Peristaltic Transport of MHD Fluid through Straight/Curved...

Peristaltic Transport of MHD Fluid through

Straight/Curved Circular Tube with Stability Analysis

By

Shagufta Yasmeen

CIIT/SP13-PMT-008/ISB

PhD Thesis

In

Mathematics

COMSATS University Islamabad

Pakistan

Spring, 2018




A Thesis Presented to


In partial fulfillment

of the requirement for the degree of

PhD Mathematics

By

Shagufta Yasmeen


Spring, 2018



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

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

Name Registration Number

Shagufta Yasmeen CIIT/SP13-PMT-008/ISB

Supervisor

Prof. Dr. Saleem Asghar

Department of Mathematics


August, 2018

Certificate of Approval

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

MHD Fluid through Straight/Curved Circular Tube with Stability Analysis” was conducted by

Ms. Shagufta Yasmeen, CIIT/SP13-PMT-008/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 (CUI), in the

partial fulfillment of the requirement for the degree of Doctor of Philosophy in the field of

Mathematics.

Student Name: Shagufta Yasmeen Signature: __________________

Examination Committee:

External Examiner 1:_______________ External Examiner 2:_______________

Prof. Dr. Tasawar Hayat Prof. Dr. Muhammad Sajid

Department of Mathematics Department of Mathematics & Statisics

Quaid-i-Azam University, International Islamic University,

Islamabad. Islamabad.

Supervisor: _________________ Head of Department: ______________

Prof. Dr. Saleem Asghar Prof. Dr. Shams ul Qamar

Department of Mathematics Department of Mathematics

COMSATS University, COMSATS University,

Islamabad. Islamabad.

Convener/Chairperson: _____________

Prof. Dr. Moiz-ud-Din Khan


COMSATS University,

Islamabad.

Author’s Declaration

I Shagufta Yasmeen, CIIT/SP13-PMT-008/ISB, hereby state that my PhD thesis titled

“Peristaltic Transport of MHD Fluid through Straight/Curved Circular Tube with Stability

Analysis” 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 Ph.D. degree.

Date: _______________

Shagufta Yasmeen


Plagiarism Undertaking

I solemnly declare that research work presented in the thesis “Peristaltic Transport of MHD Fluid

through Straight/Curved Circular Tube with Stability Analysis” 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 student are placed who submitted plagiarized thesis.

Date: ________________

__________________________

(Shagufta Yasmeen)

CIIT/SP13-PMT-008/ISB.

Certificate

It is certified that Shagufta Yasmeen, CIIT/SP13-PMT-008/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:

Prof. Dr. Shams ul Qamar


CUI, Islamabad.

To my Mother,

and memory of my Father

Acknowledgement

Almighty Allah is gracious, who gave me the courage and patience to complete this uphill task.

His blessings on me are countless and one of the most precious is that I born in a Muslim family,

who is firm believer in last prophet Muhammad (PBUH). He is the one who taught this world

humanity and the importance of knowledge and exploring the universe for wellbeing of the

human kind. No doubt this universe is full of mysteries and complicated intricacies, being

Muslim and follower of such a great personality, it is duty and moral obligation to dwell

ourselves into the quest of knowledge.

It was a great privilege for me to work under the kind supervision of Eminent Professor Dr.

Saleem Asghar. His contributions in the field of mathematics in this country are acknowledged

by many and needs no introduction. It was like a dream come true for me to work with him. His

keen interest, provision of impetus for rational thinking and logics of exposition served as a

beacon light for me at every instant. His constructive criticism and critical reviews helped me to

elevate mine concepts and complete this dissertation. Without his advice and sincere support, I

might not have been able to complete this demanding task. I have a profound regard for him and

I also pray for his long, healthy and active life.

Many people around me have motivated, contributed and facilitated me in one or another way to

complete this assignment. It might not be possible to acknowledge everyone; however, it’s

essential to mention a few over here.

Many thanks to Professor Dr. Shams ul Qamar, who being the HOD of Mathematics department

has maintained a conducive and peaceful learning environment. I express my gratitude to Dr.

Moiz-ud-din Khan, Chairman Mathematics department, for ensuring and providing a great state

of research facilities.

I am also thankful to COMSATS University Islamabad for providing me In-house study leave.

I am in debt to whole team, who worked with me especially Dr. Hafiz Junaid Anjum for

transformation of ideas, augmentation of graphical presentation and dedication in completion of

the task. It was pleasant experience with him. Ms Tayyaba Ehsan, who encouraged me through

her valuable suggestions, also deserves space in the list.

Special thanks to my brother Tanvir, father (late), mother and siblings for their moral support

and encouragement.

I am obliged to my husband Javaid for his backing and rationalization throughout my studies.

The most beautiful possession, my son Ebrahim, who sacrificed and suffered a lot at such tender

age due to my hectic routine and busy schedule but always found him cool, calm and collected

anxious to see me with a completed job.

My sincere apologies for any omission bearing in mind the limited space.

Fortunately, it happened to be such a wonderful experience and will perhaps open window of

new opportunities and helpful changing my career.

Shagufta Yasmeen


Abstract

Peristaltic Transport of MHD Fluid through Straight/Curved Circular

Tube with Stability Analysis

The phenomenon of transport of viscoelastic material in human body is a hot topic of research in

biomechanical engineering. This topic has stimulated the attention of engineering scientists,

modelers, numerical simulants and mathematical biologists. Biorheological flows are now

experimentally exemplified by blood, bile, mucus, digestive fluids, synovial lubricants etc. On

the other hand the peristaltic pumping of fluid transport is due to a wave contraction/expansion

traveling along the length of a distensible channel/tube containing liquids. This mechanism

generally appears from a region of lower to higher pressure. Many muscles possess such inherent

characteristics. Specific examples for physiological processes may include eggs motion in the

female fallopian tube, spermatozoa transportation, transport of bile in duct, intrauterine liquid

transportation in the uterine cavity, blood circulation in small vessels, gall bladder with stones,

locomotion of animals like larvae of certain insects and earthworms, urine passage from kidney

to bladder, transport of embryo in non-pregnant uterus etc. Many biomedical systems for

instance dialysis machine, blood pumps machine, heart lung machine and roller and finger

pumps operate under peristalsis in order to cater the specific day to day requirements. Such

activity is also quite prevalent in industrial technologies for example chamber fuel control,

micro-pumps in pharmacology and toxic waste conveyance in chemical engineering. Continuous

efforts for refinements in designs sure acquire much complex models of peristalsis through non-

Newtonian materials. Motivated by this fact, the primary motto here is to model and analyze the

peristaltic motion of non-Newtonian materials in a channel. Further the magnetomaterial aspect

in such consideration has relevance with blood dynamics, blood pump machines and

magnetohydrodynamic (MHD) peristaltic compressor. Blood flow rate is greatly affected under

magnetic field. With this motivation, this thesis is organized as follows. First chapter provides

literature review. Second chapter contains some standard definitions and flow equations.

Dimensionless numbers and expressions for Jeffrey fluid are included. Third chapter examines

Peristaltic flow of viscous liquid in curved tube. Fluid is electrically conducting. Relevant

problem is modeled. Lubrication approach is followed for the equations in tractable form.

Resulting problem is solved for asymptotic analytic solution. Axial velocity, pressure gradient

and pressure rise per wavelength are analyzed in detail. Comparison with existing studies is

presented. Major findings are given in conclusions. Material of this chapter is published in

“Results in Physics 7 (2017) 3307-3314”, [Error! Reference source not found.].

Chapter four introduces existence of Hartman boundary layer for peristaltic flow in curved tube.

Equations for Jeffrey liquid in curved tube are modeled. Applied magnetic field is taken large.

Both quantitative and qualitative approaches are utilized. Examination of Hartman layer is made

by considering two term analytic solution using matched asymptotic technique. Traditional

analysis of obtaining analytic results is extended for dynamical system of problem in order to

understand the flow behavior. Nonlinear autonomous differential equations are established.

These equations characterize path of fluid particles. Equilibrium plots give a complete

description of various flow patterns developed for complete range of flow variable in contrast to

existing reported studies which describes flow patterns at some particular value of parameter.

Bifurcation diagrams are displayed. The observations of this chapter are published in

“Communications in Nonlinear Science and Numerical Simulation”, [Error! Reference

source not found.].

Chapter five is prepared for flow of dusty liquid bounded by a stretching sheet. Constant strength

of applied magnetic field is chosen. Exact analytic solution to derived problem is constructed. A

comparative study with existing numerical solution is made. Skin friction and velocity results are

given. Physical interpretation to influential variables in obtained solutions is assigned. Findings

of this chapter are published in “Mathematical Problems in Engineering 2307469 (2017)5,

[Error! Reference source not found.].

TABLE OF CONTENTS

1. Introduction ....................................... Error! Bookmark not defined.

1.1 Introduction: ........................................................ Error! Bookmark not defined.

2. Preliminaries ..................................... Error! Bookmark not defined.

2.1 Flow models ........................................................ Error! Bookmark not defined.

2.1.1 Axisymmetric flow model .................................. Error! Bookmark not defined.

2.1.2 Axisymmetric flow in curved configuration ....... Error! Bookmark not defined.

2.1.3 Equations for Jeffrey liquid ................................. Error! Bookmark not defined.

2.2 Maxwell’s equations ........................................... Error! Bookmark not defined.

2.2.1 Gauss’ law for electricity .................................... Error! Bookmark not defined.

2.2.2 Gauss’ law for magnetism ................................... Error! Bookmark not defined.

2.2.3 Faraday’s law ...................................................... Error! Bookmark not defined.

2.2.4 Ampere-Maxwell law ......................................... Error! Bookmark not defined.

2.3 Ohm’s law ........................................................... Error! Bookmark not defined.

2.4 Hartmann boundary layers .................................. Error! Bookmark not defined.

2.5 Non-dimensional numbers: ................................. Error! Bookmark not defined.

2.5.1 Reynolds number: ............................................... Error! Bookmark not defined.

2.5.2 Hartmann number: .............................................. Error! Bookmark not defined.

2.5.3 Wave number ...................................................... Error! Bookmark not defined.

2.5.4 Amplitude ratio ................................................... Error! Bookmark not defined.

2.6 Dynamical system ............................................... Error! Bookmark not defined.

2.6.1 Fundamentals of stability .................................... Error! Bookmark not defined.

2.6.2 Autonomous system ............................................ Error! Bookmark not defined.

2.6.3 Equilibrium solution ........................................... Error! Bookmark not defined.

2.6.4 Asymptotically stable equilibrium ...................... Error! Bookmark not defined.

2.6.5 Stable equilibrium ............................................... Error! Bookmark not defined.

2.6.6 Nondegenerate (Hyperbolic) equilibrium point .. Error! Bookmark not defined.

2.6.7 Types of Equilibrium points ............................... Error! Bookmark not defined.

2.6.8 Linearization ....................................................... Error! Bookmark not defined.

2.6.9 Bifurcation Theory .............................................. Error! Bookmark not defined.

2.6.10 Bifurcation diagram (Equilibria curves) ............. Error! Bookmark not defined.

2.6.11 Bifurcation Point ................................................. Error! Bookmark not defined.

2.6.12 Hartman-Grobman Theorem ............................... Error! Bookmark not defined.

3. MHD peristaltic flow in a curved circular tubeError! Bookmark not

defined.

3.1 Introduction ......................................................... Error! Bookmark not defined.

3.2 Mathematical Formulation .................................. Error! Bookmark not defined.

3.3 Asymptotic solution ............................................ Error! Bookmark not defined.

3.4 Results and discussions ....................................... Error! Bookmark not defined.

3.5 Concluding remarks ............................................ Error! Bookmark not defined.

4. Analysis of Hartmann boundary layer in peristaltic transport

of Jeffrey fluid: Quantitative and qualitative approaches Error!

Bookmark not defined.


4.2 Mathematical description .................................... Error! Bookmark not defined.

4.3 Asymptotic analytical solutions (Quantitative results)Error! Bookmark not defined.

4.4 Dynamical systems (Qualitative solution) .......... Error! Bookmark not defined.

4.5 Analysis ............................................................... Error! Bookmark not defined.

4.5.1 Analytical results................................................. Error! Bookmark not defined.

4.5.2 Dynamical system (Qualitative results) .............. Error! Bookmark not defined.


5. An exact solution for MHD boundary layer flow of dusty fluid

over a stretching surface .............. Error! Bookmark not defined.


5.2 Formulation ......................................................... Error! Bookmark not defined.

5.3 Exact Analytical solution .................................... Error! Bookmark not defined.

5.4 Dynamical system formalism .............................. Error! Bookmark not defined.


6. Conclusions ........................................ Error! Bookmark not defined.

7. References ........................................... Error! Bookmark not defined.

LIST OF FIGURES

FIG. 3. 1: TOROIDAL COORDINATE SYSTEM , ,r ERROR! BOOKMARK NOT DEFINED.

FIG. 3. 2: PRESSURE GRADIENT VALUES /dp dz , (PLOTTED AS A FUNCTION OF THE AXIAL

LOCATION z (A) AT DIFFERENT TIMES t (B) FOR MAGNETIC PARAMETER M AND (C) FOR

AMPLITUDE RATIO ....................................... ERROR! BOOKMARK NOT DEFINED.

FIG. 3. 3: THE LINE PLOTS OF PRESSURE RISE, P DEPENDING ON F FOR (A) CURVATURE RATIO,

/a R (B) THE MAGNETIC PARAMETER M AND (C) THE AMPLITUDE RATIO .ERROR!

BOOKMARK NOT DEFINED.

FIG. 3. 4: AXIAL VELOCITY PROFILES, AS A FUNCTION OF RADIUS VECTOR, r (A) FOR CURVATURE

RATIO, /a R (B) FOR POLAR ANGLE, 0 (C) FOR POLAR ANGLE, 2 (D) FOR

POLAR ANGLE, 0, ,2 (E) FOR MAGNETIC PARAMETER M (F) FOR AMPLITUDE RATIO, .

........................................................................ ERROR! BOOKMARK NOT DEFINED.

FIG.4. 1: EFFECTS OF (A) MAGNETIC FIELD PARAMETER M , (B) AMPLITUDE RATIO AND (C) THE

JEFFERY FLUID PARAMETER 1 , ON THE PRESSURE RISE PER UNIT WAVELENGTH P AS

DEFINED IN EQ. ERROR! REFERENCE SOURCE NOT FOUND.ERROR! BOOKMARK NOT

DEFINED.

FIG.4. 2: VARIATION OF THE (A) MAGNETIC FIELD M , (B) AMPLITUDE RATIO AND (C) THE

JEFFERY FLUID VARIABLE 1 , ON THE PRESSURE GRADIENT /dp dz AS DEFINED IN EQ.

ERROR! REFERENCE SOURCE NOT FOUND.. .. ERROR! BOOKMARK NOT DEFINED.

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FIG.4. 3: VARIATIONS OF (A) MAGNETIC FIELD PARAMETER M , (B) AMPLITUDE RATIO AND (C)

THE JEFFERY FLUID PARAMETER 1 , ON THE AXIAL VELOCITY w IN EQ.

ERROR! REFERENCE SOURCE NOT FOUND.. .. ERROR! BOOKMARK NOT DEFINED.

