Peristaltic Transport of Fluid Flow: A Dynamical System

Peristaltic Transport of Fluid Flow: A Dynamical

System Approach

By

Tayyaba Ehsan

CIIT/FA14-PMT-006/ISB

PhD Thesis

In

Mathematics

COMSATS University Islamabad

Pakistan

Fall, 2020

ii


Peristaltic Transport of Fluid Flow: A Dynamical

System Approach

A Thesis Presented to


In partial fulfillment

of the requirement for the degree of

PhD Mathematics

By

Tayyaba Ehsan


Fall, 2020

iii

Peristaltic Transport of Fluid Flow: A

Dynamical System Approach

A Post Graduate Thesis submitted to the Department of Mathematics as partial

fulfillment of the requirement for the award of Degree of PhD in Mathematics.

Name Registration Number

Tayyaba Ehsan CIIT/FA14-PMT-006/ISB

Supervisor

Prof. Dr. Saleem Asghar

Department of Mathematics


January, 2021

iv

Certificate of Approval

This is to certify that the research work presented in this thesis, entitled “Peristaltic

Transport of Fluid Flow: A Dynamical System Approach” was conducted by Ms.

Tayyaba Ehsan, Reg.No. CIIT/FA14-PMT-006/ISB, under the supervision of Prof.

Dr. Saleem Asghar. No part of this thesis has been submitted anywhere else for any

other degree. This thesis is submitted to the Department of Mathematics, COMSATS

University Islamabad, in the partial fulfillment of the requirement for the degree of

Doctor of Philosophy in the field of Mathematics.

Student Name: Tayyaba Ehsan_ Signature: _______________

Examination Committee:

External Examiner 1: _______________ External Examiner 2: ______________

Prof. Dr. Muhammad Ayub Prof. Dr. Muhammad Sajid

Department of Mathematics Department of Mathematics & Statistics

Quaid-i-Azam University, International Islamic University,

Islamabad. Islamabad.

Supervisor: __________________ Head of Department: _______________

Prof. Dr. Saleem Asghar Dr. Abdullah Shah

Department of Mathematics Department of Mathematics

COMSATS University, COMSATS University,


Chairperson: _____________ Dean: _____________________

Prof. Dr. habil Shamsul Qamar Prof. Dr. Aftab Khan

Department of Mathematics Dean of Faculty of Sciences

COMSATS University, COMSATS University,


v

Author's Declaration

I Tayyaba Ehsan, CIIT/FA14-PMT-006/ISB, hereby state that my PhD thesis titled

“Peristaltic Transport of Fluid Flow: A Dynamical System Approach” is my own work and

has not been submitted previously by me for taking any degree from this University i.e.

COMSATS University Islamabad or anywhere else in the country/world.

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

the right to withdraw my PhD degree.

Date: _______________

Tayyaba Ehsan


vi

Plagiarism Undertaking

I solemnly declare that research work presented in the thesis titled “Peristaltic

Transport of Fluid Flow: A Dynamical System Approach” is solely my research work

with no significant contribution from any other person. Small contribution/help

wherever taken has been duly acknowledged and that complete thesis has been written

by me.

I understand the zero tolerance policy of HEC and COMSATS University Islamabad

towards plagiarism. Therefore, I as an author of the above titled thesis declare that no

portion of my thesis has been plagiarized and any material used as reference is

properly referred/cited.

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

even after award of PhD Degree, the University reserves the right to withdraw/revoke

my PhD degree and that HEC and the university has the right to publish my name on

the HEC/university website on which names of students are placed who submitted

plagiarized thesis.

Date: _______________

Tayyaba Ehsan


vii

Certificate

It is certified that Tayyaba Ehsan, CIIT/FA14-PMT-006/ISB has carried out all the

work related to this thesis under my supervision at the Department of Mathematics,

COMSATS University Islamabad and the work fulfills the requirement for award of

PhD degree.

Date: _______________

Supervisor:

Prof. Dr. Saleem Asghar


CUI, Islamabad

Head of Department:

Dr. Abdullah Shah


CUI, Islamabad.

viii

To my Parents,

And my Supervisor

ACKNOWLEDGEMENTS

All Praise and Glory is to Allah, The Greatest, All Perfection, All Greatness. Benediction

upon Prophet Muhammad (P.B.U.H), The Exalted, Best of Mankind, and Ever Enlighten-

ing.

My immense gratitude and appraisal to my eminent professor, supervisor, Prof. Dr. Saleem

Asghar, beacon of hope and mentor. I am greatly indebted to his guidance, dynamism,

vision, unwavering enthusiasm and support without which, achievement of this milestone

was a grim possibility.

My sincere appreciation to Dr. Junaid Anjum for constant motivation, enthusiasm and his

superluminal mind. Without his guidance and persistent help this thesis would not have

been possible.

To my parents and siblings who have always been so encouraging and a source of unfailing

support. Their cooperation and motivation has been the driving force for who I am today.

I am highly obliged to Dr. Abdullah Shah, Head of the Department of Mathematics, for his

kind cooperation, support and generous attitude.

I attribute my deepest appreciation to all my friends and colleagues special mention of Ms.

Shagufta who always been a source of encouragement for me. Her moral support helped

me complete my thesis.

Tayyaba Ehsan


ix

ABSTRACT

Peristaltic Transport of Fluid Flow: A Dynamical System Approach

Theoretical studies for peristalsis have been investigated extensively in the literature. The

velocity field, pressure rise per wave length and volume flow rate are determined from the

stream function. The augmented flow, bolus and trapping is discussed for some fixed values

of flow parameters.

In this thesis, the qualitative behavior of the flow field (state variables) is determined using

theory of dynamical system. The flow analysis is made through stability and bifurcation

analysis. The proposed method efficiently detects the presence of bolus. The bifurcation

points are identified at which the qualitative behavior changes. The flow behavior is char-

acterized for complete range of the parameter through bifurcation diagrams. The question

of appearance and disappearance of bolus is examined in greater depth. Contrary to general

perception; it is observed that the bolus in the flow field disappears for increasing magnetic

field for negative flux only. For other values of given volume flow rate, the bolus is pushed

towards the walls. The existence understanding of the size of bolus (visually inspecting

the contour plots of streamlines) is clarified as the area of outermost closed streamline

around an equilibrium point. The definition is supported by analytical and numerical con-

siderations with the help of examples. The concept of trapping in peristalsis is revisited; the

existing understanding being inconsistent; we propose a new approach for the identification

of bolus and trapping. The bolus can be related to existence of center (at the equilibrium

point). For trapping the center exists under the wave crest when saddles lie on the center-

line. The augmented flow is characterized as centers under the wave crest when saddles lie

above and below the centerline.

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TABLE OF CONTENTS

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Disappearance of bolus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Size of the bolus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Identification of trapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1 Peristalsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.1 Adverse and favorable pressure gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.2 Free pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.3 Retrograde pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.4 Peristaltic pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.5 Augmented pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.6 Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Types of flow patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 Bolus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.2 Backward flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.3 Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.4 Augmented flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

xi

2.4 No slip boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5 Slip boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 Non-dimensional numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6.1 Amplitude ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6.2 Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6.3 Hartmann number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6.4 Wave number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.7 Continuity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.8 Momentum equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.9 Non newtonian fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.9.1 Jeffrey’s model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.9.2 Constitutive equations in a Jeffrey fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.10 Dynamical system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.10.1 Autonomous systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.10.2 Stationary solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.10.3 Asymptotically stable equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.10.4 Stable equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.10.5 Hyperbolic (nondegenerate) equilibrium point . . . . . . . . . . . . . . . . . . . . . . . . 29

2.10.6 Types of equilibrium points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.10.6.1 Saddle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.10.6.2 Node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.10.6.3 Spiral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.10.6.4 Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.11 Dynamical system and fluid mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.12 Bifurcation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.12.1 Bifurcation parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.12.2 Bifurcation diagram (equilibria curves) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

xii

2.12.3 Bifurcation point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.12.4 Types of bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.12.5 Transcritical bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.12.6 Supercritical pitchfork bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.12.7 Subcritical pitchfork bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.12.8 Hartman-Grobman theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.12.9 Bendixson’s criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.12.10 Limit cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.12.11 Structurally stable/unstable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.12.12 Co-dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.12.13 Separatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.12.14 Local bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.12.15 Global bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.12.16 Travelling wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.12.16.1 Periodic wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.12.16.2 Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.12.16.3 Wave front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.12.17 Homoclinic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.12.18 Heteroclinic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Disappearing of a “bolus” with increasing magnetic field; Peristaltic flows. . . . 41

3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Identifying a bolus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.1 Through contour plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.2 Through the dynamical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 Bolus disappearing with increasing M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3.1 Quantitative disappearance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3.2 Qualitative disappearance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

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3.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 Peristaltic flows: A quantitative measure for the size of a bolus . . . . . . . . . . . . . . . . . 58

4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3 Analytical illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4 Numerical illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5 Identification of trapping in a peristaltic flow; A new approach using dy-

namical system theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2 Definitions of various flow patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3 Identifying Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.3.1 Present approach: Through Visual Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.3.2 Proposed Approach: Through Dynamical System . . . . . . . . . . . . . . . . . . . . . 78

5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.1 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

xiv

LIST OF FIGURES

2.1 (From [50]) The contour plots of the stream function ψ [50][eq. (11)] at

φ = 0.8 and for different values of volume flow rate q. The arrows repre-

sent the velocity vectors (in moving frame of reference) computed from the

stream function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Transcritical Bifurcation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3 Supercritical Pitchfork Bifurcation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4 Subcritical Pitchfork Bifurcation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1 A contour plot of the stream function [56, equation 19] generated using

Matlab at different values of the magnetic field parameter (a) M = 0.01, (b)

M = 1, (c) M = 6 and (d) M = 18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2 An equilibrium plot showing the evolution of the vertical coordinate of the

equilibrium point as the magnetic field M is varied. Two characteristic flow

patterns are developed corresponding to x=−0.2184 (center, shown in red)

and x = 2.9231 (saddle, shown in blue). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3 (Left) A contour plot of the stream function [56, equation 19] generated

using Matlab at M = 50,q= 0,φ1 = π/6 and λ1 = 0. The closed streamlines

are shown in black whereas the open streamlines are colored brown. (Right)

The equilibria curve, showing the evolution of the vertical component of the

equilibrium point, as the magnetic field M is varied. The qualitative nature

of the equilibria curve shown in red, corresponding to x = −0.3 is center

whereas the blue curve (x = 2.8415) represents saddles. . . . . . . . . . . . . . . . . . . . . . . 50

xv

3.4 The horizontal velocity profiles in the vertical column, x = −0.2184, at

M = 1 (dashed line), M = 6 (dash-dotted line) and M = 18 (dotted line) for

q =−0.1,φ1 = π/8 and λ1 = 0.3. The solid horizontal line marks the zero

velocity. The picture above is the zoomed view near the zero velocity level. . 52

3.5 Asymptotic velocity profiles for large M(= 100) plotted as a function of

y at x = 0 for different values of flux q = 1 (dashed), q = 0 (dotted) and

q =−1 (dash-dotted). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.6 The (unit) velocity vectors in the asymptotic state (M = 100) for q= 1 (top),

q = 0 (middle) and q =−1 (bottom). Other flow parameters are φ1 = π/8

a = 0.4,b = 0.5 and λ1 = 0.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.7 A pictorial diagram of the characteristic changes in flow dynamics if an

equilibrium point changes from being center to a saddle. . . . . . . . . . . . . . . . . . . . . . . 56

4.1 The streamlines corresponding to c = 0.21 (shown in red) and c = 0.3

(shown in black). In (a) the contour lines are plotted using Matlab whereas

in (b) the contour lines correspond to the roots yk defined in equation (4.11).

In (b), dashed and solid lines represent the roots y0 and y1, respectively. The

vertical lines in (a) mark x = −0.6639 and x = 0.6639 in order. The open

symbols in (b) show the end points of the roots y0 (squares) and y1 (trian-

gles). The point (0, 0.6667) marked with a cross represents the equilibrium

point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2 The contour plot of the stream function ψ defined in equation (4.4) for q =

0.2, φ = 0.8 generated using Matlab. In particular, the streamline marked

with a thick solid black line corresponds to the contour level c = 0.2178. . . . 65

4.3 The contour plots of the stream function ψ , defined in equation (4.4), for

different values of the parameters are generated using Matlab. In each plot,

the dashed line represents the contour level, the code identified as the out-

ermost closed streamline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

xvi

4.4 The size of the bolus sb, as defined in equation (4.3) plotted as a function of

(a) flux parameter q [50] (b) The magnetic field strength parameter M [10]. 69

5.1 The contour plots of the stream function ψ [equation 5.1] at φ = 0.4, M =

1,α = 0.1,β = 0.0, and for different values of volume flow rate q . The

arrows represent the velocity vectors (in the moving frame of reference)

computed from the stream function. Following [50], the qualitative flow

patterns are termed as (a) Backward Flow corresponding to q = −0.5, (b)

Trapping for q =−0.3 and (c) Augmented Flow resulting for q = 0.1. . . . . . . 73

5.2 The bifurcation diagram as a function of slip parameter β for the peristalsis

problem in [10]. (a)-(b) show the spatial location of the equilibria at α =

0.0, whereas (c)-(d) show the location of equilibria at α = 0.1 (see [10]

for parameter details). The solid line is used to represent saddles whereas

dashed line corresponds to centers.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3 The bifurcation diagram as a function of magnetic parameter M for the peri-

stalsis problem in [10]. (a)-(b) present the spatial location of the equilibria

at α = 0.0, whereas (c)-(d) present the location of equilibria at α = 0.1.

The solid line and dashed line presents the qualitative nature of equilibrium

points as being saddles and centers respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.4 The bifurcation diagram as a function of volume flow rate q for the peri-

stalsis problem in [10]. (a)-(b) present the spatial location of the equilibria

at α = 0.1. The solid and dashed lines are used to represent saddles and

centers respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

xvii

LIST OF TABLES

4.1 Size of the bolus, sb, as defined in (4.3), calculated for different parameter

values, both analytically and numerically. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

xviii

List Of Symbols

(X ,Y, t) Dimensional coordinate axes in fixed frame

(x,y) Dimensionless coordinate axes in wave frame

(U,V ) Dimensional velocity components in fixed frame

(u,v) Dimensionless velocity components in wave frame

a,b Amplitudes of the wave

h1,h2 Dimensionless upper and lower walls

φ1 Phase difference

∇ Gradient vector

L Velocity gradient

I Identity tensor

S Extra stress tensor

T Cauchy stress tensor

A1 First Rivilin-Erickson tensor

q Dimensionless flow rate

P Pressure

µ fluid viscosity

ψ Stream function

M Magnetic field parameter

Re Reynolds number

xix

β Slip parameter

α viscosity parameter

λ Jeffrey fluid parameter

φ Amplitude ratio

sb Size of the bolus

yu Upper branch

yl Lower branch

(xe,ye) Equilibrium point

[xA,yB] x-projection

Rem Magnetic Reynolds number

N Interaction parameter

W Simply-connected domain

p period of the periodic wave

n outward drawn normal

µt Dimensionless parameter

ls material parameter

xx

Chapter 1

Introduction

1

1.1 Introduction

Peristaltic flows result from expanding and contracting of the channel walls in the form

of sinusoidal wave propagation. This mechanism (of peristalsis) is responsible for the

transport of physiological fluid in human body and in engineering processes. Examples

are: blood circulation from heart to the body, movement of food in esophagus and the

flow of urine from ureter to the bladder [20, 21, 23, 38, 54, 60, 86, 106]. Based on the

principle of peristaltic transport, several bio-medical devices, have been invented; such as,

infusion pumps, blood pumps, peristaltic pump (used in dialysis), heart-lung machines used

to circulate blood during bypass surgeries etc. [31, 87].

1.2 Literature review

Theoretical studies of peristaltic flows in a channel and tube were first investigated by

Shapiro [92]. These investigations were based on long wavelength and small Reynolds

number approximations; which became a norm for subsequent investigations. These as-

sumptions are particularly relevant for urinary and gastrointestinal tracks of the humans.

Some essential studies, which gave rise to the later studies and laid the foundation of peri-

stalsis flow phenomena are referred to [41, 48, 49, 61, 108]. The engineering aspects of

peristalsis have been first established by Latham [57] that also validated the theoretical in-

vestigations of Shapiro [92]. Literature on peristalsis being so big cannot be referred at

length; and we will suffice to refer a few papers which, we feel, are important for our point

of view [12, 17, 18, 50, 57, 92].

