Models and analysis for curvature and wall properties

Models and analysis for curvature and wall properties effects in peristalsis

By

Anum Tanveer

Department of Mathematics Quaid-i-Azam University

Islamabad, Pakistan 2018


By

Anum Tanveer

Supervised By

Prof. Dr. Tasawar Hayat




By

Anum Tanveer

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

DOCTOR OF PHILOSOPHY IN

MATHEMATICS

Supervised By

Prof. Dr. Tasawar Hayat



Dedicated to my parents

Acknowledgement Although it is just my name mentioned on the cover, many people have contributed to this research in their own particular way and for that I want to give them special thanks. My first and foremost gratitude and thanks goes to Almighty Allah for making me enable to proceed in this dissertation successfully. Sure all praise is for HIM who created us as best of HIS creations and grant us strength, health, knowledge, ability and opportunity to achieve our goals. I also express my sincere gratitude to Holy Prophet Hazrat Muhammad S.A.W. for guide us to right path by HIS teachings of patience, motivation and immense knowledge. No research is possible without infrastructure and requisite materials and resource. At the very outset, I express my deepest thanks to Quaid-I-Azam University for all the academic support to complete my degree as a PhD student. I owe my gratitude to my esteemed supervisor Prof. Dr. Tasawar Hayat for providing me this great opportunity to do my doctoral programmed under his guidance and to learn from his research expertise. His support and advice helped me in all the time of research and writing of this thesis. Similar, profound gratitude goes to Prof. Dr. Muhammad Yousaf Malik (Chairman), Prof. Dr. M. Ayub, Prof. Dr. Sohail Nadeem and Dr. Masood Khan for their valuable support during my student carrier. I wish to express my heartiest thanks and gratitude to my parents Mr. and Mrs. Shahid Tanveer, the ones who can never ever be thanked enough for the overwhelming love, kindness and care they bestow upon me. They supported me financially as well as morally and without their proper guidance it would not been possible for me to complete my

higher education. I also owe my gratitude to my siblings Rubab, Sara, Maria, Huma and Baqir for being there for me. I would not be who am I today without you all. Special mention goes to my dear husband Dr. Taimoor Salahuddin for his guidance, support and encouragement during this work. The journey of Ph.D was just like climbing a high peak step by step. His presence taught me to get through the hardship and frustration. He helped me keep things in perspective. I am indebted to him for giving me this feeling of fulfillment in life. Completing this work would have been all the more difficult without a support and friendship. I have great pleasure in acknowledging my gratitude to my colleagues and fellow research scholars at QAU. My profound thanks and best wishes goes to my friends Ms. Sadia Ayub, Ms. Hina Zahir and Mrs. Mehwish Masood. I wish them all the luck in completion of their PhD degrees soon. I also want to take a moment to thank my colleagues Dr. Maimona Rafiq, Dr. Maria Imtiaz, Sumera Qayyum, Madiha Rashid, Sajid Qayyum and Dr. M. Waqas for their help and suggestions. I also owe great level of appreciation to office staff of mathematics Department for their correct guidance. Finally, my heart felt regard goes to my father in law and mother in law Mr. and Mrs. Prof. Salahuddin for their love and moral support. I thank the Almighty for giving me the strength and patience to work through all these years so that today I can stand proudly with my head held high.

Anum Tanveer 19-01-2018

Preface This thesis frames the topic of peristalsis due to its occurrence in biofluid mechanics and in various physiological fluid transport in living bodies. The squeezing action of muscles, transfusion of blood through pumping, primitive heart beating, coordinated contractions of ureteral walls to dispose urine, vascular motions of blood vessels, embryo pumping in tubes, locomotion of worms, myogenic cardiac enlargement, intestinal contractions, food intake to its disintegration etc involve the peristaltic pumping. The waves of constant wavelength and amplitude (periodic waves) traveling along the tube length stems the peristaltic excitation in human physiology. Keeping in mind the rhythmic and symmetrical contractions of muscles the Greek word "peristaltikos" meaning "compressing and clasping" provides basis for the word peristalsis. In addition the peristaltic movements are not limited to its natural aspect. The mechanism is equally appealing in engineering and industry. The pumping characteristic of peristalsis has key role in fabricating pumps to transport toxic liquid in order to avoid contamination of the outside environment and sanitary fluid in industrial processes. Such technique is highly advantageous in processes where the medium containing fluid is deformable under applied stresses. The captivating attributes of peristalsis find noteworthy applications in medical and industrial processes in modern industry. Ceramic, porcelain, food, paper and building industries, heart-lung machine, roller, finger and blood pumps, dialysis machines, endoscope, displacement pumps are designed on principle of peristalsis.

It should be noted that beginning from esophagus to ureteral walls the whole alimentary canal is naturally configured in curved shape. Moreover motion of fluid through wave propagation mechanism in physiological conduits, glandular ducts, industrial tubes/channels, blood arterial walls and capillaries involves curved flow peristalsis. Thus straight/planar channel assumption is found inadequate in such situations. The accurate execution of such systems require curvilinear mathematical description though they lead to complicated mathematical expressions. In addition consideration of peristalsis in respiratory, blood capillaries and cardiovascular division operates in alliance with compliant wall properties. The compliance in the boundaries specify the change in volume due to pressure. In fluid flow problems this

effect can be executed in terms of stiffness, elasticity and damping of peristaltic walls.

On account of physiological and industrial peristalsis, the objective here is to develop and analyze the fluid flows through periodic wave transport in channels. Such considerations are focused particularly for human tubular organs functioning to inspect the outcomes of different effects. Thus the mathematical challenges are performed based on essential laws and complexity in a medium is taken with reference to peristalsis. The problems are considered by keeping natural phenomenon intact and then solved through different qualitative schemes (numerical and perturbation). Thus organization of this thesis as follows.

Chapter 1 manifests the literature review based on peristaltic mechanism under different aspects and fundamental equations that will be utilized throughout the remaining chapters.

Chapter 2 aims to examine the peristaltic transport of pseudoplastic fluid under the radially imposed magnetic field and convective heat and mass conditions. The channel walls in the study satisfy the wall properties. The relevant formulation is made on the basis of long wavelength approximations. The corresponding solutions are evaluated and analyzed for both planar versus curved channel. Streamlines are developed for the fluid and curvature parameters. The contents of this chapter are published in Journal of Magnetism and Magnetic Materials 403 (2016) pp 47--59.

The objective of Chapter 3 is to analyze the peristaltic transport of incompressible fluids in a curved channel subject to the following interesting features. Firstly to examine the influence of non-uniform applied magnetic field in radial direction. Secondly to consider compliant walls of channel. Thirdly to analyze the curvature effect in flow of Carreau-Yasuda material. Fourth to examine the influence of heat transfer with viscous dissipation. Fifth to address the impact of velocity and thermal slip conditions. The relevant problems are formulated. Outcoming problems through lubrication approach are solved. Attention is focused to the velocity, temperature, heat transfer coefficient and streamlines. The contents of this chapter are published in AIP Advances 5, 127234 (2015) DOI: 10.1063/1.4939541.

Chapter 4 describes the magnetohydrodynamic peristaltic flow of Carreau fluid in a curved channel. The flow and heat transfer are discussed in presence of wall slip and compliant conditions. The generation of fluid temperature and velocity due to viscous dissipation and gravitational efforts are recorded respectively. Moreover indicated results signify activation of velocity, temperature and heat transfer rate with Darcy number. The contents of this chapter are submitted in Journal of Mechanics.

The purpose of Chapter 5 is twofold. Firstly to explore and compare the shear thinning and thickening effects in peristaltic flow of an incompressible Sisko fluid. Secondly to inspect homogeneous-heterogeneous reactions effects. Mixed convection, thermal radiation and viscous heating are present. The governing equations have been modeled and simplified using lubrication approach. The solution expressions are approximated numerically for the graphical results. The contents of this chapter are published in Journal of Molecular Liquids 233 (2017) pp 131-138.

Chapter 6 models the peristaltic flow of Sisko fluid in a curved channel. Porous medium is characterized by modified Darcy's law. Radial magnetic field is applied. Such consideration is significant to predict human physiological characteristics especially in blood flow problems. Moreover the particular features of blood flow regimes in narrow arteries and capillaries i-e., compliance and slip at the boundaries are not ignored. The whole system is set to long wavelength approximation. The detail of plotted graphs through numerical simulation is discussed. The contents of this chapter are also published in Journal of Molecular Liquids 236 (2017) pp 290-297.

Chapter 7 investigates the impact of homogeneous-heterogeneous reactions in peristaltic transport of third grade fluid in a curved channel. The third grade fluid has an ability to explore shear thinning and shear thickening effects even in steady case. The channel walls in this study satisfy the wall properties. The fluid is electrically conducting in the presence of radially imposed magnetic field. The relevant formulation is made. Solutions are computed and analyzed for various parameters of interest. The main observations are summarized in the conclusions. The contents of this chapter are published in AIP Advances 5, 067172 (2015) DOI: 10.1063/1.4923396.

Chapter 8 has been designed to explore the MHD characteristics of Jeffery nanofluid with wall properties in curved flow stream. Such consideration is more realistic and finds its importance in blood circulatory systems where both MHD and flexibility of walls play essential role. The impact of thermal radiation is not ignored since heating by radiation allows a greater speed and uniformity in reaching a set temperature due to characteristics of electromagnetic waves. Further the chemical reaction effect has been outlined in nanofluid flow up to first order. The nonlinear and coupled system is set to long wavelength and low Reynolds number assumption. The results are plotted numerically and physically interpreted in the last section. The contents of this chapter are published in Neural Computing and Applications (2016) DOI: 0.1007/s00521-016-2705-x pp 1-10.

Chapter 9 focusses its description in curved channel flow of Jeffery fluid through modified Darcy's law. In view of blood circulatory system the important aspect of wall flexibility is not ignored. Further the thermal radiation and wall slip are accounted in mathematical description of the problem. The chemical reaction effects are also present in nanofluid flow. The resulting complex mathematical system is solved efficiently through numerical approach. The flow behavior in terms of velocity, temperature, heat transfer rate and nano particle mass transfer have been emphasized in the discussion. The contents of this chapter are published in Journal of Molecular Liquids 224 (2016) 944-953.

In Chapter 10 the description of flow saturated in porous space followed by Darcy's observations are exploited to obtain mathematical model. Fluid flow comprising porous media in view of modified Darcy's law is developed. Flow stream is developed for Carreau-Yasuda nanofluid in a curved channel. Effectiveness of buoyancy is executed through mixed convection. Further thermal radiation and viscous dissipation effects are included. The graphical interpretation is made through numerical solutions. The physical significance of involved parameters is pointed out in detail. The contents of this chapter are published in Plos One (2017) DOI: 10.1371/journal.pone.0170029.

Chapter 11 discusses the effects of thermophoresis and Brownian motion in peristaltic flow of Eyring-Powell fluid in a curved channel. The channel boundaries are subject to no-slip and flexible/compliant properties. The thermal radiation is not

ignored. Mixed convection in this analysis is also accounted. The solution expressions are approximated through numerical approach. The effects of sundry parameters on quantities of interest are illustrated physically. The contents of this chapter are published in Computers in Biology and Medicine 82 (2017) pp 71-79.

Contents

1 Literature survey and fundamental equations 6

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Basics of fluid flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2.1 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2.2 Momentum conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2.3 Energy conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2.4 Concentration equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.5 Maxwells equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.6 Ohm’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.2.7 Compliant/flexible walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Radial magnetic field on peristalsis in a convectively heated curved channel 22

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Flow diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Problem development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.1 Non-dimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Solution procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4.1 Zeroth order system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.2 First order system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5.1 Velocity profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5.2 Temperature profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1

2.5.3 Concentration profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.5.4 Heat transfer coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5.5 Streamlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3 Simultaneous effects of radial magnetic field and wall properties on peristaltic

flow of Carreau-Yasuda fluid in curved flow configuration 52

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2 Flow diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52



3.4.1 Zeroth order systems and solutions . . . . . . . . . . . . . . . . . . . . . . 58

3.4.2 First order system and solutions . . . . . . . . . . . . . . . . . . . . . . . 59

3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.5.1 Velocity distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.5.2 Temperature and heat transfer coefficient . . . . . . . . . . . . . . . . . . 63

3.5.3 Streamlines pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64


4 Heat transfer analysis for peristalsis of MHD Carreau fluid in curved channel

through modified Darcy law 72

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2 Flow diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72


4.4 Solution and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78


4.4.2 Temperature distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4.3 Heat transfer rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79


2

5 Mixed convective peristaltic flow of Sisko fluid in curved channel with homogeneous-

heterogeneous reaction effects 86

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86


5.2.1 Dimensionless formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 90


5.3.1 Axial velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94



5.3.4 Homogeneous-heterogeneous effects . . . . . . . . . . . . . . . . . . . . . . 95


6 On modified Darcy’s law utilization in peristalsis of Sisko fluid 104

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104




7 Peristaltic motion of third grade fluid with homogeneous-heterogeneous re-

actions 118

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.2 Flow diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118



7.4.1 Zeroth order system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.4.2 First order system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.5.1 Velocity profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.5.2 Temperature profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.5.3 Homogeneous-heterogeneous reactions effects . . . . . . . . . . . . . . . . 131


3


8 Peristaltic flow of MHD Jeffery nanofluid in curved channel with convective

boundary conditions: A numerical study 142

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

8.2 Flow diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143


8.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

8.4.1 Axial velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148


8.4.3 Nanoparticle mass transfer distribution . . . . . . . . . . . . . . . . . . . 149



9 Numerical analysis of partial slip on peristalsis of MHD Jeffery nanofluid in

curved channel with porous space 160

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

9.2 Flow diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161


9.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

9.4.1 Axial velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166





10 Numerical simulation for peristalsis of Carreau-Yasuda nanofluid in curved

channel with mixed convection and porous space 178

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

10.2 Flow diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179


4




10.4.3 Nanoparticle volume fraction distribution . . . . . . . . . . . . . . . . . . 186



11 Peristaltic motion of Eyring-Powell nanofluid in presence of mixed convec-

tion 197

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

11.2 Flow diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197


11.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201






5

Chapter 1

Literature survey and fundamental

equations

Here our intention is to include the relevant information of existing works on peristalsis and

basic equations for fluid flow and heat and mass transport.

1.1 Background

Peristalsis is well known to physiologists due to its occurrence in digestive and reproductive

tracts. The myogenic theory of peristalsis in uterus dates back to Engelmann [1] who was first

to capture the origin of peristaltic wave train in renal pelvis and demonstrated the movement

of that the ureteral cells from one to another. Latham [2] discussed this phenomenon for

peristaltic pump. Shapiro et al. [3] found forward and backward time-mean flows in core of

tube and close to boundaries respectively. He concluded that functioning of the ureter and the

gastrointestinal system causes such motions. The pioneer efforts of Latham [2] and Shapiro et

al. [3] make a way forward for further advancements in peristaltic motion. After Eckstein [4]

and Weinberg [5] confirmation of Shapiro’s theory (long wavelength and low Reynolds number

theory), series of attempts have been made until now to make advancements in this direction.

Weinberg et al. [6] and Lykoudis [7] analyzed the ureteral physiology as peristaltic pump by

imposing different waves on ureteral walls. The impact of biomechanical forces on dynamics

of uretal muscles was reported by Fung [8] The small Reynolds number assumption leads to

6

inertia free flow. However large wavelength comprises the flow of fluid when average radius of

the tube is much smaller than peristaltic wave. Such assumptions laid strong development in

the theory of peristalsis [9−13]. The application of peristalsis on ureteral system is explored byZein and Ostrach [14]. Their work focussed the symmetric flow of viscous fluid. Later Li [15]

extended ureteral peristalsis for an axisymmetric case. For axisymmetric case the fluid transport

in circular tube is studied by Chow [16]. Here motion was initiated as Hagen-Poiseuille flow.

Meginniss [17] addressed the theory of peristaltic flow in roller pump in view of low Reynolds

number. Intestinal peristalsis with two different solutions one for peristaltic pumping (where

flow of fluid is generated in the absence of net pressure gradient) and the other for peristaltic

compression is scrutinized by Lew et al. [18] . For physical significance of peristaltic fluid

flow in veins, ducts and arteries, Lew and Fung [19] carried out an attempt by considering

small Reynolds number assumption for flow of viscous fluid in cylindrical tube. The curvature

effects and streamline phenomena have been described for viscous fluid in circular tube by

Jaffrin [20]. Such consideration provides physical basis of peristalsis in alimentary canal and

in roller pumps. Movement of spermatozoa in tube is mathematically investigated by Semleser

et al. [21]. The two dimensional peristaltic flow in a straight channel is inspected by Mitra

and Prasad [22] Nergin et al. [23] discussed variation in pressure rise per wavelength. Tube

flow with long peristaltic wave is explored by Manton [24]. Hung and Brown [25] analyzed

the solid-particle transport phenomena through peristaltic flow in a two-dimensional geometry.

His observation revealed the motion of bolus through a moving particle which makes particle

to oscillate. Liron [26] presented the detailed view of peristaltic flow in pipe and channel.

He developed the solution expression in terms of series by double expansion about wave and

Reynolds numbers and established the effective biological functioning in terms of peristaltic

flow. Solutions for peristaltic flow generated in a channel have been numerically analyzed by

Brown and Hung [27]. Srivastava and Srivastava [28] commenced a study on the pulsatile fluid

flow pattern in peripheral and core fluid regions surrounded by a non-uniform channel and tube.

They considered the perturbation solution about small amplitude parameter. The outcomes

of peristaltic flow in circular tube is reported by Srivastava and Srivastava [29]. Inertial and

streamline curvature effects in an incompressible fluid flow bounded in an asymmetric channel

has been explored by Rao and Mishra [30]. Peristaltic motion of an incompressible viscous

7

fluid in gaps between uniform and non-uniform annulus has been studied by Makheimer [31].

Viscous fluid flow with complaint walls have been analyzed by Hayat et al. [32].

Inspite of aforementioned studies focused on peristaltic fluid flow of viscous fluid, the fre-

quently witnessed natural phenomenon involve non-Newtonian fluids. Investigations on peri-

staltic carrying of chyme [33], compression and expansion of blood vessels [34] manifest that

fluids in such cases possess non-Newtonian character. For rheological complex fluids like petro-

leum, blood, shampoos, greases, muds, oils, paints, lubricants, hydrocarbons, polymer solutions,

industrial oils etc, the mathematical description followed by classical Navier—Stokes relations

are found inadequate. No doubt non-Newtonian fluids comprised nonlinear relation between

shear stress and strain rate. Thus understanding and predicting natural aspects of fluid flow

demands modelling of fluid flow problems using non-Newtonian liquids. Until now various

models have been proposed to relate non-Newtonian relationships depending on the rheolog-

ical properties. In lubricants for instance, a power law model is used in which both dilatant

and pseudoplastic behavior are addressed. Hina et al. [35] examined the wall properties in

a curved channel flow of pseudoplastic fluid with heat and mass transfer. They discussed the

shear-thinning/shear thickening effects followed through lubrication approach. Hayat et al. [36]

employed non-Newtonian Carreau-Yasuda fluid for numerical examination of peristalsis with

Hall effects. Peristalsis of third grade (differential type) fluid under long wavelength assumption

has been reported by Hayat et al. [37]. The perturbed solutions are sought in the analysis and

solution expressions are obtained in the form of series. The generalized form of Newtonian fluid

named Carreau fluid bear a tendency to show Newtonian as well as power law behavior at low

and high shear rates respectively. In fact at high shear rate ( 1) apparent viscosity of fluid

declines to exhibit power law (shear-thinning) attribute whereas apparent viscosity increases

( 1) to behave as Dilatent or shear-thickenning behavior at high shear rate. In comparison

to peristaltic flow in blood arteries both of above cases are found appropriate since blood is

inclined towards Newtonian behavior in larger arteries and show non-Newtonian characteristics

in narrow arteries. Akbar et al. [38] carried out an examination for flow of Carraeu fluid in

asymmetric channel. The flow characteristics of Carreau fluid in complaint rectangular duct

has been analyzed analytically by Riaz et al. [39]. Among non-Newtonian liquids, Sisko fluid

is another version accomplished with shear thinning and shear thickening attributes. Sisko

8

model with suitable selection of material fluid parameters can predict many typical properties

of Newtonian and non-Newtonian liquids. Zaman et al. [40] commenced a study on blood flow

in a vessel by considering blood as a Sisko fluid material. Mekheimer and Kot [41] considered

overlapping stenosis phenomenon in tapered elastic arteries through Sisko fluid model. Ali et

al. [42] examined unsteady blood flow in stenotic artery with Sisko fluid. Hayat et al. [43 44]

found porosity and magnetic field effects in flow of Sisko fluid. Jeffery fluid with the simplest

descriptive mathematical form bear tendency to relate relaxation and retardation time effects.

Ellahi and Hussain [45] described the Jeffery fluid transport in rectangular duct. In this work

the channel boundaries are subjected to partial slip effects. Narla et al. [46] studied curvature

effects on peristalsis of Jeffery nanofluid. Bhatti and Abbas [47] investigated the peristaltic

blood flow using Jeffery model by considering blood vessels as porous medium. Sheikholeslami

et al. [48] performed an analytic investigation on Jeffery-Hamel flow by Adomian decomposition

method. Viscoelastic non-Newtonian fluids perceive prominent role in physiology and industry.

The Eyring-Powell viscoelastic fluid derived by kinetic theory is beneficial in providing accurate

results of viscous fluid at low and high shear rates. Hayat et al. [49] demonstrates the effects

of fluid flow in straight channel with Eyring-Powell fluid model. Effects of chemical reaction

and convective conditions are considered. Abbasi et al. [50] explored curved channel flow of

Eyring-Powell fluid whereas Hina et al. [51] extended this work with heat and mass transfer

effects.

The interaction of peristaltic fluid flows with heat transfer effect has applications in bio-

medical sciences such as in acquiring the flow rate of blood via the initial thermal conditions

and the thermal clearance rate. The rate of blood flow can be approximated by a technique in

which heat is produced locally or injected and the thermal clearance is monitored. Particularly

the destruction of undesirable tissues, laser therapy, hemodialysis, oxygenation, thermal energy

storage, blood flow convection from the pores of tissues and hyperthermia through bioheat

transfer [52] are crucial in this direction. Radhakrishnamacharaya and Murty [53] canvassed

the heat transfer analysis with peristalsis of viscous fluid. The perturbation solution about

small wave number have been given in this work. Vajravelu et al. [54] considered heat transfer

phenomenon between two cocentric tubes containing viscous fluid with peristalsis. The solu-

tions for free convection and porosity parameters have been sought using double perturbation

9

technique. An electrically conducting fluid in the presence of imposed magnetic field is acti-

vated by MHD forces. Such forces occur in response to the interaction between induced electric

currents and applied magnetic field. Imposed magnetic field is very useful tool in several in-

dustrial and engineering processes such as metal casting, stirring, pumping, crystal growth and

cooling circuits of fast fission reactors. The magnetohydrodynamic (MHD) characteristics of

fluid flow has appreciable role in medicine. Applying magnetic field dominates the thickening

of blood viscosity and is advantageous clinically. Magnetic effects regulate the flow stream by

reducing the speed of fast moving particles and thus can be utilized in diagnosing and treat-

ment of many diseases. Hypothermia [55], tumors targeting [56], blood flow microcirculations

[57], surgical operations, intestinal disorders, cancer therapy [58], MRI, blood pumping [59] etc

are some useful applications of MHD. Involvement of MHD in human physiological systems

is practically important since blood as electrically conducting fluid shows magnetic properties

that can be utilized in treatment of diverse health issues with rare clinical disorder. Magne-

tohydrodynamically induced currents generate the electromagnetic forces and so are capable

of mechanical implementation particularly in magnetohydrodynamic sensors, electrical power

generation, magnetic drug targeting, geothermal extractions, solar power technology, space

vehicles and many others. Wang et al. [60] examined the magnetohydrodynamic aspects of

peristaltic motion induced in symmetric or asymmetric channels. Bhatti et al. [61] have talked

about the blood flow peristalsis with magnetic field and wall properties. They also examined

the slip phenomenon in the analysis. Hayat et al. [62] scruntinize the MHD effects in an in-

clined channel with Joule heating and slip conditions. The solutions are approximated through

numerical approach. Awais et al. [63] studied the convective heat transfer with magnetic field

in a symmetric channel subject to pulsatile wave. Mixed convection arrises when gravitational

effects are strong enough to promote heat transfer. Utilization of mixed convection occurs

in natural and artificial heat transfer processes like solar energy and nuclear reactors, build-

ing works, humidification/dehumidification in air-conditioning, chemical plant, solidification

processes of alloys, processing of nuclear impurities, control of chemical waste and pollutants,

design of MHD power generators etc. Mixed convection flow in an asymmetric channel with

peristalsis has been observed by Srinivas and Mathuraj [64]. In this analysis the porosity and

chemical reaction effects are also highlighted. Mixed convective flows subjected to peristaltic

10

wave transport with variable viscosity and magnetic field have been proposed by Hayat et al.

[65]. Further they revised the analysis for an incompressible Prandtl fluid [66].

Many applications in geophysical and industrial engineering involve conjugate phenomenon

of the heat and mass transfer which occurs as a consequence of buoyancy effects. The simulta-

neous effects of heat and mass transfer are found handy in the improvement of energy transport

technologies [67], metallurgy [68], blood transfusions [69], power generation [70], production of

polymers and ceramics [71], food drying [72], oil recovery [73], food processing [74], fog dis-

persion [75], the distribution of temperature and moisture in the field of agriculture [76] and

so-forth. There are some situations where transfer of heat by convection means are insufficient

in providing required heat transfer. In such cases combination of mixed convection with thermal

radiation aids in significant heat transfer. With relevance to human physiology the combined

attributes of mixed convection and thermal radiation are of prime importance specifically in

brain, liver, contraction of skeletal muscles and heart. The heat decaying aspect associated

with thermal radiation is capable of controlling the generation of excess heat inside the body

as high temperatures build serious stresses for the human body and place it in an unhealthy

condition. The maintenance of skin vasodilatation and sweating are done by mixed convection

and radiation in order to keep it at healthy level during climate changes. In such conditions

body loses heat by radiation and conduction whenever skin temperature is greater than that of

surroundings. On the other hand gain of heat by a body is noticed via radiation and conduction

in case when the temperature of the surroundings is greater than that of the skin. Having all

such in mind the representative studies for mixed convection/radiation of nonlinear fluids have

been addressed by number of researchers [77− 81].Fluid flows in glandular ducts, physiological conduits and blood flow regimes relate curved

pattern. However many of the literature available on peristalsis discussed flow pattern in

straight or planar channel which perhaps is inadequate in correct execution of natural phe-

nomenon occurring in physical and industrial processes. The curved channel flows involve

complicated mathematical description of curvilinear coordinates that is perhaps the reason of

the limited availability of literature in curved flow peristalsis. Sato at al. [82] at first developed

the curved approach and modeled a problem using curvilinear coordinates and hence arouse

the researchers to consider such realistic mechanism afterwards. Ali et al. [83] numerically an-

11

alyzed the peristaltic waves in curved channel by Shooting method. Hayat et al. [84] discussed

the compliant wall properties in curved channel flow of viscous fluid. The effect of nanopar-

ticles in curved configuration are studied by Hina et al. [85] and Noreen et al. [86]. Further

Hayat et al. [87] numerically analyzed the influence of MHD on Carreau-Yasuda fluid in curved

configuration with Hall effects.