FIG.4. 4: THE EQUILIBRIUM PLOT (I.E., LOCATION OF THE EQUILIBRIUM POINT) VIA MAGNETIC

PARAMETER M , WHEN 0.2 AND 0.15F . THE EQUILIBRIUM CURVES RED AND BLACK

CORRESPOND TO 0z AND z n RESPECTIVELY. THICK LINE SHOWS THAT THE EQUILIBRIUM

POINT IS A CENTER WHEREAS THIN LINE SHOW SADDLES, SOLID LINE SHOWS NUMERICAL

RESULTS AND THE DASHED LINES CORRESPOND TO THE ANALYTICAL RESULTS (GIVEN IN EQS.

ERROR! REFERENCE SOURCE NOT FOUND.-ERROR! REFERENCE SOURCE NOT FOUND.).

........................................................................ ERROR! BOOKMARK NOT DEFINED.

FIG.4. 5: THE CONTOUR PLOT OF THE STREAM FUNCTION, GIVEN IN EQ.

ERROR! REFERENCE SOURCE NOT FOUND.. THE FLOW PARAMETERS USED FOR THIS RESULT

ARE 5, 0.15, 0.2M F . ............................. ERROR! BOOKMARK NOT DEFINED.

FIG.4. 6: THE EQUILIBRIA PLOT FOR THE VOLUME FLUX PARAMETER F . THE LINES RED AND BLUE

CORRESPOND TO THE EQUILIBRIA POINT WITH 0z AND z n RESPECTIVELY. FOR THE

EQUILIBRIA CURVE CORRESPONDING TO 0r (SHOWN IN FIG. 4.6(B)) THE AXIAL COORDINATE

z VARIES WITH THE FLUX PARAMETER F , HENCE z -VARIATION IS SHOWN IN THE ADJACENT

FIG(S). THICK LINES REPRESENT CENTERS AND THIN LINES ARE SADDLES. ..ERROR!

BOOKMARK NOT DEFINED.

FIG.4. 7: THE CONTOUR PLOT FOR THE STREAM FUNCTION , GIVEN IN EQ.

ERROR! REFERENCE SOURCE NOT FOUND., FOR 1 0.2, 0.2, 5M AND (A) 0.2F (B)

0F (C) 0.192F AND (D) 0.2F ............. ERROR! BOOKMARK NOT DEFINED.

FIG.4. 8: THE EQUILIBRIA PLOT FOR THE OCCLUSION PARAMETER . THE LINES RED AND BLACK

CORRESPOND TO THE EQUILIBRIA POINT WITH 0z AND z n RESPECTIVELY. THE THICK

LINES REPRESENT CENTERS AND THIN LINES ARE SADDLES.ERROR! BOOKMARK NOT

DEFINED.

FIG.4. 9: THE EQUILIBRIA PLOT FOR THE JEFFREY FLUID PARAMETER 1 . THE LINES RED AND

BLACK CORRESPOND TO THE EQUILIBRIA POINT WITH 0z AND z n RESPECTIVELY. THE

THICK LINES REPRESENT CENTERS AND THIN LINES ARE SADDLES.ERROR! BOOKMARK NOT

DEFINED.

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FIG.5. 1 EFFECTS OF (A) M AND (B) ON THE VELOCITY PROFILESERROR! BOOKMARK NOT

DEFINED.

FIG.5. 2: BIFURCATION DIAGRAM (A) FOR M (B) FOR ERROR! BOOKMARK NOT DEFINED.

FIG.5. 3 PHASE PLANE DIAGRAM SHOWING DUST PARTICLE VELOCITYERROR! BOOKMARK NOT

DEFINED.

LIST OF TABLES

TABLE 4. 1: INTERVALS FOR FLOW RATE F FOR DIFFERENT VALUES OF M ERROR! BOOKMARK

NOT DEFINED.

TABLE 4. 2: INTERVALS FOR FLOW RATE F FOR DIFFERENT VALUES OF ERROR! BOOKMARK NOT

DEFINED.

TABLE 4. 3: INTERVALS FOR FLOW RATE F FOR DIFFERENT VALUES OF 1 ERROR! BOOKMARK NOT

DEFINED.

TABLE 5. 1 COMPARISON OF SKIN FRICTION COEFFICIENT FOR VIA and M WITH GIREESHA ET AL.

[68].................................................................. ERROR! BOOKMARK NOT DEFINED.

TABLE 5. 2 SKIN FRICTION COEFFICIENT AGAINST , ,and M ERROR! BOOKMARK NOT DEFINED.

LIST OF SYMBOLS

,ZR Axisymmetric coordinates

, ,r Toroidal coordinate system

b Wave amplitude

a Tube radius

S Cauchy stress tensor (2M Lt )

oB Uniform magnetic field

fC Skin friction coefficient

Re Reynolds number

R Radius of Curvature

z Axial direction

J Current density

M Magnetic parameter

, ,u v wV Velocity field vector

1 2 3, ,h h h Scale factors of toroidal coordinate system

Kinematic viscosity (2L T )

Electrical conductivity

p Pressure

H Wall of curved tube

h Wall of straight tube

c Wave speed

D

Dt Total time derivative

Wavelength

1 Ratio of relaxation to retardation time

2 Retardation time

k Wave number

Amplitude ratio

Fluid particle interaction parameter

P Pressure rise per unit wavelength

Stretched variable

o Permitivity of free space

Intermediate variable

F Dimensionless flow rate

Stream function

J Jacobian matrix

Mass concentration

Relaxation time of particle phase

K Stokes resistance

N Number density

Similarity variable

Introduction

1

Chapter 1

Introduction

Introduction

2

1.1 Introduction:

Peristalsis is a process of expanding and contracting of an extensible tube to produce

progressive waves that propagates along the length of the tube. The fluid is

transported as a result of pressure gradient arising from contraction/relaxation of

expandable walls. The physiological fluids are transported in human body through

this mechanism. Some examples manifesting this phenomena are: movement of food

bolus from mouth to esophagus, movement of chyme in the gastrointestinal tract,

transport of bile in the bile duct, transport of cilia in paramecium, urine transportation

from kidney to bladder [4], movement of ova in the fallopian tubes, embryo transport

in the uterus [5], transport of lymph in the lymphatic vessel, and blood transportation

in blood vessels, etc.

A number of important applications in medical and engineering sciences are squarely

based on this principle. Peristaltic pumps are used to pump fluids that are hazardous

for the environment. An extremely important medical discovery of finger and roller

pump is contingent upon this mechanism. Various other technical devices like

plungers, infusion pumps, dialysis machines, open-heart bypass pumps etc. are

intuitively made on this principal.

Due attention has been given in the literature, to the peristaltic pumping of fluids, in

the past via theoretical and experimental considerations. Latham [6] was first to

examine peristalsis of viscous fluids empirically analyzing the peristaltic transport of

the fluid while recording useful findings that became benchmark for further

theoretical investigations. Mathematical analysis for the consideration of peristaltic

motion is primarily based on the wave frame approach (Shapiro [7]) and the

laboratory frame approach (Fung and Yih [8]). The mathematical progress in the

peristalsis is based on the assumptions of small Reynolds number and long wave

length approximations. The approximations are well justified on the basis of the order

analysis of various terms involved in the physical problem. One such possibility has

been given by Shapiro [7] in the frame work of peristalsis in ureter and

gastrointestinal tract. Under these assumptions the analytical and experimental results

match with each other and there are a number of situations that warrants the validity

of these approximations. Shapiro [7] discussed the peristaltic pumping of viscous

fluid in a symmetrical channel and axisymmetric tube and presented the velocity

Introduction

3

profile, pressure rise per wavelength. The study reveals the occurrence of some

interesting mechanisms of reflux and trapping that has tremendous applications in

bio-science and in fact has been the subject of main motivation of subsequent studies.

The peristaltic pumping of fluids has been addressed by various other authors [6-23]

under different assumptions of small amplitude ratio, small Reynolds number, small

wave number, different geometries, different wave shapes, non- Newtonian fluid

models, permeable walls etc. We make an important remark at this point; the

preceding research is all about the straight channels and straight uniform tubes.

However, much less has been said about the peristalsis on curved tubes. Our first

focus thus remains on the peristalsis on curved tube taking advantage of the effects of

applied magnetic field.

After some understanding of peristaltic fluid flow for viscous fluid, the attention goes

on to non-Newtonian fluids. These fluids are now considered to be more realistic in

nature because of the rheology properties shown by polymeric and physiological

fluids. The constitutive equations are more complex that takes into account the rate

dependent viscosity showing viscoelastic property. For our purposes, the study of

peristalsis is needed to study the flow generated in the blood vessels, esophagus and

reproductive tracts taking the fluid as non-Newtonian in nature.

Most natural and manmade fluids are non-Newtonian in nature which is the more

specific point of view which leads to the part of the present research. The theory of

non-Newtonian fluids is a part of rheology. Rheology is the study to obtain

appropriate mathematical relations for description of the behavior of non-Newtonian

fluid flows. Nonetheless the behavior of several of these fluids has been studied in

1960‘s and 1970‘s, but the science of rheology is still in its process of development

and new phenomena are constantly being discovered. As some new technologically

significant materials are discovered acting like non-Newtonian fluids therefore

mathematicians, physicists and engineers are actively conducting research in

rheology. The non-Newtonian fluids are reflected to be more genuine. These fluids

are usually highly viscous fluids and their elastic properties are also of importance.

These fluids have time dependent viscosity. Typically non-Newtonian fluids are

polymer solutions, thermo plastics, drilling fluids, paints, fresh concrete and

biological fluids.

Introduction

4

Attention has also been focused to the peristaltic activity of non-Newtonian fluids

through their industrial and physiological importance. Sizeable information about

such flows of hydrodynamic and magnetohydrodynamic (MHD) non-Newtonian

fluids is available. We only elaborate few most recent relevant works in this direction.

Tripathi [9] studied the endoscopic effect in the peristalsis of generalized Maxwell

material via gap of concentric tubular configuration. Inner tube acts like an endoscope

while a travelling wave propagates along the surface of outer tube. Variation iteration

and homotopic procedures are implemented for solutions of problem corresponding to

large wavelength and low Reynolds number. Peristalsis of an incompressible power

law fluid in three distinct viscosity layers is explored by Pandey et al. [10]. Influence

of curvature on peristalsis of an incompressible third grade liquid in a channel is

numerically examined by Ali et al. [11]. Lubrication technique is employed for

simplification of resulting differential system. The governing flow is simulated by

shooting technique. Riaz et al. [12] explored the peristalsis of Carreau liquid in a

rectangular channel. Three dimensional flows are taken. Channel exhibits compliant

wall characteristics. Eigen function expansion and homotopic perturbation techniques

are considered for the computations.

As said earlier and having witnessed great importance; the Peristaltic transport of

fluid have attracted huge attention, it rightfully deserves, in the past few decades.

Although, a lot has been said and published in the literature; there are certain

obscured points which need to be addressed. The purpose of this study is to focus

these issues for further understanding of the peristaltic transport phenomena. To be

specific, we will be concentrating on the effects of MHD on electrically conducting

fluids. MHD peristalsis of micropolar liquid in a tube is analyzed by Wang et al. [13].

The authors performed the analysis for large wavelength. Numerical solution has been

constructed through finite difference scheme with iterative procedure. Shit and Roy

[14] examined the magnetohydrodynamic peristaltic transport of couple stress liquid

in a channel when wavelength is long. Peristalsis of fourth grade liquid in a channel

with mixed convection is addressed by Mustafa et al. [15]. Here Soret and Dufour

contributions are taken into consideration. Keller box technique is utilized for the

simulation.

Interaction of heat transfer, Hall current and porous medium in peristalsis is very

significant. Therefore advancements in the past have been made to these facts by

Introduction

5

considering non-Newtonian materials. Few recent relevant researchers covering such

salient factors may be described here. Soret effect on peristaltic motion of Maxwell

liquid in a channel has been explored by Saleem and Haider [16]. They considered the

inertia less flow and given the series solution by regular perturbation technique.

Influences of wall properties and heat transfer in the peristalsis of Burgers liquid are

examined by Mariyam et al. [17]. This attempt has been made for large wavelength

when the channel walls have different temperature. Impacts of heat transfer and

rotation in the peristalsis of fourth grade material have been addressed by Abd-Alla et

al. [18]. Initial stress and induced magnetic field are encountered. Hina et al. [19]

analyzed the curvature and heat and mass transfer effects in the peristaltic flow of

pseudoplastic liquid. The channel walls are taken flexible. Endoscopic impact on

peristalsis of MHD Jeffrey liquid in cavity is examined by Abd-Alla [20]. The cavity

is taken between two tubes. Lubrication approximations are assumed for the

development of series solutions. Peristalsis of Jeffrey liquid in a porous channel with

partial slip and heat transfer is addressed by Das [21]. Incompressible fluid in channel

is taken conducting and no Joule heating and Hall and ion-slip effects are considered.

Ramesh and Devakar [22] discussed the peristaltic activity for flow of couple stress

liquids saturating porous medium in a channel. MHD and heat transfer are also

present. Viscous dissipation, Joule heating and Hall and ion-slip effects are absent in

this study. Reynolds number is chosen small. Heat transfer analysis in peristaltic

motion of Jeffery liquid in a vertical channel with porous medium has been carried

out by Vajravelu et al. [23]. Perturbation solutions are developed. Mechanism of

peristalsis for generalized Burgers fluid in a channel is studied by Tripathi and Beg

[24]. They performed such analysis in absence of heat transfer, Hall and ion-slip,

Joule heating and convection. Hydromagnetic peristaltic flow of micropolar material

without Hall and ion-slip effects in a channel is addressed by Abd-Alla et al [25].

Micropolar material fills the porous space. Rotation is considered in this attempt. Abd

Elmaboud and Mekheimer [26] made an attempt for peristalsis of non-conducting

second order liquid filling porous space inside a symmetric geometry. Large

wavelength technique is followed. Heat and mass transfer in peristalsis of electrically

conducting Eyring-Powell liquid is examined by Shaaban and Abou-Zeid [27].

Incompressible material occupies the porous space between the coaxial tubes. Joule

heating and Hall and ion-slip are omitted. Pumping phenomenon for peristalsis of

Williamson material filling porous medium in a heated symmetric channel is analyzed

Introduction

6

by Vasdev et al. [28]. Fluid here is considered non-conducting and therefore Hall and

ion-slip and Joule heating contributions are ignored. This analysis for slight mass

transfer different boundary conditions, Hall current and Ohmic heating is

reconsidered by Eldabe et al. [29]. It should be noted that in the investigations [22-

29], the porous medium effects for non-Newtonian liquids are characterized by

classical Darcy‘s expression of viscous liquid. This is totally inadequate and such

consideration leads to serious error in the formulation. It is in view of the reason that

Darcy‘s expression depends upon the properties of liquids and thus such expression

for viscous and non-Newtonian materials are distinct. Even Darcy‘s expressions for

different non-Newtonian materials are not similar. With this view point few authors

correctly modeled the porous medium effects in peristalsis through modified Darcy‘s

law. For instance Hayat et al. [30] initially examined the peristalsis of Maxwell

material in a channel with porous medium and Hall current. This study for linearized

version of Oldroyd-B fluid (also called Jeffery linear model) in a channel with

flexible walls is examined in absence of Hall current by Hayat et al. [31]. It is

pertinent to mention here that now some authors discussed Jeffrey nonlinear fluid

model in the modeling and analysis through modified version of Darcy‘s law. In

studies [30, 31] the corresponding modified Darcy‘s law was developed. Analysis in

both attempts was presented for small amplitude ratio. Later these expressions of

modified Darcy‘s laws have been used by the researchers in the studies [32]. In [32]

Eldesoky and Mousa analyzed peristalsis of non-conducting Maxwell liquid filling

porous medium in circular tube. Effect of Hall current on peristaltic transport of

Maxwell material saturating porous medium in a channel with porous wall is

examined by El Koumy et al. [33]. Das [34] analyzed the peristalsis of Maxwell liquid

filling porous space in a compliant wall channel. Fluid is electrically conducting.

