Peristaltic Transport of non-Newtonian Fluids in a Curved Channelprr.hec.gov.pk/jspui/bitstream/123456789/7213/1/Khurram... · 2018-07-23 · Peristaltic Transport of non-Newtonian

Peristaltic Transport of non-Newtonian

Fluids in a Curved Channel

by

Khurram Javid

Department of Mathematics and Statistics

Faculty of Basic and Applied Sciences

International Islamic University,

Islamabad, Pakistan

2016



by

Khurram Javid

Supervisor by

Dr. Nasir Ali

Co Supervisor

Dr. Muhammad Sajid




Islamabad Pakistan

2016



by

Khurram Javid

A DISSERTATION SUBMITTED IN THE PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

IN

MATHEMATICS

Supervisor by

Dr. Nasir Ali

Co Supervisor

Dr. Muhammad Sajid




Islamabad Pakistan

2016

Certificate

Peristaltic Transport of non-Newtonian Fluids in

a Curved Channel

by

Khurram Javid

A THESIS SUBMITTED IN THE PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN

MATHEMATICS

We accept this thesis as conforming to the required standard

1. 2.

Prof. Dr. Saleem Asghar Prof. Dr. Masood Khan External Examiner External Examiner

3. 4.

Dr. Tariq Javed Dr. Nasir Ali Co-Supervisor Supervisor

5. 6.

Prof. Dr. M. Sajid, TI Prof. Dr. M. Arshad Zia Internal Examiner Chairman




Islamabad, Pakistan

2016

Declaration

I hereby declare and affirm that this research work neither as a whole nor as a part has been

copied out from any source. It is further declared that I have developed this research work

entirely on the basis of my personal efforts. If any part of this thesis is proven to be copied

out or found to be a reproduction of some other, I shall stand by the consequences.

Moreover, no portion of the work presented in this thesis has been submitted in support of

an application for other degree or qualification in this or any other university or institute of

learning.

Name and signature of student: .

Khurram Javid

PhD (Mathematics)

Reg. # 12-FBAS/PHDMA/S-12




Islamabad Pakistan

2016

Acknowledgements

First of all, I pay my special thanks to the creator of mankind, the everlasting Allah, who

gave us this life, taught us everything we did not know, granted us health, knowledge and

intelligence to extract the hidden realities in the universe through scientific and critical

approach. I just want to add this verse, start learning with the name of Almighty Allah and

you will find the right way you even never expected. I thank the lord Almighty with whose

kindness I have achieved this very important goal in my life. I offer salutations upon the

Holy Prophet, Hazrat Muhammad (PBUH) who has lightened the life of all mankind

with His guidance. He is a source of knowledge and blessings for the entire creations. His

teachings make us to ponder and to explore this world with directions of Islam.

I express my profound gratitude to my respectable supervisor Dr. Nasir Ali and co-

supervisor Dr. Muhammad Sajid, who helped me throughout my PhD studies to complete

my thesis. There many valuable comments and suggestions put me on the straight path

when I was led astray. I also pay my regards to all my teachers who always directed me to

right dimensions and made it possible for me to achieve an attractive goal.

My deepest gratitude goes to my family for their unflagging love and support throughout

my life; this thesis is simple impossible without them. I am indebted to my father (late)

for his care and love. I’m missing Mom (late). She always wants that I will do better and

better. I’m nothing without you my Mom (late) and father (late). Both of you are always

in our hearts. I would like to thank all my friends and family members specially Mr. Akbar

Zaman, Mr. Aamir Abaasi, Mr. Zeeshan Asghar, Mr. Zaheer Asghar, Mr. Mudassir, Mr.

Usman, Mr. Khalid Mehmood, Mr. Ali Naqi, Mr. Haroon Javed, Mr. Farrukh Javed, Umer

Rana, Mr. Sajjad Akram, Mr. Zeeshan Bashir, Mr. Syed Mughees Ali, who always helped

me during my studies in all respects. I’m grateful to them for all the emotional support,

entertainment and care they provided. I have learned a lot during group

meetings/discussions with my Supervisor and Co-Supervisor.

I extend my gratitude to my family for encouragement and support, even in the gloomiest

of times. Their prayers have always been my driving source and whose sustained hope led

me to where I am today. I am very thankful to Higher Education Commission (HEC),

Pakistan for providing me scholarship, Indigenous 5000 PhD Fellowship Program.

Khurram Javid

PIN#17-5(2Ps1-383)/HEC/Sch-Ind/2012

Dedicated to

Dr. Nasir Ali (Supervisor)

&

My Parents (late), who have always been a source of

Inspiration, Zeal and Strength for me.

All members of my family.

Preface

Synchronized contractions of the muscles that push food contents over the gastrointestinal

(GI) tract to facilitate normal digestion and the absorption of nutrients is known as

peristalsis. This phenomenon depends upon the synchronization between the muscles,

nerves and hormones in the digestive tract. Apart from that peristalsis is also involved in

urine transport from kidney to bladder, bile transfers from gall bladder into the duodenum,

the transport of spermatozoa, blood circulation in small blood vessels, the motion of chyme

in the small intestine, the mechanical and neurological aspects of reflux, transport of lymph

in the lymphatic vessels and in the vasomotion of small blood vessels such as arterioles,

venules and capillaries. Applications of peristalsis in industry include the transport of high

solids slurries, aggressive chemicals, noxious fluids (nuclear industries) and other materials

which are transported by peristaltic pumps. Hose pumps, roller pumps, tube pumps, heart-

lung machines, finger pumps, blood pumps and dialysis machines operate according to the

principle of peristalsis. The mechanism of peristalsis is studied extensively in past few

decades because of its practical importance.

Literature survey indicates that a number of theoretical and experimental studies have been

carried out dealing with peristaltic motion in straight geometries. However, less attention

is paid to the analysis of peristaltic motion in curved channel. Moreover, such studies

further narrow down for non-isothermal case. In view of the above mentioned gaps in the

literature, the aim of this thesis is mainly to investigate the peristaltic flow of non-

Newtonian fluids in a curved channel under long wavelength and low Reynolds number

assumptions. The effects of magnetic field and heat transfer on the flow are also

investigated. All problems are solved in dimensionless form with the help of analytical

(regular and singular perturbation techniques) and numerical methods (finite

difference technique, BVP4C technique, Spectral Chebyschev Collocation technique).

The thesis is organized in the following manner.

Chapter one is based on the brief introduction of peristaltic flows. Some basic definitions,

fundamental equations in curvilinear coordinates and review of existing literature on

peristalsis involving viscous and non-Newtonian fluids is presented. The important

dimensionless numbers are also defined.

In chapter two, peristaltic motion of a viscoelastic Jeffrey fluid in a curved channel under

the influence of radially-imposed magnetic field is investigated. The Jeffrey fluid model is

a fairly simple linear model using a time derivative instead of a convected derivative as

featured in the Oldroyd-B model. This model includes elastic and memory effects known

to be exhibited by gastric fluids and also certain dilute polymer solutions. The problem is

first normalized and then governing partial differential equations are reduced to a single

linear ordinary differential equation in terms of a stream function under long wavelength

and low Reynolds number approximations. Exact as well as asymptotic solutions of this

equation are obtained. The graphs of velocity profile and pressure rise per wavelength are

plotted. The streamlines are presented to discuss the trapping phenomenon. The contents

of this chapter are submitted for publication in the journal “AIP Advances”.

Chapter three deals with the peristaltic motion of an Oldroyd-B fluid in a curved channel.

The flow equation is derived under long wavelength and low Reynolds number

assumptions. Matlab built-in routine bvp4c is utilized to solve this nonlinear ordinary

differential equation. Numerical solution for axial velocity, pressure gradient, pressure rise

per wavelength and stream function are obtained for various values of Weissenberg

number. The interaction of curvature parameter with Weissenberg number is highlighted.

This study is published in “Meccanica”, 51 (2016) 87 – 98.

Chapter four examines the peristaltic transport of an Oldroyd 4-constant fluid through a

curved channel. The components of stress, based on the constitutive equation of an Oldroyd

4-constant fluid are obtained in curvilinear coordinates. The governing equation is

formulated in a wave frame of reference. The present flow model subject to long

wavelength and low Reynolds number involves viscoelastic features. Moreover, the

governing equations under such approximations are nonlinear. The resulting nonlinear

mathematical problem is solved numerically by a finite-difference method (FDM) with an

iterative scheme. Special attention is given to the flow characteristics, pumping and

trapping phenomena. A comparative study between curved and straight channels is also

included. The contents of this chapter are currently under review in “Brazilian Society of

Mechanical Sciences and Engineering”.

Chapter five looks at the peristaltic motion of an incompressible Carreau fluid in a curved

channel. The Carreau fluid model is capable of robustly predicting shear thinning, shear

thickening and relaxation effects. Numerical solution of governing boundary value

problem is presented by using a finite-difference method (FDM) with an iterative scheme.

The nonlinear boundary value problem (BVP) is also solved with an optimized spectral

Chebyschev collocation method (SCCM). An excellent correlation is observed between

the results obtained by both methods. Boundary layer formation at the channel walls is

observed for large values of Weissenberg number and for strong shear-thinning fluid. The

pumping and trapping phenomena are illustrated. The analysis presented in this chapter is

accepted for publication in “Computer Methods in Biomechanics and Biomedical

Engineering” and currently available online DOI: 10.1080/10255842.2015.1055257.

In chapter six, we explore the effects of an applied magnetic field on the peristaltic flow

of a Sisko fluid in a curved channel. The flow problem is modeled by employing long

wavelength and low Reynolds number assumptions. The solution of equation governing

the flow is constructed using the method of matched asymptotic expansion for two specific

values of power-law index. Our main focus is to highlight the boundary layer character of

the solution. The choice of Sisko model is driven firstly by its effectiveness in describing

the flow properties of shear-thinning material over four or five decades of shear rate and

secondly due to its superiority over power-law model. We found that the estimates of

boundary layer thickness at upper and lower walls in either case are different. Moreover,

the boundary layer thickness in either case is found to be inversely proportional to the

Hartmann number. The work presented in this chapter is accepted for publication in

“Meccanica”. The contents are available online DOI: 10.1007/s11012-015-0346-2.

The combined effects of fluid slippage at the channel walls, applied magnetic field and

non-Newtonian rheology on peristaltic flow in a curved channel are presented in chapter

seven. In this study, we opted for Williamson model to represent the rheology of the fluid

inside the channel. The Williamson model corresponds to fluids exhibiting strong shear-

thinning and relaxation effects. The resulting nonlinear boundary value problem (BVP) is

solved using an implicit finite difference method (FDM). An extensive quantitative

analysis is performed through numerical computations for velocity distribution, pumping

and trapping phenomena. The material presented in this chapter is published in “AIP

Advances”, 6 (2016) 025111. DOI: 10.1063/1.4942200.

Chapter eight is devoted to analyze the heat transfer in peristaltic flow of an Oldroyd 8-

constant fluid in a curved channel. The problem is modeled using fundamental laws of

mass, momentum and energy under long wavelength and low Reynolds number

assumptions. The modeled equations are simulated using a robust implicit finite difference

technique. The effects of various emerging parameters on the flow and heat transfer

characteristics are reported. The main findings of this chapter are published in

“International Journal of Heat and Mass Transfer”, 94 (2016) 500 – 508.

The study of peristaltic flow and heat transfer in a curved channel using the rheological

equation of Cross model is presented in chapter nine. The problem statement is based

upon laws of conservation of mass, linear momentum and energy. The problem is modeled

in curvilinear coordinates under long wavelength and low Reynolds number assumptions.

A well-testified finite difference method (FDM) is employed for the solution. The

influence of rheological parameters of Cross fluid, Brinkman number and curvature of the

channel on the flow and heat transfer phenomena is shown graphically and discussed in

detail. The results presented in this chapter are submitted for possible publication in

“International Journal of Heat and Fluid Flow”.

1

Contents

Chapter 1 ..........................................................................................................................10

Introduction ........................................................................................................................10

1.1 Peristalsis and its applications .............................................................................10

1.2 Review of literature .............................................................................................12

1.2.1 Peristaltic flow of Newtonian fluids ............................................................12

1.2.2 Peristaltic flow of non-Newtonian fluids .....................................................15

1.2.3 Heat and mass transfer in peristaltic flows ..................................................18

1.2.4 Peristaltic flows in a curved channel ...........................................................21

1.3 Some basic definitions, equations and terminologies .........................................23

1.3.1 Curvilinear coordinates ................................................................................23

1.3.2 Gradient, divergence and curl in curvilinear coordinates ............................27

1.3.3 Equation of continuity in curvilinear coordinates ........................................29

1.3.4 Equation of motion in curvilinear coordinates.............................................30

1.3.5 Equation of energy in curvilinear coordinates .............................................31

1.3.6 Maxwell’s equations ....................................................................................32

1.3.7 Lorentz force ................................................................................................32

1.3.8 Dimensionless numbers ...............................................................................33

2

Chapter 2 ..........................................................................................................................37

Exact and asymptotic solutions for hydromagnetic peristaltic flow of Jeffrey fluid in

a curved channel ..............................................................................................................37

2.1 Mathematical model and rheological constitutive equations ..............................38

2.2 Exact solutions ....................................................................................................47

2.3 Results and Discussion ........................................................................................50

2.4 Concluding remarks ............................................................................................61

Chapter 3 ..........................................................................................................................63

Simultaneous effects of viscoelasticity and curvature on peristaltic flow through a

curved channel .................................................................................................................63

3.1 Description of the problem ..................................................................................64

3.2 Method of solution ..............................................................................................70

3.3 Results and discussion .........................................................................................70


Chapter 4 ..........................................................................................................................79

Long wavelength analysis for peristaltic flow of Oldroyd 4-constant fluid in a

curved channel .................................................................................................................79

4.1 Description of the problem ..................................................................................79

3

4.2 Numerical method ...............................................................................................81

4.3 Results and discussion .........................................................................................84


Chapter 5 ..........................................................................................................................93

Numerical simulation of peristaltic flow of a bio-rheological fluid with shear-

dependent viscosity in a curved channel ........................................................................93

5.1 Mathematical model and rheological constitutive equations ..............................94

5.2 Validation with Spectral Chebyschev Collocation method (SCCM) ..................96

5.3 Results and interpretation ..................................................................................100

5.3.1 Flow characteristics ...................................................................................101

5.3.2 Pumping characteristics .............................................................................102

5.3.3 Trapping .....................................................................................................103

5.4 Concluding remarks ..........................................................................................110

Chapter 6 ........................................................................................................................112

Existence of Hartmann boundary layer in peristalsis through curved channel:

Asymptotic solution .......................................................................................................112

6.1 Mathematical formulation and rheological constitutive equations ...................113

6.2 Asymptotic solution ..........................................................................................115

4

6.2.1 The case for n = 1 and 𝑎 ∗≠ 0 ...................................................................116

6.2.1.1 Inner solutions ............................................................................................117

6.2.1.2 Outer solution.............................................................................................119

6.2.1.3 The case for 𝑛 = 1/2 and 𝑎 ∗≠ 0 ............................................................122


Chapter 7 ........................................................................................................................131

Simulations of peristaltic slip-flow of hydro-magnetic bio-fluid in a curved

channel ............................................................................................................................131

7.1 Description of the problem ................................................................................131

7.2 Results and discussion .......................................................................................134


Chapter 8 ........................................................................................................................145

Peristaltic flow and heat transfer in a curved channel for an Oldroyd 8–constant

fluid..................................................................................................................................145

8.1 Mathematical formulation .................................................................................145

8.2 Results and discussion .......................................................................................149


5

Chapter 9 ........................................................................................................................162

Numerical study for the flow and heat transfer in a curved channel with peristaltic

walls .................................................................................................................................162

9.1 Mathematical model ..........................................................................................163

9.2 Computation results and interpretation .............................................................164

9.2.1 Flow characteristics ...................................................................................164

9.2.2 Pumping characteristics .............................................................................166

9.2.3 Heat transfer phenomena ...........................................................................167

9.2.4. Trapping .....................................................................................................169


6

Nomenclature

a Infinite-shear rate viscosity

b Consistency index

𝑎1 Half width of the curved channel

𝑎2 Amplitude of the peristaltic wave

ia Component of the acceleration vector

*a The generalized ratio of infinite-shear rate viscosity to the

consistency index in a Sisko fluid

1 2,b b Scale factors

B applied magnetic field in the radial direction

0B Strength of magnetic field

c Wave speed

pc Specific heat at constant pressure

1

kjd Differentiation matrix of order n

, 1 4iE i Unknown constants

Re Unit vector in the radial direction

Xe Unit vector in the azimuthal direction

Ha Hartmann number

J Current density

k Dimensionless curvature of the channel

*k Thermal conductivity

7

n Power-law index

O Center of curvature of the channel

p Pressure in wave frame

P Pressure in fixed frame

𝑃∗ Modified pressure

q Time-averaged flow rate in wave frame

r Radial coordinate in wave frame

R Radial coordinate in fixed frame

𝑅∗ Dimensional radius of curvature

Re Reynolds number

x Axial coordinate in wave frame

X Axial coordinate in fixed frame

T Temperature field

0T Temperature at lower wall

1T Temperature at upper wall

)( jn xT Chebyshev polynomial

T Time

v Axial velocity component in wave frame

u Radial velocity component in wave frame

V Axial velocity component in fixed frame

U Radial velocity component in fixed frame

We Weissenberg number

8

z Heat transfer coefficient

' ',X Y Cartesian coordinates fixed at O

, , , , ,

, , , 1,2,

3,4.

j j

i i i i iC D H U M

i

j

Unknown constants

Greek Symbols

η Dimensionless radial distance in wave frame

* The ratio of the infinite-shear-rate viscosity to the zero-shear-rate

viscosity

Slip parameter

1 Coefficient of thermal expansion

λ Wavelength

' Ratio of relaxation to retardation time

2 Retardation time

1 7i i Material constants of Oldroyd 8-constants fluid

Dissipation function

Dimensionless temperature field in wave frame

𝜌 Fluid density

Wave number

Time constant

Second invariant of strain-rate tensor

Coefficient of dynamic viscosity

9

0 Zero-shear-rate viscosity

Infinite-shear-rate viscosity

Time-averaged flow rate in fixed frame

Cauchy stress tensor

S Extra stress tensor

, ,RR RX XXS S S Components of extra stress tensor in fixed frame

, ,x xxS S S Components of extra stress tensor in wave frame

First Rivlin–Ericksen tensor

Second Rivlin–Ericksen tensor

Amplitude ratio

Stream function

Under-relaxation parameter

DDt

Contravariant convected derivative

', , z ,s w Stretched variables

10

Chapter 1

Introduction

This chapter starts with a brief account of peristaltic motion and its applications. A review

of relevant literature on peristaltic motion and development of governing equations in

general curvilinear coordinates are included in the main body. The Maxwell’s equations

and dimensionless numbers appearing in the later chapters are explained in the last two

sections.

1.1 Peristalsis and its applications

The flow induced by sinusoidal contractions of a flexible wall, commonly known as

peristaltic flow, finds diverse applications in physiological and industrial domains.

Peristaltic mechanism leads to the rise of pressure gradient that eventually pushes the fluid

forward. Generally, this pumping phenomena fall by from the region of lower pressure to

the region of higher pressure. A variety of complex rheological fluids are transported due

to peristaltic pumping. In particular peristaltic motion appears in swallowing food through

oesophagus (Fig. 1.1), urine transport from kidney to bladder [1], the digestive system,

gastrointestinal tract, male reproductive tract, fallopian tube, bile duct, ovum movement in

the fallopian tube, the conveyance of spermatozoa in the human reproductive tract, roller

and finger pumps, the locomotion of worms, the motion of chyme in the small intestinal

tract [2], cardiovascular flows, the mechanical and neurological aspects of reflux, transport

of lymph in the lymphatic vessels and in the vasomotion of small blood vessels such as

11

arterioles, venules and capillaries. Embryo transport in the uterus and early-stage

embryonic heart development also utilizes peristaltic flow. In medical engineering,

peristaltic systems are utilized in diabetes pumps [3], uterine cavity [4], heart tube [5],

dialysis machines (Fig. 1.2) and pharmacological delivery systems [6]. In botanical

hydrodynamics, peristalsis arises in loam dynamics in trees and plants [7]. The mechanism

of peristaltic transport is also employed in the transport of sanitary fluids and aggressive

chemicals. The transport of corrosive fluid in the nuclear industry [8] is also achieved using

peristalsis, which provides much greater efficiency and safety than conventional methods.

Hose pumps, tube pumps, roller (Fig. 1.3) and finger pumps are also engineered on the

principle of peristalsis. In short, peristaltic motion is the nature’s way of transporting the

fluid in living systems and exploited by humans to their advantage in manufacturing of

various instruments in medical and industrial engineering. A brief literature review of the

available literature on fluid mechanics of peristalsis is provided in the next section.

Fig. 1.1: Motion of food bolus through esophagus.

(http://legacy.owensboro.kctcs.edu/gcaplan/anat2

/notes/notes8%20digestive%20physiology.htm)

Fig. 1.2: Image of blood dialysis.

(http://biology-igcse.weebly.com/dialysis.htm)

http://legacy.owensboro.kctcs.edu/gcaplan/anat2

http://biology-igcse.weebly.com/dialysis.htm

12

Fig. 1.3: Image of rotor.

(http://www.pumpindustry.com.au/peristaltic-pumps)

1.2 Review of literature

1.2.1 Peristaltic flow of Newtonian fluids

The analysis of the mechanism responsible for peristaltic transport was initiated by Latham

[9]. After that extensive literature is available on the topics that deals with the peristalsis

of the Newtonian and non-Newtonian fluids. Most of the earlier research on peristaltic flow

was confined to the analysis of urine transport in the ureter [10 – 12]. An extensive

mathematical analysis of peristaltic pumping without any relevance to physiology was

carried out by Burns and Parkes [13], Hanin [14] and Jaffrin and Shapiro [15]. Fung and

Yih [16] and Yin and Fung [17] studied the problem of unsteady peristaltic motion by

assuming the wave number to be small. A fundamental difference between the work of

Fung and Yih [16] and Shapiro et al. [18] lies in the choice of reference frame for the

analysis of peristaltic motion. The former authors carried out the analysis in fixed frame

by treating the flow to be unsteady while the later authors made the analysis in wave frame

where the flow was assumed to be steady. The seminal work carried out in Refs. [16, 17]

become a corner stone to the subsequent researches in the area. A review of early literature

on peristaltic motion was presented by Jaffrin and Shapiro [15]. Jaffrin [19] reconsidered

http://www.pumpindustry.com.au/peristaltic-pumps)

13

the peristaltic flow in a planar channel in wave frame and obtained solution for small wave

number employing regular perturbation technique. The analysis for the case when both

Reynolds number and wave number were assumed small was dealt by Zien and Ostrach

[20]. The work in Ref. [20] was extended for axisymmetric case by Li [21]. The effect of

Poiseuille flow on peristaltic transport was investigated by Mittra and Prasad [22].

Srivastava and Srivastava [23] discussed the effect of pulsatile flow on peristaltic motion

in a circular cylindrical tube. Peristaltic flow in non–uniform channel and tube was

investigated by Gupta and Seshardi [24]. Numerical simulations of peristaltic flow in

planar geometry were performed by Takabatake and Ayukawa [25] and Brown and Hung

[26]. The case of axisymmetric tube geometry was dealt by Takabatake et al. [27]. The

work in Ref. [23] was complemented by Afifi and Gad [28] for magneto–fluid filling a

porous space. Hakeem et al. [29] discussed the effects of an endoscope and fluid with

variable viscosity on peristaltic motion under low Reynolds number approximation. The

problem was analytically formulated by using a perturbation technique in terms of

dimensionless number (Weissenberg number). Peristaltic flow of a viscous incompressible

fluid through a gap between coaxial uniform and non–uniform tubes under zero Reynolds

number condition with the long wavelength approximation was analyzed by Mekheimer

[30]. He found that the magnitude of the pressure rise in the non-uniform geometry is much

smaller than the uniform geometry. Influence of slip on peristaltic transport of viscous fluid

in a channel was considered by Chu and Fang [31]. Mishra and Rao [32] examined the

peristaltic transport in a channel filled with a porous medium in the peripheral region and

a viscous liquid in the core region. They found that the peristalsis works as a pump against

greater pressure in two-layered model with a porous medium in contrast with a viscous

14

fluid in the peripheral layer. Peristaltic transport of viscous fluid through a porous medium

in an inclined planar channel was investigated by Mekheimer [33]. In another attempt [34],

he extended his results presented in [33] for magnetohydrodynamic fluid. In both Refs.

