unsteady heat and mass transfer mechanisms in

Unsteady Heat and Mass Transfer Mechanisms in

Carreau Nanofluid Flow

By

Muhammad Azam

Department of Mathematics

Quaid-i-Azam University

Islamabad, Pakistan

2018



By

Muhammad Azam

Supervised By

Prof. Dr. Masood Khan



Islamabad, Pakistan

2018



By

Muhammad Azam

A THESIS SUBMITTED IN THE PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE

OF

DOCTOR OF PHILOSOPHY

IN

MATHEMATICS

Supervised By

Prof. Dr. Masood Khan



Islamabad, Pakistan

2018

Author’s Declaration

I Muhammad Azam hereby state that my PhD thesis titled Unsteady Heat and

Mass Transfer Mechanisms in Carreau Nanofluid Flow is my own work and has not

been submitted previously by me for taking any degree from the Quaid-i-Azam

University Islamabad, Pakistan or anywhere else in the country/world.

At any time if my statement is found to be incorrect even after my graduate the

university has the right to withdraw my PhD degree.

Name of Student: Muhammad Azam

Date: 13-03-2018

Plagiarism Undertaking

I solemnly declare that research work presented in the thesis titled “Unsteady Heat and

Mass Transfer Mechanisms in Carreau Nanofluid Flow” is solely my research work with

no significant contribution from any other person. Small contribution/help wherever taken has

been duly acknowledged and that complete thesis has been written by me.

I understand the zero tolerance policy of the HEC and Quaid-i-Azam University towards

plagiarism. Therefore, I as an Author of the above titled thesis declare that no portion of my

thesis has been plagiarized and any material used as reference is properly referred/cited.

I undertake that if I am found guilty of any formal plagiarism in the above titled thesis even

afterward of PhD degree, the University reserves the rights to withdraw/revoke my PhD degree

and that HEC and the University has the right to publish my name on the HEC/University Website

on which names of students are placed who submitted plagiarized thesis.

Student/Author Signature: a a

Name: Muhammad Azam

Acknowledgements

I begin by praising the “Almighty Allah”, the Lord of the whole world who has expertized me the potential and ability to

complete this dissertation. I invoke peace for Hazrat Muhammad (PBUH) the last prophet of Allah, who is forever a torch

bearer of guidance for humanity as a whole.

I express my sincere and respectful admiration to my worthy supervisor Prof. Dr. Masood Khan for his scholarly guidance,

mentorship and vast knowledge that helped me to embark upon this highly important work. He sets high standards for his

students and he not only encourages but also guides them to meet those standards. It was a great privilege and honor for me

to work under his kind supervision. I am ever indebted and obliged to him.

I want to convey my deepest thanks and compliments to my honorable father Malik Rahim Bakhsh Khakhi, my respected

mother, my respected sisters, my gentle brothers Malik Muhammad Hashim Khakhi and Malik Muhammad Qasim

Khakhi for their endless love, prayers, encouragement, cordial cooperation and continuous support. Their efforts and

prayers flourished me throughout my life. I would never be able to pay back the love and affection showered upon me by

them. I cannot forget their kind care and their interest in my success.

I would like to extend my gratitude to Chairman Department of Mathematics Prof. Dr. Tasawar Hayat (Distinguished

National Professor) for providing a good research environment in department of Mathematics, Quaid-i-Azam University,

Islamabad. I strongly want to mention here that his research attitude motivated me to complete my PhD research work in

time.

I would also like to express my gratitude to Prof. Dr. Sohail Nadeem for providing me useful suggestions. The discussions

with them helped me to sort out the technical details of my work. I am thankful to him for being so cooperative, kind and

helpful.

I want to express my deepest thanks to my respected teachers Dr. Zaheer Abbas and Dr. Muhammad Sajid for kind

guidance and spending their precious time in my PhD dissertation.

I would like to convey my sincere and respectful gratitude to Dr. Asif Munir, Dr. Rabia Malik and Dr. Waqar Azeem

Khan for their kind support and guidance that helped me to complete this important work.

At this great occasion I want to remember my these honorable teachers Dr. Muhammad Ayub, Dr. Malik Muhammad

Yousaf, Dr. Khalid Saifullah, Dr. tahir Mahmood, Dr. Ghulam Mustafa, Dr. Khalid Pervez, Dr. Muhammad Ramzan, Dr.

Nasir Ali, Dr. Nargis Khan, Dr. Tayyab Kamran, Dr.Umar Hayat, Mr. Shahzad Shabbir and Mr. Abdul Majeed Khan.

I have been lucky enough to have good friends in my academic and social life, and cannot forget their role in my education

and university life. I want to express my unbound thanks to all my friends and colleagues especially Dr. Saeed Ahmed, Dr.

Fahad Munir Abbasi, Dr. Ashfaq Ahmed, Dr. Khalid Mahmood, Dr. Zawar Hussain, Dr. Taseer Muhammad, Dr. Zakir

Hussain, Dr. Muhammad Zubair, Dr. Muhammad Waqas, Dr. Shahid Farooq, Dr. Fahim Ud din, Dr. Tehseen Abbas, Dr.

Jamil Ahmad, Kamal Badshah (Late), Latif Ahmed, Muhammad Irfan, Humaira Sardar, Nadia bibi, Rana Anjum Saeed,

Arsalan Aziz, Bilal Ahmad, Khursheed Faiq, Khalil Choudhary, Arif ullah Khan, Arif Hussain, Sajid Qayyum, Muhammad

Ijaz Khan, Muhammad Waleed Khan, Muhammad Khan, Zafar Iqbal, Muhammad Awais, Muhammad Abbas, Zulfiqar Ali,

Asad ullah, Shahzad Nadeem. Thank you for the good time we have all together.

Muhammad Azam

Abstract

This thesis reports on results of the research project on the mathematical modeling and numerical study of a non-Newtonian

fluid. Particularly, the subject matter of this thesis concerns with the unsteady flow, heat and mass transfer of Carreau fluid

in the presence of nanoparticles. One of the most important developments in the recent decades is the vast utilization of

nanofluids in the engineering applications. The main aim of this research was the study of Carreau nanofluid flow using the

Buongiorno’s model that incorporates the effects of thermophoresis and Brownian motion. We focus on different types of

flow phenomena over moving surfaces via numerical approach. The problems studied here incorporate the effects of

magnetic field, heat generation/absorption, suction/injection, melting phenomenon, variable thermal conductivity and

nonlinear thermal radiation in different geometries. The governing partial differential equations are altered into ordinary

differential equations by adopting suitable transforming variables and then solved numerically by utilizing two different

numerical methods namely shooting RK45 and bvp4c Matlab package. In special cases, our numerical results are validated

with previously published data and achieved to be in excellent agreement.

The present thesis concentrates on the unsteady flows of non-Newtonian Carreau rheological model. The problem

considered here include the unsteady flow and heat transfer to Carreau fluid, the study of Carreau nanofluid in unsteady

heat and mass transfer, unsteady wedge flow of Carreau nanofluid, unsteady analysis of melting heat transfer in Carreau

nanofluid with heat generation/absorption, unsteady flow of Carreau nanofluid in expanding/contracting cylinder, stagnation

point flow in Carreau nanofluid in expanding/contracting cylinder, unsteady analysis of Carreau nanofluid past radially

stretching surface. To gauge and establish the physical aspects of the obtained results, a few of the velocity, temperature

and concentration profiles are displayed through figures with detailed discussion. Additionally, the local skin friction,

Nusselt and Sherwood numbers are calculated in tabular form. One key observation is that the temperature field enhances

for growing values of thermophoresis and Brownian motion parameters. Additionally, temperature as well as nanoparticles

concentration fields depreciate by increasing the melting parameter in both shear thickening and shear thinning liquids.

Furthermore, temperature gradient is a growing function of the wedge angle parameter. However, temperature ratio

parameter results in an enhancement in the temperature and its related thermal boundary layer thickness.

Contents

1 Introduction 5

1.1 The Background and Objectives of the Research . . . . . . . . . . . . . . 5

1.2 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Fundamental Laws and Solution Methodology 17

2.1 Fundamental Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Conservation Law of Mass . . . . . . . . . . . . . . . . . . . . . . 17

2.1.2 Conservation Law of Momentum . . . . . . . . . . . . . . . . . . 18

2.1.3 Conservation Law of Energy . . . . . . . . . . . . . . . . . . . . . 18

2.1.4 Conservation Law of Concentration . . . . . . . . . . . . . . . . . 19

2.2 Carreau Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Nanouid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.2 Buongiorno Model . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Unsteady Flow of Carreau Fluid towards a Permeable Surface 24

3.1 Governing Equations and Mathematical Formulation . . . . . . . . . . . 25

3.2 Flow Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 Parameters of Engineering Interest . . . . . . . . . . . . . . . . . 29

3.3 Discrete Scheme and Solution Methodology . . . . . . . . . . . . . . . . . 29

3.4 Validation of Numerical Results . . . . . . . . . . . . . . . . . . . . . . . 32

3.5 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 32

4 Unsteady Heat and Mass Transfer Mechanisms in MHD Carreau Nanouid

Flow 50

4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51


5 On Unsteady Falkner-Skan Flow of MHD Carreau Nanouid Past a

Static/Moving Wedge 74

5.1 Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75


6 E⁄ects of Melting and Heat Generation/Absorption on Unsteady Falkner-

Skan Flow of Carreau Nanouid over a Wedge 104

6.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2 Numerical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7 Unsteady Heat and Mass Transfer in Carreau Nanouid Flow over Ex-

panding/Contracting Cylinder 125 7.1 Description of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.2 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127


8 Unsteady Stagnation Point Flow of MHD Carreau Nanouid over Ex-

panding/Contracting Cylinder 149

8.1 Mathematical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

8.2 Discussion of Numerical results . . . . . . . . . . . . . . . . . . . . . . . 156

9 Unsteady Axisymmetric Flow and Heat Transfer in Carreau Fluid past

a Stretched Surface 172

9.1 Formulation of the Flow Problem . . . . . . . . . . . . . . . . . . . . . . 172

9.2 Results Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

10 E⁄ects of Magnetic Field and Partial Slip on Unsteady Axisymmetric

Flow of Carreau Nanouid over a Radially Stretching Surface 185

10.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 186

10.2 Numerical Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

10.3 Code Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

10.4 Discussion of Numerical Results . . . . . . . . . . . . . . . . . . . . . . . 194

11 Conclusions and Recommendations 218

11.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

11.2 Future Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . 221

12 Bibliography 222

Chapter 1

Introduction

1.1 The Background and Objectives of the Research

From the last two decades, the nanouid technology is a main focus of many investigators to

control the heat and mass transfer mechanisms in various energy systems. The earliest studies

of thermal conductivity improvement were presented by Masuda et al: [1]: The wide range of

current and future applications of nanouid technology can be studied in the recent book [2]: The

concept of nanouid was suggested by Choi and Eastmann [3]: Noble characteristics of nanouids

having long term stability, higher thermal conductivity, homogeneity with the base uid makes the

nanouid broad range applicable uid in di⁄erent arena [4] such as electronics, power generation,

nuclear systems, chemical production and medical elds. Di⁄erent theoretical and experimental

models have been reported by several researchers to investigate the anomalous enhancement

in the thermal conductivity. Transport characteristics of nanouids can be studied by a model

devised by Buongiorno [5]. Buongiorno reported a theme which ignores the limitations of

homogeneous and dispersion themes. He suggested the seven slip mechanisms that relate the

parallel velocity between the nanoparticles and baseuids. They are gravity, thermophoresis,

Magnus e⁄ect, uid drainage, Brownian di⁄usion, di⁄usiophoresis and inertia. Finally, he decided

that Brownian di⁄usion and thermophoresis are the concrete slip mechanisms. In view of these

facts, great number of attempts on nanouids have been reported by several authors. Khan and

Pop [6] reported a study on nanouid ow over stretched surface. They pointed out, reduced

number is a depreciating function of each non-dimensional number. Bachok et al: [7]

investigated unsteady ow of nanouid.

Their study revealed that multiple solutions exist both for shrinking and stretching cases. The

problem of energy conversion of mixed convection ow of nanouid with the e⁄ects of thermal

radiation and magnetic eld was considered by Hsiao [8]: He concluded that the Hartmann

number reduces the speed of uid in the boundary layer. Turkyilmazoglu [9] presented the

analytical study of nanouid models. He noted the improvement in heat transfer rate when

di⁄usion parameter is enhanced in multi-phase model. Serna [10] discussed the study of nanouid

with mass and heat transfer. He noticed the enhancement in heat transfer rate with respect to

base uid.

The study of generalized Newtonian uids has been gained a keen interest of the authors due to

their wide range applications in di⁄erent arena from last few years. The power law viscosity model

is treated as the simplest generalized uid model. The important class of the generalized uid are

the Ellis model, Sisko model, Cross model and Carreau viscosity model. From the above

mentioned models, the Carreau viscosity model is the most important class of generalized

Newtonian uid which overcomes the limitations of power law uid. The Carreau model was rst

devised by Carreau [11]: The study of electrophoresis of a sphere in a spherical cavity by using

the Carreau uid was considered by Hsu et al: [12]: Ali and Hayat [13] worked on peristaltic

motion in an asymmetric channel by using the Carreau uid. They demonstrated the analytical

solutions of the considered work. Tshehla [14] investigated the thin lm ow of Carreau uid over

an inclined surface. He reported the analytical as well as numerical solutions of the assumed

study. Olajuwon and Ishola [15] examined the inuence of thermal di⁄usion and thermal radiation

on magnetic Carreau uid over a vertical porous surface. Their study predicted that heat transfer

rate was depreciated with the enhancment of thermal di⁄usion and Schmidt numbers. Hayat et

al: [16] presented the analytic solutions for the ow of Carreau liquid past a convectively heated

surface. Their investigation revealed that the impact of power law index on the velocity and

temperature are quite opposite.

The investigation on Falkner-Skan wedge ow has a prime importance in both practical and

theoretical works. Especially, such types of ows occurs in enhanced oil recovery, geothermal

industries and ground water population. A lot of studies related to FalknerSkan wedge ow can be

seen in the books by Schlichting and Gersten [17] and Leal [18]: The inuence of suction/injection

on convective wedge ow with uniform heat ux was studied by Yih [19]: His numerical study

pointed out that ow separation only occurred for the case when pressure gradient m = 0: A

numerical study of MHD micropolar uid ow over a wedge was presented by Ishaq et al: [20]: They

observed that skin friction coe¢ cient grows with growing wedge angle parameter. Alam et al:

[21] examined the unsteady MHD convective ow of micropolar uid over a porous wedge. Their

numerical results showed that heat transfer rate is a growing function of the unsteadiness

parameter. Munir et al: [22] studied the viscous dissipation impact in Sisko uid ow along a wedge

with convective condition. They noted that the velocity boundary layer thickness enhances for

decreasing values of wedge angle parameter. Raju and Sandeep [23] worked on MHD Falkner-

Skan ow of Carreau uid along a wedge considering the cross di⁄usion e⁄ects. They resulted that

rate of heat transfer is more in accelerating case when compared to decelerating case.

Turkyilmazoglu [24] analyzed the slip e⁄ects on wedge ow of viscous uid. He concluded that the

velocity slip reduces the velocity boundary layer

thickness.

On the other hand, heat transfer in melting phenomena has achieved much concentration due

to its wide range utilizations incorporating thawing of magma solidication and frozen ground,

semi-conductor material, etc. Recently investigators have devoted to produce more e¢ cient and

low cost energy storage devices. Such devices are a› iated with waste heat recovery, power and

solar energy. Late heat energy storage, chemical thermal energy storage and sensible heat energy

storage are usually three ways to store energy. Themal energy can be stored by way of melting

phenomenon and latent heat. The involvement of such application is in preparation of semi-

conductor devices, welding of manufacturing process and magma solidication. Epstein and Cho

[25] mentioned melting phenomena in their study. This invention has been spread out by many

authors.

Mixed convection ow over a vertical surface embedded in porous medium with melting was

discussed via Cheng and Lin [26]. They predicted that velocity gradient is reduced for increasing

values of melting parameter. Ishaq et al: [27] persued the investigation on melting phenomenon

on laminar ow over moving surface. They found that the melting phenomenon depreciated the

local Nusselt number. Melting analysis in stagnation study with micropolar uid was boosted up

via Yacob et al: [28]. They analyzed that multiple solutions are possible for shrinking case. Hayat

et al: [29] discussed the stagnation point ow with melting couple stress uid. Their study revealed

that velocity as well as temperature eld enhances for growing the melting process.

Prasannakumara et al:

[30] conducted a numerical analysis for melting and radiative heat transfer in dusty uid. They

noticed that the velocity as well as temperature depreciated for growing values of magnetic

parameter in the presence of melting process. Kameswaran et al: [31] examined the convective

and melting heat transfer with variable permeability. They concluded that heat transfer is a

growing function of the melting parameter. Revised model was adopted to analyze the melting

heat transfer in wedge ow of second nanouid via Hayat et al: [32]. They observed that uid ow is

an enhancing function of the wedge angle

parameter.

The exploration of ow mechanisms due to expanding/contracting bodies has been a topic of

emerging research from the last few years. The construction of pipe and channel has practical

applications in industrial arena like as chimney stacks, cooling towers and formulation of heat

exchanger tubes. The earliest work of unsteady ow over a pipe with expanding or contracting

wall was presented by Uchida and Aoki [33]: An analysis of unsteady ow over a permeable

expanding pipe was carried out by Goto and Uchida [34]: Bujurke et al: [35] investigated the

unsteady ow of expanding/contracting surface by using the proposed series method. They also

used the pade approximation for the considered study. Khellaf and Lauriat [36] demonstrated a

numerical study of Carreau uid ow over a rotating vertical cylinder. They pointed out that

depreciation in apparent viscosity causes of oscillatory ows. Majdalani et al: [37] persued the

work on viscous ow between contracting or expanding walls. They showed that higher

contraction velocity corresponds to larger pressure drop. A study of unsteady viscous ow past a

stretching expanding cylinder was induced by Fang et al: [38]: The result declared that the

velocity decreases faster for a greater Reynolds number. The ow outside the stretching cylinder

was examined by Wang [39]. He also conducted a comparative study between the numerical and

asymptotic solutions. A numerical analysis on mixed convection ow over a vertical cylinder was

carried out by Lok et al: [40]: The study of viscous ow past a contracting or expanding cylinder

was considered by Fang et al: [41]: Their study showed that unsteadiness parameter controlled

the ow eld. Patil et al: [42] utilized the implicit nite di⁄erence method to present the numerical

solution for the unsteady mixed convection ow over a cylinder. It was noticed that suction

parameter reduces the velocity proles. Si et al: [43] studied the unsteady viscous ow over a

porous cylinder. They investigated that axial velocity is an enhancing function of the velocity ratio

parameter.

Zaimi et al: [44] considered a numerical study of unsteady viscous nanouid ow over contracting cylinder. Their investigation showed that the magnitude of the Sherwood number depreciated by improving values of unsteadiness parameter. Abbas et al: [45] analyzed the impact of partial slip on heat transfer in a cylinder. They examined that dual solutions are possible for the shrinking case. Imtiaz et al: [46] worked on Casson nanouid over a cylinder with convective conditions. Their study revealed that uid ow is enhancing function of the mixed convection parameter.

It is renowned fact that the investigation on axisymmetric ow due to radially stretching surface

is a valuable topic for authors. Sakiadis [47] seems to be the rst who modeled 2D axisymmetric

boundary layer equations. Ariel [48] considered the problem of axisymmetric ow due to a radially

stretching surface by using the second grade uid.

The numerical study of axisymmetric ow of Carreau liquid by way of Galerkin Least Square

technique was conducted by Martins et al: [49]: The analytic and numerical solutions were

presented by Rashidi et al: [50]: They presented the analytical solutions via HAM and numerical

solutions by way of shooting RK4. The analytical study of unsteady axisymmetric ow induced by

a radially stretching surface was considered by Sajid et al: [51]: Their investigation revealed that

the e⁄ects of Eckert number and Prandtl number on temperature eld are quite opposite. Analytic

study in axisymmetric ow considering second grade uid with unsteadiness regimes were

conducted by Ahmad et al: [52;53]: They observed that the velocity proles as well as boundary

layer thickness are enhancing function of the dimensionless time. Abbas et al: [54] reported the

problem of unsteady MHD ow of a rotating uid. They demonstrated the similar solutions by way

of keller-box method. They examined that temperature is growing function of the rotating uid

parameter. Sahoo [55] analyzed the e⁄ects of Joule heating and viscous dissipation on second

grade uid considering the partial slip. His investigation revealed that velocity and themal

boundary layer grow due to partial slip. Khan and Shahzad [56] assumed a problem of

axisymmetric ow in Sisko uid. Their study predicted that an enhancment in power law index

reduces the velocity eld. Mustafa et al: [57] worked on axisymmetric ow of nanouid. They noted

that heat transfer rate depreciates due to increase of thermophoresis parameter. Weidman [58]

derived dual solutions for the rotational axisymmetric ow. He also presented the stability analysis

of solution.

The intension of present thesis is explore the numerical solutions of unsteady phenomenon with

Carreau uid in the presences of nanoparticles. We consider the Buongiornos nanouid model to

incorporate the e⁄ects of Brownian motion and thermophoresis. The major focus of this study is

to study heat and mass transfer in Carreau nanouid ow past distinct moving surfaces numerically

by utilizing the numerical approaches namely bvp4c matlab package and shooting RK45.

1.2 Structure of the Thesis

There are eleven chapters including di⁄erent aspects regarding the unsteady boundary layer ow, heat

and mass transfer of Carreau nanouid and summarized as follows:

Chapter 1 contains the historical background, objective of the thesis as well as structure of the thesis.

Chapter 2 indicates some fundamental laws and basic denitions related to thesis.

Chapter 3 expresses an exposition to unsteady ow of Carreau uid over a permeable surface. The

complex highly nonlinear partial di⁄erential equations have been reduced to highly nonlinear

ordinary di⁄erential equations by employing suitable transformations. The numerical solutions of

resulting system are achieved by way of bvp4c Matlab package. The e⁄ects of involved

parameters on the velocity and temperature eld are explored in detail. The numerical

computations for the local skin friction coe¢ cient as well as local

Nusselt number are executed. This work has been published in Results in Physics, 6 (2016)

11681174.

The unsteady heat and mass transfer of Carreau nanouid with magnetic eld e⁄ects are dealt in

Chapter 4. The Buongiornos nanouid model is adopted to involve the inuences of thermophoresis

and Brownian motion. The problem under investigation is governed by a nonlinear di⁄erential

system which is altered into nonlinear ordinary di⁄erential system by the application of

appropriate transformations and then solved numerically by adopting bvp4c technique. The

impact of involved parameters on velocity, temperature and nanoparticles concentration elds

are demonstrated graphically. Additionally, drage force, heat and mass transfer rates are

provided in tabular form. Moreover, a comparative study of present numerical analysis with the

existing one is conducted in limiting cases. This piece of work has been published in Journal of

Molecular Liquids, 225 (2017) 554562.