Velocity and thermal slip conditions are considered. Few more representative studies

in this direction can be consulted by references [35-40] and several attempts therein.

It is seen that much attention in past has been devoted to studies on peristalsis in

planar channel. This consideration is not adequate in many cases of fluid

transportation in ducts and conduits which are of curved shape. Initially Sato et al.

[41] discussed peristalsis of viscous liquid in a curved channel. Long wavelength and

stream function formulation are adopted by Ali et al. [42] when the authors extended

the results of Sato et al. [41]. Ali et al. [43] examined heat transfer effect also in

Introduction

7

peristalsis of viscous liquid through curved channel. Hayat et al. [44] addressed

compliant wall effect on peristaltic motion of an incompressible liquid in a curved

channel satisfying no-slip conditions. Kalantari et al. [45] analyzed influence of

radially imposed MHD on peristalsis of Phan-Thien-Tanner (PTT) fluid in curved

configuration. Finite Difference Method (FDM) has been utilized for the solution of

nonlinear system of equations. Ali et al. [46] examined influence of heat transfer on

peristaltic activity of Oldroyd 8-constant fluid in a curved geometry. Numerical

method is utilized. Finite difference technique (FDM) along with an iterative

procedure is adopted for the numerical solutions. Furthermore information on

peristalsis of Newtonian and non-Newtonian fluids through curved channels can be

seen by the studies (see refs. [47-58]). It is pertinent to mention here up till now there

is no study which can describe the peristalsis in a curved tube. This consideration

seems more realistic when physiological and industrial process is accounted. With this

view point, the main theme of present thesis is to address peristaltic flow of viscous

and non-Newtonian fluids in curved tube. Constitutive relations of Jeffrey liquid are

employed for the non-Newtonian fluid.

The Peristalsis phenomenon remains under attention since last four decades.

Peristalsis study reveals many important flow features like volume flow rate, pressure

rise per unit wavelength, force of friction and trapping. The process of trapping has a

wide range of effects in the transportation of the physiological fluids, the embryo

movement, movement of pill etc. Streamline patterns lead to the study of trapping.

The region where streamlines are closed, guarantee the existence of eddy. The study

of eddy transportation or flow patterns require more appreciative enhancement.

Therefore, the analysis of streamlines pattern and their topological aspects through

bifurcation diagrams are the need of the study. This study comes under the umbrella

of dynamical system. Dynamical system actually describes the qualitative behavior of

the flow.

The qualitative theory of the linear and non-linear differential equations is important

to analyze and understand the qualitative behavior of flow structure by dynamical

system theory and helps to get the quantitative estimates. The literature review shows

that the qualitative approach to study the flow features by analyzing the streamlines

patterns is very just presented [59-60]. Jimenez [59] was first who studied the

toplogical aspects of bifurcation diagrams of codimension one and two for two

Introduction

8

dimensional planar channel by considering the assumption of long wavelength and

small Reynolds. Later Asghar [61] extended his work by considering the slip effects

and mixed convection. They discussed the effects of the surface slip and mixed

convection on streamlines patterns and their bifurcations for the peristaltic transport

of a Newtonian fluids. The nonlinear autonomous differential equations were

established and the method of dynamical systems was used to discuss the local

bifurcations and their topological changes. All types of bifurcations were also

discussed graphically.

To apply the method of dynamical systems we obtain the expression for the stream

functions in the wave frame under the assumption of long wavelength and low

Reynolds number. For the discussion of the path of the particle in wave frame a

system of nonlinear autonomous differential equations is obtained and the methods of

dynamical systems are used to discuss bifurcations and their topological changes.

Moreover, we presented an exact analytic solution for boundary layer flow of dusty

fluid over a stretching surface in the presence of applied magnetic field. We

emphasize that an approximate numerical solution of this problem was available in

the literature but no analytical solution was presented before this study. All types of

bifurcation diagrams are conferred to get the information with respect to all values of

characterizing parameters known as fluid particle interaction parameter, magnetic

field parameter, and mass concentration of dust particles.

Preliminaries

9

Chapter 2

Preliminaries

Preliminaries

10

This chapter is designed to give some related concepts and notions about research

problems which are included in this dissertation. Furthermore, the description of

dimensionless parameters is given which are appearing in the model problems. A

short overview about Jeffrey liquid is also included.

2.1 Flow models

Our intention here is to include related equations and standard definitions for

peristaltic flows with different geometries.

2.1.1 Axisymmetric flow model

We choose cylindrical coordinates , ,r z and velocity , ,r zu u uV . For no

tangential component 0u and the physical quantities do not dependent upon . For

incompressible liquid one has

10 ,

uzru

rr r z

(2.1)

2

2 2

1 1,r r r r r r

r z r

u u u u u uu u r f

t r z r r r r r r z

(2.2)

2

2

1 1,z z z z z

r z z

u u u u uu u r f

t r z r z r r r z

(2.3)

in which rf and zf are the components of body force.

2.1.2 Axisymmetric flow in curved configuration

For this type of geometry we choose toroidal coordinates , ,r z R which were

first used by Dean 1927 [62]. For an incompressible fluid and velocity field

, ,r zu u uV , we have

1 1 sin cos

0,sin sin sin

ru v R w u v

r r r R r z R r R r

(2.4)

Preliminaries

11

' ' ' ' ' ' '2 '2 ''

' ' ' ' ' ' ' '

' ' '

' ' ' ' '

' '' '

' 2 ' ' '

sin

sin sin

sin

1 cos 1

( )( )

sin sinsin

u u u u w R u v w pu

t r r R r z r R r r

v v u

r R r r r r

R u wR w R r

R r z z r

,rf

(2.5)

' ' ' ' ' ' ' ' '2'

' ' ' ' ' ' '

' ' ' '

' ' ' ' ' '

''

'

' 2 ' ' '

' '

cos

sin sin

sin

1 sin 1

( )

cos

sin sin

v v v v w R v u v wu

t r r R r z r R r

p v v u

r r R r r r r

vR w

zR

R r z R r w

r z

,f

(2.6)

' ' ' ' ' ''

' ' ' ' ' '

' '' ' '

'

' ''

' ' ' ' '

'

' ' '

sin

si( )( )

1 1

( )

1 1 1

(

nsin cos

sin

sinsin

sin )

w w v w w R wu

t r r R r z R p

R r zw u v

R r

w uR w

r r r R r z

wR

r r R r

''

'cos .z

vw f

z

(2.7)

2.1.3 Equations for Jeffrey liquid

Expression of Cauchy stress tensor T for an incompressible material is

,p T I S (2.8)

in which an extra stress tensor S for Jeffrey material obeys

2

1

,1

D

Dt

11

AS A (2.9)

Preliminaries

12

where grad gradt

1

A V V , is the first Rivilin Erikson tensor, I the identity

tensor, p the pressure, the dynamic viscosity, 2 is the retardation time and 1

the ratio of relaxation to retardation time. The velocity for present flow is

, , , , , , , , .u r z t v r z t w r z t V (2.10)

We can write now as

1

1grad ,

1

u u v w

r r r z

v v u v

r r r z

w w w

r r z

V (2.11)

12

1 1 12 ,

12

u v u v w u

r r r r r z

v u v v u w v

r r r r r r z

u w v w w

z r z r z

1A (2.12)

12

1 1 12

u v u v w u

t r t r r r t r z

D v u v v u w v

Dt t r r r t r r t r z

u w

t z r

1

V V V

AV V V

V

,

12

w v w

t r z t z

V V

(2.13)

1

.r ze e er r z

(2.14)

The components of extra stress tensor S are

2

1

21 ,

1rr

uS

t r

V (2.15)

2

1

11 ,

1r r

v u vS S

t r r r

V (2.16)

Preliminaries

13

2

1

1 ,1

rz zr

w uS S

t r z

V (2.17)

2

1

2 11 ,

1

v uS

t r r

V (2.18)

2

1

11 ,

1z z

w vS S

t r z

V (2.19)

2

1

21 .

1zz

wS

t z

V (2.20)

Scalar forms of equation of motion are

1

,r rrrr rzS S SS SP

ut r r r z r

V (2.21)

1

,r z r rS S S S SPv

t r r r z r

V (2.22)

1

.zzr zz zrSS S SP

wt r r r z r

V (2.23)

2.2 Maxwell’s equations

Four fundamental laws lead to such equations.

2.2.1 Gauss’ law for electricity

The electric flux E leaving a volume is proportional to the charge inside i.e.,

0

,

E (2.24)

in which 0 is the permittivity of free space.

2.2.2 Gauss’ law for magnetism

Total magnetic flux B around a closed surface is zero i.e.,

0. B (2.25)

Preliminaries

14

2.2.3 Faraday’s law

The voltage E induced in a closed loop is proportional to the magnetic flux rate of

change t

B that loop encloses i.e.,

.t

BE (2.26)

2.2.4 Ampere-Maxwell law

The magnetic field B around a closed loop is proportional to J plus displacement

current t

E that the loop encloses i.e.,

0 ,o

t

EB J (2.27)

in which 0 is the magnetic constant and J for current density.

2.3 Ohm’s law

Here

, J E V B (2.28)

where B designates total magnetic field and Lorentz force is equal to J ×B .

2.4 Hartmann boundary layers

Hartmann layers are the most important feature of magneto hydrodynamic (MHD)

flows. These layers were first discovered theoretically and experimentally by Julius

Hartmann [63]. These layers are developed when fluid motion in present of magnetic

field is taken.

2.5 Non-dimensional numbers:

Non-dimensionalisation is fairly simple but very important technique. It helps us to

simplify differential equation by reducing the number of parameters by rescaling

individual values. The dimensionless parameters are measure of significance of flow

aspects. Here we present definition of some dimensionless numbers which will be

employed in the subsequent chapters.

Preliminaries

15

2.5.1 Reynolds number:

Ratio of inertial to viscous forces is characterized as Reynolds number.

Mathematically we can write as

inertial forces

Re .viscous forces

2.5.2 Hartmann number:

Ratio of magnetic forces to viscous forces is known as Hartmann number i.e.,

.M BL

(2.29)

Here signifies the fluid dynamic viscosity, the electrical conductivity and L the

characteristic length.

2.5.3 Wave number

Wave number can be stated as the ratio of tube average radius a to the wavelength of

peristaltic wave .

.a

k

(2.30)

2.5.4 Amplitude ratio

Ratio of peristaltic wave amplitude to tube radius is amplitude ratio. It is symbolized

by

.b

a (2.31)

2.6 Dynamical system

2.6.1 Fundamentals of stability

The word stable derives from the Latin adjective ―stabileim‖ which means being able

to stand firmly. It is roughly characterized by the response of a system to small

disturbances in its state. A system, whose stability to be studied, is perturbed a little

Preliminaries

16

from its equilibrium. If the perturbed motion stays within acceptable limits of the

equilibrium position then it is said stable otherwise unstable. Here we present some

standard definitions relevant in this direction.

2.6.2 Autonomous system

A system of differential equations is autonomous when it is not an explicit function of

independent variable t , i.e.,

n, .d

fdt

X

X X X R (2.32)

Trajectories here do not change with time.

2.6.3 Equilibrium solution

Solution curves of Eq. (2.32) are sometime called equilibrium solutions, stationary

solutions, constant solution, singular points, rest points, critical points or fixed points

when system is at rest. Mathematically

0 0such that 0.f X X X (2.33)

2.6.4 Asymptotically stable equilibrium

Any stationary solution 0X for which small perturbation vanishes when time goes to

infinity is called asymptotically stable i.e., 0lim .t

t

X X

2.6.5 Stable equilibrium

When there is little response to small perturbation in 0X as time approaches infinity

the stationary solution is stable otherwise the stationary solution is unstable.

2.6.6 Nondegenerate (Hyperbolic) equilibrium point

For this type no eigen value of Jacobian matrix has real part.

2.6.7 Types of Equilibrium points

For linear system of equations t X AX , the qualitative analysis of trajectories in

the vicinity of stagnation point.

Preliminaries

17

Saddle: When the eigenvalues corresponding to a linear system are real and are of

opposite signs then the stagnation point is saddle. A saddle point is always unstable in

nature.

Node: When the eigenvalues corresponding to a linear system are real and are of

same signs then the stagnation point is node. There are types of nodes:

a) When both eigenvalues are negative then the node is stable because all the

trajectories will approach to the stagnation point as time evolves.

b) When both eigen values are negative then the node is unstable because the

solution explodes as time evolves.

Spiral: Eigenvalues are complex conjugates then trajectories are harmonic

oscillations. These oscillations grow when real part of eigenvalues is positive and

decay when real part of eigenvalues is negative.

Center: In this case eigenvalues are purely imaginary and the trajectories are

periodically closed curves around the stagnation point.

2.6.8 Linearization

The linearization of a nonlinear system

*,f X X (2.34)

is

. 0

X A X X (2.35)

Here 0A X

is jacobian matrix at equilibrium point.

2.6.9 Bifurcation Theory

Bifurcation theory is about the changes in the structure of a dynamical system when

parameter values are changed. These structural changes are basically topological or

qualitative changes in the behavior of dynamical system. The parameter whose values

are being changed is called the bifurcation parameter.

Preliminaries

18

2.6.10 Bifurcation diagram (Equilibria curves)

This is the graphical representation of solution of autonomous system for loci of

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

curves.

2.6.11 Bifurcation Point

A bifurcation point or branch point (with respect to parameter) is a solution of Eq.

(2.32), where the number of solutions changes when parameter passes some specific

value.

2.6.12 Hartman-Grobman Theorem

If Jacobian matrix 0A X has no eigenvalues with zero real part, then the family of

trajectories near a singular point 0X of a nonlinear system f

*X = X and those of

the locally linearized system have the same topological structure; which means that in

a neighborhood of 0

X there exists a homeomorphism which maps trajectories of the

non-linear system into trajectories of the linear system. (Bakker [64])

MHD peristaltic flow in a curved circular tube…

19

Chapter 3

MHD peristaltic flow in a curved circular

tube


20

3.1 Introduction

This chapter deals with the peristaltic flow of viscous fluid in a slightly curved tube

under the influence of applied magnetic field. The flow is three dimensional. The

governing equations are formulated using Dean‘s analysis [62] and an asymptotic

solution for the three dimensional Navier-Stokes equations is presented under the

assumption of small inertial forces Re 0 and long wavelength / 0a

approximation. The effects of curvature on the fluid flow are investigated that takes

into account the small perturbation from the straight tube. The approximate analytic

solution is obtained using regular perturbation method in terms of small curvature.