[33, 34], the effects of numerous emerging parameters on basic features of the peristaltic

motion were discussed extensively. Effects of suction on peristaltic flow of an

incompressible viscous fluid in a channel were examined by Elshehawey and Husseny [35].

They reported the analytical solution of the problem by using perturbation technique for

the case of small amplitude ratio. Elshehawey et al. [36] discussed the problem of

peristaltic transport of an incompressible viscous fluid in an asymmetric channel through

a porous medium and found an explicit form of stream function using the Adomian

decomposition method. Hayat et al. [37] extended the analysis of Ref. [36] using partial

slip condition. They showed that trapping is reduced in the presence of surface slip. Mishra

and Rao [38] studied the peristaltic flow in an asymmetric channel with asymmetry

generated by different amplitudes of the peristaltic waves in addition to different phases.

Eytan et al. [39, 40] investigated the effect of peristalsis in embryo transport within the

uterine cavity. The phenomenon of trapping was discussed and their outcomes provide

information on the flow phenomena and possible trajectory for an embryo implantation at

the uterine wall. Hayat et al. [41] highlighted the effect of endoscope on the peristaltic flow

of a Newtonian fluid in a tube and obtained the analytical solutions for velocity profile and

longitudinal pressure gradient. Pandey and Tripathi [42] investigated the mechanism of

peristaltic transport of a viscous fluid with the magnetic effects through a cylindrical tube

of finite length. Their emphasis was to discuss the mechanical efficiency of a peristaltic

pump and reflux phenomena. Nadeem et al. [43] examined the peristaltic flow of a viscous

http://www.sciencedirect.com/science/article/pii/S0895717707003779#b12

15

fluid through rectangular channel with complaint walls under the assumptions of long

wavelength and low Reynolds number approximations. They solved the governing

equations by using the Eigen function expansion method.

1.2.2 Peristaltic flow of non-Newtonian fluids

In all the studies mentioned in the previous sub-section the fluid is assumed to obey

Newton’s law of viscosity. This assumption has restricted the applications of many of the

above investigations to relatively simple physiological and industrial flows. More

generally the working fluids in such systems exhibit non-Newtonian characteristics i.e.

they do not obey Newtonian law of viscosity. This observation has therefore motived many

researchers to study peristaltic flow of non-Newtonian fluids both analytically and

numerically, and in a limited fashion, experimentally. Raju and Devanathan [44] explored

peristaltic dynamics of viscoelastic fluids with fading memory. The constitutive equation

of power-law fluid in a tube was used and series solution was obtained in terms of small

amplitude of the wave. The peristaltic mechanism of an incompressible linear viscoelastic

fluid in the case of a plane flow was studied by Bohme and Friedrich [45]. The aim was to

discuss the pressure-discharge characteristics of the peristaltic pump and the pumping

efficiency. Siddiqui and Schwarz [46] studied the peristaltic flow of the second order fluid

model through an axisymmetric conduit. A perturbation series was used to obtain explicit

forms for the velocity field and a relation between the pressure gradient and the flow rate

in terms of dimensionless parameters such that Reynolds number, the dimensionless non-

Newtonian parameters and the occlusion. Gupta and Seshadri [47] investigated the

peristalsis phenomena of spermatic fluid in vasdeferens by representing the geometry of

16

the vasdeferens as a nonuniform channel/tube. They concluded that the pressure rise is

much less in non-uniform geometry compared to that in uniform geometry. They further

provided a comparison of their theoretical results with experimental data. Srivastava and

Srivastava [48] modeled the peristaltic flow in the vasdeferens by assuming it to be a non-

uniform diverging channel and a tube. Their analysis was more realistic because the non-

Newtonian (power-law) fluid model was used in the analysis. It was shown by them that

the magnitude of pressure rise is smaller for Newtonian fluid under the given set of

conditions. Li and Brasseur [49] analyzed non-steady peristaltic transport in finite-length

tube and concluded that non-integral number of waves generates fluctuations in pressure

and shear stress. Misra and Pandey [50, 51] studied a mathematical model for oesophageal

swallowing of a food bolus by considering a power-law fluid through a circular tube of

finite length where the single wave is propagated along the wall. They provided a

comparison between the effects of single wave transport and train wave transport.

Srivastava and Srivastava [52] modeled blood as a Casson fluid flowing inside small

capillaries and blood vessels. Peristaltic transport of non-Newtonian fluid in a diverging

tube with different wave forms is studied by Hariharan et al. [53]. Lubrication theory was

employed to study the peristaltic transport of non-Newtonian power-law and Bingham

fluids. The chyme flow in small intestine using power-law fluid was modeled by Lew et

al. [54]. Characteristic of the velocity profile and the effectiveness of the peristaltic

carrying and compression were discussed. Asghar et al. [55] studied the boundary layer

structure in the magnetohydrodynamic peristaltic flow of Sisko fluid in a straight channel

under the effects of strong and weak magnetic fields. They employed singular perturbation

technique to show the existence of a Hartmann boundary layer for the strong magnetic field



17

at the location of the two plates of the channel. Hayat et al. [56] observed the influence of

an endoscope on the peristaltic flow of an incompressible and electrically conducting

Jeffrey fluid through tubes. Their analysis was based on the exact analytical expressions

for velocity profile and pressure gradient under long wavelength assumption. A

perturbation and numerical study of peristaltic flow of a Johnson-Segalman viscoelastic

fluid in an asymmetric channel was also carried out by Hayat et al. [57]. Bég et al. [58]

examined the effects of radial magnetic field on peristaltic transport of Williamson

viscoelastic fluid in an asymmetric channel. They computed approximate solution of the

governing nonlinear ordinary differential equations by using the multi-step differential

transformation method (MDTM). Pandey and Tripathi [59] discussed the peristaltic flow

of micropolar fluid in a circular cylindrical tube of finite length. The influence of coupling

number on flow characteristics was shown and discussed under long wavelength and low

Reynolds number approximations. Pandey and Tripathi [60] also analyzed the peristaltic

transport of a Casson fluid in a finite channel. The pressure distribution was shown for both

integral and non-integral number of waves at different instant of time along the

oesophageal length. Kothandapani and Srinivas [61] reported the study of peristaltic flow

using Jeffrey model. The impact of relaxation/retardation time on peristalsis phenomena

was highlighted and discussed in detail. Hayat et al. [62] examined the peristaltic motion

of an Oldroyd-B fluid through planar channel. The effects of emerging parameters on the

flow phenomena was explained in detail through graphs. Ali et al. [63] theoretically

analyzed peristaltic model of an incompressible Oldroyd 4-constant fluid in a planar

channel. They solved the governing nonlinear ordinary differential equation by using a

finite difference technique combined with an iterative method. Later, Ali et al. [64]

18

numerically investigated the peristaltic flow of an Oldroyd 8-constant fluid in circular

cylindrical tube. The interaction of shear-thinning and shear-thickening effects with

peristaltic transport was discussed in detail. Peristaltic motion of a Giesekus fluid in a

planar channel was modeled by Ali and Javed [65]. They reported the lesser magnitude of

the longitudinal velocity at the channel center for Giesekus fluid in comparison with

Newtonian fluid. An analytical solution for peristaltic flow of a Jeffrey fluid in an

asymmetric channel utilizing long wavelength and low Reynolds number approximations

was obtained by Adb-Alla et al. [66]. Hydromagnetic peristaltic transport of a Carreau fluid

in a channel with different wave forms under long wavelength approximation was studied

by Hayat et al. [67]. Pressure rise along with pumping phenomena was described in detail

through graphs. Ali et al. [68] analyzed the peristaltic motion of a non-Newtonian fluid

(Maxwell fluid) through a channel having compliant walls. Analytic solution was obtained

by using perturbation technique. The investigation of peristaltic transport of a couple stress

fluid through a porous channel, when Reynolds number is small and wavelength is large,

using appropriate analytical and numerical methods was carried out by Maiti and Misra

[69].

1.2.3 Heat and mass transfer in peristaltic flows

Peristaltic flow with heat transfer has many applications in the biomedical sciences. It is

observed that numerous industrial processes require the deep knowledge of heat transfer

and the corresponding thermal coefficients. These include condensation, crystallization,

evaporation and other boiling operations. For instance, the production of orange juice

concentrate, concentrated 2 4H SO and distilled water is based on the evaporation

19

technique. In physiology evaporation technique is used to investigate the thermal properties

of tissues/cells. Moreover, thermodynamical aspects of blood may influence the processes

like oxygenation and hemodialysis when blood is drawn out of the body. The application

of heat transfer in techniques like laser therapy and cryosurgery for treatment of malicious

canner cells have also stimulated much of interest in study of thermal modeling in tissues.

The interaction of peristalsis with heat transfer have also demonstrated several interesting

aspects of bolus dynamics in gastro-intestinal tract since the thermal properties of fluid

may affect the bolus transport. There is obvious involvement of mass transfer in all such

processes. When simultaneous effects of heat and mass transfer are considered, the

complicated relationships occur between the fluxes and the driving potentials. The energy

flux is induced by temperature gradient as well as composition gradients and mass flux can

be produced by temperature gradient. Mass transfer phenomenon is important in the

diffusion of nutrients from the blood to the neighboring tissues. In the literature very few

studies were found regarding the heat/mass transfer effects in peristaltic flows. Tang and

Shen [70, 71] studied the problem related to peristaltic flow of a heat-conducting fluid

through a cylindrical periodic domain. It was shown that the Oberbeck-Boussinesq (OB)

equations has unique solution. Additionally, comparison of numerical results with exact

solution was reported. The regularity of the solution for peristaltic transport of a heat-

conducting fluid through a flexible tube was discussed by Tang and Rankin [72].

Radhakrishnamacharya and Murty [73] studied the heat transfer on the peristaltic transport

in a non-uniform channel. Srinivas and Kothandapani [74] investigated the peristaltic

transport of viscous incompressible fluid in an asymmetric channel with heat transfer. The

linearized energy and momentum equations were solved to discuss the effects of Hartmann



20

number, Eckert number, channel width and phase angle on temperature and heat

coefficient. Vajravelu et al. [75] discussed the peristaltic flow and heat transfer in a vertical

porous annulus under long wavelength approximation. Analytical solution based on

perturbation technique was used to discuss some significant effects of heat transfer on

peristalsis. The influence of heat transfer and magnetic field on peristaltic transport of a

Newtonian fluid in a vertical annulus was discussed by Mekheimer and elmaboud [76].

Nadeem and Akbar [77] investigated the influence of heat transfer on a peristaltic transport

of Herschel–Bulkley fluid in a non-uniform inclined tube under the long wavelength and

low Reynolds number approximations. The exact solution of the governing equations that

describe the flow phenomena was reported. Heat and mass transfer in peristaltic flow of a

third order fluid in a diverging tube was studied by Nadeem et al. [78]. Two analytical

methods namely, perturbation technique and homotopy analysis method were used to find

the analytical solutions. Akbar and Nadeem [79] further analyzed the influence of heat

transfer on a peristaltic flow of Johnson Segalman fluid in a non-uniform tube. Solution

expressions based on perturbation method were obtained for stream function, temperature,

pressure gradient and heat transfer cofficients. Physical behavior of the emerging

parameters was displayed through graphs. Tripathi [80] analyzed a mathematical model of

swallowing of food bolus through the oesophagus under the influence of heat transfer.

Numerical discussion about the effects of heat transfer on the reflux and trapping

phenomena of peristaltic motion was provided. Moreover, a comparative study about the

integral and non-integral number of waves was presented. Hayat and Hina [81] examined

the effects of heat and mass transfer on the peristaltic transport of a Maxwell fluid in a

porous channel with compliant walls. Hina et al. [82] investigated the influence of heat and






21

mass transfer on the peristaltic transport in the presence of a chemical reaction. Srinivas

and Muthuraj [83] discussed the magnetohydrodynamic mixed convective peristaltic flow

through a vertical asymmetric channel with porous space in the presence of a chemical

reaction. Non-isothermal peristaltic transport of Phan-Thien-Tanner (PTT) fluid in the

presence of magnetic field was studied by Hayat et al. [84]. A series solution was presented

for small Weissenberg number.

1.2.4 Peristaltic flows in a curved channel

In addition to long wavelength and low Reynolds number approximations, (which amount

to inertial effect omission and the wavelength being sufficiently long compared with the

channel width) peristaltic flow models frequently adopt a relatively simple geometry. This

assumption may allow certain physical characteristics to still be captured, however since

invariably physiological vessels/ducts are curved to a certain extent the flows within them

cannot be fully simulated without some incorporation of this geometrical feature. In this

context a number of discrete fluid mechanical approaches are available to incorporate

curvature into physiological models. These include vorticity formulations wherein “Dean

flows” can be simulated and vortex patterns related to centrifugal force effects can be

computed [85 – 87], although computational effort is significant. Another approach is to

analyze the effects of radius of curvature parameter on flow patterns. This second

methodology is very appropriate for peristaltic transport and circumvents the need for

intense computational mesh sizes and facilitates analytical and simpler numerical

computations. A seminal study in this regard was reported by Sato et al. [88] who

considered explicitly the effects of curvature on peristaltic transport of Newtonian fluid

22

albeit under the long wavelength approximation. They observed using a stream function

formulation that the pressure-flow characteristic is linear, and its gradient is enhanced

marginally with an increasing channel curvature. They further noted that reflux close to the

outer wall exhibits greater strength than near the inner wall and that the trapped bolus of

fluid has two asymmetrical components, with the outer one growing and the inner one

depleting as the channel curvature rises. Ali et al. [89] studied the peristaltic flow of a

viscous fluid in a curved channel under long wavelength and low Reynolds number

approximations. Ali et al. [90] further investigated peristaltic motion of a third grade fluid

in a curved channel. Peristaltic transport of an incompressible third grade fluid in a curved

channel under the influence of induced magnetic field was analyzed by Hayat et al. [91].

A parametric analysis is carried out to explain the effects of various emerging parameters

on flow and pumping characteristics. Hina et al. [92] discussed the peristaltic transport of

Johnson Segalman fluid through a curved channel with wall properties. The governing

equations are linearized by using long wavelength and low Reynolds number

approximations. Solution of these governing equations was obtained numerically by using

a shooting method. In addition they also found the analytical solution by using perturbation

technique for small Weissenberg number. In another study, Hina et al. [93] studied the

effects of wall properties on the peristaltic flow of an incompressible Pseudoplastic fluid

under long wavelength and low Reynolds number approximations. Abbasi et al. [94]

modeled the governing equations for peristaltic flow of an Eyring-Powell fluid in a curved

channel and obtained the series solution of the equations governing the flow. A comparison

of results between planar and curved channels was also made. The effects of fractional

parameters of second grade fluid on peristaltic transport through a curved channel was

23

investigated by Narla et al. [95]. In order to obtain the solution of the governing equations

they used fractional calculus approach. Kalantari et al. [96] numerically examined the

peristaltic flow of a viscoelastic fluid (Phan-Thein-Tanner) in a two dimensional curved

channel under the influence of radial magnetic field.

The first rigorous attempt to model heat transfer in peristaltic flow of Newtonian fluid in a

curved channel was presented by Ali et al. [97]. They showed that temperature of the fluid

inside the channel increases by increasing curvature of the channel. Effects of wall

properties with heat and mass transfer on the peristaltic motion of a third grade fluid were

studied by Hayat et al. [98]. The influence of heat and mass transfer on the peristaltic

transport of Johnson Segalman fluid in a curved channel with flexible walls was examined

by Hina et al. [99]. Ramanamurthy et al. [100] analyzed the unsteady peristaltic flow of a

viscous fluid under the influence of heat transfer through a curved channel in a laboratory

frame of reference. Lubrication theory and low Reynolds number approximations were

utilized to linearize the governing equations. Nadeem et al. [101] examined the peristaltic

transport of Williamson fluid in the presence of nanoparticles through a curved channel

with compliant walls. Highly nonlinear partial differential equations were reduced under

long wavelength and low Reynolds number assumptions. Homotopy perturbation method

was used for the analytical solution.

1.3 Some basic definitions, equations and terminologies

1.3.1 Curvilinear coordinates

In geometry, curvilinear coordinates are coordinates system for Euclidean space in which

the coordinate lines may be curved. The curvilinear coordinates may be derived from a set

24

of Cartesian coordinates by using a transformation that is locally invertible at each point.

This mean that one can convert a point given in a Cartesian coordinate system to its

curvilinear coordinates and vice versa. The name curvilinear coordinates, coined by the

French Mathematician Lame, derives from the fact that the coordinate surfaces of the

curvilinear system are curved. Well known examples of the curvilinear coordinate systems

in three-dimensional Euclidean space 3R are Cartesian, cylindrical and spherical polar

coordinates.

An orthogonal curvilinear system is a system for which the coordinate surfaces are

mutually perpendicular. For the cylindrical system (Fig. 1.4), the coordinate surfaces are r

= constant, θ = constant and z = constant. These three coordinate surfaces intersect through

a given point at right angles. The three curves of intersection of the coordinate surfaces in

pair intersect at right angles. These curves are called coordinate lines or directions. We

draw unit basis vectors tangent to the coordinate directions at two different points (say 1P

and 2P as shown in Fig. 1.4) on the cylinder. For the cylindrical system (Fig. 1.4), we

might call them ,re e and ze . These basis vectors form an orthogonal triad like ,i j and

.k We refer to such a coordinates systems as curvilinear coordinate systems when the

coordinate surface are not planes and the coordinate lines are curves other than straight

lines.

25

Fig. 1.4: Cylindrical coordinates systems with coordinates , , .r z

We consider the orthogonal curvilinear coordinates 1 2 3, ,q q q which can be related to the

Cartesian coordinates 1 2 3, ,x x x as

1 1 1 2 3

2 2 1 2 3

3 3 1 2 3

, , ,

, , ,

, , .

q q x x x

q q x x x

q q x x x

(1.1)

In suffix notation Eq. (1.1) can be written as

, , 1, 2,3.i i jq q x i j (1.2)

We assume the Eq. (1.2) has unique inverse i.e.

, , 1, 2,3.i i jx x q i j (1.3)

In vector notation the above equation becomes

.jqx x (1.4)

26

If 2q and 3q are kept constant, the vector 1qx x describes a curve in a space which is

the coordinates curves, 1q . 1q

x is the tangent vector to this curve. The corresponding

basis vector in the direction of increasing 1q reads:

1

1

.q

q

1

x

ex

(1.5)

If we let 1

1

,bq

x then we see that

1

1

,bq

1

xe (1.6)

and in the same way

2

2

,bq

2

xe (1.7)

3

3

,bq

3

xe (1.8)

with 2

2

,bq

x and 3

3

.bq

x

Since ,jqx x we can write the line element as

1 2 3

1 2 3

,d dq dq dqq q q

x x xx (1.9)

which in view of (1.6) – (1.8) becomes

1 1 2 2 3 3 ,d b dq b dq b dq 1 2 3x e e e (1.10)

from Eq. (1.10), we get

27

2 2 2 2 2 2

1 1 2 2 3 3 .d d b dq b dq b dq x x (1.11)

Further, the volume element (Fig. 1.4) is given by

1 2 3 1 2 3.dV b b b dq dq dq (1.12)

The expression of 1q surface element of the volume element dV (i.e. the surface element

perpendicular to the 1q direction) is

1 2 3 2 3 ,dS b b dq dq (1.13)

and similarly the other surface elements are:

2 3 1 3 1,dS b b dq dq (1.14)

3 1 2 1 2.dS b b dq dq (1.15)

1.3.2 Gradient, divergence and curl in curvilinear coordinates

In this subsection, we shall provide the components of gradient, divergence and curl of a

vector and rate of deformation tensor along with the expression of the divergence of a

tensor in curvilinear coordinates.

If is a scalar function, then the components of vector are:

1 11 1

2 22 2

3 33 3

1Along : ,

1Along : ,

1Along : .

qb q

qb q

qb q

(1.16)

28

The divergence of a vector V (here considered as fluid velocity) having components

1 2,u u and 3u in the direction of the increasing 1 2,q q and 3q is given by

2 3 1 3 1 2 1 2 3

1 2 3 1 2 3

1. .b b u b b u b b u

b b b q q q

V = (1.17)

The component of the curl V are:

Along 1q : 3 3 2 212 3 2 3

1,b u b u

b b q q

V (1.18)

Along 2q : 1 1 3 321 3 3 1

1,b u b u

b b q q

V (1.19)

Along 3q : 2 2 1 131 2 1 2

1.b u b u

b b q q

V (1.20)

The component of the divergence of stress tensor are given by

2 3 11 3 1 21 1 2 31

1 2 3 1 2 3

1 1

31 33 321 1 1 22 2

1 2 2 1 3 3 1 2 1 1 3 1

1

Along :

,

b b S b b S b b Sb b b q q q

qS S bS b b S b

b b q b b q b b q b b q

S (1.21)

2 3 12 3 1 22 1 2 32

1 2 3 1 2 3

2 2

32 33 32 12 2 11 1

2 3 3 2 1 1 2 3 2 1 2 2

1

Along :

,


qS S bb S b S b


S (1.22)

2 3 13 3 1 23 1 2 33

1 2 3 1 2 3

3 3

13 3 23 3 11 1 22 2

1 3 1 2 3 2 1 3 2 3 2 3

1

Along :

.


qS b S b S b S b


S (1.23)

The Cauchy-stress tensor for an incompressible fluid reads

,P I + S (1.24)

29

where I is the identity tensor and S is the extra stress tensor which for a Newtonian

fluid is given by

†,V V S (1.25)

in Eq. (1.25) † denotes the transpose.

The component of the rate of deformation tensor in curvilinear coordinates are given by

31 2 1 111

1 1 1 2 2 1 3 3

1 1,

2

uu u b b

b q b b q b b q

(1.26)

32 1 2 222

2 2 1 2 1 2 3 3

1 1,

2

uu u b b

b q b b q b b q

(1.27)

3 3 31 233

3 3 1 3 1 2 3 2

1 1,

2

u b bu u

b q b b q b b q

(1.28)

1 1 2 212 21

2 2 1 1 1 2

,b u b u

b q b b q b

(1.29)

3 32 232 23

3 3 2 2 2 3

,b ub u

b q b b q b

(1.30)

3 31 131 13

3 3 1 1 1 3

.b ub u

b q b b q b

(1.31)

1.3.3 Equation of continuity in curvilinear coordinates

The vector form of equation of continuity is

0,t

V (1.32)

where is the fluid density and t is the time. Using the formula given in section (1.3.2)

the above equation becomes

30

2 3 1 3 1 2 1 2 3

1 2 3 1 2 3

10.b b u b b u b b u

t b b b q q q

(1.33)

For the case when is constant, we get

2 3 1 3 1 2 1 2 3

1 2 3

0.b b u b b u b b uq q q

(1.34)

Eq. (1.34) is expressed in terms of curvilinear coordinates and valid for incompressible

flows.