Chapter 5 addresses the convective Falkner-Skan ow of unsteady Carreau uid over a moving/static wedge in the presence of nanoparticles and magnetic eld e⁄ects. The suitable

transforming variables are utilized to transform the nonlinear partial di⁄erential system to a semi

couple ordinary di⁄erential system. Finally, numerical solutions of these equations are obtained

by two di⁄erent numerical approaches namely shooting RK45 and bvp4c techniques. For the

validation of current attempt, comparative study between present attempt and existing study is

provided. Additionally, a comparative study of numerical values of local Nusselt and Sherwood

numbers is also presented by using these two numerical techniques. The observations of this

chapter have been published in Journal of Molecular Liquids, 230 (2017) 48 58.

In Chapter 6, we analyze the inuences of heat generation/absorption and melting phenomena on

unsteady wedge ow of Carreau nanouid. Additionally, zero nanoparticle mass ux is assumed at

the boundary. Furthermore, ow is induced by a nonlinear stretching wedge. Mathematical

formulation is constructed with the aid of momentum, energy and concentration equations

utilizing appropriate transforming variables. The numerical solutions for the resulting

transformed system are presented for both cases of shear thickening liquid as well as shear

thinning liquid. For numerical results, a numerical method namely bvp4c function is used. The

numerical computations of drag, heat and mass transfer are also executed. It is worthy to

mention that Carreau uid reduced to viscous uid for n = 1 and We = 0: The results of current

study are published in International Journal of Heat and Mass Transfer, 110 (2017) 437446 .

The numerical investigations in chapter 7 focused on heat and mass transfer of Carreau nanouid past expanding/contracting cylinder considering temperature dependent thermal conductivity. Moreover, the more generalized convective heat and mass conditions are utilized. Additionally,

recently devised model for nanouid is deliberated. Instead of solving directly the partial di⁄erential equations, we have adopted suitable transformations on these equations and

altered them into ordinary di⁄erential equations then solved numerically (i.e., by the bvp4c method). The impact of contributed parameters in the assumed study is discussed in debth. The

aforementioned study has been published in Journal of Molecular Liquids, 231 (2017) 474484 .

Chapter 8 concentrates on the study of unsteady stagnation point ow of Carreau nanouid past

expanding/contracting cylinder with thermal radiation and magnetic eld e⁄ects. Additionally, zero

nanoparticle mass ux condition is assumed. Mathematical model is progressed with the help of

motion, energy and concentration equations. The resulting nonlinear study is computed for

numerical solutions. The numerical results are plotted and discussed in depth for the controlling

parameters. A comparison of current investigation and existing data is provided. Numerical

computations for drag force and heat transfer are also executed. These investigations have been

published in International Journal of Mechanical Sciences, 130 (2017) 64 73.

The numerical analysis for unsteady axisymmetric ow of Carreau uid past a radially stretching

surface is considered in Chapter 9. The underlying nonlinear problem is solved numerically.

Impact of controlling parameters on the velocity as well as temperature proles are shown

graphically and discussed. Additionally, the numerical results for the drag force and heat transfer

are presented in form of tables. The numerical analysis is presented for shear thinning and shear

thickening uids. These numerical investigation is performed by using bvp4c method. The contents

of this chapter has been published in Thermal Science, doi:org/10.2298/TSCI160807132K.

Chapter 10 examines the inuences of unsteady partial slip and magnetic eld on axisymmetric ow

of Carreau nanouid over a radially stretching surface with convective condition. Buongiorno nano

model is utilized to consider the Brownian motion and thermophoresis e⁄ects. A set of semi

coupled nonlinear ordinary di⁄erential equations are obtained by utilizing suitable transforming

variables. Numerical solutions are derived by two di⁄erent numerical methods namely shooting

RK45 and bvp4c Matlab package. Moreover, a comparative study with the existing literature and

current investigation is presented. The numerical results for the velocity, temperature and

nanoparticles concentration eld are graphically demonstrated and the inuences of relevant

parameters are presented in detail. The observations of this chapter are published in Results in

Physics, 7(2017) 2671-2682 .

Chapter 2

Fundamental Laws and Solution

Methodology

In this part of the thesis, the fundamental laws, some relevant denitions and solution methodogy are stated.

2.1 Fundamental Laws

2.1.1 Conservation Law of Mass

It can be stated as mass neither be created nor be destroyed. The mathematical expression for the

compressible uid, it can be written as

r fV = 0; (2.1)

where V is the uid velocity, t the time and f the uid density.

2.1.2 Conservation Law of Momentum

It states that the total linear momentum for the system is conserved. Mathematically, it is

expressed as

fai = rp + divS+ fB; (2.2)

in which ai the acceleration vector, f the uid density, S the extra stress tensor, p the pressure and B the

body forces per unit mass.

2.1.3 Conservation Law of Energy

It states that total energy of the system is conserved. Mathematically, it can be stated as

divq; (2.3)

where T the temperature, cf the specic heat and q the energy ux which is dened as

q = krT; (2.4)

in which k represents the thermal conductivity. Using Eqs. (2.3) and (2.4), the energy equations can be

demonstrated as

(2.5)

2.1.4 Conservation Law of Concentration

The concentration equation depends on the Ficks laws and it can be presented as

V rC = r J: (2.6)

in which J shows the normal mass ux which is dened as

J = DrC: (2.7)

Using Eqs. (2:6) and (2:7), the concentration equation can be indicated as

(2.8)

2.2 Carreau Fluid

Carreau uid is an important class of generalized Newtonian uid. The Cauchy stress tensor for the

generalized Newtonian Carreau uid [67;68] can be written as

= pI + A1 with (2.9)

:

(2.10)

In many physical problems, we can consider = 0. Thus, Eq. (2:9)2 can be reduced 1

as

: 2 n21 = 0[1 + ( ) ] : (2.11)

2.2.1 Nanouid

Best liquid cooling can be achieved by suspension of tiny size nanoparticles (1-100 nm size) into

the base uid. These uids are referred to nanouids. The idea of nanouid was devised by Choi.

2.2.2 Buongiorno Model

Buongiorno showed seven slip scheme that discuss a parallel velocity among nanoparticles and

base uid. Inertia, magnus e⁄ect, gravity, uid drainage, di⁄usiophoresis, thermophoresis and

Brownian Di⁄usion are the mechanisms. He decided that thermophoresis and Brownian Di⁄usion

are important mechanisms.

2.3 Solution Methodology

The non-linear ordinary di⁄erential equations with the boundary conditions are solved

numerically by way of bvp4c package. This method is based on the collocation method for

boundary value problem in the form

y0 = f(x; y; p); a x b; (2.12)

having the boundary conditions

g(y(a); y(b); p) = 0; (2.13)

where p is a vector of unknown parameters. The approximate solution S(x) is a continuous

function which is a cubic polynomial on each subinterval [xn;xn+1] of a mesh a = x0 < x1 < x2 < :::::::

< xn = b satisfying the boundary conditions

g(S(a);S(b)) = 0: (2.14)

This solution also satises the di⁄erential equation system at mid point and end points of each

subinterval

S0(xn) = f(xn;S(xn)); (2.15) S0((xn + xn+1)=2) = f((xn + xn+1)=2;S((xn + xn+1)=2)); (2.16)

S0(xn+1) = f(xn+1;S(xn+1)): (2.17)

The above conditions conclude in a system of nonlinear algebraic equations for the coefcients

dening S: In comparison to shooting method, the solution y(x) is approximated over the whole

interval [a;b] and the subsidiary conditions are taken into account every time. The nonlinear

algebraic system is solved iteratively by linearization. It is important to mentioned that this

approach relies upon the linear equation solver of the MATLAB rather than its initial value

problem codes. The basic method of bvp4c is the Simpsons method which can be seen in a

number of codes. It can be seen that with the modest consideration, S(x) is the fourth order

approximation to an isolated solution y(x) which implies ky(x) S(x)k Ch4 where C is the constant

and h is the highest of the step sizes hn = xn+1 xn: After the computation of S(x) on a mesh with

the help of bvp4c, it can be solved at any x or a set of x in the interval [a;b]: The boundary value

problem codes demand users to provide a guess for the required solution. The guess includes a

guess for an initial mesh that depicts the behavior of the required solution. The codes then use

the mesh so as to obtain a required solution with the modest mesh points. The residual r(x) for

such an approximation in the ordinary di⁄erential equation systems is dened as

r(x) = S0(x) f(x;S(x)): (2.18)

It implies that S(x) is the exact solution of the perturbed ordinary di⁄erential equations

S0(x) = f(x;S(x)) + r(x): (2.19)

Similarly, g(S(a);S(b)) is the residual in the boundary condition. If the residual is small then it

means S(x) is close to y(x). It is important to state here that bvp4c depends on algorithms that

are plausible even though the initial mesh is very poor, yet furnish the correct results as h ! 0

[60].

Chapter 3

Unsteady Flow of Carreau Fluid towards a Permeable Surface

The main objective of this chapter is to present numerical study of unsteady twodimensional

boundary layer ow and heat transfer of an incompressible Carreau uid over a permeable time

dependent stretching sheet. Using suitable transformations, the time dependent partial

di⁄erential equations are converted to non-linear ordinary differential equations. The numerical

solutions of these non-linear ordinary di⁄erential equations with associated boundary conditions

are determined by using the bvp4c function in MATLAB. The numerical results are investigated

for the emerging parameters namely, the unsteadiness parameter, mass transfer parameter,

Prandtl number, power law index and Weissenberg number. It is important to state that both

thermal and momentum boundary layer thicknesses diminish with improving unsteadiness and

mass transfer parameters. A comparison with the available published literature in limiting cases

is performed and found to be in good agreement.

3.1 Governing Equations and Mathematical Formu-

lation

Consider a laminar two-dimensional unsteady ow and heat transfer with Carreau uid in the region

y > 0 over an unsteady moving surface. The physical model and coordinate system are shown in

Fig. 3:1: The Cartesian coordinates x and y are chosen in such a way that x axis is along the

stretching sheet and y axis is normal to it. The ow is generated due to the permeable stretching

sheet by applying two equal and opposite forces along x axis. It is assumed that the surface is

moving with the velocity Uw(x;t) and that the mass ux velocity is Vw(t): It is assumed that T1 (Tw

> T1). The viscous dissipation e⁄ects are neglected in heat transfer process here. For the unsteady

2D ow, the velocity as well as temperature elds are suggested in a way

V = [u(x;y;t);v(x;y;t);0]; T = T(x;y;t): (3.1)

3.2 Flow Geometry

Fig. 3:1. Physical model and coordinate system.

Under these assumptions, the basic boundary layer equations of the problem under consideration can be

written as [59].

(3.4)

along with the boundary conditions u = Uw(x;t); v = V w(t) ; T = Tw(x;t) at y = 0; (3.5)

u ! 0; T ! T 1 as y ! 1: (3.6)

wit u hand v the velocity components along the x and y directions, respectively. Further, T and

are the uid temperature and kinematic viscosity, respectively, the thermal di⁄usivity with

uid density, k thermal conductivity of the uid and Cp the specic heat.

Further, we consider the stretching velocity Uw(x;t); mass uid velocity Vw(t) and the surface

temperature Tw(x;t) in a way:

(3.7)

where ct < 1 with c and a are constants having dimensions (time) 1. Additionally, V0 shows

uniform suction/injection velocity (V0 < 0 for injection and V0 > 0 for suction). The e⁄ective

stretching rate increases or decreases with time since c > 0 or c < 0;

respectively.

The following suitable transformations are used for the present case:

: (3.8)

By introducing the above transformations, the momentum and energy equations (3:3) and (3:4);

respectively, are transformed into following ordinary di⁄erential equations

n 3 1 + nWe2(f 00)2 1 + We2(f 00)2 2 f000 + ff00 (f0)2 A[f0 + f00] = 0; (3.9)

2

00 + Pr(f0 2f0 ) Pr (3.10)

where prime shows di⁄erentiation with respect to the local Weissenberg

number, Pr the Prandtl number and the unsteadiness parame-

ter. The altered conditions are

f(0) = S; f0(0) = 1; (0) = 1; (3.11)

f0(1) ! 0; (1) ! 0; (3.12)

where is a mass transfer parameter (S < 0 for injection and S > 0 for suction). It should

be noted that the results for viscous uid can be achieved by putting n = 1 and We = 0 in Eq. (3:9):

3.2.1 Parameters of Engineering Interest

From the engineering point of view, the quantities of interest in this problem are the drag force Cfx

and heat transfer Nux

(3.13)

where the wall shear stress w and the wall heat ux qw are dened as

: (3.14)

Thus substituting Eq. (3:8) into Eq. (3:14) and using Eq. (3:13) the following expressions can be attained

Re Re 1=2 Nux = 0(0); (3.15)

where Rex = xUw is the local Reynolds number.

3.3 Discrete Scheme and Solution Methodology

The non-linear ordinary di⁄erential equations (3:9) and (3:10) subject to the boundary

conditions (3:11) and (3:12) are solved numerically by utilizing numerical technique known as

bvp4c function in MATLAB. This method is based on the collocation method for solving boundary

value problem of the form

y0 = f(x;y;p); a x b;

with the general nonlinear, two-point boundary conditions

(3.16)

g(y(a);y(b);p) = 0; (3.17)

where p is a vector of unknown parameters. The approximate solution S(x) is a continuous

function which is a cubic polynomial on each subinterval [xn;xn+1] of a mesh a = x0 < x1 < x2 < :::::::

< xn = b satisfying the boundary conditions

g(S(a);S(b)) = 0: (3.18)

This solution also satises the di⁄erential equation system at mid point and end points of each

subinterval

S0(xn) = f(xn;S(xn)); (3.19)

S0((xn + xn+1)=2) = f((xn + xn+1)=2;S((xn + xn+1)=2)); (3.20)

S0(xn+1) = f(xn+1;S(xn+1)): (3.21)

The above conditions conclude in a system of nonlinear algebraic equations for the coefcients

dening S: In comparison to shooting method, the solution y(x) is approximated over the whole

interval [a;b] and the subsidiary conditions are taken into account every time. The nonlinear

algebraic system is solved iteratively by linearization. It is important to mentioned that this

approach relies upon the linear equation solver of the MATLAB rather than its initial value

problem codes. The basic method of bvp4c is the Simpsons method which can be seen in a

number of codes. It can be seen that with the modest consideration, S(x) is the fourth order

approximation to an isolated solution y(x) which implies ky(x) S(x)k Ch4 where C is the constant

and h is the highest of the step sizes hn = xn+1 xn: After the computation of S(x) on a mesh with

the help of bvp4c, it can be solved at any x or a set of x in the interval [a;b]: The boundary value

problem codes demand users to provide a guess for the required solution. The guess includes a

guess for an initial mesh that depicts the behavior of the required solution. The codes then use

the mesh so as to obtain a required solution with the modest mesh points. The residual r(x) for

such an approximation in the ordinary di⁄erential equation systems is dened as

r(x) = S0(x) f(x;S(x)): (3.22)

It implies that S(x) is the exact solution of the perturbed ordinary di⁄erential equations

S0(x) = f(x;S(x)) + r(x): (3.23) Similarly, g(S(a);S(b)) is the residual in the boundary condition. If the residual is small then it

means S(x) is close to y(x). It is important to state here that bvp4c depends on algorithms that

are plausible even though the initial mesh is very poor, yet furnish the correct results as h ! 0

[60].

3.4 Validation of Numerical Results

Table 3:1 shows a comparison of the numerical results of the skin-friction coe¢ cient f00(0) for

di⁄erent values of the unsteadiness parameter A when n = 1;We = 0 and S = 0 are xed with

published results of Sharidan et al: [61]; Chamkha et al: [62] and Mukhopadhyay et al: [63]: On

the evident of Table 3:1, the results are found in outstanding agreement. Table 3:2 also

represents a comparison of the present numerical results of the Nusselt number 0(0) for di⁄erent

values of Prandtl number Pr when n = 1;We = 0;S = 0 and A = 0 are xed with available published

results of Grubka and Bobba [64] and Chen [65]: From Table 3:2, it is clear that the results are

found in excellent agreement.

3.5 Numerical Results and Discussion

The non-linear di⁄erential equations (3:9) and (3:10) with the associated boundary conditions

(3:11) and (3:12) are solved numerically by using the numerical technique namely bvp4c

function in MATLAB. The numerical results are obtained for di⁄erent values of the emerging

parameters namely, unsteadiness parameter A, power law index n, Prandtl number Pr; local

Weissenberg number We; and mass transfer parameter S. The impact of these parameters on

the velocity and temperature proles are shown graphically. The numerical values of the local skin-

friction coe¢ cient Re1=2 Cfx and the local Nusselt number Re 1=2 Nux are also tabulated in Table

3:3 for various values of emerging parameters. Furthermore, the numerical calculations for the

Nusselt number 0(0) are presented in Table 3:4 for various values of physical parameters. In

order to check the accuracy of the present computed results with available published data, a

comparison is performed between current computed results and available literature in limiting

cases.

The variations of various values of the unsteadiness parameter A, mass transfer parameter S and

Weissenberg number We on the local skin friction coe¢ cient Re1=2 Cfx and the local Nusselt

number Re 1=2 Nux for both shear thinning (0 < n < 1) and shear thickening (n > 1) uids are

depicted in Table 3:3: It is obvious that by increasing the values of the unsteadiness parameter

A and mass transfer parameter S, the magnitude of the the local skin friction coe¢ cient Re1=2 Cfx

increases in both shear thinning and shear thickening uids. It is interesting to note that by

increasing the values of the Weissenberg number We, the magnitude of the local skin friction

coe¢ cient decreases for the shear thinning uid and increases for shear thickening uid. From Table

3:3; it can be seen that the local Nusselt number is an increasing function of the unsteadiness

parameter A and mass transfer parameter S both for shear thinning and shear thickening uids.

However, on incrementing the values of the Weissenberg number We, the local Nusselt number Re 1=2 Nux

decreases in shear thinning uid but increases in shear thickening uid. The numerical results of the local Nusselt number Re 1=2 Nux for several values of the Prandtl number Pr, unsteadiness parameter A and mass transfer

parameter S are tabulated in Table 3:4 both for shear thinning and shear thickening uids. From Table 3:4; it is depicted that the local Nusselt number Re 1=2 Nux is an increasing function of the Prandtl number Pr; unsteadiness parameter A and mass transfer parameter S both for shear thinning uid as well as shear thickening uid. Fig. 3:2 is constructed to represent the comparison of velocity proles between Khan and Hashim [59] and the present study. On the basis of this Fig., it can be seen that the result are in good agreement.

The behavior of unsteadiness parameter A on the velocity eld f 0( ) in the presence and absence

of mass transfer parameter S is displayed in Figs. 3:3(a) and 3:3(b) for both shear thinning and

shear thickening uids. It is observed that when the unsteadiness parameter A increases, the

velocity eld f 0( ) and momentum boundary layer thickness decrease in both the shear thinning

and shear thickening uids. However, it is observed that the momentum boundary layer thickness

is larger in case of the shear thickening uid as compared to the shear thinning uid. The

temperature eld ( ) for di⁄erent values of the unsteadiness parameter A is shown graphically in

Figs. 3:4(a) and 3:4(b): From these Figs., it is noted that the impact of increasing the values of

unsteadiness parameter A is to diminish the temperature eld ( ) and thermal boundary layer

thickness. Physically, when unsteadiness enhances the sheet looses more heat due to which

temperature diminishes.

The behavior of temperature eld ( ) for di⁄erent values of the Prandtl number Pr is displayed in

Figs. 3:5(a) and 3:5(b): From these Figs., it is noticed that an increase in the values of Prandtl

number Pr results in a decrease in temperature eld ( ) and thermal boundary layer thickness both

in shear thinning and shear thickening uids. This is because of the fact that the uid with higher

Prandtl number possesses low thermal conductivity and consequently reduces the conduction

and the thermal boundary layer

thickness.

Figs. 3:6(a) and 3:6(b) elucidate the inuence of the power law index n on the velocity proles f 0(

) and temperature proles ( ); respectively. These Figs. put in conformation that the velocity eld f

0( ) is an increasing function of the power law index n while temperature eld ( ) is a decreasing

function of it.

Figs. 3:7(a) and 3:7(b) are plotted to illustrate the inuence of the local Weissenberg number We

on the velocity eld f 0( ) and the temperature eld ( ) for both cases of shear thinning and shear

thickening uids. From these Figs., it is noticed that the velocity eld f 0( ) decreases by uplifting the

Weissenberg number We in shear thinning uid and opposite behavior has been seen in shear

thickening uid. As far as the temperature eld is concerned, it increases by increasing the values

of the Weissenberg number We in shear thinning uid: However, quite the opposite behavior is

noticed for the shear thickening uid. By the denition of the Weissenberg number, it is the ratio

of the relaxation time of the uid and a specic process time. It improves the thickness of the uid

and that is why the velocity of the uid diminishes. High Weissenberg ows means long relaxation

time in which the velocity of the uid vanishes at the wall and away from the wall the particles

move long distances within one relaxation time and the particles close the wall move short

distance.

Figs. 3:8(a) and 3:8(b) are drawn to analyze the e⁄ects of mass transfer parameter S on velocity

eld f 0( ) and temperature eld ( ) for both cases of shear thinning and shear thickening uids. These

Figs. indicate that the velocity and temperature decrease by increasing the values of mass

transfer parameter S. It is important to mention here that mass transfer reduces the momentum

boundary layer as well as thermal boundary

layer thicknesses.

Table 3:1: A comparison of numerical results for f00(0) for di⁄erent values of the unsteadiness parameter A

when n = 1;We = 0 and S = 0 are xed.

A Sharidan et al: [61] Chamkha et al: [62] Mukhopadhyay et al: [63] Present study

0:8 1:261042 1:261512 1:261479 1:261043

1:2 1:377722 1:378052 1:377850 1:377724

Table 3:2 : A comparison of numerical results for 0(0) for di⁄erent values of the

Prandtl number Pr when n = 1;We = 0;S = 0 and A = 0 are xed.

Pr Grubka and Bobba [64]

Chen

[65] Present study

0:72 1:0885 1:08853 1:088915

1:00 1:3333 1:33334 1:333333

3:00 2:5097 2:50972 2:509698

10:0 4:7969 4:79686 4:796853

Table 3:3: Numerical values of the local skin friction Re1=2Cfx and the local Nusselt number Re 1=2 Nux for

various A; S; We and n when Pr = 0:72 is xed.

