermal di⁄usivity, as we enlarged
Pr the thermal di⁄usivity of the Burgers liquid decline due to which temperature of the Burgers liquid
drops. Furthermore, Biot number is mathematically the ratio of convection of the Burgers liquid at the
surface to conduction of the Burgers liquid within the surface of a body. As the Biot number e⁄ect
(convection at the surface) increases, temperature of the Burgers liquid at the surface rises.
To exhibit the e⁄ect of 1 and 2 on non-dimensional concentration of the Burgers liquid, we have
plotted gures 6:3(a;b). We can observed from graphical illustrations as we enhances 1 and 2 the
concentration of the Burgers liquid declines. This is due to increase in the
relaxation times 1;2.
Figures 6:4(a;b) portray the variation of the concentration of the Burgers liquid in response for
distinct vales of 3 and , respectively. It is anticipated from these gures that there is a rise in the
concentration of the Burgers liquid diminishes with the increase in Deborah number and stretching
parameter. Moreover, for increasing values of stretching parameter stretching along y direction
increases which causes the concentration to increase.
Figures 6:5(a;b) are presented to interpret the variations of the concentration of Burgers liquid
for various values of k1 and Sc. Figure 6:5(a) examines that the concentration prole decreases by
uplifting the k1. Physically, this is because of the fact that in homogenous processes reactants are
consumed when a chemical process occur due to which concentration of Burgers liquid decline.
94
Furthermore, as expected, it is clear that with augmented values of Sc the concentration of Burgers
liquid enhances. Mathematically, Sc is the ratio of to DB. Consequently, higher values of Sc resemble
higher rate of viscous di⁄usion due to which concentration Burgers liquid enhances.
Table 6.1 depicts a comparison for f00(0) and g00(0) with previously published results in the
absence of nonlinear mechanisms. One can detect from these tables that outstanding agreement
between present outcomes and existing ones were achieved. This agreement demonstrates validity
of current work along with the tremendous accuracy of HAM. Tables 6.2 demonstrate the impact of
numerous emerging parameters on temperature gradient. We can detect from table that temperature
gradient is enriched with escalation in Pr. Additionally, we perceived that rise in thermal radiation
parameter, Biot number and temperature ratio parameter declines the heat transfer rate.
0 2 4 6 8 10 12 14 0 2 4
6 8 10 12 14 η η
Figs. 6.1: ( ) for augmented values of Rd (panel-a) and f (panel-b).
95
0 2 4 6 8 10 12 14 0 2 4
6 8 10 12 14 η η
Figs. 6.2: ( ) for augmented values of Pr (panel-a) and (panel-b).
0 2 4 6 8 10 12 14 0 2 4
6 8 10 12 14 η η
Figs. 6.3: ( ) for augmented values of 1 (panel-a) and 2 (panel-b).
96
0 2 4 6 8 10 12 14 0 2 4
6 8 10 12 14 η η
Figs. 6.4: ( ) for augmented values of 3 (panel-a) and (panel-b).
0 2 4 6 8 10 12 14 0 2 4
6 8 10 12 14 η η
Figs. 6.5: ( ) for augmented values of k1 (panel-a) and Sc (panel-b).
97
Table 6.1: The comparison for f00(0) and g00(0) for distinct values of when 1 = 2 = 0 are
xed.
HPM result [31] f00(0)
HPM result [31] g00(0)
Exact result [31] f00(0)
Exact result [31] g00(0)
Present result
f00(0) Present result
g00(0)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.0
1.02025
1.03949
1.05795
1.07578
1.09309
1.10994
1.12639
1.14248
1.15825
1.17372
0.0
0.06684
0.14873
0.24335
0.34920
0.46520
0.59052
0.72453
0.86668
1.01653
1.17372
1.0
1.020259
1.039495
1.05794
1.075788
1.093095
1.109946
1.126397
1.142488
1.158253
1.173720
0.0
0.066847
0.148736
0.243359
0.349208
0.465204
0.590528
0.724531
0.866682
1.016538
1.173720
1.0
1.02026
1.03949
1.05795
1.07578
1.09309
1.10994
1.12639
1.14249
1.15826
1.17372
0.0
0.06685
0.14874
0.24336
0.34921
0.46521
0.59053
0.72453
0.86668
1.016538
1.17372
Table 6.2: The temperature gradient for Rd; f; and Pr when 1 = 0:5; = 0:5; 2 = 0:2 and 3 = 0:45 are
xed.
Rd
Pr f
-ReNux
0.2
0.305214
0.4
0.294211
0.6
0.21419
0.2
0.345497
98
0.4
0.332753
0.6
0.32934
0.9
0.365217
1.3
0.420642
1.7
0.451723
0.2 0.467381
0.4 0.43638
0.6 0.419536
Chapter 7
Features of Brownian Motion and
Thermophoresis for 3D Burgers Fluid Flow
This chapter peruses the heat and mass transfer characteristics of 3D ow of Burgers liquid over a
bidirectional stretched surface by utilizing nanoparticles. The convective boundary and nanoparticles
mass ux conditions are considered. Additional, the most newly proposed relation for nano uid is
deliberated according to this condition we assumed that concentration gradient of the nanoparticles
is zero at the wall. The set of transformation is presented to alter the PDEs into ODEs then series
solutions are attained by employing the HAM. The e⁄ects of various controlling parameters to the
heat/mass transfer mechanisms are presented through sketches and scrutinized in detailed. The
99
analytical out comes for the wall temperature gradient are calculated and presented through tables.
It is seen that the enlarging values of Nb lead to an attenuation in concentration of Burgers nanoliquid.
Likewise, it is noticed that the concentration eld fall o⁄ hastily corresponding to Deborah number ( 3)
in comparison to
Brownian motion parameter.
7.1 Mathematical Description
We consider three-dimensional steady forced convective ow of an incompressible Burgers nanoliquid
over a bidirectional stretched surface with velocities u = ax and v = by, where a and b are taken as
constants. Fluid ows over the region z> 0. The mass ux of the Burgers nanoliquid is assumed to be
zero at the wall. Moreover, an assumption is made that the temperature of the Burgers nanoliquid is
augmented by the heated liquid under the sheet. Here viscous dissipation is neglected. The equations
governing the 3D ow with heat/mass transport mechanisms can be expressed by Eqs. (2.13), (2.15)
and (2.49-2.51) (cf. Chapter 2)
100
101
(7.3)
. (7.4)
The conditions of the present problem are as follows
,
(7.5)
as z!1: (7.6)
The expression for qr is simpli ed as
(7.7)
In view of Eq. (7:7), energy equation (7:3) becomes
(7.8)
Introducing the dimensionless velocities, temperature, concentration and variable as :
102
(7.9)
Upon the substitution of the similarity transformations (7.9), the above governing equations
(7:1) (7:2);(7:4) and (7:8) along with the boundary conditions (7:5) and (7:6) reduce to
f000 + (f + g)f00 f02 + 1 2(f + g)f0f00 (f + g)2f000 + 2[(f + g)3fiv
(7.10)
g000 + (f + g)g00 g02 + 1 2(f + g)g0g00 (f + g)2g000 + 2[(f + g)3giv
(7.11)
hf1 + Rd (1 + ( f 1) )3g 0i0 + Pr(f + g) 0
+PrNb 0 0 + PrNt 02 + Pr = 0; (7.12)
(7.13)
0 (0) = [1 (0)]; Nb 0(0) = Nt 0(0); (7.14)
, ! 0 as !1; (7.15)
where Nb = DBC1 is Brownian motion parameter and thermophoresis
103
parameter while the remaining dimensionless parameters 1; 2; 3, ;f and Rd are de ned in the previous
chapters (cf. Chapter 3 and Chapter 6).