The peristaltic flow in porous and nanofluids have been addressed in the papers [25, 26, 28,

29, 37, 46, 52, 65, 73, 80, 83, 100, 104]. Peristalsis for electrically conducting fluid in the

presence of Lorentz force has been discussed in the following studies: [1, 15, 34, 43, 55,

68, 69, 79, 80]. Recent applications of bio-medical importance are the targeted drug deliv-

ery in human body and a mean of safe transport of medical substances (e.g. spermatozoa)

2

to a desired location [14, 16, 51, 55, 98]. More recently, the qualitative behavior of the fluid

flow is investigated using the theory of dynamical systems. These studies are reported in

the papers [12, 17, 18, 101, 102, 107].

Useful applications of peristalsis in bio-sciences and human physiology developed keen in-

terest, in the subject, leading to many theoretical and experimental investigations in New-

tonian and non-Newtonian fluids. Some papers in Newtonian and non-Newtonian fluids

are presented in [2, 3, 11, 28, 32, 83, 85, 89, 97]. In these studies, analytical results

are obtained in various geometries taking different properties of the medium. The gov-

erning mathematical model, comprising of Navier Stokes equations and continuity equa-

tion, is simplified using the long wavelength and small Reynolds number approximation

[27, 42, 57, 59, 63, 92, 112]. The analytical solution is found in terms of stream function

which provides quantitative information regarding the velocity, pressure rise per unit wave-

length and the volume flow rate. These studies are presented in [4, 7, 9, 39, 83]. Physically

important features like ‘bolus’, ‘trapping’, ‘backward’ and ‘augmented flow’ are presented

graphically in terms of streamlines for particular choice of characterizing parameter. We

observe that the flow behavior in peristalsis strongly depends upon the governing parame-

ters of the problem; and the solution topology changes even for a small variation of these

parameters. In most of the literature, the behavior is determined by plotting the solution

(streamlines) for some discrete/specific value of these parameters and the solution topology

is decided visually. A prior knowledge of the solution behavior for some arbitrary value

of the parameter is not predicted in advance. In this approach, one may find the behavior

for these parameters alone without knowing what happens next door. This way, some be-

havior may be discover and the remaining behaviors may remain dormant. An answer to

these observations lie in dynamical system theory, main focus of this thesis, which has been

investigated from the theory of dynamical system using stability analysis and bifurcation

theory. In fact the bifurcation diagram will predict the behavior for a continuous value of

the parameter.

3

As the flow patterns change due to a change in flow parameters (e.g. magnetic field

strength, volume flow rate, viscosity or others), it is desirable to determine the specific

flow configurations such as ‘backward’, ‘trapping’ or ‘augmented flow’. One must need to

know these behaviors without actually experimenting it through a specific choice of these

parameters and repeating the whole process of drawing the streamlines from the expression

of the stream function. In advance knowledge of these parameters can be known through

the bifurcation diagram and this shall remain a focus of our attention in this thesis. As

said earlier, the flow behavior in peristalsis strongly depends upon the governing parame-

ters of the problem. And the solution topology changes even for a small variation of these

parameters. In most of the literature, the behavior is determined by plotting the solution

(streamlines) for some discrete value of these parameters and the solution topology is de-

cided visually. A prior knowledge of the solution behavior at arbitrary value is not predicted

beforehand. Also, that we may discover some behaviors while a few may remain dormant.

An answer to these observations lies in dynamical system theory and will be explored from

the bifurcation theory.

The theory of dynamical systems helps to find and understand the qualitative behaviors of

the solutions of nonlinear equations, nonlinear dynamical system and nonlinear physics.

In recent years, it has made a big way in control theory, engineering problems, biological

sciences, plasma physics and for looking the qualitative behavior of the complex situations

in Fluid Mechanics. The equations of motion of some physical system are modeled in the

setting of dynamical system in the phase space. The qualitative behavior of the system is

represented by the phase portrait corresponding to various states of the system. The equilib-

rium solutions are determined and its stability is discussed around the critical numbers. If

the differential equations involve the controlling parameter, the qualitative behavior and the

phase portraits also changes with the change of the parameter. These qualitative changes in

which the stability of equilibrium solution change from one topological surface to another

are studied by the bifurcation theory. The bifurcation diagram helps to determine the qual-

4

itative behavior for the continuous range of one parameter while keeping other parameters

fixed (in most cases).

Peristaltic transport phenomenon shows an interesting topological behavior when the char-

acterizing parameters are varied. The qualitative behavior is studied by drawing the phase

portraits and equilibria curves using the dynamical system theory. From literature review,

we come to know that there are but few research articles in which peristalsis transport phe-

nomena has been addressed using the theory of dynamical system. The fundamental work

on separation was done in [19, 24, 30, 91]. Qualitative analysis was done by number of

authors [12, 13, 17, 18, 50, 101, 102, 107]. Jimenez and Sen applied the dynamical system

theory to study the flow dynamics of Newtonian fluids in a symmetric channel [50]. The

authors of [17, 18] extended the work of [50] by incorporating slip effects and heat trans-

fer. Shear thickening and shear thinning fluids characterized by power law fluid model in a

channel and tube was further explored in [12, 103] using the dynamical system theory. The

phase portraits and equilibria curves are drawn to study the flow dynamics. Further, the au-

thors of [88, 105] investigated the streamlines patterns and equilibria curves for peristaltic

transport of nanofluids.

We observe that the work accomplished, in finding the qualitative behavior, needs further

improvements and clarifications in terms of the definitions used and the conclusions made.

We will diversify the whole idea in three directions of physical and analytical importance

which will hopefully go a long way to understand the phenomena and improve upon the

existing ideas. These include (a) Disappearance of bolus, (b) the size of the bolus and (c)

identification of trapping.

1.3 Disappearance of bolus

The bolus and its size are important in transport of physiological fluids and the movement

of food in esophagus. The boluses are made because of the circulatory flow patterns that

helps to transport the matter forward. Thus, the size of the bolus remains a central point

5

in almost all the discussions of peristalsis making the phenomena to be more crucial to

understand. The disappearance of bolus is even more important; since it changes the flow

pattern in a more subtle way with greater consequences in peristaltic transport. For exam-

ple, the flow can turn from the trapping to augmented flow or the backward flow. Realizing

the importance of bolus; we have concentrated on the reasons and conditions which are

responsible for the reduction of the size and its ultimate disappearance. What has been ob-

served is: the applied magnetic field is an important factor that reduces the size of the bolus.

For instance, [56] in their study of peristaltic transport of a Jeffrey fluid in the presence of

magnetic field report the formation of a bolus for a particular choice of parameters given

in Figures 6 to 11. [71] and [99] report similar observations in Figure 13 and Figure 5,

respectively. Some of the recent papers on peristalsis which also include results regarding

the formation of a bolus include [4, 7, 36, 39, 44, 75, 76, 83, 84, 90, 93, 94] and [22]. Al-

though the above said studies (and many others) give results showing the presence of bolus

for a particular choice of the parameters value, the mathematical framework used for the

identification of the bolus is not given in any of the reported studies. It appears to us that

the formation of a bolus is identified visually through contour plots (generated using some

mathematical packages) of the stream function i.e. checking if the contour plots include

any closed contours (streamlines). The identification of the presence of a bolus, through

visual analysis of contour plots, may have some serious implementation and reliability is-

sues. To start with, the whole process lacks robustness in a sense that in order to find a

combination of flow parameters that give rise to a bolus, one may have to generate several

contour plots corresponding to different values of various flow parameters. There is also

no guarantee that one can locate all the possible occurrences of a bolus by only analyzing

the contour plots. One should also be critical about the level curves plotted in a particular

contour plot. It might be the case that the mathematical package used for generating con-

tour plots did not include all of the level curves, due to the default settings of the package,

which can lead to false conclusions about the present flow patterns. Furthermore, in some

6

situation, it might be possible that the characteristic length scales of a flow pattern is very

small, hence it will not be visible on the plotted scale of the contour plot. As the presently

adopted technique, as discussed above, for the identification of a ‘bolus’ is prone to some

serious mathematical or human errors, one naturally asks if there is a better and mathe-

matically sound way of identifying a bolus? This is one of the motivations for the present

work. In this thesis, we show how mathematical analysis based on dynamical systems can

be used to identify a ‘bolus’.

The study of peristaltic flows in the presence of applied magnetic field is of great im-

portance due to its application in many clinical processes. For instance, in targeted drug

delivery [16], the motion of magnetic nanoparticles, used as drug carriers, is controlled

by externally applied magnetic field. It is also used to control the urethral flows [14, 67]

and to treat prostate cancer [51] by means of magnetic nanoparticle thermotherapy. Med-

ical procedures performed to remove any foreign object such as hair pin, toy or safety

pin, accidentally swallowed by a subject, make use of externally applied magnetic field

to steer such objects inside the body [40]. It is therefore quite important to understand

the effects of applied magnetic field on the peristalsis and the consequent effects on the

submerged particles. In this regards, a number of studies have been performed focusing

on the qualitative and quantitative flow investigations in terms of flow velocities, pressure

gradients and the discussion on developed flow patterns. Some of the recent studies include

[6, 8, 35, 53, 107, 109, 110].

In the peristaltic flows, under the presence of magnetic field [71, 99], it is commonly be-

lieved that the bolus, when formed, disappears when the magnetic field strength is in-

creased. However, in all of the reported studies, the claim that the bolus disappears with

increasing magnetic field is based on the visual analysis of the contour plots. For instance

[56] in their Figure 7 show contour plots at different values of the magnetic field parameter

(in an increasing order) and conclude that the bolus disappears with increasing magnetic

field based on the observation that closed streamlines disappear as magnetic field is in-

7

creased. The claim may be questioned on the fact that the contour plots may not include

all the level curves (we will discuss this in detail in section 3.3) or there might be some

flow patterns whose length scale is small and hence not visible in the given contour plot.

The conclusions based merely on the contour plots makes it weak as the contour plots may

not reveal all of the flow patterns due to default setting of the mathematical package or the

length scale issues as discussed in the preceding paragraph. Similar conclusions are also

made in [99] which also base their conclusion on the visual analysis of the contour plots.

Another issue with the aforementioned analysis is that we may not be able to find a critical

value of the magnetic field parameter at which the bolus disappears.

From the above discussion, it is clear that at present, the claim: ‘the bolus disappears

with increasing magnetic field’ which is solely based on the visual analysis of the contour

plot stands weak and requires more mathematical investigations in support of the claim or

otherwise. We analyze the aforesaid claim by means of the mathematical framework, in

terms of dynamical systems, that we propose for the identification of a bolus. The focus

of the present investigation is to determine whether the bolus, if formed, disappears with

increasing magnetic field. Through physical and mathematical explanation, we establish

the disappearance of bolus with increasing magnetic field (under certain flow conditions)

and also discuss various situations under which the claim may not be true i.e. the bolus

does not disappear with increasing magnetic field.

A confusion regarding physical description of a ‘bolus’ exists within wider research com-

munity investigating peristaltic flows. According to [92], a bolus is a circulatory flow pat-

tern which can be viewed through closed streamlines in a domain region. In our literature

survey and personal communication with some of the authors, it appears that each contour

level is also referred to as a ‘bolus’, not in agreement with the definition given by [92]. In

this thesis we adhere to the definition of [92] and consider whole of the circulatory pattern

as ‘one’ bolus.

8

1.4 Size of the bolus

After realizing importance of the bolus through its identification; the determination of the

size of the bolus is equally important. It is observed, in the literature, that size of the

bolus has been seen heuristically and casually. Generally, the size is recognized visually

from the stream function drawn from the analytical expression of stream function. No

comprehensive way, with a sound mathematical base is available. We feel that such an

important aspect needs a strong mathematical foundation which we present here. Some

background literature highlighting the need of such a study is now presented.

An important qualitative feature, as reported by [92], is the formation of a circular pattern

by means of some closed streamlines. This circular flow pattern can develop at different

locations and is usually referred to as a ‘bolus’. In some of the reported studies [e.g.

10, 39, 72] each contour level, within the same circulating flow pattern is referred to as

a separate bolus. We believe that a bolus, according to [92], refers to the whole of the

circulatory flow pattern, hence in this thesis, consistent with the terminology of [92], we

will consider whole of the circulation to be one bolus. The formation of a bolus can be of

practical use in applied medical science as it provides a circular region that moves with the

speed of the wave, hence the phenomenon can be used as a mean of safe transportation of

medical substances.

In the reported literature [4, 7, 22, 70], the formation of a bolus is usually determined by

means of contour plots which show the topological streamline patterns. We believe that

identifying the existence of a bolus through visual inspection of contour plots created with

the aid of some mathematical softwares such as Mathematica, Maple, Matlab or any other,

can be misleading as some of the geometric features might not be visible in a contour plot

due to the scaling issues i.e. if the length scale of a particular flow pattern is relatively

smaller than the length scale of the plot. Hence, a contour plot generated using some

mathematical package may not reveal all of the characteristic patterns present in the flow.

It is therefore important to include (at least) some of the contour lines from the set of

9

contour lines representing a particular flow pattern in order to show all of the present flow

patterns.

The contour plots are also used to analyze the characteristic features of a bolus such as

the size of a bolus [4, 5, 7, 10, 64, 82] for particular values of the flow parameters. For

instance, [4, 10, 22, 47] reported that the size of a bolus decreases as M is increased. It

appears to us that these conclusions were based on the visual inspection of the contour

plots at different values of M. Other examples include [36] and [45] in which the effects of

a particular flow parameter on the size of a bolus were investigated through visual analysis

of the contour plots. The contour lines plotted by different softwares can be different, based

on the default setting of the software, hence people may have different explanation of the

same flow viewed through contour plots generated using different softwares.

There does not exist any definitive measure for the size of a bolus, hence the reported

findings as mentioned above are based on the relative judgment derived from the graphical

plots generated through some mathematical packages. Furthermore, in all of the above

mentioned studies, a quantification for the size of a bolus is not given. We emphasize on

the fact that a bolus represents a circulatory flow pattern, hence its size must have some

associated quantitative measure based on which one can analyze how the quantitative size

of a bolus is affected when a particular flow parameter is varied. In particular, the term

‘bolus’ refers to a circulatory flow pattern developed in some region of the flow domain.

If a bolus exists, it can be used for the safe transport of a pill or other medical substance

of interest e.g. spermatozoa. The research investigations of peristaltic flows, therefore,

include the identification of flow conditions under which a bolus is formed. Furthermore,

if a bolus appears, it is also desirable to know how its existence is affected by the change of

certain flow parameters i.e. whether the bolus disappears or changes size when a particular

flow parameter is changed.

10

1.5 Identification of trapping

Peristaltic transport is an interesting problem of fluid mechanics in which the flow is gen-

erated through pressure gradients resulting from successive contraction and expansion of

the boundary, as in squeezing of a ketchup bottle or swallowing of food by anaconda. The

identification of trapping is another important area of attention which is squarely addressed

in the study of peristalsis. Although it is very essential but lack some understanding in its

definition and explanation. Various authors describe it with varying degree of exactness.

Trapping is somewhat more important; because the flow makes a circulatory region that

propagates forward with the speed of the wave, hence providing a mean of safe transporta-

tion.

The reported studies discussed above, have indeed improved our understanding of the peri-

staltic flow. We have now good information about quantitative flow behavior in terms of

flow velocities, developed pressure and the flow rate. However, little work has been done

in investigating qualitative flow behavior in terms of the flow patterns depending upon

the characterizing parameters of the problem. For example, in the reported studies (cited

above) the existence of bolus or occurrence of trapping is determined for some favorable

choice of the characterizing parameter. The reported studies are deficient in the sense: it

does not provide the information about the complete range of the parameter for which the

solution behavior remains the same. Further, what is the value of the parameter beyond

which the topological behavior may change? This important question is addressed using

bifurcation theory. As of now, it is decided through visual inspection of contour plots and

is thus based on hit and trial principle. This approach can thus characterize the flow for

a particular choice of parameter but is unable to classify the qualitative flow behavior for

complete range of parameter values.

Peristaltic transport behavior can be analyzed through streamlines and bifurcations with

the help of dynamical systems. This is a recent development in peristaltic flow research

enabling us to analyze the qualitative flow behavior in a peristaltic flow. Jimenez and Sen

11

presented the qualitative flow analysis for the peristaltic flow problem covering all these

aspects in [50]. This can be regarded as the most fundamental research in exploring the

strength of dynamical system to analyze the qualitative flow behavior in peristalsis that can

adequately answer the points raised in the preceding paragraph. Following this work, sim-

ilar studies have been carried out by other authors [12, 17, 18, 101, 102, 107]. Some other

fundamental work related to the use of dynamical system in fluid mechanics can be seen in

the references [19, 24].