Parameters Re1=2 Cfx Re 1=2 Nux

A S We n = 0:5 n = 1:5 n = 0:5 n = 1:5

0 0:1 1 0:984502 1:105730 1:095643 1:135326

0:5

1:123568 1:287783 1:299407 1:337392

1

1:246418 1:459715 1:467672 1:506087

2

1:449686 1:767831 1:753517 1:792850

1 0:1

1:246418 1:459715 1:467672 1:506087

0:3

1:335645 1:565328 1:528805 1:574201

0:5

1:432546 1:678373 1:592750 1:645860

0:7

1:537409 1:798792 1:659714 1:721167

0:3 1 1:335645 1:565328 1:528805 1:574201

3 1:057255 1:778164 1:460781 1:602252

5 0:924956 1:924107 1:422661 1:616516

7 0:845930 2:034999 1:397207 1:625560

Table 3:4: Numerical values of the local Nusselt number Re 1=2 Nux for various Pr;

A; S and n when We = 3 is xed.

Parameters

Re 1=2 Nux

Pr A S n = 0:5 n = 1 n = 1:5

0:72 0:4 0:3 1:242221 1:349189 1:402305

1

1:532060 1:655670 1:713346

3

3:095507 3:271185 3:338009

7

5:382384 5:586108 5:654414

1 0

1:336608 1:475306 1:542443

0:8

1:699923 1:814883 1:867066

1:2

1:851349 1:959863 2:008210

2

2:120781 2:219770 2:262867

0:8 0 1:583452 1:672845 1:715929

0:5 1:783838 1:916890 1:975349

1 2:021771 2:198410 2:272115

1:5 2:308189 2:517657 2:604050

Fig: 3:2 : A comparison of velocity proles f0( ) for di⁄erent values of the power law index n

when We = 3 and S = A = 0 are xed.

Fig. 3:3 : Velocity proles f 0( ) for di⁄erent values of the unsteadiness parameter A:

Fig. 3:4 : Temperature proles ( ) for di⁄erent values of the unsteadiness parameter A:

Fig. 3:5 : Temperature proles ( ) for di⁄erent values of the Prandtl number Pr:

Fig. 3:6 : Velocity proles f 0( ) and temperature proles ( ) for various power law index n:

Fig. 3:7 : Velocity proles f 0( ) and temperature proles ( ) for various Weissenberg number We:

Fig: 3:8 : Velocity proles f 0( ) and temperature proles ( ) for various mass transfer parameter S:

Chapter 4

Unsteady Heat and Mass Transfer

Mechanisms in MHD Carreau

Nanouid Flow

In this chapter, we study the unsteady heat and mass transfer mechanisms in a

magnetohydrodynamic (MHD) Carreau nanouid ow induced by a permeable stretching surface.

The Buongiornos model is used to incorporate the e⁄ects of Brownian motion and

hg n0 10

thermophoresis. The local similarity transformations are employed to alter the leading partial

di⁄erential equations to a set of ordinary di⁄erential equations. The resulting non-linear ordinary

di⁄erential equations are solved numerically by an e⁄ective numerical approach namely bvp4c

function in MATLAB to explore the e⁄ects of physical parameters. The velocity, temperature and

nanoparticle concentration proles have been calculated for both shear thinning and shear

thickening uids. A very good agreement is noticed between the present results and previous

published works in some limiting cases. It is important to mention here that the dimensionless

temperature and nanoparticle concentration are higher in the presence of magnetic eld. The

analysis further reveals that the local Nusselt number and local Sherwood number are decreasing

functions of the thermophoresis parameter.

4.1 Problem Formulation

In this investigation, heat and mass transfer analysis for the unsteady two-dimensional boundary

layer ow of Carreau nanouid induced by a permeable stretching sheet in the presence of external

time dependent magnetic eld is considered. The coordinate system is selected in such a manner

that x axis is measured along the stretching sheet and y axis is normal to it and the ow is occupied

above the sheet y > 0. Two equal and opposite forces are spontaneously implemented along x

axis so that the sheet is stretched with the velocity Uw(x;t) along x axis. The temperature Tw(x;t)

and concentration Cw(x;t) at the surface of sheet are considered to be higher than the ambient

temperature T1 (Tw > T1) and ambient concentration C1 (Cw > C1); respectively.

An external time dependent magnetic eld B(t) is applied in the positive y- direction. The induced

magnetic eld is considered to be very small as compared to the external applied magnetic eld and

is therefore neglected. It is further assumed that the surface is permeable having mass uid

velocity Vw(t) with Vw(t) < 0 for suction and Vw(t) > 0

for injection.

For the unsteady two-dimensional ow, the velocity, temperature and concentration elds are assumed

to be of the form

V = [u(x;y;t);v(x;y;t);0]; T = T(x;y;t); C = C(x;y;t): (4.1)

Under the above assumptions and after applying the usual boundary-layer analysis, the basic

boundary layer equations governing the conservations of mass, momentum, energy and

nanoparticle concentration for the Carreau nanouid in the presence of time dependent magnetic

eld can be expressed as (cf. Chapter 2)

(4.2)

(4.3)

(4.4)

: (4.5)

The boundary conditions for the physical problem are given by

u = Uw(x;t); v = Vw(t) T = Tw(x;t); C = Cw(x;t) at y = 0; (4.6) u ! 0; T ! T 1; C! C1 as y ! 1; (4.7)

where DT the thermophoresis di⁄usion coe¢ cient, DB the Brownian di⁄usion coe¢ cient, the

kinematic viscosity, the e⁄ective thermal di⁄usivity; T uid temperature, C the nanoparticle

concentration and = (c)p =(c)f the ratio of heat capacity of nanoparticle material to heat capacity

of base uid. Also, the stretching velocity Uw(x;t); surface temperature Tw(x;t), surface

nanoparticles concentration Cw(x;t) , mass uid velocity Vw(t) and time dependent magnetic eld

B(t) are considered to be the form

(4.8)

where 1 ct > 0 with a and c are positive constants possessing the dimensions (time) 1. Here B0

represents the intensity of magnetic eld and V0 denotes a uniform suction/injection

velocity.

To convert the governing equations into ordinary di⁄erential equations, we introduce the following

local similarity transformations

,

(4.9)

where denotes the stream function that satises the equation of continuity with u = @@y

and .

Thus the transformed non-linear momentum, energy and concentration equations can be written

as

n 3 1 + nWe2 (f 00)2on1 + We2 (f 00)2o 2 f000 + ff00 (f0)2 Ahf0 + f00i M2f0 = 0;

2

(4.10)

A 0 0 + Nt 0 2 = 0; (4.11) 00 + Pr(f0 2f0 ) Pr ( 0 + 3 ) + Pr Nb

2

00 + PrLe(f0 2f0 ) Pr (4.12)

with the altered conditions

n

f(0) = S; f0(0) = 1; (0) = 1; (0) = 1; (4.13)

f0(1) ! 0; (1) ! 0; (1) ! 0; (4.14)

where prime denotes di⁄erentiation with respect to : In the above equations, Pr

Prandtl number, unsteadiness parameter, M = q aB0 magnetic parameter,

Le = DB , Lewis number, , thermophoresis parameter,

Weissenberg number, mass transfer parameter and Nb = DB(Cw C1) Brown-

ian motion parameter.

Cfx, Nux and Shx are the quantities of physical interest which are dened as

(4.15)

where w, qw and qm are given by

:

(4.16)

In view of Eqs. (4:9) and (4:16); we obtain

Re Re 1=2 Nux = 0(0); Re 1=2 Shx = 0(0);

(4.17)

where Re = xUwrepresents the local Reynolds number.


The system of partially coupled non-linear ordinary di⁄erential equations (4:10) (4:12) with the

boundary conditions (4:13) and (4:14) has been solved numerically by using an e⁄ective

numerical technique known as bvp4c function in Matlab. For the accuracy and verication of the

present results, comparisons of the skin friction and Nusselt number are made with available

results in the literature. In the rst step, the obtained results of wall shear stress for particular

values of the unsteadiness parameter are compared with those reported by Sharidan et al: [61];

Chamkha et al: [62] and Mukhopadhyay and Gorla [63] (see Table 4:1): In the next step, we

compared our achieved results of wall temperature gradient with those obtained by Grubka and

Bobba [64]; Chen [65] and Sharma [66] for selected values of the Prandtl number in limiting

cases (see Table 4:2): An excellent agreement with the results of the aforesaid researchers is

noticed which surely sets a benchmark of quality of our numerical approach.

Table 4:3 is portrayed to indicate the inuence of unsteadiness parameter A, magnetic parameter

M; mass transfer parameter S and Weissenberg number We on the local skin-friction coe¢ cient

for both shear thinning (0 < n < 1) and shear thickening (n > 1) uids. From this Table, it is revealed

that the magnitude of the the local skin friction coe¢ cient Re1=2Cfx increases by increasing the

values of unsteadiness parameter A, magnetic parameter M and mass transfer parameter S both

for shear thinning and shear thickening uids. The magnitude of skin friction coe¢ cient is a

decreasing function of the Weissenberg number in shear thinning uid but opposite trend has

been noticed for shear thickening uid. It is also observed that the magnitude of the the local skin

friction coe¢ cient in shear thinning uid is comparatively less than that for shear thickening uid.

Table 4:4 is constructed to demonstrate the inuence of unsteadiness parameter A, the Prandtl

number Pr; the thermophoresis parameter Nt, the Brownian motion parameter Nb and the Lewis

number Le on the local Nusselt number Re 1=2 Nux for shear thinning and shear thickening uids.

On the evident of Table 4:4, an enhancement in the unsteadiness parameter and Prandtl number

grows the local Nusselt number both in shear thinning uid as well as shear thickening uid. It is

also examined that the local Nusselt number is a decreasing function of the thermophoresis

parameter, Brownian motion parameter and Lewis number in shear thinning and shear

thickening uids. Table 4:5 provides a sample of our numerical results of the reduced Sherwood

number 0(0) for selected values of the unsteadiness parameter, Prandtl number; thermophoresis

parameter, Brownian motion parameter and Lewis number when S = 0:3; M = 0:2 and We = 2:0

are xed. From this Table, it can be seen that the local Sherwood number Re 1=2 Shx enhances by

uplifting the unsteadiness parameter, Prandtl number, Lewis number and Brownian motion

parameter in both cases. It is also noted that rise in thermophoresis parameter depreciates the

mass transfer rate in shear thinning and shear thickening uids.

In order to obtain a clear sight on the physics of the problem, a parametric study is conducted

and the achieved numerical results are demonstrated with the help of graphical illustrations. The

inuence of unsteadiness parameter A on the velocity f 0( ), temperature ( ) and nanoparticle

concentration ( ) is depicted in Figs. 4:1(a) to 4:1(c). These Figs. reveal that an increment in the

values of the unsteadiness parameter depreciates the velocity, temperature and nanoparticle

concentration both for shear thinning and shear thickening uids. Additionally, momentum,

thermal and concentration boundary layer thicknesses diminish by uplifting the unsteadiness

parameter. It is further seen that the velocity of shear thinning uid is lower than that of shear

thickening uid, showing smaller boundary layer thickness. While quite the opposite is true in case

of temperature and concentration elds. From physical point of view, when the unsteadiness

enhances the sheet looses more heat due to which temperature diminishes. Figs. 4:2(a) to

4:2(c) are plotted to examine the inuence of the mass transfer parameter S on the velocity,

temperature and concentration proles. From these Figs., it is observed that the inuence of the

mass transfer parameter is similar qualitatively as that of the unsteadiness parameter.

The behavior of the magnetic parameter M on velocity f 0( ), temperature ( ) and nanoparticle

concentration ( ) is displayed in Figs. 4:3(a) to 4:3(c), respectively. These Figs. reveal that large

values of the magnetic parameter depreciate the velocity and enhance the temperature and

nanoparticle concentration in shear thinning and shear thickening uids. Further, the magnetic

parameter depresses the momentum boundary layer thickness and improves the thermal and

concentration boundary layer thicknesses. According to physical point of view, the magnetic

parameter is the ratio of electromagnetic force to the viscous force and so large values of

magnetic parameter implies that the Lorentz force enhances that generates more resistance to

the transport phenomena due to which velocity of the uid decreases. Consequently, the

momentum boundary layer thickness is a decreasing function of the magnetic parameter. As, the

Lorentz force possesses a resistive nature which opposes the motion of the uid and consequently,

heat is produced which enhances the thermal and concentration boundary layer thicknesses.

Figs. 4:4(a) to 4:4(c) demonstrate the impact of the local Weissenberg number on the velocity,

temperature and concentration proles. From these Figs., it is examined that for large values of

the local Weissenberg number, the velocity of uid diminishes in shear thinning uid but opposite

trend has been revealed in shear thickening uid. It is also observed that the temperature and

nanoparticle concentration enhance by uplifting the Weissenberg number in shear thinning uid

but opposite behavior has been noted in shear thickening uid. Same pattern has been revealed

for momentum, thermal and concentration boundary layer thicknesses. In fact, Weissenberg

number is the ratio of the relaxation time of the uid and a specic process time. In simple steady

ow, the Weissenberg number is dened as the shear rate times the relaxation time. It enhances

the thickness of the uid, so the velocity diminishes with the enhancement of the Weissenberg

number We:

Figs 4:5(a) and 4:5(b) represent the variation of dimensionless temperature and nanoparticle

concentration in response to a change in the values of Brownian motion parameter Nb: It is seen

that the dimensionless temperature enhances by uplifting the

Brownian motion parameter but opposite behavior has been examined for the dimensionless nanoparticle concentration. Additionally, thermal boundary layer thickness is a rising function while concentration boundary layer thickness is a diminishing function of the Brownian motion parameter both for shear thinning and shear thickening uids. According to the denition of the Brownian motion, on increasing the Brownian motion parameter, the intensity of this chaotic motion enhances the kinetic energy of the nanoparticles and as a result nanouid temperature rises. The e⁄ects of thermophoresis parameter Nt on the dimensionless temperature and nanoparticle concentration proles for both shear thinning and shear thickening uids are displayed through Figs.

4:6(a) and 4:6(b): The same qualitative behavior appears for the temperature eld while quite the opposite is noted for the nanoparticle concentration eld.

Fig. 4:7 has been prepared to illustrate the inuence of Lewis number Le on nanoparticle

concentration proles. It is observed that the Lewis number signicantly a⁄ects the nanoparticle

concentration distribution. The dimensionless nanoparticle concentration and concentration

boundary layer thickness decline with an increase of Lewis number. This is due to the fact that

mass transfer rate enhances as the Lewis number increases.

For a base uid of certain momentum di⁄usivity, a higher Lewis number possesses low Brownian

di⁄usion coe¢ cient which must result in a shorter penetration depth for the nanoparticle

concentration boundary layer thickness.

4:1: A comparison with previously published data for the values of

when We = S M = 0 and n = 1:

f00(0)

A

Sharidan et al:

[61]

Chamkha et al:

[62]

Mukhopadhyay and Gorla

[63]

Present

results

0:0

1:0000

0:2

1:06801

0:4

1:13469

0:6

1:19912

0:8 1:261042 1:261512 1:261479 1:26104

1:2 1:377722 1:378052 1:377850 1:37772

1:4

1:43284

2:0 1:587362

1:58737

Table 4:2 : A comparison with previously published data for the values of 0(0)

when We = S = M = A = Nt = Nb = 0 and n = 1:

Pr Grubka and Bobba [64]

Chen

[65] Sharma [66] for N = 801 Present results

0:72 1:0885 1:08853 1:0885 1:088915

1:00 1:3333 1:33334 1:3332 1:333333

3:00 2:5097 2:50972 2:5092 2:509698

10:0 4:7969 4:79686 4:7945 4:796853

4:3: Numerical values of Re1=2Cfx for various values of A; M; S; We and n.

Parameters Re1=2 Cfx

A M S We n = 0:5 n = 1:5

0:0 1:0 0:1 2:0 1:13241 1:71221

0:7

1:23365 1:93379

1:4

1:32256 2:14489

2:0

1:39088 2:31709

0:2 0:0

0:92112 1:26421

2:0

1:57986 2:89595

4:0

2:37069 5:79944

6:0

3:06526 9:13554

3:0 0:0

1:94652 4:21849

0:4

2:12414 4:44478

0:8

2:32373 4:68066

1:2

2:54517 4:92590

0:2 1:0 2:48264 3:87916

4:0 1:64297 4:90481

8:0 1:32583 5:59364

10:0 1:23789 5:83974

4:4 : Numerical values of Re 1=2 Nux for various values of A; Pr; Nt; Nb Le

and n when S 0:1; M = 2 and We = 3:0.

Parameters Re 1=2 Nux

A Pr Nt Nb Le n = 0:5 n = 1:5

0:0 0:72 0:1 0:2 1:0 0:54784 0:90770

0:7

1:06246 1:26144

1:4

1:33549 1:49685

2:0

1:52255 1:66539

0:2 1:0

0:94664 1:25473

3:0

1:69602 2:06537

5:0

2:01331 2:31754

7:0

2:12231 2:34720

0:5 0:2

0:60934 0:82132

0:4

0:60014 0:80888

0:6

0:59125 0:79685

0:8

0:58265 0:78523

0:5 0:1

0:60376 0:81463

0:3

0:58769 0:79120

0:5

0:57214 0:76854

0:7

0:55710 0:74663

0:2 1:0 0:59566 0:80282

5:0 0:57436 0:76956

10:0 0:56573 0:75771

64

15:0 0:56127 0:75192

Table 4:5 : Numerical values of Re 1=2 Shx for various values of A; Pr; Nt; Nb Le and n when S 0:3;

M = 0:2 and We = 2:0.

Parameters Re 1=2 Shx

A Pr Nt Nb Le n = 0:5 n = 1:5

0:0 0:72 0:1 0:2 1:0 0:73051 0:84434

0:7

0:99767 1:08562

1:4

1:18466 1:26507

2:0

1:32104 1:39726

0:2 1:0

1:04482 1:16416

3:0

2:36841 2:55438

5:0

3:55169 3:77015

7:0

4:68030 4:91941

0:5 0:2

0:34440 0:40443

0:4

0:21771 0:19676

0:6

0:75643 0:77136

0:8

1:27311 1:32096

0:5 0:1

1:97165 2:07690

0:3

0:00363 0:04216

0:5

0:39780 0:46495

0:7

0:56610 0:64542

0:2 1:0 0:48991 0:48728

5:0 1:77541 1:92596

10:0 3:50088 3:68669

66

15:0 4:89905 5:09781

4:1 E⁄ect of the unsteadiness parameter A on the velocity f 0( ), temperature ( ) and

( ) proles.

η η

4:2 E⁄ect of the mass transfer parameter S on the velocity f 0( ), temperature ( ) and

( ) proles.

η η

4:3 E⁄ect of the magnetic parameter M on the velocity f 0( ), temperature ( ) and

( ) proles.

Fig:4:4 : E⁄ect of the Weissenberg number We on the velocity f 0( ), temperature ( ) and concentration

( ) proles

η η

Fig. 4:5 : E⁄ect of the Brownian motion parameter Nb on the temperature ( ) and concentratio

( ) proles.

η η

Fig. 4:6 : E⁄ect of the thermophoresis parameter Nt on the temperature ( ) and concentration

( ) proles.

Fig. 4:7 : E⁄ect of the Lewis number Le on the concentration ( ) proles.

Chapter 5

On Unsteady Falkner-Skan Flow of

MHD Carreau Nanouid Past a

Static/Moving Wedge

The aim of present chapter is to explore the numerical solutions for the unsteady

twodimensional Falkner-Skan ow of MHD Carreau nanouid past a static/moving wedge in the

presence of convective boundary condition. The e⁄ects of Brownian motion and

thermophoresis are taken into account. The local similarity transformations are utilized to

alter the leading time dependent non-linear partial di⁄erential equations to a set of ordinary

di⁄erential equations. The obtained non-linear ordinary di⁄erential equations are solved

numerically by the two di⁄erent numerical techniques namely shooting method with Felhberg

formula and Newtons Raphson as well as bvp4c function in MATLAB to explore the impacts of

pertinent parameters. A comparison is presented between the current study and published

works and found to be in outstanding agreement. It is important to mention that an increment

in the wedge angle parameter depreciate the heat and mass transfer rate both for shear

thinning and shear thickening uids. Furthermore, the thermal boundary layer thickness is an

increasing function of the generalized Biot number in shear thinning and shear thickening uids.

Additionally, temperature is enhanced by growing the Brownian motion and the

thermophoresis parameters.

5.1 Model Development

Let us consider the unsteady two-dimensional Falkner-Skan ow of an incompressible Carreau

nanouid past a static/moving wedge in the presence of external time dependent magnetic eld

and convective boundary condition. It is assumed that the uid ow is induced by a stretching

wedge with the velocity as well as the free stream velocity

where a; b; c and m are positive constants with 0 m 1: It

should be noted that Uw(x;t) > 0 corresponds to a stretching wedge surface velocity and

Uw(x;t) < 0 compares to a contracting wedge surface velocity (see Fig. 5:1).

Geometry of the Problem

Fig. 5:1: Physical description of ow problem.

The wedge angle is assumed to be = : An external time dependent magnetic eld

is applied normal to the wedge surface. It is also considered that the lower

surface of the wedge is heated by convection from a hot uid of temperature Tw(x;t) which

provides a heat transfer coe¢ cient hf: It is further assumed that the surface temperature

Tw(x;t) and concentration Cw(x;t) at the surface of sheet are considered to be higher than the

ambient temperature T1 (Tw > T1) and ambient concentration C1 (Cw > C1) respectively. The

combined e⁄ects of thermophoresis and Brownian motion are taken into account due to the

nanoparticles.

For the unsteady two-dimensional wedge ow, the velocity, temperature and concentration elds

are assumed to be of the form (cf. Chapter 2)


Under the above aforesaid assumptions and after applying the usual boundary-layer analysis,

the basic boundary layer equations governing the conservations of mass, momentum, energy

and nanoparticle concentration for the Carreau nanouid in the presence of time dependent

magnetic eld can be expressed as [69]

(5.2)

(5.4)

(5.5)

along with the boundary conditions

(i) static wedge

u ! Ue; T ! T 1; C! C1 as y ! 1; (5.7)

(ii) moving wedge

(5.8)

u ! Ue; T ! T 1; C! C1 as y ! 1: (5.9)

where DB Brownian di⁄usion coe¢ cient, thermal di⁄usivity; T the uid temperature, = (c)p

=(c)f ratio of heat capacity of nanoparticle to heat capacity of base uid, DT the thermophoresis

di⁄usion coe¢ cient and C the nanoparticle concentration . Also, the stretching velocity Uw(x;t);

free stream velocity Ue(x;t); surface temperature Tw(x;t), surface nanoparticles concentration

Cw(x;t) and time dependent magnetic eld B(t) are assumed to be the form

(5.10)

where T0 and C0 represent the initial reference temperature and concentration, respec-

tively.