Capillaries, filters, water flow, petroleum reservoirs, manufacture process, chemical reactors

etc are some biological and engineering systems in which fluid flow experience resistance due

to pores or voids. The porous media involved in such systems increases the contact surface

area of liquid(fluid) and solid surface. Usually porous space occurs in response to variation

in media structure such as erosion, deposition, expanding or shrinkage that offer resistance to

flow and thus affects the transport properties of the media. Darcy [88] was first to experimen-

tally analyzed the flow of fluid saturating porous media. Polubarinova-Kochina [89] presented

a filtration theory of liquids through porous media. Johnson and Dunning [90] gave capillary

properties of the liquids using flow of homogeneous fluid in porous medium and concluded that

fluid distribution and saturation is affected by wettability. Liu and Masliyah [91] developed the

non-Newtonian fluid flow in porous medium with the consideration of Herschel—Bulkley and

Meter fluid. The volume averaging technique is employed to achieve the governing equations

for this analysis. It is well admitted fact that the pressure drop at low Reynolds number in

porous media follows Darcy law (simple relation between velocity and pressure gradient). Since

fluid flows in pores of capillaries and tubes involve low Reynolds number having porous space,

thus fluid flows in porous media is found appealing in scientific and technological disciplines

like metallurgy and earth science. However the surface tension forces have key role in such

flows which appears to be neglected in Darcy law and thus needs modified Darcy law. Fur-

ther application of Darcy law is confined to deep porous media whereas circulation of small

capillaries, water flow in grounds also found to possess porous media. At present amount of

literature is available that addresses porous media referring Darcy’s law. Abbasi et al. [92]

investigated peristaltic fluid flow through a porous medium in view of Darcy law. Bhatti and

Abbas [93] carried out an investigation on blood flow characteristics of Jeffery fluid saturating

porous medium. Ramesh [94] developed the peristaltic transport couple stress fluid containing

porous medium in an asymmetric channel. Further he extended this work for inclined MHD

12

and heat and mass transfer effects [95]. Velocity slip and chemical reaction aspects in peri-

staltic flow through porous medium has been reported by Machireddy and Kattamreddy [96].

Abd-Alla et al. [97] presented rotation effects in peristaltic flow with porous medium. They

considered micropolar fluid with an external magnetic field. Mekheimer et al. [98] studied slip

flow in porous medium with reference to surface acoustic wavy wall. Pascal and Pascal [99]

considered the non-Darcian flows through porous media for execution of non-linear effects in

power law fluids. Mathematical expressions are obtained using Darcy-Forchheimer equation in

case of high Reynolds number flow and modified Darcy’s law has been accounted for non-linear

rheological effects of power law fluid. Tan and Masuoka [100 101] further examined Stokes first

problem using Oldroyd-B and second grade fluid in porous half space through modified Darcy’s

law. Hayat et al. [102] talked about the oscillatory flow in a porous medium for Oldroyd-B

fluid model. They also carried out such investigation for third grade fluid [103 104].

Many theoretical and experimental efforts have been reported by the scientists in the past

to enhance thermal conductivity of base fluids like water, oil, ethylene glycol etc. A technique

was proposed to increase the thermal properties of normal coolants by allowing the suspension

of millimeter and micrometer-sized particles preserving high thermal conductivity. However the

method was not much advantageous since it produced high pressure drop, flow clogging at one

section and corrosion of the heat exchanging components. In recent time with an improvement

and advancement in technology colloidal suspensions of nanometer-sized particles (100 nm)

with considerable heat transfer properties have been reported. Such technique with an addition

of tiny solid particles in conventional fluids respond in prominent heat transfer due to high

thermal conductivity of nanofluid (resulting fluid). The preparation of nanofluids is due to

addition of materials like metals, non-metals, carbides and hybrid etc into water, oil or glycols.

Such fluid has great significance in biochemistry, medicine and engineering industry. Usually

dispersion of nanoparticles in base fluid creates greater (nearly double) heat transfer through

convection and conduction. Choi [105] commenced the idea of nanofluid and examined that

an enhancement of thermal properties of base fluid is due to the combination of base fluid

with nanoparticles. One of the attractive feature of nanofluid phenomenon is the fact that any

settling motion due to gravitational effects are excelled by Brownian and thermal agitation. This

aspect clears the fact that the theoretical aspect of nanofluid exists only when nanoparticles are

13

considered small enough. Buongiorno [106] presented a model to emphasize heat enhancement

of nanofluid as main outcome Brownian or thermal agitation. Thermophoresis and Brownian

diffusion in flow of nanofluid are significant since they facilitate manufacturing of optical fibres,

polymer separation, drug discovery and fluctuations in stock market. Many researchers argued

the nanofluid characteristics followed through the studies of Choi [105] and Buongiorno [106].

Also better accuracy of heat transfer rate can be found using single phase approach. Refs.

[107 − 114] give the comprehensive study on nanofluid characteristics relating thermophoresisand Brownian diffusion (Buongiorno model).

Constructive/generative and destructive chemical reactions are two general categories of

chemical reaction. The relevance of chemical reaction in digestive physiology is witnessed to

create or break the bonds between chemical substances associated with body. The climate

changes on the surface of earth are also in view of chemical reactions by using constructive or

destructive forces. Weathering, erosion, building of sand deltas, mountains and earth quakes

are some examples. Further based on the physical state (i.e., color, shape, length, size, weight,

distribution, appearance, language, income, disease, temperature, radioactivity, architectural

pattern, etc.) the materials in chemical reaction are characterized through homogeneous and

heterogeneous reactions. The former occurs in a single phase (gaseous, liquid, or solid) whereas

the later as components of two or more phases. Homogeneous reactions are theoretically simple

when compared with heterogeneous reactions since reacting product depends only on the nature

of reacting species. On the other hand the heterogeneous reactions preserve practical impor-

tance as it relates dependence of product on nature of two or more different reacting species.

Such reactions are witnessed in batteries, corrosion phenomenon and electrolytic cells. Also

certain chemically reacting processes comprised homogeneous and heterogeneous reactions. In

such cases the catalyst (agent) is used to accelerate the reaction speed so that reaction con-

tinued to the desired limit. Chaudhary and Merkin [115] examined a simple isothermal model

for homogeneous-heterogeneous reactions in boundary-layer flow. They focused the case when

diffusion coefficients of the reactant and autocatalyst are different and found the dominance of

surface reaction and homogeneous reaction. Merkin [116] presented an asymptotic review on

isothermal homogeneous-heterogeneous reactions. He considered homogeneous reaction as cu-

bic autocatalysis where heterogeneous reaction is represented by a first-order process. Das and

14

Chaudhury [117] addressed the heterogeneous catalytic activity of nanoparticle by time distri-

bution formalism. Imtiaz et al. [118] analyzed the homogeneous-heterogeneous reaction effects

in curved stretching surface. They underlined the study using homotopy analysis method. Ef-

fects of induced magnetic field and homogeneous—heterogeneous reactions for Casson fluid has

been canvassed by Raju et al. [119]. The flow characteristics of Williamson fluid by stretch-

ing cylinder has been explored by Malik et al. [120]. They examined the flow pattern by

Keller box technique. In spite of vast literature is available on homogeneous-heterogeneous

reactions, scarce information on peristalsis with homogeneous and heterogeneous reactions is

noticed [121 123].

Many experiments have been carefully performed in reference to water and mercury to justify

adherence of fluid with the boundary. However experimental as well as theoretical results are

not found accurate since adhesion condition is found significant even when fluid does not wet

the wall of surface. The slip effect is necessary due to non-continuum effect for peristalsis in

microchannels or nanochannels such as blood flow domains [124] where the mean path length

is comparable to separation between channel walls. Slip conditions at the boundary manifests

the linear relation between velocity and shear stress of the considered fluid and found actively

involved in paints, polishing of artificial heart valves, emulsions and polymer industry. However

less devotion towards slip relative to peristalsis is shown in the literature. Bhatti et al. [125]

proposed the endoscopy analysis and studied slip effects on blood flow. Yildirim and Sezer

[126] conducted homotopy perturbation method to discuss peristaltic motion of viscous fluid

with partial slip. Kumar et al. [127] analyzed the peristalsis in an asymmetric channel with slip

effect. Johnson-Segalman fluid flow with peristalsis in an asymmetric channel comprising slip

effects has been reported by Das [128]. Saravana et al. [129] captured heat and mass transfer

effects through peristaltic motion of non-Newtonain fluid with slip conditions. Jyothi and Rao

[130] captured the slip flow characteristics in an electrically conducting Williamson fluid by

employing perturbation technique. Hayat et al. [131−133] focussed on effects of slip conditionsunder peristaltic flow of viscous and Jeffery nanofluids respectively. Ellahi and Hussain [134]

examined partial slip effect on MHD Jeffery fluid with peristalsis.

Heat transfer in fluid flow analysis by physical movement of particles is referred as con-

vection. The macroscopic fluid transportation produces development in heat transfer that can

15

be witnessed in number of physical processes like thermal storage, gas turbines, nuclear fluid

transport etc. The conduction refers to transfer of heat between a solid boundary and static

fluid. The corresponding conditions on the boundary in conduction are found with the aid of

Fourier law of heat conduction. Whereas in case of moving fluid both conduction and convection

attributes of heat transfer are active and boundary conditions in such case needs modification.

The combination of Fourier law of heat conduction and the Newton law of cooling provide the

required form. Such boundary conditions are named "convective type boundary conditions"

[135 136]. However scarce information is available towards convective conditions in peristalsis

(see refs. [137− 140]).Ability of a vessel boundaries to resist any change toward its original dimensions under

applied distending force is termed as compliance. This effect usually occurs in response to

pressure gradient and it has particular significance in human physiology. The stretching (com-

pliance) of blood veins and arteries in response to pressure has larger effect on blood pressure.

Some dysfunction produces reduction in compliance (stiff arteries) and causes hypertension,

diabetes and smoking in patients. Thus the compliant wall properties in terms of rigidity, elas-

ticity and damping are major aspects associated with membrane problem. Until now number

of researchers addressed compliant wall properties with peristalsis. The ability of compliance

to reduce drag in compliant coatings has fascinated many scientists and engineers. The hu-

man tabular organs are capable of recoil back to their original position due to compliance.

Information available in literature on peristaltic flows in different geometries with Newtonian

and non-Newtonian fluids in compliant walls channel is vast. Impacts of wall properties on

peristalsis in channel is studied by Mitra and Prasad [141]. They found the existence of mean

flow reversal at the center and boundaries of channel. Camenschi [142] and Camenschi and

Sandru [143] worked on viscous fluid flow in pipes of small radii with elastic characteristics.

The stability analysis of flow between compliant channel has been discussed by Davies and

Carpenter [144]. Heat transfer attributes of Newtonian fluid in a channel with wall properties

has been presented by Radhakrishnamacharya and Srinivasulu [145] Hayat et al. [146] and Ali

et al. [147] presented the influence of compliant boundaries on peristaltic activity of Jhonson-

Segalman and Maxwell fluids respectively. MHD peristaltic motion with flexible boundaries

in a heated channel containing viscous fluid was examined by Srinivas et al. [148]. Srinivas

16

and Kothandapani [149] also investigated theoretically the peristaltic flow in a flexible channel

with heat and mass transfer effects. Mustafa et al. [150] compared the analytical and numeri-

cal results in nanofluid flow between compliant boundaries with slip conditions and heat/mass

transfer. Srinivas et al. [151 152] examined peristaltic flows satisfying wall properties with heat

and mass transfer effects. Hayat et al. [153] numerically analyzed the radiative heat transfer

effects in peristaltic transport of Sutterby fluid with compliant walls. Hina et al. [154] captured

the results of heat and mass transfer in peristaltic flow of Johnson—Segalman fluid in curved

compliant channel.

1.2 Basics of fluid flow

For real fluid flow situations the proper description of motion must not violate the conservation

principles based on the physical laws. Three basic laws related to fluid motion are conservation

of mass, momentum and energy respectively. The physical motion of fluid is governed by

utilizing these concepts in terms of continuity, momentum and energy equations. Further if the

medium containing fluid is assumed conserved, an additional equation namely; concentration

equation gives the complete flow pattern.

1.2.1 Mass conservation

Its mathematical form is

+∇ (V) = 0 (1.1)

with ∇ the gradient operator, the density, the time and V the velocity of fluid. For

incompressible fluid =constant and one arrives at

∇V = 0 (1.2)

1.2.2 Momentum conservation

Conservation of momentum stems from the Newton’s seconds law of motion which relates

magnitude of force to the product of mass and acceleration. Thus for unit volume the "flow

field" in terms of velocity profile is described by this basic equation. The force components

17

consists of surface (∇τ ) and body (f) forces. Symbolically

V

=∇τ+f (1.3)

in which τ (= −I+ S) the Cauchy-stress tensor, the fluid pressure, I the identity tensor andS the extra stress tensor (varies with considered fluid model). The momentum equation adds

further terms and takes up the extended form as additional effects such as mixed convection,

magnetohydrodynamics and porosity are accounted. For such cases:

f = [ ( − 0) + ( −0)] + J×B+R (1.4)

where symbolizes the gravity, the thermal and concentration expansion coefficients

respectively, the fluid temperature, the fluid concentration, 0 the temperature of the

wall, 0 the concentration of the wall, J the current density, B the magnetic field and R the

resistance in medium.

1.2.3 Energy conservation

The fundamentals of energy equation traced basis on first law of thermodynamics applied on

control volume. Its mathematical form is

= −∇q+ (1.5)

In above equation (= ) demonstrates the internal energy with the specific heat at fixed

pressure, q(= −1 grad ) presents the heat flux with 1 the thermal conductivity of fluid and is the source term governing energy transport. The factor is responsible for the modification

of heat transport characteristics with velocity as well as heating and cooling in surface. Hence

thermal radiation, thermophoresis, Brownian diffusion and viscous dissipation are considered

through this term.

18

1.2.4 Concentration equation

During motion of fluid in various engineering processes, notable mass transport takes place.

Such transport of mass comprised mixing of polluting chemicals to subject matter. This ar-

resting action of mass transfer must be recorganised through conservation of mass within the

fluid flow. Two active mechanisms convection and molecular diffusion are playing part behind

this transportation of mass. For the mass concentration of fluid per unit volume the mass

equation in vector notation has the following form:

= ∇2 + (1.6)

In above equation stands for mass diffusion coefficient and be any source term that is

capable of imposing a marked impression on concentration of fluid. In this thesis, will

be subjected to report chemical reaction, thermophoresis and Brownian diffusion. Moreover,

the concentration equation is modified for the consideration of homogeneous-heterogeneous

reactions aspects in Chapters 5 and 7 only.

1.2.5 Maxwells equations

The combined impression of electric and magnetic fields follows four elementary laws of elec-

tromagnetism known as Maxwells equations. These are:

Guass’ law of electricity

∇E =

0 (1.7)

Guass’ law of magnetism

∇B =0 (1.8)

Faradays law

∇×E = −B

(1.9)

19

Ampere-Maxwell law

∇×B =J+ 0E

(1.10)

Here gives the density of charge particle, the electric constant, 0 the permittivity of free

space, B(= B0(applied magnetic field) +B1(induced magnetic field)) the total magnetic field,

J the current density and E the electric field strength.

1.2.6 Ohm’s law

The Ohm’s law in absence of Hall and ion-slip effects has the form:

J = (E+V×B) (1.11)

where (= 2) shows the electrical conductivity.

1.2.7 Compliant/flexible walls

The measure of ability of an object (organ/vessel/medium/artery/tract) to recoil back towards

its original dimensions when disturbing source is ejected. From medical point of view, such

characteristics of walls allow exchange of water and nutrients in blood, oxygen and carbondiox-

ide in lungs and systolic and diastolic pressure variation in cardiac physiology. Thus a wall

with flexible, stretchable, damping and elastic nature is called compliant wall. However the

rigid wall conditions are extensively considered in the theory of peristalsis which remains valid

when disturbance generated due to pressure is small enough to be neglected. But the assump-

tion remains nomore valid in underlying human physiology where radius of channel/duct/tube

wall is assumed to be thin (approx. 005cm or less), or the wall is itself deformable, then the

compliant wall assumption leads to better exposition of results.

The mathematical form for compliant characteristics is given as:

() = − 0

=

∙−∗

2

2+∗1

2

2+

0

¸ (1.12)

20

where represents the operator that signify the motions of stretched membrane, ∗ the elastic

tension, ∗1 the mass per unit area, 0the coefficient of viscous damping and 0 the outside

pressure due to tension in the muscles. Since walls are inextensible, therefore 0 = 0 is assumed

throughout the thesis.

21

Chapter 2

Radial magnetic field on peristalsis

in a convectively heated curved

channel

2.1 Introduction

The prime focus of this chapter is to address the combined effects of heat and mass transfer

in peristaltic flow with convective effects. The channel walls are curved and flexible. Mag-

netic field is applied towards radial direction to enhance the amplitude of wave (used in ECG

for synchronization purposes). The shear-thinning and thickening effects are highlighted for

pseudoplastic fluid. Inertial effects are neglected in view of small Reynolds number. Long

wavelength assumption has been utilized. The graphical illustrations are compared with planar

case and non-symmetric response of involved parameters is captured opposite to the planar

case. Moreover results obtained for curved channel are more clear.

2.2 Flow diagram

Consider a curved channel of half width coiled in a circle with centre and radius ∗ enclosing

an incompressible pseudoplastic fluid. The flow is initiated by sinusoidal waves that travel with

velocity along the peristaltic walls. The walls of the channel flexible and inextensible. The

22

flow stream is developed such that is the axial coordinate and the radial coordinate (see

Fig. 21). Further a uniform magnetic field is externally imposed. The shape of waves may be

defined by

= ±( ) = ±∙+ sin

2

(− )

¸ (2.1)

where serves the wave’s amplitude, the wavelength, the time and ± the displacementsof the upper and lower walls respectively. The magnetic field is taken in the form given below

B = (0

+∗ 0 0) (2.2)

where 0 shows the magnetic field strength. Ohm’s law leads to the following expression

J×B = (0 −20

( +∗)2 0) (2.3)

Here V = (( )( )( )) represents the velocity field for the present flow with

corresponding velocity components ( ) and ( ) respectively.

23

2.1. Physical picture of problem

2.3 Problem development

The basic equations for an incompressible fluid in the presence of magnetic field, viscous dissi-

pation and Soret effect are:

∇V = 0 (2.4)

V

= ∇τ + J×B (2.5)

= 1∇2 + τ L (2.6)

= ∇2 +

(∇2 ) (2.7)

J = (V ×B) (2.8)

24

An extra stress tensor for an incompressible pseudoplastic fluid is

τ = −I+ S (2.9)

S+ 1S

+1

2(1 − 1)(A1S+ SA1) = A1 (2.10)

with 1 and 1 as the relaxation times in pseudoplastic fluid. Also

A1 = gradV+ (gradV)

S

=

S

− (gradV)S− S(gradV) (2.11)

In above equations

=

+

+ ∗

+∗demonstrates the material time derivative, ,

the fluid temperature and concentration respectively, the thermal diffusion ratio, the

mean temperature of fluid and A1 the first Rivlin Erickson tensor.

The conservations principles of mass and linear momentum for the problem under consid-

eration yield the following set of equations

+

∗

+∗

+

+∗= 0 (2.12)

∙

+

+

∗ +∗

− 2

+∗

¸= −

+

1

+∗

{( +∗)}

+∗

+∗

−

+∗ (2.13)

∙

+

+

∗ +∗

+

+∗

¸= − ∗

+∗

+

1

( +∗)2

©( +∗)2

ª+

∗

+∗

− 20

( +∗)2 (2.14)

∙

+

+

∗ +∗

¸ = 1

∙2

2+

1

+∗

+

2

2

¸+ ( − )

+

µ

+

∗

+∗

−

+∗

¶ (2.15)

25

∙

+

+

∗ +∗

¸ =

∙2

2+

1

+∗

+

2

2

¸+

∙2

2+

1

+∗

+

2

2

¸

(2.16)

The components and are

+ 1

½µ

+

+

∗ +∗

¶ − 2

− 2 ∗

+∗

¾+1

2(1 − 1)

½4

+ 2

µ

+

∗

+∗

−

+∗

¶¾= 2

(2.17)

+ 1

½µ

+

+

∗ +∗

¶ −

µ

−

+∗

¶− ∗

+∗

¾+1

2(1 − 1)( + )

µ

+

∗

+∗

−

+∗

¶=

µ

+

∗

+∗

−

+∗

¶ (2.18)

+ 1

½µ

+

+

∗ +∗

¶ − 2

µ

−

+∗

¶− 2

µ∗

+∗

+

+∗

¶¾+1

2(1 − 1)

½2

µ

+

∗

+∗

−

+∗

¶+ 4

µ∗

+∗

+

+∗

¶¾= 2

µ∗

+∗

+

+∗

¶ (2.19)

The corresponding conditions on the boundaries comprised of no-slip condition, compliant wall

properties and convective conditions as follows:

= 0 at = ± (2.20)

1

= −1( − 0) at = ± (2.21)

= −2( − 0) at = ± (2.22)

with 1 and 2 as the heat and mass transfer coefficients and 0 0 the temperature and

26

concentration of the upper and lower walls respectively. The related equation of motion for the

compliant walls is

() = − 0

= −∗ 2

2+∗1

2

2+

0

(2.23)

The continuity of stresses implies that at the fluid-walls interfaces the pressure exerted on the

walls must be equal to that exerted on the fluid at = ± Utilization of stress continuityconcepts and −component of momentum equation the required condition of compliance can

be calculated as

∗

+∗

() =

∗

+∗

= −

∙

+

+

∗ +∗

+

+∗

¸+

∗

+∗

+

1

( +∗)2

©( +∗)2

ª− 20

( +∗)2 at = ± (2.24)

2.3.1 Non-dimensionalization

Utilizing the definitions of the dimensionless variables:

∗ =

∗ =

∗ =

∗ =

∗ =

∗ =

∗ =

∗ =

=

∗

= − 0

0 ∗ =

2

∗ =

∗1 =

1

∗1 =

1

=

∗

=

− 0

0(2.25)

the continuity equation is satisfied identically and we have the following set of stress components

and non-dimensional equations

+ 1

½µ

+

+

+

¶ − 2

− 2

+

¾+1

2(1 − 1)

½4

+ 2

µ

+

+

−

+

¶¾= 2

(2.26)

27

+ 1

½µ

+

+

+

¶ −

µ

−

+

¶−

+

¾+1

2(1 − 1)( + )

µ

+

+

−

+

¶=

µ

+

+

−

+

¶ (2.27)

+ 1

½µ

+

+

+

¶ − 2

µ

−

+

¶− 2

µ

+

+

+

¶¾+1

2(1 − 1)

½2

µ

+

+

−

+

¶+ 4

µ

+

+

+

¶¾= 2

µ

+

+

+

¶ (2.28)

Re

∙

+

+

+

− 2

+

¸= −

+

∙1

+

{( + )}+

+

−

+

¸

(2.29)

Re

∙

+

+

+

+

+

¸= −

+

+

1

( + )2

©( + )2

ª+

+

+2

( + )2 (2.30)

Re

∙

+

+

+

¸ =

∙( − )

+

µ

+

+

−

+

¶¸1

Pr

∙2

2+

1

+

+ 2

2

2

¸ (2.31)

Re

∙

+

+

+

¸ =

1

∙2

2+

1

+

+ 2

2

2

¸

+

∙2

2+

1

+

+ 2

2

2

¸ (2.32)

The non-dimensional form of wall surface and boundary conditions have the form

= 1 + sin 2 (− ) (2.33)

= 0 at = ± (2.34)

28

+1 = 0 at = ± (2.35)

+2 = 0 at = ± (2.36)

+ [1

3

3+2

3

2+3

2

] =

1

( + )2

£( + )2

¤+2

( + )2

−Re∙

+

+

+

+

+

¸+

+

at = ± (2.37)

In writing the above equations the asterisks have been suppressed for simplicity and the symbols

specify the following dimensionless quantities: the wave number, Re the Reynolds number,

the curvature parameter, Pr the Prandtl number, the Eckert number, the Hartman

number ( = 1− 3) the non-dimensional elasticity parameters, the Brinkman number, the amplitude ratio parameter, 1 2 the heat and mass transfer Biot numbers respectively,

the Soret number and the Schmidt number with definitions

=

Re =

Pr =

1 =

=

2

0 =

1 = − ∗3

3 2 =

∗13

3 3 =

30

2 =

=0

0 2 =

20

1 =1

1 2 =

2

(2.38)

Introducing the stream function ( ) and using dimensionless variables, Eqs. (226)−(237)under long wavelength ( 1) and low Reynolds number (Re → 0) assumptions yield the

following expressions:

= −

=∗

+∗

= 0 (2.39)

− +

+

1

( + )2

£( + )2

¤+2

( + )2= 0 (2.40)

2

2+

1

+

−

µ −

+

¶= 0 (2.41)

2

2+

1

+

+

µ2

2+

1

+

¶= 0 (2.42)

29

= 1 + sin 2 (− ) (2.43)

= 0 at = ± (2.44)

+1 = 0 at = ± (2.45)

+2 = 0 at = ± (2.46)

+ [1

3

3+2

3

2+3

2

] =

1

( + )2

£( + )2

¤+2

( + )2at = ±

(2.47)

with

− (1 − 1)

µ −

+

¶= 0 (2.48)

−12(1−1)(+)

µ −

+

¶+1

µ −

+

¶+

µ −

+

¶= 0 (2.49)

+ 21

µ −

+

¶− (1 − 1)

µ −

+

¶= 0 (2.50)

On combining Eqs. (2.39)-(2.40) for stream function we get

∙1

( + )2

£( + )2

¤¸+2

µ1

( + )

¶= 0 (2.51)

= −µ −

+

¶"1−

µ −

+

¶2#−1 (2.52)

with = (21 − 21) the pseudoplastic fluid parameter and subscripts as the partial derivatives.