[12, 17, 18, 50] make use of dynamical systems to identify parameter values that will result

in particular flow type. Their analysis depends on the availability of analytical equilibrium

points which is not always possible to obtain. Furthermore, extension of their work for

cases where more than two flow parameters exist is not trivial.

In a dynamical system, a bolus is in fact an elliptic equilibrium point i.e. a center. Trapping

is identified by the existence of a bolus (centers) on either side of the central line along with

a pair of saddles on the central line. In augmented flow, boluses form on either side of the

central line (as in trapping) but the accompanied pair of saddles lie above (below) the cen-

tral line. Motivated by this, we propose a new approach, based on the dynamical system,

for identification of parameter values for which trapping occurs. The presented method is

based on a sound mathematical frame work in contrast to the presently adopted approach

of inspecting contour plots of stream function in a hit and trial fashion. Furthermore our

method characterizes the qualitative flow behavior for the whole range of parameter values

unlike the presently adopted approach which characterizes flow behavior for a particular

value of the parameter. In contrast to the approach adopted in [12, 17, 18, 50], our method-

ology is not dependent on the availability of analytical equilibrium points. Furthermore, it

works for any number of flow parameters present in the governing model. The proposed ap-

proach is applied on a problem from the literature [10] for demonstration. The occurrence

of bolus, trapping, and augmented flow has been explained for the magnetic parameter,

viscosity, slip parameter and volume flow rate (see[10] for physical details).

12

Another important aspect of this study relates with the physical concepts of Bolus, Trap-

ping, Augmented and Backward flows. Some conflicts and discrepancies have been noted

in the literature in identifying these characteristic flow patterns. For example [10] refers the

flow as Trapping which is not consistent with [50, 92] and possibly others. We believe that

a clear understanding of these concepts is essential to put the record straight in the study

of peristalsis. Following the terminologies used by [92] and [50] for characterizing various

peristaltic flow patterns, we explicitly define these basic quantities to remove the confusion

present in the literature.

The thesis is structured as follows: Chapter 1 contains the Literature review.

Chapter 2 provides some basic definitions and flow models. Dimensionless numbers and

expressions for non Newtonian Jeffrey fluid are included. A short overview of dynamical

system is presented.

In Chapter 3, we present some results from [56] to elaborate the conventional contour plot

approach for identifying a bolus. We then propose a new mathematical framework, based

on the dynamical system, to characterize the qualitative behavior of the flow and to iden-

tify the existence of bolus. The claim of disappearance of bolus with increasing magnetic

field (in the literature) is investigated in depth. It is shown that the bolus, if it appears will

only vanish with increasing magnetic field M for negative values of flux q < 0. But it does

not disappear, (q ≥ 0), even for large M (as indicated by various authors in the literature)

but is only pushed towards the boundary. The content of this chapter is published in The

European Physical Journal Plus.

In Chapter 4 we propose a definition for accurately calculating the size of a bolus. The def-

inition is motivated by the appearance of closed streamlines around an elliptical stagnation

point or a center. This assertion is established by an example [50]. At some distance away

from the center, there will be a hyperbolic stagnation point (saddle points) determining the

boundary of the bolus. The area of the outermost closed loop is defined as the size of the

bolus. The content of this chapter is published in Physica A: Statistical Mechanics and

13

its Applications.

In Chapter 5, we propose a new approach, for the identification of peristaltic flow features—

‘bolus’, ‘backward flow’ ‘trapping’ and ‘augmented flow’. (a) The bolus is associated with

the existence of a center (an elliptic equilibrium point) (b) trapping occurs when center ex-

ists under the wave crests and a pair of saddles on the central line (c) the flow is augmented

when a center exist under the wave crust and a pair of saddle lie above and below the central

line. The contents of this chapter are reported in Physics of Fluids.

14

Chapter 2

Preliminaries

15

This chapter consists of some basic definitions and concepts used in the development of

the research presented in the thesis. The constitutive equations of non-Newtonian Jeffrey

fluid are included. Some relevant terminologies are explained. A brief review of dynamical

system theory is also given.

2.1 Peristalsis

The word “peristalsis” originates from the Greek word “peristaltikos” that means compress-

ing and clasping. It is a process in which the fluid is transported forward by compression

and decompression of an extensible tube to produce progressive waves that travels along

the length of the tube. The physiological fluids are transported in human beings through

this phenomena e.g. movement of food bolus from mouth to esophagus, transportation of

urine from kidney to bladder, movement of ova in the fallopian tubes, embryo transport in

the uterus and blood circulation in veins and arteries etc.

2.1.1 Adverse and favorable pressure gradients

In the context of Peristaltic flows in channels; the negative pressure gradient is favorable

pressure gradient whereas positive pressure gradient is adverse pressure gradient. The neg-

ative pressure gradient is along the wave propagation and helps the fluid flow. The adverse

pressure gradient has the opposite effect that resists the flow. The concept has a deep

meaning in the peristalsis flow particularly in physiological fluid flow problems. The ad-

verse pressure gradient can cause the reflux; a medical condition to be avoided for the

health reasons. While discussing our problems in the thesis; we observe that the volume

flow rate has a significant effect in generating the favorable or adverse pressure gradient.

The discussion will be particularly carried out to determine the range of the volume flow

rate parameter in determining a desired flow situation. We would also see that the bolus or

the closed streamlines situation lies in between the two extreme situations of favorable and

adverse pressure gradients.

16

2.1.2 Free pumping

Free pumping exists when the dimensionless time mean flow is positive and pressure dif-

ference at the ends of a tube or channel is zero.

2.1.3 Retrograde pumping

When the pressure difference is positive and the dimensionless time mean flow rate is

negative then the pumping is called retrograde pumping or backward pumping.

2.1.4 Peristaltic pumping

When both the pressure difference and the time mean flow rate are positive, then the pump-

ing is called as the peristaltic pumping.

2.1.5 Augmented pumping

Augmented pumping occurs when the pressure difference is negative and the time mean

flow rate is positive.

2.1.6 Trapping

Trapping is a phenomenon in which streamlines have the same shape as that the boundary

wall; however, under certain conditions, some streamlines split, and an internal circulating

bolus is formed. This bolus advances as a whole with the wave speed.

2.2 Types of flow patterns

2.2.1 Bolus

It refers to a circulatory flow pattern present in the domain (present under the wave crests).

17

−1

0

1

−1

0

1

0−1

0

1

−π/2 −π/4 π/4 π/2

q = −0.4

y

y

y

q = −0.2

q = 0.2

(c) Augmented Flow

(b) Trapping Flow

x

(a) Backward Flow

Figure 2.1: (From [50]) The contour plots of the stream function ψ [50][eq. (11)] at φ = 0.8

and for different values of volume flow rate q. The arrows represent the velocity vectors

(in moving frame of reference) computed from the stream function.

18

2.2.2 Backward flow

A flow type in which the whole of the fluid moves in a direction opposite to that of the

wave is termed as backward flow.

2.2.3 Trapping

In the phase plane diagram (Figure 2.1) the trapping can be visualized as the flow situation

where the flow along the central line is completely blocked by the boluses present under

the wave crests on either side of the central line of the channel.

2.2.4 Augmented flow

In this type of flow, boluses appear under the wave crests but (additionally) a stream of

fluid flows along the center line of the channel, in the same direction as that of the wave.

From dynamical system point of view; the augmented flow will consist of two saddle points

and one center in each upper and lower half of the channel. The saddles form the hetero-

clinic orbits on either side of the center line. The flow passing between the orbits along the

center line is the augmented flow.

2.3 Approximations

In peristaltic transport shape of the wave on the surface of the flexible channel or tube is

not known in advance. It constitutes unsteady boundary value problem. The modelling

of peristalsis can be performed considering wave train moving with a constant velocity

on the boundary. Normally, the problem is studied in the moving frame of reference by

immobilizing the boundary using the travelling wave transformation. That is, it is more

convenient to work in moving frame of reference, because it renders the problem as steady

state. Geometrically, two parameters are of importance. First is amplitude ratio; that is,

ratio of the amplitude of the wave to the radius of the tube or the height for a planar channel.

The range of the amplitude ratio must be between zero to one. The zero value is for straight

19

tube or channel and the value one will make a complete constriction. Second parameter is

the wave number which is the ratio of radius/height to wavelength. In order to simplify the

governing equations, without losing the physics of the problem, the small amplitude and

long wave length approximations are generally applied. This makes the asymptotic method

applicable. Also, it goes difficult to find the solutions in the generalized sense. Generally,

the peristaltic problems are three dimension, but due to complexity of this situation, it is

considered as two dimensional because it does have the experimental support.

2.4 No slip boundary condition

The boundary conditions are essential for the well posed boundary value problems to find

the velocity field and the pressure appearing in the Navier Stokes equations and continuity

equation. For the fluid-solid interface; the no-slip condition, for impermeable boundary is

given by

u.n = 0 on ∂Ω, (2.1)

where u is the velocity and n is outward unit normal. This suggests that the fluid molecules

at the boundary attains the velocity of the boundary. It may be mentioned that the no-slip

boundary condition is generally considered as hypothesis but is most widely used. It has

prompted many researchers to report slip boundary condition on the fluid-solid interface

[58, 96, 111].

2.5 Slip boundary condition

The velocity of the fluid at the static boundary is not equal to zero. The slip boundary

condition (Navier slip condition ) is given by [77].

µt [u− (u.n)n]+ (1−µt) [T.n− (n.T.n)n] = 0, (2.2)

20

where T is the stress tensor, n is outward drawn normal, u is velocity and µt is a non-

dimensional parameter. The slip length is a parameter that can be given by

u = lsn.[(∇u)+(∇u)T ]. (2.3)

Slip length can be thought of the length that is extended to outside the flow domain where

the no slip condition is satisfied. Here ls is assumed to be a material parameter.

2.6 Non-dimensional numbers

Non-dimensionalisation is important in physics and engineering processes. It helps us for

the simplification of differential equation by reducing the number of parameters by re-

scaling the dimensional variables. The non-dimensional quantities are particularly useful

when the parameters can assume large and small values. In the limiting cases of small and

large values the model equations can be simplified by order analysis. Another benefit is

that we can work mathematically without the fear of dimensions chosen and the laboratory

models can be extended to any scale. Here we define some dimensionless numbers which

will be appearing in the later chapters.

2.6.1 Amplitude ratio

This is the ratio between amplitude of wave to the radius/height for the peristaltic phe-

nomenon on tubes/channels. The amplitude ratio determines the amount of occlusion. Very

small amplitude ratio corresponds to small amplitude in comparison to the radius or height.

2.6.2 Reynolds number

It is defined as the ratio of inertial forces to the viscous forces and is commonly denoted

by Re. The small Reynolds number corresponds to laminar flow and very large number

corresponds to turbulence. For Re << 1 the viscous force dominate the inertial forces

and for Re >> 1, the inertial force dominates the viscous force. After certain value of

21

the Reynolds number the laminar flow may become turbulent after passing the transitory

domain.

Re =Inertial f orcesviscous f orces

(2.4)

2.6.3 Hartmann number

In electrically conducting fluid Lorentz force (body force) is developed when the fluid is

subjected to applied magnetic field. The study of such fluids is called Magnetohydrody-

namics (MHD). The Lorentz force appears as a body force in the momentum equation

and is resistive in nature that slows down the velocity field. The Hartmann number is a

non-dimensional number defined as the ratio of Lorentz force to viscous force. Another,

parameter of interest is interaction parameter (N) which is the ratio of Lorentz force to in-

ertial force. The magnetic Reynolds, Rem, number is defined as the ratio between advection

and diffusion of the magnetic field of lines in the magnetic transport equation. The induced

magnetic field is normally taken as negligible for Rem << 1. For large magnetic Reynolds

number the magnetic lines of force are frozen in the moving fluid. The Hartmann number

can thus be represented mathematically as

M =

√σ

µBL, (2.5)

where µ is the fluid dynamic viscosity, σ is the electrical conductivity and L is the charac-

teristic length.

2.6.4 Wave number

This is a relation between the radius or width of the tube or channel to the wave length of

the peristaltic wave. Geometrically, it relates the slope with the curvature made on the wall

by the wave.

22

2.7 Continuity equation

Total mass inside a control volume V (C.V.) is given by∫

V ρdV . The change of mass per

unit time isddt

∫V

ρdV =∫

V

∂ρ

∂ tdV . (2.6)

The rate of mass entering the (C.V.) through control surface (C.S.) is∫

S ρ u.dS. The rate of

mass change inside C.V. is given by∫V

∂ρ

∂ tdV +

∫S

ρ u.dS = 0, (2.7)

or using Gauss divergence theorem, we obtain∫V

∂ρ

∂ tdV +

∫V

∇.(ρ u)dV = 0, (2.8)

or ∫V

[∂ρ

∂ t+∇.(ρ u)

]dV = 0. (2.9)

Since dV is arbitrary, we obtain [∂ρ

∂ t+∇.(ρ u)

]= 0, (2.10)

or [∂ρ

∂ t+ u.∇ρ +ρ∇.u

]= 0, (2.11)

orDρ

Dt+ρ∇.u = 0, (2.12)

whereDDt

=∂

∂ t+ u.∇. (2.13)

For incompressible fluidDρ

Dt= 0. (2.14)

Thus the continuity equation (incompressible fluid) is

∇.u = 0. (2.15)

23

2.8 Momentum equation

Momentum in the C.V. is given∫

S ρ udV . The rate of momentum entering the C.S. is

∫S

ρ u u.dS. (2.16)

Total rate of change of momentum in C.V. is∫V

∂

∂ t(ρ u)dV +

∫V

∇.(ρ u u)dV =∫

V

[ρ

∂ u∂ t

+ u∂ρ

∂ t+ u∇.(ρ u) + ρ(u.∇)u

]dV

=∫

V

[u(

∂ρ

∂ t+ ∇.(ρ u)

)+ ρ

(∂ u∂ t

+ u.∇u)]

dV

=∫

Vρ

DuDt

dV .

The total forces applied on the C.S. are viscous forces, pressure and gravitational forces.

The viscous force on surface dS is τ(n)dS, where τ(n) is stress vector in the direction normal

to C.S. The gravitational force is ρ gdV on volume dV and the pressure is −pndS on dS

The total force is then ∫S(−pn+ τ)n dS+

∫V

ρ gdV . (2.17)

Newton’s second law of motion to C.V. gives

∫V

ρDuDt

=∫

S(−pn+ τ)n dS+

∫V

ρ gdV , (2.18)

∫V

ρDuDt

=∫

V(−∇p+ρ g)dV +

∫S

τndS. (2.19)

Now

τn = nxτ

x +nyτy +nzτ

z,

where

τx = τxx i+ τxy j+ τxzk,

τy = τyx i+ τyy j+ τyzk,

τz = τzx i+ τzy j+ τzzk.

24

Therefore∫S

τndS =

∫S

[nx(τxx i+ τxy j + τxzk

)+ ny

(τyx i+ τyy j + τyzk

)+ nz

(τzx i+ τzy j + τzzk

)]dS,

(2.20)

=∫

V

[∂

∂x

(τxx i + τxy j + τxzk

)+

∂

∂y

(τyx i + τyy j + τyzk

)+

∂

∂ z

(τzx i + τzy j + τzzk

)]dV ,

(2.21)

=∫

V

[i(

∂τxx

∂x+

∂τyx

∂y+

∂τzx

∂ z

)+ j(

∂τxy

∂x+

∂τyy

∂y+

∂τzy

∂ z

)+ k(

∂τxz

∂x+

∂τyz

∂y+

∂τzz

∂ z

)],

(2.22)

(2.23)∫

V∇.τ = 0,

where

τ =

τxx τxy τxz

τyx τyy τyz

τzx τzy τzz

(2.24)

is stress tensor. Reverting back to the balance of forces∫V

ρDuDt

dV =∫

V(−∇p+ρ g+∇.τ)dV . (2.25)

Giving the momentum equation as

ρDuDt

= (−∇p+ρ g+∇.τ) . (2.26)

2.9 Non newtonian fluids

Although a vast majority of the fluid behave like Newtonian fluid, there are fluids that

do not strictly follow the Newtonian constitutive equations. Examples are polymeric liq-

uids, suspensions, blood, physiological fluid etc. These fluids are called non-Newtonian

fluids. The non-Newtonian fluids can be broadly classified into three categories, namely,

generalized Newtonian fluid, linear viscoelastic fluids and nonlinear viscoelastic fluids. In

generalized Newtonian model the stress is nonlinearly proportional to rate of strain. There

25

are no time derivatives of either stress or rate of strain. In linear viscoelastic model, the

constitutive equation is linear in stress and rate of strain involving time derivatives of stress

and rate of strains. The time derivatives can be of stress or/and rate of strain. The most

generalized constitutive equation involves nonlinear relationship and the time derivatives

of stress and strain rate. These fluids are called nonlinear viscoelastic fluids. The most sim-

ple linear viscoelastic model is Maxwell fluid that involves time derivative of stress in the

constitutive relationship. However, no time derivative of strain rate is involved. Another

important linear viscoelastic model is Jeffrey’s model that involves time derivatives of both

stress and strain rate. The Jeffrey’s model has thus three constant parameters namely the

ambient shear rate viscosity and the time constants–stress relaxation and retardation time.