To convert the basic governing equations of the problem into ordinary di⁄erential equations, we

employ the following local similarity transformations

,

(5.11)

where indicates the stream function that satises .

Thus, the transformed non-linear momentum, energy and concentration equations can be

written as

2

00 +Pr(f0 2f0 ) Pr)f 0 + 3 g+PrNb0 0 +PrNt 0 = 0; (5.13)

00 + PrLe(f0 2f0 ) Pr )( (5.14)

and the altered conditions form

f(0) = 0; f0(0) = ; 0(0) = (2 )1=2 f1 (0)g; (0) = 1; (5.15)

f0(1) ! 1; (1) ! 0; (1) ! 0; (5.16)

where Pr = Prandtl number, unsteadiness parameter, magnetic

parameter, Le = DB Lewis number, thermophoresis parameter,

Weissenberg number, Nb = DB(Cw C1) Brownian motion parameter,

wedge angle parameter, velocity ratio parameter and Re 1=2 Biot number.

Cfx, Nux and Shx are concrete parts of ow which are

(5.17)

where w, qw and qm are

:

(5.18)

Using Eqs. (5:11); (5:17) and (5:18), we obtain the following non-dimensional expressions

n 1 (2 )1=2 Re1=2 Cfx = f00(0) 1 + We2(f00(0))2 2 ; (2 )1=2 Re 1=2 Nux =

0(0);

(2 )1=2 Re 1=2 Shx = 0(0); (5.19)

where Re = xUe indicates the local Reynolds number.


The system of locally-similar and partially coupled non-linear ordinary di⁄erential equations

(5:12) (5:14) with the associated boundary conditions (5:15) and (5:16) have been solved

numerically by employing two e¢ cient numerical techniques namely the shooting method

with Felhberg formula and Newtons method as well as bvp4c function in Matlab. A

comprehensive numerical computation is performed for di⁄erent values of the pertinent

parameters namely the local Weissenberg number We; the power law index n; the

unsteadiness parameter A; the wedge angle parameter ; the generalized Biot number ; the

magnetic parameter Ha; the Prandtl number Pr; the Lewis number Le; the Brownian motion

parameter Nb; the thermophoresis parameter Nt and the velocity ratio parameter :

To prove the authenticity of the achieved numerical results, a comparison with the existing

literature is also conducted in limiting cases. The obtained results of the skin friction coe¢ cient

for selected values of the wedge angle parameter are also compared with those reported by

Rajagopal et al: [70]; Kuo [71] and Ishaq et al: [72] (see Table 5:1): An outstanding agreement

with the results of the aforesaid authors is noticed.

Table 5:2 is portrayed to demonstrate the impact of the unsteadiness parameter A, the wedge

angle parameter ; the magnetic parameter Ha; the velocity ratio parameter and the local

Weissenberg number We on the local skin-friction coe¢ cient for both shear thinning (0 < n <

1) and shear thickening (n > 1) uids. On the basis of this Table, it is noticed that the local skin

friction coe¢ cient enhances by enhancing the values of the unsteadiness parameter and the

magnetic parameter in shear thinning and shear thickening uids. It is also observed that the

local skin friction coe¢ cient is a decreasing function of the wedge angle parameter and the

velocity ratio parameter both for shear thinning and shear thickening uids. It is further

revealed that the local skin friction coe¢ cient depresses by uplifting the values of the local

Weissenberg number in shear thinning uid but opposite trend is seen in shear thickening uid.

Table 5:3 is constructed by using two di⁄erent numerical techniques to illustrate the impact of

the unsteadiness parameter A, the wedge angle parameter and the Prandtl number Pr on the

local Nusselt number 0(0) when Ha = We = 2:0; = 0:1; = 0:2;Le = 1:0 and Nt = Nb = 0:2 are xed.

On the evident of Table 5:3, it is revealed that an enhancement in the unsteadiness parameter

and the Prandtl number grows the local Nusselt number both for shear thinning and shear

thickening uids but the local Nusselt number is a diminishing function of the wedge angle

parameter in both cases.

Table 5:4 indicates a sample of obtained numerical results of two di⁄erent numerical

techniques for the reduced Sherwood number 0(0) for di⁄erent values of the unsteadiness

parameter, the wedge angle parameter and the Prandtl number when Ha = We = 2:0; = 0:1;

= 0:2; Le = 1:0 and Nt = Nb = 0:2 are xed. From this Table, it can

be seen that the local Sherwood number grows by uplifting the unsteadiness parameter, and

the Prandtl number in both cases. It is also observed that an enhancement in the wedge angle

parameter depreciates the mass transfer rate both for shear thinning and shear thickening

uids.

To get a denite perception of the current problem, the velocity, temperature and

concentration proles are demonstrated graphically for both shear thinning and shear

thickening uids through Figs. 5:2 to 5:9. The e⁄ects of the unsteadiness parameter A on the

velocity f 0( ), temperature ( ) and concentration ( ) proles are presented through Figs. 5:2(a)

to 5:2(f). From these Figs., it can be seen that an increment in the values of the unsteadiness

parameter improves the velocity proles and depreciates the temperature as well

concentration proles both for shear thinning and shear thickening uids. The value = 0

corresponds to the static wedge and > 0 corresponds to the stretching wedge. Thus, it is

further found that the velocity of the uid is low for static wedge when compared to the

stretching wedge but qualitatively opposite trend is noticed in temperature and concentration

proles. Additionally, the momentum, thermal and concentration boundary layer thicknesses

diminish by growing the unsteadiness parameter in all cases. Physically, when unsteadiness

enhances then sheat looses heat due to which temperature of the uid decreases.

Figs. 5:3(a) to 5:3(d) are plotted to examine the inuence of the wedge angle parameter on

the velocity and temperature proles in shear thinning and shear thickening uids. From these

Figs., it can be seen that the impact of the wedge angle parameter is similar qualitatively as

that of the unsteadiness parameter. From physical point of view, the wedge angle parameter

indicates the pressure gradient. Thus, positive values of the wedge angle parameter

correspond a favorable pressure gradient which grows the ow. Also, = 0 corresponds to wedge

angle of zero degree (ow past a at plate) and = 0 relates to wedge angle of 900 degree

(stagnation point ow).

The variation of the velocity, temperature and nanoparticle concentration proles is

represented through Figs. 5:4(a) to 5:4(f) for di⁄erent values the local Weissenberg number

We regarding the shear thinning and shear thickening uids. These Figs. exhibit that the velocity

proles increase while temperature and concentration proles decrease by uplifting the values

of the local Weissenberg number in shear thinning uid but opposite behavior is noticed in

shear thickening uid. It is also observed that momentum, thermal and concentration boundary

layer thicknesses are diminishing function of the local Weissenberg number in shear thinning

uid however quite opposite trend is the true in shear thickening uid.

In Figs. 5:5(a) to 5:5(f); the e⁄ects of the magnetic parmeter Ha on the velocity, temperature

and nanoparticle concentration proles are depicted for shear thinning and shear thickening

uids. These Figs. reveal that the inuence of the magnetic parameter is similar qualitatively as

that of the unsteadiness parameter and the wedge angle parameter.

The impact of the generalized Biot number on the temperature distribution is presented

through Figs. 5:6(a) to 5:6(b) for both cases. An observation of these Figs. makes it clear that

the temperature and thermal boundary layer thickness are the growing function of the

generalized Biot number in shear thinning and shear thickening uids. The generalized Biot

number indicates that the ratio of internal thermal resistance of a solid to boundary layer

thermal resistance. When = 0; the surface of the wedge is totally isolated. It means, the

internal thermal resistance of the surface of the wedge is extremely high and there is no

occurrence of convective heat transfer from the surface of the wedge to the uid far away from

wedge.

The thermophoresis parameter Nt has a valuable importance for investigating the

temperature and nanoparticle concentration distributions in nanouid ow. The inuence of the

thermophoresis parameter on the temperature and nanoparticle concentration is elucidated

through Figs. 5:7(a) to 5:7(d) for both the cases. These Figs. reveal that the temperature and

nanoparticle concentration enhance by uplifting the thermophoresis parameter. From a

physical perspective, the thermophoresis force enhances with the enhancement of Nt which

tend to move nanoparticles from hot region to cold region and hence enhances the magnitude

of the temperature and nanoparticle concentration proles. Additionally, the thickness of the

thermal and concentration boundary layers is large for slightly improved values of the

thermophoresis parameter.

Figs. 5:8(a) to 5:8(d) present the variation of the temperature and concentration proles for

distinct values of the Brownian motion parameter Nb: It is depicted that non-dimensional

temperature increases by uplifting the Brownian motion parameter and a decrement is

observed in concentration proles. Furthermore, thermal boundary layer thickness is an

enhancing function of the Brownian motion parameter. According to the denition of the

Brownian motion, by enhancing the Brownian motion parameter, the intensity of this chaotic

motion increases the kinetic energy of the nanoparticles and as a result nanouid temperature

rises.

To illustrate the inuence of Lewis number Le on nanoparticle concentration proles, Figs.

5:9(a) to 5:9(b) is presented. It is noticed that the Lewis number signicantly a⁄ects the

nanoparticle concentration proles. From these Figs., it is observed that the nanoparticle

concentration and the concentration boundary layer thickness diminish by increasing the

values of the Lewis number. In fact, the mass transfer rate increases as Lewis number

increases. For a base uid of certain momentum di⁄usivity, a higher Lewis number possesses

low Brownian di⁄usion coe¢ cient which must result in a shorter penetration depth for the

nanoparticle concentration boundary layer thickness.

Table 5:1: A comparison of numerical results of f00(0) for di⁄erent values of the

wedge angle parameter when We = = Ha = 0 and n = 1:

Rajagopal et al: [70] Kuo [71]

Ishaq et al:

[72] Present results

0:0

0:469600 0:4696 0:4696005

0:1 0:587035 0:587880 0:5870 0:5870353

0:3 0:774755 0:775524 0:7748 0:7747546

0:5 0:927680 0:927905 0:9277 0:9276800

1:0 1:232585 1:231289 1:2326 1:2325880

Table 5:2: Numerical values of (2 )1=2 Re1=2 Cfx for various values of A; ; Ha; ; We and n.

Parameters (2 )1=2 Re1=2 Cfx (bvp4c)

(2 )1=2 Re1=2 Cfx( shooting

)

A

Ha

We n = 0:5 n = 1:5 n = 0:5 n = 1:5

0:0 0:3 2:0 0:2 2:0 1:501681 2:718339 1:50169 2:718339

0:6

1:555973 2:870351 1:555973 2:870351

1:2

1:607149 3:018818 1:607152 3:018816

2:0

1:671137 3:211355 1:671139 3:211365

0:2 0:0

1:573757 2:933024 1:573761 2:933024

0:4

1:501426 2:713382 1:501426 2:713382

1:0

1:378037 2:358776 1:378037 2:358776

1:6

1:228293 1:962934 1:228294 1:962934

0:1 0:0

0:6029617 0:7104673 0:603061 0:710478

1:0

1:022373 1:473568 1:022374 1:473571

3:0

2:025839 4:504285 2:025839 4:504285

5:0

2:839416 8:123942 2:839416 8:123943

2:0 0:2

2:04486 4:571969 2:044864 4:571970

0:1

1:929815 4:140212 1:929819 4:140515

0:1

1:686243 3:293043 1:686243 3:293033

0:2

1:55629 2:879192 1:556292 2:879191

0:2 2:0 1:55629 2:879192 1:556292 2:879191

4:0 1:24985 3:252493 1:24985 3:252493

6:0 1:094487

89 3:51069 1:094487 3:510643

8:0 0:9952928 3:710955 0:9952928 3:710955

Table 5:3: Numerical values of (2 )1=2 Re 1=2 Nux for various values of A; ; and Pr when Ha = We

= 2:0; = 0:1; = 0:2;Le = 1:0 and Nt = Nb = 0:2 are xed.

Parameters (2 )1=2 Re 1=2 Nux (bvp4c) (2 )1=2 Re 1=2 Nux ( shooting )

A

Pr n = 0:5 n = 1:5 n = 0:5 n = 1:5

0:0 0:3 3:0 0:120902 0:119587 0:120546 0:119235

0:2

0:121954 0:121059 0:121594 0:120703

0:3

0:122370 0:121612 0:122372 0:121613

0:0

0:132315 0:131485 0:132295 0:131485

0:4

0:118856 0:118125 0:118856 0:118125

1:0

0:094818 0:094282 0:094818 0:094282

1:0 0:092941 0:092298 0:092955 0:092321

2:0 0:094304 0:093730 0:092955 0:093731

Table 5:4: Numerical values of (2 )1=2 Re 1=2 Shx for various values of A; ; and Pr when Ha = We =

2:0; = 0:1; = 0:2;Le = 1:0 and Nt = Nb = 0:2 are xed.

Parameters (2 )1=2 Re 1=2 Shx(bvp4c) (2 )1=2 Re 1=2 Shx(Shooting)

A

Pr n = 0:5 n = 1:5 n = 0:5 n = 1:5

0:0 0:3 3:0 2:108938 1:839832 2:109110 1:839937

0:2

2:354029 2:109225 2:354195 2:109341

0:3

2:470564 2:236245 2:470565 2:236247

0:0

2:541684 2:306940 2:541691 2:306942

0:4

2:446169 2:212109 2:446170 2:212112

1:0

2:291078 2:060343 2:291080 2:060350

1:0 1:387905 1:262716 1:388639 1:263683

2:0 1:906415 1:719841 1:906941 1:719879

0 1 2 3 4 0 1 2 3 4 η η

0 1 2 3 4 0 1 2 3 4 η η

0 1 2 3 4 0 1 2 3 4 η η

Fig. 5:2 : E⁄ects of the unsteadiness parameter A on the velocity f 0( ), temperature ( ) and

nanoparticle concentration ( ) proles.

0 1 2 3 4 0 1 2 3 4 η η

0 1 2 3 4 0 1 2 3 4 η η

Fig. 5:3 E⁄ects of the wedge angle parameter on the velocity f 0( ) and temperature ( )

proles.

0 1 2 3 4 0 1 2 3 4 η η

η η Fig. 5:4 E⁄ects of the Weissenberg number We on the velocity f 0( ), temperature ( ) and


0 1 2 3 4 0 1 2 3 4 η η

η η Fig: 5:5 E⁄ects of the magnetic parameter Ha on the velocity f 0( ), temperature ( ) and


0 1 2 3 4 0 1 2 3 4 η η

Fig. 5:6 : Impact of the Biot number on the temperature ( ) proles.

η η

η η Fig. 5:7 E⁄ects of the thermophoresis parameter Nt on the temperature ( ) and nanoparticle

concentration ( ) proles.

0 1 2 3 4 0 1 2 3 4 η η

η η

Fig. 5:8 : E⁄ects of the Brownian motion parameter Nb on the temperature ( ) and nanoparticle concentration

( ) proles.

η η

Fig. 5:9 : Impact of the Lewis number Le on the nanoparticle concentration ( ) proles.

Chapter 6

E⁄ects of Melting and Heat

Generation/Absorption on Unsteady

Falkner-Skan Flow of Carreau

Nanouid over a Wedge

This chapter focuses on the numerical investigation of the melting and heat

generation/absorption phenomena in unsteady Falkner-Skan wedge ow of Carreau nanouid. A

zero nanoparticle mass ux condition at the boundary is implemented. Flow is induced due to

stretched wedge in the presence of the Brownian motion and thermophoresis e⁄ects. Suitable

transformations are utilized to attain non-linear ordinary di⁄erential equations. The resulting

non-linear ordinary di⁄erential equations are then solved numerically through bvp4c Matlab

package. E⁄ects of several emerging parameters on the temperature and nanoparticles

concentration proles are explored and discussed. The reduced Nusselt number is also calculated

and examined. A comparison is presented between the current results and available data and

found to be in outstanding agreement.

Our study predicts that the temperature and nanoparticles concentration proles depreciate by enhancing values of the melting parameter both for shear thinning and shear thickening uids.

6.1 Problem Description

We have considered the unsteady two-dimensional Falkner-Skan ow of an incompressible

Carreau nanouid over a wedge. It is assumed that uid ow is caused by stretching wedge with the

velocity : The free stream velocity for the present problem

is where m; a; b and c are positive constants with 0 m 1: It

should be noted that Uw(x;t) > 0 corresponds to a stretching wedge surface velocity and Uw(x;t)

< 0 corresponds to a contracting wedge surface velocity. The wedge angle is supposed to be = :

On perspective of White [73]; positive values of indicates that when the pressure gradient is

favorable or negative then the ow will be accelerating along the surface. On the other hand,

negative values of shows that the pressure gradient is adverse and the ow will be decelerating.

Additionally, = 0 (m = 0) corresponds to boundary layer ow over a horizontal at plate and = 1 (m

= 1) relates to boundary layer ow near the stagnation point of a vertical at plate. We have chosen

Tm and T1 such that T1 > Tm where Tm is the melting temperature and T1 the ambient uid

temperature. We incorporate the e⁄ects of thermophoresis and Brownian motion due to

nanoparticles. The e⁄ects of heat generation/absorption are also taken into account. For the

transient two-dimensional ow, the velocity, temperature and nanoparticles concentration elds

are again chosen in the form ( cf. Chapter 2)


Using the aforesaid assumptions along with boundary layer approximations the system of

equations governed by the conservations of mass, momentum, energy and nanoparticle

concentration for the Carreau nanouid ow is given by [69]

(6.2)

(6.5)

with BCs

(6.6)

: (6.7)

u ! Ue; T ! T 1; C! C1 as y ! 1: (6.8)

Q0 heat source/sink, DT thermophoresis di⁄usion coe¢ cient, = (c)p =(c)f ratio of heat capacity of

nanoparticle to heat capacity of base uid, DB Brownian di⁄usion coe¢ cient, thermal

di⁄usivity; = kinematic viscosity, Cp specic heat,

f

J uid latent heat and cs surface heat capacity. The condition (6:7) conveys that heat conducted

with melting surface is melting heat and addition of sensible heat required to grow solid

temperature T0 to melting temperature Tm.

The compatable variables can be written as

,

(6.9)

In view of Eq. (6:9); Eqs. (6:3) to (6:8) can be of the form

1 2

(6.12)

00 + f0 Pr A0 + Nb 0 0 + Nt 0 + Q = 0; (6.11)

f0(0) = s; Prf (0) + M 0(0) = 0; (0) = 0; Nb 0(0) + Nt 0 (0) = 0; (6.13)

f0(1) ! 1; (1) ! 1; (1) ! 1: (6.14)

From above, Weissenberg number, unsteadiness

parameter, thermophoresis parameter, Pr = Prandtl number, Nb = DB(C1 Cm) Brownian motion

parameter, velocity ratio parameter, melting parameter, wedge angle

parameter, heat source/sink

parameter and Sc = DB Schmidt number.

Cfx, Nux and Shx are given as

(6.15)

with w, qw and qm are dened as

:

(6.16) Non-dimensional form of (6:15) is

n 1 (2 )1=2 Re1=2 Cfx = f00(0) 1 + We2(f00(0))2 2 ; (2 )1=2 Re 1=2 Nux =

0(0);

(2 )1=2 Re 1=2 Shx = 1(0); (6.17)

where Re = xUe is the local Reynolds number.

6.2 Numerical interpretation

The aim of this portion is to express the e⁄ects of utilized parameters on ( ), ( ) and

to some physical parameters which depend upon spatial/temporal variables. In fact the current model

belongs to a local model.

Computations of (2 )1=2 Re1=2 Cfx for some values of We; A; M; and s with

Nt = 0:1; Pr = 2:5; Sc = 2:0;Q = 0:1 and Nb = 0:2 (look Table 6:3) are executed. From table, (2

)1=2 Re1=2 Cfx enhances for uplifting values of unsteadiness and wedge angle parameters.

1 (0). Comparison analysis is made with availble data in special cases. The obtained results of

surface drage for altering values of wedge angle parameter are compared to the work of Ishaq et

al: [72]; Rajagopal et al: [70] and Kuo [71] (look Table 6:1) and see good agreement. Also,

comparison of 0(0) with published study is made (look Table

6:2): Again, we achieved good agreement. Keep in mind, variables in Eq. (6:12) suggest

Additionally surface drag depreciates by enhancing velocity ratio and melting parameters. Also,

surface drag declines for growing values of Weissenberg number in shear thinning liquid but

opposite trend in shear thickening liquid. Computations for 0(0) have been executed to study

impact of wedge angle parameter, Weissenberg number, unsteadiness parameter, power law

index, Prandtl number, velocity ratio parameter, melting parameter, heat source/sink parameter,

Schmidt number, Brownian motion and thermophoresis parameters. From table 6:4; j 0(0)j is a

growing function of wedge angle parameter, heat generation parameter, Schmidt number and

Prandtl number. However, j 0(0)j is a declining function of melting, thermophoresis and

unsteadiness parameters. Computations of 0(0) for parameters Sc;A;Q; Nb; M; Nt;Pr and with Pr

= 2:5, s = 0:2 and We = 2:0 (look Table 6:5) are performed. From Table 6:5; 0(0) grows with

growing heat source/sink parameter, Schmidt number and wedge angle parameter. Also 0(0)

diminishes for improving of melting, Brownian motion and unsteadiness parameter.

Figs. 6:1(a) to 6:1(d) reveal behavior of melting parameter on ( ) and ( ). These

Figs. shows that ( ) and ( ) depreciate for growing melting parameter. Figs. 6:1(a) and 6:1(b) state that temperature curves cut each other about = 2 and onward show opposite behavior. Also, this critical value / reverse trend of temperature is gained earlier for shear thinning liquid as compared with shear thickening liquid. Furtthermore, thermal boundary layer thickness is uplifting function of melting parameter and reverse is true for concentration boundary layer thickness. To examine the inuence of the heat generation/absorption parameter Q on the dimensionless temperature and nanoparticles concentration, Figs. 6:2(a) to 6:2(d) are portroyed to analyze heat source/sink e⁄ects on ( ) and ( ). Note that that ( ), ( ) and their related thermal and concentration thicknesses are upgrading function of heat sorce/sink parameter. Note Q = 0 relates to no heat source/sink, Q > 0 corresponds to heat generation and Q < 0 relates to absorption. Also ( ), ( ) and their related thermal and concentration thicknesses are higher in heat generation when compared to heat absorption. Figs. 6:3(a) to 6:3(d) presents the variation of the temperature and nanoparticles concentration proles for distinct values of the wedge angle parameter : From these Figs., it is turned out that the temperature of the uid enhances by increasing the values of the wedge angle parameter in the presence of melting e⁄ects but opposite e⁄ects have been noticed in nanoparticles concentration proles. Figs. 6:3(a) and 6:3(b) show that behavior of ( ) for improving wedge angle parameter is opposite from melting parameter. Actually, wedge angle parameter

indicates pressure gradient. Positive values of wedge angle parameter regards a favorable pressure gradient which improves ow. Also, = 0 corresponds to ow past a at plate and = 1 relates stagnation point ow. Brownian motion and thermophoresis are concrete parmeters of nanouid. The variation in ( )

with thermophoresis parameter is studied via Figs. 6:4(a) and 6:4(b). These Figs. show that

nanoparticles concentration and related thickness grow by growing thermophoresis parameter.