The local Nusselt number Nux; taking into account the thermal radiation, which of practical
importance in heat transfer phenomenon, is given by
. (7.16)
By inserting Eq. (7.9), the local Nusselt number is transformed as
Re : (7.17)
7.2 Homotopic Solutions
The set of coupled non-linear di⁄erential Eqs. (7.10)-(7.13) subject to conditions (7.14) and (7.15) are
solved analytically by utilizing the HAM. To emanate with such method, it is indispensable to de ne
the initial guesses (f0;g0; 0; 0) and linear operators ($f;$g;$;$ ). For such an analytic solution, the initial
guesses (f0;g0; 0; 0) and linear operators ($f;$g;$;$ ) for the resultant equations of Burgers nanoliquid
in 3D are chosen as follows:
(7.18)
$f [f( )] = f000 f; $g [g( )] = g000 g0;
$ $ [ ( )] = 00 0: (7.19)
104
7.3 Analysis
This segment presents the predominant emphasis of this chapter is to scrutinize the characteristics of
new mass ux conditions for 3D ow of Burgers uid over a stretched surface by
utilizing nanoparticles and thermal radiation.
The impact of Nb and Nt on the temperature of Burgers liquid is portrayed through gures
7:1(a;b). One can detect from the graphical illustrations that greater values of Nb and Nt have the
tendency to boost up the temperature of the Burgers nanoliquid. It is predicted from the plots that
rise in the Nt assisted to augment the thermal di⁄usion coe¢ cient and thereby generating more heat
to the liquid due to which temperature of the Burgers liquid improves. Moreover, larger the Nb has
greater Brownian di⁄usion coe¢ cient and lesser viscous force that enriched the temperature of the
Burgers liquid.
The in uence of the Deborah numbers 1 and 2 on the concentration of the Burgers liquid is
elucidated through gures 7:2(a;b). As expected, it is clear from the diagrams that with rising values of
the Deborah numbers the concentration of the Burgers liquid enriches. This response of the Burgers
liquid appear due to to the fact that as we escalate the Deborah numbers the
relaxation times 1;2 improve.
Figures 7:3(a;b) are delineated the visualize the variation of concentration of Burgers liquid in
response to variation in the values of 3 and , respectively. It can be detected from sketches that
concentration of Burgers liquid in 3D diminished for extending values of the both Deborah
number 3 and Biot number .
105
Figures 7:4(a;b) are arranged to envisage characteristics of Nb and Nt on the concentration of
Burgers nanoliquid. We can directly observed from sketched that the concentration of Burgers
nanoliquid boosts up for greater values of Nt whereas the contrary behavior is perceived for the Nb.
As we augmented Nt the thermophoresis force boosts up due to nanoparticles travels from heated
Burgers nanoliquid to cold Burgers nanoliquid areas and hence rises the concentration of the Burgers
nanoliquid . Furthermore, superior values of the resists the motion of the nanoparticle due to which
concentration of the Burgers liquid decline.
Figures 7:5(a;b) expose features of Pr and Le on the concentration of the Burgers nanoliquid.
It is obvious from sketches 7:5(a;b) that escalating the Pr and Le contract the concentration of the
Burgers nanoliquid. Furthermore, the concentration of Burgers nanoliquid is decline when we rise
values of Le as Le has inverse relation proportional to DB. From the mathematical point of view the
Brownian di⁄usion coe¢ cient higher for weaker Lewis number and due to which concentration of the
nanoliquid decline
The heat transfer mechanisms for the Burgers nanoliquid over stretched surface is presented by
table 7.1. We can perceive from table that temperature gradient rises with aggregate values of
generalized Biot number, however it declines for augmented values of Nt. Fascinatingly, table
7.1 displays that Nb has no role for the Burgers nanoliquid.
106
0 2 4 6 8 0 2 4 6 8 10 η η
Figs. 7.1: Features of ( ) through Nt (panel-a) and Nb (panel-b).
0 2 4 6 8 10 0 2 4 6 8 10 η η
Figs. 7.2: Features of ( ) through 1 (panel-a) and 2 (panel-b).
107
0 2 4 6 8 10 0 2 4 6 8 10 η η
Figs. 7.3: Features of ( ) through 3 (panel-a) and (panel-b).
0 2 4 6 8 10 0 2 4 6 8 10 η η
Figs. 7.4: Features of ( ) through Nb (panel-a) and Nt (panel-b)..
108
0 2 4 6 8 10 12 0 2 4 6 8 10
η η
Figs. 7.5: Features of ( ) through Pr (panel-a) and Le (panel-b).
109
Table 7.1: The temperature gradient for Rd; ;Pr;Nt;Le and Nb when 1 = 0:5; = 0:5; 2 = 0:2 and 3 = 0:45
are xed.
Rd Pr Nb Nt Le -Re
0.2 0.3 1.4 0.5 0.2 1.0 0.405714
0.4 0.394711
0.6 0.21419
0.2 0.165797
0.4 0.282753
0.6 0.36938
0.9 0.275217
1.3 0.320642
1.7 0.351723
0.1 0.369381
0.6 0.36938
0.9 0.36938
0.1 0.37051
0.6 0.364689
0.9 0.360979
0.9 0.370125
1.3 0.369789
1.7 0.369536
Chapter 8
110
Characteristics of Thermophoresis Particle
Deposition on 3D Flow of Burgers Fluid
The characteristics of thermophoresis e⁄ects has been carried out in this chapter to explore the
forced convective ow of three-dimensional steady Burgers uid over a bidirectional stretched surface.
Additionally, the heat transfer mechanism is reviewed by utilizing thermal radiation and heat
generation/absorption. The problem is developed and transformed into ODEs which is solved by
employing the HAM. The achieved results are sketched and discussed physically for exciting physical
parameters. We perceive from the graphical illustrations that concentration of Burgers liquid decay
for boosts up values of thermophoretic parameter. Further, it is noticed that the concentration of the
Burgers liquid fallo⁄s rapidly with thermophoretic parameter in comparison to Schmidt number.
8.1 Formulation of the Problem
Let us deliberate the 3D motion of a forced convective ow of Burgers uid over a bidirectional stretched
surface with velocities u = ax and v = by. The x and y axes are directed along continuous stretched
sheet, z coordinate measured normal to them and the ow takes place in z> 0. We also assumed in this
research work that the temperature of Burgers liquid at stretched surface take values Tw, similarly
concentration of Burgers nanoliquid at sheet is Cw, while as z tends to in nity we have T1 and C1;
respectively. By utilizing the overhead assumption resultant equations (2.13), (2.15) and (2.49)-(2.51)
in presence of the thermophoretic e⁄ect take the form:
111
112
(8.3)
(8.4)
u = ax; v = by; w = 0; T = Tw , C = Cw at z = 0; (8.5)
as z!1; (8.6)
where (Tr;k ) are reference temperature and thermophoretic coe¢ cient.