Also heat transfer rate can be defined through the following expression

=

¯

¯→

(2.53)

30

2.4 Solution procedure

Now and in small fluid parameter can be written as follows:

= 0 + 1 + (2.54)

= 0 + 1 + (2.55)

= 0 + 1 + (2.56)

= 0 + 1 + (2.57)

The solution of concentration can be obtained in the form

=

2(2 +2( − )( + ) ln(−+

))[2{−2 +2( − )( + ) ln(

+

− )}

+2( − )(−){−1 +2( + ) ln( +

+ )}+2()( + )

− ln( + )22()(2 − 2) + ln( − )22()(

2 − 2)− (−)( − )

+ ln( + )22(−)(2 − 2)− ln( + )22(−)(2 − 2)

−( + )(){−1 +2( − ) ln( +

− )}] (2.58)

Upon substitution of () and (−) from Eq. (2.45) above expression reduces to

=−

2(−2 +2( − )( + ) ln(+− ))

[2{−2 +2( − )( + ) ln( +

− )}

+(1 −2){−( − )(−)(−1 +2( + ) ln( +

+ ))

+( + )()(−1 +2( − ) ln( +

− ))}] (2.59)

where the values of () and (−) can be obtained from Eq. (2.56).

31

2.4.1 Zeroth order system

∙1

( + )

{( + )20

¸+2

∙0 +

¸= 0 (2.60)µ

2

2+

1

+

¶0 −0

µ0 −

0 +

¶= 0 (2.61)

0 +10 = 0 at = ± (2.62)

0 = 0 at = ± (2.63)

+ [1

3

3+2

3

2+3

2

] =

1

( + )2

£( + )20

¤+2 0

( + )2at = 0

(2.64)

0 = −0 +0 +

with solutions

0 =( + )1+

√1+2

1

1 +√1 +2

+( + )1−

√1+2

2

1−√1 +2+ 3 +

1

223 + 4 (2.65)

0 = −( + )−2√1+2

4(1 +2)32

[{22(2 + 2p1 +2) +2(2 +

p1 +2)}

+21( + )4√1+2{2(−2 +

p1 +2) + (−2 + 2

p1 +2)}] +2

+1 ln( + ) +122 ln( + )2 (2.66)

Heat transfer coefficient is

0 = 0()

= ( + )−1−2

√1+2

2(1 +2)[21(1 +2)( + )2

√1+2

+{222(2 +p1 +2) + 22(2 + 2

p1 +2)

−21( + )4√1+2{2(−2 +

p1 +2) + 2(−1 +

p1 +2)}}

+4122(1 +2)( + )2

√1+2

ln( + )] (2.67)

32

2.4.2 First order system

Here we have

∙1

( + )

{( + )21

¸+2

∙1 +

¸= 0 (2.68)µ

2

2+

1

+

¶1 −

½1

µ0 −

0 +

¶¾+

½0

µ1 −

1 +

¶¾= 0 (2.69)

1 +11 = 0 at = ± (2.70)

1 = 0 at = ± (2.71)

1

( + )2

£( + )21

¤+2 1

( + )2= 0 at = ± (2.72)

1 = −1 +1 +

+

µ−0 +

0 +

¶3 (2.73)

The solution expressions at this order are given by

1 = − 1

4√1 +2(3 + 42)( + )

[−32( + )−3√1+2{24 + 2(1 +

p1 +2) +2(4 + 3

p1 +2)}

−31( + )3√1+2{−24 + 2(−1 +

p1 +2) +2(−4 + 3

p1 +2)}

+(3 + 42)( + )

√1+2

2{32124

p1 +2 + 4(−1−2 +

p1 +2)( + )21}

+(3 + 42)( + )−

√1+2

2{31224

p1 +2 + 4(1 +2 +

p1 +2)( + )22}

−2p1 +2(3 + 42)( + )33] +4 (2.74)

33

1 = 2 +1

8[621

22

4

( + )2+

42( + )−2−4√1+2

(3 + 42)2(1 + 2√1 +2)

{(48 + 48p1 +2)

+46(16 + 5p1 +2) + 42(41 + 35

p1 +2) +4(181 + 117

p1 +2)}

+41( + )−2+4

√1+2

(3 + 42)2(−1 + 2√1 +2)

{(−48 + 48p1 +2) + 46(−16 + 5

p1 +2)

+42(−41 + 35p1 +2) +4(−181 + 117

p1 +2)}

+4( + )2√1+2{−11{

2(−2 +√1 +2) + 2(−1 +√1 +2)

(1 +2)32

}

+3122{6

4 +2(20− 17√1 +2)− 12(−1 +√1 +2)

2(3 + 42)(−1 +√1 +2)( + )2}}

+4( + )−2√1+2{−22{

2(2 +√1 +2) + 2(1 +

√1 +2)

(1 +2)32

}

−3212{64 +2(20 + 17

√1 +2) + 12(1 +

√1 +2)

2(3 + 42)(1 +√1 +2)( + )2

}}

+81 ln( + ) + 8(21 +12)2 ln( + )2] (2.75)

34

and the heat transfer coefficient is

1 = 1()

=

4( + )3[−621224 + 41( + )2 − 42( + )−4

√1+2

(3 + 42)2{48(1 +

p1 +2)

+46(16 + 5p1 +2) + 42(41 + 35

p1 +2) +4(181 + 117

p1 +2)}

+21

32

2( + )−2√1+2

(3 + 42)(1 +√1 +2)

{64 + 12(1 +p1 +2) +2(20 + 17

p1 +2)}

−2231

2( + )2√1+2

(3 + 42)(−1 +√1 +2)

{64 − 12(−1 +p1 +2) +2(20− 17

p1 +2)}

+41( + )4

√1+2

(3 + 42)2{48(−1 +

p1 +2) + 46(−16 + 5

p1 +2)

+42(−41 + 35p1 +2) +4(−181 + 117

p1 +2)}

+4p1 +2( + )2+2

√1+2{−11

(1 +2)32

{2(−2 +p1 +2) + 2(−1 +

p1 +2)}

+312

2

2(3 + 42)(−1 +√1 +2)( + )2{64 +2(20− 17

p1 +2)− 12(−1 +

p1 +2)}}

−4p1 +2( + )2−2

√1+2{−22

(1 +2)32

{2(2 +p1 +2) + 2(1 +

p1 +2)}

− 3212

2(3 + 42)(1 +√1 +2)( + )2

{64 +2(20 + 17p1 +2) + 12(1 +

p1 +2)}}

+8(21 +21)2( + )2 ln( + )] (2.76)

In above expressions the values of constants 0( = 1− 6) 0( = 1− 7), ( = 1−10)and 0( = 1 2) depend upon . Their values are given below:

= 83{32sin 2(− )− (1 +2) cos 2(− )} 1 =

26(5 − 4)

2 = −( + )√1+2

( − )√1+2

26(5 + 4) 3 =

2

4 = ( − )√1+2 − ( + )

√1+2

5 = ( − )√1+2

+ ( + )√1+2

35

6 = ( − )2√1+2

+ ( + )2√1+2

1 = −( − )−2√1+2

( + )−2√1+2

4√1 +2(3 + 42)6

(6 +7)

2 =( + )−2

√1+2

4√1 +2

[324

(3 + 42)( + )2− 31222( + )−2+2

√1+2

(1 +2 +p1 +2)

+32122( + )−2+4

√1+2

(1 +2 −p1 +2) +

315( + )−2+6√1+2

3 + 42

+( − )−2

√1+2

( + )2√1+2

(3 + 42)6(6 +7)] 3 = 0

4 = 8(1 +p1 +2) +4(11 + 6

p1 +2) +2(19 + 15

p1 +2)

5 = 8(−1 +p1 +2) +4(−11 + 6

p1 +2) +2(−19 + 15

p1 +2)

6 =( + )2

√1+2

( − )2[−324 + 31222( − )2

√1+2

(3 + 42)(1 +2 +p1 +2)

−32122( − )4√1+2

(3 + 42)(1 +2 −p1 +2)− 315( − )6

√1+2

]

7 =( − )2

√1+2

( + )2[324 − 31222( + )2

√1+2

(3 + 42)(1 +2 +p1 +2)

+32122( + )4

√1+2

(3 + 42)(1 +2 −p1 +2)− 315( + )6

√1+2

]

36

1 =−43

[9( − )−1−2

√1+2

(1 +2)32+

10( + )−1−2√1+2

(1 +2)32− 812

2 ln( + )

+

+41122 ln( − )2 +

8122 ln( − )

− − 41122 ln( + )2]

2 =

41[10( + )−1−2

√1+2

(1 +2)32− 812

2 ln( + )

+ − 41122 ln( + )2 +

1

3( + )

+1 ln( + )

3{9( − )−1−2

√1+2

(1 +2)32+

10( + )−1−2√1+2

(1 +2)32− 812

2 ln( + )

+

+41122 ln( − )2 +

8122 ln( − )

− − 41122 ln( + )2}]

3 =2

2 − 2+1 ln(

−

+ )

4 = 22 − 2(1 +p1 +2)(−2 +( − )) +2(6 + 4

p1 +2 −1(2 +

p1 +2)( − ))

5 = ( + )(6− 4p1 +2 +(−2 +

p1 +2)

6 = ( + )(−6− 4p1 +2 +(2 +

p1 +2)

7 = 24 +25 + 2(−1 +p1 +2)(−2 +( + ))

8 = ( − )(6− 4p1 +2 +(−2 +

p1 +2)

9 = 224 − 21( − )4√1+2{24 + 2(−1 +

p1 +2)(−2 +( − )) +28}

10 = 217( + )4√1+2

+ 22{−24 + 2(1 +p1 +2)(−2 +( + )) +26}

1 =

41[10( + )−1−2

√1+2

(1 +2)32− 812

2 ln( + )

+ − 41122 ln( + )2 +

1

3( + )

+1 ln( + )

3{9( − )−1−2

√1+2

(1 +2)32+

10( + )−1−2√1+2

(1 +2)32− 812

2 ln( + )

+

+41122 ln( − )2 +

8122 ln( − )

− − 41122 ln( + )2}]

37

2 =

41[−812

2 ln( + )

+ +

10( + )−1−2√1+2

(1 +2)32+

1

3( + )

+1 ln( + )

3{9( − )−1−2

√1+2

(1 +2)32+

10( + )−1−2√1+2

(1 +2)32

+41122 ln( − )2 +

8122 ln( − )

− }]

+( + )(6− 4p1 +2 +(−2 +

p1 +2)

2.5 Discussion

This subsection gives physical interpretation of mathematical results. Graphical illustrations

are made for comparison between straight and curved channels.

2.5.1 Velocity profile

The velocity profile reduces with dominance in the values of Hartman number (see Figs.

22( )) both in planar and curved channels. Since application of magnetic field provide

activates frictional forces to obstruct the flow even for high pressure gradient. Also more

accurate results are recorded for planar channel. Velocity of fluid is tilted towards centre of

channel as we move from curved (small ) to straight channel (large ) and has greater impact

for small values of as noticed from Fig. 23.

2.5.2 Temperature profile

Figs. 24 − 28 manifest the oscillatory response of temperature profile since convective effectsuch that convective heating is taken on upper half and cooling is taken on the lower channel

half. Decline in fluid temperature is observed with varying values of Hartman number as

observed from Figs. 24( ) in both straight and curved channels. It is in view of the fact

that larger values of produce strong magnetic field that uses fluid heat for the generation of

current in motors and so magnetic field appears to be a retarding force which causes temperature

decay. Larger values of Biot number respond in temperature development (see Figs. 25( ))

Reduction in temperature is noticed near upper half in view of heating while opposite impact

is seen near lower wall. Since thermal conductivity of fluid decreases with an increase in Biot

38

number which in turn lessens the fluid temperature. Also the temperature distribution is

non-symmetric near centre of the channel. Decreasing influence of shear thinning/thickening

parameter has been noticed towards temperature as fluid viscosity enhances with it. Moreover

the remarkable influence is observed near upper wall of the channel (see Figs. 26( )). From

Figs. 27( ) an enhancement in temperature profile is captured for larger values of Brinkman

number in view of stronger viscous dissipation effects. The curvature parameter aids in

development of fluid temperature as depicted in Fig. 28. Moreover the temperature profile

becomes non-symmetric as planar channel is approached (large ).

2.5.3 Concentration profile

Figs. 29( ) are drawn to capture results of wall parameters on concentration distribution.

With an increase in wall elastance and mass characterizing parameters 1 and 2 respectively

the concentration declines. Whereas increasing damping parameter 3 produces viscosity en-

hancement and hence the fluid concentration. Clinically the elastic arteries deoxygenated blood

nutrients from blood veins and carries it towards lungs by peristaltic pumping and so reduces

the blood concentration in the heart. Also the graphical results of Fig. 29 depict negative con-

centration for some values in accordance with clinical results. Since higher values of Schmidt

number cause decline in mass diffusion and less diffused particles make fluid less dense to en-

hance fluid concentration and so it reduces with an increase in (see Figs. 210( )). Decline

in concentration has been observed through drawn results of Figs. 211( ) in planar and

curved channels with an increase in . the reason behind this impact is that shear-thinning ef-

fect gets dominating as increases that prevents the rise in concentration profile. An increase in

concentration profile is observed with larger values of Hartman number as presented in Figs.

212( ). Since growing values of makes flow resistive and thus concentration rises. However

more signified results are captured for planner case. An increase in curvature lessens the con-

centration profile as planar channel is reached (large ) (see Fig. 213). In view dual behavior

of Biot number i-e., heating at one and cooling at the other wall corresponds to a little variation

in concentration for an increase in heat transfer Biot number (see Figs. 214( )) Whereas

larger mass transfer Biot number elevates as observed from Figs. 215( ) since the mass

transfer Biot number is inversely relate to thermal diffusivity. Thus an increase in 2 declines

39

thermal diffusion and the less diffused fluid particles perceive generation of concentration.

2.5.4 Heat transfer coefficient

Behavior of heat transfer coefficient towards involved parameters is analyzed here Involve-

ment of sinusoidal peristaltic wave train results in an oscillatory outcomes of heat transfer

coefficient. Fig. 216 shows decline in absolute heat transfer coefficient with Hartman number

due to resistive character of magnetic field. Heat is generated inside the channel with an

increase in Biot number 1 so impact of 1 on heat transfer distribution is increasing (see

Fig. 217). The sketched results of Fig. 218 perceive increasing response of heat transfer

distribution with growing values of

2.5.5 Streamlines

The variation in the streamline pattern in response to embedded parameters has been plotted

in this subsection via Figs. 219− 221( ). The results shown in Figs. 219( ) shows thatthe size of trapped bolus and number of streamlines enhances with an increase in . With

an increase in Hartman number number of circulations and as well as bolus size reduces as

observed from Figs. 220( ). Figs. 221( ) accounts for a little impact on bolus size as fluid

parameter is increased. Moreover the observed results show that the number of circulations

also remain nearly unchanged when the fluid parameter is increased.

40

2.2(a) 2.2(b)Figs. 22: Plot of velocity for Hartman number with = 02 = 01, = 003 = 02,

1 = 002 2 = 001 and 3 = 001 () = 3 () = 40

2.3

Fig. 23: Plot of velocity for curvature parameter with = 02 = 01, = 02,

1 = 002 2 = 001 3 = 001 and = 05

41

2.4(a) 2.4(b)Figs. 24: Plot of temperature for Hartman number with = 02 = 01, = 02,

1 = 002 2 = 001 3 = 001 = 3 1 = 4 and = 002 () = 2 () = 10

2.5(a) 2.5(b)Figs. 25: Plot of temperature for Biot number 1 with = 02 = 01, = 02

1 = 002 2 = 001 3 = 001 = 25 = 3 and = 04 () = 3 () = 100

42

2.6(a) 2.6(b)Figs. 26: Plot of temperature for fluid parameter with = 02 = 01, = 02

1 = 002 2 = 001 3 = 001 = 25 1 = 4 and = 3 () = 3 () = 100

2.7(a) 2.7(b)Figs. 27: Plot of temperature for Brinkman number with = 02 = 01, = 02,

1 = 002 2 = 001 3 = 001 = 25 = 03 and 1 = 6 () = 3 () = 100

43

2.8 2.9(a)

2.9(b)

Fig. 28: Plot of temperature for curvature parameter with = 02 = 01, = 02,

1 = 002 2 = 001 3 = 001 = 25 = −03 = 3 and 1 = 6

Figs. 29: Plot of concentration for wall parameters 0( = 1 2 3) with = 02 = 01,

= 02 = 25 = 002 1 = 4 2 = 5 = 1 = 2 and = 3 () = 3 ()

= 100

44

2.10(a) 2.10(b)Figs. 210: Plot of concentration for Schmidt number with = 02 = 02 = 01,

= 25 = 002 1 = 001 2 = 001 3 = 001 1 = 4 2 = 5 = 1 and = 3 ()

= 3 () = 100

2.11(a) 2.11(b)Figs. 211: Plot of concentration for fluid parameter with = 02 = 02 = 01,

= 05 = 2 1 = 001 2 = 001 3 = 001 1 = 4 2 = 5 = 1 and = 3 ()

= 3 () = 100

45

2.12(a) 2.12(b)Figs. 212: Plot of concentration for Hartman number with = 02 = 02 = 01,

= 1 = 03 1 = 002 2 = 001 3 = 001 1 = 4 2 = 5 = 2 and = 2 ()

= 3 () = 100

2.13

Fig. 213: Plot of concentration for curvature parameter with = 02 = 02 = 01,

= 1 = 03 1 = 002 2 = 001 3 = 001 1 = 4 2 = 5 = 3 and = 2 and

= 25.

46

2.14(a) 2.14(b)Figs. 214: Plot of concentration for Biot number 1 with = 02 = 02 = 01,

= 25 = 002 1 = 002 2 = 001 3 = 001 2 = 4 = 1 = 1 and = 2()

= 3 () = 100

2.15(a) 2.15(b)Figs. 215: Plot of concentration for Biot number 2 with = 02 = 02 = 01,

= 25 = 002 1 = 002 2 = 001 3 = 001 1 = 4 = 1 = 1 and = 2()

= 3 () = 100

47

2.16 2.17Fig. 216: Plot of heat transfer coefficient for Hartman number with = 02 = 01,

1 = 002 2 = 001 3 = 001 1 = 4 = 3 = 2 and = 002

Fig. 217: Plot of heat transfer coefficient for Biot number 1 with = 02 = 01,

1 = 002 2 = 001 3 = 001 = 25 = 3 = 2 and = 002

2.18

Fig. 218: Plot of heat transfer coefficient for curvature parameter with = 02 = 01

= 3 1 = 4 1 = 002 2 = 001 3 = 001 = 03 and = 25

48

2.19(a) 2.19(b)Figs. 219: Streamlines for curvature parameter with = 01 = 0 = −002 1 = 015

2 = 005 3 = 0001 and = 5 () = 3 () = 35

2.20(a) 2.20(b)Figs. 220: Streamlines for Hartman number with = 01 = 0 = −003 1 = 01

2 = 02 3 = 01 and = 2 () = 4 () = 5

49

2.21(a) 2.21(b)Figs. 221: Streamlines for fluid parameter with = 01 = 0 = 5 1 = 015 2 = 005

3 = 0001 and = 3 () = 0 () = 0002

50

2.6 Concluding remarks

The peristaltically induced flow in a curved channel containing pseudoplastic fluid has been

considered here. Wall properties and convective effects are highlighted. The flow stream is

linearized via radially imposed magnetic field. The attractive features are listed below:

• The velocity as well as temperature profile increases with an increase in elastic tension ormass per unit area. Whereas these quantities bear decreasing response upon larger values

of damping parameter in both straight and curved channels.

• With an enhancement in shear-thinning/thickening effects rise in fluid velocity and tem-perature is observed. Whereas results for concentration are opposite.

• The non-symmetric behavior of velocity is noted with curvature parameter with maxi-mum velocity in the planar channel.

• reduction in temperature is observed for Biot number .

• Fluid’s velocity and temperature decreases for larger Hartman number while concentrationrises with .

• Rise in and weaken the concentration of fluid.

• Heat transfer coefficient is more pronounced in shear-thinning when compared with shear-thickening fluids.

• The size of bolus and streamlines bear opposite impact near upper and lower walls of thechannel.

51

Chapter 3

Simultaneous effects of radial

magnetic field and wall properties on

peristaltic flow of Carreau-Yasuda

fluid in curved flow configuration

3.1 Introduction

This chapter models magnetohydrodynamic peristaltic flow of Carreau-Yasuda material in a

curved configuration with wall slip and compliant characteristics. Magnetic field is exerted in

radial direction. Viscous dissipation in heat transfer process is accounted. Nonlinear system is

modeled invoking long wavelength and low Reynolds number. The series solutions for velocity,

temperature, heat transfer coefficient and stream function are developed and examined.

3.2 Flow diagram

Fig. 31 presents the flow of Carreau—Yasuda fluid bounded in a curved channel of radius ∗

and thickness 2. A magnetic field is applied in radial direction. The peristaltic wave travelling

with velocity along the compliant boundaries generates the flow. The fluid is flowing in axial

direction with velocity ( ) and corresponds to radial direction with corresponding

52

velocity ( ). The wave geometry is represented by

= ±( ) = ±∙+ sin

2

(− )

¸ (3.1)

The fluid is electrically conducting with applied magnetic field B in the radial direction via the

strength 0 i.e.,

B =0

+∗ (3.2)

where is the unit vector in the radial direction. Employing Ohm’s law we obtain the following

expression for Lorentz force

J×B = −20

( +∗)2 (3.3)

in which is the velocity component in axial direction and the corresponding unit vector.

53

3.1. Schematic picture of the problem


The problem is formulated using conservation principles of mass, momentum and energy as

follows:

∇V = 0 (3.4)

V

= ∇τ + J×B (3.5)

= 1∇2 + τ L (3.6)

The considered problem for MHD Carreau-Yasuda fluid with viscous dissipation leads to the

following set of equations:

54

Continuity equation:

+

∗

+∗

+

+∗= 0 (3.7)

Radial and axial components of momentum equation:

∙

+

+

∗ +∗

− 2

+∗

¸= −

+

1

+∗

{( +∗)}

+∗

+∗

−

+∗ (3.8)

∙

+

+

∗ +∗

+

+∗

¸= − ∗

+∗

+

1

( +∗)2

©( +∗)2

ª+

∗

+∗

− 20

( +∗)2 (3.9)

Energy equation comprising viscous dissipation effect:

∙

+

+

∗ +∗

¸ = 1

∙2

2+

1

+∗

+

2

2

¸+ ( − )

+

µ

+

∗

+∗

−

+∗

¶ (3.10)

The stress tensor S for Carreau—Yasuda fluid model is:

S = ()A1 (3.11)

where the apparent viscosity () can be calculated by the given relation:

() = ∞ + (0 − ∞)[1 + (Γ)1 ]

−11 (3.12)

where =p2(2), D = 1

2[gradV + gradV ] and V = [( ) ( ) 0] Also

=

+

+ ∗

+∗is the material time derivative for curved channel flow and 0 the temperature

at the channel walls.

55

The boundary conditions for the presented flow comprising velocity and thermal slip effect

and compliant nature of the walls are described through the expressions

± 1

= 0 at = ± (3.13)

± = 0 at = ± (3.14)

∗

+∗

() =

∗

+∗

∙−∗

3

3+∗1

3

2+ 0

2

¸

= −∙

+

+

∗ +∗

+

+∗

¸+2∗

+∗

∙∗

+∗

+

+∗

¸− 20

( +∗)2+

( +∗)2

∙( +∗)2(

∗

+∗

−

+∗)

¸+

( +∗)2

∙

( +∗)2

¸at = ± (3.15)

Here 1 and indicate the thermal and velocity slip parameters at the upper and lower walls

of the channel respectively. In above expressions 0 and ∞ are the zero and infinite shear-rate

viscosities, 1, and Γ are the Carreau—Yasuda fluid parameters. At high shear rate range

the viscous effects can be defined through viscosities 0 and ∞ along the channel boundaries

whereas within the channel the parameters 1, and Γ predict the shear thinning/thickening

behavior. Infact these parameters regulate the fluid behavior in the non-Newtonian regime be-

tween these two asymptotic viscosities. In Carreau-Yasuda fluid model numerous concentrated

polymer solutions can be produced corresponding to the specific values of the parameters i.e.,

1 = 2 and ∞ = 0. Usually 1 = 2 governs the Carreau fluid model. The generalized parameter

1 was introduced later by Yasuda.

The relevant momentum equations (38) and (39) can be generalized in the form of single

dependent variable ∗ the stream function by the definition as follows:

= −

=∗

+∗

Defining the following dimensionless variables:

56

∗ =

∗ =

∗ =

∗ =

∗ =

∗ =

∗ =

=

− 0

0 ∗ =

2

=

∗

∗ =

=

Re =

Pr =

1 =

=

2

0 =

1 = − ∗3

3 2 =

∗13

3 3 =

30

2 2 =

20

∗ =

∗1 =

1 (3.16)

Eqs. (38)− (310) under long wavelength and low Reynolds number yield

= 0 (3.17)

− +

+

1

( + )2

£( + )2

¤+2

( + )2= 0 (3.18)

2

2+

1

+

+

µ− +

+

¶= 0 (3.19)

with the dimensionless conditions

= 1 + sin 2 (− ) (3.20)

± 1

= 0 at = ± (3.21)

± = 0 at = ± (3.22)

+ [1

3

3+2

3

2+3

2

] =

1

( + )2

£( + )2

¤+2

( + )2at = ±

(3.23)

Here asterisks have been suppressed for mathematical simplification of problem and and

1 represent the velocity and thermal slip parameters respectively. Combining Eqs. (317) and

(318), we obtain the following expression

∙1

( + )2

£( + )2

¤¸+2

µ1

( + )

¶= 0 (3.24)

57

=

µ− +

+

¶ ∙1 +1

(1− )(− 1)1

µ− +

+

¶1¸ (3.25)

Here =∞0and = Γ

represent the viscosity ratio parameter and Weissenberg number

respectively. It should be noted that for = 1 or = 0 the equations for the viscous fluid

are recovered. Moreover for = 0 the problem reduces to that of hydrodynamic case. The

value of Yasuda parameter 1 = 1 is taken in this problem.