1.3.4 Equation of motion in curvilinear coordinates

The motion of an incompressible fluid is governed by the equation

,d

dt

V+ F (1.35)

where F is the body force per unit mass and ddt

is the material derivative. Using the

formula given in section 1.3.2, we get

Along 1q :

31 2 1 12 2 1 1 1 1 3 3

1 2 1 2 1 3 3 1 1 1

uu u u ub u b u b u b u

t b b q q b b q q b q

21 11 2 3 11 3 1 21 1 2 31

1 2 3 1 2 3 1 2 2

1 S bk b b S b b S b b S

b b b q q q b b q

31 33 31 22 2

1 3 3 1 2 1 1 3 1

,S S bb S b

b b q b b q b b q

(1.36)

Along 2q :

32 1 2 23 3 2 2 2 2 1 1

2 3 2 3 1 2 1 2 2 2

uu u u ub u b u b u b u


31

32 22 2 3 12 3 1 22 1 2 32

1 2 3 1 2 3 2 3 3


b b b q q q b b q

33 312 2 11 1

2 1 1 2 3 2 1 2 2

,S bS b S b

b b q b b q b b q

(1.37)

Along 3q :

3 3 31 21 1 3 3 3 3 2 2

1 3 3 1 2 3 2 3 3 3

u u uu ub u b u b u b u


13 33 2 3 13 3 1 23 1 2 33

1 2 3 1 2 3 1 3 1


b b b q q q b b q

23 3 11 1 22 2

2 3 2 1 3 2 3 2 3

.S b S b S b

b b q b b q b b q

(1.38)

In above equations ,i iju S and , , 1,2,3iF i j are the components of velocity, stress tensor

and body force, respectively.

1.3.5 Equation of energy in curvilinear coordinates

The energy equation is based on the first law of thermodynamics and for incompressible

fluids with constant physical properties it is of the form

* 2 ,p

dTc k T

dt (1.39)

where pc is the specific heat at constant pressure, *k is thermal conductivity, is the

coefficient of dynamic viscosity and is dissipation function. In curvilinear coordinates

it can be written as

32

*

3 2 3 3 11 2

1 1 2 2 3 3 1 2 3 1 1 1 2 2 2

1 2

3 3 3

.

p

u b b b bu uT T T T k T Tc

t b q b q b q b b b q b q q b q

b b T

q b q

(1.40)

The dissipation function represents the rate at which mechanical energy is consumed

in the process of deformation of the fluid due to viscosity. It is given by

, , 1, 2.iij

i

uS i j

q

(1.41)

In curvilinear coordinates the above expression expands to

1 1 1 2 2 2

1 1 1 22 2

1 1 2 1 2 3 1 2 2

1 1 1 1.q q q q q q

u u u uS S b u S

b q b q q b q b q

(1.42)

1.3.6 Maxwell’s equations

The set of equations that are utiltzed in magnetohydrodynamics are

0 0 , , 0, 0.mt t

E BB J + E J B (1.43)

The first two equations are respectively Amphere’s law and Fraday’s law of induction

while the third and fourth one repects the mathematical statements of Gauss’ laws for

electric and magnetic fields. In above equations E and B are the total electric and magnetic

fields and J is the current density.

1.3.7 Lorentz force

The entire combination of electric and magnetic forces on a point charge due to

electromagnetic fields is called Lorentz force. Mathematically, it is defined as

33

F = J B, (Lorentz force per unit volume) (1.44a)

where

. J = E V B (Ohm’s Law) (1.44b)

where is the electric conductivity.

1.3.8 Dimensionless numbers

1.3.8.1 Reynolds number

The Reynolds number is defined as “ratio of the inertial force to the viscous force” and

usually denoted by Re.

Re .inertial force

viscous force (1.45)

Mathematically, it is defined as

Re ,aveV D

(1.46)

where aveV is the average velocity. Osborne Reynolds, an English researcher and

mathematician, was the first to recognize the difference between three following flow

characteristics. These three characteristics, denoted as laminar, transitional and turbulent

flow, respectively. The flow is laminar if the Reynold number is “small enough” and

viscous force is dominant. The flow is turbulent if the Reynold number is “large enough”

and inertial force is dominant. For Reynolds numbers between these two limits, the flow

might switch in between laminar and turbulent conditions. Such flow, which represents the

34

onset of turbulence, is called transition. The small value of the Reynolds number

corresponds to creeping flow.

1.3.8.2 Wave number

The ratio of the width of the channel to the wavelength of the propagating wave along the

channel wall is known as wave number.

1.3.8.3 Weissenberg number

The Weissenberg number (We) is a dimensionless number used in the study of viscoelastic

flows. The dimensionless number compares the viscous forces to elastic forces. It can be

variously defined, but it is usually given by the relation of stress relaxation time of the fluid

and a specific process time. For instance, in simple steady shear, the Weissenberg number,

often abbreviated as Wi or We, is define as

.

viscous forceWe

elastic force (1.47)

1.3.8.4 Hartmann number

The ratio of electromagnetic force to the viscous force introduced by Hartmann is called

Hartmann number.

1.3.8.5 Brinkman number

The Brinkman number is a dimensionless number related to the heat conduction from a

wall to a flowing viscous fluid, commonly used in polymer processing. It is the ratio

35

between heat produced by viscous dissipation and heat transported by molecular

conduction, i.e. the ratio of viscous heat generation to external heating. The higher the

value of it, the lesser will be the conduction of heat produced by viscous dissipation and

hence larger the temperature rise.

1.3.8.6 Amplitude ratio

The ratio of amplitude of the peristaltic wave to the width of the channel is known as

amplitude ratio.

1.3.8.7 Heat transfer coefficient

The heat transfer coefficient, in thermodynamics and in mechanics is the proportionality

constant between flux and the thermodynamic driving force for the flow of heat (i.e. the

temperature difference, :T

,Q

zT

(1.48)

where Q is the amount of heat transfer (heat flux), z is the heat transfer coefficient, and

T is difference in temperature between the solid surface and surrounding fluid area.

1.3.8.8 Slip parameter

The condition which requires that the relative velocity of the fluid and surface must be zero

at every point of the contact is called no-slip condition. Specifically, in engineering

problems, the no-slip condition does not generally hold in its actuality. While in the case

36

of numerous polymeric fluids with high atomic weight, the particle closer to the surface

show slip or stick-slip on the surface. To handle the problem, Navier (1823) recommended

the general slip condition that the difference of fluid velocity and the velocity of the surface

is proportional to the shear stress at that surface. The coefficient of proportionality is called

the slip length. The normalized slip length is called slip parameter and is denoted by in

the present thesis.

37

Chapter 2

Exact and asymptotic solutions for

hydromagnetic peristaltic flow of Jeffrey fluid

in a curved channel

In this chapter, a theoretical study is conducted for peristaltic flow of a conducting non-

Newtonian (viscoelastic) fluid in a curved channel under the influence of a radially

imposed magnetic field. The robust Jeffrey model is employed to simulate rheological

characteristics. Assuming the flow to be laminar, incompressible and two-dimensional, the

governing partial differential equations are reduced to a single linear ordinary differential

equation in terms of a stream function under long wavelength and low Reynolds number

approximations. Exact as well as asymptotic solutions of this equation are obtained. The

asymptotic solution is obtained for small and large values of the combined parameter,

which is the product of Hartmann number and ratio of relaxation to retardation time, using

singular perturbation methods. It is found from both exact and asymptotic solutions, that

for strong magnetic field or for large values of ratio of relaxation to retardation time, a thin

boundary layer exists at the channel walls. The thickness of the boundary layer is found to

be inversely proportional to the product of Hartmann number and ratio of relaxation to

retardation time. Based on the exact solution, an extensive analysis is performed to probe

38

the effects of curvature of the channel, ratio of relaxation to retardation time and Hartmann

number on significant phenomena of pumping and trapping related to the peristaltic

motion. The study is relevant to magnetohydrodynamic control of physiological transport

phenomena.

2.1 Mathematical model and rheological constitutive equations

Let us consider a curved channel of width 12a filled with an incompressible Jeffrey fluid.

Let the center and radius of curvature of the circle, in which the channel is coiled, be

designated by O and *R , respectively. A schematic diagram of the flow geometry is

illustrated in Fig. 2.1 for the case when wavelength of the wave and half width of the

channel are of comparable magnitude. The fluid is assumed stationary and flow is solely

generated by progressive waves passing along the channel walls. The system comprising

of fluid inside the channel is under isothermal conditions. The assumption of originally

stationary fluid without imposed pressure gradient is valid for peristaltic mechanism in the

passage of urine from kidney to bladder, the ejection of semen from the male reproductive

organs, movement of chyme in gastro-intestinal tract etc. However, it might not be suitable

for peristaltic mechanism involved in the blood circulation in small blood vessels. Such

assumption is already made in number of available studies on peristalsis [12]. Moreover,

the mechanical properties of the channel wall are not incorporated in the present model.

The elastic/viscoelastic nature of the flexible boundaries is usually incorporated in the

analysis of peristalsis by making use of dynamic boundary conditions [22]. Literature

survey indicate that such studies are usually carried out in fixed frame of reference without

using long wavelength and low Reynolds number assumptions. However, since it is

39

intended to analyze the present flow problem in wave frame of reference under long

wavelength and low Reynolds number assumptions therefore we shall avoid making use of

such conditions. Moreover, under long wavelength and low Reynolds number assumptions

the dynamic boundary conditions lose all the important parameters characterizing the

interaction between flexible wall and the fluid motion. We employ a curvilinear coordinate

system ,R X to analyze the flow in which R is orientated along the radial direction and

X is along the direction of flow. This curvilinear coordinate system is related with the

Cartesian coordinate system ' ',X Y fixed at O through the following transformations.

' *

*

' *

*

cos , (a)

sin . (b)

XX R R

R

XY R R

R

(2.1)

Now the equation of the upper wall in ' ',X Y system is

2

''2 '2 * * 1

1 2 '

2cos tan .

YX Y R a a R ct

X

(2.2)

Using Eq. (2.1) in (2.2), we get

1 2

2sin .R a a X ct

(2.3)

Denoting left hand side of the above equation by 1H yields the equation of upper wall i.e.

1 1 2

2, sin ,H X t a a X ct

(Upper wall) (2.4)

40

where is the wavelength, 2a is the amplitude and t is the time. Similarly, the equation

of lower wall reads

2 1 2

2, sin .H X t a a X ct

(Lower wall) (2.5)

In view of Eq. (2.1), the scale factors 1 2,b b and 3b turn out to be *

*1 21,R R

b bR

and 3 1.b

Let , ,V X R t and , ,U X R t be the velocity components along R and X directions,

respectively. Therefore

= , , , , , , 0 .V X R t U X R t V (2.6)

The constitutive law for a Jeffrey fluid is

,P I S (2.7)

where the extra stress tensor S satisfies [61]

2'+ .

1+

S (2.8)

In Eq. (2.8), ' is the ratio of relaxation to retardation time, 2 is the retardation time, and

denotes material time derivative of . The Jeffrey model is appropriate for simulating

a wide range of liquids including intestinal suspensions and certain polymers. It uses a

formulation which falls between the more common Maxwell and Kelvin-Voigt models in

rheology and is also appropriate for semi-solid materials, as elucidated by Prasad et al.

41

[102]. The fluid in the channel is electrically-conducting and subjected to a magnetic field

of strength 0B in the radial direction, therefore from Maxwell’s electromagnetic field

equations, we obtain

*

0

*,

B R

R R

RB e (2.9)

where 0B is the characteristic magnetic induction in the limit ,R and Re is the basis

vector in the radial direction. It is emphasized that the magnetic field given by Eq. (2.9) is

solenoidal. We compute current density J using Ohm’s law. In low magnetic Reynolds

number approximation induced magnetic field and electric field are neglected in

comparison to the applied magnetic field and current density. Therefore, in the present case

Ohm’s law states

, = BJ V (2.10)

where is the electrical conductivity. In view of Eqs. (2.6) and (2.9), the above equation

gives

*

0

*.

B RU

R R

ZJ = e (2.11)

It is evident from (2.11) that J is in the z-direction. From Eqs. (2.10) and (2.11), the

Lorentz force is

2 2

0

2*

,B UR

R R

XJ B e (2.12)

where Xe is the basis vector in the azimuthal direction [96]. It is now easy to identify from

the above discussion that 1 2 1 2 1, , , , 0,q R q X u V u U F

42

2 2

022 *

.B UR

FR R

Therefore, from Eqs. (1.34), (1.36) and (1.37) we can

write

* * 0,U

R R V RR X

(2.13)

2 1

,

RR

XXRX

V V R U V U PV R R S

t R R R X R R R R R R

SRS

R R X R R

(2.14)

2

2

2 2

0

2

U R R 1+ + = + R + R

R + R R + R R + R R + R

RR+ .

R + R R + R

RX

XX

U U U UV PV S

t R X X R

B US

X

(2.15)

In the laboratory (fixed) frame the flow is unsteady. However, if observed in a coordinate

system moving at the wave speed, c, the flow domain may be treated as steady. We

therefore employ the following transformation to switch from the fixed frame ,R X to

the wave frame ,r x which is moving with speed c.

, , , , .x X ct r R u U c v V p P (2.16)

Thus, Eqs. (2.13) – (2.15) in the wave frame become:

* * 0,u

r R v Rr x

(2.17)

2

1

,

rr

xxrx

R u c u cv v v pc v r R S

x r r R x r R r r R r

SRS

r R x r R

(2.18)

43

2 22

0

2 2

R + c + c R+ + = +

+ R + R + R

R + c1 R+ R + .

+ R+ R + Rrx xx

u u vu u u pc v

x r r x r r x

B ur S S

r r xr r

(2.19)

The equations of wall surface in wave frame are

1 1 2

2sin ,r h a a x

(Upper wall) (2.20)

1 1 2

2sin .r h a a x

(Lower wall) (2.21)

The above equations can be made dimensionless by defining the following non-

dimensional variables and parameters:

2

*

2 1 22 0 1 2

2 2 2, , , ,Re , , ,

, , , , , ,

r u v ca a ax x u v p p

a c c c

c h ha Rk We Ha B h h

c a a a a

S S

(2.22)

and bars are dropped for simplicity. In view of (2.22), Eqs. (2.17) – (2.19) take the

following form

0,u

k v kx

(2.23)

21 1 1

Re

,xxx

k u uv v v pv k S

x k x k k

SkS

k x k

(2.24)

44

2 22

2 2

1 1Re = +

+11+ ,x xx

k u u vu u u k pv

x k x k k x

Ha k ukk S S

k xk k

(2.25)

where

2 2

2' 2

2,

1

v v v uk vS We v

x k x

(2.26)

2'

1 1

1x

u u k v v uS We v

k k x k

2

2 2

1,

v u uv

k

(2.27)

2'

2 1

1xx

k u v k u vS We

k x k k x k x

2 2 22

22,

k u k u uk u uk uv u

k x k x x k xk

(2.28)

where k is the dimensionless radius of curvature and 2We is the Weissenberg number.

Defining the stream function by the relations:

, ,k

u vk x

(2.29)

it follows that continuity (mass conservation) Eq. (2.23) is satisfied identically and Eqs.

(2.24) and (2.25) after using long wavelength and low Reynolds number approximations

reduce to

45

0,p

(2.30)

2

211 0,

( ) ( )x

p Ha kk S

x k k k

(2.31)

where

2

' 2

0, ( )

1 11 , ( )

1+

0. ( )

x

xx

S a

S bk

S c

(2.32)

It is remarked here that the long wavelength and low Reynolds number assumptions are

valid in many processes of physiological fluid transport due to peristalsis. For example, in

small intestine, ureter and in many other ducts where the bio-fluid is transported via

peristaltic activity, the wavelength of the wave is large in comparison with the radius of

the vessel. In addition to that the flow in such ducts due to peristalsis can be treated as

creeping i.e. the Reynolds number for such flow is vanishing small. A specific example

where the above mentioned assumptions hold true is the movement of chyme in small

intestine. We refer the reader to numerous articles [41 – 44, 89 – 101] regarding the

applications of long wavelength and low Reynolds number assumptions in peristaltic

flows.

Inserting Eq. (2.32b) into Eq. (2.31), we get

2

2

' 2

1 1 11

1+ ( )

pk

x k k k

2

1 0.( )

Ha k

k

(2.33)

46

Elimination of pressure between Eqs. (2.30) and (2.33), yields the following compatibility

equation:

2 2 2

2

' 2

1 1 11 1 0.

1+ ( ) ( )

Ha kk

k k k

(2.34)

It is noted that as a consequence of long wavelength and low Reynolds number assumption,

Eq. (2.34) does not involve the parameter 2 . Eq. (2.34) is subject to the usual no-slip

conditions at the walls i.e.

1

2

1, at 1 sin , ( )

1, at 1 sin , ( )

h x a

h x b

(2.35)

where 2

1

aa

is the amplitude ratio. However, these two conditions are not sufficient to

obtain a unique solution of Eq. (2.34). The additional boundary conditions are

1

2

, at 1 sin , ( )2

, at 1 sin . ( )2

qh x a

qh x b

(2.36)

The above boundary conditions follow from the definition of flow rate q in the wave

frame. The dimensionless mean flow rate Θ, in laboratory frame, and q in wave frame are

related according to the following expression [89]:

2.q (2.37)

According to definition of flow rate [89]

1

2

2 1 .

h

h

q d h h

(2.38)

47

The above expression furnish the conditions 1 2q

h and 2 2q

h . It is pointed out

that for flow problem under consideration, either the pressure difference across one

wavelength or the relative flow rate in the wave frame ,q (or the absolute flow rate in the

laboratory frame, Θ) must be prescribed. In the present analysis, we followed later

approach and prescribed q as a constant.

The dimensionless pressure rise over one wavelength is defined by [89 – 101]

2

0

.dp

p dxdx

(2.39)

2.2 Exact solutions

An exact solution of Eq. (2.34) with boundary conditions (2.35) and (2.36) emerges as:

2 2' '

2

1 11 2

3 42'

212 1 ,

2

k kk E Ek k E k E

k

(2.40)

where 1 2 3, , E E E and 4E are unknown constants, defined by the following expressions:

2 2' '

2 2' '

2 2' '

4 2 1 2 1'

1 2 1 1

2 2 2 1 2 1' ' 3

2 2 2 1

22 1 1'

1 2 2 1

12 ,

12 2

2

4 1

k k

k k

k k

E k h k h k h q

E k k h h k h k

k h k h k k h k h k

48

2 2

' '2 2 1 2 1' 2 2

1 2 12k k

k k h h k h k q

2 2 2' ' '

2'

2 2' '

2'

2'

2 22 1 2 1 2 1' '

1 2 1 2

2 1

1

2 1 1 1 1'

3 2 1 1

1

2 1 2 1'

4 1 2

2 2 2 1

,

11 1 2 ,

11 1

k k k

k

k k

k

k

h k k k h k h k k h k

h k q

E k h k h k h q

h k hE k k h k h

2'

2 2 2' ' '

11

2 1

22 21 1 1' 2 ' 2

2 1 1 2

2 ,

2 1 1

k

k k k

kh q

h k h k

k k h k h k k h h k

2 2 2

' ' '

222 1 2 1 2 1'

1 1 2 12 .k k k

h k k h k h k h k

(2.41)

It is remarked here that there is no singularity in this solution and it is well defined for all

values of '. However, we are only concerned with the case ' 0. This is because '

is the combined parameter representing the product of two non-negative parameters '

and Ha. An asymptotic solution of Eq. (2.34) for large values of Hartmann number/ratio

of relaxation to retardation time employing the singular perturbation technique is found to

be [55]:

49

2 2 2

1asymptotic 1 1

1 2 1 2 1 2

1 1 11 1

2 2 2 2

hq q qh kh k k

h h k h h h h k

2 2 2 2 22

1 2 1 1 1 21

1 2 1 2

11

2

k h k h k h h k h k h qkh

k k k k k h h k h h

1 2

1 2

2 2

1 2 ,

kh k k kh

k h k hk h k he e

k k

(2.42)

where '2 ' 2'

1 , 1 Ha

and 1. The above solution is only valid for large

values of '. It cannot be obtained directly from the exact solution (2.40) in the limit

when ' is large. The character of solution (2.40) is clearly of the boundary layer type due

to the presence of exponential terms. Graphical illustrations of the solution also delineate

such boundary layer character. Eq. (2.42) further suggests that the boundary layer growth

is of '

1 .1

OHa

This indicates that boundary layer thickness decreases by

increasing either magnetic field or the ratio of relaxation to retardation time. In such cases,

the variation in velocity is confined within thin layers near the walls (Hartmann-Stokes

layers). These have been addressed in some detail by Ghosh et al. [103], albeit for non-

deformable channels. The Jeffrey model contains two parameters namely; ' (the ratio of

relaxation to retardation time) and 2 (the retardation time). However, under the

assumptions that flow is creeping and wavelength of the peristaltic wave is large in

comparison with the channel radius, the parameter 2 vanish form the governing

equations. Therefore, it also does not appear in the exact as well as asymptotic solution.

Previous available studies, for instance Refs. [61, 80] also support our affirmation. The

50

asymptotic solution presented in Eq. (2.42) is compared with the exact solution given by

Eq. (2.40) in Fig. 2.2. This figure clearly demonstrates an excellent correlation between

both the solutions.

2.3 Results and Discussion

In this section, graphical results are presented in order to illustrate the effects of

dimensionless radius of curvature k , Hartmann number Ha and ratio of relaxation to

retardation time ' i.e. viscoelastic parameter, on flow characteristics, pumping and

trapping phenomena associated with the peristaltic motion.

The profiles of velocity at three different cross-sections, namely 2

x (panels (a), (b)),

0x (panels (c), (d)) and 2

x (panels (e), (f)) for different values of ' and for two

values of flow rate Θ = 4 (left panels), Θ = 2 (right panels) are shown in Fig. 2.3. The

parameter ' is the ratio of relaxation time to the retardation time. Larger values of '

correspond to the case of fluid with larger relaxation time or equivalently fluid with smaller

retardation time. In fact, for larger values of ' the characteristic time scale of the flow

(which in the present problem is 1 /a c ) is much less than the relaxation time of the fluid.

In such case, the elastic effects dominate over the viscous effects. It is observed that the

magnitude of flow velocity is suppressed near the plane 𝜂 = 0 with increasing ' . This

behavior is typical of Jeffrey viscoelastic fluids wherein greater elastic effects lead to a

flow retardation. Moreover, for a Newtonian fluid ( ' = 0), the velocity curves are not

symmetric about 𝜂 = 0 and the maximum of these profiles occur below 𝜂 = 0. This is

51

perhaps due to the fact that we have taken the dimensionless radius of curvature k = 2.

However, for the same values of k an increase in ' shifts the maximum velocity towards

the upper wall. In fact the role of ' here is to counteract the effects of k. On one hand by

decreasing k the velocity profiles becomes asymmetric while on the other hand these

asymmetric profiles can be made symmetric by increasing ' . If one keeps increasing '

, the velocity curves again become asymmetric and their maxima shift toward the upper

wall. A comparison of left and right panels indicates an increase in magnitude of velocity

by increasing flow rate (Θ). The above results are physically realizable because for large

values of ' the internal molecular configuration of the fluid resists the flow due to

peristaltic motion resulting in the damping of magnitude of flow velocity in the lower part

of the channel. To maintain the given flow rate, the relatively small velocity in the upper

part of the channel will increase. A combination of both effects leads to the shifting of

maxima in velocity curves toward the upper wall.

The effects of Hartmann number on velocity profile at three different cross-sections

2x (panels (a), (b)), 0x (panels (c), (d)) and

2x (panels (e), (f)) for two

different flow rates, viz Θ = 4 (left panels), Θ = 2 (right panels) are shown in Fig. 2.4. The

Hartmann number is the ratio of electromagnetic force to the viscous force. Larger values

of Hartmann number correspond to the case of strong imposed magnetic field. This figure

indicates a deceleration in the flow by increasing Hartmann number, a phenomenon

characteristic of the Lorentzian magnetic drag associated with a radial magnetic field [103].

The profiles of velocity at x = 0 reveal further interesting characteristics in the flow. We

observe that for Ha = 0 and k = 2, the velocity curve is asymmetric and its maximum lies

below 0 . However, an increase in Ha counteracts the curvature effect (k) and makes

52

the profile symmetric. For large values of Ha the profile again becomes asymmetric with

its maximum now appearing above 0 . Thus it would appear that there is a competition

between k and Ha ( ' ). For small values of k the effects of curvature are dominant.