In fact, thermophoresis force increases with increment of Nt which has tendency to move

nanoparticles from hot section to cold section. The variation in ( ) with Nb can be studied by Figs.

6:5(a) and 6:5(b): Greater values of Nb resulted in depreciation of nanoparticles concentration.

Appearance of nanoparticles resulted in Brownian motion and decrement of nanoparticles

concentration thickness.

Variation in ( ) with Schmidt number can be visualized through Figs. 6:6(a) and 6:6(b). It can be

seen that the nanoparticle concentration and the associated thickness decline for improving

Schmidt number. Actually, Schmidt number is connected with molecular di⁄usivity. Small

molecular di⁄usivity relates for larger values of Schmidt number.

Table 6:1: Computations of f00(0) for some values of when We = s = M = 0 and n = 1.

Rajagopal et al: [70] Kuo [71] Ishaq et al: [72] Present results

0:0

0:469600 0:4696 0:469600

0:1 0:587035 0:587880 0:5870 0:587035

0:3 0:774755 0:775524 0:7748 0:774754

0:5 0:927680 0:927905 0:9277 0:927680

1:0 1:232585 1:231289 1:2326 1:232588

Table 6:2: Computations of (2 )1=2 Re 1=2 Nux for some values of Pr and

when n = 1 and We = Nt = Nb = M = Q = s = Sc = 0.

= 0 = 0:3

Pr

White


White


0:1 0:1980 0:19803 0:2090 0:20908

0:3 0:3037 0:30372 0:3278 0:32783

0:6 0:3916 0:39168 0:4289 0:42892

0:72 0:4178 0:41809 0:4592 0:45955

1:0 0:4696 0:46960 0:5195 0:51952

2:0 0:5972 0:59723 0:6690 0:66904

6:0 0:8672 0:86728 0:9872 0:98727

10:0 1:0297 1:02975 1:1791 1:17913

Table 6:3: Computations of (2 )1=2 Re1=2 Cfx for some values of A; ; M; s and We when Pr = 2:5; Q =

0:1; Sc = 2:0; Nb = 0:2 and Nt = 0:1:

Parameters (2 )1=2 Re1=2 Cfx

A

M s We n = 0:5 n = 1:5

0:0 0:3 0:1 0:1 1:0 0:6841268 0:7302197

0:1

0:7192146 0:7705566

0:2

0:7531583 0:8100987

0:1 0:0 0:1 0:1 1:0 0:4814324 0:5001569

0:5

0:8378397 0:9153261

1:0

1:066367 1:217256

0:1 0:4 0:0 0:1 1:0 0:8008611 0:8662401

0:2

0:7649436 0:827903

0:4

0:7390557 0:8001039

0:1 0:4 0:1 0:2 1:0 0:7237269 0:7755364

0:3

0:6586342 0:6983139

0:4

0:5860308 0:614419

0:1 0:4 0:1 0:2 2:0 0:6645587 0:8180135

3:0 0:6075858 0:8589482

4:0 0:5617959 0:8954975

Table 6:4: Execution of j 0(0)j for some values of A;;Q;M;Nt;Nb;Sc and Pr when We = 2:0 and s = 0:2.

Parameter

s

j 0(0) j

A

M Q Nt Nb Sc Pr n = 0:5 n = 1:5

0:0 0:3 0:1 0:2 0:1 0:1 0:4 3:0 1:699029 1:668373

0:1

1:668964 1:631376

0:2

1:645636 1:599644

0:1 0:0 0:1 0:2 0:1 0:1 0:4 3:0 1:623009 1:615755

0:5

1:694837 1:641467

1:0

1:745374 1:663133

0:1 0:4 0:0 0:2 0:1 0:1 0:4 3:0 1:853189 1:803236

0:2

1:543115 1:500851

0:4

1:32914 1:292791

0:1 0:4 0:1 0:0 0:1 0:1 0:4 3:0 0:8250023 0:7682432

0:2

1:682385 1:636523

0:25

2:028409 1:999439

0:1 0:4 0:1 0:1 0:1 0:1 0:4 3:0 1:172218 1:114301

0:15

1:165083 1:107476

0:2

1:15808 1:100781

0:1 0:4 0:1 0:1 0:13 0:1 0:4 3:0 1:167921 1:110191

0:2

1:167921 1:110191

0:25

1:167921 1:110191

0:1 0:4 0:1 0:1 0:13 0:15 0:5 3:0 1:172307 1:114261

1 17

0:6

1:176800 1:118444

0:7

1:181348 1:122692

0:1 0:4 0:1 0:1 0:13 0:15 0:3 5:0 1:475772 1:40614

Table 6:5: Execution of 0(0) for values of A; ; M; Q; Nt; Nb; Sc and Pr when We = 2:0; Pr = 2:5

and s = 0:2.

Parameters 0(0)

A

M Q Nt Nb Sc n = 0:5 n = 1:5

0:0 0:3 0:1 0:2 0:1 0:1 1:0 1:452907 1:421229

0:1

1:409644 1:370983

0:2

1:367099 1:319744

0:1 0:0 0:1 0:2 0:1 0:1 1:0 1:360534 1:347744

0:5

1:435153 1:383125

1:0

1:483122 1:406795

0:1 0:4 0:0 0:2 0:1 0:1 1:0 1:573564 1:523059

0:2

1:301582 1:259996

0:4

1:117226 1:082107

0:1 0:4 0:1 0:0 0:1 0:1 1:0 0:7781625 0:725616

0:2

1:422987 1:37731

0:3

1:9515 1:93039

0:1 0:4 0:1 0:1 0:1 0:1 1:0 1:048148 0:9949226

0:2

2:106888 1:999188

0:3

3:175939 3:012551

0:1 0:4 0:1 0:1 0:2 0:1 1:0 2:106888 1:999188

0:2

1:053444 0:999594

0:3

0:7022959 0:666396

0:1 0:4 0:1 0:1 0:2 0:2 1:0 1:053444 0:999594

11 9

2:0 1:1014 1:044821

3:0 1:144063 1:085535

6:1: M on ( )and ).

0 1 2 3 4 5 6 7 8 9 10 11

12 0 1 2 3 4 5 6 7 8 9

10 11 12 η η η η

6:2: Q on ( ) and ).

6:3: on ( ) and ):

0 1 2 3 4 5 6 7 0 1 2 3 4

5 6 7 η η

Fig: 6:4: E⁄ects of Nt on ( ).

0 1 2 3 4 5 6 7 0 1 2 3 4

5 6 7 η η Fig. 6:5: E⁄ects of Nb on ( ).

0 1 2 3 4 5 6 7 0 1 2 3 4

5 6 7 η η

Fig. 6:6: E⁄ects of Sc on ( ).

Chapter 7

Unsteady Heat and Mass Transfer in Carreau Nanouid Flow over Expanding/Contracting Cylinder

In this chapter, we introduce the more general convective heat and mass conditions in the

unsteady ow of Carreau nanouid over an expanding/contracting horizontal cylinder in the

presence of temperature dependent thermal conductivity. Appropriate transformations are

used to alter the non-linear partial di⁄erential equations into ordinary di⁄erential equations.

Numerical solutions of the resulting system are calculated by an e⁄ective numerical approach

namely bvp4c function in Matlab. E⁄ects of distinct parameters on velocity, temperature and

nanoparticle concentration are analyzed. Numerical results of Nusselt and Sherwood

numbers are also computed in tabular form. Present study reveals that the velocity,

temperature and nanoparticle concentration are depreciating functions of unsteadiness

parameter. It is further noticed that the rate of heat and mass transfer is reducing for growing

values of the thermal conductivity parameter in both cases of shear thinning (0 < n < 1) and

shear thickening (n > 1) uids. In addition, on increasing the values of thermal Biot number,

the heat transfer rate enhances but opposite behavior is noticed in mass transfer rate.

7.1 Description of the Problem

Let us consider an unsteady two-dimensional laminar boundary layer ow of an incompressible

Carreau nanouid over an expanding/contracting cylinder with time dependent

radius a(t) = a0p1 Ht; where t is the time, a0 the positive constant and H the constant of

expansion or contraction strength. For positive value of H, the cylinder radius reduces with

time and the cylinder is contracting; however, for negative value of H, the cylinder radius

increases with time and the cylinder is expanding. We further consider the temperature

dependent thermal conductivity and the convective conditions on the surface of the cylinder.

Nanouid model consisting of thermophoresis and Brownian motion is adopted. Let the x and

r axis are taken along the axial and radial directions, respectively, as shown in Fig. 7:1. Heat

and mass transfer analysis is carried out under the convective surface conditions.

7.2 Physical Model

Fig. 7:1: Geometry of the problem.

The thermal conductivity of the uid can be considered in the following expression

(7.1)

where the small parameter represents the variable thermal conductivity parameter, k1

the thermal conductivity of the uid far away from the surface of cylinder and T = Tf T1 the uid

temperature di⁄erence. For the unsteady 2D convective ow, velocity, temperature and

concentration elds are suggested in a way

V = [v(r;x;t); 0; u(r;x;t)]; T = T(r;x;t); C = C(r;x;t): (7.2)

Under these assumptions and after employing the usual boundary-layer analysis, the basic

partial di⁄erential equations for the Carreau nanouid in the presence of temperature

dependent thermal conductivity can be written as [74]

(7.3)

(7.4)

: (7.6)

The boundary conditions for the physical problem are given by

(7.7)

u ! 0; T ! T 1; C! C1 as r! 1; (7.8)

where hf is the wall heat transfer coe¢ cient, km the wall mass transfer coe¢ cient, Dm the

molecular di⁄usivity of the species concentration, DT the thermophoresis di⁄usion coe¢ cient,

DB the Brownian di⁄usion coe¢ cient, Cp the specic heat and the density of uid and = (c)p =(c)f

ratio of e⁄ective heat capacity of nanoparticle to heat capacity of base uid .

To alter the above governing equations of the present problem into ordinary di⁄erential

equations, we utilize the following local similarity variables

, (7.9)

Thus, Eq. (7:3) is satised automatically and the non-linear momentum, energy and concentration equations can be expressed as

1 + nWe2(f 00)2 1 + We2(f 00)2 n23

f000

(f 0)2 Aff0 + f00g = 0;

(7.10)

2

( 00 + 0)(1 + ) + ( 0)2 + Prff0 A0g + Pr Nb0 0 + Nt 0 = 0; (7.11)

0o = 0; (7.12)

and the boundary conditions become

2 (1 (1)); (7.13)

f0( ) ! 0; ( ) ! 0; ( ) ! 0 as ! 1;

(7.14)

where is Weissenberg number, Sc = DB Schmidt number, unsteadiness

parameter, thermophoresis parameter, Nb = DB(Cw C1) Brownian motion

parameter, thermal Biot number, Pr = Prandtl

number and concentration Biot number.

From application point of view, the important mechanisms of ow are Cfx, Nux and Shx are dened as

(7.15)

where rx, qw and qm given as

:

Using Eqs. (7:9); (7:15) and (7:16), we get the following resulting expressions

: (7.17)


In general, it is di¢ cult to nd the exact solution of the partially coupled non-linear differential

equations Eqs. (7:10) (7:12) along with the boundary conditions Eqs. (7:13) and (7:14).

Numerical results of non-linear system have been presented to analyze the inuences of several

parameters on velocity, temperature and nanoparticle concentration proles. The variations of

the skin-friction coe¢ cient, Nusselt and Sherwood numbers have been also computed for the

involved parameters. A comparative study to available data is performed for special case. The

achieved numerical results of skin friction coefcient for a selected value of unsteadiness

parameter is compared with those studied by Fang et al: [75] (see Table 7:1) and found to be in

good agreement.

Tables 7:2 to 7:4 are constructed to illustrate the inuences of Schmidt number Sc; unsteadiness

parameter A; Weissenberg number We; thermal conductivity parameter ; thermophoresis

parameter Nt; thermal Biot number 1; concentration Biot number 2; Brownian motion parameter

Nb on skin friction coe¢ cient, Nusselt and Sherwood numbers. On the behalf of these Tables, it

is clear that on growing the values of the unsteadiness parameter A, the magnitude of the skin

friction coe¢ cient is depreciated. Additionally, an increment in the values of Weissenberg

number We decreases the magnitude of skin friction coe¢ cient in shear thinning liquid but

reverse trend can be seen in shear thickening liquid. From these Tables, it is depicted that an

enhancement in the unsteadiness parameter A resulted in the depreciation of the Nusselt

number as well as Sherwood number for both cases. It is noticed that rate of heat and mass

transfer diminishes by growing the thermal conductivity parameter both in shear thickening and

shear thinning uids. On increasing the values of thermal Biot number, the heat transfer rate

enhances but opposite behavior is noticed in mass transfer rates. For larger values of

concentration biot number, heat transfer rate diminished but opposite e⁄ects are observed in

mass transfer rate. Both Nusselt and Sherwood numbers depreciated with the increase of

thermophoresis parameter. Note that Nusselt number decreased by growing Brownian motion

parameter but reverse behavior has been noticed for Sherwood number. Additionally, it is also

noticed that heat and mass transfer rates are enhancing function for Schmidt number.

In order to understand the e⁄ects of various parameters on the velocity, temperature and

nanoparticle concentration proles by way of shear thickening and shear thinning uids, a graphical

representation is given through Figs. 7:2 to 7:9: Figs. 7:2(a) to 7:2(c) elucidate the inuences of

unsteadiness parameter on velocity, temperature and nanoparticle concentration proles. From

these Figs., it is obvious that velocity, temperature and concentration elds are the declining

functions of the unsteadiness parameter. Additionally, the momentum, thermal and

concentration boundary layer thicknesses are also the decreasing function of the unsteadiness

parameter for both the cases. Physically, when unsteadiness enhances the sheet looses more

heat due to which temperature diminishes. Figs. 7:3(a) and 7:3(b) are plotted to represent the

inuences of Weissenberg number in velocity and temperature elds. These Figs. reveal that the

velocity prole depreciates by increasing Weissenberg number for shear thinning liquid but

enhances in shear thickening liquid but reverse behavior can be noticed in temperature proles.

According to the denition of Weissenberg number, it is the ratio of the relaxation time of the uid

and a specic process time. It grows the thickness of uid and that is why velocity of the uid

depreciates. To examine thermal conductivity parameter e⁄ect on temperature proles, Figs.

7:4(a) and 7:4(b) are plotted. These Figs. reveal that the temperature grows with growing

thermal conductivity parameter. The value = 0 relates to the constant conductivity of the uid.

Hence, it is resulted that the surface temperature is larger for temperature dependent thermal

conductivity when compared to the constant thermal conductivity. As thermal conductivity relies

on temperature so thermal boundary layer thickness grows as average uid thermal conductivity

enhances hence magnitude of temperature eld improves by improving thermal conductivity

parameter.

The variation in temperature and nanoparticle concentration proles is presented through Figs. 7:5(a)

and 7:5(b) for distinct values of themal Biot number 1. These

Figs. reveal that the temperature and concentration proles increase by uplifting thermal Biot

number 1: Also, the thermal and concentration boundary layer thicknesses are the growing

functions of the thermal Biot number 1: The thermal Biot number 1 indicates the ratio of internal

thermal resistance of a solid to boundary layer thermal resistance. When the value 1 = 0; the

surface of the cylinder is totally insolated. It means the internal thermal resistance of the surface

of the cylinder is very high and there is no convective heat transfer from cylinder surface to the

cold uid far away from the cylinder. The inuences of concentration Biot number 2 on the

nanoparticle concentration proles can be seen in Fig. 7:6: It is observed that the concentration

prole is an enhancing function of the concentration Biot number 2:

The Brownian motion and thermophoresis parameters are two important parameters to

investigate the temperature and nanoparticle concentration proles in nanouid ow.

Figs. 7:7(a) to 7:7(d) are plotted to analyze the inuences of the thermophoresis parameter Nt on temperature and nanoparticle concentration proles. From these facts, it turns out that temperature as well as nanoparticle concentration proles enhance with the increase of the thermophoresis parameter. In fact, temperature di⁄erence between ambient and surface enhances for higher thermophoresis which improves the temperature and concentration of the uid. Figs. 7:8(a) to 7:8(d) are sketched to examine behavior of Brownian motion parameter Nb on temperature and nanoparticle concentration proles. On basis of theses Figs., it is depicted that temperature proles increase while the nanoparticle

concentration proles diminish by growing the Brownian motion parameter Nb in shear thinning and shear thickening uids.

Fig. 7:9 elucidates the impact of Schmidt number Sc for nanoparticle concentration eld. It is

noted that nanoparticle concentration proles and concentration boundary layer thickness

diminish by enhancing the Schmidt number Sc. Actually, Schmidt number Sc is the ratio of

viscosity to mass di⁄usivity. When Schmidt number enhances then mass di⁄usivity diminishes and

results in reduction in uid concentration.

Table 7:1: A comparison of numerical computation of of f00(1) with those of Fang et

al: [75] for Re = 1 and A = 0 when n = 1 and We = 0:

A Fang et al: [75] Present study

0 1:17775 1:17884

Table 7:2: Numerical computations of for distinct values of emerging para-

meters.

Parameters

A We n = 0:5 n = 1:5

2:0 0:5 2:30866 2:52842

1:0

1:58778 1:72134

0:0

1:15362 1:22891

1:0 1:0 1:47697 1:82282

2:0 1:33325 1:99166

3:0 1:25631 2:11923

Table 7:3: Numerical computations of 0(1) for distinct values of emerging parameters when Pr = 0:72

and We = 0:5 are xed.

Parameters

0(1)

A

1 2 Nt Nb Sc n = 0:5 n = 1:5

2:0 0:1 0:1 0:2 0:1 0:2 1:0 0:095352 0:095374

1:0

0:093197 0:093224

0:0

0:085072 0:085161

1:0 0:0 0:1 0:2 0:1 0:2 1:0 0:093717 0:093743

0:5

0:090917

0:090954

1:0

0:088312

0:088357

1:0 0:2 0:1 0:2 0:1 0:2 1:0 0:092573 0:092603

0:2

0:172344

0:172449

0:3

0:241799

0:242006

1:0 0:2 0:1 0:3 0:1 0:2 1:0 0:092550 0:092580

0:4

0:092529

0:092560

0:5

0:092510

0:092541

1:0 0:2 0:1 0:2 0:5 0:2 1:0 0:092498 0:092529

1:0

0:092404

0:092435

1:5

0:092307

0:092340

1:0 0:2 0:1 0:2 0:1 0:5 1:0 0:092492 0:092523

1:0

0:092357

0:092389

1:5

0:092218

0:092251

1:0 0:2 0:1 0:2 0:1 0:2 3:0 0:092593 0:092623

138 5:0 0:092603

0:092633

7:0 0:092608

0:092638

Table 7:4: Numerical computations of 0(1) for distinct values of emerging parameters when Pr

= 0:72 and We = 0:5 are xed.

Parameters 0(1)

A

1 2 Nt Nb Sc n = 0:5 n = 1:5

2:0 0:1 0:1 0:2 0:1 0:2 1:0 0:184916 0:184983

1:0

0:178061 0:178143

0:0

0:153860 0:154115

1:0 0:0 0:1 0:2 0:1

1:0 0:177901 0:177987

0:5

0:178019 0:178103

1:0

0:178127 0:178211

1:0 0:2 0:1 0:2 0:1

1:0 0:177949 0:178034

0:2

0:175599 0:175692

0:3

0:173575 0:173674

1:0 0:2 0:1 0:3 0:1

1:0 0:254651 0:254817

0:4

0:324610 0:324874

0:5

0:388678 0:389050

1:0 0:2 0:1 0:2 0:5

1:0 0:166994 0:167125

1:0

0:153498 0:153683

1:5

0:140226 0:140464

1:0 0:2 0:1 0:2 0:1 0:5 1:0 0:179613 0:179691

1:0

0:180168 0:180243

1:5

0:180353 0:180428

1:0 0:2 0:1 0:2 0:1 0:2 3:0 0:189744 0:189773

140 5:0 0:193151 0:193165

7:0 0:194824 0:194833

1 2 3 4 1 2 3 4 η η

Fig. 7:2 : Inuences of A on f 0( ), ( ) and ( ).

1 2 3 4 1 2 3 4 η η

Fig. 7:3 : Inuences of We on f 0( ) and ( ).

1 2 3 4 1 2 3 4 η η

Fig. 7:4 : Inuences of thermal conductivity parameter on the temperature ( ) proles.

η η

Fig. 7:5 : Inuences of the thermal Biot number 1 on the temperature ( ) and nanoparticle concentration

( ) proles.

Fig. 7:6 : Inuences of the concentration Biot number 2 on the nanoparticle concentration ( ) proles.

η η

η η Fig. 7:7 : Inuences of the thermophoresis parameter Nt on the temperature ( ) and nanoparticle


η η

η η

Fig. 7:8 : Inuences of the Brownian motion parameter Nb on the temperature ( ) and nanoparticle


Fig. 7:9 : Inuences of Sc on ( ).

Chapter 8

Unsteady Stagnation Point Flow of MHD Carreau Nanouid over Expanding/Contracting Cylinder

The unsteady magnetohydrodynamic (MHD) stagnation point ow of Carreau nanouid over an

expanding/contracting cylinder in the presence of nonlinear thermal radiation is investigated

numerically in this chapter. Recently devised model for nanouid namely Buongiornos model

involving Brownian motion and thermophoresis is considered in the present problem.

Additionally, zero nanoparticle mass ux condition is considered. Mathematical problem is

developed with the help of momentum, energy and nanoparticle concentration equations using

suitable transformation variables. The numerical results for the transformed nonlinear ordinary

di⁄erential equations are presented for both cases of stretching and shrinking cylinder in shear

thinning as well as shear thickening uids. For numerical computations, an e⁄ective numerical

solver namely bvp4c package is used. E⁄ects of involved controlling parameters on the velocity,

temperature and nanoparticle concentration are examined. Numerical computations for the skin

friction coe¢ cient and Nusselt number are also executed. It is interesting to note that the

temperature and nanoparticle concentration are higher in shrinking cylinder case when

compared to stretching cylinder case. Additionally, the rate of heat transfer (Nusselt number) is

a decreasing function of the unsteadiness, radiation and thermophoresis parameters in

stretching and shrinking cylinder both for shear thickening and shear thinning uids.

