Analysis of fluid flows having MHD and

thermal radiation impacts over a stretchable

cylindrical pipe under the influence of suction
and injection

By

Ayman Fatima

(Registration No: 00000400214)

A thesis submitted in partial fulfillment of the requirements for the degree of

Master of Science in Mathematics

Supervised by:
Dr. Rai Sajjad Saif

Department of Mathematics
School of Natural Sciences
National University of Sciences and Technology (NUST)
Islamabad, Pakistan
(2024)
 DEDICATION

I am grateful to Allah Almighty who gifted me a life and guided me in every

stage of life. Secondly, I want to dedicate this work to my supervisor Dr.
Rai Sajjad Saif who always supported me, mentored me and helped me for
the work I have done. This devotion is not completed without my parents,
who have been a constant source of strength throughout this program and
always encouraged me to accomplish my dreams.
 ACKNOWLEDGEMENTS

In the name of Allah, the beneficent and the most Merciful. It’s esteem to Allah
Almighty, who gave me a supremacy and guidance for this work. Indeed, without His
priceless help and blessings, I could have done nothing.
“My parents are my strength” and obviously it is not completed without them. I am
thankful to my parents for their moral and financial support for fulfilling my dreams
by distinguishing my potential.
I am indebted to my supervisor Dr. Rai Sajjad Saif for his moral support and
sympathetic behaviour. This was his courage and strong words which became the
reason of my survival during my struggling phase of this period. His critical and
vigilant scrutiny with instructive interpretations assisted me in assembling my work
in a very short time. It was not possible for me to complete my work without his
supervision and involvement.
I would acknowledge Dr. Rizwan Ul Haq and Dr. Khurheed Muhammad for being
on my guidance and evaluation committee. Finally, I want to convey my gratefulness
to all the friends who have provided valuable support to my study.
May Allah bless all.
Contents

INDEX OF TABLES IV

INDEX OF FIGURES VI

1 Prologue 1
1.1 Introductory definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Compressible and incompressible flow . . . . . . . . . . . . . . . 1
1.1.2 Steady and unsteady flow . . . . . . . . . . . . . . . . . . . . . 1
1.1.3 Laminar and turbulent flow . . . . . . . . . . . . . . . . . . . . 2
1.1.4 Newtonian and non-Newtonian fluids . . . . . . . . . . . . . . . 2
1.1.4.1 Visco-elastic fluids . . . . . . . . . . . . . . . . . . . . 4
1.1.5 Walters-B fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.6 Thermal Radiation . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.7 Magnetohydrodynamics (MHD) . . . . . . . . . . . . . . . . . . 5
1.1.8 Suction and Injection . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Boundary layer flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Momentum boundary layer . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Thermal boundary layer . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 Concentration boundary layer . . . . . . . . . . . . . . . . . . . 6
1.3 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Mass conservation law . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.2 Momentum conservation law . . . . . . . . . . . . . . . . . . . . 8
1.3.3 Energy convservation law . . . . . . . . . . . . . . . . . . . . . 8
1.4 Nanofluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.1 Buongiorno model . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Involved dimensionless parameters . . . . . . . . . . . . . . . . . . . . . 11
1.5.1 Hartmann number . . . . . . . . . . . . . . . . . . . . . . . . . 11

II
 1.5.2 Lewis number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5.3 Prandtl number . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5.4 Weissenberg number . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 Methods for solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6.1 NDSolve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6.2 Optimal homotopy analysis method (OHAM) . . . . . . . . . . 14
1.7 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Numerical examination of Buongiorno nanofluid flow over a stretch-

able cylindrical pipe with effects of magnetohydrodynamics, thermal
radiation, and suction/injection 19
2.1 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Results And Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Velocity’s radial component f (η) vs involved parameters . . . . 26
2.2.2 Velocity’s axial component f ′ (η) vs involved parameters . . . . 26
2.2.3 Concentration Profile ϕ(η) vs involved parameters . . . . . . . . 27
2.2.4 Temperature Profile θ(η) vs involved parameters . . . . . . . . . 28
2.2.5 Physical quantities vs involved parameters . . . . . . . . . . . . 29
2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Influences of magnetohydrodynamics, suction/injection and thermal

radiation phenomena for flow of Walters-B fluid across a stretchable
cylindrical pipe using OHAM 34
3.1 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Method For Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.1 Convergence and Error Analysis for OHAM Solution . . . . . . 39
3.3 Findings and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.1 Radial component of velocity f (η) vs associated parameters . . 42
3.3.2 Axial component of velocity f ′ (η) vs associated parameters . . . 42
3.3.3 Fluid’s temperature θ(η) vs associated parameters . . . . . . . . 43
3.3.4 Physical quantities vs associated parameters . . . . . . . . . . . 44
3.3.5 Validation of current analysis . . . . . . . . . . . . . . . . . . . 46
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

BIBLIOGRAPHY 48
List of Tables

2.1 Variation in skin friction coefficient, local Nusselt number and Sherwood
number for distinct data point of involved parameters . . . . . . . . . . 32

3.1 Fluctuation in skin friction coefficient and local Nusselt number with
distinct values of associated parameters . . . . . . . . . . . . . . . . . . 46
3.2 Present results of skin friction coefficient compared to Pillai et al. [53],
Nandeppanawar et al. [54] and Abid et al [40] for different values of W e
with γ = M = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

IV
List of Figures

1.1 Visual representation of laminar and turbulent flow . . . . . . . . . . . 3

1.2 Boundary layer flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Core model and coordinate system . . . . . . . . . . . . . . . . . . . . 21

2.2 f (η) for distinct values of γ and λ . . . . . . . . . . . . . . . . . . . . . 23
2.3 f ′ (η) for distinct values of γ for both suction and injection . . . . . . . 23
2.4 f ′ (η) for distinct values of M for both suction and injection . . . . . . 23
2.5 ϕ(η) for distinct values of γ for both suction and injection . . . . . . . 24
2.6 ϕ(η) for distinct values of Le for both suction and injection . . . . . . . 24
2.7 ϕ(η) for distinct values of N b for both suction and injection . . . . . . 24
2.8 ϕ(η) for distinct values of N t for both suction and injection . . . . . . . 24
2.9 θ(η) for distinct values of γ for both suction and injection . . . . . . . . 25
2.10 θ(η) for distinct values of N b for both suction and injection . . . . . . . 25
2.11 θ(η) for distinct values of N t for both suction and injection . . . . . . . 25
2.12 θ(η) for distinct values of Rd for both suction and injection . . . . . . 25
2.13 θ(η) for distinct values of P r for both suction and injection . . . . . . 26

3.1 Base model and coordinate system . . . . . . . . . . . . . . . . . . . . 36

3.2 Fluctuation in ET,k corresponds to order of approximations k when
W e = 0.00 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Fluctuation in ET,k corresponds to order of approximations k when
W e = 0.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Fluctuation in ET,k corresponds to order of approximations k when
W e = 0.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5 Fluctuation in ET,k corresponds to order of approximations k when
W e = 0.20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.6 Fluctuation in f (η) with various values of W e and λ . . . . . . . . . . 40
3.7 Fluctuation in f ′ (η) with various values of both suction and injection . 41
3.8 Fluctuation in f ′ (η) with various values of γ for both suction and injection 41

V
3.9 Fluctuation in f ′ (η) with various values of M for both suction and injection 41
3.10 Fluctuation in f ′ (η) with various values of W e for both suction and
injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.11 Fluctuation in θ(η) with various values of both suction and injection . 42
3.12 Fluctuation in θ(η) with various values of and Rd for both suction and
injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
 Abstract

Engineers and scientists have recently shown considerable interest in studying the
fluid flow and heat transfer around elongating cylinders either for Newtonian or non-
Newtonian fluids. The interest in this topic originates from its widespread application
in several industrial sectors, such as drawing of wire, cooling of metallic sheets, spinning
of metal, etc. Similarly, Magnetohydrodynamics is the knowledge of the interaction
between fluids that carry electricity and magnetic fields. Magnetic fields have an impact
on a broad range of natural and man-made flows. Since the MHD boundary layer
flow has many applications in different mechanical and chemical sectors, it has been
researched extensively over a surface that stretches either linearly or nonlinearly. In
industry, magnetic fields are commonly applied to heat, lift, stir, and pump liquid
metals. Moreover, in recent years, the study of combined heat and mass transfer
mechanisms involving nanoparticles has drawn significant interest from researchers,
largely due to its wide range of applications in engineering and biomedicine, such
as cooling infinite metallic plates, cancer therapy, and various industrial processes.
Traditional fluids including glycol, motor oil, water, and air mixes often have limited
heat conductivity. This is made better by incorporating solid particles into the fluids
to create "nanofluid". Last but not least, The research of suction and injection across
a stretched surface is crucial in fluid dynamics, due to its vast variety of industrial
and technical applications. These processes are critical for improving systems such
as polymer extrusion, aerodynamic heating, and chemical processing. Suction and
injection considerably alter boundary layer flow, consequently affecting heat and mass
transfer rates, which are crucial for maintaining optimum operating conditions and
enhancing system efficiency.
 Preface

The first chapter covers some fundamental definitions. The boundary layer flow
description, distinct phenomena, and a few associated dimensionless numbers used in
this thesis are explained in this chapter. To identify the gaps in the body of available
knowledge, a comprehensive evaluation of the literature is conducted as well as the
mathematical methods employed are also explained at the end.
The second chapter of this thesis includes a numerical examination of Buongiorno
nanofluid flow over a stretching cylinder with effects of magnetohydrodynamics, ther-
mal radiation, and suction/injection. According to prior research, no investigation has
ever been conducted on the Buongiorno Nanofluid model across a stretchable cylin-
drical pipe with the impact of heat radiation phenomena, suction/injection, and mag-
netohydrodynamics. This paper investigates the Buongiorno nanofluid model’s two-
dimensional magnetohydrodynamic flow in the presence of heat radiation effects over
a stretching cylinder. Suction and injection effects have also been considered in this
investigation. The governing equations have been solved using the method of ND solve
by the software MATHEMATICA. The study is broken up into distinct components.
After reviewing the relevant literature, the first section develops the mathematical for-
mulation by using appropriate transformations which transform the governing partial
differential equations (PDEs) into ordinary differential equations (ODEs). The third
segment uses graphical representation to highlight noteworthy discoveries. Lastly, the
conclusion of the whole research has been discussed.
The third chapter of this thesis is about the influences of magnetohydrodynam-
ics, suction/injection and thermal radiation phenomena on the flow of Walters-B fluid
across a stretchable cylindrical pipe using OHAM. Previous information on the subject
revealed that there has never been a study on the Walters-B fluid across a stretchable
cylindrical pipe influenced by magnetohydrodynamics, suction/injection, and thermal
radiation phenomena. The research is splitted into separate sections. Following the
literature review, the first section develops the mathematical formulation along with
boundary layer approximation and the governing partial differential equations (PDEs)
are modified into ordinary differential equations (ODEs) by utilizing appropriate trans-
formations. The third section explains how the governing equations are solved, pro-
viding accurate and efficient solutions, using the optimal homotopy analysis technique
[49, 50, 51, 52]. The next section emphasizes prominent findings via graphical portrayal.
The final section is structured to present the conclusion from the entire analysis.
Chapter 1

Prologue

Some fundamental definitions and introductions are covered in this chapter. The
boundary layer flow description, distinct phenomena, and a few associated dimen-
sionless numbers used in this thesis are explained in this chapter. To identify the gaps
in the body of available knowledge, a comprehensive evaluation of the literature is
conducted as well as the mathematical methods employed are also explained at the
end.