Results are discussed graphically noting that the velocity field is much more sensitive

to the curvature of tube in comparison to the pressure gradient. The peristaltic

magnetohydrodynamic (MHD) flow in a straight tube is the limiting case of this

study. Detailed analysis of the results is presented in the section 3.4.

3.2 Mathematical Formulation

MHD peristaltic flow of viscous liquid in curved tube is considered (see Fig. 3.1).

Average tube radius is denoted by a and R the radius of curvature of tube. Notice

that particular case of infinitely large radius of curvature (i.e., R ) represents the

straight tube geometry. The cylindrical coordinates cannot be used to represent the

curved tube due to the additional curvature parameter; hence we use the orthogonal

curvilinear coordinate system, as introduced in Dean [62]. A sketch of the coordinate

system given in Fig. 3.1 shows the circular cross section of the tube lying in a plane.

A point P within the cross section of tube is specified by , ,P r z . Here angle

is measured from the vertical axis that is orthogonal to the line OC joining the center

of curvature of the tube to the center of the circular cross section. The coordinate

z R represents the axial distance along the centerline of tube, of the cross-

sectional plane to the point C . The corresponding velocity is ( , , )u v w V . Here u

represents the velocity in the radial r direction, v the velocity along direction

(perpendicular to u ) and w component of velocity in the axial direction

(perpendicular to both u and v ).


21

Fig. 3.1: Toroidal coordinate system , , .r

Electrically conducting fluid is taken for constant strength. Then Lorentz force

reduces to

2 2

0 00 0

1 sin 1 sin

B u B wr z

r r

R R

J ×Β , (3.1)

0B is the characteristic magnetic induction and 0 0and r z unit vectors in andr z

directions and we write metric scale factors 1 2 31, , and 1 sin

rh h r h

R

.

Displacement of tube walls is given by the sinusoidal wave. Here wave shape is

2

, sinH z t a b z ct

, (3.2)

in which wave half amplitude is b and the wavelength. We prefer to work in the

frame of reference moving with the wave speed c ; hence the walls are stationary in

this frame of reference.

The no slip boundary conditions are

0 atu v w r H . (3.3)


22

For an infinitely long tube, with small curvature, the motion can be treated as

independent of , once the flow is fully developed. In curved geometries, the fluid

motion is not unidirectional, so we consider three dimensional forms of the governing

equations as follows:

1 1 sin cos

0 ,sin sin sin

r u v R w u v

r r r R r z R r R r

(3.4)

' ' ' ' ' ' '2 '2 ''

' ' ' ' ' ' ' '

' ' '

' ' ' ' '

' '' '

' 2 ' ' '

sin

sin sin

sin

1 cos 1

( )( )

sin sinsin

u u u u w R u v w pu

t r r R r z r R r r

v v u

r R r r r r

R u wR w R r

R r z z r

2 '

0

',

1 sinr

B u

R

(3.5)

' ' ' ' ' ' ' ' '2'

' ' ' ' ' ' '

' ' ' '

' ' ' ' ' '

''

'

' 2 ' ' '

' '

cos

sin sin

sin

1 sin 1

( )

cos

sin sin

v v v v w R v u v wu

t r r R r z r R r

p v v u

r r R r r r r

vR w

zR

R r z R r w

r z

,

(3.6)

' ' ' ' ' ' ' ' ''

' ' ' ' ' '

'

' ''

' ' ' ' ''

'

'

' ' '

sin cos

sin sin

s

( )

1

( )1

( )

1

in

sins

1 1

(

in

w w v w w R w w u vu

t r r R r z R r

w

r R rR p

R r z r r uR w

z

w

r r R r

2 ''' 0

''co .s

sinin

)1 s

B

R

wvR w

rz

(3.7)


23

To make Eqs. (3.4)-(3.7) dimensionless, we use the wave speed c , as the

characteristic speed, tube radius a and wavelength as the characteristic length scale

in the radial and axial direction respectively, and / c as a reference time. In terms of

these characteristic variables, the scaling for pressure comes out to be 2/c a ,

2

, , , , , , ,u v w r t c z a p

u v w r t z pkc kc c a c

(3.8)

in which a

k

is the wave number.

Using the dimensionless variables defined above, the governing Eqs. (3.4)-(3.7) in

their dimensionless form read

1 sin 1 cos

0sin sin sin

ru u v v R w

r r R r r R r R r z

, (3.9)

2

2 2

2

2

2

2

/

/ sinRe

1 sin

/ sin

1 cos 1

/ sin

/ sin/

/ sin / sin

w R au u u u uu

t r r R a r zk

v w

r k R a r

p v v uk

r r R a r r r r

uk R a w

R a zk

wzR a r R a rr

2 2

/,

/ sin

R ak M

R a r

(3.10)


24

2

2

2

2

2

2

2

/

/ sinRe

1 cos

/ sin

1 sin 1

/ sin

/ cos/

/ sin/ sin

w R av v v v vu

t r r R a r zk

uv w

r k R a r

p v v uk

r r r r rR a r

vk R a w

zR ak

R a rk wzR a ra r z

,

(3.11)

2

2

/

/ sinRe

sin cos

/ sin

1

/ sin/ 1

/ sin/ sin

1 1 1/ cos

/ sin

w R aw w v w wu

t r r R a r z

w u v

R a r

w

r R a rR a p

R a r z r r uk R a w

z

w vk R a w

r r R a r z

2

/.

/ sin

R a wM

R a r

(3.12)

The dimensionless constants in Eqs. (3.10)-(3.12) are defined as

0Re , ,ca

k M B a

(3.13)

where Re is the Reynolds number representing the ratio of the inertial forces to the

viscous forces, M is the magnetic field parameter, defined as the ratio of the magnetic

forces to the viscous forces.

The non-dimensional wall equation is

, 1 sin 2 ,H z t z t (3.14)


25

where /b a is amplitude ratio.

As we are interested in pressure driven peristaltic motion in which the wavelength is

much larger compared to the characteristic length scale i.e., the tube radius, so we

employ the long wavelength and small Reynolds number approximation. Under these

assumptions, dimensionless Eqs. (3.10)-(3.12) take the following forms:

0 ,p

r

(3.15)

0 ,p

(3.16)

2

/ 1 sin

/ sin / sin

/1 1 cos.

/ sin / sin

R a p w w

R a r z r r r R a r

R aw wM w

r r R a r R a r

(3.17)

3.3 Asymptotic solution

In the present work, we restrict ourselves to the case when the ratio of the tube radius

to the radius of curvature is small i.e., / 1.a R

Thus we write

0 1/ ...,w w a R w (3.18)

0 1/ ...,F F a R F (3.19)

0 1/ ...dp dpdp

a Rdz dz dz

. (3.20)

Incorporating the assumed solution (3.18)-(3.20) in the modeled Eqs. (3.15)-(3.17),

we obtain the zeroth order system as follows:

2 2

20 0 0 002 2 2

1 1,

dp w w wM w

dz r r r r

(3.21)

0 0.r H

w

(3.22)


26

The zeroth order problem Eqs. (3.21)-(3.22) represent the problem of peristalsis in a

straight circular tube which does not contain the curvature effects. If we assume

symmetry, we have 0w independent of . Symmetry condition is given by

0

0

0 .r

w

r

(3.23)

Hence, Eqs. (3.21)-(3.22) become

220 0 0

02

00

0

1,

0 , 0 .r H

r

dp w wM w

dz r r r

ww

r

(3.24)

Analytical solution of Eq. (3.24) is given by

0 0

0 2 2

0, / /.

0,

BesselI Mr dp dz dp dzw r

BesselI MH M M

(3.25)

Here 0,BesselI Mr is the modified Bessel function of first kind and zero order.

Notice that the zeroth order problem is the same as that of a straight tube with MHD

[65]. Hence the higher order corrections will bring in the curvature effects to the flow

dynamics.

Again, collecting higher order terms, the first order problem is found as

2 1 0 01

1,sin sin

0,

BesselI Mrdp dp dpw r

dz dz MBesselI MH dz , (3.26)

with boundary conditions,

1 0 ,r H

w

(3.27)

where 1,BesselI Mr is the modified Bessel function of first kind and first order and

2 22

2 2 2

1 1.

r r r r

Analytical solution to the first order system is given by


27

2 2 11

1sin ,

4

dpw r H g r

dz (3.28)

where

3 3

0 3

3 3

0 3

8 1, 0,g /

8 0,

8 1, 0,1/ .

8 0,

BesselI MH H M BesselI MHrr dp dz

M BesselI MH

BesselI Mr r M BesselI MHdp dz

M BesselI MH

Notice that the first order solution is dependent on the polar angle , in contrast to the

zeroth order solution. Thus curvature effects are included.

Combining the zeroth order solution (3.25) and the first order solution (3.28) using

Eq. (3.18), the final solution takes the following form

0 0

2 2

22 2 1

0, / /, , ,

0,

1/ sin / ,

4

BesselI Mr dp dz dp dzw r z t

BesselI MH M M

dpa R r H g r O a R

dz

(3.29)

where

3 3

0 3

3 3

0 3

8 1, 0,g /

8 0,

8 1, 0,1/ .

8 0,

BesselI MH H M BesselI MHrr dp dz

HM BesselI MH

BesselI Mr r M BesselI MHdp dz

M BesselI MH

The obtained solution (3.29) involves two unknown quantities i.e., 0 /dp dz and

1 /dp dz that will be calculated using the dimensionless volume flow rate as an input

parameter. In order to calculate /dp dz , we use the non-dimensional volume flow rate

,2

2

0 0

1, , ,

2 2

H z tF

F w r z t rdrda c

. (3.30)

As the axial velocity, w is given in the form of series expansion(3.18), we expand the

volume flux F , as well, in terms of the small parameter /a R i.e.,


28

0 1/ ,F F a R F (3.31)

where pressure gradient takes the form

3

0 1

4

2 0, 16.

2 1, 0,

F M BesselI MH Fdp a

dz R HH BesselI MH MBesselI MH H

(3.32)

In the above relation, we choose 1 0F so that 0F F . It is cautioned that this

particular choice of 0F and

1F may not work well if the higher order corrections are

included (detail is discussed in the later section). Therefore, Eq. (3.32) takes the

following form

32 0,.

2 1, 0,

FM BesselI MHdp

dz H BesselI MH MBesselI MH H

(3.33)

Substituting /dp dz from Eq. (3.33) in Eq. (3.29), one obtains

3 3

3 3

2 0,

2 1, 0,

2 0,

2 1, 0,

8 1, 0,

4 2 1, 0,

8 1, 0,

4 2

FMBesselI Mrw

H BesselI MH MBesselI MH H

FMBesselI MH

H BesselI MH MBesselI MH H

BesselI MH H M BesselI MHa F

R r BesselI MH MBesselI MH H

BesselI Mr r M BesselI MHF

H BesselI

sin .

1, 0,MH MBesselI MH H

(3.34)

Eq. (3.34) gives the axial velocity w , up to the first order, in the laboratory frame of

reference. Pressure rise per unit wavelength in terms of dimensionless variables is

1

0

dpP dz

dz

. (3.35)

3.4 Results and discussions

Here we discuss some important findings for pressure gradient, pressure rise per unit

wavelength, and axial velocity. Also investigated are the effects curvature ratio /a R ,


29

polar angle , amplitude ratio and the magnetic field parameter M on flow

dynamics and characteristics.

Fig. 3.2(a) gives the line plot of the pressure gradient /dp dz as defined in Eq. (3.33)

along the tube, as a function of the axial coordinate z , at different time instants. For a

particular instant 0t , the curve shows a periodic behavior in z -axis with two

distinct peaks corresponding to the maximum pressure gradient / 100dp dz , occur at

0.75, 1.75z and the minimum pressure gradient / 10dp dz is at 0.25, 1.25z .

These peaks, in pressure gradient /dp dz , arise due to the sinusoidal nature of

boundary. As the pressure gradient /dp dz is inversely proportional to tube radius r ,

the maximum occurs at the axial location, 0.75z corresponding to the wave

compression where the tube radius is minimum. Similarly, the minimum /dp dz is at

0.25z which corresponds to the wave expansion where the tube radius is

maximum. Furthermore the variation of pressure gradient along the wave exhibits

same pattern in the laboratory frame of reference as in wave frame of reference.

In Fig. 3.2(b), we show variation of M on / .dp dz Clearly /dp dz is enhanced for

larger magnetic field parameter M . The Lorentz force, associated with the magnetic

field parameter M (as defined in Eq. (3.1) has a retarding effect on the flow

velocities. The flow velocities decrease when M increases and consequently value of

pressure gradient /dp dz enhances p w

z z

. Notice that increment in values of

pressure gradient /dp dz is not uniform in z . For instance when the magnetic

parameter M increases from 2M (dotted line) to 6M (dashed line), the increase

in the values of /dp dz at 0.75z is approximately twice more than the increment in

/dp dz at 0.25z . This is so because the Lorentz force is inversely proportional to

tube radius (see Eq. (3.12)). Hence maximal effect is seen at the locations ( 0.75z )

which corresponds to the wave compression and the effects are minimal at wave

expansion location 0.25z .

Effects of amplitude ratio , as defined in Eq. (3.14), on the pressure gradient /dp dz

is very similar to that of magnetic field parameter M shown in Fig. 3.2(c). The

pressure gradient /dp dz shows an increasing pattern through larger amplitude ratio


30

with maxima occurring near axial location 0.75, 1.75z corresponding to the

wave compression as observed in the case of varying magnetic field parameter M .

Additionally there also exists a region in vicinity of the wave expansion

(corresponding to 0.25, 1.25, 2.25z ), where the values of pressure gradient /dp dz

have decreasing trend with the increasing values of amplitude ratio . This is so

because the tube radius increases near the wave expansion region in contrast to wave

compression region where the tube radius decreases as the amplitude ratio is

increased. Hence with increasing , the magnitude of Lorentz force decreases near to

wave expansion region and it increases in neighborhood of wave compression region.