8.1 Mathematical Analysis

The problem of unsteady two-dimensional MHD stagnation point ow of an incompressible Carreau

nanouid due to expansion/ contraction of a permeable horizontal

cylinder with time dependent radius a(t) = a0p1 Ht in the presence of nonlinear thermal

radiation is considered. Here a0 is the positive constant having dimension length, t the time and

H the constant of expansion or contraction strength. The radius of cylinder depreciates with time

for positive values of H while it grows with time for negative values of H. The x and r axes are

choosen along the axial and radial directions, respectively. It is assumed that the cylinder is

shrinking or

stretching with time dependent velocity which is linearly pro-

portional to the axial distance from the origin with c as a positive constant having dimension

(time) 1. Recently devised model for nanouid incorporating the e⁄ects of thermophoresis and

Brownian motion is adopted. The surface of the cylinder is at constant temperature Tw and

concentration Cw while the ambient uid temperature is T1, where we assume Tw > T1:

Additionally, zero nanoparticle mass ux condition is considered. A non-uniform transverse

magnetic eld of strength

is applied in the radial direction, where B0 is a constant related

to magnetic eld strength. The magnetic Reynolds number is taken to be small enough so that the

induced magnetic eld can be neglected.

Additionally, the stagnation point is considered at r = b0 and x = 0 with free stream velocity

with a as a positive constant having dimension (time) 1:

For the unsteady two-dimensional stagnation point ow, the velocity, temperature and nanoparticles

concentration elds are selected in the form (cf. Chapter 7)

V = [v(r;x;t); 0; u(r;x;t)]; T = T(r;x;t); C = C(r;x;t): (8.1)

Utilizing the boundary-layer analysis and under the aforesaid assumptions, the governing

equations for the Carreau nanouid in the presence of time dependent magnetic eld and nonlinear

thermal radiation can be demonstrated as [76]

(8.2)

: (8.5)

The associated boundary conditions for the considered problem can be written as

(8.6)

(8.7)

where (u;v) represent the velocity components in axial and radial directions, respectively, the

density of uid, the thermal di⁄usivity with k the thermal conductivity, Cp the specic heat,

DB the Brownian di⁄usion coe¢ cient, = (c)p =(c)f the ratio of heat capacity of nanoparticle to heat

capacity of base uid, DT the thermophoresis di⁄usion coe¢ cient and S > 0 the dimensionless

suction parameter.

Using the Rosseland approximation subject to the radiation [77]; the simplied structure of radiative

heat ux can be indicated as

(8.8)

where and k represent the Stefan-Boltzmann constant and the mean absorption

coe¢ cient, respectively.

In view of Eq. (8:8), the energy equation (8:4) with nonlinear thermal radiation can be written in

the following form

:

The non-dimensional suitable variables can be represented in the following manner

, (8.10)

with T = T1 (1 + ( w 1) ) and is the temperature ratio parameter.

Substituting Eqs. (8:10) into Eqs. (8:3);(8:5) and (8:9); we get following nonlinear ordinary di⁄erential equations

n 3 1 + nWe2(f 00)2 1 + We2(f 00)2 2 f000

Aff00 + f0 1g M2 Re(f0 1) = 0; (8.11)

2 00 + (1 + PrRef PrA) 0 + Nb0 0 + Nt 0

(8.13)

with the corresponding boundary conditions as obtained from Eqs. (8:6) and (8:7) in the

form

f(1) = S; f0(1) = ; (1) = 1; Nb 0(1) + Nt 0(1) = 0; (8.14)

f0( ) ! 1; ( ) ! 0; ( ) ! 0 as ! 1: (8.15)

The non-dimensional constants appearing in Eqs. (8:11) (8:15) are the local Weissenberg number We, the unsteadiness parameter A, the Prandtl number Pr, the Schmidt number Sc, the

thermophoresis parameter Nt, the Brownian motion parameter Nb, the Reynolds number Re, the temperature ratio parameter w, the magnetic parameter M, the radiation parameter NR and

the velocity ratio parameter . Note that < 0 corresponds to shrinking case and > 0 relates to stretching case. They are respectively dened as

Pr =

Re = : (8.16)

From application point of view, the important mechanisms of ow and heat transfer are the local

skin friction coe¢ cient Cfx and local Nusselt number Nux which are dened as

(8.17)

where rx and qw indicate the wall shear stress and wall heat ux respectively, which are dened as

Using Eqs. (8:10); (8:17) and (8:18), the local skin friction coe¢ cient and Nusselt number can be

expressed as

(8.19)

8.2 Discussion of Numerical results

The numerical investigation for unsteady MHD stagnation point ow of Carreau nanouid over

contracting and/ or expanding cylinder in the presence of nonlinear thermal radiation is

performed. The numerical computations have been executed using an e⁄ective numerical

approach namely bvp4c Matlab package for several values of involved parameters namely the

Reynolds number Re; suction parameter S; velocity ratio parameter ; power law index n;

magnetic parameter M; local Weissenberg number We; thermophoresis parameter Nt; Brownian

motion parameter Nb; radiation parameter NR; temperature ratio parameter w; unsteadiness

parameter A and Prandtl number Pr. For the justication of present numerical analysis, the

numerical results of skin friction coe¢ cient are compared with those reported by Lok and Pop

[78] for various values of the Reynolds number Re and suction parameter S with = 0:5; We = M

= A = 0 and n = 1

(see Table 8:1): It is noticed that the current numerical analysis is in excellent agreement with the

existing study. Note that the Carreau uid reduces to viscous uid when

We = 0 and n = 1:

The numerical results for the drag force and heat transfer rate have been reported for both the

cases of stretching an shrinking cylinder through Tables 8:2 and 8:3: On the basis of Table8:1;

the drag force is improving function of Reynolds number and magnetic parameter in shrinking

cylinder both for shear thickening (n > 1) and shear thinning (0 < n < 1) uids but opposite

behavior can be observed in stretching cylinder for both shear thickening and shear thinning uids.

Additionally, drag force is a depreciating function of the unsteadiness parameter in shrinking

cylinder both for shear thickening and shear thinning uids but reverse behavior can be seen in

stretching cylinder. Additionally, the local skin friction coe¢ cient decreases in shrinking cylinder

and increases in stretching cylinder for growing values of the Weissenberg number in shear

thinning uid and opposite e⁄ects can be seen in shear thickening uids. On the basis of Table 8:2;

it can be examined that rate of heat transfer is a growing function of the Reynolds number and

temperature ratio parameter in shrinking and stretching cylinder. It is also observed that the rate

of heat transfer is higher in stretching cylinder when compared to shrinking cylinder. Additionally,

rate of heat transfer (Nusselt number) is a depreciating function of unsteadiness parameter,

radiation parameter and thermophoresis parameter in stretching as well as shrinking cylinder.

To see the physical implication of parameters on velocity, temperature and nanoparticles

concentration elds in stretching or shrinking cylinder for both shear thickening and shear thinning

uids, Figs. 8:2 to 8:9 are plotted. Figs. 8:2(a) and 8:2(b) elucidate the variations of the

Weissenberg number We on the velocity eld. From these Figs., it can be seen that the velocity of

the uid grows in shrinking cylinder but depreciates in stretching cylinder for shear thinning uid.

But opposite mechanism can be noticed in shear thickening uid. Additionally, the momentum

boundary layer thickness decreases in case of shrinking cylinder but increases for the case of

stretching cylinder in shear thinning uid. Actually, Weissenberg number is the ratio of relaxation

time to a specic process time. For steady ow, Weissenberg number is the shear rate times the

relaxation time. It enhances the uid thickness so velocity of the uid depreciates with an increase

in Weissenberg number for the stretching cylinder.

Figs. 8:3(a) to 8:3(f) describe the e⁄ects of Reynolds number Re on the velocity, temperature

and nanoparticles concentration elds for stretching or shrinking cylinder. From these Figs., it is

clear that the velocity of the uid is enhancing function of the Reynolds number in shrinking

cylinder but decreasing in stretching cylinder. Additionally, the temperature, nanoparticles

concentration and their associated thicknesses are decreasing by increasing the values of the

Reynolds number in stretching or shrinking cylinder. Furthermore, temperature and

nanoparticles concentration are higher in shrinking cylinder when compared to stretching

cylinder.

Figs. 8:4(a) and 8:4(b) show the variation of velocity proles in stretching and shrinking cylinder under the inuence of magnetic parameter M. From these Figs., it can be depicted that the

velocity proles grow in shrinking cylinder but depreciate in stretching cylinder by increment of magnetic parameter. According to the denition of magnetic parameter which is the ratio of

electro magnetic force to viscous force and resist the uid ow due to which velocity of the uid in stretching case is decreased.

To analyze the e⁄ects of temperature ratio parameter w; on the temperature eld, Figs. 8:5(a) and

8:5(b) are plotted. Here, it is easy to understand that temperature of the uid increases in

shrinking as well as stretching cylinder with the increment of temperature ratio parameter both

in shear thickening and shear thinning uids. It can also be seen that thermal boundary layer

thickness also grows for improving the values of temperature ratio parameter in Pseudoplastic

and dilatant uids. Enhancing values of temperature ratio parameter corresponds to higher wall

temperature as compared to ambient uid temperature. Consequently, temperature of uid grows.

Variation in temperature with radiation parameter NR can be seen via Figs. 8:6(a) and 8:6(b):

On the evident of these Figs., it is analyzed that the temperature and its associated thermal

boundary layer thickness diminish in stretching as well as shrinking cylinder for improving values

of radiation parameter in dilatant and Pseduplastic uids.

The e⁄ects of thermophoresis parameter Nt on the temperature and nanoparticle concentration

proles are examined through Figs. 8:7(a) to 8:7(d): From these Figs., it is concluded that the

temperature and nanoparticle concentration elds enhance in stretching and shrinking cylinder by

growing the values of thermophoresis parameter.

Furthermore, the concentration boundary layer thickness is an enhancing function of the

thermophoresis parameter. Actually, temperature di⁄erence between ambient and surface

grows for higher thermophoresis which enhances the temperature and concentration of the uid.

Figs. Figs. 8:8(a) to 8:8(d) are demonstrated to analyze the inuence of Nb on the nanoparticle

concentration proles. On the basis of these Figs., it is noticed that the nanoparticles

concentration eld diminishes and its related concentration boundary layer thickness depreciates

by growing Brownian motion parameter in stretching as well as shrinking cylinder for both cases.

The Brownian motion takes place due to the presence of nanoparticles and resulted in the

depreciation of nanoparticles concentration thickness.

E⁄ects of Sc on ( ) are studied through Figs. 8:9(a) and 8:9(b). From these Figs., it is turned out

that the nanoparticle concentration eld and its associted concentration boundary layer thickness

decrease by growing the Schmidt number in stretching as well as shrinking cylinder. Basically,

Schmidt number is the ratio of viscosity to mass di⁄usivity. When Schmidt number grows then

mass di⁄usivity decreases and results in reduction in uid concentration.

Table 8:1: Comparison of results for f00(1) when = 0:5; We = M = A = 0 and n = 1:

S = 0:5

S = 1:5

Lok and Pop [78] present results Lok and Pop [78] Present results

Re Analytical Numerical Numerical Analytical Numerical Numerical

0:5 1:7297 1:6800 1:680280 2:1975 2:1794 2:179547

1:0 2:4038 2:3709 2:370896 3:4421 3:4406 3:440553

5:0 6:3380 6:3331 6:333064 12:6513 12:6579 12:657932

10:0 10:5448 10:5456 10:545636 23:9478 23:9515 23:951491

50:0 41:4347 41:4368 41:436822 113:9897 113:9900 113:990023

100:0 79:1703 79:1714 79:171389 226:4949 226:4950 226:495004

Table 8:2: Numerical results for Cfx Re when S = 0:3.

Parameters n = 0:5 n = 1:5

Re A M We = 0:5 = 1:5 = 0:5 = 1:5

1:0 2:0 1:0 1:0 4:072165 1:646093 5:340360 1:919324

5:0

6:650944 2:822618 10:385717 3:866903

10:0

9:148424 3:799534 15:102559 5:642660

20:0

13:725999 5:428607 23:016200 8:580177

0:5 3:0 1:0 1:0 4:911180 1:775970 5:802778 1:972830

2:5

4:226755 1:572896 5:102722 1:754663

2:0

3:570414 1:382194 4:424683 1:546423

1:5

2:957964 1:207223 3:781615 1:352006

0:5 2:0 1:0 1:0 3:570414 1:382194 4:424683 1:546423

2:0

3:932211 1:538365 5:353769 1:771076

3:0

4:367887 1:740416 6:613375 2:086045

4:0

4:811610 1:955973 8:056295 2:454863

0:5 2:0 1:0 2:0 3:463014 1:303699 4:726351 1:631564

3:0 3:412258 1:259473 4:938895 1:698062

4:0 3:381591 1:231275 5:107484 1:752386

5:0 3:360695 1:211532 5:249413 1:798580

Table 8:3: Numerical results for Nu when We = 1:0; S = 0:3; M = 1:0; Pr = 2:0 and Sc = 1:0:

Parameters n = 0:5 n = 1:5

Re A w NR Nt Nb = 0:5 = 1:5

=

0:

5 = 1:5

1:0 2:0 1:2 1:0 0:1 0:2 6:099582 6:440049 5:875745 6:464151

5:0

9:253103 10:05011 8:263307 10:19348

10:0

12:89588 13:88797 11:16809 14:16776

20:0

19:7177 20:82972 16:86407 21:34186

0:5 3:0 1:2 1:0 0:1 0:2 7:774547 7:921721 7:661618 7:934702

2:5

6:739985 6:913142 6:627457 6:924961

2:0

5:693704 5:899982 5:583537 5:910499

1:5

4:633061 4:881312 4:5284 4:890432

0:5 2:0 1:3 1:0 0:1 0:2 5:7828 5:993942 5:666841 6:005023

1:4

5:880585 6:095903 5:759083 6:107524

1:5

5:987075 6:205775 5:860432 6:217896

1:6

6:10223 6:32345 5:970979 6:336019

0:5 2:0 1:2 2:0 0:1 0:2 5:40174 5:592873 5:30962 5:601635

3:0

5:29789 5:480221 5:214611 5:488125

4:0

5:244568 5:421379 5:166432 5:428787

5:0

5:212116 5:38517 5:137331 5:392254

0:5 2:0 1:2 1:0 0:1 0:2 5:693704 5:899982 5:583537 5:910499

0:2

5:648656164

5:853341 5:539047 5:863806

0:3

5:604019 5:807118 5:494971 5:817531

0:4

5:55979 5:761311 5:451306 5:771671

1 1.5 2 2.5 3 1 1.5 2 2.5 3 η η

Fig. 8:2: Variation of velocity f 0( ) with di⁄erent values of We.

1 1.5 2 2.5 3 3.5 4 1 1.5 2 2.5 3 3.5

4 η η

η η

1 2 3 4 1 2 3 4 η η

Fig. 8:3: Variation of velocity f 0( ); temperature ( ) and nanoparticle concentration ( ) with di⁄erent values

of Re.

1 1.5 2 2.5 3 1 1.5 2 2.5 3 η η

Fig. 8:4: Variation of velocity f 0( ) with di⁄erent values of M.

η η

Fig. 8:5: Variation of temperature ( ) with di⁄erent values of w.

η η

Fig. 8:6: Variation of temperature ( ) with di⁄erent values of NR.

η η

Fig. 8:7: Variation of temperature ( ) and nanoparticle concentration ( ) with di⁄erent values of

Nt:

1 2 3 4 5 1 2 3 4 5 η η

1

0.9

0.8

0.7 0.6

0.5

0.4

0.3

0.2 0.1

0 1

λ = -0.5 λ = 1.5

5

1

0.9

0.8

0.7 0.6

0.5

0.4

0.3

0.2 0.1

0 1

( Fr Fra

b) n = 1 ame 003 28 May 2017 N Fmream00e40 0228 M28ayM2a0y1270 17N

A = -2.0,

.5

We = 1.0, M =

λ = -0.5 λ =

1.5

1.0, s = 0. 3, Re

= 0.5

5

0.03.45 0.5 0.410.4 0.03.340.04.4 0.3 0.39 0.03.023.3080.3.2

2 2 21.44 1.55 3 3

1η3.4η 6 V1

= -2.0,

4 4 1.6 5 5 We = 1.0, M = 1.0, s = 0.3, Re =

0.5

0.03.012.3070.2.1 0.36 0.1 0.1 0.3 0 1.42

1 0 0

1 1 A

4 5 1 . 4 8 1 . 5

η 2 3 4

N t = 1 . 1 , 1 . 2 , 1 . 3 , 1 . 4

N R = 1 . 0 , P r = 2 . 0 , S c = 1 . 0 , θ w = 1 . 2 , N b = 0 . 2

η 2 3 4

N t = 1 . 1 , 1 . 2 , 1 . 3 , 1 . 4

N R = 1 . 0 , P r = 2 . 0 , S c = 1 . 0 , θ w = 1 . 2 , N b = 0 . 2

Fig. 8:8: Variation of nanoparticle concentration ( ) with di⁄erent values of Nb.

1 2 3 4 5 1 2 3 4 5 η η

Fig. 8:9: Variation of nanoparticle concentration ( ) with di⁄erent values of Sc.

Chapter 9

Unsteady Axisymmetric Flow and Heat Transfer in Carreau Fluid past a Stretched Surface

The current chapter concentrates on numerical analysis in axisymmetric of unsteady Carreau uid

towards radially surface. Numerical computations are expressed for shear thickening liquid and

shear thinning liquid. The modeled equations are altered into ordinary di⁄erential system by

adopting suitable variables. The numerical solution is delibrated via bvp4c package. Numerical

computations for the drag force and heat transfer rate are developed for steady and unsteady

cases. Note that the magnitude of drag force and heat transfer rate for steady case is less than

that for unsteady case.

9.1 Formulation of the Flow Problem

The problem of axisymmetric ow of Carreau uid towards radially moving surface is assumed. The

surface is stretched along radial direction. The surface is in plane z = 0 and ow appears in half

plane z > 0. we assume cylindrical polar coordinate system (r;;z) and ow appears in rotational

symmetry. Tw(r;t) and T1 are taken as surface temperature and ambient uid temperature such

that Tw > T1:

The temperatue and velocity elds are taken in a way

T = T(r;z;t); V = [u(r;z;t); 0; w(r;z;t)]: Based on aforesaid consideration,

the governing equations stated as

(9.1)

(9.2)

(9.4)

u = Uw(r;t); w = fw(t) ; T = Tw(r;t) at z = 0; (9.5)

u ! 0; T ! T 1 as z! 1; (9.6)

where u and w denote the velocity components along r and z directions, respectively, t; ; ; k; Cp are the time, kinematic viscosity, uid density, thermal conductivity of the uid and the specic heat, respectively.

We assumed that the stretching velocity Uw(r;t); surface temperature Tw(r;t) and mass uid velocity

fw(t) are of the following form:

(9.7)

where Et < 1 with E and c are positive constants having dimensions reciprocal of time, W0 is a

uniform suction/injection velocity (W0 > 0 for suction and W0 < 0 for injection).

The particular form for the mass uid velocity fw(t); surface temperature Tw(r;t) and stretching velocity

Uw(r;t) are chosen to employ the following suitable transformation:

(r;z;t) = r2Uw Re (9.8)

where is the Stokes stream function having the property , the di-

mensionless temperature, Rer = rUw the local Reynolds number and the independent variable, respectively.

Thus the velocity components are

u = Uwf 0( ); w = 2Uw Rer 1=2 f( ): (9.9)

In view of the above transformations, governing equations (9:3) and (9:4) along with the boundary

conditions (9:5) and (9:6) are reduced to the following non-dimensional form

1 + nWe2(f 00)2 1 + We2(f 00)2 n23 f000 + 2ff00 (f0)2 Ahf0 + f00i = 0; (9.10)

2

00 + Pr(2f0 f0 ) Pr n 1

f(0) = S; f0(0) = 1; (0) = 1; (9.12)

f0(1) ! 0; (1) ! 0; (9.13)

where prime denotes di⁄erentiation with respect to the local Weissenberg

number, Pr the Prandtl number, the dimensionless parameter which measures

the unsteadiness and the constant mass transfer parameter with S > 0 for suction and

S < 0 for injection and the Eckert number.

The physical quantities of prime engineering interest are the local skin friction coefcient Cfr and the

local Nusselt number Nur which are given by

(9.14)

where w and qw are the wall shear stress and wall heat ux, respectively, having the

following expressions

: (9.15)

From Eqs. (9:8) and (9:14); nally Eq. (9:15) converted as

Re Rer 1=2 Nur = 0(0): (9.16)

9.2 Results Presentation

The concentration of current analysis is to study the problem of axisymmetric of Carreau uid

towards unsteady radially surface. The system of Eqs. (9:10) and (9:11) with conditions Eqs.

(9:12) and (9:13) are executed with the help of bvp4c solver. The e⁄ects of interesting

parameters like Prandtl number, mass transfer parameter, unsteadiness parameter, power law

index, Eckert number and local Weissenberg number are examined. The variations in velocity and

temperature elds are also measured graphically. The variations of Re and Rer 1=2 Nur are

discussed in tabular way via Tables 9:1 and 9:2 for steady and unsteady cases. Note that Eq.

(9:8) produces such parameters which depends upon spatial/temporal quantities. It means

current problem has local approximation.

Table 9:1 indicates that Re is a growing function of mass transfer parameter

for both steady and unsteady cases. Additionally Re depreciates by growing values of local

Weissenberg number for steady and unsteady cases. Furthermore, Re for the steady case

is less than that for the unsteady case. Table 9:2 reveals that mass transfer parameter and Prandtl

number enhance the Rer 1=2 Nur for steady and unsteady cases. However, on uplifting values of

local Weissenberg number, the Rer 1=2 Nur decline in shear thinning liquid but uplifts in shear

thickening liquids for both cases. Also Rer 1=2 Nur for steady case is less than that for unsteady

case. Furthermore Rer 1=2 Nur is a declining function of Eckert number.