1.1 Introductory definitions

1.1.1 Compressible and incompressible flow

Such a type of fluid flow where density can change significantly, usually because of high
speed or pressure changes is called compressible flow. One example of compressible
flow is the airflow surrounding a jet plane traveling at high speed, when the air is
compressed and its density changes. Meanwhile, an incompressible flow is a kind of
fluid flow where the fluid’s density stays almost constant, even if the pressure changes.
Its example includes water flowing through a garden hose, where the water’s density
does not change as it moves.

1.1.2 Steady and unsteady flow

Steady fluid flow describes a condition where the physical properties of the fluid, such
as density, velocity, and temperature, do not change over time at any specific point
in the flow field. This means that if you were to observe the fluid at a fixed location,
you would see consistent values for these properties. However, these properties can
differ from one location to another within the flow. In other words, while the fluid’s

1
characteristics remain stable over time at each individual point, they can vary spatially
across the flow field. Mathematically,
∂ξ
= 0, (1.1)
∂t
and unsteady flow is a flow in which physical quantities may exhibit variations with
time. Mathematically,
∂ξ
̸= 0, (1.2)
∂t
where ξ is any physical quantity of fluid and represnt time.

1.1.3 Laminar and turbulent flow

A type of flow where the fluid moves in smooth, parallel layers with little to no mixing
between them. The flow is orderly, and the velocity of the fluid at any point is relatively
constant over time. One of its examples is water flowing slowly through a straight,
smooth pipe, where the layers of water slide past each other in an orderly fashion.
Whereas, turbulent flow is characterized by chaotic and irregular fluid motion. The
fluid velocity at a point can vary rapidly and unpredictably. Water rushing down a
rocky river, creating lots of swirls and waves, is an example of turbulent flow , which
depict in Figure 1.1.

1.1.4 Newtonian and non-Newtonian fluids

Newtonian fluids are those fluids that follow Newton’s law of viscosity. The shear stress
and the rate of deformation deformation in these fluids show a linear relationship. Their
viscosity remains constant regardless of the applied shear rate. Mathematically, it can
be expressed as
du
τ =µ , (1.3)
dy
where τ , µ, and du
dy
represent the shear stress, dynamic viscosity, and strain rate in
a unidirectional flow, indicating the rate of deformation in one dimension. Newtonian
fluids include substances such as water, air, glycerin, gasoline, and alcohol.
Unlike Newtonian fluids, non-Newtonian fluids do not adhere to Newton’s law of
viscosity. When shear is applied their viscosity changes either decreasing or increasing
depending on the specific properties of the fluid. For a unidirectional flow situation,
most non-Newtonian fluids obey the following power-law form:
du
τ =η , (1.4)
dy

2
Figure 1.1: Visual representation of laminar and turbulent flow

3
where η is apparent fluid which is given by:
 n−1
du
η=λ (1.5)
dy

here λ is termed as consistency index and n denotes flow index.

1.1.4.1 Visco-elastic fluids

Visco-elastic fluids are a type of non-Newtonian fluid that display both viscous and
elastic properties during deformation. These fluids possess properties of both liquids
and solids, including viscosity, elasticity, and the capacity to deform. At low stresses
or slow deformation rates, viscoelastic fluids act like viscous fluids. Conversely, when
subjected to rapid deformation, they exhibit solid-like behavior and tend to return to
their original shape after the stress is removed.

1.1.5 Walters-B fluid

The Walters-B fluid model is a type of non-Newtonian fluid model that explains how
fluids with both viscous and elastic properties (known as visco-elastic fluids) behave.
It is employed in sectors of the economy where it is crucial to process materials like
paints, polymers, and other complex fluids. Is given by
 
dA1
τ = µA1 − k0 − (∇V)A1 − A1 (∇V) t∗
(1.6)
dt

where k0 is the fluid’s elasticity, µ is the dynamic viscosity, A1 = (∇V) + ∇Vt is the
tensor of strain rate, and dtd is the material derivative.

1.1.6 Thermal Radiation

Thermal radiation is the process by which a body emits energy as electromagnetic
waves due to its temperature. This emission is governed by the Stefan-Boltzmann law,
which states that the radiative heat flux (or power per unit area) emitted by a surface is
proportional to the fourth power of its absolute temperature. In mathematical terms,
the law is expressed as:
E = σT 4 (1.7)
where E is the radiant energy emitted per unit area, σ is the Stefan-Boltzmann
constant, and T is the absolute temperature of the body. This relationship illustrates
how significantly the emitted radiation increases with temperature.

4
1.1.7 Magnetohydrodynamics (MHD)
Magnetohydrodynamics (MHD) studies the behavior of electrically conducting fluids
in the presence of a magnetic field. The basic governing equation for MHD flow is the
Navier-Stokes equation, which is modified by including the Lorentz force term. This
force arises from the interaction between the fluid’s motion and the magnetic field. The
modified Navier-Stokes equation can be expressed as:
 
∂v
ρ + v · ∇v = −∇p + µ∇2 v + J × B (1.8)
∂t
where v is the fluid velocity, p is the pressure, µ is the dynamic viscosity, J is the
current density, and B is the magnetic field.

1.1.8 Suction and Injection

Suction and injection involve the removal (suction) or addition (injection) of fluid at a
boundary, used to control the boundary layer properties. The boundary condition for
velocity at the surface can be expressed as:
u = uw at y = 0
where uw represents the velocity of suction or injection at the wall.

1.2 Boundary layer flows

When a fluid interacts with a solid surface in a fluid flow field, a boundary layer is
created. This phenomenon happens due to the adherence of the fluid directly to the
solid surface, which causes the fluid to have zero velocity at the wall, in accordance
with the no-slip requirement. As the distance from the solid surface increases, the
speed of the fluid gradually rises until it matches the velocity of the surrounding flow
beyond the boundary layer. The region of the fluid that has the greatest impact from
frictional forces exerted by the solid surface is referred as the boundary layer. Outside
of this narrow area, the fluid maintains its initial free stream velocity, as seen in Figure.
1.2

1.2.1 Momentum boundary layer

In order to fulfill the no-slip criterion, fluid particles that are in direct contact with the
solid surface possess a velocity of zero. Subsequently, they effect the velocity of fluid
particles of adjacent layer and as result they further effect the velocity of adjacent fluid
particles and so on. This process of slowing down of velocity takes place at a certain
distance from a surface which is termed as momentum boundary layer.

5
1.2.2 Thermal boundary layer
When fluid flows over a surface that is at a higher temperature than the surrounding
fluid, a thermal boundary layer forms near the surface. This layer is a thin region
where the temperature of the fluid increases due to the heat conducted from the hotter
surface. Within this boundary layer, the temperature gradient is highest closest to
the surface and decreases with distance from the surface. As one moves away from
the heated surface, the temperature of the fluid gradually drops and asymptotically
approaches the temperature of the surrounding fluid. This phenomenon is crucial for
understanding the heat transfer between the surface and the fluid, influencing various
applications such as cooling systems, heat exchangers, and aerodynamic heating. The
behavior and thickness of the thermal boundary layer are important for optimizing
thermal performance and ensuring efficient heat transfer in engineering and industrial
processes.

1.2.3 Concentration boundary layer

The concentration boundary layer is defined as the region near a surface within which
the concentration of a nanofluid reaches 99 percent of the concentration observed in
the free stream. This layer forms due to the interaction between the nanofluid and the
surface, where the concentration of nanoparticles is significantly affected by factors such
as diffusion and convective transport. Within this boundary layer, the concentration
gradient is pronounced, and it gradually transitions from the surface concentration to
match the bulk concentration of the nanofluid in the free stream. Understanding the
behavior of this boundary layer is crucial for optimizing the performance of nanofluids
in various applications, such as enhanced heat transfer and fluid flow management in
engineering systems.
The pictorial representation is given below and it is sourced from the internet.

6
 Figure 1.2: Boundary layer flow

1.3 Conservation laws

The conservation laws in mechanics establish the governing nonlinear differential equa-
tions for each flow field. The following subsection provides an overview of these fun-
damental principles mass, momentum , energy and concentration :

1.3.1 Mass conservation law

The principle of mass conservation asserts that mass cannot be created or destroyed
within a closed system. This fundamental concept in fluid dynamics is mathematically
expressed through the continuity equation. The general form of the continuity equation
for a fluid flow is given by:

∂ρ ⃗
+ ∇ · (ρV) = 0, (1.9)
∂t
where ρ represents the fluid density, V is the velocity vector of the fluid, and ∇
⃗ is
the gradient operator. This equation states that the rate of change of density within
a volume, combined with the divergence of the mass flux, must be zero to satisfy the
conservation of mass.
In the special cases where the flow is steady (i.e., no time-dependent changes) and
incompressible (i.e., the density ρ remains constant), the continuity equation simplifies
significantly. For such flows, the equation reduces to:

⃗ · V = 0.
∇ (1.10)

7
 This simplified form indicates that the divergence of the velocity field is zero, mean-
ing that the fluid flow is incompressible and the volume of fluid entering any region is
equal to the volume exiting it.