Fig. 3. 2: Pressure gradient values /dp dz , (plotted as a function of the axial location z (a) at

different times t (b) for magnetic parameter M and (c) for amplitude ratio

Fig. 3.3 shows pressure rise variation P , as defined in Eq. (3.35), along the tube as

a function of the flow rate F . In all results, we see a linear relationship with a

negative slope between p and F (consistent with the previously reported results

e.g. Shapiro [7], Hayat et al. [65] and Siddiqui [66]). We see that for the upper half

0 0.5 1 1.5 2 2.520

40

60

80

100

120

0 0.5 1 1.5 2 2.50

20

40

60

80

100

0 0.5 1 1.5 2 2.50

20

40

60

80

dp

dz

(c)

dp

dz

(b)(a) z z

z

,= 0:2

t = 0

M = 2

t = 0

,= 0:2

M = 4

t = 0; 0:1; 0:2

dp

dz

M = 2; 4; 6

,= 0:2; 0:3;

0:4


31

plane, adverse pressure gradient 0P acts in the opposite direction of the

peristalsis 0F . On the other hand for the lower half plane, there is a favorable

pressure gradient 0P in direction of peristalsis. In Fig. 3.3(a), we show results

for curvature ratio /a R , ranging from 0 to 0.6 with a uniform increment of 0.2. It

appears as though the curvature ratio has no influence on pressure rise p that is

why all the curves overlap. It is found experimentally that volumetric flow rate

decreases in case of flow through a curved tube when compared with flow rate in case

of straight tube, if all other factors like length and radius of the tube, applied pressure

gradient, and the fluid viscosity remains constant. This is so because for flow along a

curved tube, some energy which fluid possesses through the applied pressure gradient

is consumed in driving secondary flow, as discussed below, hence more energy is

dissipated and the flow rate decreases. However, this effect does not appear in the first

order correction for small /a R as discussed in Leal [67]. To predict this effect, we

must proceed to the higher order corrections in asymptotic expansion(3.20). The

problem that we encounter in evaluation of higher order correction is the

incorporation of flux constraint in order to determine the solution fully. Recall that the

solution obtained (Eq. (3.29)) is in terms of unknown pressure gradients 0 /dp dz and

1 /dp dz corresponding to the zeroth and first orders problems respectively. Hence we

have two (or more for higher order corrections) unknown quantities that are to be

determined from a single volume flux constraint. We worked our way around this

issue by expanding volume flux F in terms of /a R (0 1

aF F F

R ) and assuming

0 1, 0F F F . This remedy, however, will not be so curing if higher order

corrections are added in expansion. Because in that case, by adopting this strategy, we

will be losing all the contributions of curvature to the pressure gradients (see Eq.

(3.29)).

In Fig. 3.3(b) we have plotted the pressure rise P against F for different magnetic

parameter M . As seen in the results presented in Fig. 3.2(b), the values of pressure

gradient /dp dz enhances through larger M . Hence the pressure rise per unit

wavelength P shows an increasing trend via M . Thus with larger values of M , the

slope of pumping curve increases. The effects of amplitude ratio , as defined in Eq.


32

(3.14), on the pressure rise per unit wavelength is displayed in Fig. 3.3(c). Obviously

adverse pressure gradient is higher for more occluded waves (larger values of ). It is

consistent with the results presented in Fig. 3.2(c), where the values of the pressure

gradient /dp dz are observed to be increasing with larger values of .

Fig. 3. 3: The line plots of pressure rise, P depending on F for (a) curvature ratio, /a R

(b) the magnetic parameter M and (c) the amplitude ratio .

We present the axial velocity profiles (i.e., the fluid velocity along the curved axis) as

a function of the radial coordinate r in Fig. 3.4. Effects of curvature ratio /a R are

shown in Fig. 3.4(a). A distinctive feature of flow along curved geometries compared

to the flow along a straight tube is existence of the secondary flow arising due to the

presence of centrifugal forces. For flows along the curved tube, in addition to the flow

in the axial direction, there exists motion in cross sectional plane (commonly termed

as ―secondary flow‖) driven essentially by the centrifugal force acting on the fluid as

it moves in a curved path along the tube axis. We therefore see in Fig. 3.4(a) that the

axial velocities w for the flow in a curved tube ( / 0a R ) are different (and

somewhat smaller) compared to the axial velocities in the straight tube ( / 0a R ).

The axial velocities w , for the flow in a curved tube, can be smaller (or greater) than

the axial velocities w for the flow in a straight tube depending upon the nature of the

-1 -0.5 0 0.5 1-400

-200

0

200

400

-1 -0.5 0 0.5 1-1000

-500

0

500

1000

-1 -0.5 0 0.5 1-500

0

500

(a) (b)

(c)

"P6

"P6"P6

a=R = 0; 0:2; 0:4; 0:6

M = 2; 3; 4; 5

,= 0:2; 0:4; 0:5

F

F

F

M = 3

a=R = 0:1

M = 2

,= 0:2

a=R = 0:1

,= 0:2


33

secondary flow (as elaborated in Leal [67]). For 0 , the radial vector traverses

the inner wall of tube whereas 2 represents the outer wall of the tube.

Secondary flow (as shown in Leal [67] ) has such a pattern that on the inner wall, flow

is being pushed towards the center of the tube; hence the velocity curves (Fig. 3.4(b))

are skewed to the left (towards the center 0z ) whereas on the outer wall, flow is

being pushed towards the boundary away from center as a consequence of which the

velocity curves (Fig. 3.4(c)) are skewed to the right (away from center 0z ).

Furthermore, there doesn‘t exist any secondary flow on 0, , and 2 . Hence the

axial velocity w for these values of are the same as for straight tube (see Fig.

3.4(d)).

Variations of magnetic field M are illustrated in Fig. 3.4(e). It shows that axial

velocities decrease for magnetic field M. These results are consistent with

observations presented in Fig. 3.2(b). Notice that the velocities show an increasing

trend near the tube wall ( 1r ). It indicates that velocity gradients are increasing in

the boundary layer for higher magnetic field. This is consistent with experimental

observation of Malekzadeh [68]. Effects of amplitude ratio (shown in Fig. 3.4(f))

are very similar to that of magnetic field M (See Fig. 3.4(c)). Increasing gives

higher Lorentz force (as explained earlier). Hence the velocity gradients decrease in

interior region and it increases in the boundary layer in a similar way as for the

increasing magnetic field M .


34

Fig. 3. 4: Axial velocity profiles, as a function of radius vector, r (a) for curvature ratio, /a R

(b) for polar angle, 0 (c) for polar angle, 2 (d) for polar angle, 0, ,2 (e)

for magnetic parameter M (f) for amplitude ratio, .

0 0.5 1 1.50

0.5

1

1.5

2

0 0.5 1 1.50

0.5

1

1.5

2

0 0.5 1 1.50

0.5

1

1.5

2

r

M = 2

a=R = 0:1

,= 0:1

? = 0; 5:=4; 3:=2; 7:=4

r

r

w w

w

a=R = 0; 0:05; 0:1

(c)

(b)(a)

M = 2

? = :=4

,= 0:1

? = :=3; :=2; 3:=4

M = 2

a=R = 0:1

,= 0:1

0 0.5 1 1.50

0.5

1

1.5

2

0 0.5 1 1.50

0.5

1

1.5

2

0 0.5 1 1.50

0.5

1

1.5

2

r

r

w

M = 2

a=R = 0:1

,= 0:1

r

,= 0:1

? = :=3

a=R = 0:1

w

(d) (e)

(f)

? = 0; :; 2:

w

M = 2

? = :=4

a=R = 0:1

,= 0; 0:1; 0:2; 0:3

M = 1; 2; 3; 4


35

3.5 Concluding remarks

Key findings of presented analysis are as follows.

1. The results for straight tube [7] can be recovered for zero curvature and no

magnetic field. Moreover the results of study [65] are retained for zero

curvature.

2. Both pressure gradient and pressure rise per unit wavelength are increasing

functions of M .

3. Axial velocity is decreased by larger M .

4. It is noticed that pressure gradient /dp dz (and hence the pressure rise per unit

wavelength p ) are influenced by the tube curvature but these effects are not

embedded into the first order solution (as shown in Leal [67]). However the

curvature effects on axial velocity w exists even in first order solution.

Secondary flow resulting from centrifugal forces acting on fluid moving along

a curved path affects the flow velocities.

Analysis of Hartmann boundary layer in peristaltic….

36

Chapter 4

Analysis of Hartmann boundary layer in

peristaltic transport of Jeffrey fluid:

Quantitative and qualitative approaches


37

4.1 Introduction:

Peristalsis of Jeffrey fluid in axisymmetric tube is studied when the fluid is subjected

to strong magnetic field. Mathematical analysis is made under the assumption of

small Reynolds number and long wave length approximations. An alternate approach

of lubrication theory can also be applied for the order of the dimensions involved in

most of the peristalsis problems. Both the approaches lead to the same mathematical

expressions. The magnetic field is investigated in the boundary layer called the

Hartman boundary layer and its impact on the boundary layer thickness. In order to

explore the effects of magnetic field in the boundary layer, we need to seek

asymptotic analytical solution. We notice that this approach facilitates to unveil the

effects of strong magnetic field explicitly and determines the boundary layer thickness

mathematically. The phenomena of controlling the boundary layer thickness are of

central importance in fluid mechanics both theoretically and practically. For large

magnetic field, the perturbation analysis becomes singular in nature which is a

diversion from the usual regular perturbation analysis for peristaltic problems.

Boundary value problem is solved analytically using singular perturbation approach

together with higher order matching technique. Stream function, velocity, and

pressure rise are calculated. In addition to the analytical solution, we explore the

qualitative behavior of the flow using the theory of dynamical systems. First of all,

non-linear autonomous differential equation is established. Qualitative analysis of

solution has been carried out for magnetic parameter, amplitude ratio, flow rate and

the Jeffrey fluid parameter. The concomitant change of stability is given through

topological flow patterns. Equilibrium plots (bifurcation diagrams) give a complete

description of the various flow patterns developed for the complete range of a flow

parameters in contrast to most of the studies that describe the flow patterns at some

particular value of a parameter. The study helps to analyze the behavior of the fluid at

the critical points that represent steady solution. The mix of the analytical and

qualitative approach will help to broaden the scope and understanding of peristaltic

transport of fluid in channels, tubes and curved tubes for the present case. The whole

discussion has been carried out for the non- Newtonian Jeffrey fluid. Comprehensive

findings and discussion has been given in section 4.5.


38

4.2 Mathematical description

An incompressible and conducting Jeffrey liquid in circular tube of radius a is

considered. Sinusoidal waves with speed c travel along the walls of tube. Amplitude

and wavelength of such waves are denoted by b and . These waves are responsible

for flow induction. The equations are presented as

0 ,U W U

R Z R

(4.1)

1

,RR RZ

SPU W U RS S

t R Z R R R Z R

(4.2)

2

0

1,

RZ ZZ

PU W W RS S B W

t R Z Z R R Z

(4.3)

where , , , 0, , ,U R Z t W R Z tV . Wall shape is described by

2

, cos .h Z t a b Z ct

(4.4)

For Jeffrey liquid

2

1

,1

D

Dt

1 1S A A (4.5)

where represents dynamic viscosity, 1 the ratio of relaxation to retardation times

and 2 the retardation time. Also

,T

grad grad 1A V V (4.6)

,D

Dt t

1 1 1A A V A (4.7)

whereT in the superscript represents the transpose and R Z

e eR Z

(

Re and

Ze are unit vectors in R and Z directions).


39

The coordinates ,r z the velocities ,u w and pressure p in wave frame are

, ,

, , , , , , , , , ,

, , , , .

z Z ct r R

u r z t U R Z t w r z t W R Z t c

p r z t P R Z t

(4.8)

where , ,R Z t , ,U W and P show coordinates, velocities and pressure in lab

frame.

Using average tube radius a and the wavelength as the characteristic length scale in

the radial and axial directions respectively. We have dimensionless variables

22 2 2

, , , , , , .r z u w h a p ct

r z u w h p ta c c a c

(4.9)

We introduce the stream function ,r z as

1

, ,k

u wr z r r

(4.10)

where 2 a

k

is the dimensionless wave number. Incompressibility constraint is

automatically justified whereas other expressions give

3 21 1 1Re

,

rr rz

p kk rS k S

r z r r r z r z r r r z

kS

r

(4.11)

2

1 1 1 1Re

11 ,

rz

zz

pk rS

r z r r r z r z z r r

k S Mz r r

(4.12)

with,

2

1

2 1 1 11 ,

1rr

ckkS

a r z r r r z r r z

(4.13)


40

2

21

1

1 1 11 ,

1 1rz

r r rckS

a r z r r r zk

z r z

(4.14)

2

1

2 1 1 11 .

1zz

ckkS

a r z r r r z z r r

(4.15)

2

2

1

2 1 1 11 ,

1

ckkS

a r r r r r z r z

(4.16)

In Eqs. (4.11) and (4.12), Reca

denotes Reynolds number and

2 22 0B a

M

Hartmann number. Considering Re small and 0k , Eqs. (4.11) and (4.12)

are reduced to

0 ,p

r

(4.17)

2 3

2

3 2 2 3

1

1 1 1 1 11 .

1

pM

z r r r r r r r r

(4.18)

Eq. (4.17) implies that p p r i.e., pressure only depends on the axial coordinate z ,

so p p z . Differentiate Eq. (4.18) with respect to r , we get

2 3 3

2

13 2 2 3 3

1 1 1 11 0 ,M

r r r r r r r r r r

(4.19)

with boundary conditions

0

at 010

r

r r r

, and at ,11

F

r h

r r

(4.20)

where 1 cos , /h z b a , is the dimensionless equation of the tube wall and

2 F the dimensionless flux.


41

4.3 Asymptotic analytical solutions (Quantitative results)

In this section, we discuss analytical solution of Eqs. (4.19)-(4.20) which describes

the considered problem of peristaltic flow of Jeffrey liquid through strong magnetic

field M . Exact solution of Eqs. (4.19)-(4.20) is possible with Mathematica and is

calculated as

2 2

1 1 1

2

1 1 1

2

1

2

1 1 1

1, 1 (0, 1 ) 1

0, 1 1 2 1, 1,

2 (1, 1 )

0, 1 1 2 1, 1

r h BesselI hM r Bessel I hM hM F

BesselI hM h M hBesselI hM

h F rBesselI M r

BesselI hM h M hBesselI hM

(4.21)

where 10, 1BesselI hM and 11, 1BesselI Mh are modified Bessel

functions of order zero and one respectively.

Corresponding pressure gradient is

3 2

1 1

1 1 1

1 0, 1 (2 ).

( 1 (0, 1 ) 2 (1, 1 ))

M BesselI hM F hdp

dz h hM BesselI hM BesselI hM

(4.22)

The obtained expression for velocity is

1 1 1

1 1

2

1

1 1 1

2 1, 1 2 1 0, 1

2 1 0, 1

1 0, 1.

1 0, 1 2 1, 1

hBesselI hM FM BesselI hM

FM BesselI rM

M h BesselI rMw

h hM BesselI hM BesselI hM

(4.23)

In the recent past, a similar study performed by Nasir et al. [65] reported the same

exact solution. However, their expression for pressure gradient /dp dz has mistaken

due to some calculation error or a typo. Although, the exact solution of Eqs. (4.19)-

(4.20) is possible as given in (4.21) but the form of solution is not suitable to extract

any useful information regarding the flow behavior in the Hartmann boundary layer

for large magnetic field M . Hence we calculate the asymptotic solution for large M

in order to get valuable insight to the characteristic flow behavior in the Hartmann

boundary layer.


42

We apply the perturbation technique to get the outer solution away from the boundary

and inner solution near the boundary and which are later matched at the edge of the

boundary layer, hence producing a uniformly valid composite solution.