We intend to develop the solution by perturbation technique. For that we expand the quantities

as follows:

= 0 +1 + (3.26)

= 0 +1 + (3.27)

= 0 +1 + (3.28)

= 0 +1 + (3.29)

3.4.1 Zeroth order systems and solutions

Here we have

∙1

( + )

{( + )20

¸+2

∙0 +

¸= 0 (3.30)µ

2

2+

1

+

¶0 +0

µ−0 +

0 +

¶= 0 (3.31)

= 1 + sin 2 (− )

0 ± 10

= 0 at = ± (3.32)

0 ± 0 = 0 at = ± (3.33)

58

+ [1

3

3+2

3

2+3

2

] =

1

( + )2

£( + )20

¤+2

( + )2at = ±

(3.34)

0 = −0 +0 +

The solutions of stream function and temperature at this order are

0 =( + )1+

√1+2

1

1 +√1 +2

+( + )1−

√1+2

2

1−√1 +2

+ 3 +1

223 + 4 (3.35)

0 = 2 + 1( + ) +122( + )2 − ( + )−2

√1+2

4(1 +2)32

[22(2 + 2p1 +2

+22 +2p1 +2) +21( + )4

√1+2

(−2 + 2p1 +2 − 22

+2p1 +2)] (3.36)

Heat transfer coefficient at this order is

0 = 0()

3.4.2 First order system and solutions

At this order the corresponding systems are:

∙1

( + )

{( + )21

¸+2

∙1 +

¸= 0 (3.37)

µ2

2+

1

+

¶1 +1

µ−0 +

0 +

¶+0

µ−1 +

1 +

¶= 0 (3.38)

= 1 + sin 2 (− )

1 ± 11

= 0 at = ± (3.39)

1 ± 1 = 0 at = ± (3.40)

1

( + )2

£( + )20

¤+2

( + )2= 0 at = ± (3.41)

59

1 = −1 +1 +

+ (1− )(− 1)(−0 +0 +

)2

The solution expressions are

1 =−22( + )−2

√1+2

( − 1)(− 1)(2 + 22 + 2√1 +2 + 32

√1 +2)√

1 +2(8 + 92)

−21( + )2

√1+2

( − 1)(− 1)(−2− 22 + 2√1 +2 + 32

√1 +2)√

1 +2(8 + 92)

+( + )

√1+2

1( + )

1 +√1 +2

+( + )−

√1+2

2( + )

1−√1 +2

+ 3 +1

223

+4 (3.42)

1 = 2 − 22( + )−2√1+2

(2 + 22 + 2√1 +2 +2

√1 +2)

2(1 +2)32

−11( + )2√1+2

(−2− 22 + 2√1 +2 +2

√1 +2)

2(1 +2)32

−32(− 1)( + )−1−3

√1+2

(1 + 3√1 +2)(8 + 92)2

[456 + 64 + 64p1 +2 +

82(23 + 19p1 +2) + 44(41 + 24

p1 +2)]−

122

2(− 1)( − 1)(32 + 8√1 +2)( + )−1−

√1+2

(1 +√1 +2)(8 + 92)

+

2122(− 1)( − 1)(32 − 8√1 +2)( + )−1+

√1+2

(−1 +√1 +2)(8 + 92)−

31(− 1)( + )−1+3√1+2

(1 + 3√1 +2)(8 + 92)2

[−456 − 64 + 64p1 +2

+82(−23 + 19p1 +2) + 44(−41 + 24

p1 +2)] + 1( + )

+2( + )2(21 +12) (3.43)

Heat transfer coefficient is as follows:

1 = 1()

60

Here the values 0( = 1− 4) 0( = 1− 4) 0( = 1 2) and 0( = 1− 2) are constants(with respect to ) given as follows:

= 83{32sin 2(− )− (1 +2) cos 2(− )} 1 =

26(5 − 4)

2 = −( + )√1+2

( − )√1+2

26(5 + 4) 3 =

2

4 = ( − )√1+2 − ( + )

√1+2

5 = ( − )√1+2

+ ( + )√1+2

1 = −( + )√1+2

( − )√1+2

2 2 =

2

3 = ( − )√1+2 − ( + )

√1+2

4 = ( − )√1+2

+ ( + )√1+2

1 = ( + ) +122( + )2 − ( + )−2

√1+2

4(1 +2)32

[22(2 + 2p1 +2

+22 +2p1 +2) + 21( + )4

√1+2

(−2

+2p1 +2 − 22 +2

p1 +2)] + ( − )

√1+2

+ ( + )√1+2

2 =−43

[( − )−1−2

√1+2

(1 +2)32+( + )−1−2

√1+2

(1 +2)32− 812

2 ln( + )

+

+( − )√1+2

+ ( + )√1+2

+ 4122 ln( − )2

+812

2 ln( − )

− − 4122 ln( + )2]

61

1 =22( + )−2

√1+2

(2 + 22 + 2√1 +2 +2

√1 +2)

2(1 +2)32

−11( + )2√1+2

(−2− 22 + 2√1 +2 +2

√1 +2)

2(1 +2)32

+82(23 + 19p1 +2) + 44(41 + 24

p1 +2)]−

122

2(− 1)( − 1)(32 + 8√1 +2)( + )−1−

√1+2

(1 +√1 +2)(8 + 92)

+

2122(− 1)( − 1)(32 − 8√1 +2)( + )−1+

√1+2

(−1 +√1 +2)(8 + 92)−

31(− 1)( + )−1+3√1+2

(1 + 3√1 +2)(8 + 92)2

(−456 − 64 + 64p1 +2)

2 =1

22

2(− 1)( − 1)(32 + 8√1 +2)( + )−1−

√1+2

(1 +√1 +2)(8 + 92)

+22( + )−2

√1+2

(2 + 22 + 2√1 +2 +2

√1 +2)

2(1 +2)32

−11( + )2√1+2

(−2− 22 + 2√1 +2 +2

√1 +2)

2(1 +2)32

+212

2(− 1)( − 1)(32 − 8√1 +2)( + )−1+

√1+2

(−1 +√1 +2)(8 + 92)

−

31(− 1)( + )−1+3√1+2

(1 + 3√1 +2)(8 + 92)2

[−456 − 64 + 64p1 +2

+2( + )2(21 +12) + 82(−23 + 19

p1 +2)

+44(−41 + 24p1 +2)]

The resulting functions of and are then used to sketch the plots. The physical interpretation

of the embedded parameters have been examined in the next section.

3.5 Discussion

Influence of pertinent variables on the hydromagnetic flow of Carreau-Yasuda fluid are plotted

in the form of graphs. In this section the velocity, temperature, heat transfer coefficient and

streamlines are discussed physically.

62

3.5.1 Velocity distribution

The graphical results captured in Figs. 32 − 38 exhibit the variation in velocity with theinvolved parameters. Fig. 32 illustrates the increasing behavior of velocity with wall elastic

parameters 1 2 whereas damping constant 3 causes reduction in velocity. The results

are in contrast to blood vessels in which increase in elasticity (1) or expansion of mass per

unit area (2) enhances the blood velocity. However damping (3) enhances the resistance

to flow of blood and velocity inside the channel reduces. The results drawn in Fig. 33 show

oscillatory behavior for the increasing values of Weissenberg number The velocity profile

has increasing impact near lower wall whereas the decreasing behavior is noted near the upper

wall of channel. Fig. 34 shows reduction in velocity via increasing values of Hartman number

due to opposing nature of applied magnetic field. Impact of velocity slip parameter on

velocity profile is decreasing (see Fig. 35) and the maximum impact is noted near the centerline

of curved channel. From Figs. 36 and 37 similar results are recorded for the increasing values

of viscosity ratio and shear thinning/thickening parameters and Both parameters show

decrease in velocity near lower wall whereas the velocity changes behavior at upper wall and

these show opposite results. In fact near negative half of channel the viscosity and shear

thickening effect dominant which reduce the velocity whereas these effects are submissive near

positive side (see Figs. 36 and 37). The results for the variation in curvature parameter are

prepared in Fig. 38. The graph indicates the decrease in velocity along lower wall whereas the

behavior changes near centerline of the channel opposite effect is seen along upper wall.

3.5.2 Temperature and heat transfer coefficient

Figs. 39 − 319 give the physical interpretation of embedded parameters on temperature dis-tribution and heat transfer coefficient An increase in temperature distribution is noted in

the channel for wall elastic and mass characterizing parameters 1 and 2. However damping

coefficient 3 reduces temperature similar to velocity profile (see Fig. 39). The response of

temperature distribution towards growing values of thermal slip parameter 1 is increasing (see

Fig. 310). Fig. 311 eludicates that temperature of fluid decreases near lower wall and it

increases near upper wall of curved channel. The variations in Brinkman number exhibit

increase in temperature distribution. It is due to the fact that comprises viscous dissipa-

63

tion effects which strengthen the temperature distribution (see Fig. 312). The velocity slip

parameter enhances the fluid temperature more efficiently near the central part of channel

(see Fig. 313). The results observed from Figs. 314 and 315 established opposite influence

on temperature distribution for curvature parameter and Hartman number . Fig. 314

clearly indicates that curved situation preserves larger heat generation inside the channel when

compared with planer channel. Further the Hartman number opposes the velocity of fluid

particles since the slow moving particles have less molecular vibrations as well as temperature.

The outcomes of thermal and velocity slip parameters on magnitude of heat transfer coeffi-

cient are found opposite (see Figs. 316 and 317). Larger values of 1 enhance the transfer

of heat as observed from Fig. 316. On the other hand the larger values of causes reduction

in transfer of heat inside the channel (see Fig. 317). From Figs. 318 and 319 the results on

the heat transfer coefficient for the Hartman number and Wessienberg number are

similar both these parameters show decay in heat transfer coefficient .

3.5.3 Streamlines pattern

Under the considered analysis some of the streamlines deviate from the boundary wall and align

themselves in the internally circulating shape like bolus. This captivating phenomenon is an

interesting feature of peristalsis. The results plotted in Figs. 320− 322 study the streamlinepattern with the variation in curvature parameter , Hartman number and velocity slip

parameter Figs. 20 (() and ()) show that the increasing reduce the size of bolus and

enhances the number of streamlines. Figs. 321 (() and ()) reveal that increasing values of

enhance the number of circulating streamlines in the upper half of channel. The bolus size

also enhances. However size of bolus reduces and the pattern of streamlines remain unchanged

in the upper half of channel. Larger values of reduce trapping bolus size near lower half

where the number of streamlines in both upper and lower halves of the channel remain nearly

unchanged (see Figs. 322 (() and ()).

64

3.2 3.3Fig. 32: Variation in velocity for wall parameters 1 2 3 with = 01 = 01, = 02,

= 3 = 01 = 02 = 05 = 01 = 05 and = 25

Fig. 33: Variation in velocity for Weissenberg number with = 01 = 01, = 02,

= 01 = 02 = 05 = 3 1 = 003 2 = 002 = 05 and 3 = 001

3.4 3.5Fig. 34: Variation in velocity for Hartman number with = 01 = 01, = 02,

= 02 = 01 = 02 = 3 = 05, 1 = 003 2 = 002 and 3 = 001

Fig. 35: Variation in velocity for velocity slip parameter with = 01 = 01, = 02,

= 05 = 02 = 01 = 3 = 05, 1 = 003 2 = 002 and 3 = 001

65

3.6 3.7Fig. 36: Variation in velocity for viscosity ratio parameter with = 01 = 01, = 02,

= 05 = 02 = 01 = 3 = 05, 1 = 003 2 = 002 and 3 = 001

Fig. 37: Variation in velocity for fluid parameter with = 01 = 01, = 02, = 05

= 02 = 01 = 3 = 02, 1 = 003 2 = 002 and 3 = 001

3.8 3.9Fig. 38: Variation in velocity for curvature parameter with = 01 = 01, = 02,

= 05 = 02 = 01 = 05 = 02, 1 = 003 2 = 002 and 3 = 001

Fig. 39: Variation in temperature for wall parameters 1 2 3 with = 01 = 01

= 01, = 02, = 3 = 02 = 05 = 02 = 4 1 = 01 = 05 and = 25

66

3.10 3.11Fig. 310: Variation in temperature for thermal slip parameter 1 with = 01 = 02,

= 03, = 01 = 3 = 02 = 05 = 02 = 4 = 05, = 25

1 = 003 2 = 002 and 3 = 001

Fig. 311: Variation in temperature for Weissenberg number with = 01 = 02,

= 03, = 01 = 3 = 02 = 05 1 = 01 = 05, = 25 = 4

1 = 003 2 = 002 and 3 = 001

3.12 3.13Fig. 312: Variation in temperature for Brinkman number with = 01 = 02, = 03,

= 3 = 02 = 05 1 = 01 = 02, = 25 = 02 = 01 1 = 003 2 = 002

and 3 = 001

Fig. 313: Variation in temperature for velocity slip parameter with = 01 = 02,

= 03, = 3 = 02 = 05 1 = 01 = 02, = 25 = 02 1 = 003 2 = 002

and 3 = 001

67

3.14 3.15Fig. 314: Variation in temperature for curvature parameter with = 01 = 02, = 03,

= 01 = 01 = 05 1 = 01 = 02, = 25 = 02 1 = 003 2 = 002 and

3 = 001

Fig. 315: Variation in temperature for Hartman number with = 01 = 02, = 03,

= 01 = 01 = 05 1 = 01 = 02, = 3 = 02 1 = 003 2 = 002 and

3 = 001

3.16 3.17Fig. 316: Variation in heat transfer coefficient for thermal slip parameter 1 with = 01

= 02, = 01 = 01 = 05 = 05 = 02, = 3 = 001 1 = 03 2 = 02 and

3 = 01

Fig. 317: Variation in heat transfer coefficient for velocity slip parameter with = 01

= 0, 1 = 01 = 01 = 05 = 05 = 02, = 3 = 001 1 = 03 2 = 02 and

3 = 01

68

3.18 3.19Fig. 318: Variation in heat transfer coefficient for Hartman number with = 01 = 0,

= 01 1 = 01 = 01 = 05 = 02, = 3 = 001 1 = 03 2 = 02 and

3 = 01

Fig. 319: Variation in heat transfer coefficient for Weissenberg number with = 01

= 0, = 05 1 = 01 = 01 = 05 = 05 = 02, = 3 = 01 1 = 03

2 = 02 and 3 = 01

3.20(a) 3.20(b)Figs. 320: Streamlines pattern for curvature parameter when = 01 = 0, = 5

= 01 = 05 = 01 1 = 03 2 = 02, 3 = 01 with (a) = 3 (b) = 5

69

3.21(a) 3.21(b)Figs. 321: Streamlines pattern for Hartman number when = 01 = 0, = 5 = 01

= 05 = 01 1 = 03 2 = 02, 3 = 01 with (a) = 5 (b) = 8

3.22(a) 3.22(b)Figs. 322: Streamlines pattern for velocity slip parameter when = 01 = 0, = 5

= 01 = 05 = 5 1 = 03 2 = 02, 3 = 01 with (a) = 001 (b) = 002

70


Peristaltic transport of Carreau-Yasuda fluid in curved channel with velocity and thermal slip

effects is studied. The channel boundaries satisfy compliant property. Major findings are:

• The velocity and temperature profiles via wall properties are similar in a qualitative sense.

• The Weissenberg number has dual impact on velocity and temperature.

• The velocity and temperature are decreasing functions of Hartman number.

• Opposing impacts for velocity slip is noted for temperature and velocity distributions.

• Temperature is increasing function of thermal slip parameter.

• Curvature parameter has greater impact on temperature in curved geometry.

• Minute response of velocity slip is recorded on streamline pattern.

71

Chapter 4

Heat transfer analysis for peristalsis

of MHD Carreau fluid in curved

channel through modified Darcy law

4.1 Introduction

This chapter is developed to investigate heat transfer on peristaltic flow of Carreau fluid in

a curved channel with rhythmic contraction and expansion of waves along the walls (similar

to blood flow in tubes). Magnetic field is imposed in radial direction. Heat transfer aspect

is further studied with viscous dissipation effect. The curved channel walls are subjected to

flow and thermal slip conditions. In addition the flow stream comprised porous medium whose

mathematical form is followed by modified Darcy law. Numerical solutions for velocity and

temperature are obtained. The striking features of flow and temperature characteristics for

the involved sundry parameters are examined by plotting graphs. An excellent agreement for

results is found when compared with existing literature in a limiting sense.

4.2 Flow diagram

Here we modelled the mathematical description of flow and heat transfer analysis by considering

an incompressible Carreau fluid in a curved channel of thickness 2 Curved channel is configured

72

in a circle of radius ∗. Fluid saturates a porous space inside the compliant boundaries. In

addition fluid is set to electrical conduction in radial direction (see Fig. 4.1) such that:

B = (0

+∗ 0 0) (4.1)

where the drop of induced magnetic field is followed by small Reynolds number. The Ohm’s

law aids in providing the required term that must be added in flow governing equation

J×B = (0 −20

( +∗)2 0) (4.2)

The geometry of the curved channel is aligned such that gravitational effects are active. The

dynamics of fluid is activated through the peristaltic waves travelling along the channel bound-

aries. Thus the relative positions of the channel walls in radial direction is the combination of

peristaltic wave and half width of the channel i.e.,

= ±( ) = ±∙+ sin

2

(− )

¸ (4.3)

73

Fig. 4.1. Pictorial representation of problem.


Continuity, momentum and energy conservation principles lead to

+

∗

+∗

+

+∗= 0 (4.4)

∙

− 2

+∗

¸= −

+

1

+∗

{( +∗)}

+∗

+∗

−

+∗+ (4.5)

∙

+

+∗

¸= − ∗

+∗

+

1

( +∗)2

©( +∗)2

ª+

∗

+∗

+ ( − 0) + − 20

( +∗)2 (4.6)

74

∙

¸= 1∇2 + τ gradV (4.7)

The Cauchy stress tensor τ and extra stress tensor for Carreau fluid model is:

τ = −p+S (4.8)

S = ()A1 (4.9)

in which the apparent viscosity () can be obtained through the following relation:

() = ∞ + (0 − ∞)[1 + (Γ)2]

−12 (4.10)

where =p2(D2) and D = 1

2[gradV+gradV ] 0 and ∞ the zero and infinite shear-rate

viscosities and and Γ the Carreau fluid parameters. For ∞ = 0 Eqs. (4.9) and (4.10) give

S = 0[1 + (Γ)2]

−12 A1 (4.11)

Also

∇2 = ( ∗

+∗)2

2

2+

1

+∗

{( +∗)}

The parameters in governing equations symbolizes the velocityV, the temperatures at the lower

and upper channel walls 0 1 the temperature of fluid and the Darcy resistance in porous

medium = ( 0). By modified Darcy law the pressure drop and velocity are related

∇ = −∗

K¯

[0[1 + (Γ)2]

−12 ]V (4.12)

where K¯demonstrates the permeability and ∗ the porosity of porous medium. The above

generalized form is capable of recovering the results of Darcy law by assuming = 1. Further

resistance in flow having porous space can be encountered as pressure gradient. Thus Eq. (4.12)

can be written as:

R =−∗K¯

[0[1 + (Γ)2]

−12 ]V (4.13)

The extra stress components and of in Carreau fluid can be obtained using Eq.

(4.11). Further the relevant boundary conditions are of considerable importance owing to flow

75

analysis in practical situations. For the present flow situation the slip boundary conditions

in terms of velocity and temperature are utilized. The slip boundaries play a significant role

in polishing valves of artificial heart and internal cavities. Moreover wall properties at the

boundaries have the following form:

± = 0 at = ± (4.14)

±

=

⎧⎨⎩ 1

0

⎫⎬⎭ at = ± (4.15)

∗

+∗

∙−∗

3

3+∗1

3

2+ 0

2

¸ = −

∙

+

+∗

¸+

1

( +∗)2

©( +∗)2

ª+

∗

+∗

+ ( − 0)− 20

( +∗)2+ at = ± (4.16)

Consider stream function ( ) and the non-dimensional quantities as follows:

∗ =

∗ =

∗ =

∗ =

∗ =

= − 0

1 − 0 ∗ =

∗ =2

=

=

2(1 − 0)

Re =

Pr =

1

=2

(1 − 0) = Pr =

=

∗

1 = − ∗3

3 2 =

∗13

3 3 =

30

2 =

K¯

∗2

2 =20

=

=

=

Γ

(4.17)

= −∗

=

+

∗

Here denotes the Darcy number and the slip parameters for velocity and temperature

respectively.

Approximation of long wavelength is commonly utilized. Sure in such cases channel half

76

width is small in comparison to wavelength. For instance = 125 and = 801 in chyme

transport though small intestine justifies this assertion. Thus = 0156 Thus dropping

asterisks one arrives at

= 0 (4.18)

− +

+

1

( + )2

£( + )2

¤+

+1

"1 +2

(− 1)2

µ−

2

2+

1

+

¶2#= 0 (4.19)

2

2+

1

+

+

µ−

2

2+

1

+

¶2 "1 +2

(− 1)2

µ−

2

2+

1

+

¶2#= 0

(4.20)

= 1 + sin 2 (− ) (4.21)

−± = 0 at = ± (4.22)

±

=

⎧⎨⎩ 1

0

⎫⎬⎭ at = ± (4.23)

+

∙1

3

3+2

3

2+3

2

¸ =

1

( + )2

£( + )2

¤+

+1

"1 +2

(− 1)2

µ−

2

2+

1

+

¶2# at = ± (4.24)

Combination of Eqs. (4.18) and (4.19) yield

∙1

( + )

£( + )2

¤+ ( + )

¸+

"( + )

"1 +2

(− 1)2

µ−

2

2+

1

+

¶2##= 0 (4.25)

=

µ−

2

2+

1

+

¶"1 +2

(− 1)2

µ−

2

2+

1

+

¶2# (4.26)

77

Note that for = 1 or = 0 the results of the viscous fluid containing porous space can be

recovered as a special case of above problem.

4.4 Solution and discussion

The problem for curved channel seems more complex than the planar case. As a result the

solutions of such problems cannot be obtained exactly. However technological advancement

has provide variety of built-in softwares that can provide the best possible approximation to

such problems numerically. The considered problem is one such example whose complexity

in terms of solutions is encountered via built-in command NDSolve in computational software

. Therefore in this section the numerical solutions corresponding to axial velocity

, temperature and heat transfer rate have been plotted. Particularly the developments

of , and with variations in Grashof number , wall compliant parameters 1 2 3

Darcy number , curvature parameter , Brinkman number , Hartman number , velocity

and thermal slip parameters respectively, fluid parameter and Weissenberg number

are physically emphasized.


Velocity in response to variation of different involved parameters are recorded in this subsection.

This objective is achieved through Figs. 42 (−) The captured results of Fig. 42() manifestdecreasing behavior of towards an increase in . Magnetic field is beneficial in treating

diseases like joint problems, migraine headaches, cancer, depression etc. An increase in

shows velocity enhancement since more pores aid the flow speed (see Fig. 42()). In reference

to blood flow the pores in walls of blood capillaries allow exchange of water, oxygen and other

significant nutrients between the blood and tissues. The axial velocity is found an increasing

function of velocity slip parameter (see Fig. 42()). The wall membrane parameters 1 and 2

cause an enhancement in since elasticity and mass per unit area play effective role in flow and

perfusion of blood in blood veins and arteries whereas damping (3) resists such movements and

so decreasing response via 3 is noticed (see Fig. 42()). The results of Fig. 42() correspond

to the dual response of on − rises near positive part of the channel whereas it falls

78

near negative part. An increase in curvature tends to reduce the velocity (see Fig. 42()).

Moreover the inclination of the graph is reduced as is increased (planer channel). The axial

velocity is an increasing function of Grashof number (see Fig. 42()). From Fig. 42() minute

effect of Weissenberg number is captured in absence of slip velocity ( = 0)

4.4.2 Temperature distribution

Variation in temperature distribution through embedded parameters is discussed via Figs.

43( − ). Reduction in is noticed for larger magnetic field (see Fig. 43()). Impression

of towards is found increasing. In fact from Fig. 43(b) the heat generation has specific

attributes associated with viscous dissipation. The thermal slip elevates the temperature (see

Fig. 43()). Actually in slip flow the shear work in response to slip at the wall is included to

calculate the heat flux from the wall. Porosity in a medium is capable of providing speedy flow

and the fluid particles moving with greater speed generate heat. Hence an increase in is noticed

when is enhanced (see Fig. 43()). Elasticity of walls comprised temperature enhancement

while damping acts alternately for (see Fig. 43()). An increase in corresponds to

decline in . However greater impact is seen near negative part of the curved channel (see Fig.

43()). Increasing fluid parameter elevates the fluid temperature (see Fig. 43()). Decrease

in temperature is noticed for planer channel case ( →∞) (see Fig. 43()).

4.4.3 Heat transfer rate

This subsection explains variation of absolute heat transfer rate effected through Figs. 44(−). Refer to peristaltic wave train along the curved channel boundaries the results are found

oscillatory. Enhancement of is shown in Fig. 44() for larger values of. Dominant behavior

of is examined with higher in Fig. 44(). The results drawn in Fig. 44() has increasing

response of towards . Moreover the heat transfer rate rises for larger (see Fig. 44()).

79

4.2(a) 4.2(b)

4.2(c) 4.2(d)

80

4.2(e) 4.3(f)

4.3(g) 4.2(h)Figs. 42: Axial velocity () variation with = 02 = 01 = 02

81

4.3(a) 4.3(b)

4.3(c) 4.3(d)

82

4.3(e) 4.3(f)

4.3(g)

Figs. 43: Temperature () variation with = 02 = 01 = 02

83

4.4(a) 4.4(b)

4.4(c) 4.4(d)

Figs. 44: Heat transfer coefficient () variation with = 01 = 02

84


Here effort is made to explore the flow and heat transfer analysis in MHD flow of Carraeu fluid

through porous medium filling curved flow configuration. The viscous dissipation and mixed

convection are also highlighted. The striking attributes of this study are summarized below:

• The non-symmetric velocity is found an increasing function of

• Fluid particles for higher gravity move with greater speed in downward direction andhence rise in is noticed with .