1.3.2 Momentum conservation law

The principle of momentum conservation states that the net force acting on a unit
volume of fluid, which includes both body forces and surface forces, is equal to the rate
of change of linear momentum within that volume. This principle can be expressed
mathematically by the following vector equation:

DV
ρ ⃗ +∇
= −∇p ⃗ · τ + ρF, (1.11)
Dt
where ρ is the fluid density, V is the velocity vector, and Dt
D
represents the material
derivative, which accounts for changes in velocity due to both time and spatial varia-
tions. The term −∇p ⃗ represents the force per unit volume due to pressure gradients,
where p is the fluid pressure. The term ∇ ⃗ · τ denotes the divergence of the stress tensor
τ , which accounts for internal forces arising from viscosity and shear stress within the
fluid. Finally, ρF represents the body forces per unit volume, such as gravitational
forces, acting on the fluid element.
This equation encapsulates how the combined effects of pressure gradients, viscous
stresses, and external body forces contribute to the overall momentum changes within
the fluid, providing a fundamental basis for analyzing fluid flow and dynamics.

1.3.3 Energy convservation law

The principle of energy conservation states that the change in kinetic and internal
energy within a control volume over time is equal to the net rate of heat added to
the system plus the rate at which work is done by the fluid on its surroundings. This
principle can be mathematically represented by the energy equation:

dT
ρCp ⃗ · (κ∇T
=∇ ⃗ ) + ϕ, (1.12)
dt
whereρ is the fluid density, Cp is the specific heat capacity at constant pressure,
which indicates the amount of heat required to raise the temperature of a unit mass
of the fluid by one degree, T denotes the temperature of the fluid, κ represents the
thermal conductivity, a measure of the fluid’s ability to conduct heat, ∇ ⃗ ) is the
⃗ · (κ∇T
term that accounts for heat conduction within the fluid, reflecting how heat diffuses
through the fluid, ϕ denotes the viscous dissipation function, which represents the rate

8
of conversion of mechanical energy into thermal energy due to viscous effects within
the fluid.
This equation highlights how the internal energy of the fluid changes over time due
to heat conduction and viscous dissipation. It provides a fundamental framework for
analyzing thermal effects in fluid flow, including heat transfer, temperature distribu-
tion, and energy dynamics in various engineering and scientific applications.

1.4 Nanofluids
Nanofluids are advanced fluids created by suspending nanoparticles, typically ranging
from 1 to 100 nanometers in size, within a base fluid. These nanoparticles can be
metallic, such as gold or silver, or non-metallic, such as silicon dioxide or alumina.
The primary goal of incorporating these nanoparticles into base fluids like water, oil,
propylene glycol, or ethylene glycol is to significantly enhance the thermal conductiv-
ity of the base fluid, which is usually quite low. By dispersing nanoparticles within
these fluids, the overall thermal conductivity is improved, making the nanofluids more
effective in heat transfer applications.
The enhancement in thermal conductivity of nanofluids can be attributed to several
key mechanisms. One such mechanism is the increased surface area-to-volume ratio of
nanoparticles due to their nanoscale size, which leads to improved thermal interactions
between the nanoparticles and the base fluid. Additionally, the significant Brownian
motion of nanoparticles at the nanoscale contributes to enhanced thermal conductivity
by increasing the fluid’s capacity to transfer heat. The interface between the nanopar-
ticles and the base fluid also creates additional thermal pathways, further improving
heat transfer. The effectiveness of these mechanisms depends on the size, shape, and
surface properties of the nanoparticles, as well as the characteristics of the base fluid.
Nanofluids have found applications in a wide range of industries due to their en-
hanced thermal properties. In the field of thermal power generation, they are used
to improve the efficiency of heat exchangers and cooling systems, leading to better
performance and reduced energy consumption. In medical devices, nanofluids play a
role in thermal therapies and diagnostics, where precise temperature control is cru-
cial. The pharmaceutical industry benefits from nanofluids in drug delivery systems
and processes that require precise thermal management. As electronic devices become
more compact, efficient thermal management is essential, making nanofluids valuable
in cooling systems for microelectronics to prevent overheating. In the paper industry,
nanofluids are used to enhance heat transfer in drying processes, improving the effi-
ciency of paper production. They also find applications in heat exchangers, where their
improved heat transfer rates can reduce the size and cost of these systems.
Despite their potential, several challenges need to be addressed for the practical ap-
plication of nanofluids. Ensuring the long-term stability of nanoparticle suspensions is

9
crucial to prevent issues such as particle settling and aggregation, which can reduce the
effectiveness of the nanofluid. Additionally, the manufacturing and cost of producing
nanoparticles and their dispersion into base fluids can be expensive, prompting ongoing
research to develop more cost-effective methods. The potential toxicity of nanoparti-
cles and their environmental impact also require thorough investigation to ensure safe
usage.
Various models have been proposed to understand and predict the thermal prop-
erties of nanofluids. These models focus on the interaction between nanoparticles and
the base fluid, considering factors such as particle size, concentration, and fluid dy-
namics. Both computational and experimental studies are continually refining these
models, providing deeper insights into the mechanisms that drive thermal conductivity
enhancement. In summary, nanofluids represent a promising area of research with the
potential to revolutionize heat transfer technologies across multiple sectors. While their
ability to enhance thermal conductivity offers new possibilities for improving efficiency
and performance, addressing challenges related to stability, cost, and safety is essential
for fully realizing their benefits.

1.4.1 Buongiorno model

Buongiorno’s model provides a theoretical framework for understanding nanofluids by
treating nanoparticles as a separate phase within a base fluid. It enhances the classical
Navier-Stokes and energy equations to account for the effects of Brownian diffusion
and thermophoresis. Brownian diffusion describes the random motion of nanoparticles
due to collisions with fluid molecules, while thermophoresis represents their movement
induced by temperature gradients. This model indicates that the inclusion of nanopar-
ticles leads to enhanced thermal conductivity and improved convective heat transfer in
nanofluids. Such enhancements are vital for optimizing heat transfer in various engi-
neering applications, including heat exchangers, solar collectors, and cooling systems.
The following are the energy and transport equations for nanoparticles as described
in the Buongiorno nanofluid model. These equations provide a framework for under-
standing how nanoparticles affect fluid dynamics and thermal performance.

dT ∇T · ∇T
= α∇2 T + τ DB ∇C · ∇T + τ DT (1.13)
dt T∞
and  
dC DT
= DB ∇2 C + ∇2 T (1.14)
dt T∞
(ρc)p
where τ = (ρc) f
is the ratio of effective heat capacity of nanoparticle material and
the base fluid, DB and DT are the Brownian and thermophoretic diffusion coefficients
respectively, and T∞ is the surrounding temperature of the fluid.

10
1.5 Involved dimensionless parameters

1.5.1 Hartmann number

The Hartmann number (Ha), also referred to as the magnetic parameter (M ), was
introduced by Julius Hartmann. It characterizes the ratio of magnetic forces to vis-
cous forces in electrically conducting fluids. This dimensionless number is essential for
analyzing the influence of magnetic fields on fluid dynamics, particularly in fields like
plasma physics and magnetohydrodynamics (MHD).
Mathematically, the Hartmann number is expressed as:
√
B0 L σ
M= √ , (1.15)
µ
where B0 denotes the strength of the magnetic field, L is a characteristic length
scale, σ is the electrical conductivity of the fluid, and µ is the dynamic viscosity.
This dimensionless number helps to understand how magnetic fields affect the flow
of conducting fluids. A higher Hartmann number signifies that the magnetic forces
are dominant compared to the viscous forces, which can significantly alter the fluid’s
flow characteristics. It is particularly crucial in the study of MHD, where it assists
in evaluating and predicting the behavior of plasma and other conductive fluids un-
der magnetic fields. This understanding is vital for various applications, including
astrophysical phenomena, industrial processes involving molten metals, and advanced
cooling systems.

1.5.2 Lewis number

The Lewis number (Le) is a dimensionless parameter that quantifies the ratio of thermal
diffusivity to mass diffusivity in a fluid. It is used to characterize fluid flows where heat
and mass transfer occur simultaneously, providing insights into the relative thickness
of thermal and concentration boundary layers.
Mathematically, the Lewis number is defined as:
α
Le = , (1.16)
D
where α represents the thermal diffusivity of the fluid, and D denotes the mass
diffusivity.
Thermal diffusivity (α) measures the rate at which heat spreads through the fluid,
while mass diffusivity (D) describes the rate at which mass is transported. The Lewis
number indicates how these two processes compare: a Lewis number greater than one

11
suggests that thermal diffusion is more rapid than mass diffusion, resulting in thinner
thermal boundary layers compared to concentration boundary layers. Conversely, a
Lewis number less than one implies that mass diffusion dominates over thermal dif-
fusion, leading to thicker thermal boundary layers relative to concentration boundary
layers.
Understanding the Lewis number is crucial in applications involving simultaneous
heat and mass transfer, such as in chemical reactors, environmental engineering, and
various industrial processes.

1.5.3 Prandtl number

The Prandtl number (Pr) is a dimensionless quantity that characterizes the relative
importance of momentum diffusivity (kinematic viscosity) compared to thermal dif-
fusivity in fluid flow. It provides insight into the relative thickness of the velocity
boundary layer and the thermal boundary layer in a fluid.
Mathematically, the Prandtl number is defined as:
ν
Pr = , (1.17)
α
where ν denotes the kinematic viscosity of the fluid, and α represents the thermal
diffusivity.
Kinematic viscosity (ν) measures the fluid’s resistance to shear stress, which affects
the momentum transport within the fluid. Thermal diffusivity (α) describes the rate
at which heat is conducted through the fluid. The Prandtl number thus represents the
ratio of these two properties: a higher Prandtl number indicates that momentum diffu-
sivity is more significant relative to thermal diffusivity, resulting in a thinner thermal
boundary layer compared to the velocity boundary layer. Conversely, a lower Prandtl
number suggests that thermal diffusion dominates over momentum diffusion, leading
to a relatively thicker thermal boundary layer.
The Prandtl number is crucial in various engineering applications, including heat
exchangers, cooling systems, and aerodynamic analysis, where understanding the bal-
ance between momentum and heat transfer is essential for optimizing performance and
designing effective systems.