Introducing dimensionless velocities, temperature, concentration and variable as:
(8.7)
The substitution of the above introduced similarity transformations (8:7) into Eqs. (8:1) leads
to the governing ordinary di⁄erential equations which are given as below:
f000 + (f + g)f00 f02 + 1 2(f + g)f0f00 (f + g)2f000 + + 2[(f + g)3fiv
(f + g)2f00(f00 + g00) 2(f + g)2 f00 2 2(f + g)(f0 + g0)f0f00
(8:6)
2g0(f + g)3f000] + 3[2(f00 + g00)f00 (f + g)fiv] = 0; (8.8)
(8.9)
113
Prf0 + Pr = 0; (8.10)
00 Scf0 + Sc(f + g) 0 Sc 0 0 + Sc 00 = 0; (8.11)
f = 0; f0 = 1; g = 0; g0 = ; = 1; = 1 at = 0; (8.12)
f0 ! 0; f00 ! 0; g0 ! 0; g00 ! 0; ! 0 , ! 0 as !1;
(8.13)
where is the thermophoretic parameter. Additionally, the other non-dimensional
parameters 1; 2; 3; Pr; ;Rd and Sc are de ned in the preceding chapters (cf. chapter 3 and chapter 5).
From the engineering and industrial point of view, the essential features of heat/mass transport
mechanisms can designated as
at z = 0: (8.14)
The above dimensionless variables reduce in the following form
Re Re (8.15)
8.2 The Analytic Series Solution
The subsequent set of Eqs. (8.8)-(8.11) with conditions (8.12) and (8.13) are elucidated analytically by
utilizing the HAM. The series solutions for the heat transfer and mass di⁄usion mechanisms are
obtained selecting the suitable initial guesses (f0;g0; 0; 0) and linear operators ($f;$g;$;) are chosen as:
f0( ) = 1 e ; g0( ) = (1 e ); 0( ) = e ; 0( ) = e (8.16)
114
3 3
$$
$ $ (8.17)
8.3 Graphical Results and Discussion
In segment of dissertation an extensive analytical computations have been performed with varying
impacts of the embedding parameters.
To investigate the behavior of thermal layer for the Burgers liquid, the temperature of the Burgers
liquid for numerous values of Rd and is sketch in gures 8:1(a;b). We can witnessed from these drafts
that temperature of the Burgers liquid boosts up for enlarged values of thermal radiation. As we rise
Rd the surface heat ux enhances due to which temperature of Burgers liquid boosts up. To visualize
the variation in the temperature of the Burgers liquid with the enlarged sets of values of is clari ed in
sketched 8:1(b). One can detect from the graphical illustrations that temperature of the Burgers liquid
augmented function for (> 0). Because heat source mechanism (> 0) contributes excess amount of
heat to the liquid due to which the temperature of the Burgers liquid improves.
In gures 8:2(a;b) the variation in temperature of the Burgers liquid for numerous values of the Pr
is presented for Pr < 1 and Pr > 1, respectively. One can detected from these plots that the
temperature of the Burgers liquid declines for augmented values of Pr for both cases (i:e:;Pr < 1 and
Pr > 1). From mathematical point of view Pr is ratio of momentum to thermal di⁄usivity, so as we
enhances Pr thermal di⁄usivity of the Burgers liquid decline due to which temperature of the Burgers
liquid drops. One can also detected from these sketches that the temperature of the Burgers liquid is
superior in case of Pr < 1 as compared to Pr > 1.
115
Figures 8:3(a;b) show concentration distribution variation for the ratio of and 1. It is anticipated
from gure 8:3(a) that the concentration distribution diminishes as increases.
Additionally, it is can be detected from these sketches that the concentration of the Burgers liquid grow
for superior values of 1.
Figures 8:4(a;b) illustrate the features of concentration of the Burgers liquid in response to change
in the 2 and 3. It is illustrated through gure 8:4(a) that concentration Burgers liquid boosts up with the
escalation in the 2. Furthermore, it is observed that the concentration of the Burgers liquid diminishes
as 3 is augmented.
The in uence of Pr and on the concentration of the Burgers liquid is elucidated through gures
8:5(a;b). An observation of gure 8:5(a) makes it clear that the concentration of the Burgers liquid is
a diminishing functions of Pr. Additionally, we witnessed that as Pr boosts up concentration gradient
at the surface rises. Also, from gure 8:5(b) we can perceive that the concentration of the Burgers
liquid decline adjacent the surface while the contrary behavior is witnessed as z tends to in nity with
growing value of heat source parameter.
The impact of Sc and on the concentration of the Burgers liquid is depicted through gures 8:6(a;b).
It is anticipated from these gures tha the concentration of the Burgers liquid fallo⁄s with accumulative
values of Sc and . Physically, Sc is ratio of momentum to molecular di⁄usivity, as we augmented the Sc
molecular di⁄usion of the Burgers liquid decline due to which temperature of Burgers liquid
diminishes.
1
0.9
0.8
0.7
0.6
2 4 6 8
η
: Impact of ( ) via
116
0 2 4 6 8 0 2 4 6 8 10 η η
Figs. 8.1: Impact of ( ) via Rd (panel-a) and (panel-b).
1
0.9
0.8
0.7
0.6
2 4 6 8
η
: Impact of ( ) via
117
0 6 8 0 2 4 η
Figs. 8.2 Pr for Pr < 1 (panel-a) and Pr > 1 (panel-b).
0 2 4 6 8 0 2 4 6 8 η η
Figs. 8.3: Impact of ( ) via (panel-a) and 1 (panel-b).
1
0.9
0.8
0.7
0.6
2 4 6 8
η
: Impact of ( ) via
118
0 6 8 0 2 4 η
Figs. 8.4 2 (panel-a) and 3 (panel-b).
0 2 4 6 8 0 2 4 6 8 η η
Figs. 8.5: Impact of ( ) via Pr (panel-a) and (panel-b).
1
0.9
0.8
0.7
0.6
2 4 6 8
η
: Impact of ( ) via
119
0 6 8 0 2 4 η
Figs. 8.6 Sc (panel-a) and (panel-b).
120
Chapter 9
Melting Heat/Mass Transfer in Generalized
Burgers Fluid Flow
The motivation behind this research work is the theoretical study of heat transfer and mass
di⁄usion mechanisms on generalized Burgers uid over a stretched sheet. Features of melting heat
mechanism and nonlinear radiation have been deliberated in energy equation. Additionally, this
research work characterize magnetic eld along y axis of Burgers liquid. The resultant problem in
dimensional form is reduced to a dimensionless expression by implementation of suitable similarity
transformations. The resulting dimensionless problem governing the generalized Burgers is
interpreted analytically by means of HAM. The impacts of numerous ow parameters like A, Ha2, M,
Rd, m, Pr and Sc on ow and heat/mass transport features are computed and presented graphically.
Here, we can noticed that the velocity of generalized Burgers liquid is sensitive to changes in the
melting parameter. It is concluded from the graphical sketches that as we improved the melting
parameter the velocity of generalized Burgers liquid is boosts up.
9.1 Problem Development
We deliberate steady 2D forced convective heat transfer and mass di⁄usion mechanisms of an
electrically conducting generalized Burgers liquid. We have applied magnetic eld of strength B0 normal
to the sheet. We have also ponder a small magnetic Reynolds number due to which impact of induced
magnetic eld can be neglected. Additionally, we have ponder melting mechanism and nonlinear
121
thermal radiation in the energy equation to visualize the impact of these phenomenon on the heat
transfer mechanisms of the generalized Burgers liquid. We also assumed in this research work that
temperature of surface is Tm while the ambient temperature is T1 with Tm >T1. The concentration of
the generalized Burgers liquid at sheet is assumed to Cm while C1 is the ambient concentration. By
utilizing the aforementioned assumptions, the governing equations (2.2), (2.6), (2.9), (2.69) and (2.70)
(cf. Chapter 2) can be illustrated by:
(9.1)
(9.3)
(9.4)
122
with
u = Uw(x) = cx; T = Tm; C = Cm ;
at y = 0; (9.5)
. (9.6)
Here D is the mass di⁄usivity, h latent heat of liqui, (T0;cs) temperature and heat capacity of solid
surface, respectively.