• Wall compliance causes an enhancement of and since elasticity in various human

organs aids breathing and fluid flow.

• Velocity and temperature rise in response to corresponding slip parameters (as shear stresseffects are accounted with slip at the vessel walls).

• Generation of and is captured with larger .

• Weissenberg number preserve dual impact of velocity and temperature decays for .

• Reduction in , and is noticed with an increase in curvature (as movement of fluid

particles in curved channel requires more energy).

• Addition of pores favours and .

85

Chapter 5

Mixed convective peristaltic flow of

Sisko fluid in curved channel with

homogeneous-heterogeneous

reaction effects

5.1 Introduction

This chapter emphasizes for the simultaneous effects of shear thinning and shear thickening in

mixed convective peristaltic flow of Sisko fluid. The channel boundaries are considered curved

in shape with compliant properties. The severity of gravitational effects are retained in the flow.

Viscous dissipation and thermal radiation are retained. In addition homogeneous-heterogeneous

reactions are also examined. The simplified system is then addressed numerically. The results

signify the pronounced behavior of velocity and temperature corresponding to shear thinning.

Concentration bears opposite response for homogeneous and heterogeneous parameters.


Mathematical formulation for an incompressible Sisko fluid (comprising shear thinning and

shear thickening effects) bounded in a curved channel of inner radius ∗ and separation 2

86

has been modelled in this section. The dynamics of flow geometry is configured in such a way

that peristaltic wave while propagating along the channel boundaries in radial direction ()

generates the flow (see Fig. 51). In addition the channel geometry is such that the gravitational

effects are not ignored. The relevance of compliance in terms of wall’s stiffness, elasticity and

damping is also present. Further the flow subject to a simple homogeneous and heterogeneous

reaction model comprised of two chemical species and (having concentration values and

respectively) is taken in to account.

Thus the following expression gives the relative position of the wall surface in radial direction

as sum of separation between channel and sinusoidal peristaltic wave

= ±( ) = ±∙+ sin

2

(− )

¸ (5.1)

5.1. Physical description of problem.

The homogeneous-heterogeneous reaction effects are assumed to have the form:

+ 2 → 3 = 2

87

However corresponding to catalyst surface the single, isothermal, first order chemical reaction

preserves the following mathematical form:

→ =

For the two reactions the same temperature value is assumed where and denote the rate

constants.

The mathematical description of the problem relates conservation principles of mass, mo-

mentum and energy. Thus in present case one obtains

+

∗

+∗

+

+∗= 0 (5.2)

∙

− 2

+∗

¸= −

+

1

+∗

{( +∗)}

+∗

+∗

−

+∗ (5.3)

∙

+

+∗

¸= − ∗

+∗

+

1

( +∗)2

©( +∗)2

ª+

∗

+∗

+ ( − 0) + (− 0) (5.4)

∙

¸= 1∇2 −

µ−16∗ 303∗

¶+ τ gradV (5.5)

where the homogeneous-heterogeneous reaction effects can be encountered through the following

equations [21,27]:

=

¡∇2¢− 2 (5.6)

=

³∇2

´+

2 (5.7)

The Cauchy stress tensor τ and extra stress tensor for Sisko fluid model are:

88

τ = −I+ S (5.8)

S =h+ |Π|−1

iA1 (5.9)

where Π = 12(A21) is the second invariant of symmetric part of velocity gradient. In Sisko fluid

model , and (≥ 0) represent the shear rate viscosity, consistency index and power-law indexrespectively. Involvement of power index provides an edge to Sisko fluid as shear thinning

( 1), shear thickening ( 1). Newtonian fluid ( = 1 = 0 = or = 0 = )

behavior can be considered using this model. Also

∇2 = ( ∗

+∗)2

2

2+

1

+∗

{( +∗)}

The quantities in above mentioned flow governing equations represent the the thermal expansion

coefficient the concentration expansion coefficient , the temperatures at the left and right

channel walls 0 1 the concentrations at both the walls, the uniform concentration of reactant

as 0, the homogeneous diffusion coefficient , the heterogeneous diffusion coefficient ,

the Stefan-Boltzmann constant ∗, the mean absorption coefficient ∗, the temperature of fluid

and the stress components whose values can be evaluated using Eq. (59). Note

that in writing Eq. (55) the Rosseland approximation for radiative heat flux is used to obtain

the corresponding radiation term.

For considered problem, the boundary conditions include the no-slip condition, prescribed

surface temperature, relevant conditions corresponding to homogeneous-heterogeneous reac-

tions and the compliant properties of wall as follows:

= 0 at = ± (5.10)

=

⎧⎨⎩ 1

0

⎫⎬⎭ , at = ± (5.11)

89

→ 0 at = −,

− = 0 at = (5.12)

→ 0 at = −,

+ = 0 at = (5.13)

Physically compliant effect demonstrates the decomposition of applied pressure in terms of wall

elastic tension (∗) mass per unit area (∗1) and viscous damping (0) i-e.,

=

∙−∗

2

2+∗1

2

2+ 0

¸

which after using Eq. (54) becomes

∗

+∗

∙−∗

3

3+1

3

2+ 0

2

¸ = −

∙

+

+∗

¸+

1

( +∗)2

©( +∗)2

ª+

∗

+∗

+ ( − 0) + (− 0) at = ± (5.14)

5.2.1 Dimensionless formulation

Non-dimensionalizing the above flow stream and invoking the definition of stream function

∗ ( ) will lead to required set of equations through

∗ =

∗ =

∗ =

∗ =

∗ =

=

∗

= − 0

1 − 0 ∗ =

∗ =2

∗ =

=

202

=

2(1 − 0)

=

02

∗ =

³

´−1Re =

1 =

∗ =

0 ∗ =

0, Pr =

1 =

=2

(1 − 0) = Pr =

16∗ 303∗

=

1 = − ∗3

3 2 =

∗13

3 3 =

30

2 =

(5.15)

= −∗

=

+

∗

90

where the above symbols exhibit the definitions of following physical parameters: the radia-

tion parameter the Grashof numbers of temperature and concentration, the Eckert

number, the Brinkman number, 1 the ratio of diffusion coefficient , the strength mea-

suring parameters (for homogeneous and heterogeneous reactions respectively), the Schmidt

number and ∗ the Sisko fluid parameter

Large wavelength has special relevance in physiological processes since it has major role

in the blood vessels. Properties of blood can be predicted using wave propagation in vessels.

Wavelength must be long while amplitude should be small comparatively to avoid the bending

rigidity of vessel wall and nonlinear convective fluid inertia. In skin laser treatment the waves

with long wavelength are increasingly being used for the removal of fine lines and wrinkles

as photons of larger wavelengths reduces the energy of radiation that produces the biological

damages. Also in radar systems the large wavelengths are typically used to detect severe

weather conditions since longer wavelengths absorb the intervening particles and a distant

thunderstorm behind a closer thunderstorm will appear on the radar screen with its proper

intensity. Implication of simplified assumptions of long wavelength and low Reynolds number

[7] and dropping the asterisks we get the required set of equations as follows:

= 0 (5.16)

− +

+

1

( + )2

£( + )2

¤+ ( +) = 0 (5.17)

2

2+

1

+

+

µ− +

+

¶+Pr

µ2

2

¶= 0 (5.18)

1

µ2

2+

1

+

¶−2 = 0 (5.19)

1

µ2

2+

1

+

¶+2 = 0 (5.20)

where from Eq. (59)

=

µ− +

+

¶"1 + ∗

µ− +

+

¶−1#

91

and dimensionless conditions

= 1 + sin 2 (− ) (5.21)

= 0 at = ± (5.22)

=

⎧⎨⎩ 1

0

⎫⎬⎭ at = ± (5.23)

= 1 = −

− = 0 at = (5.24)

= 0 = − 1

+ = 0 at = (5.25)

+

∙1

3

3+2

3

2+3

2

¸ =

1

( + )2

£( + )2

¤+( +) at = ±

(5.26)

From application point of view the diffusion coefficients of chemical species and are of

comparable size that leads to the assumption that diffusion coefficients are equal. The equality

of diffusion coefficients and i.e., 1 = 1 leads to the following relation:

+ = 1 (5.27)

Moreover upon elimination of pressure between Eqs. (516) and (517) the stream function equa-

tion can be obtained. Thus combining Eqs. (519) and (520) for homogeneous-heterogeneous

reactions and Eqs. (516) and (517) for stream function, the above system takes the form:

∙1

( + )

£( + )2

¤+ ( + )( +)

¸= 0 (5.28)

2

2+

1

+

+

µ− +

+

¶+Pr

µ2

2

¶= 0 (5.29)

1

µ2

2+

1

+

¶−(1− )2 = 0 (5.30)

= 1 + sin 2 (− ) (5.31)

= 0 at = ± (5.32)

92

=

⎧⎨⎩ 1

0

⎫⎬⎭ at = ± (5.33)

= 1 = −

− = 0 at = (5.34)

+

∙1

3

3+2

3

2+3

2

¸ =

1

( + )2

£( + )2

¤+( +) at = ±

(5.35)

=

µ− +

+

¶"1 + ∗

µ− +

+

¶−1#

where the lower case letters represent partial derivatives. It is remarkable to notice that above

system is capable of recovering shear thinning/thickening and Newtonian behavior by suitable

values of the parameters. However in this chapter shear thinning and shear thickening effects

will be emphasized and compared through graphs.


For the above problem the obtained system of equations is highly nonlinear and coupled. Hence

we seek an approximation to the solution by an appropriate numerical method.

built-in routine NDSolve provides such tool of numerical computation for solving ODEs espe-

cially in bounded domains where minimum to maximum range is attainable. Thus non-linearity

of above system is encountered directly via NDSolve to graphically sketch the axial velocity

, temperature , heat transfer coefficient and homogeneous-heterogeneous reaction effects

. This section is made to explore shear thinning/thickening and homogeneous-heterogeneous

reaction effects. More specifically comparison of results towards shear thinning and shear thick-

ening has been made for , and with variation in heat and mass transfer Grashof numbers

and , Sisko fluid parameters and ∗ wall parameters 0( = 1 2 3) curvature pa-

rameter , Brinkman number and radiation parameter . Homogeneous-heterogeneous

reactions are discussed for Schmidt number and strength governing parameters and

of homogeneous and heterogeneous reactions respectively.

93

5.3.1 Axial velocity

This subsection highlights shear thinning ( 1) and thickening ( 1) effects on axial

velocity towards change in embedding parameters via Figs. (52 − 56( ) & 57) The

velocity variation is found to preserve an increasing response towards with both shear

thinning and thickening effects as depicted in Figs. 52 ( ). Also the captured results show

non-symmetric behavior when shear thickening is addressed ( 1) (see Fig. 52 ()). However

tends to effect alternately i.e., an increase in reduces the velocity for both cases (shear

thinning/thickening) as seen from Figs. 53 ( ). Since includes homogenous/heterogeneous

reaction effects. The graphical results are found new in this case. The wall elasticity causes an

uplift in corresponding 1 as well as for 1 (see Figs. 54 ( )). Clinically blood is

capable of showing thinning/thickening behavior at low and high shear rates. So when walls

of blood are elastic (capillaries) exchange of water, oxygen and other nutrients becomes easy.

Whereas the damping tend to respond oppositely (see Figs. 54( )). The outcomes of Figs.

55( ) exhibit contradictory behavior of with variation in ∗ for 1 and 1. It is

noticed that increases when thinning effect dominates whereas opposite effects are developed

with the dominance in thickening effect. The larger values of curvature lowers the velocity

and graphs become flatten as planar channel is approached ( → ∞) in both cases (see Figs.56( )). An increase in causes reduction in velocity (see Fig. 57). Also more clear results

are obtained in shear thinning regime.


This subsection relates the physical description of temperature towards variation in embedding

parameters in both cases (shear thinning and thickening) through Figs. (58− 511( )). Theimpression of on is increasing since attributes to promote heat transfer by the fluid (see

Figs. 58( )). Also results are more signified in the case when 1 (see Fig. 58()). The ∗

with its increasing values produces reduction in (see Figs. 59( )). Radiation produces heat

decay as it transforms thermal energy to electromagnetic energy. Thus decline in is noticed

for an increase in (see Figs. 510( )). Curvature causes temperature to reduce in both

shear thinning (Fig. 511( )) and thickening (Fig. 511()) cases.

94


The absolute heat transfer rate corresponding to 1 and 1 under the influence of

, and is prepared in this subsection via Figs. (512 − 513 ( )). The dual responseof graphs is sketched towards as an approval of peristaltic waves travelling along the channel

boundaries. The viscous dissipation () and thermal radiation () tend to leave opposite

indication on heat transfer rate (see Figs. 512 513( )) since viscous dissipation elevates heat

transfer whereas radiation reduces it.

5.3.4 Homogeneous-heterogeneous effects

This subsection encounters the homogeneous-heterogeneous reaction effects on concentration

through graphical illustrations depicted in Figs. 514 − 517. Since left channel wall is fixedat constant value, the graphical variation is found only at the right channel wall. The growth

of homogeneous reaction (growing ) reduces where an increase in heterogeneous reaction

enhances (see Figs. 514 & 515). The decline in is captured as gets larger (see Fig.

516). The reason may lie in the fact that density of fluid particles reduces with an increase in

Schmidt number. Moreover conversion of flow stream from curved to planer regimes (small to

large ) produces decay in (see Fig. (517)).

95

5.2(a) 5.2(b)

Figs. 52: Velocity variation with when = 02 = 01 = 02 = 02 = 03

= 3 ∗ = 04 1 = 3 = 001 2 = 002 = 07 where () = 0, () = 3

5.3(a) 5.3(b)Figs. 53: Velocity variation with when = 02 = 01 = 02 = 02 = 03

= 3 ∗ = 04 1 = 3 = 001 2 = 002 = 05 where () = 0, () = 3

96

5.4(a) 5.4(b)Figs. 54: Velocity variation with 1 2 3 when = 02 = 01 = 02 = 02

= 03 = 3 ∗ = 05 = 05 = 07 where () = 0, () = 3

5.5(a) 5.5(b)Figs. 55: Velocity variation with ∗ when = 02 = 01 = 02 = 02 = 03

= 3 = 05 1 = 3 = 001 2 = 002 = 07 where () = 0, () = 3

97

5.6(a) 5.6(b)Figs. 56: Velocity variation with when = 02 = 01 = 02 = 02 = 08

= 05 ∗ = 05 1 = 3 = 001 2 = 002 = 07 where () = 0, () = 3

5.7

Fig. 57: Velocity variation with when = 02 = 01 = 02 = 02 = 03 = 3

∗ = 04 1 = 3 = 001 2 = 002 = 07 and = 05

98

5.8(a) 5.8(b)Figs. 58: Temperature variation with when = 02 = 01 = 02 = 02 = 03

= 3 ∗ = 04 1 = 3 = 001 2 = 002 = 02 = 15 where () = 0, () = 3

5.9(a) 5.9(b)Figs. 59: Temperature variation with ∗ when = 04 = 01 = 02 = 02 = 03

= 3 = 2 1 = 01 3 = 001 2 = 002 = 02 = 15 where () = 0, ()

= 3

99


= 3 = 2 1 = 3 = 001 2 = 002 ∗ = 04 = 15 where () = 0, () = 3


= 02 = 2 1 = 3 = 001 2 = 002 ∗ = 04 = 15 where () = 0, () = 3

100

5.12(a) 5.12(b)Figs. 512: Heat transfer variation with when = 01 = 01 = 02 = 03

= 02 = 3 1 = 01 3 = 001 2 = 002 ∗ = 04 = 15 where () = 0, ()

= 3

5.13(a) 5.13(b)Figs. 513: Heat transfer variation with when = 01 = 01 = 02 = 08

= 2 = 3 1 = 01 3 = 001 2 = 002 ∗ = 04 = 15 where () = 0, () = 3

101

5.14 5.15Fig. 514: Concentration variation with when = 02 = 01 = 02 = 15 = 02

and = 3

Fig. 515: Concentration variation with when = 02 = 01 = 02 = 05 = 02

and = 3

5.16 5.17Fig. 516: Concentration variation with when = 02 = 01 = 02 = 02 = 08

and = 3

Fig. 517: Concentration variation with when = 02 = 01 = 03 = 02 = 08

and = 05

102


Mixed convective peristaltic flow of Sisko fluid is carried for shear thinning and thickening

effects. Viscous dissipation, thermal radiation and homogeneous-heterogeneous reaction effects

are also outlined. The specific attributes associated with this study are summarized below:

• The Grashof numbers for heat and mass transfer on velocity are reverse.

• Larger curvature lead to reduction in velocity, temperature and concentration

• Velocity is greater in case of shear thinning (n0) and less in shear thickening (n1)materials when compared with Newtonian (n=1) fluid.

• Role of material parameter on velocity in shear thinning and thickening situations isopposite.

• Velocity is decreasing function of index parameter.

• Elasticity and damping have opposite effects on velocity.

• Concentration for homogeneous reaction is different when compared with heterogeneousreaction.

• Thermal radiation and Brinkman number effects on temperature and heat transfer rateare opposite.

103

Chapter 6

On modified Darcy’s law utilization

in peristalsis of Sisko fluid

6.1 Introduction

The significant impact of porous medium in a curved channel of small radius has been employed

in this chapter via modified Darcy’s law. Sisko fluid fills the curved channel. The flow stream

is developed by peristaltic wave train along the curved walls of the channel. These impacts

contribute in the field of medicine especially in ureteral peristalsis and blood flows in arteries.

The flow stream is regulated by imposing magnetic field towards radial coordinate. In addition

viscous dissipation in energy equation and chemical reaction in concentration equation are

highlighted. The boundaries of the channel are considered flexible with dominating slip effect.

The numerical technique has been employed to execute the coupled system and plotting graphs.

Decreasing impact of magnetic field is captured as an approval of its resistive characteristics

while velocity and temperature show enhancement with an increase in .


Mathematical description for peristaltic fluid flow of an incompressible Sisko fluid in a curved

channel has been formulated here. Chemical reaction impact is captured. Pressure is decom-

posed to get wall flexibility aspects. Moreover fluid is electrically conducting with an applied

104

magnetic field as follows:

B = (0

+∗ 0 0) (6.1)

Note that dropping of an induced magnetic field is in view of small magnetic Reynolds number.

Further by using Ohm’s law one gets

J×B = (0 −20

( +∗)2 0) (6.2)

The relative displacement of the curved channel walls in radial direction can be taken as sum

of half channel width and sinusoidal peristaltic wave

= ±( ) = ±∙+ sin

2

(− )

¸ (6.3)

The considered kinetics can be govern through the mass, momentum and energy conserva-

tion principles. Thus one gets

+

∗

+∗

+

+∗= 0 (6.4)

∙

− 2

+∗

¸= −

+

1

+∗

{( +∗)}

+∗

+∗

−

+∗+ (6.5)

∙

+

+∗

¸= − ∗

+∗

+

1

( +∗)2

©( +∗)2

ª+

∗

+∗

+− 20

( +∗)2

(6.6)

∙

¸= 1∇2 + τ gradV (6.7)

= ∇2 +

¡∇2¢− ( − 0) (6.8)

The stress tensor for Sisko fluid model is:

τ = −I+ S (6.9)

105

S =h+ |Π|−1

iA1 (6.10)

∇2 = ( ∗

+∗)2

2

2+

1

+∗

{( +∗)}

Here Π = 12(A21) serves the second invariant of symmetric part of velocity gradient, the

chemical reaction parameter, the thermal diffusion ratio, the shear rate viscosity, and

the consistency index, (≥ 0) the power-law index and the stress components.

An interesting feature of Sisko fluid is related to power-law index as it constitutes shear

thinning ( 1) and thickening ( 1) as well as Newtonian fluid ( = 1 = 0 = or

= 0 = ) characteristics. Further the values of stress components can be evaluated via Eq.

(610). Here = ( 0) gives the resistance offered by porous medium whose mathematical

form is developed using modified Darcy law that relates pressure and velocity as follows:

∇ = −∗

K¯

h+ |Π|−1

iV (6.11)

with usual meaning of parameters. Additionally this generalized version of Darcy law is its

capable of recovering the outcomes for Darcy law by specific values of parameters ( = 1 or

= 0). As pressure gradient is a reason for providing resistance in fluid flow saturating porous

space therefore Eq. (611) can be written as:

R =−∗K¯

h+ |Π|−1

iV (6.12)

The slip and compliant wall conditions for the considered problem give the following expressions:

± 1 = 0 at = ± (6.13)

± 2

=

⎧⎨⎩ 1

0

⎫⎬⎭ at = ± (6.14)

± 3

=

⎧⎨⎩ 1

0

⎫⎬⎭ at = ± (6.15)

106

∗

+∗

∙−∗

3

3+∗1

3

2+ 0

2

¸ = −

∙

+

+∗

¸+

1

( +∗)2

©( +∗)2

ª+

∗

+∗

+ − 20

( +∗)2 at = ± (6.16)

Dimensionless formulation of above flow stream with involvement of stream function ∗ ( )

gives

∗ =

∗ =

∗ =

∗ =

∗ =

=

∗

= − 0

1 − 0 ∗ =

∗ =2

∗ =

∗ =

³

´−1Re =

, Pr =

1 2 =

20

=

=

=2

(1 − 0) = Pr =

=

2

1 = − ∗3

3 2 =

∗13

3 3 =

30

2 =

K¯

2∗

=0

(1 − 0) =

− 0

1 − 0 (6.17)

= −∗

=

+

∗

(6.18)

The definitions of physical parameters are same as previous chapters whereas represents the

chemical reaction 0( = 1 2 3) the slip and ∗ the Sisko fluid parameters respectively

Through verified assumptions of long wavelength and low Reynolds number simplified ex-

pressions are obtained as follows:

= 0 (6.19)

− +

+

1

( + )2

£( + )2

¤+2

µ1

( + )2

¶+1

"1 + ∗

µ−

2

2+

1

+

¶−1#= 0 (6.20)

2

2+

1

+

+

µ− +

+

¶= 0 (6.21)

107

1

µ2

2+

1

+

¶+

µ2

2+

1

+

¶− = 0 (6.22)

where from Eq. (610)

=

µ− +

+

¶"1 + ∗

µ− +

+

¶−1#

and dimensionless conditions

= 1 + sin 2 (− ) (6.23)

− ± 1 = 0 at = ± (6.24)

± 2

=

⎧⎨⎩ 1

0

⎫⎬⎭ at = ± (6.25)

± 2

=

⎧⎨⎩ 1

0

⎫⎬⎭ at = ± (6.26)

+

∙1

3

3+2

3

2+3

2

¸ =

1

( + )2

£( + )2

¤+2

µ1

( + )2

¶+

1

"1 + ∗

µ−

2

2+

1

+

¶−1#= 0 (6.27)

From (619) pressure is found independent of and dropping asterisks is for simplicity. On

combining Eqs. (619) and (620) to eliminate pressure the stream function equation can be

obtained. Thus we have

∙1

( + )

£( + )2

¤+2

µ1

( + )

¶¸+

"( + )

(1 + ∗

µ−

2

2+

1

+

¶−1)#(6.28)

with

=

µ− +

+

¶"1 + ∗

µ− +

+

¶−1#

108


This section emphasize the graphical description of resulting ODEs through built-in routine

NDSolve in. The technique is simple and advantageous since it provides accurate

graphical picture within small CPU time (5-25 mins) per evaluation. Thus elementary features

of axial velocity, temperature, concentration and heat transfer coefficient are presented here.

Axial velocity towards variation in different involved parameters is highlighted through

Figs. (61 − 67) in this subsection Velocity is found a decreasing function of Sisko fluidparameter ∗ as seen from Fig. 61 Similar to blood flow in capillary tubes addition of pores or

an increase in porosity () generates velocity of fluid particles as shown in graphical results

of Fig. 62 The magnetic field causes flow regulation in linearize form by providing sufficient

obstruction to flow kinetics in turbulent region, therefore decrease in velocity is noted for an

increase in Hartman number (see Fig. 63). The effect is often utilized in surgeries to reduce

blood reduction. When we move from curved (small ) to straight (large ) channel fluid

velocity reduces since higher velocities are required to move in curved channel (see Fig. 64)

Larger causes velocity enhancement as depicted in Fig. 65. Slip effect of velocity occurs

as a result of adherence to the walls thus little impact of slip is noted on overall velocity of

fluid particles velocity rises with an increase in slip effect (see Fig. 66). With relevance

to clinical physiology where compliance causes the blood flow in veins and respiratory tract to

enter oxygen to body, the compliant effects of wall such as elastance 1 and mass per unit area

2 produce velocity generation. However damping parameter 3 obstruct such movements and

so velocity reduces upon increase in 3 (see Fig. 67).

Development in temperature upon larger values of pertinent parameters is plotted graphi-

cally via Figs. (68−614) here. The Sisko fluid parameter ∗ tends to increase the velocity (seeFig. 68). Elevation in is sketched with higher values of since it generates internal heat

(see Fig. 69) Since adding more pores to system makes flow easier for fluid particles flowing

through porous space and movement of these particles generate heat and thus temperature of

fluid rises with an increase in (see Fig. 610). Decline in fluid temperature is noted upon

increase in (see Fig. 611). The drawn results of Fig. 612 show decline of temperature

with increasing curvature . The results captured in Fig. 613 show rise of with thermal slip

effect. Further temperature rises for higher elasticity parameters (1 2) and falls for damping

109

coefficient 3 (see Fig. 614).

The outcomes of concentration with development in involved parameters has been sketched

in Figs. (615 − 622). Decline in concentration is noticed for larger ∗ and (see Figs.

615−616). Chemical reaction effects creates concentration drop as it moves from constructive(( 0) to destructive ( 0) region. However signified results are found foe destructive case

(see Fig. 617). An enhancement in concentration is captured for rise in curvature parameter

(see Fig. 618). The decreasing impression of is developed on since larger weakens

density and increases diffusion of fluid particles (see Fig. 619). The slip produces disturbance

of fluid concentration at relative fixed positions ( = 0 1) near the walls. Hence decreasing

response of concentration slip is observed for an increase in 3 (see Fig. 620) Growing values

of reduces viscosity and so fall in concentration is seen from Fig. 621 The plotted results

of Fig. 622 perceive decreasing impact of 1 2 on whereas impact of 3 is reverse.