However, the effects of curvature can be minimized by increasing either Ha or ' . For

large values of Ha or ' the profiles again becomes asymmetric. These observations may

have interesting implications, where it is desired to minimize the effects of curvature. In

such situations the magnetic parameter can be used as a control parameter against the

effects of curvature. This may hold significant potential in gastric disease treatments such

as magnetic endoscopy [104]. The effects of Hartmann number on flow velocity are

physically justified on the following grounds. First it is naturally anticipated that in a

curved channel the fluid accumulates toward the lower wall and moves with the velocity

which is not symmetric about 𝜂 = 0. The Lorentzian force due to magnetic field acts as a

resistance to the flow and its magnitude is proportional to the transverse velocity (see Eq.

2.12), hence amplitude of the flow velocity is suppressed near the plane 𝜂 = 0 in the lower

part of the channel. In order to compensate the decrease in the prescribed flow rate, the

flow velocity in upper part of channel will increase. During this process the maximum in

velocity curves also shift toward the upper wall.

Figure 2.5 illustrates the effects of curvature on velocity for small values of Ha and ' .

As expected it is observed that for small values of k the profile is asymmetric and the

maximum arises beneath 0. For large values of k the velocity profile regains its

symmetrical shape. Such observations are also reported by Goldstein regarding Poiseuille

flow of Newtonian fluid in a curved channel [105].

53

The pressure rise-flow rate plots based on Eq. (2.39) for different values of ' and Ha are

shown in Figs. 2.6 and 2.7, respectively. The following observations can be drawn for these

figures:

In the pumping region (Δp >0, Θ>0), Δp decreases by increasing ' while it

increases by increasing Ha. In this region, peristalsis has to work against the

pressure rise in order to maintain the prescribed flux. Our analysis, reveals that the

non-Newtonian characteristics of the fluid can be tuned to reduce the pressure-rise

in the pumping region; a solution which can be predicted using Newtonian model.

The free pumping flux i.e. Θ for Δp = 0 decreases by increasing ' and Ha. Δp =

0, the flow is solely due to peristaltic waves.

In the co-pumping region (Δp<0, Θ>0), Δp increases (decreases) by increasing '

(Ha). In this region the pressure rise per wavelength assists the flow due to

peristalsis.

It is anticipated that in the wave frame the streamlines have a shape similar to the channel

walls as the walls are stationary. However, under some certain conditions some streamlines

may split to enclose a volume of fluid called bolus in closed streamlines. Therefore some

circulating regions occur which move with the speed of peristaltic wave. Such a

phenomenon of bolus formation is called trapping. Moreover, if such a circulating region

is symmetric about the central plane 𝜂 = 0, then we designate it as a symmetric bolus of

fluid. To illustrate the trapping phenomenon the streamlines pattern for different values of

' , Ha and k are shown in Figs. 2.8 – 2.10, respectively. It is evident through Fig. 2.8 that

for small values of k i.e. 2k and ' 0 the bolus is not symmetric about 0. For this

choice of parameters, the bolus is displaced towards the upper wall. However, for ' 3.2

54

, it nearly becomes symmetric and then again becomes asymmetric for large values of '.

This evolution in bolus symmetry demonstrates the non-trivial influence of rheological

material property on the peristaltic flow. Elasticity of the fluid clearly exerts a significant

effect on the trapping phenomenon. Newtonian models cannot capture this level of detail.

It would appear that further investigations, perhaps involving normal stress differences in

the viscoelastic flow, may provide a deeper understanding of the delicate interplay between

bolus transformation and non-Newtonian behavior.

The effects of Hartmann number (Fig. 2.9) on streamlines are similar to the effects of ' .

Figure 2.10 illustrates the effects of channel curvature on the lower trapping limit. The

lower trapping limit is the smallest values of for which the trapping occurs. Below this

value of no circulating zone can be identified in the flow field. It is observed that lower

trapping limit increases with increasing the curvature of the channel.

55

Fig. 2.1: Schematic diagram of the curved moving boundary problem.

Fig. 2.2: Comparison between the exact solution (bubbles line) and asymptotic solution

(solid line) of u(η) at cross section x = π with k = 2.5, Θ = 0 and = 0.4.

56

Fig. 2.3: Variation of u for different values of ' at cross section 2

x (panels (a), (b)), 0x

(panels (c), (d)) and 2

x (panels (e), (f)) with Ha = 1, = 0.4 and k = 2. Left panels correspond to

Θ = 4 while right panels are for Θ = 2.

57

Fig. 2.4: Variation of u for different values of Ha at cross section 2

x (panels

(a), (b)), 0x (panels (c), (d)) and 2

x (panels (e), (f)) with ' = 1, = 0.4 and k =

2. Left panels correspond to Θ = 4 while right panels are for Θ = 2.

Fig. 2.5: Variation of u for different values of k at a cross section 2

x with

Ha = 0.2, ' = 0.5, Θ =1.64 and = 0.4.

58

Fig. 2.6: Variation of 𝛥𝑝 for different

values of ' with Ha = 1, k = 2 and

= 0.4.

Fig. 2.7: Variation of 𝛥𝑝 for different values of Ha

with ' = 2, k = 2 and = 0.4.

59

Fig. 2.8: Streamlines for (a) ' = 0, (b) ' = 1.5, (c) ' = 3.2, (d) ' = 5, (e) ' = 10 and

(f) ' = 15 with k = 2. The other parameters chosen are Ha = 0.5, Θ = 1.5 and = 0.8.

60

Fig. 2.9: Streamlines for (a) Ha = 0, (b) Ha = 0.5, (c) Ha = 0.875, (d) Ha = 1, (e) Ha =

1.5 and (f) Ha = 3 with k = 2. The other parameters chosen are ' = 0.5, Θ = 1.5 and

= 0.8

Fig. 2.10: Streamlines for (a) k = 2 (b) k→∞ with ' = 0.2. The other parameters chosen

are Ha = 0.5, Θ = 1 and = 0.8.

61

2.4 Concluding remarks

A theoretical study has been performed to investigate the combined effects of magnetic

field and viscoelasticity on peristaltic flow through a curved channel. An exact solution of

the derived differential equation has been obtained. An asymptotic formula for stream

function for large values of magnetic parameter or Jeffrey rheological parameter (ratio of

relaxation to retardation time) is also obtained using a singular perturbation method.

Graphical results are presented to illustrate the effects of the emerging parameters on

velocity profile, pressure rise per wavelength and streamlines pattern. The following

important conclusions are drawn from this study.

For large values of magnetic field (Ha) or ratio of relaxation to retardation

time ' , a thin boundary layer exists at both walls of the channel. The

thickness of the boundary layer is inversely proportional to '1 .Ha i.e.,

it is of the order'

1 .1Ha

This result emphasis the role of strong

magnetic field to control the thickening of boundary layer in peristalsis and

allows the possibility to confine the non-Newtonian effects in the vicinity

of the walls of the channel.

The effect of curvature (k) is to cause the velocity profile to be asymmetric

whereas the effects of ' and Ha are to counter act the effects of k. For a

fixed small value of k an increase in Ha or ' aids the velocity profile in

regaining a symmetric shape. However, for large values of Ha or ' the

velocity profile again become asymmetric with the maximum lying above

0.

62

The pressure rise per wavelength in the pumping region decreases by

increasing ' and k while it increases for large values of Ha.

For small values of k the bolus is not symmetric about 𝜂 = 0. However, an

increase in Ha or ' helps the bolus to regain its symmetric shape. For large

values of Ha or ' the bolus again becomes asymmetric.

63

Chapter 3

Simultaneous effects of viscoelasticity and

curvature on peristaltic flow through a curved

channel

The main aim of this chapter is to analyze peristaltic flow of an Oldroyd-B fluid in a curved

channel. Assuming the flow to be incompressible, laminar and two-dimensional, the

governing partial differential equations are reduced under long wavelength and low

Reynolds number approximations into a single nonlinear ordinary differential equation in

the stream function. Matlab built-in routine bvp4c is utilized to solve this nonlinear

ordinary differential equation. The solution thus obtained is used to investigate the effects

of curvature of the channel and Weissenberg number on important phenomena of pumping

and trapping associated with peristaltic motion. It is found that for small values of

Weissenberg number, the effects of curvature are dominant. However, for large values of

Weissenberg number, viscoelastic effects counteract the effects of curvature and help the

flow velocity and circulating bolus of fluid to regain their symmetry.

64

3.1 Description of the problem

We have already explained the geometry in chapter 2. The momentum equation which

governs the flow without magnetic field is given by

.d

dt

V (3.1)

The extra stress tensor S for an Oldroyd-B fluid satisfies [62]

1 21+ 1+ .D D

Dt Dt

S (3.2)

In the above equation 1 is the relaxation time, DDt

is the contravariant convected

derivative defined by

,TD

Dt t

S SV S LS SL

for a contravariant

tensor of rank 2

(3.3)

and

.D

Dt t

Tb bV b Lb bL for a contravriant vector (3.4)

In view of Eq. (1.24), we can write Eq. (3.1) as

.d

Pdt

V

S (3.5)

Now to find the governing equations, we need to eliminate S between Eqs. (3.2) and (3.5).

To this end, we apply the operator 11+D

Dt

to the momentum equation (3.5) and get

65

1 1 11+ 1+ 1+ .D d D D

PDt dt Dt Dt

VS (3.6)

Following Harris [106], we use the commutativity of the operators and DDt

i.e.

.D D

Dt Dt

(3.7)

Therefore, Eq. (3.6) becomes

1 1 11+ 1+ 1+ ,D d D D

PDt dt Dt Dt

VS (3.8)

which in view of Eq. (3.2) takes the form

*

1 21+ 1+ ,D d D

PDt dt Dt

1

VA (3.9)

where *P is the modified pressure.

For two-dimensional velocity field defined by (2.6), Eq. (3.9) yield the following

equations:

2 * * 2 *

2 2* * * ** *

22 * 2 * 2 2 2

1 2 22 * 2 * * ** *

2 2 22

2 2

3 2 2

2 2 22

2

U V V R U V R U R U V VV

R R R t R R X R R X R X R R RR R R R

V R V U U R U U U V UV U U V

R R R X R R t X R R R R R RR R R R

V V VV V

R t

2* 2 * * 2 * 2

2

2* * 2 * 2*

2 22

R U V R UV V R V R VV U U

R t R R R X X R R X R R XR R

66

2* 3 * 3 * 2 *2 2 * 3

2 2 3* 2 * 2 2 * 2* *

* 2 * 2 * 2 *

2 3 3 2 2** * * * *

2

*

3 3

4 6 4 3 2

2 2

R U R U R U R U U R V UU

R R X R R R X R X t X R R R XR R R R

R V U R V U V R U V R U V V V

X R X R R X R R X R RR R R R R R R R R R

V V

R R R R

2 2 3 3 * 3 2

* 2 2 3 2 * 2 *

2* 2 * 3 * *

2 2 2* 2 ** * *

3* 2 * 2

* 2 * 2

2 22 2 2

2 2 2

2

V V V V V R U V VV

R R R R R R t R R R X R R R t

R U V R V V R V R U VV U

R X R R R X t t R R R XR R R R R R

R U V R U V R

R R R X R R X X

2 2* 2 *2

3* *

2 2 3*2 2 * 2 * 3 * 3

3 2 * 2 * 2 * 3*

3

2

V V R V

R R R X X XR R

R V R V V R V R VV U

X R R R X R R t X R R XR R

*1,

P

R

(3.10)

2* 2 * 2 * 2

2* * * 2 * 2 **

2* 3 2 2 * 2 * 2 2

2

12 2 * 2 * * 2* *

12

32 2

2

U R U U U U U U U R U R VV V

t R R X R R R R R R R R R X R R R XR R

R V U U U U R U U R U UU V

X R R R t R R X t R R X X tR R R R

R

* 2 * 2 2 * 22

2 2 2* 2 * ** * *

3 * 3 2 * 2 * 2

2 2 22 * 2 * ** *

* *

3*

2 2 2 2 2

1 2 1

3 2

UV U R UV U UV U UV V U V R U VV

R R R X X R R R R R R X XR R R R R R

U R U U U R U U R U U U

R t R R R X R R R X R X R R R R X tR R R R

R U RU

XR R R

3 2* 2 * 2 * 3

2 * 2 * 2 * 2*

3 2* 3 3 * 3 *2 2

3 2 3* 3 3 * 2 2* * *

2 2

62 2

U U R U U R U U R U

R X R R R X R R X X R R X tR

R U U V U U R U R V UV V V

R R X R X R R R X XR R R R R R

67

22 2 * 3 * 2

2 * * 2 2 * 2 **

2* 2 * 3 * 3 *2 *

2 3 3* 2 * 2* * *

2 2 22

3 9

U V U V U V U V R V V R U V

R R R R R R R R R R R R R X R R X R XR R

R V R V R V R U V R VV U V

R X R R R X R R R X X X XR R R R R R

2*2 2 *2 2 * 2 * *

3 3 * 2 ** *

3 3 1,

RR V R V R U V R P

X t X t R R R X R R XR R R R

(3.11)

It is pointed out here that Eqs. (3.10) and (3.11) do not contain term involving 2 . Thus

using the approach of Harris [106], one would get the same governing equations for

Oldroyd-B and Maxwell fluids for peristaltic flow in a curved channel.

Employing the transformations given in Eq. (2.16) to switch from the laboratory frame to

the wave frame.

Eqs. (3.10) and (3.11) can be put in the following form

2 * * 2 *

2 2* * * ** *

2 2 2*2 * 2

1 2 22 * 2 * ** *

2 22

*

3 2 2

2 2 2 2

u c R u cv v v R u R u v vv c

r R r x r R x r R x r x r R rr R r R

u c R u c u c u c vv R v u u uc v

r r R x r R x x r R rr R r R

u c v vv

r R r

* *2 2 22

22 * 2*

2 22

R v u c R v u cv v v vcv c

r r x r R r x x xr R

68

2* 2 * 2

2

* 2 * 2

** 3 * 3 * 2 2 * 3

2 2 2* 2 * 2 2 2 * 2* *

* 2 * * 22

2 3 3 ** * *

2

33

4 6 4

R v R vu c u c

r R x r R x

R u cR u R u cR u u R v uu c

r R x r r R x r x x r R r xr R r R

R v u R u R uv v

x r x r R x rr R r R r R

*

2 2* *

2 *2 2 3 3 3 2

* * 2 2 3 2 * 2 *

2* *2 * 3

2 2 2* 2* * *

3 2

2 2 12 2 2 2 2

222

v R u v vv

r x r rr R r R

R u cv v v v v v v v vv c c

r R r r R r r r r r x r R r x r R r x

R u c cR u cv R v c v vv

r x r R r x x xr R r R r R

*

*

3 2 2* 2 * 2 * 2 *2

3* 2 * 2 * *

2 2 3*2 2 * 2 * 3 * 3

3 2 * 2 * 3 * 3*

32

2

R u v

r R r x

R u v R u v R v v R v

r R r x r R x x r R r x x xr R

R v R v v R v R vv c u c

x r R r x r R x r R xr R

*1,

p

r

(3.12)

2* 2 * 2 * 2

2* * * 2 * 2 **

12

R u c u c u cu u u u u R u R vc v v

x r R x r R r r R r r r R x r R r xr R

23 2* 2 * 2 * 2 222

12 2 * 2 * 2 * 2 2* *

* * 22 22

2 2* 2 * ** *

32 2

2 2 2 2 2 2

u c u cR v u u R u R u uc c u c u c cv

x r R r x r R x r R x xr R r R

R u c v R u c v u c v u c v u cu u u v vv

r R r x x r r R r r R xr R r R

2*

2*

* * *3 3 2 2 * 2

2 2 2 33 * 2 * ** * *

3* * 2 * 2 *

2 * 2 * 2 **

2 31

22 2

R u c v

xr R

R u c R u c R u cu u u u R u u c u u

r r R r x r R r x r x r R r r x x xr R r R r R

R u u R u u R u u Rc

r x r R r x r R x x r Rr R

2 33 * 3

33 * 3 *

23 * 3 *2 2 2

2 3 23 * 2 2 * * 2* * *

22 * 3 * 2 *

2 * 2 *

2

2 26 22

2

u c vu R u

x r R x r R

u c u cv u u R u R v u v u v vv v

r x r R r x x r r R r r r R rr R r R r R

u v R v v R u v R v

r r r R r x r R x r x r

22 * 3 * 3

2 * 2 * 2*

v R v R vu c

r x r R r x r R r xR

69

2*2 * *2 2 * 2 * *

3 3 3 2 * 2 ** * *

3 9 3 1.

R u v R v v cR v R u v R P

x x x x r R r x r R xr R r R r R

(3.13)

For the subsequent analysis, we first non-dimensionalized the above equations using the

dimensionless variables (2.22) and then employ long wavelength and low Reynolds

number assumptions to get

0,p

(3.14)

3 22 3 2

12 3 21 1 1

k kk We

k k

,p

kx

(3.15)

where 2

11

cWe

is the Weissenberg number and is defined through Eq. (2.29).

Eliminating pressure between Eqs. (3.14) and (3.15), we get the following compatibility

equation

32 3

12 31 1

k kk We

k k

2 2

21 0.

(3.16)

We mention here that under long wavelength assumption, Eq. (3.16) corresponds to

peristaltic flow of Maxwell fluid due to vanishing of the terms involving retardation

constant. But still it is capable of predicting simultaneous effects of viscoelasticity and

70

curvature of the channel. Such an equation cannot be obtained when peristaltic flow is

considered in straight channel or tube which is evident by taking limit of Eq. (3.16) when

k→∞. In that case Eq. (3.16) reduces to the corresponding equation of peristaltic transport

of Newtonian fluid in a straight channel.

3.2 Method of solution

Due to nonlinear nature of Eq. (3.16), an exact solution is difficult to obtain. Therefore, we

opted to go for a numerical solution. To this end, we employed Matlab built-in routine

bvp4c for solving nonlinear ordinary differential equation (3.15). In the limit when k→∞

or 1We = 0, our result reduce to the corresponding results for a Newtonian fluid in a straight

channel.

3.3 Results and discussion

In this section graphical results are displayed for various values of Weissenberg number

1We and dimensionless radius of curvature (k) in order to analyze flow characteristics,

pressure gradient, pumping and trapping phenomena associated with peristaltic motion.

Figure 3.1(a) shows the velocity profile u at cross–sections 2

x (narrow part the

channel) for different values of We when Θ = 4 and k = 2. We observe from this figure that

for 1We = 0 and k = 2, velocity profile is not symmetric about η = 0 and maximum in it lies

71

below η = 0. A symmetry about η = 0 is observed for 1We = 0.25. A further increase in

1We shifts the maximum in velocity towards the upper wall.

The velocity profiles for different values of 1We , by taking Θ = 2 and keeping other

parameters same as in Fig. 3.1(a), are shown in Fig. 3.1(b). This figure depicts the same

behavior as predicted by Fig. 3.1(a) but for very large values of 1We compared with those

chosen in Fig. 3.1(a). Figure 3.1(c) illustrates velocity profiles at x = 0 (undisturbed part

of the channel) by keeping other parameters same as chosen in Fig. 3.1(a). Again this figure

shows similar behavior of velocity profile as observed in Fig. 3.1(a). Figure 3.1(d) exhibits

asymmetry and shift of maximum in velocity towards the upper wall for various values of

1We greater in comparison with those chosen in Fig. 3.1(c). Similar observations can be

drawn from Fig. 3.1(e) and 3.1(f).

The effects of dimensionless radius of curvature (k) on velocity profile of Oldroyd-B fluid

are shown through Fig. 3.2. A striking observation is made from this figure i.e. velocity

becomes asymmetric and maximum in it shifts towards the upper wall for small values of

k. This observation to contrary to that what is observed for a Newtonian fluid. For a

Newtonian fluid, a decrease in k shifts the maximum in velocity towards the lower wall. In

general, it is concluded from Figs. 3.1 and 3.2 are that viscoelastic fluid driven by

peristaltic waves in a curved channel behaves in a very different way than that of a

Newtonian fluid. Perhaps the role of Weissenberg number here, which characterizes the

viscoelastic fluid is to counter the effects of curvature and make the velocity symmetric

72

even for very small values of k. For large values of 1We the viscoelastic effects dominate

the effect of curvature and shift the maximum in velocity profile towards the upper wall.

The plots of pressure gradient dp

dx over one wave length for various values of 1We and

k are shown in Figs. 3.3 and 3.4, respectively. We observe from Figs. 3.3 that dp

dx

increases in the wider part of the channel while it decreases in the narrow part of the

channel by increasing 1We . Opposite trend can be observed by increasing k as evident from

Fig. 3.4.

Figures 3.5 and 3.6 are plotted to see the variation of pressure rise per wavelength p

against dimensionless mean flow rate Θ for various values of 1We and k, respectively.

Following interesting observations can be made from these figures.

In pumping region (Δp >0, Θ > 0) there exists a critical value of flow rate Θ, below

which Δp decreases and above which it increases, by increasing 1We . Thus pressure

resistance for a viscoelastic fluid is lesser in magnitude than that for a Newtonian

fluid.

The situation is different in free pumping (Δp = 0) and co-pumping region (Δp <

0; Θ > 0). Here Δp increases by increasing 1We .

Δp in pumping region increases in going from curved to straight channel below a

certain critical value of Θ. Above this critical value a reverse trend is observed. This

reverse trend also prevails in free pumping and co-pumping regions.

73

The streamline patterns for different values of 1We and k are shown in Figs. 3.7 and 3.8. It

is observed that the presence of curvature in channel destroys the symmetry of circulating

bolus of the fluid. However, as expected the circulating bolus of fluid regain its symmetry

for large values of k.

The effects of 1We on streamline patterns are quite interesting. Here, as observed for

velocity profile there is competition between 1We and k. For small values of 1We the

effects of k are dominant and bolus is shifting towards the upper wall. However, an increase

in 1We counters the effects of k and as a result bolus regain its symmetry. A further increase

in 1We dominates the effects of k thus making the bolus asymmetric and at the same time

shifts it towards the lower wall.

Figure 3.9 is plotted to see the effects of curvature on lower trapping limit (i.e. maximum

value of Θ for which trapping occurs). It is observed that lower trapping limit increases by

increasing curvature of the channel.

A general observation after examining streamlines plots is that for a symmetric channel

mixing phenomenon is strong due symmetric nature of the circulating region. However,

for a curved channel due to shift of bolus in lower half of the channel, there is no mixing

of fluid in the upper half of the channel.

74

Fig. 3.1: Variation of u for different values of 1We at cross section 2

x (panels

(a), (b)), 0x (panels (c), (d)) and 2x (panels (e), (f)) with = 0.4 and k = 2. Left

panel correspond to Θ = 4 while right panel are for Θ = 2.

75

Fig. 3.2: Variation of u for different values of k at a cross section 2

x with 1We = 5, Θ = 1.64

and = 0.4.

Fig. 3.3: Variation of dpdx

over one wavelength

for different values of 1We with k = 2, = 0.4

and Θ = 0.

Fig. 3.4: Variation of dpdx

over one wavelength for

different values of k with 1We = 5, = 0.4 and Θ = 0.

76

Fig. 3.5: Variation of 𝛥𝑝 for different values

of 1We with k = 2 and = 0.4.

Fig. 3.6: Variation of 𝛥𝑝 for different values of

k with 1We = 5 and = 0.4.

Fig. 3.7: Streamlines for (a) 1We = 0, (b) 1We = 2, (c) 1We = 5 and (d) 1We = 8 with k = 2. The other

parameters chosen are Θ = 1.5 and = 0.8.

77

Fig. 3.8: Streamlines for (a) k = 2, (b) k = 3.5, (c) k = 5 and (d) k→∞ with 1We = 5. The other

parameters chosen are Θ = 1.5 and = 0.8.

Fig. 3.9: Streamlines for (a) k = 2 and (b) k→∞ with 1We = 0.2. The other parameters

chosen are Θ = 1 and = 0.8.