Where qr characterizes radiative heat ux and is simpli ed as
(9.7)
In view of Eq. (9.7), energy equation (9.3) becomes
: (9.8)
We are interested to nd solution of overhead research problem, therefore, we present the following
transformations
(9.9)
123
Making use of the transformations (9:9), Eq. (9:1) is identically satis ed and Eqs. (9:2); (9:4) and (9:8)
are transformed as
f000 (f0)2 + ff00 + f2f000 + 2ff0f00 1 + 2 3f2f002 2f(f0)2f00 + f3fiv
+ 3 ffiv + (f00)2 + 4 f2fv 2ff0fiv 2ff00f000 + f0(f00)2
Ha2 2ff000 1ff00 + f0 + A2 + Ha2A = 0; (9.10)
hf1 + Rd ( m + (1 m) )3g 0i0 + Prf 0 = 0; (9.11)
00 + Scf 0 = 0; (9.12)
f0(0) = 1; Prf(0) + M 0(0) = 0; (0) = 0; (0) = 0; (9.13) f0(1) ! A; f00 ! 0; f000 ! 0; (1) ! 1;
(1) ! 0: (9.14)
Where 4 = 4a2 is the Deborah number, the magnetic parameter,
the ratio of stretching rates, the temperature ratio parameter, the
dimensionless melting parameter. Additionally, mathematically M is the ratio of Stefan numbers
and cf(T1 Tm) for solid and liquid phases, respectively, while the remaining
h
dimensionless parameters 1; 2; 3, Rd, Pr, and Sc are de ned the previous chapters (cf. Chapters 4 and
5).
124
The most signi cant quantities from the engineering and industrial point of view are Nux and Shx
which are Characterize as
at y = 0: (9.15)
Dimensionless forms of Nux and Shx are
Re Re (9.16)
9.2 The Analytic Solution
In this section we are interested in nding series solutions for the considered problem. Therefore, the
initial guesses (f0; 0; 0) and linear operators ($f;$;$ ) can be written as
f0( ) = A + (1 A) 1 e (9.17)
Pr
$ $ [ ( )] = 00 0; $ [ ( )] = 00 0: (9.18)
9.3 Results and Discussion
This segment is prepared to extract fascinating insights regarding the features of numerous physical
parameters on the generalized Burgers liquid.
The variation of velocity and temperature of the generalized Burgers liquid with for distinct values
of the stretching ratio parameter A is illustrated through gures 9:1(a;b). We infer from these gures
that there is a rise in the velocity pro le with each incremented value of A. It is straightforwardly
125
appeared from gure 9:1(a) that the uid velocity builds up for the stretching ratio parameter for (A> 1
and A< 1). Further, from gure 9:1(b) we can observe that the non-dimensional temperature of
generalized Burgers liquid boosts up with enhancement of A.
The variation of velocity and temperature of the generalized Burgers liquid is presented through
graphical illustrations 9:2 (a;b) for distinct values of Ha. We infer from these gures 9:2 (a) that the
velocity pro le diminishes as the magnetic parameter is augmented. This happens because of the way
that the Lorentz force rises which produces resistance to motion of generalized Burgers liquid due to
which of velocity liquid declines. Further, one can detected from the sketches that the temperature
of the generalized Burgers liquid diminishes as the magnetic parameter is enlarged. Actually, Lorentz
force rises with the augmented values of magnetic parameter which generates resistive forces to the
motion of the liquid due to which energy is converted into heat.
The features Pr on the velocity and temperature of the generalized Burgers liquid is interpreted
through sketches 9:3 (a;b). An observation of gure 9:3 (a) makes it clear that the velocity of the
generalized Burgers liquid is decreasing functions of Pr. Moreover, one can detect from graphical
illustrations that temperature of generalized Burgers liquid enhances for larger values of Pr.
The impact of M on velocity and temperature of generalized Burgers liquid is portrayed through
gures 9:4 (a;b). An increase in the velocity pro le is revealed through gure 9:4 (a) for increasing values
of the melting parameter. Physically, melting parameter boosts up the molecular motion which
enhances the ow. Moreover, it is observed that the temperature of the generalized Burgers liquid
diminishes for larger values of the melting parameter. Physically, as we augmented the melting
parameter molecular moment improves due to which energy is dissipated and reduction in liquid
temperature.
126
Figures 9:5 (a;b) interpret the impact of Rd and m on the temperature distribution. One can detect
from these sketches that with the rise of radiation parameter, temperature of generalized Burgers
liquid reduces. Also, from gure 9:5 (b) we can perceive from the plots that the temperature of
generalized Burgers liquid declines near the plate while the opposite trend is detected as z tends to in
nity with increasing value of m. Physically, temperature of the melting surface rises with the rise in
temperature ratio parameter which tends to augment in molecular movement which nally results into
dissipation in energy and the fall in liquid temperature.
To visualize features of concentration of generalized Burgers uid for distinct values of M and Sc
are plotted in gures 9:6 (a;b). These gures reveal that the concentration of the generalized Burgers
uid decreases for augmented values of M. The variation in concentration of the generalized Burgers
liquid with the increment for a few sets of values of the Schmidt number is illustrated in gure 9:6 (b).
We can perceive that temperature of generalized Burgers liquid declines with growing value of Sc.
Physically, for the greater values of Sc means that mass di⁄usivity of the liquid particles decays due to
which concentration the generalized Burgers liquid decline.
Table 9.1 is presented to visualize the heat/mass transfer mechanisms at the boundary of the
stretched. One can detect from the table that the magnitudes of temperature and concentration of
liquids are augmented with aggregate values of A and Pr whereas, it is decreased for greater values
of the magnetic, melting and radiation parameters.
127
0 2 4 6 8 10 12 0 2 4 6 8 10
12 η η
Figs. 9.1: E⁄ects of f0 ( ) and ( ) for distinct values of A.
0 2 4 6 8 10 12 0 2 4 6 8 10
12 η η
Figs. 9.2: E⁄ects of f0 ( ) and ( ) for distinct values of Ha.
128
0 2 4 6 8 10 12 0 2 4 6 8 10
12 η η
Figs. 9.3: E⁄ects of f0 ( ) and ( ) for distinct values of Pr.
0 2 4 6 8 10 12 0 2 4 6 8 10
12 η η
Figs. 9.4: E⁄ects of f0 ( ) and ( ) for distinct values of M.
129
0 2 4 6 8 10 12 0 2 4 6 8 10
12 η η
Figs. 9.5: E⁄ects of ( ) for distinct values of Rd and m.
0 2 4 6 8 10 12 0 2 4 6 8 10
12 η η
Figs. 9.6: E⁄ects of ( ) for distinct values of M and Sc.
Table 9.1: Variations of the Re and Re for numerous sets of physical parameters
A;Ha;Pr;M and Rd when 1 = 0:5; 2 = 0:2; 3 = 0:45, 4 = 0:1;Sc = 0:2 and m = 0:3 are xed.