The model reduces to Newtonian model when the two time constants approaches zero.

We in this thesis will be using Jeffrey’s model which is linear viscoelastic fluid. We will

thus be concentrating on the modelling of Jeffrey’s fluid only.

2.9.1 Jeffrey’s model

We will include related equations and definitions for peristaltic flows with different geome-

tries.

2.9.2 Constitutive equations in a Jeffrey fluid

The Jeffrey fluid model is a relatively simple model and its rheology is different from New-

tonian fluid. The Jeffrey fluid model predicts relaxation/retardation time effects. Cauchy

stress tensor T in a Jeffrey fluid is defined by the following expression

T =−PI+S, (2.27)

in which the extra stress tensor S satisfies

S =µ

1+λ1

[A1 +λ2

dA1

dt

], (2.28)

A1 = L+L>, (2.29)

26

where I is the identity tensor, P is the pressure, µ is the viscosity of the fluid, λ1 is the ratio

of relaxation to retardation times, λ2 is the retardation time and A1 is the Rivilin Ericksen

tensor.

For two-dimensional flow the velocity field vector V takes the form as

V = [U (X ,Y, t) ,V (X ,Y, t) ,0] . (2.30)

Using equation (2.30) one can write

L = gradV =

∂U∂X

∂U∂Y 0

∂V∂X

∂V∂Y 0

0 0 0

, L> = (gradV)> =

∂U∂X

∂V∂X 0

∂U∂Y

∂V∂Y 0

0 0 0

, (2.31)

A1 =

2∂U

∂X∂U∂Y + ∂V

∂X 0

∂V∂X + ∂U

∂Y 2∂V∂Y 0

0 0 0

, (2.32)

dA1

dt=

2(

∂

∂ t +U ∂

∂X +V ∂

∂Y

)∂U∂X

(∂

∂ t +U ∂

∂X +V ∂

∂Y

)(∂U∂Y + ∂V

∂X

)0(

∂

∂ t +U ∂

∂X +V ∂

∂Y

)(∂V∂X + ∂U

∂Y

)2(

∂

∂ t +U ∂

∂X +V ∂

∂Y

)∂V∂Y 0

0 0 0

.(2.33)

The components of extra stress tensor S are therefore given by

SXX =2µ

1+λ1

[1+λ2

(∂

∂ t+U

∂

∂X+V

∂

∂Y

)]∂U∂X

, (2.34)

SXY =µ

1+λ1

[1+λ2

(∂

∂ t+U

∂

∂X+V

∂

∂Y

)](∂V∂X

+∂U∂Y

), (2.35)

SYY =2µ

1+λ1

[1+λ2

(∂

∂ t+U

∂

∂X+V

∂

∂Y

)]∂V∂Y

, (2.36)

SY X = SXY , SXZ = SY Z = SZX = SZY = SZZ = 0. (2.37)

The momentum equation for two-dimensional flow of Jeffrey fluid is given by

ρ

(∂U∂ t

+U∂U∂X

+V∂U∂Y

)=

∂P∂X

+∂

∂X(SXX)+

∂

∂Y(SXY )+ρfX , (2.38)

ρ

(∂V∂ t

+U∂V∂X

+V∂V∂Y

)=

∂P∂Y

+∂

∂X(SXY )+

∂

∂Y(SYY )+ρfX . (2.39)

27

2.10 Dynamical system

Let the two ordinary differential equation is represented by x1(t) and x2(t), that governs

any physical or chemical law; may be written as

x1 = F1(x1,x2),

x2 = F2(x1,x2). (2.40)

In vector representation, the above equation can be written as

X = F(X). (2.41)

2.10.1 Autonomous systems

The system (2.41) is termed as autonomous because it does not depends explicitly on time.

The dynamical system is represented by equation (2.41). This system has three main con-

stituents time, state space, law of evolution. For X = F(X), time is continuous, F defines

the evolution and it is independent of time. The state space is also called as phase plane.

2.10.2 Stationary solutions

Stationary solutions are the points, in the phase plane, where the system is at rest. These

are also defined as constant solutions, steady states, stagnation points, critical points, equi-

librium points, singular points, rest points etc. Mathematically at X = X0

F(X0) = 0. (2.42)

2.10.3 Asymptotically stable equilibrium

The equilibrium solutions are asymptotically stable when small perturbation vanishes as

time goes to infinity.

28

2.10.4 Stable equilibrium

The stationary solution is stable if the response to a small perturbation remains small as the

time evolves. It is unstable as the deviation grows.

2.10.5 Hyperbolic (nondegenerate) equilibrium point

The stationary solution is nondegenerate (hyperbolic) when the Jacobian matrix has no

eigenvalues with zero real part.

2.10.6 Types of equilibrium points

For linear system of equations X = AX, the qualitative nature of trajectories in the neigh-

borhood of equilibrium point.

2.10.6.1 Saddle

When the eigenvalues that corresponds to a linear system are real and of opposite signs

then the equilibrium point is saddle. A saddle point is always unstable.

2.10.6.2 Node

When the eigenvalues are real and are of same signs then the equilibrium point is node. It

is stable when both the eigenvalues are negative because all the trajectories approach to the

equilibrium point as the time evolves. It is unstable when both the eigenvalues are positive

as the solution blow up when time goes to infinity.

2.10.6.3 Spiral

When eigenvalues are complex conjugates then trajectories are periodic solutions. These

solutions blow when real part of eigenvalues is positive and decay when real part of eigen-

values is negative.

29

2.10.6.4 Center

When the eigenvalues are purely imaginary, the trajectories are periodic solutions and they

revolve around the equilibrium point.

2.11 Dynamical system and fluid mechanics

Peristalsis is a mean to transport liquid and gases in a flexible/duct by contracting travel-

ling waves. In biological system, the physiological fluid is transported by the contraction

of muscles in rhythmic fashion. For example, the movement of food through esophagus

and other anatomical passages happens due to peristaltic wave motion in a natural way.

Peristaltic pumps are emulated on the phenomenon of peristalsis. These are designed on

macroscopic and microscopic levels. The macroscopic pumps include rotary and linear

peristaltic pumps. The flexible tube configuration is important in a way that: the increase

of pumping cycle or the diameter increases the flow rate. A big advantage of peristaltic

pumps is that it isolates the fluid from the environment completely.

We briefly discuss the qualitative behavior of the fluid flow problem using the theory of

dynamical system - the concept used in this thesis. The solutions of governing differential

equations give the topological properties of the flow patterns by the phase diagrams drawn

in the phase plane. The topological behavior of the Navier Stokes equations can be studied

near the stationary/equilibrium points. The streamlines of the flow problem near the sta-

tionary point can then be represented by the phase plane trajectories in the phase space of

the dynamical system.

The change is a natural phenomenon that we experience in everyday life. This change of

state can be qualitative or quantitative. The quantitative change corresponds to a small re-

versible perturbation, whereas, qualitative change corresponds to a permanent change of

state that is not reversible. The study of dynamical system helps to understand; how to

control the system to a desired state. The state of the system is characterized by one or

more parameters called as controlling parameters. The qualitative behavior of the dynam-

30

ical system switches, from stable to unstable motion, stationary state to motion, regular

to irregular, from steady to unsteady motion etc. This change of behavior can be due to

physical, chemical or biological parameter.

The state of the system is defined by the functions called state functions. The dynamical

system is represented by a system of first order differential equations. The state space is

called phase plane and the the solution set in it is called phase diagram. The dynamical

system analysis is initiated from the equilibrium solution or the stationary solution given

by the equations

f (x,y,α,β ,q,M,φ) = 0, (2.43)

g(x,y,α,β ,q,M,φ) = 0. (2.44)

And the differential equation representing the trajectories determine the behavior of the

solution. The stationary solution will be stable if the small perturbation of the stationary

solution goes to zero as the time elapse to infinity. The solution that is not stable is unstable.

However, the stability is of local nature. In this thesis, we will mainly concern with linear

stability analysis that is close to the stationary point and is called local stability. The local

stability and the qualitative behavior can be determined by the eigenvalue of the dynamical

system. An important qualification of the equilibrium is hyperbolic or nonhyperbolic or

degenerate. The behavior at hyperbolic equilibrium can be classified as node (stable or un-

stable), unstable saddle or stable/unstable spiral. The corresponding eigenvalue have one

common property that the real part is nonzero. The stable/unstable branches from the topo-

logical point of view are termed as stable/unstable manifold. The exception occurs when

eigenvalue are equal, one of the eigenvalue is zero or the eigenvalue are purely imaginary.

These exceptional cases are very important that gives rise to the change of the behavior

when the characterizing parameters of the dynamical system are varied. The change of the

behavior with these parameters occurs at the bifurcation point located in the bifurcation di-

agrams. Since in this thesis, we will be discussing local stability ‘the theorem of Hartman

Grobman’ for hyperbolic equilibria ensures the local behavior is qualitatively the same that

31

we find from the linearization of the nonlinear system. In order to look for a global picture,

one solution of importance is limit cycle which also falls in the category of stable orbits.

The periodic orbits with infinite time period have tremendous applications and substantial

manifestations in fluid mechanics and plasma physics. The travelling wave solution may

lead to such orbits called heteroclinic and homoclinic orbits. From the wave representa-

tions these correspond to shock waves and solitary waves respectively. Therefore a few

words about these will not be out of place.

Peristalsis is a basic mechanism by which biological fluids are transported in human body.

A number of devices have been invented, which are based on this principle. The peristaltic

flow presents itself as an augmented flow, backward flow, trapping and formation of bo-

luses. Analysis of some recent literature on peristalsis shows that these phenomena are

some what miss interpreted and we feel that some amount of clarity is needed. We try to

explain (here) some of the grey areas to place these concepts in the right perspective. We

consider the work of [92] as authentic for interpreting and defining these concepts.

We present a model problem explaining the concepts as we perceive them. We remember

that the flow behavior in peristalsis strongly depends upon the governing parameters of the

problem. And the solution topology changes even for a small variation of these parameters.

In most of the literature, the behavior is determined by plotting the solution (streamlines)

for some discrete value of these parameters and the solution topology is decided visually.

A prior knowledge of the solution behavior at arbitrary value is not predicted before hand.

Also that, we may see some behaviors while a few may remain dormant. An answer to

these observations lie in dynamical system theory. The solution behavior is found by in-

vestigating critical points and the stability considerations. The bifurcation diagram helps

to determine continuous domain of one parameter keeping other parameters fixed.

The theory of dynamical system determines the quantitative and qualitative change of the

system through the variation of the parameter that controls the behavior of the state. The

qualitative change in the behavior through variation of the parameter defines the bifurca-

32

tion. An important aspect of the dynamical system is to avoid the undesirable qualitative

change that may occur in the system. For example, stability to instability, regular to irregu-

lar or regular to chaos or turbulent. A suitable choice of the parameter may then determine

the required flow behavior.

2.12 Bifurcation

Most often the dynamical system involves one or more parameters. The position of station-

ary points and the qualitative behavior will depend upon these parameters. The variation of

the parameter leads to different qualitative behaviors when passing through some threshold

values. For example, the real part of the eigenvalue of the Jacobean matrix may change

from negative to positive while taking up zero value momentarily. All these situations cor-

respond to different solution behaviors. Means to say the qualitative or topological behavior

switches between different regimes with the variation of these parameters.

2.12.1 Bifurcation parameter

It defines the changes in the topological structure of a dynamical system when parameter

values are varied. These changes are structural or qualitative changes in the behavior of

dynamical system. The parameter whose values are being varied is called the bifurcation

parameter.

2.12.2 Bifurcation diagram (equilibria curves)

This is the graphical representation of solution of autonomous system for spatial location

of stagnation points as a function of parameter. These diagrams are also called equilibria

curves.

33

2.12.3 Bifurcation point

A bifurcation point (branch point) with respect to parameter is a solution of equation (2.42)

where the number of solutions changes when parameter passes some specific value.

2.12.4 Types of bifurcation

Appearance and variation of the defining parameters of the system varies the stationary

point, its location in the phase plane and the qualitative behavior of the solution. The bi-

furcation diagram shows the graph of the equilibria with respect to the parameter. The

qualitative behavior or the number of solutions may change at some critical value of this

parameter. The point is called bifurcation point.

There are many classifications of the bifurcations. Some of these are turning point (fold

bifurcation), pitchfork, transcritical, supercritical and subcritical bifurcations. Hopf bifur-

cation switches the equilibria to periodic oscillations. The mentioned bifurcations are local

in nature and are generally referred as stationary bifurcations. Another, important class of

bifurcation corresponds to periodic orbits. Periodic orbits with infinite period allow homo-

clinic and heteroclinic bifurcations. The corresponding orbits are solitary wave (solitonic

solution/pulse) and shock wave structures (wave front/kinks). This behavior has special

significance in the travelling wave solution of the partial differential equations. In the sense

of phase plane there are two centers and one saddle such that the orbit connects itself at the

saddle, the orbit is called homoclinic. Similarly, if the orbit connects two distinct saddles,

it constitutes heteroclinic orbit. From mathematical point of view, if the travelling wave

reaches a constant state at both ± ∞ it represents a homoclinic orbit and if it reaches two

distinct states it corresponds to heteroclinic orbit. The homoclinic and heteroclinic orbits

have a special interest in peristalsis wave propagation in fluid mechanics where the flow

behaviors are determined by the streamlines generating due to the presence of saddles and

centers. The saddles presenting inset and outset branches in the flow domain while centers

are the emergence of closed streamlines signifying the boluses.

34

−10 −5 0 5 10−10

−5

0

5

10

λ0

y

Figure 2.2: Transcritical Bifurcation.

2.12.5 Transcritical bifurcation

y = λ0y− y2. (2.45)

The stagnation points are y = 0 and y = λ0. The red lines show the stable branches while

the blue lines shown the unstable branch of the problem. The branch y = 0 loses stability

at the bifurcation point (0,0). Concomitantly there is an exchange of stability at the other

branch.

2.12.6 Supercritical pitchfork bifurcation

This is the elementary example of supercritical pitchfork bifurcation.

x = λ0x− x3. (2.46)

There exist three equilibrium points/stagnation points x = 0,x =±√

λ . The red lines show

the stable branches whereas the blue lines show the unstable branch of the dynamical sys-

tem. The shifting of stability is shown in Figure 2.3. The differential equation is symmetric

35

−10 −5 0 5 10−4

−3

−2

−1

0

1

2

3

4

λ0

x

Figure 2.3: Supercritical Pitchfork Bifurcation.

that is invariant with respect to x and it can be found in the stagnation points. The symmetry

plays a decisive role in the bifurcation theory.

2.12.7 Subcritical pitchfork bifurcation

This is the basic example of subcritical pitchfork bifurcation.

x = λ0x+ x3. (2.47)

Figure 2.4 depicts the stability behavior of the equilibrium points. The red lines represent

stable stagnation points whereas the blue lines show the unstable branch of the dynamical

system. The stability is lost at the bifurcation point (0,0) locally. There is no exchange of

stability when compared with supercritical bifurcation.

2.12.8 Hartman-Grobman theorem

The family of trajectories near a stagnation point/equilibrium point of a nonlinear sys-

tem and those of the locally linear system have the same structure (topological ); if Jaco-

36

−10 −5 0 5 10−4

−2

0

2

4

λ0

x

Figure 2.4: Subcritical Pitchfork Bifurcation.

bian matrix has no eigenvalues with zero real part. That means in a vicinity of stagnation

point/equilibrium point there lies a homeomorphism which maps trajectories of the nonlin-

ear system into trajectories of the linear system [19].

2.12.9 Bendixson’s criterion

Let x = M(x,y) and y = N(x,y) be an analytical dynamical system and let W be a simply-

connected domain of the phase portrait on which the divergence of the vector field is not

identically zero and independent of sign. Then there exists no closed orbits lying in W [19].