78


A mathematical model is presented to explore the simultaneous effects of curvature of

channel and viscoelasticity on peristaltic transport. The problem is governed by fourth

order nonlinear ordinary differential equation which is solved numerically using Matlab

built-in routine bvp4c. The effects of various emerging parameters on basic features of

peristalsis are explained through various plots. The main points of the conducted study can

be summarized as follows:

An increase in curvature results in asymmetric velocity profiles with maxima lying

below 𝜂 = 0.

The effects of Weissenberg number are to counter the effects of curvature and thus

making the velocity profiles symmetric. For large values of 1We , the viscoelastic

effects dominate resulting in asymmetric velocity profiles whose maxima lie above

𝜂 = 0.

Δp decreases by increasing 1We or k below a certain critical values of Θ. Above

this value a reverse trend is observed.

The circulating bolus of fluid becomes asymmetric by increasing curvature of

channel. However, an increase in 1We counteract the effects of curvature and helps

the bolus to regain its symmetrical shape.

For large value of 1We viscoelastic effects dominate resulting in asymmetry of

bolus.

The mixing phenomena in a straight channel is stronger than that in a curved

channel.

79

Chapter 4

Long wavelength analysis for peristaltic flow

of Oldroyd 4-constant fluid in a curved

channel

This chapter presents the analysis of peristaltic transport of an Oldroyd 4-constant fluid in

a curved channel. The governing equation in curvilinear coordinates is a fourth order

nonlinear ordinary differential equation. Long wavelength and low Reynolds number

approximations are employed. The solution of the resulting nonlinear equation is obtained

using finite difference method (FDM) combined with an iterative scheme. The effects of

material parameters on the velocity profile and the pressure gradient are investigated in

detail. Pumping and trapping phenomena are analyzed. A comparative study between

curved and straight channels is also included.


The essential difference between chapter 2 and 4 lies in the choice of non-Newtonian

model used to describe the rheology of the fluid in the channel. Here, it is assumed that

fluid inside the channel obeys the constitutive equation of an Oldroyd 4–constant model.

For an Oldroyd 4–constant fluid [63] the extra stress tensor S satisfies the equation

80

1 3 2tr 1 .D D

Dt Dt

SS S (4.1)

In above equation 3 is the relaxation time and tr denotes the trace.

It should be pointed out that the model (4.1) includes the Oldroyd-B model (for 3 = 0),

the Maxwell model (for 2 = 3 = 0) and the Newtonian fluid model (for 1 = 2 = 3

= 0) as the limiting cases.

Following the similar procedure as described in chapter 2, we get

22

2 3 2 2

2 22

1 3 2

11 2 1

11 ,

11 2 1

x

We Wek

Sk

WeWek

(4.2)

where 33 .

cWe

a

Inserting Eq. (4.2) into Eq. (2.31) (with Ha = 0), we get

22

2 3 2 22

2 22

1 3 2

11 2 1

1 11 0.

( ) 11 2 1

We Wekp

kx k k k

WeWek

(4.3)

Eliminating of pressure between Eqs. (2.30) and (4.3) yield the following compatibility

equation

81

22

1 2 22

2 22

2 2

11 2 1

1 11 0,

( ) 11 2 1

kk

k k

k

(4.4)

where 1 2 3We We and 2 1 3We We . The above equation is subject to the boundary

conditions given in Eqs. (2.35) and (2.36). Moreover, present analysis is carried out in

absence of imposed magnetic field.

4.2 Numerical method

In this section, we briefly describe a suitable numerical technique for solution of Eq. (4.4)

subject to boundary conditions (2.35) and (2.36). Due to non-linear nature of Eq. (4.4),

direct application of finite difference method is not appropriate. For this reason, iterative

methods are commonly used. In such methods an iterative procedure is constructed in such

a way that the original non-linear boundary value problem is converted into a linear one at

(n+1)th iterative step. We for this particular problem propose the following iterative

procedure:

22

24 1 3 1 2 1

4 3 2

n n n

g gg k k

gk g g k

k

2 22 2

2 21

2 20,

n

g g g gg k k g k k

k k

(4.5)

82

11

1

11

2

, 1, at ,2

, 1, at ,2

nn

nn

qh

qh

(4.6)

where

2

2

1 2

2

2

2 2

1

1 2

,

1

1 2

n

n

n

n

k

g

k

(4.7)

and the index (n) denotes the iterative step. It is now clear that above boundary value

problem is linear in 1n . Employing central difference formulae

1 11

1 1 ,2

n nn

i i

(4.8a)

1 1 12 1

1 1

2 2

2,

n n nn

i i i

(4.8b)

1 1 1 13 1

2 1 1 2

3 3

2 2,

2

n n n nn

i i i i

(4.8c)

1 1 1 1 14 1

2 1 1 2

4 4

4 6 4n n n n nn

i i i i i

. (4.8d)

Eqs. (4.5) and (4.6) can be put in the form

1 1 2 1 3 1 4 1 5 1

2 1 1 2 0. 2,3,..., 2n n n n n

i i i i i i i i i ia a a a a i M

(4.9)

83

1 11 1 1

1

1 11 1 1

2

, 1, at ,2 2

, 1, at ,2 2

n nn i i

n nn i i

qh

qh

(4.10)

where,

1 2

4 3

1, ,

i

i i i

k g ga a g k

2 223

2 2

1,i i i

i i

g g Ha ka g k k

k k

2 224

2 22

1,i i i

i i

g g Ha ka g k k

k k

2 225

2 22

1.i i i

i i

g g Ha ka g k k

k k

(4.11)

Now at a fixed cross-section and for M uniformly discrete points , 2,3,..., 2i i M with

a grid size , Eq. (4.9) with boundary conditions (4.10) define a system of linear

algebraic equation which is to be solved for each iterative (n+1)th step. In the present case,

we have used Gaussian elimination method for solving such a system.

It is now clear that above boundary value problem is linear in 1n . Inserting finite-

difference approximations of 1n and its derivative in Eq. (4.5) and boundary conditions

(4.6), a system of linear algebraic equation can be obtained and solved for each iterative

step (n+1). In this way numerical values of 1n at each cross-section can be obtained. Of

course some suitable initial numerical values of n are specified at each cross-section to

start the iterative procedure. Unfortunately increasing number of iteration a convergent

84

solution is not always possible, especially when initial numerical values of are not given

carefully. In such circumstances the method of successive under-relaxation is used. In this

method the estimated value of at (n+1) iterative step i.e. 1n is refined to get the

convergent value of at the same step. This can achieved by the following formula

1 1 , 0,1 ,n n n n (4.12)

where is an under-relaxation parameter. We should choose so small that convergent

iteration is reached. The iteration in this problem are carried out to calculate the value of

convergent to eight places of decimal.

The above described method has been already used by many authors. We refer the reader

to articles [63, 120] for further details about this technique.


Eq. (4.4) indicates that it is characterized by three parameters namely dimensionless radius

of curvature ( k ) and two material constants of Oldroyd 4-constant fluid i.e. 1 and 2.

Therefore it is quite likely that all these parameters affects the flow, pumping and trapping

characteristics associated with the peristaltic motion significantly. To see the influence of

these parameters on aforementioned characteristics, we have plotted Figs. 4.1 – 4.10.

The effects of material parameter 1 on velocity u are shown in Fig. 4.1 when k = 2.

This figure indicates a decrease in magnitude of velocity by increasing 1. Figure 4.2

shows that the magnitude of velocity profile u enhances by increasing 2.

85

Asymmetry in profiles of u owes to the small values of k . It is interesting to observe

that for 2 0.5 the maximum value of the velocity profile u lies below the centerline

0. However it shifts towards the upper half of the channel by increasing 2. Such

effects are not significant in Fig. 4.1 where 2 is kept fixed and 1 is increased. Thus in

contrast with Newtonian fluid, the viscoelastic fluid bears the potential to counter the

effects of curvature. The effect of k on u is shown in Fig. 4.3. As expected smaller

values of k results in asymmetric velocity profiles with maxima lying below 0.

Figures 4.4 – 4.6 illustrate the effects of 1 2, and k on pressure rise per wave length

.p Figure 4.4 shows that p in pumping region 0, 0p shows increasing

trend. The free pumping flux i.e. 2q for 0p is also found to increase by

increasing 1 . However in co-pumping region 0, 0p a reverse trend is observed.

The effects of 2 on p is quite opposite to that of 1 in all three regions. The effects of

1 and 2 on p in a straight channel is already reported in Ref. [63]. A comparison of

above stated results with those in Ref. [63] shows qualitative agreement. Figure 4.6 shows

the profile of p for three values of k . This figure reveals that p in pumping region

decreases in going from curved to straight channel.

Thus it may be concluded that the variation of pressure rise per wavelength with 1 and

2 is qualitatively similar for both straight and curved channels. For a fixed prescribed

flow rate, the effect of curvature is just to increase the magnitude of p in the pumping

86

region. Thus to maintain the same flux as that in the straight channel, peristalsis has to

work against greater pressure rise in the curved channel.

The streamline pattern for different values of 1 2, and k are shown in the Figs. 4.7 – 4.10.

Figure 4.7(a) shows asymmetric pattern of streamlines for 1 0 and k = 2 with

circulating bolus of fluid appearing near the upper wall. This bolus consists of two

approximately equal halves but clearly it is not symmetric about the centerline. However,

for 1 0.36 , the bolus becomes approximately symmetric about the centerline (Fig.

4.7c). An increase in 1 from 0.36 to 2 destroys the symmetry of the circulating bolus

about the centerline and in this case the size of the upper half increases and it pushes the

lower half toward the lower wall of the channel (Fig. 4.7d). Also further increase in 1

increases the size of upper half of the circulating bolus.

On the other hand it is observed from Fig. 4.8(a) that a circulating bolus of fluid consisting

of two halves with upper half larger in size as the lower one exists when 1 0.85 and k =

2. By increasing 2 from 0.85 to 4 the bolus approximately becomes symmetric about the

centerline (Fig. 4.8c). Finally it again becomes asymmetric when 2 5 (Fig. 4.8d). Hence

Figs. 4.7 and 4.8 conclude that 1 and 2 significantly counter the effects of curvature and

help the bolus to regain its symmetric shape. Effect of k on streamline pattern is analyzed

for an Oldroyd 4–constant fluid in Fig. 4.9. We can see that for smaller values of k i.e. for

a curved channel the bolus/circulating region is not symmetric about the centerline. It is

concentrated in the upper half of the channel. However the bolus tends to region its

87

symmetric shape when k increased. In the limit when k →∞ the well-known symmetric

shape of the bolus is observed.

Figure 4.10 reflects the effects of dimensionless radius of curvature ( k ) on lower trapping

limit. I.e. minimum value of for which circulating region exists. This figure indicates

that lower trapping limit increases in going from straight channel to curved channel.

88

Fig. 4.1: Variation of u for different values of

1 2 0.5 with = 0.4, Θ = 1.8 and k = 2.


2 1 0.5 with ϕ = 0.4, 𝛩 = 1.8 and k = 2.

Fig. 4.3: Variation of u for different values of k with ϕ = 0.4, 𝛩 = 1.8, 1 0.2 and 2 0.3 .

Fig. 4.4: Variation of 𝛥𝑝 for different values of 1

with 2 0.5 , k = 2 and ϕ = 0.4.

Fig. 4.5: Variation of 𝛥𝑝 for different values of 2

with 1 0.5 , k = 2 and ϕ = 0.4.

89

Fig. 4.6: Variation of 𝛥𝑝 for different values of 1 0.1 with 2 0.2, k = 2 and ϕ = 0.4.

Fig. 4.7: Streamlines for 1 1 1(a) 0, (b) 0.325, (c) 0.36 and 1(d) 2, with k = 3. The other

parameters chosen are 2 2 , Θ = 1.5 and = 0.8.

90

Fig. 4.8: Streamlines for 2 2 2(a) 0.85, (b) 2.6, (c) 4 and 2(d) 5, with k = 3. The other parameters

chosen are 1 0.85 , Θ = 1.5 and = 0.8.

91

Fig. 4.9: Streamlines for (a) k = 2, (b) k = 3.5, (c) k = 5, and (d) k → ∞ with 1 = 0.5 and 2 = 0.7.

The other parameters chosen are Θ = 1.5 and = 0.8.

Fig. 4.10: Streamlines for (a) k = 2.5 and (b) k → ∞ with 1 = 0.1 and 2 = 0.2. The other

parameters chosen are Θ = 1 and = 0.8.

92


The peristaltic transport of an Oldroyd 4-constant fluid in a curved channel is analyzed for

some physical quantities of interest. The following points are worth mentioning.

Effect of 1 on the velocity u is quite opposite to that of 2 .

Asymmetric in velocity profile is observed through curvature.

Viscoelastic effects i (i = 1, 2) counter the effect of curvature.

The pressure rise per wavelength in pumping region increases with

increasing 1 whereas it decreases via 2 and k .

For smaller values of k the bolus is not symmetric about 0. However an

increase in 1 and 2 help the bolus to regain its symmetry.

93

Chapter 5

Numerical simulation of peristaltic flow of a

bio-rheological fluid with shear-dependent

viscosity in a curved channel

The theme of this chapter is to analyze peristaltic motion of a non-Newtonian Carreau fluid

in a curved channel under long wavelength and low Reynolds number assumptions, as a

simulation of digestive transport. The flow regime is shown to be governed by a

dimensionless fourth order nonlinear ordinary differential equation subject to no-slip at the

walls. A well-tested finite difference method (FDM) based on an iterative scheme is

employed for the solution of the boundary value problem (BVP). The important

phenomena of pumping and trapping associated with the peristaltic motion are investigated

for various values of rheological parameters of Carreau fluid and curvature of the channel.

An increase in Weissenberg number is found to generate a small eddy in the vicinity of the

lower wall of the channel which is enhanced with further increase in Weissenberg number.

For shear-thinning bio-fluids (power-law rheological index, n < 1) greater Weissenberg

number displaces the maximum velocity towards the upper wall. For shear-thickening bio-

fluids, the velocity amplitude is enhanced markedly with increasing Weissenberg number.

94

5.1 Mathematical model and rheological constitutive equations

In contrast to previous chapter, here the analysis of peristaltic motion in a curved channel

is carried out for Careau fluid. The geometry and the underlying assumptions are same as

described in chapter 2 and 4. Therefore, the continuity and momentum equations for the

present problem are as in Eqs. (2.30) and (2.31). For the elimination of ,xS we start with

the definition of extra stress for Carreau fluid [67] i.e.

1

2 2

0 1 .

n

xS

(5.1)

In the above equation, is the infinite-shear-rate viscosity, 0 is the zero-shear-rate

viscosity, is the time constant, n is the power-law index and is defined as

1

: .2

(5.2)

Using the definition of and velocity field, the components of extra stress in wave frame

are

1* * 2 2 2

1* * 2 2 2

1* * 2 2 2

2 1 1 , ( )

11 1 , ( )

2 1 1 , ( )

n

n

x

n

xx

vS We a

uu k vS We b

k k

k u vS We c

k x k

(5.3)

where

22 2

211 2 2

2 2 .2

uv u k v k u v

k k k x k

(5.4)

95

In above equations 1

cWea

is Weissenberg number (a measure of the product of the

relaxation time and the deformation rate), *

0

is the dimensionless ratio of the

infinite-shear-rate viscosity to the zero-shear-rate viscosity and δ is the wave number. Since

in general 0 , therefore the ratio * is chosen less than one in the present analysis.

The definition of stream function enables us to write (5.3) and (5.4) after using the long

wavelength and low Reynolds number approximations as

1 2

* * 2 2 22

0, ( )

11 1 1 , ( )

0, ( )

x

n

x

S a

S We bk

S c

(5.5)

therefore Eq. (5.5b) become

12 22

* * 2

2

1 11 1 1 1

n

xS Wek k

2

2.

(5.6)

Intersection of Eq. (5.6) into Eq. (2.31) (with Ha = 0) readily yields

12 22

2 * * 2

2

2

2

1 11 1 1

( )

11 0.

n

pk We

s k k k

k

(5.7)

96

Elimination of the pressure between Eqs. (2.30) and (5.7) yields the following

compatibility equation:

12 22

2 * * 2

2

2

2

1 11 1 1

( )

11 0.

n

k Wek k

k

(5.8)

The boundary conditions to be satisfied by Eq. (5.8) are already given in Eqs. (2.35) and

(2.36). Eq. (5.8) subject to boundary conditions (2.35) and (2.36) is solved using the finite

difference numerical technique which has been already explained in detail in chapter 4.

The validation of finite difference solution is achieved using spectral Chebyschev

collocation method (SCCM). The procedure of the method is explained in the next section.

5.2 Validation with Spectral Chebyschev Collocation method (SCCM)

The nonlinear boundary value problem defined by Eq. (5.8) subject to boundary conditions

(2.35) and (2.36) is also solved with an optimized spectral Chebyschev collocation method

(SCCM). In this approach an expansion in terms of special polynomial functions is utilized

to obtain a steady or un-steady solution [107]. This technique has recently been applied

successfully to simulate many non-linear transient multi-physical transport phenomena

including rocket gel propulsion heat transfer [108], electrohydrodynamic pumps [109],

transient heat conduction in tissue [110], magnetohydrodynamic blood flow in curved

vessels [111] and peristaltic propulsion [112]. In the present study the Chebyshev

polynomial series is used in the direction. The principal advantage of SCCM lies in the

97

accuracy achievable for a given number of unknowns. For problems with solutions which

are sufficiently smooth, SCCM demonstrates exponential rates of convergence and

accuracy. To optimize the present method, the code has been tested for convergence with

respect to the spatial resolution. The solutions converge in 30 iterations with a Newton-

Raphson method, executed on an SGI Octane desktop dual processor machine [113].

Numerical solutions are found to be independent of the number of collocation points for a

sufficiently large number of collocation points. N = 70 yields the optimal convergence and

very high accuracy (up to 610 ) and is therefore implemented in all the computations. In

SCCM, we seek an approximate solution, which is a global Chebyshev polynomial of

degree N defined on the re-mapped interval [-1, 1]. We discretize the interval by using

collocation points to define the Chebyshev nodes in [-1, 1], namely

cos , 0,1,2,..... .j

jx j N

N

(5.9)

The derivatives of the functions at the collocation points are given by:

.2,1,)()(0

nxfdxfN

j

j

n

kjj

n (5.10)

where n

kjd represents the differentiation matrix of order n and are given by

,,....1,0,),()(4

0

1

,0

1 NjkxTxTc

n

Nd jn

N

n

k

n

l

n

oddlnl l

njkj

(5.11)

,,....1,0,),()()(2

0

2

,0 1

22

2 NjkxTxTc

lnn

Nd jn

N

n

k

n

l

n

evenlnl

njkj

(5.12)

98

Here )( jn xT are the Chebyshev polynomial and the coefficients j and lc are defined

as

1,....2,11

,02

1,....2,11

,02

1

),coscos()( 1

Nl

Norlc

Nj

NorjxnxT ljjjn

. (5.13)

As described above the Chebyshev polynomials are defined on the finite interval [-1, 1].

Therefore to apply Chebyshev spectral method to the nonlinear boundary eqns (5.1) – (5.2),

we make a suitable linear transformation and transform the physical domain [0, ) to

Chebyshev computational domain [-1, 1]. We sample the unknown function w at the

Chebyshev points to obtain the data vector, T

Nxwxwxwxww ]),(......),(),(),([ 210 . The

next step is to find a Chebyshev polynomial F of degree N that interpolates the data, i.e.,

( ) , 0,1,... ,j jF x w j N and obtain the spectral derivative vector w by differentiating F

and evaluating at the grid points, i.e. ' '( ), 0,1,... .j jw F x j N This transforms the

nonlinear differential equations into a system nonlinear algebraic equations which are

solved by Newton’s iterative method starting with an initial guess. To circumvent difficulty

near the point of inflection for the steady solution, the arc-length method has been used. In

this technique, the arc-length “s” plays a central role in the formulation. Further details are

documented in Bég et al. [114]. Appropriate convergence criteria are prescribed. Once the

stream function is computed, an outer loop allows the computation of velocity components

as per the definitions in eqn. (2.29), viz , .k

u vk x

To verify the accuracy

of the finite difference method (FDM) computations, we compare the solutions obtained

with the spectral method (SCCM), for the second of these velocity components i.e. v in

99

Tables 5.1 – 5.3. Very close correlation is achieved. In both cases the solutions are stable

and converge quickly. Confidence in both codes is therefore high. Furthermore, the

solutions given in Tables 5.1 – 5.3 provide a benchmark for other researchers who may

wish to extend the present formulation and validate other algorithms.

η We = 0.5, n = 0.01 We = 1, n = 0.01 We = 5, n = 0.01

u u u

FDM SCCM FDM SCCM FDM SCCM

-0.8338 -1.0000 -1.0000 -1.0000 -1.00000 -1.0000 -1.0000

-0.5111 -0.2163 -0.2164 -0.235 -0.23497 -0.3291 -0.3292

-0.0269 -0.0190 -0.0191 -0.08336 -0.08335 -0.1917 -0.1918

0.5111 -0.4224 -0.4222 -0.3756 -0.37558 -0.2572 -0.2574

0.8338 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000

Table 5.1: FDM and SCCM numerical solutions compared for u with three different

values of We for shear thinning fluids (n < 0) when k = 2.5, * = 0.0061, Θ = 1.5,

32

x and = 0.4.

Η We = 0.5, n = 1.5 We = 1, n = 1.5 We = 5, n = 1.5

u u u


-0.8338 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000

-0.5111 -0.2621 -0.2622 -0.2705 -0.2707 -0.2763 -0.2764

-0.0269 0.05928 0.05926 -0.0717 -0.0718 -0.0797 -0.0799

100

0.5111 -0.434 -0.43398 -0.4409 -0.4411 -0.448 -0.4478

0.8338 -1.0000 -1.00000 -1.0000 -1.0000 -1.0000 -1.0000


values of We for shear thickening fluids (n > 0) when k = 2.5, * = 0.0061, Θ = 1.5,

32

x and = 0.4.

η n = 0.01, We = 7 n = 0.2, We = 7 n = 1.5, We = 7

u u u


-0.8338 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000

-0.5111 -0.3821 -0.3822 -0.2871 -0.2873 -0.2762 -0.2763

-0.0269 0.02104 0.02105 -0.1234 -0.1235 -0.07977 -0.07980

0.5111 -0.1354 -0.1356 -0.3026 -0.3027 -0.4485 -0.4487

0.8338 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000


values of n when k = 2.5, * = 0.0061, Θ = 1.5, 3

2x and = 0.4.

5.3 Results and interpretation

In this section, we present graphical illustrations of some important features of the

peristaltic motion such as flow characteristics, pumping characteristics and trapping

phenomenon for various values of the parameters k, n and We. The flow field produced by

101

peristaltic activity is illustrated in section 5.3.1. The pressure-flow relations are explained

in section 5.3.2. Section 5.3.3 includes a discussion of the trapping phenomenon.

5.3.1 Flow characteristics

In Fig. 5.1 the axial velocity 𝑢(𝜂) for both shear-thinning (panel (a)) and shear-thickening

(panel (b)) is plotted against η for some specific values of Weissenberg number (We). It is

observed that for shear-thinning fluids (n < 1) an increase in We shifts the maximum in

velocity profile above the centerline 𝜂 = 0. However, such an effect of We on 𝑢(𝜂) for

shear-thickening fluids is not obvious as evident from Fig. 5.1(b). Here slight shift of

maximum in the velocity 𝑢(𝜂) toward the lower wall is observed. Moreover, it is also

observed that for shear-thickening (dilatant) fluids, the amplitude of velocity profile 𝑢(𝜂)

increases by increasing We. It is further noted that for strong shear-thinning fluids (power-

law index near to zero) with larger relaxation times, the gradient of velocity sharply

changes near the channel walls indicating there the development of a thin boundary layer.