A Pr Ha M Rd Re Re
130
0.0 1.2 0.5 0.2 0.3 0.576975 0.323019
0.1
0.611021 0.373850
0.3
0.656129 0.403057
0.2 0.8
0.521174 0.374226
1.0
0.584995 0.398601
1.1
0.610947 0.399281
0.2
0.656947 0.430217
0.3
0.645173 0.401147
0.4
0.635131 0.377842
0.0
0.626802 0.428702
0.1
0.531021 0.343850
0.3
0.500043 0.323551
0.0 0.626802 0.428702
0.1 0.624517 0.399831
0.2 0.613513 0.357622
Chapter 10
2D Analysis for Generalized Burgers Fluid
Flow in Presence of Nanoparticles
This chapter aims to examine the two-dimensional forced convective ow of generalized Burgers
liquid under the impacts of nano-sized material particles. Utilizing appropriate similarity
131
transformations the coupled PDEs are transformed into set of ODEs. The analytic results are carried
out through HAM to investigate features of several engineering parameters for temperature of
generalized Burgers liquid. In this research work we have presented the results graphically and also
debated these sketched intensely. The presented results show that magnitude of - 0 (0) and - 0 (0)
diminish with each increment of Nt while incremented values of Nb lead to a quite the opposite e⁄ect
on - 0 (0) and - 0 (0).
10.1 Problem Formulation
Let us ponder the 2D forced convective ow of generalized Burgers nano uid over a stretched sheet of
constant surface temperature Tw and concentration Cw. The generalized Burgers nanoliquid lies above
the y axis and ow is persuaded due to stretched sheet. The uniform temperature and concentration
of the generalized Burgers nano uid as y tends to in nity are T1 and C1, respectively. By utilizing
overhead assumptions, the governing equations (2.2), (2.13), (2.14), (2.69) and (2.70) can be
illustrated with the subsequent relations:
(10.1)
132
(10.3)
(10.4)
with
u = Uw = ax; v = 0; T = Tw; C = Cw at y = 0; (10.5)
as y!1: (10.6)
By introducing the following dimensionless quantities, the overhead problem can reduced in simpler
form
(10.7) So, overhead
problem reduces to
133
f000 + ff00 (f0)2 + 1 2ff0f00 f2f000 + 2 f3fiv 2f(f0)2f00 3f2f002
+ 3 (f00)2 ffiv + 4 f2fv 2ff0fiv 2ff00f000 + f0(f00)2 = 0; (10.8)
00 + PrNb 0 0 + Prf 0 + PrNt( 0)2 = 0;
(10.9)
(10.10)
f = 0; f0 = 1; = 1; = 1 at = 0; (10.11)
f0 ! 0; f00 ! 0; f000 ! 0; ! 0; ! 0 as !1: (10.12)
The dimensionless parameters 1; 2; 3; 4; Pr;Nt;Nb and Le are de ned in the previos chapters (cf. Chapter
3 and Chapterp 9).
The quantities of engineering and industrial interest are Nux and Shx, which are Characterize as
at y = 0: (10.13)
In view of Eq. (10.7)
Re Re (10.14)
134
10.2 Homotopic Solutions
For the series solution of this research work involving Eqs. (10:8) (10:12) we employ HAM. The
functional values at origin are (f0; 0; 0) and operators ($f;$;$ ) for HAM solutions are chosen as
$ (10.16)
The overhead operators ful ll the properties given as follows:
$f[G1 + G2e + G3e ] = 0; $ [G4e + G5e ] = 0;
$ [G6e + G7e ] = 0;
(10.17)
in the overhead equations Gi (i = 1 7) are characterize the constants.
Table 10.1: To obtain a good approximation up to some extend for di⁄erent orders when the
convergence during the implantation of HAM is achieved, when 1 = 0:5; 2 = 0:2;Le = 1:1;Nt = 0:1; 3 =
0:45;Nb = 0:1; 4 = 0:1 and Pr = 1.
approximations -f 00(0) - 0(0) - 0(0)
f0( ) = 1 e ; 0( ) = e ; 0( ) = e ; (10.15)
$f [f( )] =
d3
d 3 d
d f ( );
d2 $ [ ( )] = 2
d 1 ( );
135
1
5
10
15
20
25
30
35
0:970712
0:920563
0:910284
0:909913
0:910203
0:910300
0:910313
0:910313
0:484510
0:535665
0:539382
0:538080
0:537679
0:537604
0:537600
0:537600
0:074521
0:250488
0:274686
0:272165
0:271603
0:271490
0:271481
0:271481
10.3 Interpretation of Results
The transformed set of Eqs. (10:8) (10:12) are highly non-linear equations. Graphical analysis of this
research work contains transport of heat/mass phenomenon.
Figures 10:1(a) and 10:1(b) explored the variation of 1 and 2 on the velocity, temperature and
concentration generalized Burgers liquid versus the similarity variable . It is expected from these
sketches that the temperature of the generalized Burgers liquid boosts up with the augmented values
of Deborah numbers 1 and 2 while the analogous behavior is detected for the concentration of the
generalized Burgers liquid. Moreover, the velocity pro le and associated boundary layer thickness
diminish as the Deborah numbers 1 and 2 are augmented. It is illustrated through gures 10:2(a) and
10:2(b) that physical pro les possess a reverse behavior when compared with gures 10:1(a) and
10:1(b) for increasing values of the Deborah numbers
3 and 4.
Figures 10:3(a) and 10:3(b) reveal graphical interpretation of the temperature and concentration
distributions for numerous values of Pr. From the mathematical point of view Pr is ratio of momentum
136
to thermal di⁄usivity so, as we augmented the Pr the thermal di⁄usivity decline due to which
temperature of the generalized Burgers liquid drop. Additionally, we can detect from the graphical
illustrations that concentration gradient at surface increases when the Pr boosts up. It is also
witnessed that liquid who retain low Pr fallo⁄ more gradually as compared to liquids with greater Pr.
The e⁄ects of the Nb on the temperature generalized Burgers liquid and nanoparticle fraction are
shown in gure 10:4(a). One can perceive from the graphical illustrations that as Nb rises temperature
gradient at surface declines. Physically, Brownian movement of nanoparticles is simply the result of
all impulses of liquid molecules on surface of particles. The velocity of the liquid molecules depend on
the temperature and these liquid have high velocities. Actually, the velocities of molecules de ne the
temperature of homogeneous liquid. Additionally, the liquid particles which possesses higher
temperature generally have greater velocities. Therefore, features of random motion of particles are
more signi cant at high temperature. The collision of molecules is random and is of order
femtoseconds. The impact of Nt on temperature ( ) and concentration ( ) distributions is shown in
gure 10:4(b). From these sketches that enrichment in Nt results in rise temperature of generalized
Burgers liquid. From mathematical point of view Nt depends on the temperature gradient due to which
small nanoparticles tend to scatter quicker in warmer regions and slower in cooler regions. The
combined e⁄ect of scattering of the nanoparticles is their apparent movement from warmer to cooler
regions. Due to this migration of particles, greater concentration of liquid particles is in the colder
region. Therefore, the temperature of generalized Burgers liquid rises. Moreover, we can perceived
that as Nt rises the concentration gradient at surface drops as Nt rises.
Figures 10:5(a) and 10:5(b) illustrate the variation of - 0 (0) and - 0 (0) in response to change in
Nb. We ca detected from the graphical illustrations that heat transfer mechanisms on the surface of
137
sheet declines while the opposite behavior is observed for each incremented value of the Brownian
motion parameter. This observation designates that raise in the Brownian motion parameter favor
the di⁄usion of mass. This results in an increase in the concentration gradient on the surface. Figures
10:6(a) and 10:6(b) show the in uence of Nt on - 0 (0) and - 0 (0). These graphs show that the - 0 (0)
and - 0 (0) both decrease as the thermophoresis parameter increases. Furthermore, we have observed
from table 10:1 that a similar behavior is obtained for the - 0 (0) and the - 0 (0) for increasing values
of Nb and Nt:
0 2 4 η 6 8 10 0 2 4 η 6 8 10
Figs. 10.1: Diagram of f0 ( ), ( ) and ( ) via 1 (panel-a) and 2 (panel-b).