2.12.10 Limit cycle

Periodic solutions of a dynamical system emerged as closed trajectories in the phase plane

and must divide the plain into two regions (outer and inner). If there exist a neighborhood

of closed trajectory that does not contain another closed one, such trajectory is called a limit

cycle. It is stable if all the neighboring paths approach it as time goes to infinity, otherwise

it is unstable.

37

2.12.11 Structurally stable/unstable

Consider the dynamical system defined over a domain

y = f (y). (2.48)

The dynamical system is structurally stable if a minuscule change of f (y) leaves the topo-

logical structure unaltered in the given domain, otherwise it is said to be structurally unsta-

ble.

2.12.12 Co-dimension

It is defined as the number of parameters that must be diversified for the bifurcation to

exist. It is the smallest dimension of the parameters space that holds the bifurcation in all

its means. Bogdanov-Takens bifurcation is an example of co-dimension two, saddle-node,

hopf, transcritical and pitchfork bifurcations are all examples of co-dimension one.

2.12.13 Separatrix

The circulation on a simple closed trajectory determined by the junction of homoclinic

connection and the stagnation point at the origin is called separatrix. The separatrix splits

the phase space in to attracting basins. The two inset branches of a saddle point in a

homoclinc orbit are the examples of separatrix.

2.12.14 Local bifurcation

A local bifurcation exist if the topological structure of phase plane is altered locally. These

bifurcations are studied with local analysis. The bifurcation changes the topological struc-

ture of the phase plane only in the vicinity of the stagnation point (nonhyperbolic in nature).

The steady state bifurcation occurs when the eigenvalue is zero but if the eigenvalues are

purely imaginary, this is a Hopf bifurcation.

38

2.12.15 Global bifurcation

When large invariant sets (periodic orbits) strike into the stagnation points and the topolog-

ical structure of the trajectories in the phase plane is affected. The change is not restricted

to small neighborhood as in local bifurcation rather it is extended to arbitrarily large dis-

tances. Such type of bifurcations is global bifurcations. Saddle connections and multiple

limit cycles are common examples.

2.12.16 Travelling wave

Travelling wave propagates in a specific direction with constant velocity while retaining its

shape. In biology, impulses and peristaltic waves are represented by travelling waves. In

fluid dynamics, the travelling waves are categorized as periodic wave, solitary wave and

shock wave. The solutions of travelling wave can be represented by u(x, t) = F(ξ1), where

ξ1 = (x− ct), the space and time variables are given by x and t and c is the velocity of

propagation of the wave. When c = 0, it represents the stationary wave. These waves have

the properties that differentiate them from each other.

2.12.16.1 Periodic wave

The travelling wave having the property F(ξ1, p) = F(ξ1) is called the periodic wave with

period p.

2.12.16.2 Pulse

The wave that attains the constant state such that u(∞) = u1 = u(−∞), is called as pulse.

2.12.16.3 Wave front

If the constant states are such that u(∞) = u1 and u(−∞) = u2; u1 is not equal to u2. The

wave is called wave front. These definitions from the dynamical system point of view; can

be seen as follows.

39

2.12.17 Homoclinic orbits

A homoclinic orbit joins a saddle point/node to itself. Or in other words; it is the trajec-

tory of the dynamical system in the phase space that connects saddle point to itself. The

wave type is also called pulse (mathematically defined as above) or solitary wave and the

corresponding solution is called solitonic. To connect the two definitions; we say that in

order to have a solitary solution, there need to be a homoclinic orbit in the phase space at

the critical point.

2.12.18 Heteroclinic orbits

A heteroclinc orbits connects two distinct saddles. Or it may connect node to a saddle

or saddle to a node. From the earlier definition it is also called a wave front. The wave

structure is kink or shock.

40

Chapter 3

Disappearing of a “bolus” with increasing magnetic field;

Peristaltic flows

41

3.1 Introduction

In this chapter, we address two important questions regarding peristaltic flows: (1) How

to identify a ‘bolus’–a circulatory flow pattern– and (2) Does the bolus disappear with

increasing the magnetic field M? We propose a method based on the dynamical system

which efficiently detects the presence of a bolus. The proposed method is also used to

calculate the critical value of a parameter at which the qualitative flow behavior changes.

Furthermore, the method characterizes the qualitative flow behavior for the complete range

of the parameter values instead of some discrete values. The bolus, if it exists, disappears

with increasing the magnetic field M only if the flux is negative. Otherwise, the bolus is

pushed towards the boundary but does not disappear even for large M.

3.2 Identifying a bolus

In this section, we first elaborate the conventional approach of identifying a bolus through

visual analysis of contour plots. The limitations and drawbacks of this technique, as men-

tioned in the Introduction, are discussed in detail with examples from [56]. We then propose

an alternative method based on the dynamical systems to identify the existence of a bolus.

The advantages of the proposed methods over the conventional approach are discussed in

detail.

3.2.1 Through contour plots

The results presented in this section represent the solution, reported in [56], of an incom-

pressible viscous fluid. The flow, in a two dimensional channel, develops due to propaga-

tion of sinusoidal waves at the boundaries. The mathematical model governing the consid-

ered physical problem includes the two dimensional continuity and Navier Stokes equation

along with moving wall boundary conditions at the top and the bottom. The author has also

investigated the effects of the magnetic field on the developed peristaltic flow. The solution

42

is obtained in terms of stream function ψ , by making use of the long wavelength and the

low Reynolds number approximation (see [56] for details). As the subject of this chapter

is of generic nature concerning the graphical analysis technique used in peristaltic flow

investigations, we have therefore omitted the mathematical details of the governing prob-

lem and the adopted solution methodology. We refer keen readers to [56] where detailed

mathematical analysis is presented. We have two fold objective of using results of [56];

one is to highlight the issues in the graphical analysis technique adopted in previously re-

ported studies; second is to validate our proposed technique for the identification of a bolus.

As mentioned in the Introduction, the reported studies of peristalsis do not specify any

mathematical methodology to identify the presence of a bolus. It seems that the identifi-

cation of a bolus is carried out by mere visualization of contour plots generated through

some mathematical package. For illustration, we reproduce the results of [56, Figure 7] by

generating contour plots in Matlab using their stream function. The parameters that appear

in the stream function are: flux q, amplitudes of the waves a, b, Jeffrey fluid parameter λ1

and the phase shift φ1 between the waves at the top and the bottom boundary. Figure 3.1

gives the contour plots at q =−0.1,a = 0.4,b = 0.5,λ1 = 0.3,φ1 = π/8 for different values

of the magnetic field parameter M. It can be seen in Figure 3.1 (a-c) that there are some

closed streamlines indicating the existence of a bolus. [56] have shown similar contour

plots in their work (Figure 7) and reported the existence of a bolus at M = 0, M = 2 and

M = 4. However, it is not clear how the author chose this particular set of parameter values

(q = −0.1,a = 0.4,b = 0.5,λ1 = 0.3,φ1 = π/8,M = 0,2 and 4), giving rise to a bolus,

from a wide range of possible combinations of parameter values. Furthermore, it is also

not clear whether this is the only flow condition under which a bolus is formed or if there

exists other combinations of parameter values giving rise to a bolus.

One of the drawbacks of this approach, for identifying a bolus through contour plots, is that

we may not see all the flow patterns in a contour plot generated through some mathematical

43

-2 0 2

-1

0

1

-2 0 2

-1

0

1

-2 0 2

-1

0

1

-2 0 2

-1

0

1

(b)

(c) (d)

(a)

M = 1

M = 6 M = 18

M = 0.01

a = 0.4, b = 0.5, q = −0.1φ = π/8, λ1 = 0.3

Figure 3.1: A contour plot of the stream function [56, equation 19] generated using Matlab

at different values of the magnetic field parameter (a) M = 0.01, (b) M = 1, (c) M = 6 and

(d) M = 18.

44

packages for two reasons: (1) the mathematical package may plot certain level curves based

on the default setting of the software (2) the flow patterns with relatively small length scales

may not be visible on the contour plot. For example, consider the results of [56, Figure 7]

which are reproduced here in Figure 3.1. In [56] it is concluded that the bolus disappears

for M = 6 based on the fact that the contour plot corresponding to M = 6 [56, Figure 7(d)]

does not have any closed streamlines. Interestingly, when we reproduce the same results of

[56] using Matlab, the closed streamlines do appear (shown in Figure 3.1(c)) on contrary

to the observations of [56]. It is this technical issue that we want to emphasize i.e. in a

contour plot generated through some mathematical package, the contour levels are picked

up at random and may not necessarily reveal all of the important flow parameters. Hence

for completeness one must include some contour levels from the set of contour curves

representing a particular flow pattern to get a complete picture in a contour plot.

It is observed that no closed streamlines are present at M = 18, hence it seems right to say

that the bolus disappears with increasing M as claimed in [56]. However, the identification

of the critical value of M at which this qualitative flow behavior changes is yet to be deter-

mined. It is indeed a cumbersome job, if one resorts to the visual analysis of contour plots,

for the identification of the critical M as one may have to generate several contour plots

(with great care) at different values of M.

It is evident from the above discussion that a mathematical framework is required for the

identification of a bolus with an objective that this method will eliminate the need of test-

ing various combinations of flow parameters to determine the existence of a bolus. Fur-

thermore, the method should also be able to identify all the possible flow conditions under

which a bolus can form. In the following section, we propose one such method which not

only meets the above mentioned objectives but also gives the critical value of M (or any

other parameter) at which the flow behavior changes.

45

3.2.2 Through the dynamical system

A ‘bolus’, as it is defined, is in fact a circulatory flow pattern which is identified by close

streamlines in the contour plot of a stream function. This (or any other) qualitative flow

behavior can be successfully analyzed by the consideration of dynamical system [50]. It is

known from the basic continuum mechanics that if x represents the instantaneous spatial

location of a particle then its velocity is calculated as x = v(x, t), where dot represents

the temporal derivative. For a two-dimensional flow, following [50], we use the velocity

expressions in terms of the stream functions to define a dynamical system, i.e.

dxdt

=∂ψ

∂y, (3.1)

dydt

= −∂ψ

∂x, (3.2)

for a known stream function ψ . For the results presented in Figure 3.1, we have used

the stream function reported by [56] (equation 19), hence the dynamical system for that

particular stream function is given by

(3.3)

dxdt

=Nq + 2tanh

(N(h1−h2)

2

)N (h1 − h2)− 2tanh

(N(h1−h2)

2

)+

(q + h1 − h2)sech(

N(h1−h2)2

)sinh

(N(h1+h2)

2

)N (h1 − h2)− 2tanh

(N(h1−h2)

2

) N sinh(Ny)

+(q + h1 − h2)sech

(N(h1−h2)

2

)cosh

(N(h1+h2)

2

)N (h2 − h1) + 2tanh

(N(h1−h2)

2

) N cosh(Ny) ,

46

dydt

= − ∂

∂x

(h1 + h2)(

Nq + 2tanh(

N(h1−h2)2

))2N (h2 − h1) + 4tanh

(N(h1−h2)

2

) +Nq + 2tanh

(N(h1−h2)

2

)N (h1 − h2)− 2tanh

(N(h1−h2)

2

)y

+(q + h1 − h2)sech

(N(h1−h2)

2

)sinh

(N(h1+h2)

2

)N (h1 − h2)− 2tanh

(N(h1−h2)

2

) cosh(Ny)

+(q + h1 − h2)sech

(N(h1−h2)

2

)cosh

(N(h1+h2)

2

)N (h2 − h1) + 2tanh

(N(h1−h2)

2

) sinh(Ny)

,(3.4)

where h1 = 1+acosx and h2 =−d−bcos(x+φ1) are the wall equations (top and bottom,

respectively), a, b are the wave amplitudes, φ1 is the phase shift between the waves, N2 =

M2 (1+λ1), M is the magnetic field parameter and λ1 is the Jeffrey fluid parameter (see

[56] for details). The equilibrium points of the above dynamical system can be calculated

(exactly or numerically) by solving the following system of equations,

(3.5)

Nq + 2tanh(

N(h1−h2)2

)N (h1 − h2)− 2tanh

(N(h1−h2)

2

)+

(q + h1 − h2)sech(

N(h1−h2)2

)sinh

(N(h1+h2)

2

)N (h1 − h2)− 2tanh

(N(h1−h2)

2

) N sinh(Ny)

+(q + h1 − h2)sech

(N(h1−h2)

2

)cosh

(N(h1+h2)

2

)N (h2 − h1) + 2tanh

(N(h1−h2)

2

) N cosh(Ny) = 0,

47

(3.6)

− ∂

∂x

(h1 + h2)(

Nq + 2tanh(

N(h1−h2)2

))2N (h2 − h1) + 4tanh

(N(h1−h2)

2

)+

Nq + 2tanh(

N(h1−h2)2

)N (h1 − h2)− 2tanh

(N(h1−h2)

2

)y

+(q + h1 − h2)sech

(N(h1−h2)

2

)sinh

(N(h1+h2)

2

)N (h1 − h2)− 2tanh

(N(h1−h2)

2

) cosh(Ny)

+(q + h1 − h2)sech

(N(h1−h2)

2

)cosh

(N(h1+h2)

2

)N (h2 − h1) + 2tanh

(N(h1−h2)

2

) sinh(Ny)

= 0.

Furthermore, the qualitative behavior of the equilibrium points, obtained viz the solutions

of the above system of equations, can be studied through the eigenvalues of the Jacobian

based on the Hartman-Grobman theorem for local linearization near the critical points [81].

For the classification of the critical points we adopt Bakker’s notation [19] i.e. an equilib-

rium point is called a ‘center’ if the eigenvalues form a complex conjugate pair. In graphical

representation a center is identified with closed streamlines as for a ‘bolus’, hence the ex-

istence of a center is linked to the existence of a bolus.

For the dynamical system given in equations (3.3) to (3.4), corresponding to the stream

function of [56, equation 19], we have computed the critical points and their evolution

with the magnetic field parameter M, numerically. The results are presented in Figure 3.2

in which two equilibria curves are present corresponding to x = −0.2184 and x = 2.9231

with the qualitative nature being center and saddle, respectively. From this equilibria plot,

it can be seen that corresponding to M = 0.01 centers are formed at (−0.2184,−0.9231)

and (−0.2184,0.7297) whereas saddles appear at (2.9231,−0.3174) and (2.9231,0.3193)

which can be verified through the contour plots, corresponding to the same flow parameters,

given in Figure 3.1(a). Furthermore, the equilibria curve also shows that the center (or

bolus) disappears at M = 17.5 again in agreement with the contour plot given in Figure

3.1(d).

48

0 5 10 15 20−1.5

−1

−0.5

0

0.5

1

1.5

M

y

a = 0.4, b = 0.5, q = −0.1φ1 = π/8, λ1 = 0.3

red: x= −0.2184blue: x= 2.9231

Figure 3.2: An equilibrium plot showing the evolution of the vertical coordinate of the

equilibrium point as the magnetic field M is varied. Two characteristic flow patterns are

developed corresponding to x = −0.2184 (center, shown in red) and x = 2.9231 (saddle,

shown in blue).

49

0 50 100 150−2

−1

0

1

2

−4 −2 0 2 40.4

0.6

0.8

1

1.2

1.4

M

red: x= −0.3000blue: x= 2.8415

yy

x

M = 50

a = 0.3, b = 0.4, q = 0

φ1 = π/6, λ1 = 0

Figure 3.3: (Left) A contour plot of the stream function [56, equation 19] generated using

Matlab at M = 50,q = 0,φ1 = π/6 and λ1 = 0. The closed streamlines are shown in black

whereas the open streamlines are colored brown. (Right) The equilibria curve, showing

the evolution of the vertical component of the equilibrium point, as the magnetic field M

is varied. The qualitative nature of the equilibria curve shown in red, corresponding to

x =−0.3 is center whereas the blue curve (x = 2.8415) represents saddles.

From the above discussion, it is evident that the ‘dynamical system approach’ for identi-

fication of a bolus is advantageous to the conventional ‘contour plot approach’ for three

reasons. Firstly, the dynamical system approach is robust in the sense that it does not work

on ‘hit and trial’ principle as in the contour plot approach where one has to plot and analyze

contour plots corresponding to various flow parameter values. Secondly, the dynamical sys-

tem approach gives a complete spectrum of the flow behavior for all values of a particular

parameter in contrast to the contour plot approach which only identifies the existence of a

bolus for a particular flow parameter values. The third advantage that the dynamical sys-

tem approach has over the contour plot approach is that it also locates the critical value

of a parameter after which the flow behavior changes. It is this particular feature of the

dynamical system approach that we use in the following section to investigate whether a

bolus disappears with increasing magnetic field, M?