In the middle most region, the velocity varies linearly with 𝜂 . A similar behavior is

exhibited by the velocity profile as illustrated in Fig. 5.2, where once again it is evident

that the velocity profile 𝑢(𝜂) exhibits boundary layer character for strong shear-thinning

fluids and this feature diminishes as fluid behavior changes from shear-thinning to shear-

thickening i.e. the shearing effects extend to the whole flow domain from the walls. For

shear-thickening fluids no such boundary layer character of velocity is observed even for

large values of We. Thus it may be concluded that the existence of a thin boundary layer is

a typical characteristics of shear-thinning fluids with greater relaxation time. The

development of thin boundary layer for strong shear-thinning fluids with greater relaxation

102

time can also be explained by pure shear effects with the fact that largest shear rate should

occur at the channel walls. Another important aspect which is highlighted through Fig. 5.3

for somewhat weak shear-thinning fluids (n = 0.5) is that an increase in We counter-acts

the effects of curvature and makes the velocity profiles symmetric. Such effects of We on

velocity profiles have to the knowledge of the author’s not been reported hitherto in the

literature. The effects of dimensionless radius of curvature k on 𝑢(𝜂) are illustrated in Fig.

5.4. As expected, this figure reveals that for small values of k, the velocity profiles are not

symmetric about the centerline and peak velocity in these profiles appears below η = 0.

However by increasing k the profiles become symmetric about η = 0. Curvature therefore

demonstrates a significant effect on hydrodynamics.

5.3.2 Pumping characteristics

The pressure difference across one wavelength is plotted against time-averaged flow rate

in the laboratory frame Θ (= 𝑓 + 2) in Figs. 5.5 – 5.7. These figures illustrate the effects of

Weissenberg number (We), rheological power-law index (n) and curvature of channel (k),

respectively. The present study examines three regions of pressure flow rate plots;

peristaltic pumping region (Θ > 0, Δ𝑝 > 0), free pumping region (Θ > 0, Δ𝑝 = 0) and co-

pumping region (Θ > 0, Δ𝑝 < 0).

In the peristaltic pumping region, the pressure rise per wavelength resists the flow

due to peristalsis and magnitude of such resistance decreases by increasing We for

shear-thinning bio-fluids. Conversely the resistance increases with elevation in We

for shear-thickening fluids. However, the effects of We on pressure rise (Δ𝑝) in the

peristaltic pumping region are less prominent for shear-thickening fluids in

103

comparison with shear-thinning fluids. It is further observed that Δ𝑝 is an

increasing function of power-law index and curvature of the channel in pumping

region. Evidently therefore both rheological characteristic and channel curvature

(geometrical characteristic) exert a substantial influence on pumping.

The free pumping flux region the value of Θ for Δ𝑝 = 0 shows a decreasing trend

for shear-thinning fluids when viscoelastic effects become strong. However, it

shows the opposite behavior with We for shear-thickening fluids. Moreover, it also

decreases when fluid behavior changes from shear-thickening to shear-thinning or

when curvature effects become weak (straighter channel configurations).

In the co-pumping region the pressure assists the flow due to peristalsis and the

magnitude of such assistance increases by increasing n, whereas it decreases by

increasing We and k.

5.3.3 Trapping

The pattern of streamlines for different values of We for both shear-thinning and shear-

thickening fluids are illustrated in Figs. 5.8 and 5.9, respectively. Figure 5.8 illustrates the

streamlines patterns for shear-thinning fluid. It is observed that a circulating bolus of fluid

concentrated in the upper half of the channel exists for We = 0.1. By increasing We from

0.1 to 1.4, a small eddy appears near the lower wall of the channel. With a further increase

in We from 1.4 to 2.75, the eddy in the lower half of the channel gets bigger in size whist

the circulating bolus concentrated in the upper half is depleted. This shrinking behavior of

the bolus is not observed for shear-thickening fluid by increasing We (Fig. 5.9). However,

two small eddies appear near the lower wall of the channel which merge together into a

104

single eddy for large values of We. The effects of power-law index on streamlines pattern

are shown in Fig. 5.10 for We = 2. Figure 5.10 shows a circulating bolus of fluid

concentrated in the upper half of the channel and a single eddy near the lower channel wall

for a strong shear-thinning fluid. The size of this small eddy decreases while the circulating

bolus expands when n takes the values 0.25. Increasing n from 0.25 to 1 i.e. changing the

behavior of the bio-fluid from shear-thinning to Newtonian, results in the disappearance of

the small eddy from the lower half of the channel. A further increase in n does not affect

the circulating bolus of fluid but gives rise to two small eddies near the lower channel wall.

The two eddies merge together by further increasing the value of n. The effect of curvature

of the channel on streamline patterns is illustrated through Fig. 5.11. Figure 5.11(a) reveals

the presence of a circulating bolus of fluid concentrated in the upper part of the channel for

k = 2. By increasing k from 2 to 3.5 the bolus contracts and there appears another bolus of

fluid in the lower half of the channel. The size of this bolus increases and it becomes equal

the size of bolus in the upper half as k→∞. The effect of curvature of the channel on lower

trapping limit of Θ for n = 1.5 and We = 7 is shown in Fig. 5.12. It is found that the lower

trapping limit increases by increasing the curvature of the channel.

105

Fig. 5.1: Variation of We on 𝑢(𝜂) with k = 2.5, * = 0.0061, Θ = 1.5 and = 0.4.

Fig. 5.2: Variation of n on 𝑢(𝜂) when k = 2.5, *

= 0.0061, Θ = 1.5 and = 0.4.

Fig. 5.3: Variation of n on 𝑢(𝜂) when k = 2.5,

* = 0.0061, Θ = 1.5 and = 0.4.

Fig. 5.4: Variation of k on 𝑢(𝜂) when We = 0.5, n = 0.5, * = 0.0061, Θ = 1.5 and = 0.4.

106

Fig. 5.5: The pressure rise versus flow rate with k = 2, * = 0.0061 and = 0.4 for (a) n = 0.5

and (b) n = 1.5.

Fig. 5.6: The pressure rise versus flow rate for

We = 2, k = 2, * = 0.0061 and = 0.4.

Fig. 5.7: The pressure rise versus flow rate for

We = 5, n = 1.5, * = 0.0061 and = 0.4.

107

Fig. 5.8: Streamlines for (a) We = 0.1, (b) We = 1.4 and (c) We = 2.75, with k = 2. The

other parameters chosen are * = 0.0061, Θ = 1.5 and = 0.8 for n = 0.1.

Fig. 5.9: Streamlines for (a) We = 0.1, (b) We = 1, and (c) We = 2, with k = 2. The other

parameters chosen are * = 0.0061, Θ = 1.5 and = 0.8 for n = 1.5.

108

Fig. 5.10: Streamlines for ((a) shear thinning) (a) n = 0.01, (b) n = 0.25 and (c) Newtonian

(for We = 0 & n = 1) ((d) and (e) shear thickening) (d) n = 1.3, and (e) n = 2, with k = 2.

The other parameters chosen are * = 0.0061, Θ = 1.5 and = 0.8 for We = 2.

109

Fig. 5.11: Streamlines for (a) k = 2, (b) k = 3.5 and (c) k→∞. The other parameters chosen

are We = 2, n = 0.5, * = 0.0061, Θ = 1 and = 0.8.

Fig. 5.12: Streamlines for (a) k = 2 and (b) k→∞. The other parameters chosen are We = 7,

n = 1.5, * = 0.0061, Θ = 1 and = 0.8.

110


The simultaneous effects of curvature and non-Newtonian rheology on peristaltic flow in

a curved channel under long wavelength assumption have been investigated. A finite

difference scheme combined with an iterative scheme is employed for the numerical

solution of the transformed nonlinear differential equation describing the flow regime. The

key emerging parameters dictating the hydrodynamics are shown to be the channel

curvature (k), Weissenberg number (We) and rheological power-law index (n). Extensive

computations and flow visualization are presented.

It is found that non-Newtonian rheology and curvature of the channel significantly alter the

various features of the peristaltic motion. In fact, the flow velocity exhibits a boundary

layer character for strong shear-thinning fluid with larger relaxation time. As with the flow

velocity structure in a straight channel [62] (where except in a thin layer near the channel

walls the velocity over rest of the cross-section is uniform), the velocity in a curved channel

also varies sharply in thin layers near the channel walls. However, distinct from the flow

velocity in straight channel, the velocity in curved channel varies linearly over rest of the

cross-section. Moreover, for a fixed value of flow rate the pressure rise per wavelength in

the pumping region achieves higher values for a curved channel in comparison with a

straight channel. The trapping phenomenon is found to be largely dependent on channel

curvature and non-Newtonian characteristics. In fact the circulating bolus loses its

symmetry and the lower trapping limit increases for large values of curvature. As a result

of losing symmetry, the mixing of fluid due to circulation reduces in the lower half of the

channel. The present analysis also demonstrates that stable, robust computations are readily

achieved with the finite difference code employed and future investigations will examine

111

alternative rheological material models (e.g. viscoelastic fluids with normal stress

differences [115]) for peristaltic flows in curved channels. The present computations also

provide a good benchmark for commercial computational fluid dynamics codes (e.g.

ADINA-F [116, 117]) and it is envisaged that both approaches may reveal further intricate

flow structures in peristaltic hydrodynamic simulations.

112

Chapter 6

Existence of Hartmann boundary layer in

peristalsis through curved channel:

Asymptotic solution

The main objective of this chapter is to investigate boundary layer character of the velocity

in peristaltic flow of a Sisko fluid in a curved channel under the influence of strong imposed

radial magnetic field. The Sisko fluid model falls in the category of generalized Newtonian

fluid models. The constitutive equation of Sisko model is described in terms of three

material constants namely; power-law index (n), infinite shear rate viscosity (a) and

consistency index (b). This model is capable of predicting shear-thinning and shear-

thickening effects for n < 1 and n > 1, respectively. The equation governing the flow is first

derived under the assumptions of long wavelength and low Reynolds number, and then

made dimensionless by defining appropriate parameters. In dimensionless form it contains

three dimensionless parameters namely; generalized ratio of infinite-shear rate viscosity to

consistency index, power-law index and Hartmann number characterizing strength of the

imposed magnetic field. It is found that the governing equation of flow becomes singular

for large values of Hartmann number. Asymptotic solutions representing flow velocity at

large values of Hartmann number are reported for two specific values of power-law index

(namely 𝑛 = 1 and 𝑛 = 1/2) using singular perturbation technique. The flow velocity in

113

either case exhibits qualitatively similar behavior. In fact, it exhibits boundary layer

character i.e. it varies sharply in thin layer near the walls and varies linearly over rest of

the cross-sections. This is contrary to what that is observed for flow velocity in straight

channel (where except in thin layer near the channel walls the velocity over rest of the

cross-section is uniform). The estimates of boundary layer thickness at upper and lower

walls in either case are different. Moreover, the boundary layer thickness in either case is

found to be inversely proportional to the Hartmann number.

6.1 Mathematical formulation and rheological constitutive equations

The extra stress tensor which for Sisko model satisfies [55]

-1

,n

a b

S (6.1)

in the above equation a is the infinite shear rate viscosity and b is the consistency index.

The Sisko model reduces to power-law model by setting a = 0 while the corresponding

tensor of Newtonian model can be obtained by either putting b = 0, or a = 0 and n = 1

simultaneously. The model (6.1) is capable of predicting shear-thinning and shear-

thickening effects for n < 1 and n > 1, respectively.

In components form, Eq. (6.1) gives

1

*2 ,n

x x

vS a

(6.2a)

1

*1

,n

x

uu k vS a

k k

(6.2b)

114

1

*2 ,n k u v

S ak x k

(6.2c)

22 2

11 2 22 2 .

2

uv u k v k u v

k k k x k

(6.3)

The parameter *10

01

, na ba

ac

is the generalized ratio of infinite-

shear rate viscosity to the consistency index in a Sisko fluid if 1n . In the case of

*1, 0n a denotes a viscous Newtonian fluid, whilst * 0a describes a purely power-law

fluid. Moreover, the case *1, 0n a corresponds to Newtonian fluid with viscosity

*1 a . Using the definition of stream function and employing long wavelength and low

Reynolds number approximation, we get

0,xxS (6.4a)

21

*

2

11 ,

n

xS ak

(6.4b)

0,S (6.4c)

22

2

11 .

k

(6.5)

Insertion of the expression of xS from Eq. (6.4(b)) into Eq. (2.31) yields

22

2 *

2

1

21 11

n

pk a

x k k k

115

2 2

2

11 1 0.

Ha k

k k

(6.6)

Elimination of pressure between Eqs. (2.30) and (6.6) gives

12 22 2

2 *

2 2

1 1 11 1

n

k ak k k

2 2

1 0.Ha k

k

(6.7)

The solution of the above equation is required to satisfy the boundary conditions (2.35) and

(2.36).

6.2 Asymptotic solution

Due to the nonlinear nature of Eq. (6.7), an exact solution is difficult to find. However, it

is possible to obtain an asymptotic solution of this equation for large values of Hartmann

number. To this end, we rewrite Eq. (6.7) in the form

12 2

2 *

2 2

1 1 11 1

n

k ak k k

2 2

1 0,Ha k

k

(6.8)

where 1/ and 1.Ha

The above equation in terms of small parameter ε becomes

116

12 2 2

2 *

2 2

1 11 1

n

k ak k k

2

1 0.k

k

(6.9)

Since the small parameter ε multiplies with highest derivative in Eq. (6.9), therefore Eq.

(6.9) subject to boundary conditions (2.35) and (2.36) constitute a singular perturbation

problem. The singular perturbation problems in fluid mechanics usually exhibit boundary

layer phenomena [55]. In such problems the greatest gradients in flow velocity are confined

in a thin layer (boundary layer) near the solid surface. Therefore, it is expected that solution

of Eq. (6.7) subject to boundary conditions (2.35) and (2.36) exhibits boundary character

for large values of Hartmann number. The method of matched asymptotic expansion is

used to obtain the solution. In this method an inner solution at the location of two walls

and an outer solution away from the wall while remaining inside the channel are obtained.

Then by defining some intermediate variable each inner and outer solution are matched at

the upper and lower walls to determine the unknown constants. Finally based on these

solutions a composite solution valid for small values of ε is expressed. We refer the reader

to Ref. [55] by Asghar et al. and book by Bush [118] for procedural details.

6.2.1 The case for n = 1 and 𝒂∗ ≠ 𝟎

For n = 1 and 𝑎∗ ≠ 0, Eq. (6.9) takes the following form

2 * 2 22

2

( 1) 11 1 0.

a kk

k k k

(6.10)

117

6.2.1.1 Inner solutions

To find inner solution at 1,h let us introduce the stretched variable

1 .h

s

(6.11)

In new variable, Eq. (6.10) becomes

4 3 2 22 * 4 3

1 4 3 2

1

1 2in in in

a h k ss s sh k s

2 2 2 2

2 2 22

11 1 1

1in in ink k

s s sh k sh k s h k s h k s

2

2

1

0,k

h k s

(6.12)

where in is the solution satisfying the inner boundary condition at 1.h

By taking only dominant terms in Eq. (6.12), we can write

4 2 2 2* 2 4

1 4 2

1

1 0.in ink

a h ks h k s

(6.13)

From Eq. (6.13), is to be determined by the principle of the least degeneracy. In order

to balance the terms in Eq. (6.13), we require = 1, and thus

4 22

14 20,

in in

s s

(6.14)

where 1

*1

1.

1

k

h ka

The inner solution in is subject to the following boundary conditions at 1h ,

, 0, at 0.2

inin q

ss

(6.15)

118

Now we proceed for a two terms perturbation solution of Eq. (6.14). To this end we write

0 1( ) ( ) ( ).in in ins s s (6.16)

Substituting Eq. (6.16) into Eq. (6.14) and (6.15), we obtain the leading order system as

follows:

4 220 0

14 20,

in in

s s

(6.17)

00 , 0, at 0.

2

inin q

ss

(6.18)

The solution of above system is

1 1

0 0 1 0 1( ) 1 1 .2

s sin qs C e s D e s (6.19)

We choose the constant 0C = 0 to avoid the exponential growth in above solution. Thus

we get

1

0 0 1( ) 1 .2

sin qs D e s (6.20)

At order ε we have the following system

4 221 1

14 20,

in in

s s

(6.21)

11 0, 1, at 0.

inin s

s

(6.22)

It can be easily shown that the solution of above equation is

1 1

1 1 1 1 1( ) 1 1 .s sin s s C e s D e s

(6.23)

119

Again we choose 1C 0 and write Eq. (6.23) in the form

1

1 1 1( ) 1 .sin s s D e s

(6.24)

Substitution of Eqs. (6.20) and (6.24) into Eq. (6.16) yield the two terms inner solution at

1h as follows:

1 1

2term 0 1 1 1( ) 1 1 ,2

s sin qs D e s s D e s (6.25)

where 𝐷0 and 𝐷1 are unknown constants to be determined by the matching technique.

To find the inner solution at 2 ,h let us introduce the following stretched variable

2 .r

h

(6.26)

Performing the same procedure as above, the two terms inner solution at 2h is

2 2

2term 0 2 1 2( ) 1 1 ,2

IN qH e H e

(6.27)

with 2

*2

1,

1

k

h ka

and 0H and 1H are unknown constants to be determined by the matching technique.

6.2.1.2 Outer solution

For the outer solution, we only choose terms independent of ε in Eq. (6.10) and write

2

2 22

1 1 10.

out out

k k k

(6.28)

120

To find a two terms solution of above equation we expand

0 1( ) ( ) ( ).out out out (6.29)

Substituting Eq. (6.29) into Eq. (6.28) and solving the resulting systems, we get

2 2

0 0 1 1( ) ,2 2

out k a b k c d

(6.30)

as two terms outer solution. In Eq. (6.30) 𝑎0, 𝑏0, 𝑐1, and 𝑑1 are unknown constants to be

determined by higher-order matching of two inner expansions and one outer expansion.

The higher-order matching of the inner solution at 1h and the outer solution is done by

defining an intermediate region of O and an intermediate parameter 'z by

' 1z ,h

(6.31)

and then converting the inner and outer solutions given by Eqs. (6.25) and (6.30) in the

intermediate parameter 'z and discarding the terms higher than .O In this way, we

get

2 2int

'1 11 1 0 0 1 1 1 1 0

1

z 12 2

,

out h hh kh a b kh c d k h a

O

(6.32)

int

' 1 '

0 1 1 1 11 z z 1 .2

in qD D D (6.33)

The above two solutions are matched by comparing the various power of ε to get

2

11 1 0 0 0 0

21 1

1 0 1 1 1 1

1

, 0,2 2

, .2

h qh kh a b D D

k h hD a kh c d D

(6.34)

121

The above matching procedure is repeated by taking inner solutions at 2h and thus

we write

2 .h

w

(6.35)

In the same way as done before, we convert the inner and outer solutions given by Eqs.

(6.32) and (6.35) in the intermediate parameter w and discard the terms higher than

.O Thus we get

2 2int

2 22 2 0 0 2 1 1 2 0

1

12 2

,

out h hh kh a b kh c d w k h a

O

(6.36)

1

int1

0 1

2 2

1 1 .2

IN kHq kwH w H

k h k h

(6.37)

Again, the above two solutions are matched by comparing the various power of ε to get

2

22 2 0 0 0 0

22 2

0 1 2 1 1 1

2

, 0,2 2

, .2

h qh kh a b H H

k h ha H kh c d H

(6.38)

Solving Eqs. (6.34) and (6.38), we find

2

20 0 2 2

1 2 1 2

1 2

1

1 2 1 2 1 2

21 1 21

1 1

1 1 2 1 2 1 2

1

1

1 1 2

1 11 , 1 ,

2 2

11 ,

1 11 ,

2

1

hq q qa b h kh

h h k h h k

k h k h qc

h h k h h

k h k h k hh qd kh

h h h h k

k h qD

h h

2

1

2 1 2

1 1, 1 .

k h qH

k h h k

(6.39)

122

The two terms composite solution is defined by

int int

out in in IN IN

composite ( ) ( ) ,s (6.40)

which is view of Eqs. (6.30), (6.25), (6.33), (6.27) and (6.37) becomes

2212

composite 2 2

1 2 1

11

2 2 2

k hhq qh k kh

h h k

221 2 1 21

1

1 2 1 2 1 2 1 2

1 1

2 2

k h k h k h k hhk kh

h h k h h k

1 21 21 2

1 2 1 2

11 .

h hk h k h q

e eh h k

(6.41)

The presence of exponential terms in Eq. (6.41) clearly shows that the character of solution

(6.41) is of boundary layer type. Eq. (6.41) further suggests that the boundary layer growth

is of *

11 /O a h k kHa at the upper wall while it is of *

21 /O a h k kHa

at the lower wall. This indicates that boundary layer thickness is inversely proportional to

Hartmann number whilst it is directly proportional to 𝑎∗. This boundary layer character of

solution is also confirmed through Figs. 6.1 and 6.2. In such case the variation in velocity

is confined in thin layers near the walls.

6.2.1.3 The case for 𝒏 = 𝟏𝟐⁄ and 𝒂∗ ≠ 𝟎

In this subsection, we shall find the solution of governing equation (6.9) for shear-thinning

fluid (n < 1) and large values of Hartmann number. To this end, we rewrite Eq. (6.9) as

follows:

123

2 2 2

2 *

2 2

1 11 1

( )

n

k ak k k

2

1 0.( )

k

k

(6.42)

An order of magnitude analysis reveals that Eq. (6.42) is tractable for 𝑛 < 1 which

corresponds to the case of shear-thinning fluids and as a particular choice, we take 𝑛 =

1 2⁄ . Now following the similar procedure as given in subsection 6.2.1.1, the two terms

inner solution at 1h and 2h is given by

3 3

2term 0 3 1 3( ) 1 12

s sin qs M e s s M e s

3

0 32*

3

41 ,

3 2

sM se

a

(6.43)

4 4

2term 0 4 1 4( ) 1 12

IN qU e U e

4

0 42*

4

41 ,

3 2

Ue

a

(6.44)

where 3 4

* *1 2

1 1,

k k

h k h ka a

, 0 1 0, ,M M U and 1U are unknown constants

and their values are found by using the matching procedure.

The outer solution of equation (6.42) will remain same as given in (6.30) i.e.

2 2

0 0 1 1( ) .2 2

out k a b k c d

(6.45)

In order to find unknown constants, we rewrite Eqs. (6.43) and (6.45) at 1h in terms of

intermediate parameter 'z for higher-order matching:

124

2 2int

'1 11 1 0 0 1 1 1 1 0

1

z 12 2

,

out h hh kh a b kh c d k h a

O

(6.46)

int

0' 1 '

0 3 3 1 *

3

21 z z 1

2 3

inMq

M Ma

0

1 *

4

4.