138
0 2 4 η 6 8 10 0 2 4 η 6 8 10
Figs. 10.2: Diagram of f0 ( ), ( ) and ( ) via 3 (panel-a) and 4 (panel-b).
0 2 4 η 6 8 10 0 2 4 η 6 8 10
Figs. 10.3: Diagram of ( ) and ( ) via Pr:
139
0 2 4 η 6 8 10 0 2 4 η 6 8 10
Figs. 10.4: Diagram of ( ) and ( ) via Nb (panel-a) and Nt (panel-b).
NN t t
Figs. 10.5: Diagram of - 0 (0) (panel-a) and - 0 (0) (panel-b) for di⁄erent values of Nb.
140
t
Figs. 10.6: Diagram of - 0 (0) (panel-a) and - 0 (0) (panel-b) for di⁄erent values of Nt.
Table 10.1: Variations of - 0 (0) and - 0 (0) for several sets of physical parameters when Pr;Nb;Nt
and Le when 1 = 0:5; 2 = 0:2; 3 = 0:45 and 4 = 0:1 are xed.
Pr Nt Nb Le 0(0) 0(0)
1.0 0.1 0.1 1.0 0.539254 0.222525
1.1
0.570483 0.247336
1.3
0.626940 0.296305
1.1 0.2
0.538991 - 0.099143
0.3
0.508732 -0.418895
0.4
0.479696 - 0.713186
0.2
0.552397 0.448412
141
0.3
0.534892 0.515044
0.4
0.518096 0.548069
1.0 0.539254 0.222525
1.1 0.537600 0.271481
1.2 0.536122 0.317943
Chapter 11
Conclusions, Summary and Future Work
Research is not only an ever continuing but also ever growing as well as further advancing process.
It gives birth to further research: rather it paves the way for further research. It can be said that the
end of the previous research is actually the beginning of the new research. Same is the case with this
thesis. The thesis aims at not only summarizing its own main contributions but also stimulating future
research.
11.1 Contributions of this Work
This dissertation focuses on the features of boundary layer ow, heat and mass transfer to non-
Newtonian Burgers uid induced by stretched surfaces. The Burgers liquid relation had not earlier
deliberated within the context of stretched surface; which we have done here. Particularly, Burgers
uid model for 2D and 3D while the generalized Burgers uid model for 2D were considered. The
analytical solutions were presented by utilizing HAM. To provide a better representation of the nature
142
of ow, thermal and concentration features were inspected for various industrial parameters such as
1; 2; 3; 4; Pr;Nt;Nb and Le etc. A rather lengthy summary of analytical study was provided in preceding
chapters and so we keep our discussion here brief. Thus, the main ndings of this work could be
summarized as follows:
A profound observation is that velocity of Burgers liquid diminish with augmented values of
materials parameter 2 of Burgers uid; however, quite con icting behavior is perceived for 4 of
generalized Burgers uids.
It was perceived that the velocity pro le diminished as the magnetic parameter augmented.
The temperature of the liquid was raised with the augmented values of 1 and 2 whereas contrary
behavior is perceived for 3 and 4 and quite similar behavior is detected for concentration of the
liquid.
Pr is signi cantly a⁄ected the temperature of Burgers liquid. Additionally, with rising values of
Pr the temperature of Burgers liquid was diminished.
The temperature of liquid was diminished with augmented when we involved the relaxation
time in the energy equation.
It was identi ed from the graphical illustrations that, as we augmented the heat source
parameter, the temperature of the liquid was improved while the reverse trend was noticed for
the heat sink parameter.
It was straightforwardly appeared that the temperature of the liquid build up for stronger
thermal radiation due to the enhanced surface heat ux under the in uence of thermal
143
radiation.
It was observed that the veloity pro le enhanced and the temperature distribution diminished
for augmented values of the melting parameter which happened due to enhanced molecular
motion.
The temperature of the liquid was enhanced with the rising values of Nb. Moreover, Nb did not
a⁄ect the temperature of Burgers liquid when we utilized the new mass ux
conditions.
The augmented values of f indicated a rise in the temperature of the liquid.
The temperature and concentration of Burgers liquid tended to enhance with growing values
of Nt.
For the homogeneous reaction parameter the concentration pro le declined and associated
concentration boundary layer thickness was enhanced.
11.2 Future Work
There is a great deal of scope for further work with regards to this research. Within this body of work
we have presented the mathematical modeling and analytical solutions for non-Newtonian Burgers
uid. Here we have studied theoretically the characteristics of heat/mass transport to Burgers uid
model due to stretching sheet. Certainly there is a considerable amount of work remaining in this very
interesting non-Newtonian uid model. Here we will discuss some, but not all, of the many possible
extensions. Firstly, it would be an interesting extension to study ows of Burgers and generalized
144
Burgers uids over stretching cylinder which have direct relevance to the industrial applications.
Nevertheless, this is an area of further study that needs to be addressed in order to improve our
understanding of Burgers uid in context of time-dependent ows. TFor our better understanding, the
data used is not in the experimental pro le but used on basis of problem convergences. Consequently,
this work clearly motivates the need for detailed experimental results with which to compare our
theoretical analysis.
145
Bibliography
[1] J.M. Burgers, Mechanical Considerations-Model Systems-Phenomenological Theories of
Relaxation and of Viscosity in: J. M. Burgers (2nd-Ed.), First Report on Viscosity and Plasticity,
Nordemann Publishing Company, NewYork (1935).
[2] K.R. Rajagopal and A.R. Srinivasa, A thermodynamic frame work for rate type uid models, J:Non
NewtonianFluidMech:; 88 (2000) 207-227.
[3] B.H. Tan, I. Jackson and J.D.F. Gerald, High-temperature viscoelasticity of ne-grained
polycrystalline Olivine, Phys:Chem:Miner:;28 (2001) 641-664.
[4] C.A. Tovar, C.A. Cerdeirina, L. Romani, B. Prieto and J. Carballo, Viscoelastic behavior of Arzua-
Ulloa cheese, J:TextureStud:;34 (2003) 115-129.
[5] M.C. Wang and K.Y. Lee, Creep Behavior of cement stabilized soils, Highway Res: Record;442
(1973) 58-69.
[6] A.R. Lee and A.H.D. Markwick, The mechanical properties of bituminous surfacing materials
under constant stress, J:Soc:Chem:Ind:;56 (1937) 146-156.
[7] R.N.J. Saal and J.W.A. Labout, Rheological properties of asphalts, in: F.R. Eirich (Ed.),
Rheology Theory and Applications, vol. II, AcademicPress, NewYork, (1958).
[8] K. Majidzadeh and H.E. Schweyer, Viscoelastic response of asphalts in the vicinity of the glass
transition point, Assoc:AsphaltPavingTechnol:Proc:;36 (1967) 80-105.
146
[9] A.H. Gerritsen, C.A.P.M. van Grup, J.P.J. van der Heide, A.A.A. Molenaar and A.C. Pronk,
Prediction and prevention of surface cracking in asphaltic pavements, in: Proceedings of the Sixth
International Conference, Structural Design of Asphalt Pavements, vol. I, UniversityMichigan,
(1987).