50

3.3 Bolus disappearing with increasing M

We begin this section by presenting a counter example from [56] in which bolus does not

disappear with increasing the magnetic field, M. Figure 3.3 (left) shows a contour plot

for the stream function of [56, equation 19] at M = 50,q = 0,φ1 = π/6 and λ1 = 0. In

the contour plot, open streamlines are colored brown whereas the closed streamlines are

shown in black. It can be seen that the bolus exists even at such a large magnetic field.

The equilibrium curve, given in Figure 3.3 (right), for the magnetic field parameter M also

shows that the bolus exists even at large values of the magnetic field parameter M. These

results suggest that the bolus does not always disappear with increasing magnetic field

M, hence there are certain flow conditions, that we investigate below, under which bolus

disappears with increasing magnetic field M.

In the preceding section, we have shown that a bolus is in fact an equilibrium point whose

qualitative nature, characterized through eigenvalues of the Jacobian, is a center. Hence a

bolus can disappear in two ways: (1) Quantitative disappearance i.e. the equilibrium point

no longer remains an equilibrium point (2) Qualitative disappearance i.e. the qualitative

nature of the equilibrium point changes from being center.

3.3.1 Quantitative disappearance

By quantitative disappearance, we mean that the equilibrium point, around which the bolus

is formed, no longer remains an equilibrium point due to changes in the flow dynamics

with increasing magnetic field, M. It is well understood that with increasing magnetic field

M, the velocities drop in the interior domain and increase in the boundary layer [62] which

can also be seen from the results shown in Figure 3.4.

In Figure 3.4, we show the velocity profiles at x= 0 as a function of the vertical coordinate y

for different values of the magnetic field parameter M (=1, 6 and 8 shown by dashed, dash-

dotted and dotted, respectively). It can be seen that the equilibrium point (−0.2184,0.7592)

at M = 1 (which forms a center, see Figure 3.2) moves towards the boundary as the mag-

51

−1.5 −0.5 0.5 1.5−1

−0.5

0

0.5

0

u

M = 1, 6, 18

y

x = −0.2184a = 0.4, b = 0.5, q = −0.1φ1 = π/8,λ1 = 0.3

Figure 3.4: The horizontal velocity profiles in the vertical column, x =−0.2184, at M = 1

(dashed line), M = 6 (dash-dotted line) and M = 18 (dotted line) for q = −0.1,φ1 = π/8

and λ1 = 0.3. The solid horizontal line marks the zero velocity. The picture above is the

zoomed view near the zero velocity level.

52

netic field M is increased. In particular, for M = 18, the velocity curve has dropped below

the zero line i.e. the equilibrium point vanishes and the bolus disappears. Hence a bolus

disappears quantitatively when the velocity curve falls below zero line in the asymptotic

state (M >> 1).

In order to understand the flow behavior in the asymptotic state, we look at the flux condi-

tion used in the peristaltic studies i.e.

ψ (x∗,h(x∗))−ψ (x∗,0) =∫ h(x∗)

0u(x∗,y)dy, (3.7)

where ψ is the stream function, u is the horizontal velocity, x∗ represents a particular hori-

zontal location and y is the vertical coordinate which varies from the central line to the top

boundary i.e. 0≤ y≤ h(x∗). The above equation is usually rewritten as follows:

q =∫ h(x∗)

0u(x∗,y)dy, (3.8)

by assuming flux to be zero at the central line i.e. ψ (x∗,0) = 0 and q at the top boundary

i.e. ψ (x∗,h(x∗)) = q. In peristaltic flows, q is used as an input parameter which prescribes

the volumetric flux rate.

The flow behavior in the asymptotic state (i.e. for large M) can be inferred using the flux

relation given in equation (3.8). As discussed above, the velocities in the interior domain

decrease as M is increased. In the asymptotic state, the velocity curve will not fall below

the zero line if the assumed flux q is non-negative (otherwise the integral in equation (3.8)

will be negative yielding a contradiction to the assumed non-negative value of q). However,

for negative q, the velocities in the interior domain will be negative so that the integral in

equation (3.8) remains negative. The velocity curves at x = 0 (horizontal location of the

equilibrium point) plotted as a function of the vertical coordinate at M = 100 for different

values of q is given in Figure 3.5. It can be seen that in the asymptotic state, the velocity

curve for q = 0 and q = 1 remains above the zero line whereas for q = −1 the velocity

curve falls below the zero line in agreement with the conjectures made above using the flux

integral given in equation (3.8).

53

−1.5 −0.5 0.5 1.5−1

−0.5

0

0.5

−0.01

0

0.01

u

q = 1, 0,−1

y

x = 0

a = 0.4, b = 0.5, M = 100

φ1 = π/8,λ1 = 0.3

Figure 3.5: Asymptotic velocity profiles for large M(= 100) plotted as a function of y at

x = 0 for different values of flux q = 1 (dashed), q = 0 (dotted) and q =−1 (dash-dotted).

To further support the above argument regarding the flow behavior in the asymptotic state,

we show the (unit) velocity vectors, at M = 100 for different values of the flux q (Figure

3.6). In the Figure, blue arrows indicate the velocity vectors with positive horizontal veloc-

ity u and the red arrows correspond to the velocity vectors with negative horizontal velocity

component. It can be seen that in the asymptotic state for q = 1 and q = 0, there exists flow

in the positive direction hence the velocity curves can not fall below zero line whereas for

q =−1, there is no flow in the positive x-direction which shows that the velocity curve will

lie below the zero line.

From the above discussion, we conclude that the velocity curves, in the asymptotic state,

will only fall below the zero line, if the input flux is assumed to be negative (i.e. q < 0).

Hence the quantitative disappearance of a bolus is only possible if q < 0. For a positive

flux (q > 0), the equilibrium point will not disappear, hence a bolus (if it exists) will only

disappear if the qualitative behavior of the equilibrium point changes.

54

−3 −2 −1 0 1 2 30

0.5

1

0

0.5

1

0

0.5

1

q = 1

q = 0

q = −1

(a)

(b)

(c)

a = 0.4, b = 0.5,φ1 = π/8,λ1 = 0.3, M = 100

Figure 3.6: The (unit) velocity vectors in the asymptotic state (M = 100) for q = 1 (top),

q = 0 (middle) and q =−1 (bottom). Other flow parameters are φ1 = π/8 a = 0.4,b = 0.5

and λ1 = 0.3.

55

-0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

saddle

center

Figure 3.7: A pictorial diagram of the characteristic changes in flow dynamics if an equi-

librium point changes from being center to a saddle.

3.3.2 Qualitative disappearance

In the preceding section, we have established the fact that a bolus, if formed, will disap-

pear with increasing M if the assumed flux is negative (q < 0). For a non-negative flux

(q ≥ 0), if a bolus is formed i.e. there exists an equilibrium point which makes a center,

the equilibrium point does not vanish for large M. Hence for q ≥ 0, a bolus will only dis-

appear if the qualitative nature of the equilibrium point changes. In Figure 3.7, we give a

pictorial description of the flow changes that will occur if the equilibrium point changes

its qualitative behavior from being a center to a saddle with increasing M. It is shown that

if a center changes to saddle, the vertical velocities will have to increase in order to pinch

the center at a point which will then deform to a saddle. This particular behavior is not

possible with increasing M because we know that the velocities decrease as the magnetic

field is increased, hence an equilibrium point will not change its qualitative behavior with

increasing M.

It can therefore be concluded that a bolus, when formed, will only disappear with increasing

M if the input flux is negative (q < 0). For non-negative flux (q ≥ 0), the bolus will not

56

disappear even at asymptotically large M.

3.4 Concluding remarks

Realizing the limitation and drawbacks of the conventional technique in determining the

existence and disappearance of bolus; we address these issues using dynamical system the-

ory. The bolus can be seen as a circulatory flow pattern at the equilibrium point with a

qualitative behavior as center (determined from the eigenvalues of the Jacobian matrix).

Contrary to the visual observation of bolus as contour plot in determining the disappear-

ance of bolus with magnetic field; our study proves mathematically and through physical

interpretation that a bolus, if it exists, will only vanish with increasing magnetic field M

provided the flux is negative q < 0. More importantly, the bolus does not disappear, even

for large magnetic field in the case of q ≥ 0 but is pushed towards the boundary. Further,

this study determines the flow behavior for continuous range of the parametric values of

the flow through bifurcation diagram which was only possible for discrete values in the

conventional method.

57

Chapter 4

Peristaltic flows: A quantitative measure for the size of a

bolus

58

4.1 Introduction

In this chapter, we define the size of a bolus sb (a circular flow pattern developed in peri-

staltic flows): “The area of the outermost closed streamline around an equilibrium point”.

The definition is validated analytically and numerically through some examples. Numer-

ical scheme is found to be more efficient than analytical method. We further show that

the proposed definition works better than the conventional approaches where the size of a

bolus is usually inferred by visually inspecting the contour plots of a stream function. The

proposed definition can serve as a basis for the development of future mathematical frame-

work which can be used to investigate the quantitative effects of various flow parameters

on the size of a bolus.

4.2 Definition

Let

C = c : ψ(x,y)− c = 0isaclosedcurve, (4.1)

be the collection of all the contour levels which correspond to closed streamlines around the

equilibrium point (xe,ye). Let C be any streamline defined by ψ(x,y) = c, we define x(C )

to be the projection of a streamline on the x-axis. Let l(c) be the length of the x-projection

of the streamline corresponding to ψ(x,y) = c, then cm defined as

cm = maxc∈C

l(c), (4.2)

gives the closed streamline ψ(x,y) = cm around the equilibrium point (xe,ye) with the

longest x-projection. As the (closed) streamlines are nested, hence the outermost closed

streamline is obtained for the contour level cm. The size of the bolus, for the two-dimensional

case, is defined as the area of the outermost closed streamline centered at the equilibrium

point (xe,ye) i.e.

sb =∫ xB

xA

∫ yu(x)

yl(x)dydx, (4.3)

59

where yu and yl are the upper and lower branches, respectively, of the closed streamline

centered at (xe,ye) and [xA xB] defines the x-projection (i.e. the domain) of the streamline.

4.3 Analytical illustration

As an illustration, we apply our proposed definition, given in equation (4.3), to the stream

function,

ψ(x,y) =(

1+3qh

) y2−(

1h2 +

qh3

)y3

2, (4.4)

which is borrowed from [50] and calculate the size of the bolus analytically. In the above

equation q is a constant representing the input volume flux rate,

h = 1−φ(1− cos2 x), (4.5)

is the height of the channel and φ is the ratio of the average channel height to the wave

amplitude. For convenience we transform the physical domain in (x,y) to a rectangular

box in (ξ ,η) using ξ = x and η = y/h(x). The stream function given in equation (4.4) is

accordingly transformed to,

ψ(ξ ,η) = (h+3q)η

2− (h+q)

η3

2. (4.6)

A streamline corresponding to a particular contour level c is given by ψ(ξ ,η) = c which

gives

(h+q)η3− (h+3q)η +2c = 0, (4.7)

which is rewritten as

(he +q)η3− (he +3q)η +2c = 0, (4.8)

to restrict our analysis to streamlines whose ξ -projection include the ξ component of the

equilibrium point (ξe,ηe). In the above equation he represents the height of the channel at

ξe i.e. he = 1−φ(1−cos2 ξe). The polynomial in equation (4.8) is cubic in η , hence it will

have three roots. The nature of the roots is determined by analyzing the discriminant, ∆,

of the cubic polynomial (4.8). As for a bolus, the streamlines will be closed which means

60

that the roots of the cubic polynomial given in equation (4.7) will have at least two real

roots which will together form a closed contour line. Hence we restrict ourselves to the

case ∆ > 0 which ensures that the cubic polynomial (4.7) will have three distinct real roots.

Using ∆ > 0, we obtain the following interval for the choice of c

c ∈

(−∞ c∗ =

(he +3q)3/2√27(he +q)

), (4.9)

which will give us three distinct real roots. Hence we only need to consider the streamlines

corresponding to c ∈ (−∞ c∗) to identify the ones that are closed. To check the closeness

of a streamline, we write the roots of the cubic polynomial (4.7) using the representation

given by François Viète [78], i.e.

ηk(ξ ) = 2

√h+3q

3(h+q)cos

(13

arccos

(− 3c

h+3q

√3(h+q)h+3q

)− 2πk

3

),

−π

2≤ ξ ≤ π

2, 0≤ η ≤ 1,k = 0, 1, 2,

(4.10)

which in the physical plane (x,y) reads

yk(x) = 2h

√h+3q

3(h+q)cos

(13

arccos

(− 3c

h+3q

√3(h+q)h+3q

)− 2πk

3

),

−π

2≤ x≤ π

2, 0≤ y≤ h(x),k = 0, 1, 2.

(4.11)

For a particular value of c, the three roots given above will make streamline(s) which

may be close or open. One can employ various checks to see if a streamline is closed.

For instance, integrating a conservative force potential function (e.g. gravitational poten-

tial) along the streamline, calculating the homology of the streamline using Javaplex [95],

checking if the roots are equal at the end points etc. For illustration, we check the differ-

ence in the roots at the end points to decide about the closeness of a streamline. Notice that

if a root makes a branch of a closed streamline, then it must be real which means that the

argument of arccos must be less than or equal to 1 i.e.∣∣∣∣∣− 3ch+3q

√3(h+q)h+3q

∣∣∣∣∣≤ 1, (4.12)

61

which gives

h3 +9qh2 +27(q2− c2)h+27q

(q2− c2)≥ 0. (4.13)

For a particular streamline (i.e. specifying c and q), one can find the solution interval of

the inequality given in equation (4.13) and consequently the end points of a root. For illus-

tration, we chose two streamlines corresponding to c = 0.21,0.3 (q = 0.2, φ = 0.8) shown

in red and black, respectively in Figure 4.1(a). The streamlines are plotted using Matlab

utility for plotting a contour line. The plotted streamlines enclose the equilibrium point

(0, 0.6667) marked with a cross. From the contour plot, it can be seen that the streamline

corresponding to c = 0.3 is a closed streamline whereas c = 0.21 gives an open streamline.

Below, we will extract the same information analytically by making use of the inequality

(4.13).

For c = 0.3, the inequality given in equation (4.13) is satisfied for h≥ 0.6961 which gives

−0.6639 ≤ x ≤ 0.6639, hence the x-projection of the streamline corresponding to c = 0.3

is given by the interval [−0.6639 0.6639] which also gives the end points defining the

domain of the roots. These analytical observations can be verified graphically in Figure

4.1(a) where the vertical lines marked at x = −0.6639 and x = 0.6639 coincide with the

end pints of the x-projection of the streamline corresponding to c = 0.3. Also plotted

in Figure 4.1(b) are the roots, given by equation (4.11), of the cubic polynomial defined

in equation (4.7) for c = 0.3, q = 0.2 and φ = 0.8. The dashed black and solid black

represents the roots y0 and y1 whereas the open symbols, squares and triangles, represent

the end points of the roots, y0 and y1, respectively. It is clearly seen from the Figure that

the roots y0 and y1 form the upper and lower branch, respectively, of a closed streamline

which is seen analytically from the fact that at the end points, the difference between the

roots is approximately 10−3. The third root y3 lies outside the physical domain of the

problem ( i.e. y3(x)≥ h(x) ∀x ∈ [−π/2 π/2]) and hence is disregarded. The length of the

x-projection of the streamline can be calculated using the projection interval found above

62

−

π2

−

π4

0π4

π2

0

0.5

1

−

π2

−

π4

0π4

π2

0

0.5

1

q = 0.2, φ = 0.8

c = 0.21

c = 0.3

yy

c = 0.21

c = 0.3

(b)(a)

x x

Figure 4.1: The streamlines corresponding to c = 0.21 (shown in red) and c = 0.3 (shown

in black). In (a) the contour lines are plotted using Matlab whereas in (b) the contour

lines correspond to the roots yk defined in equation (4.11). In (b), dashed and solid lines

represent the roots y0 and y1, respectively. The vertical lines in (a) mark x =−0.6639 and

x = 0.6639 in order. The open symbols in (b) show the end points of the roots y0 (squares)

and y1 (triangles). The point (0, 0.6667) marked with a cross represents the equilibrium

point.

63

(x ∈ [−0.6639 0.6639]).