The upshots of heat transfer rate via varying values of emerging parameters are drawn in

Figs. (623−626). The oscillatory impression of is an approval of peristaltic wave movementalong the channel boundaries. Rise in heat transfer rate is captured for , , 2 and

through sketched results of Figs. (623− 626) respectively.

6.1 6.2Fig. 61: Variation in verses ∗ when = 02 = 01 = 02 = 3 = 2 = 02

1 = 01 2 = 002 3 = 001 = 3 and 1 = 001.

Fig. 62: Variation in verses when = 02 = 01 = 02 = 3 = 2 ∗ = 05

1 = 01 2 = 002 3 = 001 = 3 and 1 = 001.

110

6.3 6.4Fig. 63: Variation in verses when = 02 = 01 = 02 = 3 ∗ = 002 = 02

1 = 01 2 = 002 3 = 001 = 3 and 1 = 001.

Fig. 64: Variation in verses when = 02 = 01 = 02 = 2 ∗ = 02 = 02

1 = 01 2 = 002 3 = 001 = 3 and 1 = 001.


1 = 001 2 = 002 3 = 001 = 02 and 1 = 001.

Fig. 66: Variation in verses 1 when = 02 = 01 = 02 = 3 ∗ = 02 = 02

1 = 01 2 = 002 3 = 001 = 3 and = 2.

111

6.7 6.8Fig. 67: Variation in verses ( = 1 2 3) when = 02 = 01 = 02 = 3 ∗ = 04

= 02 = 02 = 3 and 1 = 001.

Fig. 68: Variation in verses ∗ when = 02 = 01 = 02 = 3 = 2 = 2

= 02 1 = 01 2 = 002 3 = 001 = 3 1 = 001 and 2 = 002.


= 02 1 = 01 2 = 002 3 = 001 = 3 1 = 001 and 2 = 002.


= 2 1 = 01 2 = 002 3 = 001 = 3 1 = 001 and 2 = 002.

112


= 2 1 = 01 2 = 002 3 = 001 = 3 1 = 001 and 2 = 002.


= 2 1 = 01 2 = 002 3 = 001 = 3 1 = 001 and 2 = 002.

6.13 6.14Fig. 613: Variation in verses 2 when = 02 = 01 = 02 = 3 ∗ = 02 = 2

= 2 1 = 01 2 = 002 3 = 001 = 3 1 = 001 and = 02.

Fig. 614: Variation in verses ( = 1 2 3) when = 02 = 01 = 02 = 3 ∗ = 02

= 2 = 2 = 02 = 3 1 = 001 and 2 = 002.

113

6.15 6.16Fig. 615: Variation in verses ∗ when = 02 = 01 = 02 = 02 = 3 = 05

= 2 = 03 = 2 = 2 1 = 01 2 = 002 3 = 001 = 3 1 = 001 2 = 002

and 3 = 001.

Fig. 616: Variation in verses when = 02 = 01 = 02 ∗ = 02 = 3 = 05

= 2 = 03 = 2 = 2 1 = 01 2 = 002 3 = 001 = 3 1 = 001 2 = 002

and 3 = 001.

6.17 6.18Fig. 617: Variation in verses when = 02 = 01 = 02 ∗ = 02 = 3 = 05

= 2 = 02 = 2 = 2 1 = 01 2 = 002 3 = 001 = 3 1 = 001

2 = 002 and 3 = 001.

Fig. 618: Variation in verses when = 02 = 01 = 02 ∗ = 02 = 02 = 05

= 2 = 03 = 2 = 2 1 = 01 2 = 002 3 = 001 = 3 1 = 001 2 = 002

and 3 = 001.

114


= 2 = 03 = 2 = 2 1 = 01 2 = 002 3 = 001 = 3 1 = 001 2 = 002

and 3 = 001.

Fig. 620: Variation in verses 3 when = 02 = 01 = 02 ∗ = 02 = 3 = 05

= 2 = 03 = 2 = 2 1 = 01 2 = 002 3 = 001 = 3 1 = 001 2 = 002

and = 02.


= 02 = 03 = 2 = 2 1 = 01 2 = 002 3 = 001 = 3 1 = 001

2 = 002 and 3 = 001.

Fig. 622: Variation in verses ( = 1 2 3) when = 02 = 01 = 02 ∗ = 02 = 3

= 05 = 2 = 03 = 2 = 2 = 02 = 3 1 = 001 2 = 002 and

3 = 001.

115

6.23 6.24Fig. 623: Variation in verses when = 01 = 02 = 3 ∗ = 02 = 2 2 = 002

1 = 01 2 = 002 3 = 001 = 3 1 = 001 and = 02.


= 2 1 = 01 2 = 002 3 = 001 = 3 1 = 001 and 2 = 002.

6.25 6.26

Fig. 625: Variation in verses 2 when = 02 = 01 = 02 = 3 ∗ = 02 = 2

= 2 1 = 01 2 = 002 3 = 001 = 3 1 = 001 and = 02.

Fig. 626: Variation in verses when = 02 = 01 = 02 2 = 002 ∗ = 02 = 2

= 2 1 = 01 2 = 002 3 = 001 = 3 1 = 001 and = 02.

116


The specified attributes of this chapter are:

• Reduction in velocity is noticed in case of Sisko fluid parameter ∗ whereas and

increase for ∗

• Development in and reduction in is observed for .

• Impression of curvature towards velocity and temperature is decreasing

• Similar response of slip parameters is captured towards velocity and temperature.

• Rise in velocity is seen for power-law parameter .

• An uplift in velocity is captured for elasticity parameters while damping acts in reversetrend.

• Magnetic field tend to reduce the velocity and temperature of fluid.

• Increasing behavior of on temperature and heat transfer rate is captured.

• Chemical reaction aspects reduce the concentration.

• Heat transfer rate enhances with and 2

117

Chapter 7

Peristaltic motion of third grade

fluid with

homogeneous-heterogeneous

reactions

7.1 Introduction

Peristaltic flow of third grade fluid in presence of homogeneous- heterogeneous reactions is

considered. An electrically conducting fluid in a compliant wall curved channel is taken. Radial

magnetic field and dissipation are retained. Lubrication approach is adopted. Impacts of sundry

variables on velocity, temperature and concentration are addressed.

7.2 Flow diagram

Here Fig. 71 shows curved channel of half width which bend in a circle with centre O and

radius ∗ An incompressible third grade fluid fills the channel. Sinusoidal waves with velocity

propagate along the flexible walls of channel. Further we consider the flow subject to a simple

homogeneous and heterogeneous reaction model in the presence of two chemical species and

with concentrations and respectively. The equations of wall surface are described as

118

follows:

= ±( ) = ±∙+ sin

2

(− )

¸ (7.1)

Fig. 7.1. Schematic picture of the problem.

The radial magnetic field is defined by

B = (0

+∗ 0 0) (7.2)

Utilization of Ohm’s law gives the following expression

J×B = (0 −20

( +∗)2 0) (7.3)

The homogeneous-heterogeneous reaction model is considered in the form same as chapter 5:

+ 2 → 3 = 2

119

while on the catalyst surface we have the single, isothermal, first order chemical reaction:

→ =

where and are the rate constants. Both reaction processes are assumed to occur at the

same temperature.


The conservations of mass and momentum for the considered problem are expressed as follows:

+

∗

+∗

+

+∗= 0 (7.4)

∙

+

+

∗ +∗

− 2

+∗

¸= −

+

1

+∗

{( +∗)}

+∗

+∗

−

+∗ (7.5)

∙

+

+

∗ +∗

+

+∗

¸= − ∗

+∗

+

1

( +∗)2

©( +∗)2

ª+

∗

+∗

− 20

( +∗)2 (7.6)

The energy and homogeneous-heterogenoeus reaction equations are

∙

+

+

∗ +∗

¸ = 1∇2 + ( − )

+

µ

+

∗

+∗

−

+∗

¶ (7.7)

120

=

¡∇2¢− 2 (7.8)

=

¡∇2¢+ 2 (7.9)

where

∇2 = ( ∗

+∗)2

2

2+

1

+∗

{( +∗)}

Extra stress tensor for thermodynamic third grade fluid is given by

S = A1 + 1A2 + 2A21 + (A21)A1 (7.10)

in which viscosity and fluid parameters ( = 1 2) and must satisfy:

1 ≥ 0 ≥ 0 |1 + 2| ≤p24

The Rivlin-Ericksen tensors are given by

A1 = gradV + (gradV) A2 =

A1

+A1(gradV) + (gradV)

A1

The appropriate boundary conditions are described through the following expressions:

= 0 = 0 at = 0 (7.11)

= 0 = 0 at = (7.12)

∗

+∗

∙−∗

3

3+∗1

3

2+ 0

2

¸ = −

∙

+

+

∗ +∗

+

+∗

¸+

∗

+∗

+

1

( +∗)2

©( +∗)2

ª− 20

( +∗)2 at = (7.13)

121

→ 0 at = 0,

− = 0 at = (7.14)

→ 0 at = 0,

+ = 0 at = (7.15)

Here 0 serves as uniform concentration of reactant and and are the components

of an extra stress tensor S. The values of these components can be evaluated using Eq. 7.10

= 2

+ 1

(2

+

µ2

−

+∗

¶µ

+

∗

+∗

−

+∗

¶+ 4

µ

¶2)

+2

(4

µ

¶2+ 4

µ∗

+∗

+

+∗

¶+ 2

µ

+

∗

+∗

−

+∗

¶2)

+2

(4

µ

¶2+

µ

+

∗

+∗

−

+∗

¶2)

=

µ

+

∗

+∗

−

+∗

¶+ 21

µ∗

+∗

+

+∗

¶µ

− 2

+∗

¶×1

½

µ

+

∗

+∗

−

+∗

¶+

2

+∗

µ∗

+∗

+

+∗

¶¾×

(4

µ

¶2+ 4

µ∗

+∗

+

+∗

¶+ 2

µ

+

∗

+∗

−

+∗

¶2)

×µ

+

∗

+∗

−

+∗

¶

= 2

µ∗

+∗

+

+∗

¶+ 1

µ2∗

+∗

−

+∗

¶µ

+

∗

+∗

−

+∗

¶+21

(

µ∗

+∗

+

+∗

¶+ 2

µ∗

+∗

+

+∗

¶2)+

µ∗

+∗

+

+∗

¶

×2(4

µ

¶2+ 4

µ∗

+∗

+

+∗

¶+ 2

µ

+

∗

+∗

−

+∗

¶2)

+2

(4

µ∗

+∗

+

+∗

¶2+

µ

+

∗

+∗

−

+∗

¶2)

Introducing the stream function ∗ ( ) and employing the definitions of the following

122

dimensionless variables:

= −∗

=

∗

+∗∗

∗ =

∗ =

∗ =

∗ =

∗ =

∗ =

∗ =

∗ =

=

∗

= − 0

0 ∗ =

2

∗ =

∗ =

2

2 1 =

∗ =

0 ∗ =

0, (7.16)

one has

= 2

+ 1

(2

+

µ2

−

+

¶µ

+

+

−

+

¶+ 4

µ

¶2)

+2

(4

µ

¶2+ 4

µ

+

+

+

¶+ 2

µ

+

+

−

+

¶2)

+2

(4

µ

¶2+

µ

+

+ ∗

−

+

¶2)

=

µ

+

+

−

+

¶+ 21

µ

+

+

+

¶µ

− 2

+

¶×1

½

µ

+

+

−

+

¶+

2

+

µ

+

+

+

¶¾×

(4

µ

¶2+ 4

µ

+

+

+

¶+ 2

µ

+

+

−

+

¶2)

×µ

+

+

−

+

¶

= 2

µ

+

+

+

¶+ 1

µ2

+

−

+

¶µ

+

+

−

+

¶+21

(

µ

+

+

+

¶+ 2

µ

+

+

+

¶2)+

µ

+

+

+

¶

×2(4

µ

¶2+ 4

µ

+

+

+

¶+ 2

µ

+

+

−

+

¶2)

+2

(4

µ

+

+

+

¶2+

µ

+

+

−

+

¶2)

123

continuity equation is satisfied and Eqs. (75) − (710) under long wavelength and lowReynolds number assumptions yield the following expressions:

= 0 (7.17)

− +

+

1

( + )2

£( + )2

¤+2

( + )2= 0 (7.18)

2

2+

1

+

−

µ −

+

¶= 0 (7.19)

1

µ2

2+

1

+

¶ −2 = 0 (7.20)

µ2

2+

1

+

¶+2 = 0 (7.21)

= − +

+ − 2

µ −

+

¶3 (7.22)

and the dimensionless conditions yield

= 1 + sin 2 (− ) (7.23)

= 0 at = 0 (7.24)

= 0 at = 0 (7.25)

= 1 = 0

− = 0 at = (7.26)

= 0 = 0 1

+ = 0 at = (7.27)

+ [1

3

3+2

3

2+3

2

] =

1

( + )2

£( + )2

¤+2

( + )2at = 0

(7.28)

In above equations omission of asterisks is for simplicity, 1 is the ratio of diffusion coefficient

and the strength measuring parameters (for homogeneous and heterogeneous reaction

124

respectively) and other physical parameters with definitions

=

Re =

Pr =

1 =

=

2

0 =

1 = − 3

3 2 =

∗13

3 3 =

30

2 =

=

=

202

=

2 =

20

(7.29)

From application point of view the diffusion coefficients of chemical species and are of

comparable size that leads to the assumption that diffusion coefficients are equal. The equality

of diffusion coefficients and i.e., 1 = 1 leads to the following relation:

+ = 1 (7.30)

Combining Eqs. (717) and (718) and using above relation, we obtain the following systems:

∙1

( + )

©( + )2

ª¸+2

µ1

( + )

¶= 0 (7.31)µ

2

2+

1

+

¶ −

µ −

+

¶= 0 (7.32)

1

µ2

2+

1

+

¶ −(1− )2 = 0 (7.33)

= 1 + sin 2 (− )

= 0 at = 0 (7.34)

= 0 at = 0 (7.35)

= 1 = 0

− = 0 at = (7.36)

+ [1

3

3+2

3

2+3

2

] =

1

( + )2

£( + )2

¤+2

( + )2at = 0 (7.37)

125


We expand and in small Deborah number and in small homogeneous reaction

parameter as follows:

= 0 + 1 + (7.38)

= 0 + 1 + (7.39)

= 0 + 1 + (7.40)

= 0 + 1 + (7.41)

= 0 +1 + (7.42)

7.4.1 Zeroth order system

∙1

( + )

{( + )20

¸+2

∙0 +

¸= 0 (7.43)µ

2

2+

1

+

¶0 −0

µ0 −

0 +

¶= 0 (7.44)µ

2

2+

1

+

¶0 = 0 (7.45)

0 = 0 at = 0 (7.46)

0 = 0 at = 0 (7.47)

0 = 1 = 00

−0 = 0 at = (7.48)

+ [1

3

3+2

3

2+3

2

] =

1

( + )2

£( + )20

¤+2 0

( + )2at = 0

(7.49)

0 = −0 +0 +

126

and the solution at this order are

0 =( + )1+

√1+2

1

1 +√1 +2

+( + )1−

√1+2

2

1−√1 +2+ 3 +

1

223 + 4 (7.50)

0 = −( + )−2√1+2

4(1 +2)32

[{22(2 + 2p1 +2) +2(2 +

p1 +2)}

+21( + )4√1+2{2(−2 +

p1 +2) + (−2 + 2

p1 +2)}] +22

+11 ln( + ) +122 ln( + )2 (7.51)

0 = 22 + 11 ln( + ) (7.52)

Heat transfer coefficient is of the form

0 = 0()

= ( + )−1−2

√1+2

2(1 +2)[21(1 +2)( + )2

√1+2

+{222(2 +p1 +2) + 22(2 + 2

p1 +2)

−21( + )4√1+2{2(−2 +

p1 +2) + 2(−1 +

p1 +2)}}

+4122(1 +2)( + )2

√1+2

ln( + )] (7.53)

7.4.2 First order system

Here we have

∙1

( + )

{( + )21

¸+2

∙1 +

¸= 0 (7.54)µ

2

2+

1

+

¶1 −

½1

µ0 −

0 +

¶¾+

½0

µ1 −

1 +

¶¾= 0 (7.55)µ

2

2+

1

+

¶1 − 0(1− 0)

2 = 0 (7.56)

1 = 0 at = 0 (7.57)

1 = 0 at = 0 (7.58)

127

1 = 0 = 01

−1 = 0 at = (7.59)

1

( + )2

£( + )21

¤+2 1

( + )2at = 0 (7.60)

1 = −1 +1 +

− 2µ0 −

0 +

¶3

The solution expressions at this order are

1 =1

2√1 +2(3 + 42)( + )

[32( + )−3

√1+2

1 + 3√1 +2

{8(1 +p1 +2) +4(11 + 6

p1 +2)

+2(19 + 15p1 +2)}− 31( + )3

√1+2

(−1 + 3√1 +2)

{8(−1 +p1 +2)

+4(−11 + 6p1 +2) +2(−19 + 15

p1 +2)}

−(3 + 42)( + )

√1+2

2{32124

p1 +2 + 2(−1−2 +

p1 +2)( + )21}

−(3 + 42)( + )−

√1+2

2{31224

p1 +2 + 2(1 +2 +

p1 +2)( + )22}

+p1 +2(3 + 42)( + )33] + 4 (7.61)

1 = 22 +1

4[621

22

4

( + )2+

22( + )−2−4√1+2

(3 + 42)2(1 + 2√1 +2)

{(48 + 48p1 +2)

+46(16 + 5p1 +2) + 42(41 + 35

p1 +2) +4(181 + 117

p1 +2)}

+41( + )−2+4

√1+2

(3 + 42)2(−1 + 2√1 +2){(−48 + 48

p1 +2) + 46(−16 + 5

p1 +2)

+42(−41 + 35p1 +2) +4(−181 + 117

p1 +2)}

+2( + )2√1+2{−11{

2(−2 +√1 +2) + 2(−1 +√1 +2)

(1 +2)32

}

+3122{6

4 +2(20− 17√1 +2)− 12(−1 +√1 +2)

(3 + 42)(−1 +√1 +2)( + )2}}

+2( + )−2√1+2{−22{

2(2 +√1 +2) + 2(1 +

√1 +2)

(1 +2)32

}

−3212{64 +2(20 + 17

√1 +2) + 12(1 +

√1 +2)

(3 + 42)(1 +√1 +2)( + )2

}}

+411 ln( + ) + 4(21 + 12)2 ln( + )2] (7.62)

128

1 =1

8[( + )2{−6311 + 2(−1 + 22)

222 + 211(−6 + 922) +11(−2 + 822 − 6222)}+ 822+11( + )2 ln( + ){2 +11(8− 1222) + 9211 − 822 + 6222}+ 81 ln( + )

−2211( + )2 ln( + )2(2 + 311 − 322) + 2311( + )2 ln( + )3] (7.63)

and the heat transfer coefficient is

1 = 1()

=

4( + )3[−1221224 + 41( + )2 − 2

42( + )−4

√1+2

(3 + 42)2{48(1 +

p1 +2)

+46(16 + 5p1 +2) + 42(41 + 35

p1 +2) +4(181 + 117

p1 +2)}

+41

32

2( + )−2√1+2

(3 + 42)(1 +√1 +2)

{64 + 12(1 +p1 +2) +2(20 + 17

p1 +2)}

−4231

2( + )2√1+2

(3 + 42)(−1 +√1 +2){64 − 12(−1 +

p1 +2) +2(20− 17

p1 +2)}

+241( + )4

√1+2

(3 + 42)2{48(−1 +

p1 +2) + 46(−16 + 5

p1 +2)

+42(−41 + 35p1 +2) +4(−181 + 117

p1 +2)}

+4p1 +2( + )2+2

√1+2{−11

(1 +2)32

{2(−2 +p1 +2) + 2(−1 +

p1 +2)}

+312

2

(3 + 42)(−1 +√1 +2)( + )2

{64 +2(20− 17p1 +2)− 12(−1 +

p1 +2)}}

−4p1 +2( + )2−2

√1+2{−22

(1 +2)32

{2(2 +p1 +2) + 2(1 +

p1 +2)}

− 3212

(3 + 42)(1 +√1 +2)( + )2

{64 +2(20 + 17p1 +2) + 12(1 +

p1 +2)}}

+8(21 +21)2( + )2 ln( + )] (7.64)

in which appearing constants can be determined by appropriate boundary conditions through

Mathematica.

7.5 Discussion

Here analysis for velocity , temperature , concentration and heat transfer coefficient

are studied graphically. In particular the behaviors of wall parameters 1 2 3, curvature

129

parameter , Brinkman number , Hartman number, Schmidt number , Deborah number

and homogeneous-heterogeneous reaction parameters and are analyzed. Moreover the

comparison of results in straight versus curved channel is also examined.

7.5.1 Velocity profile

Figs. 72( ) display the effect of wall parameters 1 2 and 3 on the velocity profile. The

graphical results show the increase in velocity with an increase in the elastic parameters 1 2

in both straight and curved channels. Physically elastic nature of wall offers less resistance to

flow of fluid However decline in velocity can be seen with an increase in damping wall parameter

3. The increasing values of Hartman number causes reduction in velocity profile (see Figs.

73( )). Here the magnetic field in radial direction acts as a retarding force for the flow of

fluid. The results sketched in Fig. 74 show increase in velocity when we move from straight

to curved geometry i.e., the increasing values of curvature parameter . Figs. 75( ) indicate

the growth of velocity profile with Deborah number Since viscosity reduces with progressing

values of which in turn enhances the velocity of fluid. Also velocity in third grade fluid is

found greater than viscous fluid ( = 0)

7.5.2 Temperature profile

Figs. 76( ) eludicate the impact of wall membrane parameters on temperature distribution.

The displayed results manifest the development of temperature with 1 2 and reduction of

temperature with 3 The variations in Hartman number lessen the magnitude of temper-

ature as observed from Figs. 77( ) for both straight and curved channels. This is due to

retarding nature of magnetic field. Figs. 78( ) report an enhancement in temperature distri-

bution with larger values of Brinkman number. In fact relates the viscous dissipation effects

which causes increase in temperature. The results displayed in Fig. 79 show that temperature

is increasing function of curvature parameter . From Figs. 710( ) increase in temperature

profile is noticed for larger Deborah number . It can be concluded that temperature in third

grade fluid is higher than viscous fluid. Also the temperature distribution is symmetric in

curved channel when compared with straight channel.

130

7.5.3 Homogeneous-heterogeneous reactions effects

Impacts of homogeneous and heterogeneous reaction parameters and are presented in the

Figs. 711− 712( ). The results displayed in Figs. 10( ) show the decreasing response ofhomogeneous reaction parameter on the concentration distribution. However the opposite

behavior for the variation of heterogeneous reaction parameter is observed in Fig. 712() It

is evident from Figs. 711& 712( ) that in both cases change in concentration is observed near

the lower boundary of the channel since upper boundary is fixed. The reduction in concentration

distribution for curvature parameter can be seen in Fig. 713. This leads to conclusion that

curved geometry reduces the concentration more than a straight channel. The variations of

Schmidt number on are depicted in the Figs. 714( ). The graphical results exhibit

decline of concentration distribution (see Figs. 714( )). It follows by the fact that density

of fluid particles reduces for larger Schmidt number. Hence it promotes the flow of fluid. Thus

the less dense fluid particles acquire high speed and possess larger molecular vibrations which

lessen the concentration of fluid.


This section displays behavior of involved variables in heat transfer coefficient In view of

sinusoidal peristaltic wave the oscillatory behavior of heat transfer coefficient is anticipated.

Figs. 715( ) show the increase in heat transfer distribution with 1, 2 where 3 lowers the

transfer of heat . Impact of Hartman number on heat transfer distribution is decreasing (see

Figs. 716( )). Figs. 717 and 718( ) are drawn to examine the effects of Brinkman number

and curvature parameter on . The sketched results show increase in the magnitude of

heat transfer distribution with both the parameters. Since Brinkman number accounts for

viscous dissipation effects that are responsible for temperature development. Further Figs.

719( ) illustrate the decrease in absolute value of heat transfer coefficient with Deborah

number . It is anticipated that curved channel is responsible for the reduction of heat transfer

coefficient more efficiently.

131

7.2(a) 7.2(b)Figs. 72: Plot of velocity for wall parameters 1 2 3 with = 01 = 01, = 02,

= 002 and = 25 () = 3 () = 100

7.3(a) 7.3(b)Figs. 73: Plot of velocity for Hartman number with = 01 = 0, = 01, 1 = 04

2 = 02 = 002 and 3 = 03 () = 3 () = 100

132

7.4 7.5(a)

7.5(b)

Fig. 74: Plot of velocity for curvature parameter with = 01 = 0, = 002 = 02,

= 05 1 = 04 2 = 02 and 3 = 03

Figs. 75: Plot of velocity for Deborah number with = 01 = 0, = 02, 1 = 04

2 = 02 3 = 03 and = 05 () = 3 () = 100

133

7.6(a) 7.6(b)Figs. 76: Plot of temperature for wall parameters 1 2 3 with = 01 = 01, = 02,

= 2 and = 3 () = 4 () = 100

7.7(a) 7.7(b)Figs. 77: Plot of temperature for Hartman number with = 01 = 01, = 02,

1 = 04 2 = 02 3 = 03 = 3 and = 002 () = 35 () = 40

134

7.8(a) 7.8(b)Figs. 78: Plot of temperature for Brinkman number with = 01 = 01, = 02

1 = 04 2 = 02 3 = 03 = 2 and = 002 () = 4 () = 100

7.9

Fig. 79: Plot of temperature for curvature parameter with = 01 = 01, = 01

1 = 001 2 = 002 3 = 003 = 002 = 2 and = 3

135

7.10(a) 7.10(b)Fig. 710: Plot of temperature for Deborah number with = 02 = 01, = 02,

1 = 004 2 = 003 3 = 001 = 2 and = 1 () = 35 () = 20

7.11(a) 7.11(b)Figs. 711: Plot of concentration for homogeneous reaction parameter with = 01

= 01, = 01, = 5 and = 2 () = 10 () = 50

7.12 7.13(a)

136

7.13(b)

Fig. 712: Plot of concentration for heterogeneous reaction parameter with = 01

= 01, = 01 = 1 and = 04 () = 3 () = 50

Figs. 713: Plot of concentration for curvature parameter with = 02 = 01 = 01,

= 08 = 2 and = 2

7.14(a) 7.14(b)Figs. 714: Plot of concentration for Schmidt number with = 01 = 01 = 01,

= 02 and = 04 () = 2 () = 50

137

7.15(a) 7.15(b)Figs. 715: Plot of heat transfer coefficient for wall parameters 1 2 3 with = 01

= 01, = 2 = 002 and = 1 () = 4 () = 50

138

7.16(a) 7.16(b)Figs. 716: Plot of heat transfer coefficient for Hartman number with = 01 = 01,

= 002 1 = 004 2 = 002 3 = 003 and = 1 () = 4 () = 10

7.17(a) 7.17(b)Figs. 717: Plot of heat transfer coefficient for Brinkman number with = 01 = 01,

1 = 004 2 = 002 3 = 003 = 2 and = 002 () = 4 () = 60

139

7.18 7.19(a)

7.19(b)

Fig. 718: Plot of heat transfer coefficient for curvature parameter with = 01 = 01

= 1 1 = 004 2 = 002 3 = 003 = 2 and = 002

Figs. 719: Plot of heat transfer coefficient for Deborah number with = 01 = 01

= 1 1 = 004 2 = 002 3 = 003 and = 2 () = 4 () = 50

140


Here the effects of homogeneous-heterogeneous reaction on peristaltic flow of third grade fluid

through a curved channel are studied in the presence of radially applied magnetic field. The

main points are listed below:

• Velocity increases with elasticity, curvature parameters and Deborah number. HoweverHartman number reduces the velocity.