1.5.4 Weissenberg number

The Weissenberg number (We) is a dimensionless quantity that quantifies the relative
importance of a fluid’s relaxation time compared to its flow time scale. It provides
insight into the fluid’s ability to resist deformation and return to its original shape
after experiencing stress.

12
 Mathematically, the Weissenberg number is expressed as:

tr
We = , (1.18)
tf
where tr represents the relaxation time of the fluid, and tf denotes the flow time
scale. The relaxation time (tr ) is a measure of how long it takes for the fluid to return
to its equilibrium state after being deformed, while the flow time scale (tf ) represents
the time over which the fluid experiences flow.
A high Weissenberg number indicates that the relaxation time of the fluid is long
relative to the time scale of the flow. This suggests that the fluid exhibits significant
elastic behavior, which can lead to complex flow phenomena such as shear-thickening
or shear-thinning. Shear-thickening occurs when the fluid becomes more viscous with
increasing shear rate, while shear-thinning is when the fluid becomes less viscous under
the same conditions. The specific response depends on the fluid’s rheological properties
and the nature of the applied stress.
The Weissenberg number is particularly important in the study of complex fluids,
such as polymer solutions and suspensions, where elastic effects play a significant role
in the flow behavior and can impact various industrial and scientific processes.

1.6 Methods for solution

1.6.1 NDSolve
Mathematica, also known as the Wolfram Language, provides powerful tools for solving
differential equations through its built-in function ‘NDSolve‘. This function is designed
to numerically solve a wide variety of differential equations, including ordinary differ-
ential equations (ODEs), partial differential equations (PDEs), differential-algebraic
equations (DAEs), and delay differential equations (DDEs).
Unlike ‘DSolve‘, which aims to find exact symbolic solutions to differential equa-
tions, ‘NDSolve‘ offers numerical approximations of the solutions. This makes it partic-
ularly useful for handling complex problems where symbolic solutions are either difficult
or impossible to obtain. ‘NDSolve‘ is capable of addressing both initial value problems
(IVPs) and boundary value problems (BVPs), providing flexibility in its application.
In solving initial value problems, ‘NDSolve‘ computes the solution based on initial
conditions specified at the beginning of the domain. For boundary value problems,
it finds solutions that satisfy given conditions at the boundaries of the domain. The
function employs numerical methods to approximate solutions, making it applicable to
a broad range of real-world scenarios where precise analytical solutions are not feasible.

13
 Overall, ‘NDSolve‘ is an essential tool in Mathematica for researchers and engineers
needing to model and analyze systems governed by differential equations, offering ro-
bust numerical solutions for complex and diverse applications.

1.6.2 Optimal homotopy analysis method (OHAM)

To address complex fluid dynamics problems, the Optimal Homotopy Analysis Method
(OHAM) is utilized to derive infinite series solutions. OHAM offers significant advan-
tages compared to traditional numerical and perturbation techniques by providing
analytical solutions expressed in terms of known functions, thereby avoiding the errors
typically associated with domain discretization in numerical methods. This method al-
lows for precise control over the convergence rates and accuracy of the solution through
the introduction of an auxiliary parameter, which serves as a means to fine-tune the
approximation process.
One of the key benefits of OHAM is its ability to handle a broad range of problems
by transforming them into a form where series solutions can be obtained analytically.
This approach not only improves solution accuracy but also offers insight into the
solution behavior without relying on numerical approximations that may introduce
discretization errors.
However, the effectiveness of OHAM is influenced by the choice of initial assump-
tions and the linear operators used in the method. Additionally, the method can
encounter convergence issues when dealing with very small parameter values. Despite
these challenges, careful optimization of auxiliary parameters can substantially improve
both convergence and accuracy. As a result, OHAM proves to be a valuable tool for
analyzing complex fluid behaviors, providing robust analytical insights into intricate
fluid dynamics scenarios.

1.7 Literature review

Engineers and scientists have recently shown considerable interest in studying the fluid
flow and heat transfer around elongating cylinders. The interest in this topic originates
from its widespread application in several industrial sectors, such as drawing of wire,
cooling of metallic sheets, spinning of metal, etc. Wang [1] was the pioneer to investi-
gate the phenomenon of constant flow caused by a cylinder that is elongating and sub-
merged in a fluid with high viscosity. Ishak et al. [2] conducted a study that built upon
Wang’s [1] research. They investigated the impact of heat transmission on a solid cylin-
der that is being stretched, considering both injection and suction circumstances. The
study included numerical and perturbation solutions. Later, Mukhopadhyay [3] car-
ried out an investigation on the partial-slip flow with magnetohydrodynamics (MHD)
and boundary layer effects over a stretched cylinder and derived numerical solutions

14
as well as analytical solutions for some particular cases. This research contributed to
the existing knowledge and provided a fresh perspective on these phenomena. Hashim
and Alshomrani [4] examined the energy transportation when high-temperature phase
change materials melted over a stretching cylinder. He discovered that the flow and
heat transfer characteristics are more pronounced for flow over a cylinder compared
to a flat plate. Saif et al. [5] investigated the Maxwell fluid flow over a cylindri-
cal stretching pipe, incorporating thermophoresis, Brownian motion, and melting heat
transfer effects. Their analysis reveals that higher curvature and Brownian motion pa-
rameters increase temperature distribution, while higher melting and Prandtl numbers
reduce heat and mass fluxes. One might quote a number of recent studies on stretching
cylinder in [6, 7, 8].
Magnetohydrodynamics is the knowledge of the interaction between fluids that
carry electricity and magnetic fields. Magnetic fields have an impact on a broad range
of natural and man-made flows. Since the MHD boundary layer flow has many appli-
cations in different mechanical and chemical sectors, it has been researched extensively
over a surface that stretches either linearly or nonlinearly. In industry, magnetic fields
are commonly applied to heat, lift, stir, and pump liquid metals. Many investigations
were done theoretically as well as experimentally in this direction, like, Joneidi et al.
[9] examined the heat transfer in magnetohydrodynamic flow caused by a stretched
cylinder utilising the method of homotopy analysis technique. The combined effects
of buoyancy force, convective heating, thermophoresis, magnetic field, and Brownian
motion on stagnation-point flow and heat transfer in nanofluid flow over a stretching or
contracting surface are examined by Makinde et al. [10]. Mukhopadhyay and Mandal
[11] examined the impact of velocity and thermal slip on MHD mixed convection flow
over a permeable sheet, concluding that an upsurge in the suction parameter reduces
the surface temperature. The topic of MHD mixed convection over a porous channel
in the presence of nanofluid was statistically explored by Fersadou et al. [12]. They
discovered that as the volume proportion of nanoparticles enhances, so does the rate of
heat transfer. In the presence of nonlinear thermal radiation, the properties of MHD
mixed convection flow of Ag-water nanofluid via an inclined stretched cylinder was
studied by Hayat et al. [13]. Later, by considering the effects of viscous and Joule
dissipation, Waqas et al.[14] addressed the MHD flow of micropolar fluid across a non-
linear stretchable surface. Their research shows that as the Prandtl number rises, so
does the fluid’s temperature and the corresponding thickness of the boundary layer.
Mukhopadhyay [15] explored the effects of a constant magnetic field on a stretched
cylinder having a partial-slip phenomenon at the boundary of the surface and derived
numerical solutions as well as analytical solutions for some particular cases. The influ-
ences of MHD and radiation on flow across a stretched cylinder inside a porous material
have been examined by Abbas et al. [16]. They derived analytical solutions via the per-
turbation method and numerical solutions through the Keller box technique. Recently,
multiple scholars have examined the flow of Newtonian/non-Newtonian liquids with

15
varying boundary conditions across a stretched cylinder. For additional information,
readers are directed to the works of T. Hayat et al. [17], [18], Z Asghar et al. [19], and
other similar research.
In recent years, the study of combined heat and mass transfer mechanisms involving
nanoparticles has drawn significant interest from researchers, largely due to its wide
range of applications in engineering and biomedicine such as cooling infinite metallic
plates, cancer therapy, and various industrial processes. Traditional fluids including
glycol, motor oil, water, and air mixes often have limited heat conductivity. This is
made better by incorporating solid particles into the fluids to create nanofluid. Choi
[20] coined the term "nanofluid" at the beginning, referring to a combination consisting
of solid nanoparticles suspended in a foundational liquid. Metals including Au, Cu, Ag,
Ti, Fe, and Hg are frequently used as building blocks for these nanoparticles, as well as
non-metals like Al2 O3 , CuO, SiO2 , and T iO2 . Buongiorno [21] later conducted a study
to explore nanofluids’ increased heat transfer capabilities. He noted that standard ideas
of thermal dispersion were insufficient to explain the high heat transfer coefficients re-
ported in nanofluids, which are partly due to enhanced turbulence from nanoparticle
spinning. He developed a mathematical model for heat convection in nanofluids that
integrates both thermophoresis diffusion and Brownian motion. BuKuznetsov and
Nield [22] used similarity analysis to investigate the independent convective move-
ment of nanofluids along a vertical plate, concentrating on how parameters like the
buoyancy-ratio parameter, Brownian motion, Lewis number, and thermophoresis af-
fect the flow.Shiekholeslami et al. [23] used an effective control volume finite element
technique (CVFEM) for their numerical simulations to study MHD flow and heat trans-
portation in an inclined L-shaped enclosure containing Al2 O3 nanoparticles.Malik and
Salahuddin [24] focused on the numerical analysis of Williamson fluid stagnation-point
flow over a stretching cylinder, whereas Shiekholeslami et al. [25] used the Lattice
Boltzmann method to investigate the impact of Lorentz forces on forced convection
flow of nanofluids in a porous cavity. Salahuddin et al. [26] investigated the mixed
convection flow of Williamson fluid under slip situations, using the Keller box method
for their numerical solutions. Additionally, Shiekholeslami et al. [27] addressed Darcy
and KKL models for CuO-water nanofluid flow in a porous cavity and porous me-
dia. Brownian motion and thermophoresis effects were investigated by Rehman et al.
[28] in relation to the combined effects of chemical reactions and stratification on the
electrically conducting Eyring Powell nanofluid flow around a stretching cylinder. Af-
terwards, Khan et al. [29] studied the impact of first-order chemical processes while
a cone-generated non-Newtonian Williamson fluid flows. Inspired by these applica-
tions, numerous scientists and engineers have been investigating nanofluid flows and
heat transfer from various perspectives, as seen in the works of Shiekholeslami [30],
Shiekholeslami and Ganji [31], Rehman et al. [32], Shiekholeslami and Shehzad [33],
Hosseini et al. [34], Khan et al. [35], Shiekholeslami et al. [36], and Hashim et al. [37],
among others.