3

MM

a

(6.47)

Now matching various powers of in Eqs. (6.46) and (6.47), we get

2

11 1 0 0 0 0

21 1

1 0 1 1 1 1

3

, 0,2 2

, .2

h qh kh a b M M

k h hM a kh c d M

(6.48)

Similarly, we obtain some unknown constants at 2h by using again higher-order

matching:

2

22 2 0 0 0 0

22 2

0 1 2 1 1 1

4

, 0,2 2

, .2

h qh kh a b U U

k h ha U kh c d U

(6.49)

From Eqs. (6.48) and (6.49), we get

2

20 0 2 2

1 2 1 2

1 2

1 2

1 2 3 4 1 2

21 1 21

1 1

3 1 2 3 4 1 2

1

1

3 1 2

1 11 , 1 ,

2 2

11 ,

1 11 ,

2

1

hq q qa b h kh

h h k h h k

k h k h qc

h h k h h

k h k h k hh qd kh

h h k h h k

k h qD

h h

2

1

4 1 2

1 1, 1 .

k h qH

k h h k

(6.50)

The composite solution of Eq. (6.42) is defined by

125

int int

out in in IN IN

composite ( ) ( ) ,s (6.51)

which is view of Eqs. (6.30), (6.32), (6.35), (6.38) and (6.43) yields

1 23 4

22

2composite 2 2

1 2

221 1 21

1

3 1 2 3 4

1 2

3 4 1 2

11

2 2 2

1

2 2

1

h h

hq qh k kh

h h k

k h k h k hhk kh

h h k

k h k h qe e

h h

1.

k

(6.52)

It is remarked here that asymptotic solution of Eq. (6.42) is not possible for shear-

thickening case (𝑛 > 1), in the sense that corresponding nonlinear equation can be solved

but its solution does not satisfy the required boundary conditions. This fact has also been

pointed out Asghar et al. [55] in their study on peristaltic flow of Sisko fluid in an

asymmetric channel. The non-existence of boundary layer solution in case of shear-

thickening fluid for large Hartmann number is because of the fact that in flow of such fluids

the viscous force is of comparable magnitude to the magnetic force. The solution given by

Eq. (6.52) is clearly of boundary layer type due to the presence of exponential terms. In

this case Eq. (6.52) suggests that the boundary layer thickness at upper and lower walls is

of *

1 /O a h k kHa and *

2 / ,O a h k kHa respectively. The qualitative

behavior of solution predicted on the basis of (6.52) is also confirmed through graphical

illustration given in Fig. 6.3. It is further observed through Figs. 6.2 – 6.4 that the structure

of flow velocity in a curved channel is markedly different than that in a straight channel.

As with the flow velocity structure in a straight channel [55] (where except in a thin layers

near the channel walls the velocity over rest of the cross-section is uniform), the velocity

in a curved channel also varies sharply in thin layers near the channel walls. However,

126

distinct from the flow velocity in the straight channel, the velocity in a curved channel

varies linearly over rest of the cross-section.

127

Fig. 6.1: Variation of velocity profile 𝑢(𝜂) for

k = 3, Θ = 1.5, n = 1, 𝑎∗ = 1 and x = π.

Fig. 6.2: Variation of velocity profile u(𝜂) for

k = 2.5, Θ = 1.5, Ha = 1, n = 1 and x = π.

Fig. 6.3: Variation of 𝑢(𝜂) for k = 2.5, Θ =

1.5, 𝑎∗ = 1, n = 1/2 and x = π.

Fig. 6.4: Variation of 𝑢(𝜂) for 𝐻𝑎 = 50,

Θ = 1.5, 𝑎∗ = 1, n = 1/2 and x = π.

128


A study is performed to investigate the effect of applied magnetic field on peristaltic flow

of Sisko fluid through a curved channel. The asymptotic formulae of stream function for

large Hartmann number are obtained using singular perturbation method for two cases

namely; *1, 0n a and *1 , 0,2

n a where n is the power-law index and *a is the

generalized ratio of infinite shear rate viscosity (a) to the consistency index (b). The former

case corresponds to viscous fluid with viscosity *1 a while later case describes shear-

thinning fluids with viscosity 1

* ,n

a

where is the magnitude of rate of

deformation tensor. In the former case, the boundary layer thickness is of

*

11 /O a h k kHa at the upper wall whilst at lower wall it is of

*

21 /O a h k kHa . Similarly, in the second case the layer thickness at upper and

lower walls is of *

1 /O a h k kHa and *

2 / ,O a h k kHa respectively. The

following important conclusions are drawn from this study.

In the first case *1, 0 ,n a a thin boundary layer exists at the channel

walls for large values of Hartmann number or small values of *a .

Moreover, the thickness of the boundary layer is inversely proportional to

the Hartmann number. However, it shows increasing trend with *a .

In the second case *1/ 2, 0 ,n a the qualitative behavior of flow

velocity is similar to corresponding behavior of flow velocity in case 1.

129

However, the estimates of boundary layer thickness in either case are

different.

The structure of flow velocity in a curved channel is quite different from

that in a straight channel. In a straight channel, except in thin layers near the

walls, the velocity over rest of the cross-section is uniform. The analytical

estimates of boundary layer thickness reveal that in a straight channel, the

thickness of boundary layers at the upper and lower walls is same. On the

contrary, in a curved channel the boundary layer thickness at the upper wall

is greater than that at the lower wall. This is because that in a curved channel

boundary layer thickness at upper and lower walls depends upon both the

curvature of channel and the axial coordinate x. For instance, the choice of

parameters k = 2.5, * 1, 100, 0a Ha x yields 0.014 and 0.006,

respectively as thickness of boundary layer at upper and lower walls.

The large values of the dimensionless radius of curvature produce velocity

profile exhibiting boundary layer similar to that in a straight channel.

The formation of boundary layer for large values of Hartmann number is

justified on the following grounds. In the present flow situation the

magnetic force acts as a resistance to the flow and its magnitude is

proportional to the transverse velocity (See Eq. (2.12)), hence amplitude of

flow near the channel center is suppressed. To maintain the given flow rate,

the relatively small velocity near boundaries will increase. A combination

of both effects leads to the formation of boundary layer at the channel walls.

Due to the symmetry of velocity about the centerline, the thickness of

130

boundary layer at either wall is same in a straight channel. Moreover, the

suppression of velocity amplitude due to magnetic force is uniform near the

center in a straight channel and that is why the middle-most region out of

the boundary layers moves with a uniform velocity. However, due to

asymmetry of the velocity in the curved channel the suppression of velocity

amplitude due to magnetic force is not uniform in the region outside

boundary layers. Therefore, the velocity in this region varies linearly with

radial distance. Because of asymmetry in the velocity, the boundary layer

thickness at either wall is also different in a curved channel.

It is important to mention that boundary layer phenomenon in peristaltic

flow of a Sisko fluid for large values of Hartmann number was already

reported by Wang et al. [120] and Asghar et al. [55] in a straight channel.

No other study is a variable in the literature highlighting the boundary layer

phenomena in peristalsis. In fact, the present study extends the results

reported in above investigations for a curved channel. It is worth mentioning

that our results in the limit when k are compatible with the existing

results of Wang et al. [120] and Asghar et al. [55].

131

Chapter 7

Simulations of peristaltic slip-flow of hydro-

magnetic bio-fluid in a curved channel

In the present chapter the influence of slip and magnetic field on transport characteristics

of a bio-fluid are analyzed in a curved channel. The problem is modeled in curvilinear

coordinate system under the assumption that the wavelength of the peristaltic wave is larger

in magnitude compared to the width of the channel. The resulting nonlinear boundary value

problem (BVP) is solved using an implicit finite difference method (FDM). The flow

velocity, pressure rise per wavelength and stream function are illustrated through graphs

for various values of rheological and geometrical parameters of the problem. The study

reveals that a thin boundary layer exists at the channel wall for strong magnetic field.

Moreover, small values of Weissenberg number counteract the curvature and make the

velocity profile symmetric. It is also observed that pressure rise per wavelength in pumping

region increases (decreases) by increasing magnetic field, Weissenberg number and

curvature of the channel (slip parameter).


This chapter differs from previous chapter in two aspects. In contrast to previous chapter

the non-Newtonian model chosen here is Williamson model [58]. Moreover, here the

132

analysis is carried out in presence of slip at the channel wall. Therefore, the only change in

governing equation (2.31) comes from the definition of .xS To derive the expression of

xS , we start with the definition of extra stress S for Williamson fluid [58] i.e.

1

0 1 .xS

(7.1)

In the subsequent analysis it is assumed that 0. Thus the extra stress tensor becomes

1

0 1 .xS

(7.2)

In components form, Eq. (7.2) gives

1

1

1

2 1 , ( )

11 , ( )

2 1 , ( )

xx

x

vS We a

uu k vS We b

k k

k u vS We c

k x k

(7.3)

where

22 2

211 2 2

2 2 .2

uv u k v k u v

k k k x k

(7.4)

The above expressions after using the definition of stream function and employing long

wavelength approximation reduce to

12 2

2 2

22

2

2

0, ( )

1 11 1 1 , ( )

0, ( )

11 ( )

xx

x

S a

S We bk k

S c

dk

(7.5)

133

Inserting the expression of xS from Eq. (7.5(b)) into Eq. (2.27) and eliminating

pressure, one gets

12 2

2

2 2

2

1 1 11 1 1

( )

1 0,( )

k Wek k k

Ha k

k

(7.6)

Eq. (7.6) is subject to slip conditions at the walls i.e.

1

2

1, at 1 sin , ( )

1, at 1 sin , ( )

x

x

S h x a

S h x b

(7.7)

where β is the dimensionless slip parameter. It is interesting to note that due to slip

condition at the walls, the dimensionless radius of curvature k comes in the boundary

condition and thus makes the velocity at the wall a function of curvature of the channel and

rheological parameters of the fluid. The derivation of boundary condition (7.7(a)) is

illustrated as under.

The dimensional boundary condition on U in fixed frame is

RX

lU S

at 1.R H (7.8)

In the wave frame the above condition becomes

rx

lu c S

at 1.r h (7.9)

Using the dimensionless variables (2.22) and dropping the bars, we get

1 xu S at 1.r h (7.10)

134

Now, the condition (7.7(a)) follows from (7.10) after using the definition of stream

function. The boundary conditions (2.36) will remain the same. The solution of Eq. (7.6)

subject to boundary conditions (7.7) and (2.36) is obtained using FDM.


In this section, the quantities of interest such as flow velocity, pressure rise per wavelength

and trapping phenomena are examined numerically for various values of involved

parameter. Figure 7.1 illustrates the effects of Weissenberg number on flow velocity by

keeping the other parameters fixed. It is interesting to note that for We = 0 (Newtonian

fluid), the velocity profile is not symmetric about η = 0 and maximum in it lies below η =

0. However, the maximum shifts toward the upper wall of the channel and amplitude of

the flow velocity is enhanced with increasing We. The asymmetric profile of velocity for

We = 0 and k = 3 indicates that the effect of curvature is dominant in Newtonian fluid. Such

effect of curvature diminishes and velocity profile become symmetric by slightly

increasing We. In such a case there is a balance between viscoelastic and curvature effects.

However, for large values of We the viscoelastic effects counteract the curvature effects

resulting in asymmetric velocity profile with maxima lying near the upper wall. The effects

of Hartmann number on flow velocity can be observed through Fig. 7.2. Figures 7.2 shows

that the magnitude of flow velocity decreases by increasing Hartmann number. For strong

magnetic field a thin boundary layer exist at both walls of the channel. Moreover, the peak

in velocity profile, which was originally in the lower half of the channel for small values

of Hartmann number, shifts in the upper half of the channel for large values of Hartmann

number. The boundary layer character of velocity for strong magnetic field in a straight

135

channel for peristaltic motion of a shear-thinning fluid is already highlighted through

singular perturbation technique by Asghar et al. [55]. But for a curved channel no such

results are reported yet. The results of Asghar et al. [55] indicate that for strong magnetic

field two thin sharp boundary layers are formed near both the walls. The middle region

outside both boundary layer moves with constant speed a though it were a plug flow. Our

results for a curved channel also indicate the formation of two thin boundary layer at the

channel wall. However, the middle most region outside the boundary layer does not move

with a constant speed rather the speed of fluid in this region increases linearly from a

maximum at the lower wall to a maximum at the upper wall. Figure 7.3 demonstrates the

effects of curvature on flow velocity. It is evident from this figure that for small values of

k i.e. for large channel curvature, the velocity profile becomes asymmetric. Moreover, the

peak in velocity profile shifts in the lower half of the channel. In the limiting case when k

→ ∞, the results of the straight channel are recovered. The present theoretical model of

peristalsis also bears the potential to investigate the effects of slip at the channel wall on

various flow characteristics. We have plotted Fig. 7.4 to illustrate such effects on velocity

profile. This figure reveals that velocity at both walls increases by increasing slip

parameter. Figure 7.5 demonstrates the effects of ϕ on velocity profile. It shows that not

only the amplitude of the flow velocity decreases but also the contraction of the peristaltic

wall is observed with increasing .

An important feature of peristaltic motion is pumping against the pressure rise per

wavelength. Figures 7.6 – 7.10 are plotted to highlight the effects of Weissenberg number,

Hartmann number, slip parameter, amplitude ratio and curvature of the channel on pressure

rise per wavelength (Δp). Figure 7.6 shows that for a given value of flow rate, Δp in the

136

pumping region (Δ𝑝 ≥ 0, Θ ≥ 0 ) increases by increasing Weissenberg number. Thus

shear-thinning fluid with greater relaxation time offers more resistance to peristaltic

activity in comparison with Newtonian fluid. The effect of Hartmann number on Δp in

pumping region is similar to the effect of Weissenberg number (Fig. 7.7). However, an

increase in slip parameter significantly reduces the pressure rise per wavelength in the

pumping region (Fig. 7.8). This observation may have certain implications in situations

where it is desired to reduce the magnitude of resistance offered by pressure to the

peristaltic motion. A mechanism may be devised to produce fluid slippage at the walls of

channel which could then be utilized to reduce the pressure rise wavelength. Figure 7.9

illustrates the effects of curvature of the channel on Δp. It is clear from this figure that Δp

in pumping region for a curved channel is significantly lesser in magnitude than that for a

straight channel. Figures 7.10 shows that the effects of amplitude ratio on Δp in

pumping region are similar to the effects of Weissenberg number and Hartmann number.

Figures 7.11 – 7.15 are plotted to see the influence of Weissenberg number, Hartmann

number, slip parameter, amplitude ratio and curvature of the channel on trapping

phenomenon. Figure 7.11 reveals that a bolus of fluid concentrated in the upper half of the

channel exists for We = 0. The size of this bolus reduces and it shrinks by increasing We.

There is no indication of splitting of this bolus into two halves by increasing We. However,

the splitting of bolus into two halves is observed by increasing Hartmann number as evident

from Fig. 7.12. The panel (c) of Fig. 7.12 demonstrates the splitting of bolus, which was

originally concentrated in the upper half of the channel, into two halves. A further increase

in Hartmann number increases the size of bolus in lower half of the channel. Moreover,

137

large values of Hartmann number confirm the appearance of a squeezed bolus concentrated

in the lower half of the channel.

The effects of slip parameter on trapping phenomenon are shown in Fig. 7.13. It is observed

that the size of trapped bolus reduces by increasing the slip parameter. The splitting of

bolus into two halves is also seen as a result of increasing slip parameter. It is interesting

to note that the trapped bolus disappears for large values of slip parameter. It is therefore

concluded that both Hartmann number and slip parameter have significant effects on the

trapping phenomena. In fact, the slip parameter reduces the circulation of the bolus while

magnetic parameter shifts the concentration of the bolus from the upper half of the channel

to the lower one. Figure 7.14 depicts that the symmetry of the circulating bolus about the

centerline owes to large value of the parameter k. For small values of k i.e. for greater

channel curvature, the bolus becomes asymmetric and it circulates about a point in the

upper half of the channel. Figure 7.15 shows the streamlines patterns for different values

of . Panel (a) shows a circulating bolus of fluid concentrated in the upper part of the

channel for = 0.2. It is observed that both size and circulation of bolus increase with

increasing from 0.2 to 0.7.

138

Fig. 7.1: Variation of u for different values

of We with k = 3, Ha = 0, β = 0, Θ = 1.5 and

= 0.4.


Ha with k = 3, We = 0.4, β = 0, Θ = 0.5 and =0.4.


of k with We=0.5, Ha=0, β=0,Θ = 0.5 and =0.4.

Fig. 7.4: Variation of u for different values of β

with k =2, We = 0.5, Ha = 0, Θ =1.5 and = 0.4.

Fig. 7.5: Variation of u for different values of with k =3, We = 0.5, Ha = 0, Θ =1.5 and β =0.

139


of We with k = 2, Ha = 0, β = 0 and = 0.4.

Fig. 7.7: Variation of 𝛥𝑝 for different values of Ha

with k = 2, We = 0.1, β = 0.01 and = 0.4.


of β with k = 2, Ha = 0.01, We = 0.01 and

= 0.4.

Fig. 7.9: Variation of 𝛥𝑝 for different values of k

with Ha = 0.01, We = 2, β = 0 and = 0.4.

140

Fig. 7.11: Streamlines for different values of Weissenberg number We (0, 0.125, 0.25, 1.5) with k = 2,

Ha = 0, β = 0, Θ = 1.5 and = 0.8.

Fig. 7.12: Streamlines for different values of Hartmann number Ha (0, 0.4, 1.1, 2.5) with

k = 2, We = 0 .1, β = 0, Θ = 1.5 and = 0.8.

141

Fig. 7.13: Streamlines for different values of slip parameter β (0, 0.5, 1.1, 3) with k = 2, Ha

= 0, We = 0.1, Θ = 1.5 and = 0.8.

142

Fig. 7.14: Streamlines for different values of k (2, 3.5, 5, ∞) with We = 0.1, Ha = 0, β = 0,

Θ = 1.5 and = 0.8.

Fig. 7.15: Streamlines for different values of (0.2, 0.4, 0.6, 0.7) with We = 0.1, Ha = 0, β = 0,

Θ = 1.5 and k = 2.

143


A theoretical model capable of illustrating the effects of fluid rheology, applied magnetic

field, amplitude ratio and slip at walls on peristaltic motion in a curved channel is

presented. The governing equation of the model is simulated by a suitable finite difference

technique for various value of the involved parameters. The main findings of the present

study are

For large values of applied magnetic field, the boundary layer character exhibited

by the velocity profile in a curved channel is found to be markedly different than

that in a straight channel.

The asymmetric velocity profile in a curved channel becomes symmetric by

increasing We. In fact for given value of k, a certain value of We could be found

which neutralizes the effects of curvature and produces a symmetric velocity

profile. For large values of We, the viscoelastic effects dominate and velocity

profile again become asymmetric with peak appearing in the upper half of the

channel.

The velocity at both walls of the channel increases by increasing slip parameter.

The amplitude of the flow velocity is suppressed with increasing the amplitude of

the peristaltic wave.

The pressure rise per wavelength increases by increasing Weissenberg number,

Hartmann number, amplitude ratio and curvature of the channel. However, it

reduces significantly by increasing slip parameter.

A circulating bolus of fluid concentrated in upper half of the channel exists in a

curved channel.

144

The size of trapped bolus decreases (increases) by increasing Weissenberg number

(amplitude ratio).

The bolus of fluid concentrated in the upper half of the channel shifts in the lower

half for large value of the Hartmann number.

The trapped bolus of fluid disappears for large value of the slip parameter.

145

Chapter 8

Peristaltic flow and heat transfer in a curved

channel for an Oldroyd 8–constant fluid

In this chapter, a study is carried out to investigate the heat transfer characteristics in

peristaltic flow of an Oldroyd 8-constant in a curved channel when inertial and streamline-

curvature effects are negligible. The solution of resulting nonlinear governing equations is

obtained using finite difference method (FDM) combined with an iterative scheme. The

impacts of physical parameters on the flow and heat transfer characteristics is investigated

in detail. Particular attention is given to explain the pumping and trapping phenomena in

detail. A comparative study between curved and straight channels is also made. It is found

that, the rate of heat transfer increases with increasing the curvature of the channel. The

current two-dimensional analysis is applicable in bio-fluid mechanics, industrial fluid

mechanics and some of the engineering fields.

8.1 Mathematical formulation

Consider the non-isothermal flow of an Oldroyd 8-constant fluid in a curved channel. The

geometry of the flow problem and coordinate systems are already explained in Fig. 2.1. In

contrast to previous chapters, the upper and lower walls of the channel are maintained at

146

constant temperature 1T and 0 ,T respectively. The determine equation for stream function

under long wavelength and low Reynolds number assumptions is given by Eq. (2.31).

Where the expression of xS can be derived from the constitutive equation of an Oldroyd

8-constant given by

25 3 61 2 4

27

tr tr2 2 2

tr .2

D D

Dt Dt

SS S S S S I

I

(8.1)

In Eq. (8.1), i 1,3,5,6i are relaxation times while i 2,4,7i are retardation times.

It should be pointed out that the model (8.1) includes the Oldroyd 6-constant model (for

i = 0, i = 6 – 7), the Oldroyd 4-constant model (for i = 0, i = 4 – 7), the Oldroyd-B

model (for i = 0, i = 3 – 7), the Maxwell model (for i = 0, i = 2 – 7) and the Newtonian

fluid model (for i = 0, i = 1 – 7) as the limiting cases.

Now performing the similar analysis is carried out in previous chapters, one get

22

1 2 2

2 22

2 2

11 1

11 ,

11 1

x

kS

k

k

(8.2)

where 1 and 2 are parametric constants.

Substitution of xS in Eq. (2.31) and elimination of pressure gives

147

22

1 2 22

2 22

2 2

11 2 1

1 11 0.

( ) 11 2 1

kk

k k

k

(8.3)

For the two-dimensional flow under consideration, it is instructive to define temperature

field in the form

, , .T T X R t (8.4)

The general form of energy equation is given by Eq. (1.39).

In view of Eqs. (8.1), (1.40) and (1.42), Eq. (1.39) gives

2* * 2

* *

* * 2*

1+ + =

+p

T T R U T T R Tc V k R R

t R R R X R R R R XR R

* *

* * * *+ .RR RX XX

V U R V U V R US S S

R R R R X R R R R R R X

(8.5)

To switch from fixed frame ,R X to wave frame ,r x , we employ the transformations

defined in Eq. (2.22) and write Eq. (8.5) in wave frame as given below:

2* * 2* *

* * 2*

1+ + =

+p

R u cT T T T R Tc c v k r R

x r r R x r r r R xr R

* *

* * * *+ ,rr rx xx

u cv u R v v R uS S S

r r r R x r R r R r R x

(8.6)

where the same symbol T is used for the temperature in the wave frame.

148

In view of dimensionless quantities defined in Eq. (2.22) the above equation becomes

2 22

* 2

1+ + = +

+ + +

pc k u c kc v k

k x k x k k x

+ ,

+ + + +x xx

u cv u k v v k uBr S S S

k x k k k x

(8.7)

where, 1

0 1

T TT T

and

2

*

0 1

cBrk T T

. Eq. (8.7) after using long

wavelength and low Reynolds number approximations reduce to

2

2

1 11 0.

( )xk BrS

k k

(8.8)

In summary for the problem under consideration, we have to solve

22

1 2 22

2 22

2 2

11 1

1 11 0,

( ) 11 1

kk

k k

k

(8.9)

22

1 2

22

2 2

11 1

1 11

( ) 11 1

kk Br

k k

k

22

2

11 0.

k

(8.10)

It is remarked that in the limiting case when 𝜆4−7 → 0, the form of Eq. (8.9) does not alter

except slight modifications in the expression of 𝛼1 and 𝛼2.

149

Eq. (8.9) and (8.10) are subject to the following conditions at the walls [97 – 101] i.e.

1

2

, 1, 0, at 1 sin , ( )2

, 1, 1, at 1 sin , ( )2

qh x a

qh x b

(8.11)

where 1

1

ba

is the amplitude ratio.

The heat transfer coefficient at both walls [100] is defined by

, 1,2.

i

i

h

hz i

x

(8.12)


In this section, we present graphical results illustrating the impact of material parameters,

curvature parameter and Brinkman number on flow and heat transfer characteristics. The

graphical results are produced by solving the nonlinear differential equations (8.9) and

(8.10) subject to boundary conditions (8.11) using an implicit finite difference method. The

details of this method can be found in Refs. [62, 64].

The axial velocity u at a cross-section 2

x for different values of the rheological

parameter 𝛼1 is plotted in Fig. 8.1. This figure indicates an increase in the axial velocity

by increasing 𝛼1 in the middle most region. However, the axial velocity decreases with

increasing 𝛼1 in the vicinity of the walls. It is also observed that due to smaller values of k

i.e. for large channel curvature, the profiles of axial velocity are asymmetrical with their

maximum lying below the plane 𝜂 = 0. The variation of axial velocity u at a cross-

150

section 2

x for various values of 𝛼2 is shown through Fig. 8.2. Here, the axial

velocity follow a decreasing trend by increasing 𝛼2 in the middle most region. On the

contrary, it increases with increasing 𝛼2 near the channel walls. It is also interesting to note

that larger values of 𝛼2 helps the axial velocity to regain its symmetry with respect to the

plane 𝜂 = 0. In this way, the role of 𝛼2 is to counteract the effect of curvature and help the

axial velocity to regain its symmetry. The effects of channel curvature on axial velocity at

a cross-section 2

x are shown in Fig. 8.3. The parameter k is the dimensionless radius

of curvature of the channel and is inversely related with the curvature of the channel. As

expected, the axial velocity loose its symmetry of larger values of channel curvature with

maximum in it lying below 𝜂 = 0.