• Similar response is noticed for Brinkman number and curvature parameter on the tem-perature profile and heat transfer coefficient.

• Temperature increases for Brinkman number and Hall parameter.

• The Hartman and Deborah numbers show reduction in the temperature distribution.

• Opposite effects of homogeneous and heterogeneous reaction parameters are seen on theconcentration profiles.

• The concentration distribution is decreasing function of curvature parameter and Schmidtnumber.

141

Chapter 8

Peristaltic flow of MHD Jeffery

nanofluid in curved channel with

convective boundary conditions: A

numerical study

8.1 Introduction

This chapter explores MHD peristalsis of Jeffery nanofluid in curved channel. Heat transfer

analysis comprised thermal radiation while the mass transfer has been discussed in terms of

thermophoresis, Brownian motion and chemical reaction. With reference to blood flow in circu-

lar compliant arteries the curved channel boundaries are considered flexible. Moreover thermal

field is formulated under more general approach of convective boundary conditions. Inertial

effects have been neglected by small Reynolds number and large wavelength considerations.

The detailed physical interpretation of velocity, temperature, nanoparticles mass transfer and

heat transfer rate has been presented towards different emerging parameters. The recorded re-

sults indicate non-symmetric behavior of velocity in curved channel. Further Brownian motion

and thermophoresis impacts on the temperature and mass transfer of nanoparticles are found

reverse. In addition magnetic field causes reduction in velocity and temperature of fluid.

142

8.2 Flow diagram

The problem under consideration characterizes the flow of an incompressible Jeffery nanofluid in

a channel of separation bend in a circle of inner radius . The flow generation inside the channel

is initiated by the peristaltic waves that travel along the curved boundaries. In addition the

wall’s elastic tension, mass per unit area and damping effects are not ignored for the present

study. The coordinate system is taken in such a way that denotes the radial-direction whereas

for an axial direction. Further the fluid is subject to electrical conduction via imposed magnetic

field B in radial direction by the following expression:

B = (0

+∗ 0 0) (8.1)

The required term that must be included in flow analysis can be obtained by the utilization of

Ohm’s law as follows:

J×B = (0 −20

( +∗)2 0) (8.2)

The configuration of peristaltic waves can be visualized through the expression:

= ±( ) = ±∙+ sin

2

(− )

¸ (8.3)


The present flow can be governed through the conservation principles of mass, momentum,

energy and nanoparticle volume fraction. Thus the relevant equations can be put in the following

form:

+

∗

+∗

+

+∗= 0 (8.4)

Momentum equation in component form including magnetic field and mixed convection effects

are

∙

− 2

+∗

¸= −

+

1

+∗

{( +∗)}

+∗

+∗

−

+∗ (8.5)

143

∙

+

+∗

¸= − ∗

+∗

+

1

( +∗)2

©( +∗)2

ª+

∗

+∗

+ ( − 0) + ( − )( − 0)− 20

( +∗)2 (8.6)

Energy conservation equation with viscous dissipation and radiation effects:

()

∙

¸= 1∇2 + () (∇∇ ) + ()

(∇∇ )

−

µ−16∗ 303∗

¶+ ( − )

µ

¶+

µ

−

+∗+

∗

+∗

¶ (8.7)

Conservation of nanoparticle volume fraction with chemical reaction effects:

= (∇2) +

(∇2 )−( − 0) (8.8)

in which

∇2 = ( ∗

+∗)2

2

2+

1

+∗

{( +∗)}

and =()()

ratio of heat capacity of nanoparticle material to that of fluid, the density of

fluid, the acceleration due to gravity, 0 and 0 the temperature and concentration at lower

channel wall, and the Brownian and thermophoresis diffusion coefficients respectively,

∗ the Stefan-Boltzmann constant, ∗ the mean absorption coefficient, the mean temperature

of fluid and the chemical reaction parameter. The extra stress tensor S in Jeffery fluid gives

the components and as follows:

S =

1 +

∙1 + 1

¸A1 (8.9)

144

where A1 = ∇V+(∇V) represents the first Rivlin Erickson tensor and 1 the Jeffery fluidparameters. Here

=2

1 +

∙1 + 1

¸

(8.10)

= =

1 +

∙1 + 1

¸µ∗

+∗

−

+∗+

¶ (8.11)

=2

1 +

∙1 + 1

¸µ∗

+∗

+

+∗

¶ (8.12)

Note that in writing Eq. (8.7) the Rosseland approximation for radiative heat flux is used.

The non linear boundary conditions comprising wall compliant and convective conditions are

as follows:

= 0 at = ± (8.13)

1

= −1( − 0), at = (8.14)

= −2( − 0), at = (8.15)

1

= −1(1 − ), at = − (8.16)

= −2(1 − ), at = − (8.17)

∗

+∗

∙−∗

3

3+∗1

3

2+ 0

2

¸ = −

∙

+

+∗

¸+

1

( +∗)2

©( +∗)2

ª+

∗

+∗

+ (1− 0) ( − 0) + ( − )( − 0)− 20

( +∗)2 at = ± (8.18)

Here 1and 1 are the values of temperature and concentration at upper wall of the channel

and 1 2 the heat and mass transfer coefficients.

Define the non-dimensional quantities and denote the stream function by ∗ ( ). Then

145

∗ =

∗ =

∗ =

∗ =

∗ =

=

2

=

∗

= − 0

1 − 0 =

− 0

1 − 0 ∗ =

∗ =2

=

=

2(1 − 0)

=

(1 − 0)( − )2

Re =

=

(1 − 0)

=

(1 − 0)

=2

(1 − 0) = Pr =

16∗ 303∗

=

=

1 = − ∗3

3 2 =

∗13

3 3 =

30

2 Pr =

1

∗1 =1

2 =

220

1 =1

1 2 =

2

(8.19)

= −∗

=

+

∗

Here the symbols above show the radiation parameter the local temperature Grashof num-

ber , the local nanoparticles Grashof number , the Brinkman number , thermophoresis

parameter the Brownian motion parameter the chemical reaction parameter the

heat and mass transfer Biot numbers 1 and 2 respectively and the Jeffery fluid parameter

∗1

Implication of above definitions, dropping the asterisks and followed the long wavelength

approximation yield the simplified set of non-dimensional equations and conditions as follows:

= 0 (8.20)

− +

+

1

( + )2

∙( + )2

1 +

µ−

2

2+

1

+

¶¸+( + )( +) +2

µ1

+

¶= 0 (8.21)

146

2

2+

1

+

+

1 +

µ−

2

2+

1

+

¶2+Pr

µ2

2

¶+Pr

µ2

2

¶+Pr

µ

¶= 0 (8.22)

µ2

2+

1

+

¶+

µ2

2+

1

+

¶− = 0 (8.23)

= 1 + sin 2 (− ) (8.24)

= 0 = ± (8.25)

+1 = 0 at = (8.26)

+1(1− ) = 0 at = − (8.27)

+1 = 0 at = (8.28)

+1(1− ) = 0 at = − (8.29)

+ [1

3

3+2

3

2+3

2

] =

1

( + )2

½( + )2

1 +

µ−

2

2+

1

+

¶¾+

++2

µ1

( + )2

¶ at = ± (8.30)

Combination of Eqs. (8.20) and (8.21) constitute a single equation for stream function as

follows:

∙1

( + )

½( + )2

1 + 1

µ−

2

2+

1

+

¶¾+ ( + )( +) +2

µ1

+

¶¸= 0

(8.31)

147

8.4 Discussion

The resulting system of equations together with relevant boundary conditions are coupled and

non-linear due to which exact solution seems difficult to obtain. Therefore graphs corresponding

to axial velocity , temperature , nanoparticle volume distribution and heat transfer coeffi-

cient are plotted through numerical approach via built in command NDSolve in.

More specifically outcomes of , and under the influences of varying magnetic field ,

heat and mass transfer Grashof numbers and , thermophoresis and Brownian motion

parameters and , wall compliant parameters 1 2 3 Jeffery fluid parameter cur-

vature parameter , Brinkman number , radiation parameter , heat and mass transfer

Biot numbers 1 and 2 and chemical reaction parameter will be outlined in this section.


The graphical results of axial velocity under the influence of varying magnetic field , local

temperature and nanoparticle Grashof numbers and , thermophoresis and Brownian

motion parameters and , wall compliant parameters 1 2 3, Jeffery fluid parameter

and curvature parameter have been made in Figs. 81(− ). It is noticed that for small

values of curvature parameter (in curved geometry) an increase in lowers the axial velocity

since the presence of magnetic field alters the rate of flow (see Fig. 81()). Medically damping

effect of MHD is utilized in magnotherapy to cure arthritis, migraine, cancer and depression

etc without measurable effect on the blood flow. An increase in and has opposite effects

on velocity profile i.e., an increase in enhances whereas effect of is reverse (see Figs.

81(() & ()). In addition maximum impact is seen at the core part of the curved channel.

With growing values of the fluid particles become more dense and hence velocity profile is

decreasing (see Fig. 81()). On the other hand the fluid viscosity gets stronger with larger

values of causing the decline in (see Fig. 81()). Moreover velocity is tilted towards the

negative channel wall since viscosity has effective role near lower part of the curved channel.

An increase in velocity has been noticed with larger values of 1 and 2. However 3 tends

to reduce the velocity (see Fig. 81()). Similar to blood veins the channel walls possess elastic

nature, so when the elasticity (1) of walls increases or mass per unit area (2) expands the

148

velocity (of blood) increases. In contrast the damping parameter (3) boost up the resistance

and considerable force is required to move the fluid particles between channel walls (or blood

veins) and hence velocity reduces. Larger shows an increase in (see Fig. 81()). The effect

of corresponding to small values (curved channel) is found non-symmetric about centerline of

the channel. An increasing effect on is noticed as we move from curved (small ) to straight

(large ) channel. Moreover velocity profile appears to be symmetric as straight channel is

approached (see Fig. 81()).


This subsection aims to examine the influence of magnetic field , Brinkman number , local

temperature and nanoparticle Grashof numbers and , thermophoresis and Brownian

motion parameters and , radiation parameter , wall compliant parameters 1 2

3, curvature parameter and heat transfer Biot number 1 respectively on the temperature

through Figs. 82( − ). The outcomes of are found to be well matched with . This

fact was anticipated since temperature and velocity are directly related via well known kinetic

theory. Therefore decline in temperature is noticed with , , , 3 where , and 1

2 cause an increase in similar to axial velocity (see Figs. 82((), (−) & ()) An increasein promotes the temperature of fluid (see Fig. 82()) since characterizes the viscous

dissipation which means transformation of kinetic energy to internal energy (heating up the

fluid) in response to viscosity. Increasing produces decrease in as radiation causes heat

loss (see Fig. 82()). The temperature profile is found decreasing and non-symmetric with an

increase in (see Fig. 82()). The convective heat transfer reduces the thermal conductivity

inside the channel and hence falls the fluid temperature (see Fig. 82()).

8.4.3 Nanoparticle mass transfer distribution

Figs. 83(− ) relate the nanoparticle phenomenon corresponding to magnetic field , ther-

mophoresis and Brownian motion parameters and , mass transfer Biot number 2,

curvature and chemical reaction respectively. The results of Fig. 83() indicate dominance

of nanoparticle mass transfer when increases. With an increase in the fluid viscosity

gets weaken and less viscous particles decay the mass transfer distribution of nanoparticles

149

(see Fig. 83()). The density of nanofluid particles enhances with growing values of and

so increasing behavior of is noticed towards (see Fig. 83()). An increase in volume

fraction has been noticed via increase in 2 (see Fig. 83()). The results drawn in Fig. 83()

show increasing effect of nanoparticles mass transfer as we move from curved to straight chan-

nel (small to large ). The impact of on is observed decreasing as a whole. In addition

is greater with constructive/generative chemical reaction ( 0) in contrast to destructive

chemical reaction ( 0) From physical point of view 0 yield constructive forces inside

the channel building up the fluid concentration where 0 are responsible for generation of

destructive forces that breaks the bond between the fluid particles and hence concentration

drops (see Fig. 83()). On earth’s surface chemical reaction effects are witnessed in terms of

constructive and destructive forces in response to gravity. The constructive forces build up the

existing landform (through landslide or deposition) or create new one (through floods) whereas

destructive forces affect the earth’s surface by breaking down landforms through weathering

and erosion.


This subsection deals with the impact of involved parameters on heat transfer coefficient

through Figs. 84(−). The distinctive oscillatory shape of graphs have been observed due toperistaltic wave travelling along the walls of channel. Fig. 84() exhibits increasing behavior

of on magnitude of heat transfer coefficient along the upper channel wall. The results

recorded in Figs. 84(−) comprised of decreasing response of heat transfer coefficient uponincreasing values of thermal radiation, heat transfer Biot number and curvature parameter. As

, 1 and transfer less heat from channel walls to fluid therefore the absolute value of heat

transfer appears decreasing (see Figs. 84(− )).

150

8.1(a) 8.1(b)

8.1(c) 8.1(d)

151

8.1(e) 8.1(f)

8.1(g) 8.1(h)Figs. 81: Variation in axial velocity when = 02 = 01 and = 02

152

8.2(a) 8.2(b)

8.2(c) 8.2(d)

153

8.2(e) 8.2(f)

8.2(g) 8.2(h)

154

8.2(i) 8.2(j)

Figs. 82: Variation in temperature when = 02 = 01 and = 02

155

8.3(a) 8.3(b)

8.3(c)

156

8.3(d) 8.3(e)

8.3(f)

Figs. 83: Variation in nanoparticles volume fraction when = 02 = 01 and = 02

157

8.4(a) 8.4(b)

8.4(c) 8.4(d)Figs. 84: Variation in heat transfer rate when = 01 and = 02

158


Curved channel flow of an incompressible MHD Jeffery nanofluid under the effects of compli-

ant walls has been carried out. The governing equations comprised of thermal radiation and

chemical reaction (destructive/generative) effects. The major outcomes can be summarized as

follows:

• Magnetic field due to its opposing character tends to reduce the velocity and temperatureof fluid.

• Mixed convection effects on velocity and temperature are found opposite.

• Impacts of and on velocity, temperature and nanoparticle mass transfer are reverse.

• Curvature causes non-symmetric behavior of velocity profile.

• temperature decays for higher values radiation and curvature parameters.

• Impacts of wall parameters on velocity and temperature are found alike while oppositeresponse in noticed for nanoparticle mass transfer.

• The heat transfer Biot number causes reduction in temperature while mass transfer Biotnumber shows enhancement in nanoparticle mass transfer.

• The chemical reaction effects are more pronounced in generative/constructive case.

• Absolute value of heat transfer coefficient decreases when 1 and are increased.

• Temperature and heat transfer increases via larger .

159

Chapter 9

Numerical analysis of partial slip on

peristalsis of MHD Jeffery nanofluid

in curved channel with porous space

9.1 Introduction

This chapter focuses on porosity effect in peristaltic flow in curved channel enclosing Jeffery

nanofluid. The presence of porous medium in channel is demonstrated using generalized ver-

sion of modified Darcy’s law. The gravitational effects are active enough to consider mixed

convection in flow analysis. Heat and mass transfer aspects of fluid flow are highlighted with

consideration of thermal radiation and chemical reaction effects. Further due to small separa-

tion between channel walls the dominance of slip effect not ignored. The curved channel walls

are subject to flexible/compliant wall properties. The assumed problem results in complicated

mathematical expression that are simplified using small Reynolds number and large wavelength

concepts. The non-linear and coupled system is then solved via numerical technique. In par-

ticular, variations in velocity, temperature, nanoparticles mass transfer and heat transfer rate

are discussed. Results specify amplification of fluid velocity and temperature upon increment

in values of slip and Darcy number.

160

9.2 Flow diagram

Consider the flow of Jeffery nanofluid in curved channel of same dimensions and uniform mag-

netic field as in previous chapters one has

Fig. 9.1. Schematic diagram.

J×B = (0 −20

( +∗)2 0) (9.1)

and wave geometry of the form

= ±( ) = ±∙+ sin

2

(− )

¸ (9.2)


The relevant equations for flow analysis are

Continuity equation

+

∗

+∗

+

+∗= 0 (9.3)

161

Components of momentum equation:

∙

− 2

+∗

¸= −

+

1

+∗

{( +∗)}

+∗

+∗

−

+∗+ (9.4)

∙

+

+∗

¸= − ∗

+∗

+

1

( +∗)2

©( +∗)2

ª+

∗

+∗

+ ( − 0) + ( − )( −0)− 20

( +∗)2+(9.5)

Conservation of energy and nanoparticle mass transfer with viscous dissipation, thermal radi-

ation and chemical reaction:

()

∙

¸= 1∇2 + () (∇∇ ) + ()

(∇∇ )

−

µ−16∗ 303∗

¶+

"4

µ

¶2+

µ

+

+∗+

¶2# (9.6)

= (∇2) +

(∇2 )−( − 0) (9.7)

For Jeffery fluid stress tensor has the following form:

S =

1 +

∙1 + 1

¸A1 (9.8)

with 1 and as retardation and ratio of retardation to relaxation times respectively.

∇2 = ( ∗

+∗)2

2

2+

1

+∗

{( +∗)}

For non-Newtonian Jeffery fluid the relation between pressure drop and velocity is developed

by modified Darcy’s law i.e.,

∇ = −∗K¯(1 + )

∙1 + 1

¸V (9.9)

162

with usual meanings of parameters. Note that for specific parametric values ( = 0 1 = 0)

the form of Darcy law can be recovered. As fluid flow in porous medium experience resistance

in response to pressure gradient, hence involvement of R is permissible in Eq. (99) to obtain

R =−∗

K¯(1 + )

∙1 + 1

¸V

Further the stress components of Jeffery fluid are

=2

1 +

∙1 + 1

¸

(9.10)

= =2

1 +

∙1 + 1

¸µ∗

+∗

−

+∗+

¶ (9.11)

=2

1 +

∙1 + 1

¸µ∗

+∗

+

+∗

¶ (9.12)

Further the radiation term in Eq. (96) is followed through Rosseland approximation. Slip and

compliant conditions at the walls give

± 1 = 0 at = ± (9.13)

± 2

=

⎧⎨⎩ 1

0

⎫⎬⎭ , at = ± (9.14)

± 3

=

⎧⎨⎩ 1

0

⎫⎬⎭ , at = ± (9.15)

∗

+∗

∙−∗

3

3+∗1

3

2+ 0

2

¸ = −

∙

+

+∗

¸+

1

( +∗)2

©( +∗)2

ª+

∗

+∗

+ (1− 0) ( − 0) + ( − )( − 0)

− 20

( +∗)2+2 at = ± (9.16)

Now applying dimensionless formulation and definition of stream function ∗ ( ) will

lead to the required set of equations:

163

∗ =

∗ =

∗ =

∗ =

∗ =

=

2

=

∗

= − 0

1 − 0 =

− 0

1 − 0 ∗ =

∗ =2

=

=

2(1 − 0)

=

(1 − 0)( − )2

Re =

=

(1 − 0)

=

(1 − 0)

Pr =

1

=2

(1 − 0) = Pr =

16∗ 303∗

=

1 = − ∗3

3 2 =

∗13

3 3 =

30

2 =

∗1 =1

2 =

220

∗ =

=

K¯

2∗ (9.17)

= −∗

=

+

∗

The whole system under the long wavelength and omission of asterisks takes the form:

= 0 (9.18)

− +

+

1

( + )2

∙( + )2

1 +

µ−

2

2+

1

+

¶¸+( +) +2

µ1

( + )2

¶+

1

(1 + )

= 0 (9.19)

2

2+

1

+

+

µ−

2

2+

1

+

¶2+Pr

µ2

2

¶+Pr

µ2

2

¶+Pr

µ

¶= 0 (9.20)

µ2

2+

1

+

¶+

µ2

2+

1

+

¶− = 0 (9.21)

164

= 1 + sin 2 (− ) (9.22)

−± 1

1 +

µ−

2

2+

1

+

¶= 0 = ± (9.23)

± 2

=

⎧⎨⎩ 1

0

⎫⎬⎭ at = ± (9.24)

± 3

=

⎧⎨⎩ 1

0

⎫⎬⎭ at = ± (9.25)

+ [1

3

3+2

3

2+3

2

] =

1

( + )2

½( + )2

1 + 1

µ−

2

2− 1

+

¶¾+ ++2

µ1

( + )2

¶+

1

(1 + )

at = ± (9.26)

Combining Eqs. (918)− (919) one gets single equation for stream function as follows:

∙1

( + )

½( + )2

1 + 1

µ−

2

2+

1

+

¶¾¸+

∙( + )( +) +2

µ1

+

¶+

( + )

(1 + )

¸= 0 (9.27)

9.4 Discussion

The above system give non-linear and coupled mathematical expressions whose exact solutions

seems difficult to obtain. Thus an appropriate technique either a series transform or some

numerical method should apply to encounter such difficulty in terms of solution. Here we intend

to execute the above system directly by using NDSolve built-in command of computational

software. Thus graphs are made for axial velocity , temperature , nanoparticle

mass transfer and heat transfer rate .

165


Upshots of velocity corresponding to various involved parameters have been made in this sub-

section. The parabolic representation of velocity profile is examined from Figs. 92( − ).

Heat and mass transfer Grashof numbers ( and ) increase the velocity as displayed in

Figs. 92( & ) due to loss of viscosity. Physical significance of mixed convection is found

in nuclear reactor technology and cooling processes when forced convection is inadequate to

dissipate excess energy. Velocity drops for higher values of Hartman number as viewed in

Fig. 92(). This mitigating action of magnetic field is proficient in healing of arthritis, cancer-

ous tumors and migraine without notable effect on blood flow stream. activation of velocity is

noticed with rise in (see Fig. 92()). This impact of porosity is of prime importance in clin-

ical domain since pores in blood capillaries makes water, oxygen and other nutrients exchange

between the blood and tissues. Slip causes velocity enhancement with its growing values (see

Fig. 92()). Output of 1 and 2 on velocity of fluid is increasing whereas 3 decreases

(see Fig. 92()). The impression of curvature on velocity is non-symmetric and it gets flatten

as straight channel is approached (large ) (see Fig. 92()) Thermophoresis tend to reduce

the speed of nanoparticles hence velocity reduces with rise in (see Fig. 92()). Whereas

viscosity gets weaken with growing values of Brownian motion therefore velocity increases with

(see Fig. 92()). Jeffery fluid parameter () activates flow pattern and thus velocity of

nanofluid as depicted in Fig. 92().


Temperature variations towards different parameters of interest has been plotted via Figs.

93(− ). Since increment in Brownian diffusion and thermophoresis cause energy production

so temperature of fluid rises with these parameters (see Figs. 93( & )). Decay of excess heat

is specific attribute associated with thermal radiation therefore rise in perceives decreasing

outcome of temperature (see Fig. 93()). Larger values of curvature causes fall in temperature

(see Fig. 93()). The fluid temperature deviates as a result of thermal slip and overall de-

creasing response of slip is noticed on (see Fig. 93()). Graph displayed in Fig. 93() show

rise in temperature with . This outcome was anticipated as adding more pores (large )

reduce resistive forces. The results appeared in Figs. 93( & ) show reverse impact of

166

and on temperature. Clearly heat generation is particular characteristic of . Whereas

thermal conductivity of fluid reduces with an rise in and thus fluid temperature also shows

reduction. 1 and 2 raises temperature while 3 decays it (see Fig. 93()).


In view of peristaltic travelling wave the dual behavior of heat transfer rate is captured

under involved parameters through Figs. 94( − ) since we consider single phase nanofluid.

Whereas in multiphase flows Nussult and Sherwood numbers are highlighted to study

heat transfer characteristics of fluid. As less heat is transfer from fluid to boundary as straight

channel is reached so rate of heat transfer declines with an enhancement in (see Fig. 94())

Heat transfer rate reduces for an increase in and (see Figs. 94( & )). Slip causes

decline of as shown in Fig. 94() Fall in heat transfer rate is noticed as porosity in a medium

gets enlarge (see Fig. 94()).


Figs. 95( − ) present the changes in nanoparticle mass transfer distribution towards in-

crease in involved parameters. Mass transfer show decreasing behavior with growth in values

of (see Fig. 95()). Larger Nb makes nanoparticles more dense to dominate concentration

. The effect is captured in Fig. 95() decreases as slip enhances the deviation of flow along

the channel walls (see Fig. 95()). The mass transfer distribution lessens in case when chem-

ical reaction gets stronger. The effect is influential on earth’s mantle where construction and

destruction forces retaliate natural atmospheric conditions. Fall in nanoparticle mass transfer

is captured as flow stream migrates from curved to straight regime (see Fig. 95()). Results

recorded in Fig. 95() show decrease of with an increase in Darcy number .