To look for an asymptotic solution, we assume the solution of Eqs. (4.19)-(4.20) in

the form

1/2

0 1 2

1

1, where , 0

1r

M

(4.24)

The leading order Eq. is:

2

0 0

2 2

1 10 ,

r r r r

(4.25)

whose solution is found to be

2

0 1 2 .c c r (4.26)

The first order Eq. reads

2

1 1

2 2

1 10 .

r r r r

(4.27)

The solution to the first order problem (4.27) is given as

2

1 3 4 .c c r (4.28)

Hence the two term asymptotic solution is given by (4.26) and (4.28) i.e.,

2 2

1 2 3 4 ,out r c c r c c r (4.29)

which must satisfy the boundary conditions,

1

0 , 0 at 0 .out

out rr r r

(4.30)

Thus the outer solution (4.29) satisfying the boundary conditions(4.30) becomes


43

2 2

2 4 .out r C r C r (4.31)

To discuss the solution near the boundary r h , we introduce the stretched variable

.p

h r

(4.32)

Eq. (4.19) takes the form

2 3

4 3 22 2 3 3

4

24 4

2

2 2

3 1 3 1 2 1

1 1 1 1 1

1 1 10 .

in in in

p p pp p p

in in

p pp p

in

pp

h h h

h h

h

(4.33)

Applying binomial expansion and by the principle of least degeneracy, we choose

1/ 2p . It reduces Eq. (4.33) to

4 2

4 20

in in

. (4.34)

Inner solution for the conditions

1/2

at 0 .

in

in

F

h

(4.35)

Writing

1/2

0 1 .in in in (4.36)

Leading order problem is

4 2

0 0

4 20 ,

in in

(4.37)


44

0

0at 0 .

0

in

in

F

(4.38)

Solution of Eqs. (4.37)-(4.38) is

0 3 41 1 .in F D e D e (4.39)

To avoid exponential growth, we choose 3 0D so

0 4 1 .in F D e (4.40)

The first order Eq. is

4 2

1 1

4 20 ,

in in

(4.41)

with boundary conditions

1 0

at 0.and

in

in

h

(4.42)

The solution is given by

1 7 81 1 .in h D e D e (4.43)

To avoid exponential growth, we choose 7 0D ,

1 8 1 .in h D e (4.44)

Therefore we can write the two terms inner solution at r h as

2 4 81 1 .in

terms F D e h D e

(4.45)

Clearly Prandtl‘s condition cannot be used for the higher order terms because of the

occurrence of the algebraic term which is unbounded as . Thus we apply

higher order matching technique. For this we introduce an intermediate variable in

the intermediate region O


45

, 0 1 .h r

Expanding the outer and the inner solutions in intermediate parameter and ignoring

the terms of order higher than O , one obtains

int

2 2 2 2 1/2 2 2 1/2 1/2

2 2 2 4 4 42 2 .out C h C C h C h C C h (4.46)

1/2 1/2int

1/2 1/2 1/2 1/2

4 81 1 .in F D e h D e

(4.47)

Matching Eqs. (4.46) and (4.47), we get

22

2 4 4 82 3

22, , 0, .

F hF h FC C D D

h h h

The composite solution can be written as

int

.out in in

composite (4.48)

Substituting values we obtain

2 22 2

2 3

2 2.

h r

comp

F h F h Fr r e

h h h

(4.49)

Since the velocities are computed from the stream function using the relations

1u

r z

and

1,w

r r

which give

2 2

4 2

32

1

2 2

2,

12

h r

h r

h F h F hh r e

Frh h h ru

h h Mh Fe

h r

(4.50)

2 2

2 3

2 22 1 2 1e .

h rh FF h Fw

h h h r

(4.51)

The solution (4.49) is uniformly valid in whole of the problem domain and satisfies

all the boundary conditions. The Hartmann layer is found on the surface of the tube


46

which is represented by the exponential function in Eq. (4.49) . The boundary layer is

of thickness

1

1

1O

M

which was clearly not possible to infer from the exact

solution.

Also the pressure gradient /dp dz is calculated in the form

2 4 2

1 13 3

1

2 4 4 3

1 1

3 3 3

1 1

3 2 3 2 3 3 3 3

1 1

[ 2 1 1 21

2 2

2 4 4

2 2 ].

h r

dp Mh F h h rMF

dz r h

h rMF h rM h rM e Mr Fh

Mr Fh Mr F Mr F

Mr h Mr h Mr h Mr h

(4.52)

Pressure rise per unit wavelength P is computed numerically as

2

0

.dp

P dzdz

(4.53)

The results obtained in this section are used to analyze the quantitative behavior of the

flow in terms of pressure gradients, velocity profiles and other dynamic quantities.

4.4 Dynamical systems (Qualitative solution)

In this section we use the ideas from the theory of dynamical systems to investigate

the effects of various parameters on the qualitative behavior of the flow. In the

previous studies [59-61] dynamical system has been used to characterize the flow

behavior as being augmented, backward and trapping (c.f. [59]). The flow behavior,

so obtained, primarily depends upon the choice of the parameter—the volume flow

rate F . The choice of different values of this parameter may correspond to different

behaviors by plotting the streamlines. We make use of the dynamical systems for a

slightly different but more practical point of view. In particular, we explore the

topological changes in the streamlines due to varying values of a particular flow

parameter through the equilibria curves/ bifurcation diagram. The aim is thus to work

out a particular value of the parameter that will result in a desired flow pattern. We

hope that these results will be useful in the field of medical sciences. For instance, one


47

may require knowing a combination of flow parameters (the volume flow rate F ,

magnetic fie ld M etc.) that will result in a circular flow pattern (usually referred to

as a bolus) which can be used for safe propagation of substance (a pill or an embryo).

We recall that the flow is steady in the moving frame of reference (Eq. (4.8)), the

streamlines coincide with the path lines, and hence the flow given by the stream

function Eq. (4.21) can be described by the non-linear autonomous system:

1 1 1

22

1 1

2

1 1 1

22 2

1 1

22 2

1 1

2

1 1

•

2 1 0, 1 1, 1

2 1 1, 1

2 1 0, 1 0, 1

1 0, 1

1 1, 1

2 1 1, 1 0, 1

Mh rBesselI Mh BesselI Mh

M r F BesselI Mh

FhM BesselI M r BesselI Mh

M rh BesselI Mhh

rh M BesselI Mh

Mh BesselI M r BesselI Mh

r

1

3 2

1 1 1

1 1

22

1 1 1

1 1

1 1, 1 1, 1

4 1, 1 1, 1

1 0, 1 2 0, 1

, , , , ,

h M BesselI M r BesselI Mh

h BesselI M r BesselI Mh

Mh BesselI Mh BesselI Mh

G z F M

(4.54)

1 1 1

2•

1 1 1 1

2

1 1 1

2 1

(2 (1, 1 ) 2 1 (0, 1 )

1 (0, 1 ) 2 1 (0, 1 ))

1 (0, 1 ) 2 (1, 1 )

, , , , .

h BesselI Mh M F BesselI Mh

M h BesselI M r F M BesselI M rz

M h BesselI Mh h BesselI Mh

G z F M

(4.55)

Where ,r z

represents the position vector, ,u w

the velocity vector i.e.,

1

,r u z rr z

and

1,z w z r

r r

. 1, , ,F M

are the volume flow rate,

magnetic field parameter, Jeffrey fluid parameter and the amplitude ratio respectively.


48

The axial domain is the real line ( i.e., z ) whereas in the radial direction

the domain is bounded by h z and h z . . .i e h z r h z

The equilibrium points (also referred to as the stagnation points or the critical points)

of the dynamical system (4.54)-(4.55) can be found using the solution of the nonlinear

system as follows:

1 1 1

22

1 1

2

1 1 1

22 2

1 1

22 2

1 1

2

1 1

2 1 0, 1 1, 1

2 1 1, 1

2 1 0, 1 0, 1

1 0, 1

1 1, 1

2 1 1, 1 0, 1

h hM rBesselI hM BesselI hM

M r F BesselI hM

FhM BesselI M r BesselI hM

M rh BesselI hM

rh M BesselI hM

Mh BesselI M r BesselI hM

1

3 2

1 1 1

1 1

22

1 1 1

1 1, 1 1, 1

4 1, 1 1, 10,

1 0, 1 2 0, 1

h M BesselI M r BesselI hM

h BesselI M r BesselI hM

Mh BesselI hM BesselI hM

(4.56)

1 1 1

2

1 1 1 1

2

1 1 1

2 (1, 1 ) 2 1 (0, 1 )

1 (0, 1 ) 2 1 (0, 1 )0.

1 (0, 1 ) 2 (1, 1 )

h Bes selI hM M F BesselI hM

M h BesselI M r F M BesselI M r

Mh BesselI hM h BesselI hM

(4.57)

In the previously reported studies [59-61], the analogous system (to the one given

above (4.56)-(4.57)) were easily solved analytically. On the contrary, the system

(4.56)-(4.57) is non-linear as well as non-algebraic which makes it nearly impossible

to be solved analytically. We, therefore, tend to workout solution of the system (4.56)

-(4.57) numerically leading to the bifurcation diagrams presented in the Figs 4.4-4.9.

The qualitative nature of the equilibrium points can then be investigated using the

Jacobian,

1 2

,

,

G ,G

,e e

e e

r z

r z

Jr z

at the equilibrium points ,e ez r , (Hartmann-

Grobman theorem).


49

Now we make a small digression to show that our numerical solutions of finding the

critical points and the bifurcation behavior are reliable. For that, we choose to make

the comparison with the asymptotic expressions of velocity components obtained for

large magnetic parameter Eqs. (4.50)-(4.51). To make the situation further simplified

we expand the exponential terms in Eqs. (4.50)-(4.51) up to ( )O h r . The resulting

system takes the form:

2 2

14 2

3 21 1

1

2 21 1

2 10 ,

1 121 1

h F h F hh r h r M

Frh h h r

h M h Mh Fh r M

h r

(4.58)

2 2

1 12 3

1

2 22 1 2 11 1 1 0 .

1

h FF h FM h r M

h h h rM

(4.59)

Eqs. (4.58) and (4.59) can now be solved analytically and the resulting equilibrium

points are found as:

and

2 2

1

2

2 2

2

11

12

1, 0,

2 222

11

h h h FM

r zh FFh

h h FMM

(4.60)

2 2

1

2

2 2

2

11

12

1, .

2 222

11

h h h FM

r z nh FFh

h h FMM

(4.61)

It may be remembered that these critical points have been obtained from asymptotic

results for large magnetic parameter and working up to O h r . Detailed analyses of

the results of the sections 4.3 and 4.4 will be presented in the sections 4.5.1 and 4.5.2

respectively.


50

4.5 Analysis

This section is divided into two subsections to focus on quantitative (analytical

results) and qualitative results (Dynamical system) separately. Using the solutions

obtained in sect(s) 4.3, we will discuss the quantitative flow behavior in terms of

pressure gradients, velocity profiles and other quantities of interest. The effects of

various flow parameters on these dynamical quantities and their physical significance

will be discussed in detail in sect(s). 4.5.1. Furthermore, using the information from

dynamical systems developed in sect(s). 4.4, we will discuss the qualitative behavior

of the flow i.e., identify the characteristic topological flow patterns developed under

various flow conditions. In sect(s). 4.5.2, we also discuss how these developed flow

patterns change when a particular flow parameter is varied.

4.5.1 Analytical results

The quantities of physical interest in peristalsis are pressure rise per unit wavelength,

pressure gradient, the flow velocities and the boluses. Backward flow and trapping are

two interesting consequences of the peristaltic flow as observed by Shapiro [7]. Our

intension here is to analyze pressure rise per unit wavelength, pressure gradient, and

the flow velocities for outcome of sundry variables. Fig. 4.1 shows the effects of the

Jeffery fluid parameter 1 , Hartmann number M , and amplitude ratio , on the

pressure rise per unit wave length as defined in Eq. (4.53). A linear relationship is

observed between the pressure rise variation per unit wave length P and flow rate

F . Clearly, pressure rise enhances (in an interval of F ) as the value of the magnetic

field M , the amplitude ratio and the Jeffery fluid parameter 1 are increased. In

peristalsis, the retrograde pumping 0, 0P F occurs in the second quadrant,

whereas the fourth quadrant 0, 0P F represents the augmented pumping.

Notice that there is some critical value of flux F (which in turn depends on the

parameters M , 1 and ) below which the retrograde pumping occurs. This critical

value of F defines the regime of retrograde pumping. The quantitative record of these

critical flux values for different values of M , , and 1 are given in Table 4.1. It is

seen from numerical values given in Table 4.1 that the region of retrograde pumping

decreases with increasing M and 1 whereas for , the region of retrograde

pumping increases.


51

Fig.4. 1: Effects of (a) magnetic field parameter M , (b) amplitude ratio and (c) the Jeffery

fluid parameter 1 , on the pressure rise per unit wavelength P as defined in Eq. (4.53)

Table 4. 1: Intervals for flow rate F for different values of M

M Interval for F when 0P Interval for F when 0P

5 1 0.4541F

0.4541 1F

10 1 0.4627F

0.4627 1F

15 1 0.4685F

0.4685 1F

-1 -0.5 0 0.5 1-6000

-4000

-2000

0

2000

-1 -0.5 0 0.5 1-2000

-1000

0

1000

-1 -0.5 0 0.5 1-1000

-500

0

500

r = 0:2

(a) (b)

"P6

(c)F

F

M = 5; 10;

15

61 = 0:2; 0:4;

0:6

F

M = 5

61 = 0:2

"P6 "P6

,= 0:2

61 = 0:2

M = 5

,= 0:2

,= 0:2; 0:4;

0:6


52

Table 4. 2: Intervals for flow rate F for different values of

Interval for F when 0P Interval for F when 0P

0.2 1 0.4721F

0.4721 1F

0.4 1 0.3548F

0.3548 1F

0.6 1 0.204F

0.204 1F

Table 4. 3: Intervals for flow rate F for different values of 1

1 Interval for F when 0P

Interval for F when 0P

0.0 1 0.4593F

0.4593 1F

0.2 1 0.4623F

0.4623 1F

0.4 1 0.4654F

0.4654 1F

0.6 1 0.4699F

0.4699 1F

Variations of /dp dz with different physical parameters as defined in Eq. (4.22) are

shown in Fig. 4.2. The values of pressure gradient /dp dz are small around 0z and

2z (the axial location corresponding to wave trough), whereas /dp dz is

relatively large near z (corresponding to wave crest). This is exactly what is

expected physically as the pressure gradient /dp dz is inversely proportional to the

tube diameter (see Eqs. (4.18) and (4.52)), hence the values of /dp dz are small near

wave trough where the tube diameter is least and the values of /dp dz are greatest

near the wave crest where the tube diameter is largest. For larger M values of

pressure gradient /dp dz increases owing to the increased Lorentz force. The effect

of the amplitude ratio (shown in Fig. 4.2(b)) is different to that M in the sense that

the values of /dp dz decreases with M at all axial location whereas for , there exists


53

region in which /dp dz increases with increasing . As explained above (in the

discussion of Fig. 4.2(a)) /dp dz is inversely proportional to the tube diameter. Notice

that with increasing , the tube diameter decreases at the wave trough (near 0,2z

), hence the pressure gradient /dp dz decreases locally, whereas at the wave crest

(near z ), the tube diameter increases resulting in lower pressure gradient locally.

The effect of the Jeffery fluid variable 1 (see Fig. 4.2(c), with increasing values of 1

, the flow velocities decrease, hence the magnitude of the Lorentz force reduces

resulting in large pressure gradients /dp dz .