16
 The fluids that do not comply with Newton’s viscosity law, which means their in-
teractions with stress and strain are nonlinear, are known as non-Newtonian fluids.
These fluids are of major relevance in real-world applications, playing key roles in dif-
ferent industrial processes such as polymer and food processing and pharmaceutical
manufacture. In physiology, non-Newtonian fluids are exploited in medication deliv-
ery methods and tissue engineering. Additionally, they are essential for geophysical
applications and oil prospecting. Non-Newtonian fluids may be divided into numerous
categories, including shear-thinning, shear-thickening, and visco-elastic fluids. Shear-
thinning liquids: whipped cream, paints, and ketchup, demonstrate a drop in apparent
viscosity with an increase in the rate of deformation. Conversely, shear-thickening flu-
ids: combined cornstarch and water, as well as quicksand, exhibit a rise in apparent
viscosity with increasing strain rates. Fluids that are visco-elastic, have both elastic
and viscous characteristics. Examples of visco-elastic fluids include cheese, marshmal-
low cream, gelled foods, colloids, biopolymers, biological fluids, and granular solids. A
famous property of visco-elastic fluids is the Weissenberg effect, where the fluid rises up
a rod when disturbed. Additionally, visco-elastic fluids display a recoil phenomenon,
where the particles of fluid revert to their initial state after the applied tension is re-
moved owing to the tensile forces. In particular, one visco-elastic fluid model is known
as Walters-B fluid, which falls under the category of non-Newtonian fluids. Beard
and Walters [38] created this visco-elastic fluid model, which is capable of forecasting
the characteristics of the flow of various polymer solutions, such as paints, industrial
fluids, and hydrocarbons. This fluid preserves viscous and elastic features, making
Walters-B fluid models especially suitable for modeling complicated flow behaviours
in various engineering and scientific situations. It is often used to investigate flow dy-
namics under different situations, such as inside porous media, on stretched surfaces,
or when magnetic fields are present. Hussain and Ullah [39] examined the Walters-
B fluid’s boundary layer flow around a stretched cylinder, taking into consideration
the temperature-dependent viscosity effects. Abid Majeed et al. [40] explored the
flow and heat transmission properties of Walters-B fluid around a stretched cylinder.
Later, Waqas et al. [41] studied bioconvection and transport processes in magnetized
Walters-B nanofluid over a cylindrical disk, adding nonlinear radiative heat transfer
effects. The influence of thermal radiation on the flow of Walters-B nano liquid through
a cylindrical surface under convective and uniform heat flux conditions has been re-
ported by Mahat et al. [42]. Malik and Mustafa [43] developed an analytical solution
by utilizing an optimal homotopy analysis technique for the unsteady flow of Walters-B
fluid across a deformable heated surface having a quadratic temperature profile.
The research of suction and injection across a stretched surface is crucial in fluid dy-
namics, due to its vast variety of industrial and technical applications. These processes
are critical for improving systems such as polymer extrusion, aerodynamic heating,
and chemical processing. Suction and injection considerably alter boundary layer flow,
consequently affecting heat and mass transfer rates, which are crucial for maintain-

17
ing optimum operating conditions and enhancing system efficiency. Many researchers
examined the influences of suction and injection under distinct blends of phenomena,
like, Chamkha [44] performed an analysis for temperature layers on fluid flow and
heat transfer around an extended cylinder exposed to continual fluid suction or in-
jection. Hayat et al. [45] examined heat transfer with thermal radiation and MHD
for Maxwell fluid flow within a porous channel and derived a series of solutions using
HAM.. Mansour et al. [46] examined how temperature layers paired with consistent
suction or injection affect the flow and heat transmission properties of micropolar flu-
ids around an elongating cylinder. Additionally, Shafique and Kamal [47] studied the
impact of constant suction or blowing on the flow dynamics of micropolar fluids around
a stretched cylinder. Moreover, Hayat et al. [48] explored the MHD squeezing flow of a
couple of stress nanomaterials between parallel surfaces, incorporating thermophoresis,
Brownian motion, and a time-dependent magnetic field.

18
Chapter 2

Numerical examination of Buongiorno

nanofluid flow over a stretchable
cylindrical pipe with effects of
magnetohydrodynamics, thermal
radiation, and suction/injection

In this chapter, we explore the effectiveness of the Buongiorno nanofluid model within
the context of a stretchable cylindrical pipe. The study incorporates the impacts of
thermal radiation, suction/injection, and magnetohydrodynamics on fluid dynamics.
The study examines the intricate interactions among these factors and their effects on
fluid flow and heat transfer properties. The governing partial differential equations
are derived and solved using NDSolve method by MATHEMATICA, which produces
a thorough numerical solution for a variety of cases. The current findings provide new
insights into how these factors collectively influence the skin friction coefficient, Sher-
wood number, and Nusselt number. The analysis significantly enhances the existing
knowledge of the role of Newtonian properties in nanofluids, particularly in practical
engineering applications involving stretching cylinders.

2.1 Mathematical Modeling

Let a steady, axisymmetric flow of Buongiorno nanofluid over a cylinder that is ex-
panding horizontally and has a radius R. This problem is modeled in a cylindrical
coordinate system, where the r-axis extends radially and the z-axis runs along the axis
of the stretched cylinder. The radial and axial velocity components are u and w respec-
tively. a normal magnetic field B = (B0 , 0, 0) is applied to the cylinder, and because

19
the magnetic Reynolds number is low, the induced magnetic field is disregarded. After
implementing the boundary layer approximation, the equations governing the flow are
listed below:

∂u u ∂w
+ + = 0, (2.1)
∂r r ∂z

!
∂w ∂w ∂w ∂ 2w σwB02
u +w =ν + 2 − , (2.2)
∂r ∂z r∂r ∂r ρ

!
∂ 2T
 
∂T ∂T 1 ∂T ∂C ∂T
u +w =α + + τ DB
∂r ∂z ∂r2 r ∂r ∂r ∂r
 2 ! (2.3)
DT ∂T 16σ ∗ T∞
3
∂ 2T 1 ∂T
+τ + + ,
T∞ ∂r 3κ∗ ρcp ∂r2 r ∂r
! !
∂C ∂C ∂ 2 C 1 ∂C DT ∂ 2T 1 ∂T
u +w = DB + + + , (2.4)
∂r ∂z ∂r2 r ∂r T∞ ∂r 2 r ∂r
where u(r, z) is the radial velocity component and w(r, z) is the axial velocity compo-
nent, σ is the fluid’s electrical conductivity, ν is the kinematic viscosity, α = ρCk p is
the fluid’s thermal diffusivity, and σ ∗ and κ∗ are the Stefan-Boltzmann constant and
Rosseland mean absorption coefficient respectively, τ is the ratio of the fluid’s heat
capacity to the effective heat capacity of the nanoparticle material, DB is the coeffi-
cient of Brownian diffusion and DT is the coefficient of thermophoretic diffusion. The
following are the equivalent boundary conditions:

u = uw , w = ww , T = Tw , C = Cw at r = R,
(2.5)
w → 0, T → T∞ , C → C∞ as r → ∞,
where uw = − νcl λ and ww = czl , in which the constant c is positive whose dimension
p
is [1/time] and λ is a constant which refers to injection of mass for λ < 0 and suction
of mass for λ > 0.

20
 Figure 2.1: Core model and coordinate system

Now we will translate equations 3.2 to 3.6 into dimensionless form using the follow-
ing similarity transformations:
r r
r 2 − R2 c νc T − T∞ C − C∞
η= , ψ= zRf (η), θ = ,ϕ = . (2.6)
2R νl l Tw − T∞ Cw − C∞
In the transformations above, the stream function ψ is related to the velocity com-
ponents using the following equations
r
1 ∂ψ 1 νc 1 ∂ψ cz
u=− =− Rf (η), w = = f ′ (η), (2.7)
r ∂z r l r ∂r l
where η is the similarity variable. Using the above-mentioned modifications, equation
3.2 is satisfied, and equations 3.3 and 3.5 will have the following format:

f ′2 − f f ′′ − 2γf ′′ − (1 + 2γη)f ′′′ + M 2 f ′ = 0, (2.8)

Pr f θ′ + 2γθ′ + (1 + 2γη)θ′′ + Pr Nb(1 + 2γη)θ′ ϕ′

4 (2.9)
+ Pr Ntθ′2 + Rd(2γθ′ + (1 + 2γη)θ′′ ) = 0,
3
Nt
f ϕ′ Le + 2γϕ′ + (1 + 2γη)ϕ′′ + 2γθ′ + (1 + 2γη)θ′′ = 0. (2.10)


Nb
The associated boundary conditions are,
f (0) = λ, f ′ (0) = 1, θ(0) = 1, ϕ(0) = 1, at η = 0,
(2.11)
f ′ (∞) → 0, θ(∞) → 0 ϕ(∞) → 0 at η → ∞,

21
where γ is the curvature parameter, M is the Hartmann number, Rd is the radiation
parameter, P r is the Prandtl number, Le is the Lewis number, N b is the Brownian
motion parameter, N t is the thermophoresis parameter, which are stated as follows:
r s
1 νl σB02 l 4σ ∗ T∞
3
ν ν
γ= , M= , Rd = ∗
, Pr = , Le = ,
R c ρc κk σ DB (2.12)
τ DB (Cw − C∞ ) τ DT (Tw − T∞ )
Nb = , Nt = .
ν νT∞

The physical quantities used in this inquiry, like coefficient of skin friction Cf ,
Nusselt number N u and the Sherwood number Sh, are defined by,

τrz |r=R zqs zqm

Cf = , Nu = |r=R + qr |r=R , Sh = , (2.13)
ρUs2 k(Tw − T∞ ) DB (Cw − C∞ )

where τrz is the surface shear stress, the surface mass flux is qm , and the surface heat
flux is qs . The dimensionless representations of coefficient of skin friction, local Nusselt
number, and Sherwood number using transformations 3.15 are given below:

 
−1/2 4
1/2 ′′
Rez Cf = f (0), Rez N u = − 1 + Rd θ′ (0) and Rez −1/2 Sh = −ϕ′ (0),
3
(2.14)
2
where Rez = czνl represents the dimensionless local Reynolds number.