For c = 0.21, the inequality (4.13) is satisfied for all values of h in the given problem do-

main i.e. h ∈ [0.2 1], hence the roots are real throughout the domain (x ∈ [−π/2 π/2]).

It is further noted that only y0 and y1 lie within the domain of the physical problem. The

difference in the roots, y0− y1 at the end points is approximately 0.05, hence these do not

form a closed streamline which can be verified graphically from Figure 4.1.

The mathematical analysis described above for c = 0.21, 0.3 is repeated for other values

of c which gives us the collection of closed streamlines, C = c : 0.2178≤ c≤ 0.3555.

Within C, the streamline having the longest x-projection is found for cm = 0.2178 with

length of the x-projection l(cm) = 3.0916. Figure 4.2 shows the contour plot of ψ , defined

in equation (4.4), generated using Matlab. The streamline shown with thick solid black line

corresponds to c = 0.2178 which verifies the analytical observations stated above.

Finally, the size of the bolus, defined as the area of the outer most closed streamline around

the equilibrium point (0,0.6667), is calculated using our proposed definition (4.3)

sb =∫ 1.5458

−1.5458

∫ y1(x)

y0(x)dydx, (4.14)

where y0 and y1 are defined in equation (4.11). An exact integration of the above integral

might be possible using some mathematical package that allows symbolic operations but

we have not made such an attempt and instead compute the integral numerically which

gives sb ≈ 1.0834.

As an illustration, we have shown the analytical working of the definition given in the

preceding section, for calculating the size of a bolus, but we recommend numerical imple-

mentation of the definition which is discussed in the following section.

64

−

π2

−

π4

0π4

π2

0

0.2

0.4

0.6

0.8

1

q = 0.2, φ = 0.8

x

y

Figure 4.2: The contour plot of the stream function ψ defined in equation (4.4) for q =

0.2, φ = 0.8 generated using Matlab. In particular, the streamline marked with a thick solid

black line corresponds to the contour level c = 0.2178.

65

4.4 Numerical illustration

In this section, we demonstrate the numerical use of the definition (4.3) to calculate the size

of a bolus. The physical domain is discretised, with fine resolution, and stream function

values ψi j are computed at the discrete points (xi, y j) i.e.

ψi j = ψ(xi,y j)

Using a built-in utility of Matlab [66] for computing the contour levels i.e. ‘contour’, we

obtain curves (in discrete points) corresponding to contour levels ψmin ≤ c ≤ψmax, where

ψmin and ψmax represents the minimum and maximum value of ψ respectively. The con-

tour levels are chosen, in discrete sense, from ψmin to ψmax with an increment of 10−3.

After computing the contour levels, we need to check each curve corresponding to a partic-

ular contour level to determine the closeness of that particular streamline. A streamline is

regarded as close if the distance between the first and the last point of the discrete contour

curve is less than some tolerance (set to 10−4 in our codes). From the collection of these

closed streamlines, the outermost closed streamline is chosen to be the one with the longest

x - projection. Finally, the area enclosed by the outermost closed contour line is computed

using numerical quadrature (trapezoidal or some other) which gives the size of the bolus

sb.

In Figure 4.3, we show the results computed using our numerical code, developed in Mat-

lab. In the contour plots, generated using built in Matlab function, the particular streamline

shown by thick black dashed line is the outermost closed streamline identified by our nu-

merical code. The effectiveness of the code is evident from the given contour plots for

different values of the flux parameter q. The size of the bolus sb, defined as the area of

the outermost closed streamline, is computed to be sb = 0.7854, 1.4687 and 1.0809 for the

boluses corresponding to q =−0.2,0 and 0.2, respectively.

66

−

π

4−

π

20

π

4

π

2

0

0.25

0.5

0.75

1

−

π

2−

π

40

π

4

π

2

0

0.25

0.5

0.75

1

−

π

2−

π

40

π

4

π

2

0

0.25

0.5

0.75

1

(c)

(a) (b)

q = −0.2

sb = 0.78

q = 0

sb = 1.47

q = 0.2

sb = 1.08

Figure 4.3: The contour plots of the stream function ψ , defined in equation (4.4), for

different values of the parameters are generated using Matlab. In each plot, the dashed line

represents the contour level, the code identified as the outermost closed streamline.

67

Table 4.1: Size of the bolus, sb, as defined in (4.3), calculated for different parameter values,

both analytically and numerically.

ParametersBolus size, sb

Analytical Numerical

φ = 0.8

q =−0.2 0.7827 0.7854

q = 0 1.4686 1.4687

q = 0.2 1.0834 1.0809

q = 0.2

φ = 0.1 0.4454 0.4457

φ = 0.3 0.7911 0.7909

φ = 0.5 0.9609 0.9608

A comparison of the numerical and analytical results is given in Table 4.1. We compute the

size of the bolus sb corresponding to different values of the flow parameter both analytically

and numerically. The analytical results are computed using similar analysis to what is pre-

sented in section 4.3 and for numerical results we use our Matlab code. The analytical and

numerical results are listed in Table 4.1. It can be seen that the size of bolus sb computed

analytically is approximately the same as the size computed numerically. Quantitatively,

the maximum relative error amongst all of the results given in Table 4.1 is 0.0034 which

shows that the numerical results are minutely different than the analytical results.

The results presented above shows the efficacy of our proposed definition to the size of

the bolus sb, defined in equation (4.3). As the proposed definition quantifies the size of a

bolus, it can be used to analyze the variation in bolus size with respect to a particular pa-

rameter. For illustration, we apply our definition of bolus size to the problems considered

in [10, 50]. Figure 4.4(a) gives the size of the bolus, sb, as a function of the flux param-

eter ‘q’ see [50]. In this particular example, there are three distinct regimes: (1) no bolus

−1 ≤ q < 0.3 (2) size of bolus increasing with q for −0.3 ≤ q < −0.05 (3) a decreasing

68

trend of sb for −0.05 ≤ q < 1. Figure 4.4(b) shows the size of the bolus calculated using

our proposed definition applied on the problem considered in [10]. The plotted results show

that the size of the bolus decreases with increasing M and finally the bolus disappears at

M = 2.29. The example clearly shows the effectiveness of our proposed definition, equa-

tion (4.3), for quantification of the size of a bolus. This has also improved the analysis

performed to determine the effects of a particular flow parameter on the size of a bolus.

−0.5 0 0.5 10

0.5

1

1.5

2

sb

q0 0.5 1 1.5 2

0.05

0.1

0.15

0.2

0.25

0.3

M

sb

Figure 4.4: The size of the bolus sb, as defined in equation (4.3) plotted as a function of (a)

flux parameter q [50] (b) The magnetic field strength parameter M [10].

4.5 Concluding remarks

The significant outcomes of the above study are:-

• Size of a bolus, sb is defined (for the first time) as the area of the outermost closed

streamline around an equilibrium point.

• Mathematically, this definition is implemented for calculating the area of the bolus,

69

from equation (4.3), both analytically and numerically. The results of two approaches

match each other with a negligible error.

• The issue of the bolus size has been settled by clearly defining it and calculating its

size both analytically and numerically. The variation of the size with volume flow

rate and magnetic field parameter is shown in two separate examples.

70

Chapter 5

Identification of trapping in a peristaltic flow; A new

approach using dynamical system theory

71

5.1 Introduction

In this chapter, we have identified trapping through dynamical system. Some inconsis-

tencies in the definition, of flow patterns, appear in the literature. These are resolved by

explicitly defining the flow patterns such as: ‘Bolus’, ‘Backward flow’, ‘Trapping’, and

‘Augmented flow’. The chapter starts with definitions of different flow patterns in section

5.2. The trapping is identified through dynamical system in section 5.3. These definitions

are implemented by taking an example from the literature. The conclusions has been given

in section 5.5

5.2 Definitions of various flow patterns

We begin by explaining different flow patterns that can potentially develop in a peristaltic

flow. Hence, we explicitly define terminologies related to such characteristic peristaltic

flow patterns that would otherwise be discreetly present in the reported literature[50, 92].

It is important to do so because there appears to be a repeated occurring confusion or

misunderstanding regarding terminologies, used to characterize various flow features in the

reported studies of peristaltic flows. In peristalsis, the qualitative flow pattern that develops

may change depending on the values of flow parameters such as flux q, magnetic field M,

phase difference φ1 or others (see [10, 107] for details). To illustrate various flow patterns

that may develop in peristaltic flows, we borrow the stream function ψ , which is reported

by [10],

ψ =qMycosh(hM) + y(1 + qM2β )sinh(hM)− (q + h)sinh(My)

hM cosh(hM) + (−1 + hM2β )sinh(hM)

+1

8(hM cosh(hM) + (−1 + hM2β )sinh(hM))2

((q + h)α

(ycosh(2hM)

+ y(−1 + 2h2M2 − 2Mycosh(My)

(hM cosh(hM) +

(−1 + hM2

β)

sinh(hM))

− 2hM sinh(2hM))+ 2

(hM(y + h2M2

β)

cosh(hM) +(−y + hM2 (h2 − hβ + yβ

))sinh(hM)

)sinh(My)

)),

(5.1)

72

−1

0

1

−1

0

1

−0.5 −0.25 0 0.25 0.5

−1

0

1

y

y

y

x

(b) Trapping

(a) Backward Flow

q = −0.5

q = −0.3

q = 0.1

(c) Augmented Flow

Figure 5.1: The contour plots of the stream function ψ [equation 5.1] at φ = 0.4, M =

1,α = 0.1,β = 0.0, and for different values of volume flow rate q . The arrows represent

the velocity vectors (in the moving frame of reference) computed from the stream function.

Following [50], the qualitative flow patterns are termed as (a) Backward Flow correspond-

ing to q =−0.5, (b) Trapping for q =−0.3 and (c) Augmented Flow resulting for q = 0.1.

73

where M represents the magnetic field parameter, h is the height of the channel, β is the

slip parameter, α represents the viscosity parameter, φ is the amplitude ratio and q rep-

resents the volume flow rate (see [10] for discussion on various parameters present in

equation (5.1). The above stream function represents the solution of a peristalsis prob-

lem (considered in [10]) of incompressible viscous fluid with variable viscosity in a two

dimensional channel with uniform thickness. In [10], the governing model is simpli-

fied under the assumption of long wavelength and low Reynolds number. Figure 5.1

shows the contour plots, obtained using the stream function given in equation (5.1), at

M = 1,α = 0.1,β = 0.0,φ = 0.4 and for different values of q. The qualitative flow patterns

that are observed can be classified into three distinct categories: backward, trapping, and

augmented flow (following [50]).

The important qualitative observations, from Figure 5.1, are the presence of eddies indi-

cated by closed streamlines (Figures 5.1b-c) and the flow direction shown by the drawn

velocity vectors. In Figure 5.1(a), no circulatory flow pattern exists anywhere in the do-

main and all of the flow is taking place in the backward direction. The circulatory flow

pattern is present both in Figure 5.1(b) and Figure 5.1(c); however, the characteristic flow

along the central line of the channel is significantly different i.e., a fluid stream flowing

in the forward direction is shown in Figure 5.1(c), whereas the flow along the central line

is completely blocked (Figure 5.1b) in the sense that the fluid stream incident from wave

trough can no longer flow along the central line and is diverted to flow over (under) the

vortices. These characteristic flow features motivate the terminologies: backward (Figure

5.1a), trapping (Figure 5.1b) and augmented flow (Figure 5.1c) used in [50, 92] and others

[7, 9, 10, 12, 17, 18, 101, 102, 107]. However, there are studies in which these character-

istic flows have been named contrarily e.g., an augmented flow is referred to as “trapping.”

For example, see Figure 10 (c-f) in [10], the flow qualitatively resembles that of augmented

flow but it has been termed (in [10]) as trapping.

“Bolus” is another term, that is frequently used in peristaltic flow investigations, often refer-

74

ring to a circulatory flow pattern shown in Figure 5.1 (b-c). Shapiro used the term “trapped

bolus” for referring to a flow situation similar to the one depicted in Figure 5.1(b). In this

case there are two distinct flow features: (1) presence of eddy on either side of central line

and (2) blockage of flow along the central line. Note that eddies are also present in Figure

5.1(c), but qualitatively the flow is different from the one depicted in Figure 5.1(b). In

the literature, e.g. [9, 64, 74, 82, 83], an augmented flow is being referred to as trapping,

probably because of the presence of eddies that are also present in trapping, which [92]

is referred to as “trapped bolus.” In addition, there are number of occurrences, in the re-

ported literature where an eddy is being referred to as bolus, perhaps motivated by the fact

that eddies are present in trapping. It therefore feels necessary to use different names for

a circulatory flow pattern and the flow type in which eddies on either side of the central

line grow large enough to seize the flow along the central line. With intent of facilitating

consistency in the literature, we suggest that the circulatory flow pattern be referred to as

a “bolus”, whereas the term “trapping” be used for referring to the particular flow type in

which the flow along the central line is blocked due to boluses present on either side of the

central line (Figure 5.1b). Furthermore, we support our proposal by pointing out to the fact

that in trapping, there actually exist two eddies (with opposite rotation) that come close

and block the flow along central line, but these two eddies never merge together; hence; it

seems appropriate if these are referred to as two boluses. In the light of the above discus-

sion, there is a need to explicitly define these characteristic flow patterns.

Definition 5.1. Bolus: It refers to a circulatory flow pattern present somewhere in the

domain (present under the wave crests in Figures 5.1 b-c).

Definition 5.2. Backward Flow: A flow type in which the whole of the fluid moves in a

direction opposite to that of the wave (see Figure 5.1a).

Definition 5.3. Trapping: It refers to the flow situation where the flow along the central

75

line is completely blocked by the boluses present under the wave crests on either side of the

central line (see Figure 5.1b).

Definition 5.4. Augmented Flow: In this type of flow, boluses appear under the wave

crests but (additionally) a stream of fluid flows along the center of the channel, in the same

direction as that of the wave (see Figure 5.1c).

An explanation regarding what classifies as a bolus is necessary, as in the reported liter-

ature confusion lies in defining bolus and the size of a bolus. Through literature review

[4, 7, 39, 64, 74, 82] and from private correspondence with some of the authors, it seems as

though a particular closed streamline is being regarded as a bolus. We emphasize that this

is not consistent with the definition of [92] and, of course, the definitions we propose above.

A bolus represents a circulatory flow pattern, and hence, the whole of the circulatory flow

pattern should be regarded as a bolus and not the particular closed streamline; otherwise,

there would be infinitely many boluses (corresponding to each closed streamline) within

one circulatory flow pattern. When defining the size of a bolus, it seems appropriate to con-

sider the size of the outermost closed streamline and define the size of the bolus to be the

area of the outermost closed streamline (as defined in [33]). Furthermore, when discussing

an increase/decrease or appearing/disappearing of a bolus, one should again consider the

whole circulation and not a particular closed streamline. Although the closed streamlines

may reduce in size or may disappear, the bolus can still exist.

The definitions given in this section will help remove inconsistencies in the literature re-

garding the terminologies being used for referring to a particular peristaltic flow pattern.

We now focus on the main subject of this study i.e., devising a mechanism used for the

identification of flow parameters resulting in “trapping”.

5.3 Identifying Trapping

As discussed in the Introduction as well as in section 5.2, trapping is a particular flow pat-

tern that can potentially develop in a peristaltic flow, depending on the chosen values of the

76

physical parameters present in the governing mathematical model. The peristaltic research,

therefore, includes the investigation of flow parameters to identify the parameter choices

which will result in trapping. In this section, we revisit the presently adopted approach

i.e., the visual inspection of contour plots for the identification of parameter values result-

ing in trapping. The limitations and disadvantages associated with the presently adopted

technique are discussed, motivating the need of a more robust and mathematically sound

technique for identification of trapping, which we propose in the second part of this section.

5.3.1 Present approach: Through Visual Analysis

In most of the reported studies of peristaltic flows [4, 7, 9, 39, 83], the discussion of trap-

ping is carried out mainly through contour plots of the stream function. As an illustration,

see Figure 7 in [10], the closed streamlines under the wave crests indicate the presence

of boluses on either side of the streamlines. Furthermore, the streamline coming from the

wave trough goes above (or below) the bolus, suggesting that there is not any flow along

the central line. Based on these observations, one can conclude that trapping occurs for

this choice of flow parameter values. To start with, the presently adopted approach does

not provide any mathematical method for identification of parameter values resulting in

trapping i.e., the whole approach is based on hit and trial principle. On the technical side,

the contour plots are usually obtained through some mathematical package (Matlab, Math-

ematica or others); hence, it is vital to choose appropriate contour levels ensuring all flow

patterns are being displayed. Furthermore, it lacks completeness in the sense that one may

be able to find one, two or a few collections of parameter values that would result in trap-

ping, but there might still be infinitely many other choices (of parameter values) that could

result in trapping.