Fig.4. 2: Variation of the (a) magnetic field M , (b) amplitude ratio and (c) the Jeffery fluid

variable 1 , on the pressure gradient /dp dz as defined in Eq. (4.22).

0 2 4 60

200

400

600

0 2 4 60

100

200

300

0 2 4 640

50

60

70

z

z z

(c)

(a) (b)

M = 5; 10; 15

F = 0:2

r = 0:2

61 = 0:2; 0:4;

0:6

M = 5

61 = 0:2

! dp

dz

! dp

dz

! dp

dz

,= 0:2

61 = 0:2

,= 0:2; 0:4;

0:6

M = 5

,= 0:2


54

Fig.4. 3: Variations of (a) magnetic field parameter M , (b) amplitude ratio and (c) the

Jeffery fluid parameter 1 , on the axial velocity w in Eq. (4.23).

Fig. 4.3 shows the influence of M (Fig. 4.3(a)), the amplitude ratio (Fig. 4.3(b))

and the Jeffery fluid parameter 1 (Fig. 4.3(c)) on the flow dynamics analyzed

through the axial velocities w , plotted as a function of r (at the axial location 0.2z

). For larger M the velocities in the interior domain decreases. This is because of

Lorentz force which opposes the fluid motion. Velocity gradients in boundary layer

increase because boundary layer thickness region decreases as M increases. Fig.

4.3(b) indicate wave amplitude on w . For larger the tube diameter increases

(around 0z ), hence the flow decelerates. The effect of the Jeffrey fluid parameter 1

(also referred to as the retardation time) on the flow dynamics is shown in Fig. 4.3(c).

When the ratio of relaxation time to retardation time goes to zero then Jeffery fluid

behaves like Newtonian fluid. However, it is interesting to note that the retardation

time does not contribute mathematically in the ultimate analysis of peristaltic

transport. This is because the deformations and rate of deformations are small in

peristaltic transport of fluids. This in mind, we have preferred linear visco-elastic fluid

0 0.5 1 1.5 2-1

-0.5

0

0.5

1

0 0.5 1-1

-0.5

0

0.5

1

0 0.5 1

-0.5

0

0.5

1

0.9 1 1.1

r

rr

M = 5; 10; 15

(a) (b)

(c)

F = 0:2

z = 0

61 = 0:2; 0:4;

0:6

M = 5

61 = 0:2

w w

w

,= 0:2

61 = 0:2

,= 0:2; 0:4; 0:6

M = 5

,= 0:2


55

which is the appropriate model for small deformations. Now when the relaxation time

increases (Fig. 4.3(c)) the fluid tends to be more elastic in nature, and so more time

for the stress to settle down. Stress in fluid is higher for higher values of relaxation

time and consequently there is more resistance to the flow causing a decrease of

velocity profile for increasing relaxation time. Notice that the velocities in the

boundary layer increases in contrast to velocities in the interior domain which show

decreasing pattern with increased values of 1 . This contrasting behavior is forced by

the mass conservation principle which requires increased flux in the boundary when

the flux decreases in the interior region.

4.5.2 Dynamical system (Qualitative results)

This section focuses on the equilibria curve or the bifurcation diagram with respect to

various characterizing parameters. As said earlier, the equilibrium points are

calculated numerically. Now we show the locus of various equilibria/bifurcation

diagrams as a function of certain flow parameters while keeping the other parameters

fixed. Figs. (4.4 –4.9) are the bifurcation diagrams for various values of characterizing

parameters which are traced numerically. Based on the nature of the eigenvalues,

Bakker [64], the equilibrium points are classified as being ―center‖ or ―saddle‖,

represented by a thick line or thin line respectively. The solid lines correspond to the

numerical results and dashed lines correspond to analytical results.

In Fig. 4.4, we have plotted the equilibrium points (bifurcation diagram) showing the

behavior of the solution on the bifurcation curve for varying M from 0 to 200 (with a

difference of 310 ) and keeping other flow parameters ( 10.2, 0.2, 0.15F )

fixed. The curves (in red) corresponding to 0z (wave crest) and r close to h shows

that centers will be formed at this domain location i.e., 0z and radially close to the

boundary wall ( r h ). Another equilibria curve is obtained corresponding to z n

(shown in black). Radially these equilibria lie close to the tube wall i.e., r h (see

Fig. 4.4). Eigen values of these equilibria show that these are saddle in nature.


56

Fig.4. 4: The equilibrium plot (i.e., location of the equilibrium point) via magnetic parameter M

, when 0.2 and 0.15F . The equilibrium curves red and black correspond to 0z and

z n respectively. Thick line shows that the equilibrium point is a center whereas the dashed

lines correspond to the analytical results (given in Eqs. (4.60)-(4.61)).

The analytical results of dynamical system (4.58)-(4.59) are shown by dashed lines in

Fig. 4.4. Analytically, we were able to calculate two equilibrium points,

corresponding to 0z and z n . It is encouraging to see that analytical results

match with numerical results. The analytical results are only valid for large M and

h r i.e., in the region located radially close to the top wall r h . Therefore

the equilibria curves obtained analytically (dashed lines) coincide with the equilbria

curves traced numerically (solid lines) only in the region for which h r is small

which can be seen by the logarithmic scale of h r at 0z and z . Notice that

this agreement between the numerical and analytical results is not only quantitative

(i.e., in terms of the location of the equilibrium point) but also qualitative (i.e., the

nature of the equilibrium point, centers for 0z and saddles for z n ).

0 50 100 150 200

0.8

1

1.2

1.4

10-2

10-1

10-2

10-1

(h ! r)jz=:

(h ! r)jz=0

M

r

61 = 0:2, , = 0:2,F = 0:15


57

Fig.4. 5: The contour plot of the stream function, given in Eq. (4.21). The flow parameters used

for this result are 5, 0.15, 0.2M F .

To further support the observation of the numerical results, we plot the streamlines in

Fig. 4.5 to investigate the various topological patterns present in the flow. Fig. 4.5

predicts that a center and a saddle will be formed at ,r z gives 1.1706433,0 and

0.6724,n respectively. Interestingly the contour plot given in Fig. 4.5 indicates

that various flow patterns as predicted by Fig. 4.4 are indeed present at the predicted

locations. Furthermore, all the topological flow patterns of particular interest i.e.,

saddles and centers that are formed are accurately recorded in the equilibria curve

plot. Here an important physical observation is very much evident; the contour plot

may not necessarily reveal all of the flow patterns as we only plot a certain number of

level curves in a contour plot. Hence, to analyze various flow patterns present in the

flow one should obtain equilibria curves (as given in Fig. 4.4) using the dynamical

system analysis and not by an arbitrary pick of parameter (as is generally presented in

the literature) that only explains the behavior at the isolated value of the parameter. In

0.0216280.021628

0.043257

0.0432570.064885

0.0648850.086513

0.086513

0.10814

0.10814

0.12977

0.12977

0.12977

0.15

14

0.1514

0.1514

0.1514

0.1514

0.1514

0.17

303

0.17303

0.17303

0.17303

0.17303

0.17303

0.19465

0.19465

0.194650.19465

0.21628

0.2

16

28

0.21628

-3 -2 -1 0 1 2 30

0.2

0.4

0.6

0.8

1

z

r


58

the subsequent discussion, we will refer to such Fig(s). containing equilibria curves,

as the ‗equilibria plot‘.

In Fig. 4.6, we give an equilibria plot for the volume flux F while keeping other flow

parameters fixed 10.2, 5, 0.2M . The equilibrium curves obtained

numerically show that for 0.1969F , there are no equilibrium points in the flow.

Corresponding to 0z , centers are formed which descend towards the radial axis (

0r ) as the flux parameter F is decreased. The equilibrium point corresponding to

z n , form a saddle and their radial location also show a decreasing trend with

decreasing F as observed for the equilibrium point corresponding to 0z . The last

equilibrium curve, shown in green, is such that its qualitative nature is saddle and its

radial location is fixed ( 0r ), whereas the axial location changes with the flux

parameter F , in a sinusoidal pattern as shown in Fig. 4.6(b). In particular,

for 0.15F centers and saddles are formed at, 1.0189,0 , 0.68935,n and in

agreement with the results given in contour plot Fig. 4.5.

The observations reported above can be verified from the contour plots of the stream

function given in Fig. 4.7. It can be seen in Fig. 4.6(a) that a center and a saddle is

formed at , 1.0343,0 and 0.70301,r z n corresponding to 0.2F , agreeing

with the contour plot given in Fig. 4.7. Furthermore, as the flux decreases (Fig.

4.7(b)), the equilibrium points move radially downwards consistent with the

observations made in Fig. 4.6. At 0.192F the saddle point has reached the center

of the tube 0r and its radial location remains constant for 0.1969 0.2F ,

again in agreement with the observations made in the contour plot (Fig. 4.7). These

equilibria points vanish for 0.1969F (Fig. 4.6(b)).


59

Fig.4. 6: The equilibria plot for the volume flux parameter F . The lines red and blue correspond

to the equilibria point with 0z and z n respectively. For the equilibria curve

corresponding to 0r (shown in Fig. 4.6(b)) the axial coordinate z varies with the flux

parameter F , hence z -variation is shown in the adjacent Fig(s). Thick lines represent centers

and thin lines are saddles.

Fig.4. 7: The contour plot for the stream function , given in Eq. (4.20), for

1 0.2, 0.2, 5M and (a) 0.2F (b) 0F (c) 0.192F and (d) 0.2F

The equilibria plot of the occlusion parameter , while keeping others fixed

10.15, 5, 0.2F M , is given in Fig. 4.8. It shows that there are two equilibria

-0.2 -0.18 -0.16 -0.14 -0.12-4

-2

0

2

4

-0.5 0 0.5 10

0.5

1

1.5

r

F

z

F

61 = 0:2, , = 0:2, M = 5

(b)(a)

-4 -2 0 2 40

0.5

1

-4 -2 0 2 40

0.5

1

-4 -2 0 2 40

0.5

1

-4 -2 0 2 40

0.5

1

(a)

(c)

z

z z

(d)z

F = 0:2F = 0

(b)

r r

F = !0:2F = !0:192

r r


60

curves corresponding to 0z (centers) and z n (saddles) forming near the tube

wall. As the value of is increased, the tube diameter increases at 0z (wave crest)

and decreases z n (wave trough), hence the radial location of the equilibrium

point corresponding to 0z increases with where the radial location of the

equilibrium point corresponding to z n decreases with increasing . At 0.2 ,

centers and saddles are formed at 0,1.0189 and ,0.6893n respectively, which is

once again, in agreement with the contour plot given Fig. 4.5.

Fig.4. 8: The equilibria plot for the occlusion parameter . The lines red and black correspond

to the equilibria point with 0z and z n respectively. The thick lines represent centers

and thin lines are saddles.

Fig. 4.9 gives the equilibria plots of the Jeffrey fluid parameter

1 for 5, 0.2 and 0.15M F . There are two equilibrium points

corresponding to 0z (center) and z n (saddle) both forming near the tube wall

h . With increasing 1 , the velocities in the boundary layer increases (as shown in Fig.

4.3(c)), therefore, the radial location of both of the equilibrium points show an

increasing trend with 1 .

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

61 = 0:2, M = 5F = 0:15

r

,


61

Fig.4. 9: The equilibria plot for the Jeffrey fluid parameter 1 . The lines red and black

correspond to the equilibria point with 0z and z n respectively. The thick lines represent

centers and thin lines are saddles.


Substantial outcomes of the above study are:

1. The analytic solution of problem is obtained using singular perturbation and

higher order matching technique for large magnetic field to explore the effects

of magnetic field in the boundary layer called Hartman boundary layer. The

boundary layer is found to be of thickness 1

1

1O

M

. The boundary layer

thickness decreases with increasing M and 1 .

2. Fluid velocity decreases when magnetic field is increased because of the flow

opposing nature of Lorentz force.

3. Equilibria curves/bifurcation curves are plotted using dynamical system

approach for all parameters involved. It is found that that nature of

0 0.2 0.4 0.6 0.8 1

0.8

1

1.2

1.4

r

61

M = 5, , = 0:2F = 0:15


62

equilibrium points is either center or saddle. These curves are plotted for

all continuous values of parameter rather than at some discrete points.

This gives us a complete picture of the fluid flow and the values of the

characterizing parameters for which we will have the saddle points or

centers indicating boluses.

An exact solution for MHD boundary layer flow of……

63

Chapter 5

An exact solution for MHD boundary layer

flow of dusty fluid over a stretching surface


64

5.1 Introduction:

Exact solution for steady stretched flow of dusty liquid is developed. The problem is

now fully specified in terms of characterizing parameters known as fluid particle

interaction parameter, magnetic field parameter and mass concentration of dust

particles. The solution works for all values of the characterizing parameters. Velocity

and skin friction are coefficient are examined by plots and tabulated values. We

emphasize that an approximate numerical solution of this problem was available in

the literature but no analytical solution was presented before this study. Comparison

between these solutions is in an excellent agreement.

5.2 Formulation

Here steady flow of conducting dusty fluid by a moving surface is addressed. The

surface is stretching with the velocity wu x sx , where the positive constant s

shows stretching rate. Cartesian coordinate system is selected such that x and y axes

are taken along (and normal to) the surface, respectively. Origin of the system is

located at the leading edge. The shape of dust particles is assumed to be spherical with

uniform size and constant number density. The flow of dusty fluid together with the

boundary conditions is given as (c.f. [69]):

0,u v

x y

(5.1)

2

2

2

0 ,p

u u uu v u u

x y y

Bu

(5.2)

0,p p

u v

x y

(5.3)

1

,p p

p p p

u uu v u u

x y

(5.4)

0; ( ), 0,

; 0, 0, .

w

p p

y u u x v

y u u v v

(5.5)

where , pu u and , pv v denotes x and y components of fluid and dust particles

velocity, respectively. Parameter /mN denotes the mass concentration of dust


65

particles, / Km indicates relaxation time of particle phase and , ,v and 0B

represent the density, kinematic viscosity, electrical conductivity of fluid and induced

magnetic field, respectively. Terms appearing in and are the Stokes resistance

K , the number density N and the mass of the dust particle m .

We define the following transformations [69]:

, ( ), ( ),

( ), ( ) ,p p

sy u sxf v s f

u sxg v s g

(5.6)

in which prime () represents differentiation with respect to . Introducing

transformations (5.6) in the equations (5.1)-(5.5), we get

2 0,f ff f g f Mf (5.7)

2 0,gg g f g (5.8)

(0) 0, (0) 1, ( ) 0.

( ) 0, ( ) ( ) as .

f f f

g g f

(5.9)

Here and M are the fluid particle interaction parameter and magnetic parameter,

respectively. These are defined by

2

01, .

BKM

s sm s

(5.10)

Coefficient of skin friction fC along with shear stress w are

2

0

, ( ) .wf w

w y

uC

u y

(5.11)

Using the transformation(5.6), the dimensionless skin friction coefficient is defined by

1/2 0x fRe C f , where 2 /xRe sx is the local Reynolds number.

In the next section, we will present the exact analytical solution of these equations.