2.2 Results And Discussion

The governing differential equations were numerically solved using the NDsolve method
in MATHEMATICA, which provided numerical solutions for the velocity, temperature,
and concentration profiles under the influence of the MHD, and thermal radiation.
Graphical illustrations are used to show numerical results in order to better visualize
the physical problem. Figures 2-13 represent numerical results, illustrating the impact
of various non-dimensional parameters, namely, Prandtl number (P r), Lewis number
(Le), curvature parameter (γ), Hartmann number (M ), suction/injection parameter
(λ), Brownian motion parameter (N b), thermophoresis parameter (N t), and Radiation
parameter (Rd), on velocity, temperature and concentration profile.

22
 f

1.5

1.0

Solid (λ = 0.5)
0.5
Dashed (λ = 0.0)
Dotted (λ = -0.5)
γ = 0.1, 0.2, 0.3

η
2 4 6 8 10 12
Pr = 2.0, M = 0.1, Nb = 0.1,
Rd = 0.1, Nt = 0.1, Le = 0.1
-0.5

Figure 2.2: f (η) for distinct values of γ and λ

f' f'
1.0 1.0
M = 0.1, Nb = 0.1, Pr = 2.0, Rd = 0.1, Nt = 0.1, Le = 0.1 γ = 0.1, Nb = 0.1, Pr = 2.0, Rd = 0.1, Nt = 0.1, Le = 0.1

0.8 0.8

Solid (λ = 0.1)
0.6 0.6 Dashed (λ = -0.1)
Solid (λ = 0.5)
Dashed (λ = -0.5)
0.4 γ = 0.1, 0.2, 0.3 0.4
M = 0.0, 1.0, 2.0

0.2 0.2

η η
2 4 6 8 10 12 2 4 6 8 10

Figure 2.3: f ′ (η) for distinct values of Figure 2.4: f ′ (η) for distinct values of
γ for both suction and injection M for both suction and injection

23
 ϕ ϕ
1.0 1.0
M = 0.1, Nb = 0.1, Pr = 2.0, Rd = 0.1, Nt = 0.1, Le = 2.5 M = 0.1, Nb = 0.1, Pr = 2.0, Rd = 0.1, Nt = 0.1, γ = 0.1

0.8 0.8

0.6 Solid (λ = 0.5) 0.6 Solid (λ = 0.5)

Dashed (λ = -0.5) Dashed (λ = -0.5)
0.4 γ = 0.1, 0.2, 0.3 0.4 Le= 2.0, 2.5, 3.2

0.2 0.2

η η
5 10 15 20 2 4 6 8 10 12

Figure 2.5: ϕ(η) for distinct values of Figure 2.6: ϕ(η) for distinct values of
γ for both suction and injection Le for both suction and injection

ϕ ϕ
1.0 1.0
M = 0.1, Pr = 2.0, γ = 0.1, Nt = 0.3, Rd = 0.1, Le = 1.0 Pr = 2.0, M = 0.1, Nb = 0.3, γ = 0.1, Rd = 0.1, Le = 2.0

0.8 0.8

0.6 0.6 Solid (λ = 0.3)

Solid (λ = 0.5) Dashed (λ = -0.3)
Dashed (λ = -0.1)
0.4
Nb= 1.0, 2.0, 5.0
0.4
Nt = 0.1, 0.2, 0.3

0.2 0.2

η η
2 4 6 8 10 12 5 10 15

Figure 2.7: ϕ(η) for distinct values of Figure 2.8: ϕ(η) for distinct values of
N b for both suction and injection N t for both suction and injection

24
 θ θ
1.0 1.0
M = 0.1, Nb = 0.1, Pr = 2.0, Rd = 0.1, Nt = 0.1, Le = 0.1 M = 0.1, Pr = 2.0, γ = 0.1, Nt = 0.1, Rd = 0.1, Le = 0.1

0.8 0.8

Solid (λ = 0.3)
0.6 Solid (λ = 0.5) 0.6
Dashed (λ = -0.3)
Dashed (λ = -0.5)
γ= 0.1, 0.2, 0.3 Nb= 1.5, 2.0, 2.5
0.4 0.4

0.2 0.2

η η
2 4 6 8 10 12 2 4 6 8 10 12

Figure 2.9: θ(η) for distinct values of Figure 2.10: θ(η) for distinct values of
γ for both suction and injection N b for both suction and injection

θ θ
1.0 1.0
M = 1.0, Pr = 2.0, Nb = 0.2, γ = 0.25, Rd = 0.1, Le = 0.1 M = 0.1, Nb = 0.1, Pr = 2.0, γ = 0.1, Nt = 0.1, Le = 0.1
0.8 0.8
Solid (λ = 0.1)
Dashed (λ = -0.1)
0.6 0.6
Solid (λ = 0.3)
Dashed (λ = -0.3)
0.4 Nt= 0.1, 0.4, 0.7 0.4
Rd= 0.0, 0.2, 0.4

0.2 0.2

η η
5 10 15 20 2 4 6 8 10 12
Figure 2.11: θ(η) for distinct values of Figure 2.12: θ(η) for distinct values of
N t for both suction and injection Rd for both suction and injection

25
 θ
1.0
M = 0.1, Nb = 0.1, γ = 0.1, Rd = 0.1, Nt = 0.1, Le = 0.1

0.8

Solid (λ = 0.5)
0.6
Dashed (λ = -0.5)

0.4
Pr= 1.0, 1.5, 2.0

0.2

η
2 4 6 8 10 12
Figure 2.13: θ(η) for distinct values of P r for both suction and injection

2.2.1 Velocity’s radial component f (η) vs involved parameters

The velocity’s radial component f (η) for various values of γ and λ is shown in Figure
2. Boundary conditions 3.16 clearly demonstrate that f (η), which begins at λ and rises
to its eventual predicted value f (∞), which is relatively far from the cylinder’s surface.
Thus, f (η) grows with increasing λ. Moreover, increasing the curvature parameter on
f (η) causes the profile to climb more quickly and steeply, which indicates a smaller
boundary layer.

2.2.2 Velocity’s axial component f ′ (η) vs involved parameters

Figure 3 represents the variation of curvature parameter γ and suction injection pa-
rameter λ on velocity’s axial component f ′ (η). It is clear from the graph that f ′ (η)
increases as we increase the curvature parameter γ. A higher f ′ (η) is the result of
a number of physical processes that are enhanced when the curvature parameter is
increased. These impacts include thinning of the boundary layer, increased pressure
gradients, streamline convergence, and decreased viscous effects. This indicates that
the fluid can accelerate more quickly as the surface curvature becomes more notice-
able, which raises the f ′ (η) overall. Figure 3 also demonstrates that with the increase
in suction/injection parameter, f ′ (η) decreases. When the suction/injection parameter

26
is increased, f ′ (η) decreases mainly because the fluid in the boundary layer is added or
removed, changing the momentum distribution. Fluid is removed via suction, which
lowers the available momentum and slows the flow close to the surface. Figure 4 depicts
the impact of the magnetic parameter M and suction/injection parameter λ on f ′ (η).
Figure 4 demonstrates that the increase in magnetic parameter decreases f ′ (η). The
decrease in f ′ (η) with an increase in the magnetic parameter is primarily due to the
Lorentz force, which opposes the fluid motion and dissipates kinetic energy. A reduced
f ′ (η) is the result of this force’s resistive mechanism, which also thickens the boundary
layer and decreases the fluid’s momentum.

2.2.3 Concentration Profile ϕ(η) vs involved parameters

Figure 5 displays the fluctuation in concentration profile for various measures of γ for
both suction and injection. It is evident from the figure that with the increase in
suction/injection parameter λ, concentration profile ϕ(η) decreases. When the suc-
tion/injection parameter increases, it indicates that fluid is being pulled out of the
boundary layer. As a result, the concentration of nanoparticles close to the surface is
decreased and the boundary layer is thinned. The concentration of the nanoparticles in
the boundary layer is reduced overall when the boundary layer thins because they are
pulled closer to the surface. As a result, the profile of concentration falls. Figure 5 also
demonstrates that the concentration profile rises as the curvature parameter increases.
The cylinder’s surface becomes more curved as the curvature parameter rises, which
tends to confine the flow around the cylinder more tightly. Because of this confinement,
there may be an increase in the accumulation of nanoparticles close to the surface, rais-
ing the concentration profile. Figure 6 showcases the change in concentration profile
for various values of λ and Le. The figure makes it clear that the concentration profile
drops when the Lewis number rises. As the Lewis number is defined by the ratio of
kinematic viscosity to the Brownian diffusion coefficient, the decrease in concentration
profile with the increase in Lewis number is due to the reduced effect of Brownian
motion, which limits nanoparticle diffusion causing the concentration profile to drop
more rapidly. As a result, the boundary layer becomes thinner. With less diffusion
due to the higher Lewis number, the concentration gradient becomes steeper, causing
the transition from high to low concentration to occur over a smaller distance, thus
reducing the boundary layer thickness. Figure 7 illustrates the effect of Brownian mo-
tion parameter N b and suction/injection parameter λ on concentration profile ϕ(η).
It shows that as the Brownian motion parameter increases, the concentration profile
decreases. With stronger Brownian motion, the nanoparticles diffuse more effectively
throughout the boundary layer. While this increases the spread of nanoparticles, it also
leads to a lower concentration at any given point, especially near the surface. Con-
sequently, the concentration boundary layer thickens. This improved diffusion widens
the region across which the concentration changes from high near the surface to low

27
away from it, allowing nanoparticles to move further before their concentration consid-
erably drops. Figure 8 depicts the variation of the thermophoresis parameter N t, for
both suction and injection, on the concentration profile. It shows that the concentra-
tion profile grows in tandem with an increase in the thermophoresis parameter. More
movement of nanoparticles from hotter to colder regions is facilitated by an increase in
the thermophoresis parameter, which causes more accumulation and higher concentra-
tion profiles within the boundary layer. As a result, the concentration boundary layer
becomes thicker. The increased accumulation of nanoparticles due to thermophoresis
causes the concentration to remain higher over a broader region, extending the distance
over which the concentration transitions from high near the surface to lower farther
away. This broadening of the region results in a thicker concentration boundary layer.