[10] W.R. Peltier, P. Wu and D.A. Yuen, The viscosities of the earth mantle, in: F.D. Stacey,
M.S. Paterson, A. Nicholas (Eds.), Anelasticity in the Earth, American Geophysical
Union;Colorado, 1981.
[11] D.A. Yuen and W.R. Peltier, Normal modes of the viscoelastic earth, Geophys:J:Royal
Astronom:Earth, 69 (1982) 495-526.
[12] G. Muller, Generalized Maxwell bodies and estimates of mantle viscosity, Geophys:J:R:
Astronom:Soc:;87 (1986) 1113-1141.
[13] G. Rumpker and D. Wolf, Viscoelastic relaxation of a Burgers half-space:implications for the
interpretation of the Fennoscandian uplift, Geophys:J:Int:;124 (1996) 541-555.
[14] P.N. Chopra, High-temperature transient creep in olivine rocks, Tectonophysics, 279
(1997) 93-111.
[15] J.M. Krishnan and K.R. Rajagopal, Review of the uses and modeling of bitumen from and ancient
to modern times, ASMEMech:Rev:;56 (2003) 149-214.
[16] P. Ravindran, J.M. Krishnan, and K.R. Rajagopal, A note on the ow of a Burgers uid in an
orthogonal rheometer, Int:J:Eng:Sci:;42 (2004) 1973-1985.
[17] R. Quintanilla and K. R. Rajagopal, On Burgers uids, Math:Meth:Appl:Sci:;29 (2006)
147
2133-2147.
[18] C. Fetecau, T. Hayat, and Corina Fetecau, Steady-state solutions for some simple ows of
generalized Burgers uids, Int:J:Non linearMech:;41 (5) (2006) 880-887.
[19] C. Fetecau, D. Vieru, T. Hayat and Corina Fetecau, On the rst problem of Stokes for
Burgers uid. I: The case
2183-2194.
[20] D. Vieru, T. Hayat, Corina Fetecau and C. Fetecau, On the rst problem of Stokes for
2
Burgers uid. II: The cases = 4 and
[21] M. Khan, S.H. Ali and C. Fetecau, On accelerated ows of Burgers uid. The cases The
2
cases = 4 and
[22] S. Koneru, E. Masad and K. R. Rajagopal, A thermomechanical framework for modeling the
compaction of asphalt mixes, Mech:Mater:;40 (2008) 846-864.
[23] M. Khan, S. H. Ali and H. Qi, Exact solutions of starting ows for a fractional Burgers uid between
coaxial cylinders, NonlinearAnal : RealWorldAppl:;10 (2009) 1775-1783.
[24] Corina Fetecauc, T. Hayat, M. Khan and C. Fetecau, A note on longitudinal oscillation of a
generalized Burgers uid in a cylinderical domains, J:Non NewtonianFluidMech:; 165 (2010) 350-
361.
[25] M. Khan, A. Anjum, C. Fetecau and H. Qi, Exact solutions for some oscillating motions of a
fractional Burgers uid, Math:Comput:Model:;51 (2010) 682-692.
148
[26] M. Jamil and C. Fetecau, Some exact solutions for rotating ows of a generalized Burgers uid in
cylindrical domains, J:Non NewtonianFluidMech:;165 (2010) 1700 1712
[27] M. Khan, Stokes rst problem for an MHD Burgers uid, Commun:Theor:Phys:;59
(2013) 99-104.
[28] M. Khan, R. Malik, and A. Anjum, Exact solutions of MHD second Stokes ow of generalized
Burgers uid, Appl:Math:Mech: Engl:Ed:;36 (2015) 211-224.
[29] S. Han, L. Zheng and X. Zhang, Slip e⁄ects on a generalized Burgers uid ow between two
side walls with fractional derivative, J:EgyptianMath:Soc:;24 (2016) 130-137.
[30] M. Waqas, T. Hayata, M. Farooq, S.A. Shehzad and A. Alsaedi, Cattaneo-Christov heat ux model
for ow of variable thermal conductivity generalized Burgers uid, J:Mol:Liq:, 220 (2016) 642-648.
[31] S.U.S. Choi, Enhancing thermal conductivity of uids with nanoparticles, ASME, FED,
231 (1995).
[32] H.F. Oztop and E. Abu-Nada, Numerical study of natural convection in partially heated
rectangular enclosures lled with nano uids, Int:J:HeatFluidFlow, 29 (2008) 13261336.
[33] O.D. Makinde and A. Aziz, Boundary layer ow of a nano uid past a stretching sheet with a
convective boundary condition, Int:J:Therm:Sci:;50 (2011) 1326-1332.
[34] A.V. Kuznetsov and D.A. Nield, Natural convective boundary-layer ow of a nano uid past a vertical
plate, Int:J:Therm:Sci:, 49 (2010) 243-247.
149
[35] D. Pal and H. Mondal, E⁄ects of Soret Dufour, chemical reaction and thermal radiation on MHD
non-Darcy unsteady mixed convective heat and mass transfer over a stretching sheet,
Commun:NonlinearSci:Numer:Simulat:;16 (2011) 1942-1958.
[36] D. Pal and H. Mondal, MHD non-Darcian mixed convection heat and mass transfer over a non-
linear stretching sheet with Soret Dufour e⁄ects and chemical reaction, Int:Commun:
HeatMassTransf:;38 (2011) 463-467.
[37] M. Turkyilmazoglu, Exact analytical solutions for heat and mass transfer of MHD slip ow in nano
uids, Chem:Eng:Sci:;84 (2012) 182-187.
[38] W.A. Khan, M. Khan and R. Malik, Three-dimensional ow of an Oldroyd-B nano uid towards
stretching surface with heat generation/absorption, PLOSONE;9 (8) (2014): e105107.
[39] M. Khan, R. Malik, A. Munir and W.A. Khan, Flow and heat transfer to Sisko nano uid over a
nonlinear stretching sheet, PLOSONE, 10 (5) (2015), e0125683.
[40] T. Hayat, M. Waqas, S.A. Shehzad, A. Alsaedi, On model of Burgers uid subject to magneto
nanoparticles and convective conditions, J:Mol:Liq:; 222 (2016) 181-187
[41] T. Hayat, G. Bashir, M. Waqas and A. Alsaedi, MHD 2D ow of Williamson nano uid over a nonlinear
variable thicked surface with melting heat transfer, J:Mol:Liq:; 223, (2016) 836-844.
[42] R. Cortell, Fluid ow and radiative nonlinear heat transfer over stretching sheet, J:King
SaudUniv:Sci:;26 (2013) 161-167.
[43] A. Pantokratoras and T. Fang, Sakiadis ow with nonlinear Rosseland thermal radiation,
Phys:Scr:;87(1) (2013) 015703.
150
[44] A. Mushtaq, M. Mustafa, T. Hayat and A. Alsaedi, Nonlinear radiative heat transfer in the ow of
nano uid due to solar energy: numerical study, J:TaiwanInst:Chem:Eng:; 45 (2014) 1176-1183.
[45] S. T. Hussain, Rizwan Ul Haq, N. F. M. Noor and S. Nadeem, Non-linear radiation e⁄ects in mixed
convection stagnation point ow along a vertically stretching surface, Int:J: Chem:React:Eng:;
(2016) doi: 10.1515/ijcre-2015-0177.