66

5.3 Exact Analytical solution

Based upon the solution given by Crane[70], we propose that the nonlinear coupled

equations (5.7)-(5.9) admit exponentially decaying solution of the form

3 6

1 2 4 5, ,f e g e

(5.12)

where , for 1, ,6i s i are arbitrary constants. The values of these constants can

be determined by putting the solution (5.12) in the equations (5.7)-(5.9). This yield

1 2 4 3 6 51/ , and ,(1 )

(5.13)

where the term is given as

1 .1

M

(5.14)

Substituting the values of , for 1, ,6i s i from Eq. (5.13) into Eq. (5.12), we get

closed form exact solution:

1 1

1 , 1 ,1

f e g e

(5.15)

It is worth noting that the solution given in (5.15) works for all values of , and

M . The physical quantities like the velocities of fluid and the dust particles and the

skin friction coefficient are now expressed as:

, , 0 1 .1 1

f e g e f M

(5.16)

Numerical solution of Eqs. (5.7)-(5.9) exists in the literature, but the exact analytical

closed from solution is presented here for the first time. Now using these results one

can easily interpret the effects of dusty fluid parameters on the physical quantities

without using any numerical scheme. A comparative study of present analysis with

numerical results by Gireesha et al. [71] shows excellent agreement. Note that in [71]

numerical results are due to Ruge-Kutta (RKF45) method.


67

Table 5. 1:Comparison of skin friction coefficient for via and M with Gireesha et al. [71]

M Gireesha et al.

[71]

=0

Exact

solution

Gireesha et al.

[71]

=0.5

Exact

Solution

0.2 1.000 1.000000 1.034 1.033505

0.2 1.095 1.095445 1.126 1.126114

0.5 1.224 1.224745 1.252 1.252251

1.0 1.414 1.414214 1.438 1.438101

1.2 1.483 1.483240 1.506 1.506032

1.5 1.581 1.581139 1.602 1.602540

2.0 1.732 1.732051 1.751 1.751609

We also present the graphs for velocity profiles (Fig. 5.1(a) and 5.1(b)) for different

values of parameters andM . These results are compared with [69]. This

comparison will further validate the authenticity of the two solutions. From Fig. 5.1

we note that both fluid velocity and dust particles velocity decay with increasing M .

Larger magnetic field gives rise to more Lorentz force. Such force is responsible for

reduction in velocity. Effects of fluid particle interaction , mass concentration of

dust particles and M on skin friction coefficient are given in Table 5.2. Clearly

skin friction for , and M is increased. It is due to the fact that more resistance is

noticed to this quantity through larger , ,and M .


68

Fig.5. 1 Effects of (a) M and (b) on the velocity profiles

Table 5. 2: Skin friction coefficient against , ,and M

M -f(0)

0.2 1.437591

0.5 0.5 1.0 1.471960

1.0 1.527525

0.2 1.420094

0.1 0.5 1.0 1.425950

1.0 1.431782

2.0 1.741647

0.1 0.5 5.0 2.456284

10.0 3.321646

5.4 Dynamical system formalism

In this section, we construct the non-linear autonomous system of differential

equations to study the qualitative behavior of the fluid flow. The prime objective will

be to find the critical points and their behavior thereof. The phase plane diagram will

present the complete picture regarding the flow field. The fluid behavior, as we know,

is very sensitive to the flow parameters. The behavior can drastically change over the

values of these parameters such as magnetic field M and fluid particle interaction

in the present case. This change of behavior with these parameters can be

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

f0 (5),

g0 (5)

f0 (5),

g0 (5)

(b)(a)

- = 0:2; 0:5; 1:0M = 2:0; 5:0;10

. = 0:1

M = 1:0

. = 0:1

- = 0:5

f 0(5)g0(5)

f 0(5)g0(5)


69

characterized through the bifurcation curves which will also be represented in the

following pages. In order to write the dynamical system regarding our problem; we

utilize the expression of velocity field given in then Eq. (5.6). Therefore, we can

write

1 3 , , , , ,

sy

x sxe G x y M

(5.17)

1 4

11 , , , , ,

sy

y s e G x y M

(5.18)

here 1 /x dx dt u

and 1 /y dy dt v

. 0 and 0x y .

The equilibrium points or the critical points of the dynamical system (5.17)-(5.18) are

the solution of the non-linear equations given below:

0,

sy

sxe

(5.19)

1

1 0.

sy

s e

(5.20)

Solving (5.19) and (5.20) simultaneously, we get the critical point 1 1, 0,0x y .

Jacobian can be used to investigate the qualitative nature of the equilibrium point i.e.,

1 1

1 1

3 3

,4 4

,

0,0

0.

0

0

x y

x y

s sy y

sy

G G

x yJ

G G

x y

sce c xe s

s

se

(5.21)

The characteristic equation corresponding to Jacobian matrix is

det 0.J (5.22)

The corresponding eigenvalues are s and s . Since s is positive stretching rate so

product of eigenvalues will always be negative. Hence the critical point 0,0 is


70

saddle in nature. In the Fig. 5.2(a) we have drawn equilibrium curve for magnetic

parameter M . We can see that for all values of M the bifurcation diagram is a

horizontal line along 0y . There is no change in the qualitative behavior of equilibria

for all M . This shows that the magnetic field does not change the topological

behavior of the flow field as is expected physically anticipated by the stretching of the

sheet requires with (0,0) as the center of the stretching having velocity there. Fig.

5.2(b) shows equilibria curve (bifurcation diagram) for fluid particle interaction

parameter . This graph shows that nature of equilibrium points remains same for all

variations of .

Fig.5. 2: Bifurcation diagram (a) for M (b) for

We can now repeat the whole analysis for the particle phase. The corresponding

autonomous system for particle phase is given by

2 5 , , , , ,1

sy

px u sxg sx e G x y M

(5.23)

2 6

11 , , , , .

1

sy

py v s e G x y M

(5.24)

The equilibrium points or the critical points of the dynamical system (5.23)-(5.24)

can be found using the exact solution of the non-linear equations given below:

0,1

sy

sxe

(5.25)

M

y

-

y

(a) (b)

- = 0:2, . = 0:2 M = 0:2, . = 0:2

0 0.5 1 1.5-1

-0.5

0

0.5

1

0 0.5 1 1.5-1

-0.5

0

0.5

1


71

1

1 0.1

sy

s e

(5.26)

Solving (5.25)-(5.26) simultaneously we get critical point

2 2

1 10, ln .x y

s

(5.27)

We formulate Jacobian to find the nature of critical point

2 2

5 5

6 6

,

0.

0

x y

G G

sx yJ

G G s

x y

(5.28)

The corresponding eigenvalues are ands s . So the critical point 2 2,x y is saddle

for all values of 0s . As we can see from Eq. (5.27) that dust particles velocities are

zero at 2

1 1lny

s

which is always negative. Thus, there will be no

stagnation point in the region of interest for the dusty particles. If we draw the phase

plane diagram for the dust particles; we clearly observe from Fig. 5.3 that the

horizontal velocity is always zero for sufficiently large y whereas the vertical

velocity is quite apparent (see the boundary condition(5.5)). However when we go

close to 0y and enter into the boundary layer region the fluid particle interaction

takes place and the velocity of the particles is modified; naturally depending upon the

interaction parameter . For higher values of it is influenced more strongly.


72

Fig.5. 3 Phase plane diagram showing dust particle velocity


An exact analytical solution for the flow of dusty fluid over a linearly stretching

surface is presented. This result is uniformly valid all for values of physical

characterizing parameters. Expressions of velocity and skin-friction are calculated

purely analytically. The results are compared with the existing literature [69], which

are obtained numerically giving very good agreement. This work provides a base for

further research in undertaking analytical solutions of two phase dusty fluid flow over

a stretching surface. Another important way, besides the analytical and numerical

solution is to look at the behavior of the solution without bothering about exact or

approximate solution. In relatively simpler situations of linear and non-linear

equations one may be successful to provide some plausible solutions but in highly

non-linear complex situations these methods fail to provide satisfactory solutions.

Dynamical system theory has been employed and behavior of dust fluid particles over

stretching surface is analyzed with an objective of introducing the theory for two

phase fluids. It is seen that the critical point obtained here by defining the autonomous

system is not changing its behavior. It remains saddle for all values of characterizing

parameters.

Conclusions

73

Chapter 6

Conclusions

Conclusions

74

This research work primarily deals with MHD flow in different settings. In the

process, significant progress is made in the existing literature from different

perspectives. The peristaltic flow was extensively discussed, in the literature, for

planar channels, curved channels and tubes. However, the most natural and realistic

situation of curved tube was unattended. We present a three dimensional curved

peristaltic incompressible viscous flow under the effect of magnetic field. The effects

of the small curvature are prominent since the centrifugal force plays an important

role for the curved path of the fluid. The effects of magnetic field on the curved tube

are examined at length. In order to make comparison, validity and correctness of the

analysis; we observe that Shapiro [7] and T. Hayat [65] are special cases of this study.

Reference [7] can be recovered when the tube is straight, i.e., the curvature is zero and

no magnetic field is applied; whereas, the results of [65] can be recovered by taking

curvature of the curved tube as zero. Toroidal coordinate system is used to model the

problem. Well established approximations of long wave length and small Reynolds

number are used for the way forward. It is remarked that in the first order solution of

pressure gradient /dp dz and pressure rise per unit wavelength P curvature effects

do not contribute but are present in the first order expression of axial velocity (as

shown in Leal [67]). The secondary flow, resulting from the centrifugal forces acting

on the fluid moving along a curved path, affects the flow velocities. It is noticed that

the axial velocities on the inner side of the wall ( 0 ) are small compared to

flow velocities in the straight tube whereas on the outer side of the wall ( 2 ),

the flow velocities are somewhat larger than the velocities in the straight tube. The

effects of the secondary flow disappears for 0, , and 2 , hence the fluid

velocities are unaffected and are the same as in the straight tube.

The magnetic field parameter M has an increasing effect on the pressure gradient

/dp dz and the pressure rise per unit wavelength P with a relatively stronger

influence in the region of wave contractions where the tube diameter is minimized.

The amplitude ratio affects the flow in a similar way as that of the magnetic field

parameter M . Furthermore it is noted that the axial velocities w in the interior region

of the tube, decreases due to the increase in the magnetic field M , whereas the

velocity gradients show an increasing trend with the increasing magnetic field

Conclusions

75

consistent with other reported analytical (Hayat et al. [65]) and experimental

observations (Malekzadeh et al. [68]).

Our next study concerns with MHD peristaltic transport of Jeffrey fluid that is

subjected to strong magnetic field. In most of the MHD problems the effects of

applied magnetic field are seen in the interior of flow as a whole. However, its effect

in the boundary layer is not considered in the peristaltic channel flows. Here, we

examine the Hartman boundary layer to see the effects of magnetic field in the

boundary layer and the boundary layer thickness. The exact solution of the governing

mathematical model can be computed using Mathematica. However, it is hard to find

useful information regarding the flow characteristics in the boundary layer for large

magnetic field M . The success lies in finding asymptotic analytical solution of

the problem that explains the presence of boundary layer and its thickness.

Mathematically, singular perturbation method with higher order matching

technique is used to achieve this end. The boundary layer thickness is found to be

of

1

1

1O

M

, which decreases with increasing magnetic M and visco-elastic

1 parameters. The choice of the fluid is taken as non-Newtonian Jeffrey fluid. The

parabolic nature of velocity profile decreases with increasing M which is a known

characteristic of the boundary or pressure driven flow with magnetic field. Hence, we

conclude that strong magnetic field and the Jeffrey parameter controls the boundary

layer thickness and confines the effects of non-Newtonian properties of fluid close to

the walls (in the boundary layer) of the tube allowing better and convenient passage to

the fluids away from the boundary layer.

For qualitative behavior of the solution; theory of dynamical system is employed.

Using analytical expression of stream function, a dynamical system is formed by

writing axial and radial velocities through the stream function, and writing these

velocity components as a time derivative of axial coordinate z and radial coordinate

r respectively. The qualitative behavior of the flow can then be characterized by

analyzing the eigenvalues of the Jacobian matrix [64]. The equilibrium solutions

found are either saddle or center in nature. In particular, the equilibrium solutions that

form a center produces a region where the streamlines are closed, hence a circulatory

eddy is formed centered at the equilibrium point. Such circulatory patterns usually

Conclusions

76

referred to as a ―bolus‖ in the peristaltic flows, are of great significance as these can

be used in transportation of pills or other medical substances. The equilibria diagrams

presented can be helpful in choosing the values of flow parameters that will produce a

bolus at a desired location in the flow domain. In contrast to previously reported

studies, where the qualitative behavior of the flow was given for a fixed value of the

parameter, here we give equilibria curves which characterize the flow behavior for all

values of that parameter.

In particular, the equilibria curve for magnetic field parameter M given in Fig. 4.4

shows that the bolus forming at 0z and radially near the tube wall does not dis-

appear even at large values of the parameter 200M . This observation is of

significant value as in most of the previously reported studies [72-73], it is believed

that bolus disappears with increasing M .

The equilibria curve obtained for various flow parameters show that the qualitative

flow behavior does not change with the parameter i.e., if the flow near the equilibrium

point is elliptic (center) or hyperbolic (saddle), it remains so for all values of the

parameter (see Figs. 4.4-4.7). However the location of the equilibrium point varies

when the parameter value is changed owing to changes in quantitative flow behavior

i.e., velocities and pressure gradients. This change in the location of equilibrium point

is negligible for the Jeffrey fluid parameter (shown in Fig. 4.9) consistent with the

observation made in sect. 4.5.1 i.e., the dynamic flow profile (velocity, pressure

gradients) show little dependence on the Jeffrey fluid parameter.

Our next finding is to find an exact analytical solution for a two phase stretching

problem under the applied magnetic field. Exact analytic solution and mathematical

expressions for fluid velocity and skin friction are obtained. A comparative study with

existing numerical solution is made. The results are in good agreement with the

existing study. To explore the effects of pertinent parameters; dynamical system

approach is applied to the flow of dusty liquid bounded by a stretching sheet. The

fluid behavior is analyzed by investigating the critical points and bifurcation

diagrams. The analytical results and the behavior, so obtained are compared. We

know that finding analytic solution is very difficult for the fluid flow problems

because of the high degree of non-linearity in the governing equation. In these cases,

it is desirable to find the qualitative behavior of the fluid flow using the theory of

Conclusions

77

dynamical system. A benchmark analysis of dynamical system theory has been

introduced for the stretching in the dusty field with magnetic field. This analysis can

be further extended to the dusty fluid problem where the analytical or numerical

solution is difficult if not impossible to obtain.

We believe that this work provides better insight to the peristaltic flows given the fact,

that we have modeled it for the curved tube which closely resembles the physical

geometry of the biological ducts. It may be added that we have given the solution up

to the first order correction to the proposed model and aim to introduce the higher

order correction for more accurate calculations of the curvature effects on the flow

dynamics in our future work. Furthermore, the effects of magnetic field in the

boundary layer have been investigated for the Jeffrey fluid under strong magnetic

field. Dynamical system approach has been added to draw the bifurcation diagram to

find the behavior of the flow for various parameters. This approach can be further

extended to more complicated problems to see the qualitative behavior of the fluid

flow.

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78

Chapter 7

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