In light of the above discussion, it is clear that a mathematical approach is needed for the

identification of parameter values resulting in trapping. An approach that analyses all the

parameter values and gives, in a systematic way, a complete range of parameter values for

77

which trapping occurs.

5.3.2 Proposed Approach: Through Dynamical System

The use of dynamical system is motivated by the fact that trapping in fact refers to a par-

ticular flow pattern which can be identified by analyzing streamlines through dynamical

systems. A bolus is in fact an elliptic equilibrium point (center) of a dynamical system,

which means that around this equilibrium point, streamlines form a nest of closed loops

indicating the presence of a circular flow pattern. The elliptic equilibrium point is also

accompanied by a hyperbolic equilibrium point (saddle) on either side, marking the bound-

ary of closed streamlines. Hence, in a particular peristaltic flow, trapping can be effectively

identified by analyzing the equilibrium points of the associated dynamical system.

The proposed dynamical system approach is explained by applying it on the problem given

in [10]. The starting point for the application of dynamical system is essentially to write

the first order autonomous system of ordinary differential equations. As the flow is steady

in the moving frame of reference, the streamlines coincide with the path lines, and hence,

the flow is described by the non-linear autonomous system,

x =∂ψ

∂y= f (x,y,α,β ,q,M,φ), (5.2)

y =−∂ψ

∂x= g(x,y,α,β ,q,M,φ). (5.3)

The critical points (equilibrium points) can be calculated by solving f = 0 = g (equations

5.2-5.3), simultaneously after using the expression of ψ given by equation (5.1). These

equations being nonlinear, non-algebraic in nature are solved numerically, and their qual-

itative behavior is explored by finding the Jacobian at these points (Hartmann Grobman

theorem). The list of equilibria will then be classified as being center, saddles, or nodes

depending on the eigenvalues of the Jacobian matrix.

In Figures 5.2(a-b), we have plotted bifurcation diagram for the slip parameter β at

M = 1,α = 0.0,q = −0.35, and φ = 0.4. The dashed red line in the Figure indicates

78

0 0.05 0.1 0.15 0.2−0.3

−0.15

0

0.15

0.3

β

x

0 0.05 0.1 0.15 0.2−0.5

−0.25

0

0.25

0.5

β

y

0 0.05 0.1 0.15 0.2−0.5

−0.25

0

0.25

0.5

β

y

0 0.05 0.1 0.15 0.2−0.3

−0.15

0

0.15

0.3

β

x

M = 1.0, α=0.0, q = −0.35, φ=0.4

(d)(c)

M = 1.0, α=0.1, q = −0.35, φ=0.4

(b)(a)

Figure 5.2: The bifurcation diagram as a function of slip parameter β for the peristalsis

problem in [10]. (a)-(b) show the spatial location of the equilibria at α = 0.0, whereas

(c)-(d) show the location of equilibria at α = 0.1 (see [10] for parameter details). The solid

line is used to represent saddles whereas dashed line corresponds to centers.

79

centers, whereas saddles are represented by solid blue line. A bifurcation diagram not

only gives the location of an equilibrium point but it also tells about the qualitative nature

(center, saddle or node) of the equilibrium point. Furthermore, it gives this information on

the complete range of the chosen parameter. For instance, at β = 0.0, two centers exist

at (0,±0.4362), showing that boluses exist under the wave crest. In addition, two saddles

exist at (±0.2132,0), hence, it can be concluded that trapping occurs as the boluses exist

under the wave crests and saddles, on either side of bolus, lie on the central axis. The bi-

furcation diagram shows that this behavior persists for 0 ≤ β < 0.15226, hence, trapping

occurs for all values of β in this interval. Note that our result also includes particular case,

β = 0.0 in [10].

Figures 5.2(c-d), show the bifurcation diagram for the slip parameter β at M = 1,α =

0.1,q = −0.35, and φ = 0.4. Again, it shows that centers exist under the wave crest and

saddles lie on the central line for 0 ≤ β < 0.1431, which means that trapping exists for

0≤ β < 0.1431. Our results verify the particular case (β = 0.0 ) reported in [10].

Note that with the use of our proposed approach, we were able to identify all the parameter

values for which trapping occurs. In contrast, the presently adopted approach identifies the

parameter value resulting in trapping by hit and trial and, hence, cannot analyze a parame-

ter for its complete range of values. In the following section we give some more examples

from [10] where trapping is identified by visual inspection of streamlines. For the same

examples, we use our proposed approach to find the complete range of parameter values

for which trapping occurs.

5.4 Examples

In this section, we continue working on the problem in [10] and apply our proposed dy-

namical system approach to find other possible combinations of flow parameter values that

will result in trapping.

Figure 8 in [10] shows streamlines at α = 0.0,β = 0.0,q = −0.35,φ = 0.4 for M =

80

0 0.5 1 1.5 2 2.5−0.3

−0.15

0

0.15

0.3

M

x

0 0.5 1 1.5 2 2.5−0.5

−0.25

0

0.25

0.5

M

y

0 0.5 1 1.5 2 2.5−0.3

−0.15

0

0.15

0.3

M

x

0 0.5 1 1.5 2 2.5−0.5

−0.25

0

0.25

0.5

M

y

β = 0.0, α=0.0, q = −0.35, φ=0.4

β = 0.0, α=0.1, q = −0.35, φ=0.4

(a)

(c)

(b)

(d)

Figure 5.3: The bifurcation diagram as a function of magnetic parameter M for the peristal-

sis problem in [10]. (a)-(b) present the spatial location of the equilibria at α = 0.0, whereas

(c)-(d) present the location of equilibria at α = 0.1. The solid line and dashed line presents

the qualitative nature of equilibrium points as being saddles and centers respectively.

81

−1 −0.5 0 0.5 1−0.6

−0.3

0

0.3

0.6

q

x

−1 −0.5 0 0.5 1−1.2

−0.6

0

0.6

1.2

q

y

(a) (b)

β = 0.0, M = 1.0,α = 0.1, φ = 0.4

Figure 5.4: The bifurcation diagram as a function of volume flow rate q for the peristalsis

problem in [10]. (a)-(b) present the spatial location of the equilibria at α = 0.1. The solid

and dashed lines are used to represent saddles and centers respectively.

0,1,1.5. It is clear from the plotted contour plots, for the aforementioned values of the

magnetic field parameter that the trapping indeed exists. However, it is not clear if the

trapping exists only for these values of M. In order to classify the flow behavior for

the complete range of the magnetic field M, we calculate the bifurcation diagram of M

for α = 0.0,β = 0.0,q = −0.35,φ = 0.4 and show the result in Figures 5.3(a-b). The

dashed red line shows centers, whereas the blue solid line represents saddles. As for

M < 2.29, the center exists under the wave crests (at x = 0) sandwiched between two

saddles which lie on the central line, (at y = 0), and hence, it can be concluded that for

α = 0.0,β = 0.0,q = −0.35, and φ = 0.4, trapping will occur for M < 2.29. Similarly in

[10] [Figure 8], it is shown that trapping occurs for M = 0,1,1.5 when other flow parame-

ters are chosen to be α = 0.1,β = 0.0,q=−0.35, and φ = 0.4. We calculate the bifurcation

diagram for this case and the results are shown in Figures 5.3(c-d), which shows that trap-

ping exist for M < 2.09 .

Figures 10(b,d,f) in [10] show contour plots of the stream function for different val-

82

ues of flux q = (−0.35,0.2,0.5) while keeping the other parameters fixed at M = 1,α =

0.1,β = 0.0, and φ = 0.4. Although they have referred flow as being trapping in all these

cases, however in the light of definitions given in section 5.2, trapping only occurs for

q = −0.35, whereas for q = 0.2,0.5, the fluid stream incident from wave trough contin-

ues to flow along the central line, hence should be classified as “augmented flow”. In

order to characterize the flow for the complete range of flux parameter, we have calculated,

and shown in Figure 5.4, the bifurcation diagram for q at M = 1,α = 0.1,β = 0.0, and

φ = 0.4. The equilibria curves in dashed line and solid line represent centers and saddles,

respectively. Although the centers and saddles exist for −0.4256 ≤ q ≤ 1.0, the particu-

lar combination i.e., the center being sandwiched by saddles lying on the central line only

occurs for −0.4256 ≤ q ≤ −0.1955. For q ≥ −0.1955, the saddles no longer lie on the

central line (corresponding y 6= 0), and hence, this represents augmented flow. There are

many other cases in the problem in [10] that can be analyzed using dynamical systems to

identify parameter values resulting in trapping, however, our objective here was to validate

our proposed dynamical system approach for the identification of trapping. In this regards,

we have shown that dynamical system approach very efficiently identifies the existence of

trapping, not just for one parameter value but for the complete range of that parameter.

5.5 Concluding Remarks

The terms ‘Bolus’, ‘Backward flow’, ‘Trapping’ and ‘Augmented Flow’ are explicitly de-

fined closely following the terminology of [92] and [50]. (a) Bolus is represented by a

circulatory flow pattern in peristaltic flow (b) Backward flow is a type of flow in which the

whole of the fluid moves in a direction opposite to that of the wave (c) Trapping is a flow

pattern when centers exist under the wave crests and saddles (hyperbolic equilibrium point)

appear on the central line. (d) An augmented flow is the type of flow when center is formed

under the wave crests and saddles appear above (below) the central line.

A procedure based on the dynamical systems is presented, to identify the existence of trap-

83

ping. This technique makes use of bifurcation diagrams for continuous spectrum of the

relevant parameter to identify the behavior on the branches of the equilibria. The trapping

occur when center exists on either side of the centerline and two saddles appear on the cen-

terline. In this approach, we need not to rely on the experiment to determine the trapping

but can be given directly from the bifurcation diagram.

84

Chapter 6

Conclusions and Future Work

85

6.1 Conclusions

We first recount the procedure normally adopted in the literature to find analytical solu-

tion for peristaltic transport problems. The stream function is obtained analytically us-

ing long wavelength and small Reynolds number approximation. The velocity, pressure

rise per wavelength and volume flow rate is calculated from the stream function. Impor-

tant features like bolus, trapping, augmented and backward flows are presented graphically

through streamline for some choice of the characterizing parameters. However, little use

has been made of the analytical results in finding the qualitative behavior of flow patterns.

For example, the bolus (closed streamlines) is determined for some favorable choice of the

characterizing parameter with no idea of its behavior close to this value. The analytical

approach of finding the solution is thus found to have some short comings in explaining

some important features. (a) The stability question of the fluid flow is not addressed. (b)

The range of the values of parameter for which the behavior of the flow remains the same

is not known. (c) The critical value of the parameter beyond which the topological behav-

ior changes cannot be determined. (d) The value of the parameter which corresponds to a

specific flow pattern cannot be predicted. The answer to these and some other questions

lies in the stability and bifurcation analysis.

Based on the dynamical system; a mathematical framework has been proposed to identify

the existence and disappearance of the bolus– defined as circulatory flow pattern. Theory

of dynamical system has been employed to find the equilibrium points and their dynamical

behavior. We show that a bolus is in fact an equilibrium point of the dynamical system,

whose qualitative behavior can be characterized as a ‘center’ based on the eigenvalues of

the Jacobian matrix [81]. The literature, on peristalsis, suggests that the existence of the

bolus is identified by the visual inspection of contour plots as closed streamlines.

The proposed method of identifying a bolus through dynamical systems (our method) has

several advantages over the conventional approach. Our method is robust in the sense: it is

based on the eigenvalues of the Jacobian matrix. We can find the qualitative flow behavior

86

around the equilibrium point. Whereas in the contour plot approach the qualitative flow

behavior is obtained by visual inspection of the streamlines patterns which can be mislead-

ing as the contour plot may not reveal all the flow patterns. The misconception could be

due to limited number of level curves plotted or due to the length scale of the flow pattern

being relatively small. Whereas, our method is based on the exact mathematical findings of

the eigenvalues (purely imaginary) of the Jacobian matrix– the requirement for the center

or bolus. Also, our method characterizes the flow behavior for the complete range of a

parameter (say magnetic field M) in contrast with the contour plot approach in which the

flow behavior is obtained at a specific value of the parameter. Another advantage of the

presented method is that it can identify the critical value of a parameter at which the qual-

itative flow behavior changes (e.g. the value of the magnetic field M at which the bolus

disappears).

The other objective of this study is to investigate the disappearance of a bolus with increas-

ing magnetic field M, as said and reported in the most of the studies [71, 99]. We show

that a bolus, if formed, will only disappear with increasing magnetic field M if the assumed

flux is negative q < 0. For q ≥ 0. it is concluded that the bolus does not disappear with

increasing M but instead pushed (and somewhat stretched) in the boundary layer.

This provides a subtle approach to identify bolus by identifying the critical values of the

parameter at which the qualitative behavior changes. Our investigations provide a better

understanding of the effects of magnetic field parameter M on the qualitative flow behav-

ior.

Some analytical and numerical procedures are developed to investigate characteristic fea-

tures of a bolus (circular flow pattern) quantitatively. For that, we propose the definition

(4.3), for calculating the size of a bolus, sb realizing that the size of a bolus is the area of

the outermost closed streamline around an equilibrium (or stagnation) point. The analytical

and numerical implementation of the definition is given in detail using a few examples. The

definition is robust in the sense that the area is (and should) computed using the outermost

87

closed streamline. We show that the numerically implemented scheme produces same re-

sults as analytical scheme. But the amount of the work is far less in numerical scheme than

analytical approach. We conclude that the numerical use of the definition (implemented

here) is much better option.

We believe that this work provides vital tools to investigate the features of an important flow

pattern arising in peristalsis (and other relevant disciplines). With the proposed definition,

we are able to calculate, with high accuracy, the size of a bolus which gives our definition

a great advantage over the conventional approaches where any conclusion about the size

of a bolus is made qualitatively by merely visualizing the contour plots which sometimes

conceal the presence of such boluses if the length scales of the bolus are small.

Based on the proposed definition of bolus size, sb, we have developed some numerical

tools to analyze the variation in the size of a bolus due to change in flow parameters. In

the reported literature, such conclusions regarding change in size of bolus due to changing

values of a parameter, are only made through visual inspection of contour plots. We hope

that our work will further enrich and improve the understanding of peristalsis and various

flow patterns that develop in peristaltic flows.

Another objective of the study was to develop a technique for identifying range of pa-

rameter values that would result in trapping in a peristaltic flow. While working on this,

we realized that the terminologies adopted for describing characteristic flow pattern in re-

ported studies is not consistent. To remove such confusion in the literature we define terms

‘Bolus’, ‘Backward flow’, ‘Trapping’ and ‘Augmented Flow’ closely following the ter-

minology in [92] and [50]. In light of these definitions, ‘bolus’ represents a circulatory

flow pattern, whereas flow is classified as trapping when the flow along the central line is

blocked by boluses present under the wave crests.

Trapping is an important peristaltic flow pattern which can be used for safe transportation

of medical substances; hence it is very useful to know the flow parameter values that result

in trapping. The previously adopted technique is based on hit and trial principle and can

88

only determine a particular parameter value of the flow parameter (not the whole range

of parameter values) for which trapping occurs. We have proposed a new method based

on dynamical systems, to identify the existence of trapping. This technique makes use

of bifurcation diagrams that give the location and types of equilibrium points plotted as a

function of a particular flow parameter. The trapping exists when an equilibrium point of

type center is sandwiched between equilibrium point of type saddles, lying on the central

line (y = 0). This approach enables us to find the complete range of parameter values for

which trapping occurs.

The proposed technique has been applied on the peristalsis problem in [10]. It is shown

that all the results reported in [10] are in agreement with our results obtained by dynamical

system approach. [10] have reported particular values of flow parameter for which trapping

exists. However, with our proposed dynamical system approach we were able to recover

not only the particular parameter values reported in [10] but also a whole range of other

values which could not be identified using the approach adopted in [10]. Hence, it is con-

cluded that our propose approach for identification of trapping is much more effective in

the sense that it does not require trial testing and also that it characterizes the flow behavior

for whole range of parameter values (not just one particular value).

To summarize; the disappearance of bolus, size of the bolus and identification of trapping

have been investigated for peristaltic flow problem using the theory of dynamical system.

In the process some inconsistencies are observed which are resolved for future references

and clarity. It is hoped that the research will provide further insight and will further expand

the scope of dynamical system for the qualitative behavior of the fluid flow problems.

89

Chapter 7

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90

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