The pressure rise per wavelength and flow rate relationship illustrating the pumping

characteristics is shown in Fig. 8.4 for different values of 𝛼1 . It is observed that in

peristaltic pumping region 0, 0 ,p the pressure rise per wavelength increases by

increasing 𝛼1. The curve 1 2 0.5 illustrates the pressure flow rate relationship for a

Newtonian fluid. This curve lies above all other curves in the peristaltic pumping region

thereby indicating that pressure rise per wavelength decreases in going from Newtonian to

viscoelastic fluid. The effects of 2 (Fig. 8.5) on pressure rise per wavelength in peristaltic

pumping region are quite opposite to the effects of 1. Here, the pressure rise per

wavelength decreases by increasing 2. The effects of channel curvature on pressure rise

per wavelength are displayed in Fig. 8.6. This figure reveals that ∆𝑝 decreases in the

151

peristaltic pumping region as .k Thus in the peristaltic pumping region, the flow rate

is low and pressure difference is high for straight channel as compared to curved channel.

It is worthwhile to validate our results of heat transfer before analyzing the heat transfer

characteristics. To this end, we have compared our numerical solution of temperature

distribution against the available solution given in Ref. [97]. The plots obtained via both

solutions are displayed in Fig. 8.7. This figure clearly demonstrates an excellent correlation

between both the solutions. This obviously corroborates the validity of our numerical

results of heat transfer.

Having established the validity of our solution, we shall now discuss the effects of various

emerging parameters on temperature distribution inside the channel and heat transfer

coefficient. The temperature variations in fluid inside the channel for various values of

involved parameters are shown in Figs. 8.8 – 8.11. Figure 8.8 shows the profiles of

temperature for different values of 1. This figure depicts rise in temperature of the fluid

inside the channel by increasing 1. On the contrary, Fig. 8.9 indicate a decrease in the

temperature of the fluid by increasing 2. The curvature of the channel also significantly

influence the temperature of the fluid. In fact, it is noted from Fig. 8.10 that temperature of

the fluid decreases in going from straight to curved channel. The temperature profiles for

various values of Brinkman number are shown in Fig. 8.11. From this figure, an increase

in temperature is observed by increasing Brinkman number.

Figures 8.12 – 8.15 describe the effects of various values of 1 2, ,k and Br on the heat

transfer coefficient at the upper wall. It is observed that heat transfer coefficient

periodically varies at the peristaltic wall and it shows quantitative variations by changing

152

the involved parameters. Figure 8.12 shows that the amplitude of the heat transfer

coefficient increases by increasing the rheological parameter 1 . On the contrary, as

evident through Fig. 8.13 it follows a converse trend by increasing 2 . Figures 8.14 and

8.15 depict an increase in amplitude of heat transfer coefficient by increasing k and Br. In

conclusion it is observed that both rheological and geometrical parameters of the problems

bear the significant potential to affect the heat transfer coefficient.

The patterns of streamline for different values of 1 2, and k are shown in Figs. 8.16 –

8.18. Figure 8.16 shows the streamlines patterns for different values of 𝛼1. Panel (a)

indicates asymmetric patterns of streamlines with a bolus of fluid in the upper part of the

channel for 𝛼1 = 0. By increasing 𝛼1 from 0 to 0.365, the bolus splits into two parts with

upper part bigger in size as compared to the lower part (panel (b)). A further increase in 𝛼1

increases the size and circulation of the upper part of the bolus (panel (c)). At the same

time it results in weakening of the strength of the lower part of the bolus. The effects of 2

on streamlines are shown in Fig. 8.17. Panel (a) confirms a circulating bolus of fluid

concentrated in the upper half of the channel and a single eddy in the lower half for

2 0.85. For moderate value of 2 the bolus in the upper half of the channel shrinks whilst

the single eddy in the lower half for 2 0.85 converts into a circulating bolus of fluid

(panel (b)). The size of bolus in the upper half still bigger than the size of bolus in the lower

half and panel (c) shows that the larger values of 2 result in disappearance of bolus in the

lower half of the channel. Moreover, the bolus in upper half of the channel for 2 4.4

gets squeezed for 2 16.8 . The effects of channel curvature on streamlines pattern are

153

shown in Fig. 8.18. As expected the circulating bolus of fluid is asymmetric for large values

of channel curvature. However, it becomes symmetric as .k

154


1 2 0.5 with ϕ = 0.4, Θ = 1.2 and k = 3.


of 2 1 0.5 with ϕ = 0.4, Θ = 1.2 and k = 3.

Fig. 8.3: Variation of u for different values of k with ϕ = 0.4, Θ = 1.2, 1 0.3 and 2 0.5 .

Fig. 8.4: Variation of 𝛥𝑝 for different values of 𝛼1 with

2 0.5, k = 2 and ϕ = 0.4.

Fig. 8.5: Variation of 𝛥𝑝 for different values of

𝛼2 With 1 0.5, k = 2 and ϕ = 0.4.

155

Fig. 8.6: Variation of 𝛥𝑝 for different values of 𝛼1 = 0.5 with 2 2, k = 2 and ϕ = 0.4.

Fig. 8.7: Comparison between the solution (Ref. [97]) (bubbles line) and numerical

solution (solid line) of heat transfer at cross section x = − 𝜋2⁄ , 𝛼1 = 0, 𝛼2 = 0, k = 2, Θ =

1.5 and ϕ = 0.4.

156

Fig. 8.8: Profile of temperature 𝜃(𝜂) for different

values of 𝛼1 with 2 0.5, k = 2, Br = 1.5, Θ = 1.5

and ϕ = 0.4.


values of 𝛼2 with 1 0.5, k = 2, Br = 1.5,

Θ = 1.5 and ϕ = 0.4.

Fig. 8.10: Profile of temperature 𝜃(𝜂) for

different values of k with 𝛼1 = 0.3, 2 0.5,

Br = 1.5, Θ = 1.5 and ϕ = 0.4.


values of Br with 𝛼1 = 0.3, 2 0.5, k = 2,

Θ = 1.5 and ϕ = 0.4.

157

Fig. 8.12: Variation of heat transfer coefficient z at upper wall for different values of 𝛼1.

The other parameters chosen are 2 0.5, k = 2, Br = 1.5, Θ = 1.5 and ϕ = 0.4.

Fig. 8.13: Variation of heat transfer coefficient z at upper wall for different values of 𝛼2.

The other parameters chosen are 1 0.5, k = 2, Br = 1.5, Θ = 1.5 and ϕ = 0.4.

Fig. 8.14: Variation of heat transfer coefficient z at upper wall for different values of k.

The other parameters chosen are 1 20.5, 0.5, Br = 1.5, Θ = 1.5 and ϕ = 0.4.

158

Fig. 8.15: Variation of heat transfer coefficient z at upper wall for different values of Br.

The other parameters chosen are 1 20.5, 0.5, k = 2, Θ = 1.5 and ϕ = 0.4.

Fig. 8.16: Streamlines for 1 1 1(a) 0,(b) 0.365,and ( ) 2,c with k = 2.5. The other

parameters chosen are 2 2 , Θ = 1.5 and ϕ = 0.8.

159

Fig. 8.17: Streamlines for 2 2(a) 0.85, (b) 4.4, and 2(c) 16.8, with k = 2.5. The

other parameters chosen are 1 0.85, Θ = 1.5 and ϕ = 0.8.

160

Fig. 8.18: Streamlines for (a) k = 2, (b) k = 3.5, and (c) k → ∞ with 𝛼1 = 0.25 and 𝛼2

= 0.5. The other parameters chosen are Θ = 1.5 and ϕ = 0.8.

161


This paper analyses the heat transfer characteristics of peristaltic flow of an Oldroyd 8-

constant fluid in a curved channel. The governing equations of the flow and heat transfer

are modeled in curvilinear coordinates under negligible inertial and streamline curvature

effects. Extensive numerical computational are carried out to investigate the effects of

various emerging parameters on flow and heat transfer characteristics. It is found that the

material parameters of the Oldroyd 8-constant fluid bear the potential to counteract the

effects of curvature and help the velocity profile and circulating bolus to regain their

symmetric shapes. The maximum pressure against which peristalsis has to work as a

positive displacement pump to increases with increasing the channel curvature. The

temperature of fluid inside the channel is also found to increases by increasing channel

curvature and Brinkman number. However, an increase in the material parameter of

Oldroyd 8-constant results in the cooling of fluid inside the channel.

162

Chapter 9

Numerical study for the flow and heat transfer

in a curved channel with peristaltic walls

In the present chapter, we study the flow and heat transfer characteristics of a Cross fluid

in a curved channel. The flow is induced by the peristaltic waves propagating along the

walls of the channel. The problem is modeled in curvilinear coordinates under long

wavelength and low Reynolds number assumptions. The flow regime is shown to be

governed by a dimensionless fourth order, nonlinear ordinary differential equation subject

to no-slip wall boundary conditions. The heat transfer phenomena exhibiting viscous

dissipation is described in terms of a second order linear differential equation. A well-

testified finite difference method (FDM) is employed for the solution. The influence of

rheological parameters of Cross fluid, Brinkman number and curvature of the channel is

presented in graphical form. For shear-thinning bio-fluids (power-law rheological index, n

> 1) greater Weissenberg number displaces the maximum velocity towards the upper wall.

Further, the bolus regain its symmetry for large values of the dimensionless radius of

curvature parameter. (Here, results of straight channel are discovered for larger k). The

present study also highlights that the temperature of fluid and rate of heat transfer increase

with increasing curvature of the channel.

163

9.1 Mathematical model

For a non-Newtonian Cross model [119] the extra stress tensor S is given by

0, 1

11

nn

S (9.1)

From the usual analysis as carried out previously, one gets

*

*

*

*

*

*

12 , ( )

11

1 1, ( )

11

12 , ( )

11

xx

x

vS a

nWe

uu k vS b

n k kWe

k u vS c

n k x kWe

(9.2)

and

22 2

211 2 2

2 2 ,2

uv u k v k u v

k k k x k

(9.3)

as component of extra stress S in the wave frame. After using long wavelength and low

Reynolds number approximations Eq. (9.2(b)) and (9.3) reduce to

* 2*

2

1 11 ,

11

xSn kWe

(9.4)

and

22

2

2

11 .

k

(9.5)

Inserting Eq. (9.4) into Eq. (2.31) and eliminating pressure, one get

164

* 22 *

2

11 11 0.

1( ) 1k

nk kWe

(9.6)

From the detailed derivation prevented in previous chapter the energy equation under long

wavelength was found to be

2

2

1 11 0.

( )xk BrS

k k

(9.7)

The above equation in view of Eq. (9.4) becomes

2* 2*

2

11 11 0.

1( ) 1k Br

nk kWe

(9.8)

The boundary conditions at the walls are defined in Eq. (8.13).

9.2 Computation results and interpretation

In this section, we study the graphical behavior of the solutions as well as numerical

illustrations of some important features of the peristaltic motion (flow characteristics,

pumping characteristics, heat distribution and trapping phenomenon) for various values of

the following parameters k, n, Br and We .

9.2.1 Flow characteristics

The flow velocity at a cross-section x = − 𝜋2⁄ for different values of Weissenberg number

in case for 𝑛 > 1 is shown in Fig. 9.1. The corresponding curve for a Newtonian fluid (We

165

= 0, n = 1) indicates that flow velocity attains maximum slightly below η = 0. It is observed

that velocity curves are not symmetric about η = 0 due to presence of curvature in the

channel and maxima in these curves lie below η = 0 for small values of Weissenberg

number (0 ≤ 𝑊𝑒 < 0.5). A shift in location of maximum velocity toward the upper wall

from its Newtonian value is observed for Weissenberg number equal to 0.5 which becomes

significant by further increasing Weissenberg number. It is interesting to note that

curvature of the channel compels the fluid to move with a non-symmetric velocity whose

maximum lies below η = 0. An increase curvature parameter shifts the location of

maximum velocity toward the lower wall. The role of Weissenberg number is to counteract

the effects of curvature of the channel and help the fluid velocity to regain its symmetric

shape. It is also noted that for larger Weissenberg number the location of maximum

velocity shift toward the upper wall i.e. in the upper part of the channel for which 0 < 𝜂 <

ℎ1. Figure 9.2 illustrates the effects of power-law index on flow velocity in a curved

channel for a fixed non-zero value of Weissenberg number. This figure shows that for a

fixed value of Weissenberg number an increase in power-law index shift the velocity

maximum from the lower half to the upper half of the channel. In this process the velocity

profile also attains symmetric shape. The effects of curvature on the flow velocity can be

well understood by examining Fig. 9.3. Figure 9.3 reveals that small values of

dimensionless radius of curvature parameter k (which corresponds to large channel

curvature), result in non-symmetric velocity profile about η = 0 with maximum in it

appearing at η such that ℎ2 < 𝜂 < 0. The appearance of the velocity maximum in lower

part of the channel is due to the choice of rheological parameters of Cross model which

corresponds to smaller Weissenberg number. This is in accordance with the observation of

166

previous figures. There it was seen that for strong shear-thinning fluids (𝑛 > 1) larger

values of Weissenberg number yield velocity profiles with maximum lying above η = 0.

Figure 9.3 further reveals that flow velocity regain its symmetry for large values of

dimensionless radius of curvature. The above observations may be summarized as:

Due to presence of curvature in the channel, the velocity profile losses its symmetry and

maximum in it shifts towards the lower wall. Non symmetric velocity profiles with

maximum lying in the lower half are the only admissible profiles for Newtonian fluids.

However, the corresponding velocity profile of Cross model exhibits interesting character,

due to the presence of rheological parameters n and We. In fact, for shear-thinning case n

> 1 the role of Weissenberg number is to counteract the effects of curvature and help the

fluid velocity to regain its symmetric shape.

9.2.2 Pumping characteristics

The pressure difference across one wavelength is plotted against time-averaged flow rate

in the laboratory frame Θ (= 𝑓 + 2) in Figs. 9.4 – 9.6. These figures illustrate the effects

of Weissenberg number (We), rheological power-law index (n) and curvature parameter

(k), respectively. The present study examines three regions of pressure flow rate plots;

peristaltic pumping region (Θ > 0, Δ𝑝 > 0), free pumping region (Θ > 0, Δ𝑝 = 0) and co-

pumping region (Θ > 0, Δ𝑝 < 0).

In the peristaltic pumping region for a fixed value of time-averaged flow rate, the

magnitude of pressure rise per wavelength increases by decreasing We for shear-

thinning bio-fluids. Similarly Δ𝑝 is an increasing function of power-law index (n)

167

and curvature parameter (k) for a fixed value of time-averaged flow rate in pumping

region.

The free pumping flux i.e. the value of Θ for Δ𝑝 = 0 is independent of Weissenberg

number for shear-thinning fluids. Moreover, for a fixed Weissenberg number it

increases when fluid behavior changes from Newtonian to shear-thinning fluid.

In co-pumping region the pressure assists the flow due to peristalsis. It is observed

that for a fixed value of mean flow rate, the magnitude of assistance provided by

pressure decreases with increasing Weissenberg number for shear-thinning fluids.

Similarly, by keeping Weissenberg number fixed, the magnitude of pressure rise

per wavelength increases in going from Newtonian to shear-thinning fluids for a

fixed value of mean flow rate Θ. It is also observed through Figs. 9.4 – 9.6 that

pressure rise in pumping region increases in going from curved to straight channel

irrespective of the nature of the fluid. This is quite striking observation because for

a Newtonian fluid pressure rise in pumping region decreases in going from curved

to straight channel [97]. However, pressure rise for a non-Newtonian Cross fluid

behave differently than its counterpart for a Newtonian fluid. This observation

suggests the use of non-Newtonian characteristics of the fluid to reduce the pressure

rise per wavelength in the pumping region. The Newtonian fluid does not suggest

any such solution to reduce the pressure rise.

9.2.3 Heat transfer phenomena

The influence of pertinent parameters (Weissenberg number (We), power-law index (n),

Brinkman number (Br) and curvature of the channel (k)) on the temperature distribution

168

inside the peristaltic channel at a cross-section x = − 𝜋2⁄ is graphically shown in Figs. 9.7

– 9.10. Figure 9.7 illustrates the effects of We on temperature distribution. This figure

shows a detraction in temperature for larger values of Weissenberg number for shear-

thinning fluid. An interesting pattern that is perceived from Fig. 9.8 is that the magnitude

of the temperature in the peristaltic channel enhances with a shift in the behavior of fluid

from Newtonian to shear-thinning. The effects of Brinkman number (Br) on temperature

distribution across the channel are shown in Fig. 9.9. This figure predicts a rise in

temperature by increasing Brinkman number Br. The effects of curvature of the channel on

the temperature are shown in Fig. 9.10. It is observed that temperature inside the channel

increases with an increase in the channel curvature for shear-thinning fluids.

Figures 9.11 – 9.14 describe the effects of various values of We, n, Br and k on the heat

transfer coefficient at the upper wall. It is observed that heat transfer coefficient

periodically varies at the peristaltic wall and it shows quantitative variations by changing

the involved parameters. Figures 9.11 shows that the amplitude of the heat transfer

coefficient increases for shear-thinning fluid with increasing the rheological parameter We.

On the contrary, as evident through Fig. 9.12 it follows that magnitude of heat transfer

coefficient is enhanced with increasing n. Figure 9.13 depicts a decrease in amplitude of

heat transfer coefficient by increasing k for shear-thinning fluid. However, an increasing

trend is observed by increasing Br. It is interesting to note that heat transfer coefficient is

not in phase with the motion of the peristaltic wall. Figure 9.14 indicates that heat transfer

coefficient oscillate periodically behind the phase with the peristaltic wall.

169

9.2.4. Trapping

The streamlines of the flow for different values of involved parameters are shown in Fig.

9.15 and Fig. 9.16. The effect of power-law index (n) on streamlines is shown through Fig.

9.15. Two circulating boluses of fluid, one bigger in size concentrated in upper part of

channel and the other of less strength in lower part for Newtonian fluid can be identified

from Fig. 9.15(a). With the increase of power-law index, the lower bolus decreases in size

while the upper one gets squeezed by increasing power-law index. The effect of curvature

of the channel on streamline patterns is illustrated in Fig. 9.16. As expected the bolus of

fluid concentrated in the upper part of the channel for small values of k (corresponding to

larger channel curvature) is transformed into two symmetric boluses by decreasing the

channel curvature.

170

Fig. 9.1: Variation of u for different values ofWe with n = 1.99 (for shear thinning). The other parameters

chosen are = 0.4, Θ = 1.5, 𝜇0 = 1.3, 𝜇∞= 0.05 and k = 2.

Fig. 9.2: Variation of u for different values of n with We = 0.5, = 0.4, Θ = 1.5, 𝜇0 = 1.3, 𝜇∞= 0.05 and k = 2.

Fig. 9.3: Variation of u for different values of k with n = 1.2 (for shear thinning). The other parameters

chosen are We = 0.5, = 0.4, Θ = 1.5, 𝜇0 = 1.3, 𝜇∞= 0.05 and k = 2.

171

Fig. 9.4: Variation of 𝛥𝑝 for different values of 𝑊𝑒 with n = 1.8 (for shear thinning). The other

parameters chosen are = 0.4, 𝜇0 = 1.3, 𝜇∞= 0.05 and k = 2.

Fig. 9.5: Variation of 𝛥𝑝 for different values of n with We = 0.5, = 0.4, 𝜇0 = 1.3, 𝜇∞= 0.05 and k = 2.

Fig. 9.6: Variation of 𝛥𝑝 for different values of k with n = 2 (for shear thinning). The other parameters

chosen are 𝑊𝑒 = 0.2, = 0.4, 𝜇0 = 1.3, 𝜇∞= 0.05 and k = 2.

172

Fig. 9.7: Profile of temperature 𝜃(𝜂) for different values of 𝑊𝑒 with n = 1.2 (for shear thinning). The

other parameters chosen are = 0.2, 𝜇0 = 1.3, 𝜇∞= 0.05, Br = 1, Θ = 1.5 and k = 2.

Fig. 9.8: Profile of temperature 𝜃(𝜂) for different values of n with We = 0.5, = 0.2, 𝜇0 = 1.3, 𝜇∞= 0.05,

Br = 1, Θ = 1.5 and k = 2.

Fig. 9.9: Profile of temperature 𝜃(𝜂) for different values of Br with n = 1.3 (for shear thinning). The other

parameters chosen are We = 0.5, = 0.2, 𝜇0 = 1.3, 𝜇∞= 0.05, Θ = 1.5 and k = 2.

173

Fig. 9.10: Profile of temperature 𝜃(𝜂) for different values of k with n = 1.3 (for shear thinning). The

other parameters chosen are We = 0.5, = 0.2, 𝜇0 = 1.3, 𝜇∞= 0.05, Θ = 1.5 and k = 2.

Fig. 9.11: Variation of Heat transfer coefficient z at upper wall for different values of 𝑊𝑒 with n = 1.05

(for shear thinning). The other parameters chosen are k = 1.5, Br = 1, Θ = 1.5 and = 0.2.

Fig. 9.12: Variation of Heat transfer coefficient z at upper wall for different values of n.

The other parameters chosen are 0.2,We k = 1.5, Br = 1, Θ = 1.5 and = 0.2.

174

Fig. 9.13: Variation of Heat transfer coefficient z at upper wall for different values of 𝑘 with n = 1.05

(for shear thinning). The other parameters chosen are We = 0.2, Br = 1, Θ = 1.5 and = 0.2.

Fig. 9.14: Variation of Heat transfer coefficient z at upper wall for different values of Br with n = 1.05

(for shear thinning). The other parameters chosen are k = 1.5, We = 0.2, Θ =1.5 and = 0.2.

Fig. 9.15: Streamlines for (a) Newtonian and (b) n = 5. The other parameters chosen are We = 0.3, = 0.8,

Θ = 1.5, 𝜇0 = 1.3, 𝜇∞= 0.05 and k = 2.

175

Fig. 9.16: Streamlines for (a) k = 2, (b) k = 3.5, and (c) k → ∞. The other parameters chosen are

We = 0.37, n = 1.1, = 0.8, Θ = 1.5, 𝜇0 = 1.3 and 𝜇∞= 0.05.

176


An analysis of flow and heat transfer is presented in a curved peristaltic channel under long

wavelength and low Reynolds number assumptions. The non-Newtonian rheology of the

fluid is characterized by Cross model. The flow and energy equation are numerically solved

using a finite difference method (FDM) combined with an iterative scheme. Extensive

computations and flow visualizations are presented. The whole analysis is summarized in

the following main points.

The response of flow velocity with increasing Weissenberg number depends on the

nature of the fluid. In fact, increase in Weissenberg number counteracts the effects

of curvature for shear-thinning fluids.

Contrary to the case of Newtonian fluid, the pressure rise per wavelength in

pumping region for a Cross fluid decreases in going from straight to curved

channel.

The temperature of fluid inside the channel increases in going from straight to

curved channel. Moreover, it decreasing in going from shear-thinning to Newtonian

fluids.

The heat transfer coefficient oscillates periodically behind the phase with the

peristaltic wall. Further, its amplitude increases with increasing the channel

curvature.

The size of circulating bolus of fluid in the vicinity of the lower wall decreases with

increasing power-law index.

177

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Peristaltic Transport of non-Newtonian Fluids in a Curved Channelprr.hec.gov.pk/jspui/bitstream/123456789/7213/1/Khurram... · 2018-07-23 · Peristaltic Transport of non-Newtonian