2.2.4 Temperature Profile θ(η) vs involved parameters

Figure 9 shows the influence of suction/injection parameter λ and curvature parameter
γ on temperature profile θ(η). As the suction/injection parameter increases, the tem-
perature profile drops. This occurs because injection adds colder fluid, increasing heat
transfer and lowering the surface temperature, whereas suction removes warmer fluid
from the surface and replaces it with cooler fluid from the surroundings. This figure also
shows that the temperature profile grows as the curvature parameter increases. This
is a result of how changing the curvature parameter affects the flow and heat transfer
properties surrounding the cylinder. Enhanced heat transfer, thicker thermal boundary
layers, and changed convection patterns all contribute to an increase in the tempera-
ture profile. The impact of the suction/injection parameter λ and the Brownian motion
parameter N b on the temperature profile θ(η) is shown in Figure 10. The graph makes
it evident that as the Brownian motion parameter increases, so does the temperature
profile. A higher temperature profile is the consequence of enhanced heat transfer,
elevated fluid effective thermal conductivity, and enhanced dispersion of nanoparti-
cles due to the elevation of the Brownian motion parameter. Better dispersion of the
nanoparticles throughout the fluid and more consistent heat distribution are the re-
sults of the improved Brownian motion. A thicker thermal boundary layer results from
the temperature staying elevated for a longer period of time due to the enhanced heat
transfer. This expands the region where the temperature changes from the surface to
the ambient temperature. Figure 11 demonstrates the variation in temperature profile
θ(η) for different values of thermophoresis parameter N t for both suction and injection.
As we increase the thermophoresis parameter, the temperature profile also increases.
A higher temperature profile near the surface results from the enhanced movement
and redistribution of nanoparticles, which improve heat transfer and thermal conduc-
tivity in the fluid as the thermophoresis parameter rises. Consequently, the thickness
of the thermal boundary layer increases. The temperature differential extends farther
from the surface due to the enhanced thermophoresis-driven nanoparticle movement,

28
increasing the thermal boundary layer. Figure 12 explains the impact of radiation pa-
rameter Rd on θ(η) for both injection and suction. The graph shows the growth in the
temperature profile by increasing the radiation parameter. An increase in the radia-
tion parameter leads to enhanced radiative heat transfer, which increases the amount
of heat absorbed by the fluid near the surface. This additional heat input results in a
higher temperature profile. As a result, the thermal boundary layer becomes thicker.
The increased heat absorption causes the temperature to remain elevated over a larger
region, extending the distance over which the temperature decreases from the surface
to the ambient temperature, thus thickening the thermal boundary layer. Figure 13
shows the fluctuation in Prandtl number P r and suction/injection parameter λ in the
temperature profile. It displays how the temperature profile decreases as the Prandtl
number increases. Certainly, The kinematic viscosity to thermal diffusivity ratio is
known as the Prandtl number. A greater Prandtl number suggests that momentum
diffusivity is higher than thermal diffusivity. This results in a reduced temperature
profile because it creates a narrower thermal boundary layer and less efficient heat
transfer.

2.2.5 Physical quantities vs involved parameters

Table 1 shows the variation in the expression for skin friction coefficient, Nusselt num-
ber, and Sherwood number with numerous values of involved parameters. While vary-
ing one parameter, all other parameters are taken as 0.1.
The skin friction coefficient indicates the fluid’s shear stress on the cylinder’s sur-
face. The table makes it evident that by increasing the magnetic parameter, the ex-
pression for coefficient of skin friction decreases. A greater Lorentz force acting against
the fluid flow is created when the magnetic parameter is increased, which reduces the
velocity gradient near the wall and lowers the skin friction coefficient. Consequently,
the momentum boundary layer gets thicker. The liquid close to the surface experiences
a slowdown due to the opposing Lorentz force, which prolongs the time it takes for
the velocity to reach the free-stream value. This longer transition distance between
the surface and free-stream velocities thickens the boundary layer. With the increment
of the curvature parameter, the expression for skin friction coefficient decreases. The
flow dynamics surrounding the curved surface change as the curvature parameter rises,
resulting in a decrease in the velocity gradients close to the surface. As a result, the
skin friction coefficient decreases due to a thicker boundary layer and less wall shear
stress. Also, with the rise of the suction/injection parameter, the expression for skin
friction coefficient drops. As the suction/injection parameter increases, the fluid is
drawn away from the surface, which reduces the boundary layer thickness. This reduc-
tion in boundary layer thickness can lead to a decrease in the velocity gradient near the
surface resulting in fall of the skin friction coefficient. Since the skin friction coefficient
is a measure of the velocity gradient at the wall and parameters including Lewis num-

29
ber Le, thermophoresis parameter N t, Brownian motion parameter N b, and radiation
parameter Rd mostly affect the thermal or concentration fields without significantly
altering the velocity distribution near the surface, there is no substantial change in the
skin friction coefficient when these parameters fluctuate.
Then, this table shows the fluctuation in expression for thelocal Nusselt number
for different values of involved parameters. Utilizing the local Nusselt number, one can
compute the convective heat transfer at the surface of the cylinder. The table demon-
strates that by increasing the magnetic parameter, local Nussult number decreases.
The larger Lorentz force results in decreased fluid velocity and convection, a thicker
boundary layer, and a changed temperature distribution as the magnetic parameter
increases. The local Nusselt number decreases as a result of all these factors, indi-
cating less convective heat transfer than conduction. Additionally, as the curvature
parameter is increased, the local Nusselt number grows, as this table demonstrates.
This is because increasing the curvature parameter generally results in increased heat
transfer surface area, improved convective mixing, thinner thermal boundary layers,
and better heat dissipation. A greater local Nusselt number, which denotes more ef-
ficient convective heat transport, is the result of all these factors. The local Nusselt
number decreases as the Brownian motion parameter increases. Increased Brownian
motion lowers the local Nusselt number, which results in a thicker thermal boundary
layer and perhaps reduced convective heat transfer efficiency. The local Nusselt num-
ber rises as the thermophoresis parameter increases. Increasing the thermophoresis
parameter makes it easier for nanoparticles to diffuse and move around in the fluid,
which enhances convective heat transfer and thermal conductivity. Consequently, an
increase in the local Nusselt number indicates a more effective transmission of heat
from the fluid’s surface. The thermal boundary layer gets thinner as well. Because of
the enhanced heat transfer brought about by thermophoresis, the temperature gradient
close to the surface steepens and cools more quickly away from the surface, reducing
the thickness of the thermal boundary layer. The local Nusselt number decreases as
the radiation parameter increases. As the radiation parameter increases, radiative heat
transfer becomes more significant than convective heat transfer. As a result, convection
loses ground to other heat transfer methods, which lowers the local Nusselt number and
decreases convective heat transfer efficiency. The result is a thicker thermal boundary
layer due to the decrease in convective heat transmission. The temperature gradient
near the surface becomes less steep, which causes the thermal boundary layer to spread
farther from the surface since less heat is being carried away from the surface via con-
vection. The local Nusselt number is inversely correlated with Lewis number. This
is because thicker thermal boundary layers and less temperature gradients result from
lower thermal diffusivity compared to Brownian diffusivity at higher Lewis numbers.
Convective heat transfer efficiency is lowered as a result, and the local Nusselt number
is also lowered. Finally, the effect of the injection/suction parameter on the local Nus-
selt number is displayed. The table makes it clear that raising the suction/injection

30
parameter causes the local Nusselt number to rise. A thinner thermal boundary layer,
a greater velocity gradient close to the wall, and an increased convective heat transfer
coefficient are the results of increasing the suction/injection parameter. Convective
heat transfer is improved by these elements working together, raising the local Nusselt
number.
Lastly, table 1 shows how the Sherwood number varies for different values of the
related parameters. The Sherwood number rises as the magnetic parameter does as
well. Because of better mixing and thinner boundary layers, an increase in the mag-
netic parameter frequently results in a more efficient mass transfer process. As a result,
there is an increase in the mass transfer coefficient and the Sherwood number. The
Sherwood number increases along with the curvature parameter. Better mixing and
flow patterns, a thinner boundary layer, and an improved effective surface area for
mass transfer are all brought about by a higher curvature value. The mass transfer
coefficient increases and the Sherwood number rises when all of these components come
together. The Sherwood number rises as the parameter of Brownian motion increases.
An increase in the Brownian motion parameter improves fluid mixing and particle dis-
persion. The mass transfer coefficient increases when mass movement near the surface
is more effective. Consequently, a higher Sherwood number reflects increased effec-
tive mass transfer, thinning the concentration boundary layer. The Sherwood number
decreases as the thermophoresis parameter increases. Mass transport at the surface
can be less successful and the concentration boundary layer can be disrupted by in-
creasing the thermophoresis parameter. As a result, there is a reduction in the mass
transfer coefficient and Sherwood number. The Sherwood number rises in parallel
with the radiation parameter. Raising the radiation parameter leads to better mass
transport close to the surface, which also enhances the temperature distribution and
reduces boundary layer thickness. An raised Sherwood number is the outcome of this
improved mass transfer efficiency. Lewis number increases cause Sherwood number to
climb as well, which thins the boundary layer. If the fluid dynamics are favorable, then
a higher Lewis number might result in an increase in the effective mass transfer. An
higher Sherwood number can result from modifications to the concentration gradient
and thickness of the boundary layer, which can improve the mass transfer coefficient.
As the suction/injection parameter is increased, the Sherwood number decreases be-
cause suction can break up the concentration boundary layer and lessen the effective
concentration gradient close to the surface. A lower Sherwood number indicates a less
effective mass transfer mechanism as a result of this.