167

9.2(a) 9.2(b)

9.2(c) 9.2(d)

168

9.2(e) 9.2(f)

9.2(g) 9.2(h)

169

9.2(i) 9.2(j)

Figs. 92: Variation in axial velocity when = 02 = 01 & = 02

170

9.3(a) 9.3(b)

9.3(c) 9.3(d)

171

9.3(e) 9.3(f)

9.3(g) 9.3(h)

Figs. 93: Variation in temperature when = 02 = 01 & = 02

172

9.4(a) 9.4(b)

9.4(c) 9.4(d)

173

9.4(e)

Figs. 94: Variation in heat transfer coefficient when = 01 & = 02

174

9.5(a) 9.5(b)

9.5(c) 9.5(d)

175

9.5(e) 9.5(f)Figs. 95: Variation in nanoparticle mass transfer when = 02 = 01 & = 02

176


The attractive features of this study are summarized below:

• Resistive impact of magnetic field is approved for velocity and temperature.

• Increase in favors velocity and temperature growth whereas concentration drops with

.

• Dissimilar behavior of thermophoresis and Brownian diffusions found towards and

where results for are alike.

• Curvature effects on velocity and mass transfer are similar.

• Fall in temperature is noticed with thermal radiation and curvature parameters.

• Fluid show deviation at the boundaries due to slip effect.

• Behavior of wall elastic parameters towards velocity and temperature is alike. Howeverdamping bears opposite impact.

• Chemical reaction aspects are more pronounced generative case.

• Rate of heat transfer reduces for growing values of 2 and .

177

Chapter 10

Numerical simulation for peristalsis

of Carreau-Yasuda nanofluid in

curved channel with mixed

convection and porous space

10.1 Introduction

Main theme of present chapter is to model and analyze the peristaltic activity of Carraeu-Yasuda

nanofluid saturating porous space in a curved channel. Unlike the traditional approach, the

porous medium effects are characterized by employing modified Darcy’s law for Carreau-Yasuda

fluid. To our knowledge this is first attempt in this direction for Carreau-Yasuda fluid. Heat

and mass transfer are further considered. Simultaneous effects of heat and mass transfer are

examined in presence of mixed convection, viscous dissipation and thermal radiation. The

compliant characteristics for channel walls are taken into account. The resulting complex

mathematical system is discussed for small Reynolds number and large wavelength concepts.

Numerical approximation to solutions are thus plotted in graphs and the physical description

is presented. It is concluded that larger porosity in a medium cause an enhancement in fluid

velocity and reduction in concentration.

178

10.2 Flow diagram

The mathematical description for an incompressible Carreau-Yasuda nanofluid in a channel

configured in a circle of inner radius ∗ and separation 2 is made in this section. The presence

of porous medium between the curved walls of the channel is considered. The gravitational

effects are taken into account. The dynamics of fluid inside the channel boundaries is developed

through the propagation of peristaltic waves along the channel walls (see Fig. 101). Moreover

relative to arterial like flow peristalsis the influential aspect of compliance in terms of wall’s

stiffness, elasticity and damping is not ignored. The relative positions of the curved channel

walls in radial direction can be visualized through the following expression:

Fig. 10.1. Geometry of the problem.

= ±( ) = ±∙+ sin

2

(− )

¸ (10.1)

179


The problem under consideration can be put in mathematical form via conservation principles

of mass, momentum, energy and nanoparticle volume fraction respectively. Thus one obtains

Continuity equation

+

∗

+∗

+

+∗= 0 (10.2)

-component of momentum equation:

∙

− 2

+∗

¸= −

+

1

+∗

{( +∗)}

+∗

+∗

−

+∗+ (10.3)

-component of momentum equation:

∙

+

+∗

¸= − ∗

+∗

+

1

( +∗)2

©( +∗)2

ª+

∗

+∗

+ ( − 0) + ( − )( − 0) + (10.4)

Energy equation:

()

∙

¸= 1∇2 + () (∇∇ ) + ()

(∇∇ )

−

µ−16∗ 303∗

¶+ τ L (10.5)

Nanoparticles mass transfer equation:

= (∇2) +

(∇2 ) (10.6)

The Cauchy stress tensor τ and extra stress tensor for Carreau—Yasuda fluid model are:

τ = −p+S (10.7)

S = ()A1 (10.8)

180

in which () can be obtained through the following relation:

() = ∞ + (0 − ∞)[1 + (Γ)1 ]

−11 (10.9)

where =p2(2) and D = 1

2[gradV + gradV ] The involvement of zero and infinite

shear-rate viscosities 0 and ∞ and the Carreau—Yasuda fluid parameters 1, and Γ provide

an edge to this fluid model to the associated characteristics of these five quantities. Firstly in

the range of high shear rate the dominance of viscous effects can be defined by 0 and ∞ along

the channel walls. On the other hand the shear thinning/thickening behavior can be predicted

through the parameters 1, and Γ. Actually the functioning of asymptotic viscosities (0 and

∞) is responsible for fluid regulation in the non-Newtonian arrangement. Moreover in Carreau-

Yasuda fluid model the specific values of parameters can form the numerous concentrated

polymer solutions such as 1 = 2 and ∞ = 0. Fixed value of Yasuda parameter 1 = 2

represents the Carreau model. The value of Yasuda parameter is fixed in this problem at

1 = 1. Also

∇2 = ( ∗

+∗)2

2

2+

1

+∗

{( +∗)}

The pressure drop and velocity are related by Darcy’s law. However for Carreau-Yasuda fluid

the relation is followed by newly developed modified Darcy’s law preserving following filtration

forms:

∇ = −∗

K¯

[∞ + (0 − ∞)[1 + (Γ)]

−1 ]V (10.10)

where the permeability and porosity of porous medium are represented by K¯and respectively.

The above generalized form is capable of recovering the results of Darcy law for large (→∞)or by assuming = 1. Since flow resistance containing porous space can be explained in terms

of pressure gradient, thus Eq. (1010) can be written as:

R =−∗K¯

[∞ + (0 − ∞)[1 + (Γ)]

−1 ]V (10.11)

The extra stress components and of in Carreau-Yasuda fluid can be obtained

using Eq. (108). It is remarkable to mention that the Rosseland approximation corresponding

to radiative heat flux is utilized in Eq. (105) to obtain the relevant radiation term. In considered

181

problem, the no-slip condition, prescribed surface temperature and concentration values at the

channel boundaries and the compliant properties of wall can be put in the following forms:

= 0 at = ± (10.12)

=

⎧⎨⎩ 1

0

⎫⎬⎭ , at = ± (10.13)

=

⎧⎨⎩ 1

0

⎫⎬⎭ , at = ± (10.14)

∗

+∗

∙−∗

3

3+∗1

3

2+ 0

2

¸ = −

∙

+

+∗

¸+

1

( +∗)2

©( +∗)2

ª+

∗

+∗

+ ( − 0) + ( − )( − 0) + at = ± (10.15)

Consideration of non-dimensional quantities and stream function ∗ ( ) by the defini-

tions below will lead to required set of equations as follows:

∗ =

∗ =

∗ =

∗ =

∗ =

=

∗

= − 0

1 − 0 =

−0

1 −0 ∗ =

∗ =2

=

=

2(1 − 0)

=

(1 − 0)( − )2

Re =

=

(1 − 0)

=

(1 − 0)

Pr =

1

=2

(1 − 0) = Pr =

16∗ 303∗

=

=

1 = − ∗3

3 2 =

∗13

3 3 =

30

2 =

K¯

2∗ (10.16)

= −∗

=

+

∗

in which the non-dimensional quantities above are the definitions of following physical parame-

ters: the wave number, the amplitude ratio parameter, Re the Reynolds number, Pr the

182

Prandtl number, 1, 2 3 the elasticity parameters, the radiation parameter the local

temperature Grashof number, the local nanoparticles Grashof number, the Eckert num-

ber, the Brinkman number, the thermophoresis and Brownian motion parameters

respectively the Schmidt number and the Darcy number.

Thus utilization of above parameters, omission of asterisks and long wavelength approxima-

tion yield:

= 0 (10.17)

− +

+

1

( + )2

£( + )2

¤+ ( +)

+1

∙1 +(1− )(− 1)

µ−

2

2+

1

+

¶¸= 0 (10.18)

2

2+

1

+

+

µ−

2

2+

1

+

¶2 ∙1 +(1− )(− 1)

µ−

2

2+

1

+

¶¸+Pr

µ2

2

¶+Pr

µ2

2

¶+Pr

µ

¶= 0 (10.19)

µ2

2+

1

+

¶+

µ2

2+

1

+

¶= 0 (10.20)

= 1 + sin 2 (− ) (10.21)

= 0 at = ± (10.22)

=

⎧⎨⎩ 1

0

⎫⎬⎭ at = ± (10.23)

=

⎧⎨⎩ 1

0

⎫⎬⎭ at = ± (10.24)

183

+

∙1

3

3+2

3

2+3

2

¸ =

1

( + )2

£( + )2

¤+ ( +)

+1

∙1 +(1− )(− 1)

µ−

2

2+

1

+

¶¸ at = ± (10.25)

The equation of stream function can be obtained from Eqs. (1017) and (1018) by elimi-

nating pressure. Thus one gets

∙1

( + )

£( + )2

¤+ ( + )( +)

¸+

∙1

½1 +(1− )(− 1)

µ−

2

2+

1

+

¶¾¸= 0 (10.26)

where

=

µ−

2

2+

1

+

¶ ∙1 +1

(1− )(− 1)1

µ−

2

2+

1

+

¶1¸ (10.27)

Here =∞0

and = Γdepict the viscosity ratio parameter and Weissenberg number

respectively. It can be verified that for = 1 or = 0 the results of the viscous nanofluid

with porous medium can be recovered as a special case of present problem.


The above mentioned problem results in the non-linear coupled system of equations whose

analytic solution seems difficult to obtain. However with the intense algorithmic advancement

many built-in solution softwares are available at present. is one of these. The

exact as well as numerical approximation to solution expressions can be obtained efficiently

through built-in routine NDSolve provides level of numerical computation with

its systematic algorithm selection, automatic error tracking and precision arithmetics. Here

we solve the above system numerically to skip the complexity of solutions and to obtain the

graphical results directly. Thus the graphical description of pertinent parameters towards axial

velocity , temperature , concentration and heat transfer coefficient has been made in this

section. Particularly the development of , and with the varying values of heat and mass

transfer Grashof numbers and , thermophoresis and Brownian motion parameters

184

and , wall compliant parameters 1 2 3 Darcy number , viscosity ratio parameter

fluid parameter , curvature parameter , Prandtl number Brinkman number , radiation

parameter , Weissenberg number will be emphasized via physical basis.


Developments in velocity distribution as a result of variation in different embedding parameters

are sketched in this subsection via Figs. 102(− ) The axial velocity is noticed an increasing

function of mixed convection parameters (Grashof numbers). It is due to viscosity drop (see

Fig. 102()). Mixed convection is proficient to provide energy dissipation in nuclear reactor

technology and electronic cooling processes where forced convection fails to achieve required

target. The dual response of on velocity is captured in Fig. 102(). The porosity shows an

increase in velocity since adding more pores causes flow easier in a medium. Thus increasing

behavior of is drawn through Fig. 102(). Clinically pores in walls of blood capillaries allow

exchange of water, oxygen and many other nutrients between the blood and the tissues. Growing

values of wall elastic parameters produce velocity development where damping effects oppositely

(see Fig. 102()). The thermophoresis () lowers speed of nanoparticles that in turn lowers

fluid velocity (see Fig. 102()). On the other hand viscosity gets weak with Brownian diffusion

() and so activation of is observed with (see Fig. 102()). The results of Figs. 102(()

& ()) show dual response of and on . It is seen that non-symmetric velocity rises near

positive side of channel and it reduces near negative side. The decline in velocity with an

increase in is depicted in Fig. 102(). Due to curved flow configuration the velocity preserves

non-symmetric behavior. Also becomes flatten as straight channel is obtained ( →∞).


The physical description of embedded parameters on temperature is made in this subsection

(see Figs. 103( − )). Impression of towards (see Fig. 103()). Radiation indicates

heat decay and thus decreasing response with an increase in towards is noticed from Fig.

103(). An increase in and activates energy production and thus temperature rises

(see Figs. 103(() & ()). Verification of results can be made with the study [6]. Increasing

porosity () causes rise in temperature since addition of pores causes growth in velocity and

185

hence temperature of nanofluid (see Fig. 103()). Since thermal conductivity decreases with

an increase in therefore decay in is noticed from Fig. 103(). The curvature tends

to reduce the temperature when one moves from curved to planer channel (small to large ).

Additionally greater impact is seen in case of curved channel (see Fig. 103()). The wall elastic

parameters 1 and 2 produce temperature development while temperature decays for 3 (see

Fig. 103()).

10.4.3 Nanoparticle volume fraction distribution

Figs. 104( − ) communicate the development in nanoparticle volume fraction distribution

. Decrease in is noticed with . Hence there is diffusion enhancement with (see Fig.

104()). The density of nanoparticles enhances with growth of Brownian diffusion. Increase in

is captured for larger in Fig. 104(). The characteristics of wall compliant parameters on

are found opposite from and i.e., an increase in 1 and 2 correspond decline in where

3 causes promotion of Such results are anticipated since elasticity causes deformation of

nutrients easier in case of blood veins and arteries where alternate effect of damping is recorded

clinically (see Fig. 104()). The decay of is noticed from Fig. 104() for larger . Higher

allow more pores in the medium which are responsible for diffusion of fluid and reduction of

The flow stream is converted to straight regime as we increase the value of curvature. From

Fig. 104() it is noticed that the volume fraction reduces when we move from curved to planer

regimes (small to large ).


The variation in absolute heat transfer rate under the influence of involved parameters is

prepared in this subsection via Figs. 105( − ). In favour of peristaltic waves along the

channel boundaries the dual response of graphs towards is captured. The thermophoresis

and Brownian diffusions ( ) enhance the heat transfer rate (see Figs. 105(() & ()). The

drawn results of Fig. 105() indicate dominance of with higher values of . The results

captured in Figs. 105(() & ()) have opposite responses of and towards i.e., decline

of is observed with rise in whereas enhances for an increase in .

186

10.2(a) 10.2(b)

10.2(c)

187

10.2(d) 10.2(e)

10.2(f)

188

10.2(g) 10.2(h)

10.2(i)

Fig. 102: Axial velocity variation with = 02 = 01 & = 01

189

10.3(a) 10.3(b)

10.3(c) 10.3(d)

190

10.3(e) 10.3(f)

10.3(g) 10.3(h)Figs. 103: Temperature variation with = 02 = 01 & = 01

191

10.4(a) 10.4(b)

10.4(c) 10.4(d)

192

10.4(e)

Figs. 104: Nanoparticle mass transfer variation with = 02 = 01 & = 01

193

10.5(a) 10.5(b)

10.5(c) 10.5(d)

194

10.5(e)

Figs. 10.5: Heat transfer coefficient variation with = 01 & = 01

195


Mixed convection flow bounded in curved channel with compliant boundaries is developed

for Carreau-Yasuda nanofluid. The observation is made for porous medium using modified

Darcy’s law specifically. Such conditions are applicable in blood vessels where small pores allow

exchange of water, ions, gases, lymph transport and other small molecules. An increase in

porosity signifies disease states where endothelial barrier breaks down and allow large molecules

like protein out of the vessel. In addition the thermal radiation and viscous dissipation effects

are also examined. The particular points of this study are:

• The non-symmetric is the outcome of curved channel.

• Mixed convection increases the fluid velocity.

• The fluid velocity, temperature and heat transfer rate show dominating behavior towardsDarcy number where concentration falls for .

• The opposite results of and are seen for velocity and concentration.

• Weissenberg number preserves decelerating impact on velocity whereas fluid parameters and increase .

• Reduction in , and is noticed with an increase in curvature.

• Viscous dissipation affects and positively whereas alternative results of radiation are

observed.

196

Chapter 11

Peristaltic motion of Eyring-Powell

nanofluid in presence of mixed

convection

11.1 Introduction

The peristaltic motion of Eyring-Powell fluid subject to nanofluid phenomenon in curved chan-

nel has been modeled in this chapter. The mass, momentum and energy conservation equations

comprised of viscous dissipation and mixed convection. Brownian motion and thermophoresis

effects are captured for nanofluid flow. Assuming the large wavelength of the peristaltic wave

the whole coupled system has been simplified and then solved through numerical approxima-

tion. Fluid velocity and temperature show rise in their values with an increase in Brownian

diffusion whereas fall in these quantities is noticed with thermophoresis.

11.2 Flow diagram

Consider the flow of an incompressible Eyring-Powell nanofluid in a channel coiled in circle of

radius ∗ and separation 2. Flow is induced due to advancement of peristaltic waves along

the curved channel walls. The coordinates of curved channel are chosen such that is along the

channel whereas is perpendicular to (see Fig. 11.1). In addition the gravitational effects,

197

wall compliant properties such as stiffness, elasticity and damping are present in the present

flow analysis. The configuration of wall geometry comprising the sinusoidal peristaltic waves

travelling along the channel walls with speed in the form:

Fig. 11.1. Geometry of the problem.

= ±( ) = ±∙+ sin

2

(− )

¸ (11.1)


The governing equations for the considered flow are:

+

∗

+∗

+

+∗= 0 (11.2)

198

∙

− 2

+∗

¸= −

+

1

+∗

{( +∗)}

+∗

+∗

−

+∗ (11.3)

∙

+

+∗

¸= − ∗

+∗

+

1

( +∗)2

©( +∗)2

ª+

∗

+∗

+ ( − 0) + ( − )( − 0) (11.4)

()

∙

¸= 1∇2 + () (∇∇ ) + ()

(∇∇ )

−

µ−16∗ 303∗

¶+ ( − )

+(

+

∗

+∗

−

+∗) (11.5)

= (∇2) +

(∇2 ) (11.6)

Extra stress tensor for Eyring -Powell fluid is [50]:

S =

∙+

1

sinh−1(

1)

¸A1 (11.7)

where =q

12(A1)2 and , 1 represent the material fluid parameters. Series expansion of

sinh−1 leads the following form:

sinh−1(

1)=

1− 16(

1)3 (

1)5 1 (11.8)

The relevant boundary conditions include:

= 0 at = ± (11.9)

= 0, at = ± (11.10)

= 0, at = ± (11.11)

199

∗

+∗

∙−∗

3

3+∗1

3

2+ 0

2

¸ = −

∙

+

+∗

¸+

1

( +∗)2

©( +∗)2

ª+

∗

+∗

+ ( − 0)

+( − )( − 0) at = ± (11.12)

Utilizing the definitions of non-dimensional quantities and stream function ∗ as follows:

∗ =

∗ =

∗ =

∗ =

∗ =

=

∗

= − 0

0 =

− 0

0 ∗ =

∗ =2

=

=

20

=

0( − )2

Re =

=0

=

0

Pr =

1 =

2

0 = Pr

=16∗ 303∗

=

1 = − ∗3

32 =

∗13

3 3 =

30

2

=1

1 =

2

6212 (11.13)

With symbols and as the Eyring-Powell fluid parameters.

Upon application of above mentioned definitions and the long wavelength approximation

and

= −

=

+

(11.14)

the continuity equation is identically satisfied. The remaining system under long wavelength

assumption perceive the following form:

= 0 (11.15)

− +

+

1

( + )2

£( + )2

¤+ ( +) = 0 (11.16)

200

2

2+

1

+

−

µ2

2− 1

+

¶+Pr

µ2

2

¶+Pr

µ2

2

¶+Pr

µ

¶= 0 (11.17)

µ2

2+

1

+

¶+

µ2

2+

1

+

¶= 0 (11.18)

with conditions

= 1 + sin 2 (− ) (11.19)

= 0 = ± (11.20)

= 0 = 0 at = ± (11.21)

+ [1

3

3+2

3

2+3

2

] =

1

( + )2

£( + )2

¤+ + at = ± (11.22)

Here the asterisks have been dropped for simplicity. Combination of Eqs. (11.15) and

(11.16) lead to

∙1

( + )

£( + )2

¤+ ( + )( +)

¸= 0 (11.23)

where

= −(1 +)

µ2

2− 1

+

¶+

µ2

2− 1

+

¶3

11.4 Discussion

The system is coupled and non-linear and thus difficult to solve exactly for explicit solutions.

Thus we execute this system numerically using built-in command NDSolve in .

Hence in this section the development of velocity , temperature concentration and heat

transfer distribution with rise in Eyring-Powell fluid parameters , heat and mass transfer

Grashof numbers and , thermophoresis and Brownian motion parameters and ,

wall compliant parameters 1 2 3 curvature parameter , Prandtl number , radiation

201

parameter and Brinkman number .


The velocity outcomes corresponding to emerging parameters have been sketched via graphical

illustrations in this subsection. The graphical description of Figs. 112(−) show the parabolicshape of . The decreasing impact of Eyring-Powell fluid parameters and is captured in

Figs. 112( & ) where maximum impact is seen at the centre of curved channel. Figs. 112( &

) are developed to examine the effects of heat and mass transfer Grashof numbers ( and )

on The displayed results portray the increasing effect of on where an increase in tends

to reduce (see Figs. 112( & )). Since an increase in causes reduction in viscosity whereas

with its increasing values enhances the fluid concentration. The parameters arising due to

mixed convection are found substantially useful in heating or cooling of channel walls with small

separation and in case of laminar flow to dissipate energy more actively than forced convection.

The Brownian and thermophoresis diffusion ( and ) preserve reverse effects towards . It

is observed that due to its heat absorbent characteristic reduce the fluid viscosity and thus

fluid velocity appears to be increasing (see Fig. 112()). On the other hand an increase in

reduces since it makes the nanofluid particles more dense (see Fig. 112()). The comparison

of results has been made via studies [79, 114, 132] in Table 1. Fig. 112() demonstrates the

increasing behavior of with flexible wall parameters 1 and 2 whereas opposite trend is

noticed for 3 Directed to blood flow microcirculatory systems rise in elasticity and mass per

unit area make flow easier where damping acts alternately. The reduction in velocity is noticed

with an increase in curvature parameter as depicted in Fig. 112(). Moreover the velocity

profile becomes flatten as straight channel ( →∞)is approached (see Fig. 112()).


The demonstration of physical effects of involved parameters on nanofluid temperature distri-

bution has been made through Figs. 113(− ). Similar to velocity the decreasing behavior

of is observed upon increasing and (see Figs. 113( & )). An increase in and

comprised of dissimilar outcomes of i.e., higher values of rises where acts alternately

(see Figs. 113( & )). Similar results have been reported by Hayat et al. [79, 114, 132] in

202

the limiting case (see Table 1). Larger values of causes decline in since an increase in

strengthens viscosity and reduces thermal conductivity (see Fig. 113()). Decay in tem-

perature is noticed for larger (see Fig. 113()). The elasticity and mass characterizing

parameters 1 and 2 produce enhancement of where damping coefficient 3 tends to reduce

it (see Fig. 113()). Physically 1 and 2 elevate the flow speed (cardiovascular compliance)

and particles moving with higher velocities elevate the flow whereas damping coefficient 3

reduce the speed and temperature of moving fluid. Decline in temperature is sketched in Fig.

113() as planar channel is approximated ( → ∞). Additionally greater impact is seen incase of curved channel.


Figs. 114(− ) transmit the nanoparticle concentration concerning to different parameters

of interest. The decline of concentration is captured as enhances its value whereas causes

concentration development (see Figs. 114( & )). Moreover more clear results are recorded

near positive side of curved channel. The captured results of Fig. 114() manifest the growth of

with growing values of The nanoparticles acquire higher densities with an enhancement

in Brownian motion which in turn develops the concentration of fluid The outcome appears

to be decreasing towards higher values of thermophoresis diffusion since viscosity weakens when

increases and less viscous particles diffuse out to reduce the concentration (see Fig. 114().

Increasing curvature or in other words migration of flow to planar regime ( → ∞) producesconcentration development (see Fig. 114()).

Table 1: Comparison of present result with refs. [79, 114, 132]

Authors

Present 1-4 1-4 opposite response

Hayat et al. [79] 0-3 0-30 similar response

Hayat et al. [114] 0-0.6 0.3-0.6 opposite response

Hayat et al. [134] 0-3 1-1.5 opposite response

203


The development of heat transfer rate for the considered flow analysis has been made via

Figs. 115( − ). The involvement of peristaltic wave at the curved channel boundaries

produces oscillatory graphical description of . The reduction in transfer of heat is noticed

upon increasing Brownian diffusion, thermal radiation and curvature (see Figs. 115( & )).

Since movement of fluid from curved to planar (small to large ) channel minimize heat transfer

from boundary to fluid so shows decreasing effect with an increase in curvature (see Fig.

115()) The thermophoresis promotes rate of heat transfer with its growing values (see Figs.

115()).

204

11.2(a) 11.2(b)

11.2(c) 11.2(d)

205

11.2(e) 11.2(f)

11.2(g) 11.2(h)Figs. 112: Variation in axial velocity when = 02 = 01 & = 02

206

11.3(a) 11.3(b)

11.3(c) 11.3(d)

207

11.3(e) 11.3(f)

11.3(g) 11.3(h)Figs. 113: Variation in temperature when = 02 = 01 & = 02

208

11.4(a) 11.4(b)

11.4(c) 11.4(d)

209

11.4(e)

Figs. 114: Variation in nanoparticle mass transfer distribution when = 02 = 01 &

= 02

210

11.5(a) 11.5(b)

11.5(c) 11.5(d)Figs. 115: Variation in heat transfer coefficient when = 01 & = 02

211


Magnetohydrodynamic peristaltic flow of Eyring-Powell nano material is considered in a curved

configuration. Wall compliant properties and thermal radiation aspects are highlighted. The

attractive features of this study are:

• Fluid parameters cause reduction in the velocity and temperature whereas concentrationgives oscillatory response.

• Heat and mass transfer Grashof umbers and bear opposite impression towards

velocity.

• Dissimilar response of thermophoresis and Brownian motion is noticed for , and .

• Qualitatively similar response of velocity and temperature of nanofluid towards curvatureis recorded.

• Larger values of thermal radiation and Prandtl number perceive decline in temperature.

• Wall compliant parameters produce increment in velocity and temperature.

• Heat transfer rate decreases for , and it increases for

212

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Documents

Models and analysis for curvature and wall properties