[46] M. Atlas, Rizwan Ul Haq and T. Mekkaoui, Active and zero ux of nanoparticles between a
squeezing channel with thermal radiation e⁄ects, J:Mol:Liq:;223 (2016) 289-298.
[47] T. Hayat, S. Qayyum, M. Waqas and A. Alsaedi, Thermally radiative stagnation point ow of
Maxwell nano uid due to unsteady convectively heated stretched surface, J:Mol: Liq:;224 (2016)
801-810.
[48] D. Pal and P. Saha, In uence of nonlinear thermal radiation and variable viscosity on
hydromagnetic heat and mass transfer in a thin liquid lm over an unsteady stretching surface,
Int:J:Mech:Sci:;119 (2016) 208-216.
[49] Z. Abbas, M. Sheikh and S.S. Mots, Numerical solution of binary chemical reaction on stagnation
point ow of Casson uid over a stretching/shrinking sheet with thermal radiation,
Energy, 95 (2016) 12-20.
[50] P.V. S.Narayana and D.H. Babu, Numerical study of MHD heat and mass transfer of a Je⁄rey uid
over a stretching sheet with chemical reaction and thermal radiation, J: TaiwanInst:Chem:;59
(2016) 18-25.
[51] C. Cattaneo, Sulla conduzionedelcalore, AttiSemin. Mat. Fis. Univ. Modena Reggio Emilia
151
3 (1948) 83-101.
[52] C.I. Christov, On frame indi⁄erent formulation of the Maxwell-Cattaneo model of nite speed heat
conduction, Mech:Res:Commun:;36 (2009) 481-486.
[53] M. Ciarletta and B. Straughan, Uniqueness and structural stability for the Cattaneo
Christov equations, Mech:Res:Commun:;37 (2010) 445-447.
[54] T. Hayat, M. I. Khan, M. Farooq, A. Alsaedi, M. Waqas and T. Yasmeen, Impact of Cattaneo-
Christov heat ux model in ow of variable thermal conductivity uid over a variable thicked surface,
Int:J:HeatMassTransf:;99 (2016) 702-710.
[55] J. Sui, L. Zheng and X. Zhang, Boundary layer heat and mass transfer with CattaneoeChristov
double-di⁄usion in upper-convected Maxwell nano uid past a stretching sheet with slip velocity,
Int:J:Therm:Sci:;104 (2016) 461-468.
[56] L. Liu, L. Zheng, F. Liu and X. Zhang, Anomalous convection di⁄usion and wave coupling transport
of cells on comb frame with fractional Cattaneo-Christov ux, Commun:
NonlinearSci:Numer:Simul:;38 (2016) 45-58.
[57] S. Nadeem and N. Muhammad, Impact of strati cation and Cattaneo-Christov heat ux in the ow
saturated with porous medium, J:Mol:Liq:;224 (2016) 423-430.
[58] R. Malik, M. Khan and M. Mushtaq, Cattaneo-Christov heat ux model for Sisko uid ow past a
permeable non-linearly stretching cylinder, J:Mol:Liq:;222 (2016) 430-434.
152
[59] T. Salahuddin, M.Y. Malik, A. Hussain, S. Bilal and M. Awais, MHD ow of Cattaneo Christov heat
ux model for Williamson uid over a stretching sheet with variable thickness: Using numerical
approach, J:Magn:Magn:Mater:;401 (2016) 991-997.
[60] W. A. Khan, M. Khan, A. S. Alshomrani and L. Ahmad, Numerical investigation of generalized
Fourier s and Fick s laws for Sisko uid ow, J:Mol:Liq:;224 (2016) 1016-1021.
[61] J.H. Merkin, A model for isothermal homogeneous-heterogeneous reactions in boundary layer
ow, Math:Comput:Model:;24 (1996) 125-136.
[62] M.A. Chaudhary and J.H. Merkin, A simple isothermal model for homogeneous heterogeneous
reactions in boundary layer ow: I. Equal di⁄usivities, FluidDyn:Res:; 16 (1995) 311-333.
[63] W.A. Khan and I. Pop, E⁄ects of homogeneous heterogeneous reactions on the viscoelastic uid
towards a stretching sheet, ASMEJ:HeatTransf:;134 (2012) 1-5.
[64] P.K. Kameswaran, S. Shaw, P. Sibanda and P.V.S.N. Murthy, Homogeneous heterogeneous
reactions in a nano uid ow due to a porous stretching sheet, Int:J:HeatMassTransf:; 57 (2013)
465-472.
[65] T. Hayat, M. I. Khan, M. Farooq, T. Yasmeen and A. Alsaedi, Stagnation point ow with Cattaneo-
Christov heat ux and homogeneous-heterogeneous reactions, J:Mol:Liq., 220
(2016) 49-55.
[66] Z. Abbas, M. Sheikh and I. Pop, Stagnation-point ow of a hydromagnetic viscous uid over
stretching/shrinking sheet with generalized slip condition in the presence of homogeneous
heterogeneous reactions, J:TaiwanInst:Chem:Eng:;55 (2015) 69-75.
153
[67] M. Ramzan, M. Bilal and J.D. Chung, E⁄ects of MHD homogeneous-heterogeneous reactions on
third grade uid ow with Cattaneo-Christov heat ux, J:Mol:Liq:, 223 (2016) 1284-1290.
[68] T. Yasmeen, T. Hayat, M.I. Khan and M. Imtiaz, Ferro uid ow by a stretched surface in the
presence of magnetic dipole and homogeneous-heterogeneous reactions, J:Mol:Liq:; 223 (2016)
1000-1005.
[69] M. Imtiaz, T. Hayat, A. Alsaedi and A. Hobiny, Homogeneous-heterogeneous reactions in MHD
ow due to an unsteady curved stretching surface, J:Mol:Liq:;221 (2016) 245-253.
[70] M. Epstein and D.H. Cho, Melting heat transfer in steady laminar ow over a at plate,
J:HeatTransf:, 98 (1976) 531-533.
[71] W.T. Cheng and C.H. Lin, Melting e⁄ect on mixed convective heat transfer with aiding and
opposing external ows from the vertical plate in a liquid-saturated porous medium,
Int:J:HeatMassTransf:;50 (2007) 3026-3034.
[72] A. Ishak, R. Nazar, N. Bachok and I. Pop, Melting heat transfer in steady laminar ow over a moving
surface, HeatMassTransf:;46 (2010) 463-468.
[73] N. Bachok, A. Ishak and I. Pop, Melting heat transfer in boundary layer stagnation-point ow
towards a stretching/shrinking sheet, Phys:Lett:;A, 374 (2010) 4075-4079.
[74] T. Hayat, M. Farooq, A. Alsaedi and Z. Iqbal, Melting heat transfer in the stagantion point ow of
Powell-Eyring uid, J:Thermophys:HeatTransf:;27 (2013) 761-766.
[75] S. Liao, On the homotopy analysis method for nonlinear problems, Appl:Math:Comput:;
154
147 (2) (2004) 499-513.
[76] S. Liao, Comparison between the homotopy analysis method and homotopy perturbation
method, Appl:Math:Comput:; 169 (2) (2005) 1186-1194.
[77] S. Liao, A short review on the homotopy analysis method in uid mechanics, J:Hydrodyn: Ser:B:;
22 (5) (2010) 882-88.
[78] S. Liao, On the relationship between the homotopy analysis method and Euler transform,
15 (6) 2010, 1421-1431.
[79] S. Rosseland, Astrophysik und Atom-Theoretische Grundlagen, SpringerVerlag;Berlin,
(1931) 41-44.