The variation in velocity and temperature elds corresponding unsteadiness parameter is depicted

through Figs. 9:1(a) and 9:1(b). These variations show that velocity and temperature are

declining functions of unsteadiness parameter. Also uplifting values of unsteadiness parameter

decreases momentum and thermal boundary layer thicknesses. Further, momentum boundary

layer thickness is thicker in shear thickening liquid than that of shear thinning liquid. However,

quite opposite is true for thermal boundary layer thickness. The impact of mass transfer

parameter on velocity and temperature elds is reported through Figs. 9:2(a) and 9:2(b). On

evidence of Figs. that a grow in mass transfer parameter responds a decrease in velocity and

temperature elds. However, momentum and thermal boundary layer thicknesses indicate

decreasing behavior for improving mass transfer parameter. Actually, resistance occurs to the

uid ow and uid velocity due to suction.

Ehancement of power law index enhances velocity proles and declines temperature eld as indicated via Figs. 9:3(a) and 9:3(b). Physically, enhancement of power law index helps

to decrease resistive force. Also momentum and thermal boundary layer thickness have quite

opposite behavior for growing power law index.

The variation of velocity and temperature elds with local Weissenberg number is portroyed via

Figs. 9:4(a) and 9:4(b). Elevation of local Weissenberg number is to boost velocity and

temperature elds in shear thinning liquid but opposite behavior is examined in shear thickening

liquid. However, local Weissenberg number has tendency to grow momentum boundary layer

thickness in shear thinning liquid.

Fig. 9:5 portroys the variation of Prandtl number with temperature proles. It shows that ination

in Prandtl number lowers temperature and thermal boundary layer thickness in both cases.

Actually enhancement in Prandtl number responds to low thermal conductivity and as a result

diminishes thermal boundary layer thickness.

Fig. 9:6 demonstrates the impact of Eckert number on temperature eld. It shows that temperature is

growing function of Eckert number.

Table 9:1: Variations of Re for some values of A; S; We and n.

Re

Parameters steady case (A = 0) unsteady case (A = 0:2)

S We n = 0:5 n = 1:5 n = 0:5 n = 1:5

0:2 2:0 1:119129 1:614836 1:152785 1:676683

0:4

1:347501 1:890583 1:377471 1:948162

0:6

1:613982 2:186952 1:640031 2:240451

0:8

1:913899 2:500332 1:936121 2:550076

0:2 2:0 1:119129 1:614836 1:152785 1:676683

4:0 0:950219 1:781116 0:977609 1:850448

6:0 0:861721 1:899110 0:885892 1:973780

8:0 0:805068 1:991355 0:827130 2:070245

Table 9:2: Variations of Rer 1=2 Nur for some values of We;A;Pr;S and n.

Rer 1=2 Nur

Parameters steady case (A = 0) unsteady case (A

Pr S We Ec n = 0:5 n = 1:5 n = 0:5 n = 1:5

0:72 0:2 2:0 0:2 0:92432 1:06309 0:97643 1:10823

1

1:18143 1:33451 1:23418 1:38254

3

2:59035 2:76978 2:65366 2:83765

10

6:10549 6:28025 6:19358 6:38878

0:72 0:2 2:0 0:2 0:92432 1:06309 0:97643 1:10823

0:4

1:05781 1:22960 1:10687 1:27112

0:6

1:21218 1:41117 1:25748 1:44913

0:8

1:38707 1:60545 1:42834 1:64011

0:72 0:2 2:0 0:2 0:92432 1:06309 0:97643 1:10823

4:0

0:85435 1:09335 0:91293 1:13618

6:0

0:80908 1:11055 0:87282 1:15204

8:0

0:77587 1:12190 0:84381 1:16254

0:72 0:2 2:0 0:0 1:00350 1:16080 1:05616 1:20581

1:0 0:60761 0:67225 0:65748 0:71791

2:0 0:21173 0:18371 0:25879 0:23000

3:0 0:18416 0:30484 0:13989 0:25790

0 1 2 3 4 5 6 7 0 1 2 3 4

5 6 7 η η

Fig. 9:1 : Variation of f 0( ) and ( ) with A:

0 1 2 3 4 5 6 7 0 1 2 3 4

5 6 7 η η

Fig. 9:2 : Variation of f 0( ) and ( ) with S:

0 1 2 3 4 5 6 7 0 1 2 3 4

5 6 7 η η

Fig. 9:3 : Variation of f 0( ) and ( ) with n:

0 1 2 3 4 5 6 7 0 1 2 3 4

5 6 7 η η

Fig. 9:4 : Variation of f 0( ) and ( ) with We.

Fig. 9:5 : Variation of ( ) with Pr:

Fig. 9:6 : Variation of ( ) with Ec.

Chapter 10

E⁄ects of Magnetic Field and Partial

Slip on Unsteady Axisymmetric

Flow of Carreau Nanouid over a

Radially Stretching Surface

The unsteady magnetohydrodynamic (MHD) axisymmetric ow of Carreau nanouid over a radially

stretching sheet is investigated numerically in this chapter. Recently devised model for nanouid

namely Buongiornos model incorporating the e⁄ects of Brownian motion and thermophoresis is

adopted here. Additionally, partial velocity slip and convective boundary condition are

considered. Mathematical problem is modeled with the help of momentum, energy and

nanoparticles concentration equations using suitable transformation variables. The numerical

solutions for the transformed highly nonlinear ordinary di⁄erential equations are computed. For

numerical computations, Runge-Kutta Felhberg integration scheme is adopted. E⁄ects of involved

controlling parameters on the temperature and nanoparticles concentration are examined.

Numerical execution for

Nusselt and Sherwood numbers are also performed. It is interesting to note that the strong magnetic eld grows thermal and concentration boundary layer thicknesses. Additionally, the local Nusselt and Sherwood numbers depreciate by improving values of unsteadiness parameter, magnetic parameter, velocity slip parameter and thermophoresis parameter.

10.1 Statement of the Problem

The problem of unsteady axisymmetric two-dimensional ow of an incompressible Carreau

nanouid towards radially stretching sheet in the presence of time dependent magnetic eld is

considered. The surface is stretched in radial direction having stretching

velocity . The occurance of ow is in plane z > 0. We assume cylindri-

cal polar coordinates (r;;z) for mathematical interpretation. A non-uniform transverse

magnetic eld of strength is implemented in z direction, where B0 is

a constant related to magnetic eld strength (see Fig. 10:1). The magnetic Reynolds number is

taken to be small enough so that the induced magnetic eld can be neglected. Recently devised

model for nanouid incorporating the e⁄ects of Brownian motion and thermophoresis is utilized.

Additionally, the velocity partial slip condition at the surface is also implemented. A heated uid

under the surface of the sheet with temperature Tw is used to change the temperature of the

sheet by convective heat transfer mode which provides the heat transfer coe¢ cient hf. Moreover,

the surface of the sheet is at constant concentration Cw with Cw > C1:

Fig. 1: Flow conguration.

For the unsteady two-dimensional axisymmetric ow, nanoparticles concentration, temperature and

velocity elds are choosen in the following manner (cf. Chapter 9)

C = C (r;z;t); T = T(r;z;t); V = [u(r;z;t); 0; w(r;z;t)]: (10.1)

Using boundary layer analysis and the aforesaid assumptions, the governing equations for the

Carreau nanouid in the presence of time dependent magnetic eld can be read

as

(10.2)

(10.4)

(10.5)

where (u;w) represent the velocity components in (r;z) directions, respectively, the material constant, n

the power law index, the kinematic viscosity, the uid density,

= (c)p =(c)f the ratio of the e⁄ective heat capacity of the nanoparticle to the e⁄ective heat capacity

of the base uid, the thermal di⁄usivity with k the thermal conductivity, Cp the specic heat,

DT the thermophoresis di⁄usion coe¢ cient, DB the Brownian di⁄usion coe¢ cient and t the time.

The boundary conditions for velocity, temperature and nanoparticles concentration are

(10.6) u ! 0; T ! T 1; C ! C1 as z! 1: (10.7)

where T1 and C1 are the temperature and concentration at innity, respectively. Additionally, the velocity

partial slip condition is assumed to be of the form

(10.8)

where l is the slip length having dimension of length.

The suitable variables can be expressed in a manner

(10.9)

where Stokes stream function posesses . The velocity compo-

nents are represented in following way

u = Uw f0( ); w = 2Uw Rer 1=2 f( ): (10.10)

Substitution of Eqs. (10:9) into Eqs. (10:3); (10:4) and (10:5) yealds the following nonlinear ordinary di⁄erential equations

00 + Pr (10.12)

(10.13)

n 1 f(0) = 0; f0(0) = 1 + Lf00(0) 1 + We2(f00(0))2 2 ; 0(0) = (1 (0)); (0) = 1;

(10.14)

f0(1) ! 0; (1) ! 0; (1) ! 0; (10.15)

The dimensionless parameters appearing in Eqs. (10:11) (10:13) are unsteadiness parameter A,

Schmidt number Sc, Weissenberg number We, thermophoresis parameter Nt, the generalized

Biot number ; Prandtl number Pr; Brownian motion parameter Nb, magnetic parameter M and

velocity slip parameter L. Note that L = 0 corresponds to no slip case. They are respectively dened

as

Pr = (10.16)

(10.17)

The important mechanisms of ow, heat and mass transfer are Cfr; Nur and Shr which are written as

(10.18)

where w; qw and qm are

Using Eqs. (10:9); (10:18) and (10:19); the drag, heat and mass transfer rates get the following form:

Re Rer 1=2 Nur = 0(0); Rer 1=2 Shr = 0(0):

(10.20) where Rer = rUw is the local Reynolds number.

10.2 Numerical Procedure

In general, it is di¢ cult to nd the exact solution of the system of highly non-linear ordinary

di⁄erential Eqs. (10:11) (10:13) with the boundary conditions (10:14) and (10:15). Therefore,

the partially coupled highly nonlinear ordinary di⁄erential equations involving momentum,

energy and nanoparticles concentration along with the boundary conditions are solved

numerically by utilizing the Runge-Kutta Felhberg integration scheme. This scheme is adopted to

solve the IVP in a way

: (10.21)

In this scheme, the di⁄erential Eqs. (10:11) (10:13) are rst converted into a system of seven rst

order di⁄erential equations. To solve this system by adopting RK45 scheme, we need seven initial

conditions but three initial conditions each in f( ); ( ) and ( ) are unknowns. These three end

conditions are used to develop three unknown initial conditions with the help of shooting

scheme. An important factor of this scheme is to choose the most suitable nite value of : Thus,

we have made some initial guesses 1

with the help of Newton-Raphson method for missing conditions so that the conditions f0(1) = 0; (1) =

0 and (1) = 0 are satised. In the current problem, the value of

= is taken to be 10 and step-size is taken to be = h = 0:01 with relative error 1 tolerance 10 5: Consequently, the non-linear equations and the corresponding boundary conditions are

converted into a system of rst order equations as

(10.22)

y40 = y5; y50 = Pr (10.23)

(10.24)

where the unknowns are stated as

f = y1; f0 = y2; f00 = y3; = y4; 0 = y5; = y6;

with the boundary conditions taking the following form

0 = y7; (10.25)

y5(0) = (1 y4(0)); y6(0) = 1; (10.26)

y2(1) ! 0; y4(1) ! 0;y6(1) ! 0: (10.27)

10.3 Code Validation

For the verication of current numerical study, the numerical results of skin friction coe¢ cient are

compared with those investigated by Ariel [80] for several values of velocity

slip parameter L when We = A = M = 0 and n = 1 (see Table 10:1): Additionally,

the numerical results of skin friction coe¢ cient are also compared with those reported by

Makinde et al: [81] for some values of magnetic parameter M when We = A = L = 0 and n = 1

(see Table 10:2): It is observed that present numerical investigation is in excellent agreement

with the existing literature.

10.4 Discussion of Numerical Results

A numerical analysis for unsteady axisymmetric ow of Carreau nanouid over a convectively

heated radially stretching sheet in the presence of time dependent magnetic eld is performed.

The numerical computations have been performed by adopting an e⁄ective numerical scheme

namely the shooting technique along with fourth-fth order Runge-Kutta integration scheme for

several values of involved parameters namely, magnetic parameter, local Weissenberg number,

power law index, velocity slip parameter, unsteadiness parameter, generalized Biot number,

Brownian motion parameter, Prandtl number thermophoresis parameter and Schmidt number.

The numerical execution for Re , Rer 1=2 Nur and Rer 1=2 Shr have been performed through Tables

10:3 to 10:5. From these Tables, it is noted that Re is an enhancing function of magnetic and

unsteadiness parameters. Additionally, the magnitude of local skin friction coe¢ cient decreases for

growing values of velocity slip parameter in both cases. Furthermore, Re depreciates for shear

thinning liquid but grows for shear thickening liquid. On the basis of these Tables, it is noticed that both

the local Nusselt and Sherwood numbers depreciate for enhancing the values of unsteadiness

parameter, magnetic parameter, velocity slip parameter and thermophoresis parameter in both cases.

Also Rer 1=2 Nur depreciates with the increment of Brownian motion parameter but opposite behavior

can be noticed in local Sherwood number. Additionally, the generalized Biot number is an enhancing

function of Rer 1=2 Nur but diminishing function of the local Sherwood number in both cases.

Furthermore, both the local Nusselt and Sherwood numbers decrease with growing local Weissenberg

number for shear thinning liquid but reverse behavior can be seen in shear thickening liquid. We have

also conducted a comparative study of numerical values of local skin friction coe¢ cient, local Nusselt

number and local Sherwood number between two di⁄erent numerical approaches namely shooting RK45

and bvp4c and found to be in excellent agreement. It is important to state Eq. (10:9) develope

parameters which dependon temporal/spatial quantities. It means current problem is local

approximation.

Figs. 10:2(a) to 10:2(d) depict the variation of temperature ( ) and nanoparticles concentration

( ) with di⁄erent values of unsteadiness parameter A. We see that temperature and nanoparticle

concentration uplift for improving unsteadiness parameter. Additionally, the associated thermal

and nanoparticle concentration boundary layer thicknesses are also the enhancing function of

unsteadiness parameter. Infact, enhancement in unsteadiness has the tendency to improve the

thermal as well as concentration boundary layer thicknesses.

Figs. 10:3(a) to 10:3(d) indicate the variation of velocity f0( ); temperature ( ) and nanoparticles

concentration ( ) with local Weissenberg number. On the behalf of these Figs., it is noted that

the velocity of uid decreases in shear thinning case but increases in shear thickening case for

growing values of local Weissenberg number. From these Figs., temperature as well as

nanoparticles concentration improve for improving Weissenberg number in shear thinning liquid

but reverse behavior can be noticed for shear thickening liquid. Furthermore, the associated

thermal and concentration boundary layer thicknesses uplift in shear thinning uid whereas

opposite results can be revealed in shear thickening uid.

The variation of velocity f0( ); temperature ( ) and nanoparticles concentration ( ) with di⁄erent

values of magnetic parameter M are depicted through Figs. 10:4(a) to 10:4(d) in both cases. It

is observed that the velocity of uid depreciates by improving values of magnetic parameter. On

the basis of these Figs., it can also be predicted that temperature, nanoparticle concentration

and their associated thermal and concentration boundary layer thicknesses enhance for

improving values of magnetic parameter. Note that M = 0 is the case of hydrodynamic ow and M

> 0 demonstrates the hydromagnetic ow. Infact, the strong magnetic eld grows thermal and

concentration boundary layer

thicknesses. Figs. 10:5(a) to 10:5(d)elucidate the variation of f0( ); ( ) and ( ) with di⁄erent values of velocity

slip parameter in both cases regarding shear thinning and shear thickening uids. From these Figs.,

it is clear that the velocity of uid diminishes by improving values of velocity slip parameter. From

these Figs., it can also be examined that temperature, nanoparticle concentration and their

related thermal and concentration thicknesses are increasing functions of the generalized

velocity slip parameter. Physically, with the increased velocity slip, as a result of depreciate in the

tendency of uid to remove the heat away from the plate an improve in temperature and

nanoparticles concentration is

noticed.

Figs. 10:6(a) to 10:6(d) reveal the variation of ( ) and ( ) with di⁄erent values of generalized Biot

number. On the evidence of these Figs., it is clear that the temperature, nanoparticles

concentration and their associated thermal and concentration boundary layer thicknesses grow

for improving values of generalized Biot number. The generalized Biot number is the ratio of

internal thermal resistance of a solid to boundary layer thermal resistance. When = 0; the surface

of sheet is totally insolated. The internal thermal resistance of the surface of sheet is very large

and there is no convective heat transfer from the surface of sheet to the cold uid far away from

the sheet.

Figs. 10:7(a) to 10:7(d) represent the variation of temperature ( ) and nanoparticles

concentration ( ) with di⁄erent values of Nb. On the evidence of these Figs., it is clear that the

temperature and its associated thermal boundary layer thickness grows for growing values of

Brownian motion parameter but opposite results can be observed in nanoparticle concentration

eld. Brownian motion appears due to the presence of nanoparticles and resulted in the

decrement of the nanoparticles concentration thickness.

The variation of nanoparticles concentration ( ) with di⁄erent values of thermophoresis

parameter can be visualized through Figs. 10:8(a) to 10:8(d). Noted that nanoparticle

concentration and its related concentration boundary layer thickness grow with the

enhancement of Nt. Actually, temperature di⁄erence between ambient and surface enhances for

higher thermophoresis which grows the temperature and concentration of the uid. Physically, the

thermophoresis force grows with the improvement of which tends to move nanoparticles from

hot portion to cold portion and hence enhances the magnitude of the nanoparticle concentration

prole.

The variation of nanoparticles concentration ( ) with di⁄erent values of Schmidt number can be

observed through Figs. 10:9(a) to 10:9(d). Nanoparticle concentration and its related

concentration boundary layer thickness depreciate with the enhancement of Schmidt number.

In fact, Schmidt number is the ratio of viscosity to mass di⁄usivity. When Schmidt number

increases then mass di⁄usivity decreases and results in reduction in uid concentration.

Table 10:1: Computation results of f00(0) for several values of the velocity slip

parameter L when We = A = M = 0 and n = 1:

f00(0)

L Exact [80] HPM [80]

Perturbation

[80] Asymptotic [80] Present results

0:0 1:173721 1:178511 1:173721

1:173734

0:01 1:153472 1:157311 1:153481

1:153485

0:02 1:134017 1:136998 1:134090

1:134031

0:05 1:079949 1:080820 1:081010

1:079964

0:1 1:001834 1:000308 1:009522

1:001850

0:2 0:878425 0:874453 0:930213

0:878444

0:5 0:650528 0:645304 1:201623 1:529918 0:65055

1:0 0:462510 0:458333

0:574163 0:462547

2:0 0:299050 0:296534

0:310753 0:299099

5:0 0:149393 0:148454

0:149590 0:149455

10:0 0:082912 0:082532

0:082833 0:082974

20:0 0:044368 0:044228

0:044337 0:044423

50:0 0:018732 0:018698

0:018727 0:018770

100:0 0:009594 0:009583

0:009593 0:009619

Table 10:2: A comparison of computation results of f00(0) for several values of the magnetic parameter M when We = A = L = 0 and n = 1:

M2 Makinde et al: [81] Present results

0:0 1:17372 1:17372

0:5 1:36581 1:36581

1:0 1:53571 1:53571

2:0 1:83049 1:83049

3:0 2:08484 2:08485

Table 10:3: Computation of Re for selected values of A; M; We and L:

Parameters Re (bvp4c) results Re (shooting) results

A M We L n = 0:5 n = 1:5 n = 0:5 n = 1:5

0:1 0:2 1:0 0:1 0:9752046 1:087644 0:9752046 1:087644

0:2

0:9944376 1:110897 0:9944376 1:110897

0:3

1:013416 1:134013 1:013416 1:134013

0:2 0:0 1:0 0:1 0:9834754 1:097047 0:9834754 1:097047

0:4

1:026272 1:151427 1:026272 1:151427

0:8

1:140041 1:300083 1:140041 1:300083

0:2 0:2 0:2 0:1 1:05672 1:062799 1:05672 1:062799

1:0

0:9944376 1:110897 0:9944376 1:110897

1:6

0:9270093 1:154088 0:9270092 1:154088

0:2 0:2 1:0 0:2 0:8867801 0:9594447 0:88678 0:9594447

0:3 0:800512 0:84874 0:800512 0:84874

0:4 0:729968 0:7635109 0:729968 0:7635109

Table 10:4: Computations of Rer 1=2 Nur for selected values of A; M; We; L; ; Nt and Nb when Pr =

2:5 and Sc = 2:

Parameters Rer 1=2 Nur(bvp4c) results

Rer 1=2 Nur(shooting)

results

A M We L

Nt Nb n = 0:5 n = 1:5 n = 0:5 n = 1:5

0:1 0:2 1:0 0:1 0:1 0:2 0:1 0:092321 0:092519 0:0923206 0:0925186

0:2

0:092075 0:092306 0:0920749 0:0923062

0:3

0:091793 0:092067 0:091793 0:0920672

0:2 0:0 1:0 0:1 0:1 0:2 0:1 0:092107 0:092330 0:0921072 0:0923301

0:4

0:091978 0:092235 0:0919776 0:0922349

0:8

0:091587 0:091956 0:0915874 0:0919561

0:2 0:2 0:2 0:1 0:1 0:2 0:1 0:092209 0:092221 0:0922089 0:092221

1:0

0:092075 0:092306 0:0920749 0:0923062

1:6

0:091908 0:092369 0:0919083 0:092369

0:2 0:2 1:0 0:2 0:1 0:2 0:1 0:091719 0:091887 0:0917189 0:0918872

0:3

0:091385 0:091512 0:0913845 0:0915119

0:4

0:091068 0:091168 0:0910685 0:0911679

0:2 0:2 1:0 0:1 0:1 0:2 0:1 0:092075 0:092306 0:0920749 0:0923062

0:2

0:170379 0:171183 0:170379 0:171183

0:3

0:237523 0:239105 0:237523 0:239105

0:2 0:2 1:0 0:1 0:1 0:2 0:1 0:092075 0:092306 0:0920749 0:0923062

0:3

0:092033 0:092266 0:0920326 0:0922657

0:4

0:091990 0:092225 0:0919898 0:0922246

0:2 0:3 1:0 1:2 0:1 0:2 0:1 0:092075203

0:092306 0:0920749 0:0923062

0:2 0:091131 0:091386 0:0911308 0:0913855

0:3 0:090041 0:090322 0:0900413 0:0903222

Table 10:5: Computations of Rer 1=2 Shr for selected values of A; M; We; L; ; Nt and Nb when Pr

= 2:5 and Sc = 2:

Parameters Rer 1=2 Shr(bvp4c) results

Rer 1=2 Shr(shooting)

results

A M We L

Nt Nb n = 0:5 n = 1:5 n = 0:5 n = 1:5

0:1 0:2 1:0 0:1 0:1 0:2 0:1 1:074825 1:115492 1:07482 1:11549

0:2

1:028389 1:073354 1:02839 1:07335

0:3

0:977571 1:027777 0:977571 1:02778

0:2 0:0 1:0 0:1 0:1 0:2 0:1 1:034539 1:078220 1:03454 1:07822

0:4

1:010224 1:059022 1:01022 1:05902

0:8

0:941778 1:005378 0:941778 1:00538

0:2 0:2 0:2 0:1 0:1 0:2 0:1 1:053693 1:056039 1:05369 1:05604

1:0

1:028389 1:073354 1:02839 1:07335

1:6

0:998357 1:087086 0:998357 1:08709

0:2 0:2 1:0 0:2 0:1 0:2 0:1 0:975535 1:006120 0:975535 1:00612

0:3

0:929771 0:951642 0:929771 0:951642

0:4

0:889646 0:905889 0:889646 0:905889

0:2 0:2 1:0 0:1 0:1 0:2 0:1 1:028389 1:073354 1:02839 1:07335

0:2

0:929443 0:976614 0:929443 0:976614

0:3

0:845049 0:893804 0:845049 0:893804

0:2 0:2 1:0 0:1 0:1 0:2 0:1 1:028389 1:073354 1:02839 1:07335

0:3

0:970229 1:016750 0:970228 1:01675

0:4

0:912315 0:960404 0:912315 0:960404

0:2 0:3 1:0 1:2 0:1 0:2 0:1 1:028389205

1:073354 1:02839 1:07335

0:2 1:090037 1:133641 1:09004 1:13364

0:3 1:110817 1:153984 1:11082 1:15398

0 1 2 3 4 0 1 2 3 4 η η

0 1 2 3 4 0 1 2 3 4 η η

Fig. 10:2: Variation of temperature ( ) and nanoparticles concentration ( ) proles with

di⁄erent values of A:

η η

0 1 2 3 0 1 2 3 η η

0 1 2 3 0 1 2 3 η η

Fig. 10:3:

concentration ( ) proles with di⁄erent values of We:

η η

0 1 2 3 0 1 2 3 η η

0 1 2 3 4 0 1 2 3 4 η η

Fig. 10:4:

concentration ( ) proles with di⁄erent values of M:

η η

0 1 2 3 0 1 2 3 η η

0 1 2 3 0 1 2 3 η η

Fig. 10:5 :

concentration ( ) proles with di⁄erent values of L:

0 1 2 3 0 1 2 3 η η

0 1 2 3 0 1 2 3 η η

Fig. 10:6: Variation of temperature ( ) and nanoparticles concentration ( ) proles with

di⁄erent values of :

0 1 2 3 0 1 2 3 η η

0 1 2 3 0 1 2 3 η η

Fig. 10:7: Variation of temperature ( ) and nanoparticles concentration ( ) proles

with di⁄erent values of Nb:

0 1 2 3 0 1 2 3 η η

Fig. 10:8: Variation of nanoparticle concentration ( ) proles with di⁄erent values

of Nt:

0 1 2 3 0 1 2 3 η η

Fig. 10:9: Variation of nanoparticles concentration ( ) proles with di⁄erent values of Sc:

Chapter 11

Conclusions and Recommendations

11.1 Conclusions

The objective of present thesis was to contribute a meaningful investigation

in the eld of generalized Newtonian uids especially the Carreau uid in the

presence of nanoparticles. In this thesis, we have studied theoretically the

heat and mass transfer of Carreau nanouid over di⁄erent moving surfaces.

Buongiornos nanouid model was adopted to incorporate the thermophoresis

and Brownian motion e⁄ects. Numerical solutions were achieved for the

boundary value problems and the results were compared with the existing

literature. E⁄ects of the dimesionless parameters like as the Weissenberg

number, unsteadiness parameter, Eckert number, mass transfer parameter,

Reynolds number, power law index, wedge angle parameter, Prandtl number,

magnetic parameter, Brownian motion parameter, Lewis number,

thermophoresis parameter, velocity ratio parameter, thermal Biot number,

concentration Biot number, melting parameter, Schmidt number, heat

generation/absorption parameter, radiation parameter, temperature ratio

paramere and velocity slip parameter on the velocity, temperature,

concentration elds. The skin friction coe¢ cient, Nusselt and Sherwood

numbers have also been examined. Thus, the key ndings of this investigation

are summarized as:

The dimensionless velocity as well as temperature proles were decreased for

increasing unsteadiness and mass transfer parameters.

An increase of Weissenberg number resulted in a decrement in the uid

velocity in shear thinning uid and opposite e⁄ects in shear thickening uid were

seen. However, quite the opposite was true for the temperature eld.

The dimensionless temperature was enhanced by uplifting the Brownian motion

and thermophoresis parameters.

Nusselt and Sherwood numbers were declining functions of thermophoresis

parameter.

The ow, temperature and concentration elds were greatly a⁄ected by the

Hartmann number, wedge angle and unsteadiness parameters.

Fluid velocity was smaller for static wedge when compared to the stretching

wedge. However, qualitatively quite the opposite trend was observed for the

temperature and concentration elds.

The temperature and nanoparticles concentration proles were diminished with

increment of melting parameter.

The temperature and nanoparticles concentration were growing functions of heat

source/sink parameter with melting e⁄ects.

The dimensionless temperature was enhancing function of the thermal

conductivity parameter.

The nanoparticle concentration and concentration boundary layer thickness were

decreased by increasing the values of the Schmidt number.

The momentum boundary layer thickness was controlled for stronger

magnetic eld for both cases of stretching and shrinking cylinder in shear

thinning uid as well as shear thickening uid. Further, results showed that the

e⁄ects of temperature ratio parameter was to increase the temperature and

thermal boundary layer thickness in all cases. But quite the reverse was

observed for augmented radiation parameter.

Improving values of power law index depreciated temperature eld butenhanced

velocity and its related thicknes.

The local Nusselt number and the magnitude of skin friction coe¢ cient for the

steady ow (A = 0) were less than that for the unsteady ow.

The temperature, nanoparticles concentration and their associated thermal

and concentration boundary layer thicknesses were enhanced by increasing

values of generalized Biot number in shear thickenings and shear thinning

uids.

11.2 Future Recommendations

The study undertaken in present thesis has spaned many problems regarding

unsteady ow, heat and mass transfer for Carreau uid in the presence of

nanoparticles. But several problems are still open for further investigation.

The present study can be extended for curved stretching ows as well as

rotating ows of Carreau uid. Additionally, the multiple solutions for such ows

can be investigated. The present study was restricted to numerical solutions

for unsteady ows of Carreau uid via bvp4c and shooting methods. The

numerical solutions for the extension work can be searched via nite di⁄erence

method and nite element method.

Chapter 12

Bibliography

Bibliography

[1] H. Masuda, A. Ebata, K. Teramae and N. Hishinuma, Alteration of thermal

conductivity and viscosity of liquid by dispersing ultra-ne particles, Netsu

Bussei, 7(4) (1993) 227 233:

[2] S.K. Das, S.U.S. Choi, W. Yu and T. Pradeep, Nanouids: Science and

Technology, Wiley, New Jersey, 2007:

[3] S.U.S. Choi and J.A. Eastman, Enhancing thermal conductivity of uids with

nanoparticles, ASME Pub: Fed., 231 (1995) 99 106:

[4] J. Buongiorno and L.W. Hu, Nanouid coolants for advanced nuclear power

plants, Proceedings of ICAPP, 5(5705) (2005) 15 19:

[5] J. Buongiorno, Convective transport in nanouids, J. Heat Transf., 128(3)

(2006) 240 250:

[6] W.A. Khan and I. Pop, Boundary layer ow of a nanouid past a stretching sheet,

Int. J. Heat Mass Transf., 53 (2010) 2477 2483:

[7] N. Bachok, A. Ishak and I. Pop, Unsteady boundary-layer ow and heat

transfer of a nanouid over a permeable stretching/ shrinking sheet, Int. J.

Heat Mass Transf., 55 (2012) 2102 2109:

[8] K.L. Hsiao, Nanouid ow with multimedia physical features for conjugate

mixed convection and radiation, Comput. Fluids, 104 (2014) 1 8:

[9] M. Turkyilmazoglu, Analytical solutions of single and multi-phase models for

the condensation of nanouid lm ow and heat transfer, Eur. J. Mech. B/Fluids,

53 (2015) 272 277:

[10] J. Serna, Heat and mass transfer mechanisms in nanouids boundary layers, Int. J.

Heat Mass Transf., 92 (2016) 173 183:

[11] P.J. Carreau, Rheological equations from molecular network theories, Trans.

Soc. Rheol, 116 (1972) 99 127:

[12] P. Hsu, S.H. Hung and H.Yu, Electrophoresis of a sphere at an arbitrary

position in a spherical cavity lled with Carreau uid, J. Colloid Interface Sci.,

280 (2004) 256 263:

[13] N. Ali and T. Hayat, Peristaltic motion of a Carreau uid in an asymmetric chennel,

Appl. Math. Comput., 193 (2007) 535 552:

[14] M.S. Tshehla, The ow of Carreau uid down an incline with a free surface, Int. J

. Phys. Sci., 6 (2011) 3896 3910: [15] B.I. Olajuwon and Bakai Ishola, Convection heat and mass transfer in a

hydromgnetic Carreau uid past a vertical porous plate in presence of

thermal radiation and thermal di⁄usion, Thermal Science, 15(2) (2011) 241

252:

[16] T. Hayat, S. Asad, M. Mustafa and A. Alsaedi, Boundary layer ow of Carreau

uid over a convectively heated stretching sheet, Appl. Math. Comput., 246

(2014)

12 22:

[17] H. Schlichting and K. Gersten, Boundary Layer Theory, Eighth Revised Ed.,

Springer-Verlag, Berlin, 2000:

[18] L.G. Leal, Advanced Transport Phenomena: Fluid Mechanics and Convective

Transport Processes, Cambridge University Press, New York, 2007:

[19] K.A. Yih, Uniform suction/blowing e⁄ect on forced convection about a

wedge: uniform heat ux, Acta Mech., 128 (1998) 173 181:

[20] A. Ishaq, R. Nazar and I. Pop, MHD boundary-layer ow of a micropolar uid

past a wedge with variable wall temperature, Acta Mech., 196 (2008) 75

86:

[21] M.S. Alam, T. Islam and M.M. Rahman, Unsteady hydromagnetic forced

convective heat transfer ow of a micropolar uid along a porous wedge with

convective surface boundary condition, Int. J. Heat Tech., 33(2) 2015: ;

[22] A. Munir, A. Shahzad and M. Khan, Convective ow of Sisko uid over a wedge

with viscous dissipation, J. Braz. Soc. Mech. Sci. Eng., 38 (2016) 581 587:

[23] C.S.K. Raju and N. Sandeep, Falkner-Skan ow of a magnetic-Carreau uid past

a wedge in the presence of cross di⁄usion e⁄ects, Eur. Phys. J. Plus, 131

(2016) 267:

[24] M. Turkyilmazoglu, Slip ow and heat transfer over a specic wedge: an exactly

solvable Falkner-Skan equation, J. Eng. Maths, 92 (2015) 73 81:

[25] M. Epstein and D.H. Cho, Melting heat transfer in steady laminar ow over a

ate plate, J. Heat Transf., 98 (1976) 531 533:

[26] 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 medium, Int. J. Heat Mass Transf., 50(2007) 3026 3034:

[27] A. Ishaq, R. Nazar, N. Bachok and I. Pop, Melting heat transfer in steady

laminar ow over a moving surface, Heat Mass Transf., 46(2010) 463 468:

[28] N.A. Yacob, A. Ishaq and I. Pop, Melting heat transfer in boundary layer

stagnation point ow towards a stretching/shrinking sheet in a micropolar

uids, Comput. Fluids, 47(2011) 16 21:

[29] T. Hayat, M. Mustafa, Z. Iqbal and A. Alsaedi, Stagnation-point ow of couple

stress uid with melting heat transfer, Appl. Math. Mech. -Eng. Ed., 34(2)

(2013) 167 176:

[30] B.C. Prasannakumara, B.J. Gireesha and P.T. Manjunatha, Melting

phenomenon in MHD stagnation-point ow of dusty uid over a stretching

sheet in the presence

of thermal radiation and non-uniform heat source/sink, Int. J. Comput. Methods

Eng. Sci. Mech., 16(2015) 265 274:

[31] P.K. Kameswaran, K. Hemalatha and M.V.D.N.S. Madhavi, Melting e⁄ect on

convective heat transfer from a vertical plate embedded in a non-Darcy

porous medium with variable permeability, Adv. Powd. Tech., 27 (2016)

417 425:

[32] T. Hayat, A. Shaq, M. Imtiaz and A. Alsaedi, Impact of melting phenomenon

in the Falkner-Skan wedge ow of second grade nanouid: A revised model, J.

Mol. Liq., 215(2016) 664 670:

[33] S. Uchida and H. Aoki, Unsteady ows in a semi-innite contracting or

expanding pipe, J. Fluid Mech., 88(2) (1977) 371 387:

[34] M. Goto and S. Uchida, Unsteady ow in a semi-innite expanding pipe with

injection through wall, Trans. Jpn. Soc. Aeronaut. Space Sci., 33(1990) 14

27:

[35] N.M. Bujurke, N.P. Pai and G. Jayaraman, Computer extended series solution

for unsteady ow in a contracting or expanding pipe, IMA J. Appl. Math.,

60(2) (1998) 151 165:

[36] K. Khellaf and G. Lauriat, Numerical study of heat transfer in a non-Newtonian

Carreau-uid between rotating concentric vertical cylinders, J. Non-Newtonian Fluid

Mech., 89(2000) 45 61: [37] J. Majdalani, C. Zhou and C.A. Dawson, Two-dimensional viscous ows

between slowly expanding or contracting walls with weak permeability, J.

Biomech., 35(10) (2002) 1399 1403:

[38] T. Fang, J. Zhang, Y. Zhong and H. Tao, Unsteady viscous ow over an

expanding stretching cylinder, Chin. Phys. Lett., 12 (2011) 124707:

[39] C.Y. Wang, Fluid ow due to a stretching cylinder, Phys. Fluids, 31(1988) 466

482:

[40] Y.Y. Lok, J.H. Merkin and I. Pop, Mixed convection ow near the axisymmetric

stagnation point on a stretching or shrinking cylinder, Int. J. Ther. Sci.,

59(2012) 186 194:

[41] T. Fang, J. Zhang and Y. Zhong, Note on unsteady viscous ow on the outside

of an expanding or contracting cylinder, Commun. Nonlinear Sci. Numer.

Simulat., 17 (2012) 3124 3128:

[42] P.M. Patil, S. Roy and I. Pop, Unsteady e⁄ects on mixed convection boundary

layer ow from a permeable slender cylinder due to non-linearly power law

stretching, Computers Fluids, 56 (2012) 17 23:

[43] X. Si, L. Li, L. Zheng, X. Zhang and B. Liu, The exterior unsteady viscous ow

and heat transfer due to a porous expanding stretching cylinder, Comput.

Fluids, 105

(2014) 280 284: [44] K. Zaimi, A. Ishaq and A. Ishaq, Unsteady ow due to a contracting cylinder in

a nanouid using Buongiornos model, Int. J. Heat Mass Transf., 68 (2014)

509 513:

[45] Z. Abbas, S. Rasool and M.M. Rashidi, Heat transfer analysis due to an

unsteady stretching cylinder with partial slip condition and suction, Ain

Shams Eng. J., 6 (2015) 939 945:

[46] M. Imtiaz, T. Hayat and A. Alsaedi, Mixed convection ow of casson nanouid

over a stretching cylinder with convection boundary conditions, Adv. Powd.

Tech., 27 (2016) 2245 2256:

[47] B.C. Sakiadis, Boundary layer behavior on continuous solid surface: 1.

Boundary layer equations for two-dimensional and axisymmetric ow, AIChE

J., 7 (1961) 26 28:

[48] P.D. Ariel, Axisymmetric ow of a second grade uid past a stretching sheet,

Int. J. Eng. Sci., 39 (2001) 529 553:

[49] R.R. Martins, F.S. Silveira, M.L. Martins-Costa and S. Frey, Numerical

investigation of inertia and shear thinning e⁄ect in axisymmetric ows

Carreau uids by a Galerkin Least Squares Method, Lat. Amer. Appl. Res., 38

(2008) 321 328:

[50] M.M. Rashidi, H. Shahmohamadi and S. Dinarvand, Analytic approximate

solutions for unsteady two-dimensional and axisymmetric squeezing ows

between parallel plates, Math. Probl. Eng., Doi: 10:1155 (2008) 935095:

[51] M. Sajid, I. Ahmad, T. Hayat and M. Ayub, Series solution for unsteady

axisymmetric ow and heat transfer over a radially stretching sheet,

Commun. Nonlinear Sci. Numer. Simulat., 13 (2008) 2193 2202:

[52] I. Ahmad, M. Sajid, T. Hayat and M. Ayub, Unsteady axisymmetric ow of a

second-grade uid over a radially stretching sheet, Comput. Math. Appl., 56

(2008) 1351 1357:

[53] I. Ahmad, M. Sajid and T. Hayat, Heat transfer in unsteady axisymmetric

second grade uid, Appl. Math. Comput., 215 (2009) 1685 1695:

[54] Z. Abbas, T. Javed, M. Sajid and N. Ali, Unsteady MHD ow and heat transfer

on a stretching sheet in a rotating uid, J. Tai. Instit. Chem. Eng., 41 (2010)

644 50:

[55] B. Sahoo, E⁄ects of slip viscous dissipation and Joul heating on the MHD ow

and heat transfer of a second grade uid past a radially stretching sheet, Appl.

Math. Mech. Engl. Ed., 31 (2010) 159 173:

[56] M. Khan and A. Shahzad, On axisymmetric ow of Sisko uid over a radially

stretching sheet, Int. J. Nonlinear Mech., 47 (2012) 999 1007:

[57] M. Mustafa, J.A. Khan, T. Hayat and A. Alsaedi, Analytical and numerical

solutions for axisymmetric ow of nanouid due to non-linearly stretching

sheet, Int. J.

Nonlinear Mech., 71 (2015) 22 29: [58] P. Weidman, Axisymmetric rotational stagnation point ow impining on a

radially stretching sheet, Int. J. Nonlinear. Mech., 82 (2016) 1 5:

[59] M. Khan and Hashim, Boundary layer ow and heat transfer to Carreau uid

over a nonlinear stretching sheet, AIP Advances, 5 (2015) 107203:

[60] Shampine and L.F. Kierzenka, "Solving boundary value problems for ordinary

differential equations in MATLAB with bvp4c", Tutorial Notes (2000):

[61] S. Sharidan, T. Mahmood and I. Pop, Similarity solutions for the unsteady

boundary layer ow and heat transfer due to a stretching sheet, Int. J. Appl.

Mech. Eng., 11(3) (2006) 647 654:

[62] A.J. Chamkha, A.M. Aly and M.A. Mansour, Similarity solution for unsteady

heat and mass transfer from a stretching surface embedded in a porous

medium with suction/injection and chemical reation e⁄ects, Chem. Eng.

Commun., 197 (2010) 846 858:

[63] S. Mukhopadhyay and R.S.R. Gorla, Unsteady MHD boundary layer ow of an

upper convected Maxwell uid past a stretching sheet with rst order

constructive/destructive chemical reaction, J. Naval Architect Mar Eng., 9

(2012) 123 133:

[64] L.J. Grubka and K.M. Bobba, Heat transfer characteristics of a continuous

stretching surface with variable temperature, ASME J. Heat Transf., 107

(1985) 248 250:

[65] C.H. Chen, Laminar mixed convection adjacent to vertical continuously

stretching sheets, Heat Mass Transf., 33 (1998) 471 476:

[66] R. Sharma, E⁄ect of viscous dissipation and heat source on unsteady

boundary layer ow and heat transfer past a stretching surface embedded in

a porous medium using element free Galerkin method, Appl. Math.

Comput., 219 (2012) 976 987:

[67] R.B. Bird, C.F. Curss, R.C. Amstrong and O. Hassager, Dynamics of Polymeric

Liquids, Wiley, New York, 1987:

[68] P.J. Carreau, Rheological equations from molecular network theories, Trans.

Soc. Rheol, 116 (1972) 99 127:

[69] M. Khan, Hashim, M. Hussain and M. Azam, Magnetohydrodynamic ow of

Carreau uid over a convectively heated surface in the presence of non-linear

radiation, J. Magnetism Magnetic Material, 412 (2016) 63 68:

[70] K.R. Rajagopal, A.S. Gupta and T.Y. Na, A note on the Falkner-Skan ows of a

non-Newtonian uid, Int. J. Nonlinear Mechanics, 18(4) (1982) 313 320:

[71] B.L. Kuo, Application of the di⁄erential transformation method to the solutions of

Falkner-Skan wedge ow, Acta Mech., 164(2003) 161 174:

[72] A. Ishaq, R. Nazar and I. Pop, Moving wedge and at plate in a micropolar uid,

Int. J. Eng. Sci., 44 (2006) 1225 1236:

[73] F.M. White, Viscous Fluid Flow, McGraw-Hill, New York, USA, 1991:

[74] Hashim, M. Khan and A.S. Alshomrani, Characteristics of melting heat

transfer during ow of Carreau uid induced by a stretching cylinder, Eur. Phys.

J. E, 2017 , 40 : 8:

[75] T. Fang, J. Zhang, Y. Zhong and H. Tao, Unsteady viscous ow over an

expanding stretching cylinder, Chin. Phys. Lett., 12 (2011) 124707:

[76] M. Khan , M. Azam , AS. Alshomrani. On unsteady heat and mass transfer in

carreau nanouid ow over expanding or contracting cylinder. J Mol Liq., 231

(2017) 474 84:

[77] W.J. Smith, E⁄ect of gass radiation in the boundary layer on aerodynamic

heat transfer, J. Aer. Sci., 20 (1952) 579 580:

[78] Y.Y. Lok and I. Pop, Wangs shrinking cylinder problem with suction near a

stagnation point, Phys. Fluids, 23 (2011) 083102:

[79] M. Khan, M. Azam and A.S. Alshomrani, Numerical study of unsteady

axisymmetric ow and heat transfer in Carreau uid past a stretched surface,

Ther. Sci., DOI:

10:2298=TSCI160807132K:

[80] P.D. Ariel, Axisymmetric ow due to a stretching sheet with partial slip, Int. J.

Comput. Math. Appl., 54 (2007) 1169 1183: [81] O.D. Makinde, F. Mabood, W.A. Khan and M.S. Tshehla, MHD ow of a

variable

viscosity nanouid over a radially stretching convective surface with radiative

heat, J. Mol. Liq., 219 (2016) 624 630:

unsteady heat and mass transfer mechanisms in

Documents