31
Table 2.1. Variation in skin friction coefficient, local Nusselt number and Sherwood
number for distinct data point of involved parameters
1/2
M γ Nb Nt Rd Le λ Rez Cf Rez −1/2 N u Rez −1/2 Sh
0.0 0.1 0.1 0.1 0.1 0.1 0.1 -1.0874 1.11237 -0.5748
0.5 - - - - - - -1.20887 1.08161 -0.553014
1.0 - - - - - - -1.50918 1.00854 -0.49762
1.5 - - - - - - -1.8999 0.92211 -0.428158
0.1 0.1 - - - - - -1.09258 1.11105 -0.57389
- 0.2 - - - - - -1.1292 1.13045 -0.460925
- 0.3 - - - - - -1.1659 1.15236 -0.362401
- 0.4 - - - - - -1.20248 1.17639 -0.273356
- 0.1 0.1 - - - - -1.09258 1.11105 0.170492
- - 0.2 - - - - - 1.07855 0.210089
- - 0.3 - - - - - 1.04639 0.221792
- - 0.4 - - - - - 1.01458 0.226377
- - 0.1 0.1 - - - - 1.11105 -0.57389
- - - 0.2 - - - - 1.13635 -1.41823
- - - 0.3 - - - - 1.16246 -2.309
- - - 0.4 - - - - 1.18937 -3.24835
- - - 0.1 1.0 - - - 1.39104 -0.194249
- - - - 2.0 - - - 1.62466 -0.0413631
- - - - 3.0 - - - 1.84783 0.032841
- - - - 4.0 - - - 2.07318 0.075773
- - - - 0.1 1.0 - - 1.04666 0.00787855
- - - - - 2.0 - - 1.00858 0.51814
- - - - - 3.0 - - 0.98826 0.906986
- - - - - 4.0 - - 0.975393 1.23451
- - - - - 0.1 -0.5 -0.824168 0.439966 0.0026952
- - - - - - -0.3 -0.904665 0.628322 -0.157064
- - - - - - 0.0 -1.04214 0.978426 -0.458768
- - - - - - 0.3 -1.20045 1.39653 -0.822523
- - - - - - 0.5 -1.31751 1.50448 -1.09194

2.3 Conclusion
The behavior of the Buongiorno nanofluid model over a stretching cylinder was an-
alyzed, considering the effects of heat radiation, suction/injection, and magnetohy-
drodynamics. The governing equations were numerically solved using Mathematica’s
NDSolve method, revealing intricate interactions between several non-dimensional pa-

32
rameters and their effects on concentration, temperature, and velocity profiles. These
are conclusions made from the gathered data.
• The velocity, temperature, and concentration profiles are greatly influenced by
the curvature parameter, which also generally lowers the skin friction coefficient and
increases heat and mass transmission.
• Due to enhanced mass transfer, the magnetic parameter increases the Sherwood
number while decreasing the Nusselt number and the velocity profile.
• Temperature and concentration profiles were also shown to be significantly influ-
enced by parameters like thermophoresis and Brownian motion, with Brownian motion
generally reducing the Nusselt and Sherwood numbers, while thermophoresis increases
the Nusselt number but decreases the Sherwood number.
• A rise in the temperature profile and an increase in the Sherwood number as a
result of improved radiative heat transfer and mass transfer efficiency were indicative
of radiation’s influence.
Overall, the study highlights the intricate interactions between these physical char-
acteristics and provides insightful information for improving mass and heat transfer
procedures in nanofluid applications that include stretching cylinders.

33
Chapter 3

Influences of magnetohydrodynamics,
suction/injection and thermal
radiation phenomena for flow of
Walters-B fluid across a stretchable
cylindrical pipe using OHAM

This chapter investigates the fluid behavior flowing across an elongating cylindrical
pipe, with the Walters-B fluid model capturing the fluid’s visco-elastic properties that
are crucial for describing complex flow dynamics. A constant magnetic field is applied
to study its impact on the flow. Thermal radiation is also considered to assess its
effect on the fluid’s temperature distribution. Moreover, the stretchable surface of the
cylinder that controls the flow characteristics is subjected to suction or injection. Upon
utilizing the optimal homotopy analysis method (OHAM), the governing equations are
solved, providing accurate and efficient solutions. The findings offer insights into the
interplay of viscoelasticity, curvature, MHD, suction/injection, and thermal radiation
on the flow and thermal properties of Walters-B fluid across a stretchable cylindrical
pipe, with potential applications in various industrial and engineering procedures.

3.1 Mathematical Formulation

The present problem contains Walters-B fluid’s flow across a horizontally stretching
cylinder with radius R, having prescribed (quadratic) surface temperature at the cylin-
der Tw , that exceeds the surrounding temperature just above the surface of the cylin-
der, as compared to ambient temperature T∞ . Mathematical modelling is executed in
a cylindrical coordinate system, where the z-axis runs along the stretching cylinder’s

34
axis, and the r-axis extends radially. The components of radial and axial velocity are u
and w respectively, as depicted in Figure 1. A constant magnetic field B = (B0 , 0, 0) is
applied across the radial direction, and induced magnetic field is disregarded because of
the low magnetic Reynolds number. Moreover, thermal radiation is also incorporated.
The extra stress tensor for visco-elastic (Walters-B) fluid, introduced by Beard and
Walters [21], is as follows:
 
dA1
τ = µA1 − k0 − (∇V)A1 − A1 (∇V) , T
(3.1)
dt

where µ is dynamic viscosity, V is the velocity vector, dtd is the material time derivative,
A1 is the tensor named as first Rivlin-Erickson, and k0 is the parameter for material
fluid. Using the boundary layer approximation and the aforementioned assumptions,
the equations that govern the flow are given as follows,

∂u u ∂w
+ + = 0, (3.2)
∂r r ∂z

!
∂ 2w k 0 3 ∂w ∂
 
∂w ∂w ∂w ∂w
u +w =ν + 2 + r
∂r ∂z r∂r ∂r ρ r ∂z ∂r ∂r
!
∂ 2w
 
∂w ∂ ∂w w ∂
+ r − r (3.3)
r∂r ∂z ∂r r ∂r ∂r∂z
!!
u ∂ ∂ 2w σwB02
− r 2 − ,
r ∂r ∂r ρ

! !
16σ ∗ T∞
3
 
∂T ∂T 1 ∂ ∂T 1 ∂ ∂T
u +w =α r + ∗ r , (3.4)
∂r ∂z r ∂r ∂r 3κ ρCp r ∂r ∂r
where the radial and axial velocity components are, respectively, u(r, z) and w(r, z), σ
is the fluid’s electrical conductivity, α = ρCk p is the thermal diffusivity of the fluid, σ ∗
is the Stefan-Boltzmann constant, ν is the kinematic viscosity, and κ∗ is the Rosseland
mean absorption coefficient. The associated constraints are:
 2
z
u = uw , w = ww , T = Tw = T∞ + T0 at r = R,
l (3.5)
w → 0, T → T∞ as r → ∞,
where uw = − l λ and ww = czl , which contains a positive constant c whose
p νc
dimension is [1/time] and λ relates to injection of mass when λ < 0 and suction of
mass when λ > 0.

35
 Figure 3.1: Base model and coordinate system

Now the equations 3.2 - 3.5 are transformed into dimensionless form using the
following similarity transformations.
r r  2
r 2 − R2 c νc z
η= , ψ= zRf (η), T = T∞ + T0 θ(η). (3.6)
2R νl l l
In the aforementioned transformations, the velocity components and the stream
function ψ are connected by the following equations
r
1 ∂ψ 1 νc 1 ∂ψ cz
u=− =− Rf (η), w = = f ′ (η), (3.7)
r ∂z r l r ∂r l
where the similarity variable is denoted by η, and the temperature field and dimen-
sionless velocity function are denoted by θ(η) and f (η) respectively. Using the above-
mentioned transformations, equation 3.2 is satisfied identically, and equations 3.3 and
3.4 will take the form as:

(1 + 2γη)f ′′′ + 2γf ′′ + f f ′′ − (f ′ )2 + 4Weγ(f ′ f ′′ + f f ′′′ )

(3.8)
+ (1 + 2γη)We(f f (iv) − 2f ′ f ′′′ + (f ′′ )2 ) − M 2 f ′ = 0,
4
(1 + Rd) (1 + 2γη)θ′′ + 2γθ′ + Pr f θ′ − 2f ′ θ = 0. (3.9)
   
3
The associated dimensionless constraints are,
f (0) = λ, f ′ (0) = 1, θ(0) = 1 at η = 0,
(3.10)
f ′ (∞) → 0, θ(∞) → 0 at η → ∞.

36
 q
where γ = 1
R
νl
c
denotes the curvature parameter, We = ck0
ρνl
is the Weissenberg
q
σB02 l 4σ ∗ T∞
3
number, M = ρc
is the Hartmann number, Rd = is the radiation parameter,
κ∗ k
and Pr = σ is the Prandtl number.
ν

For this investigation, the physical quantities such as skin friction coefficient Cf
(frictional drag force) and Nusselt number N u (wall heat transfer rate) are defined by

τrz |r=R zqs |r=R

Cf = , Nu = + qr |r=R , , (3.11)
ρUs 2 kT0 ( zl )2

where τrz symbolizes the shear stress and qs provides the wall heat flux. Utilizing trans-
formations 3.6, the local Nusselt number and skin friction coefficient’s dimensionless
forms are provided as follows:

 
′′ ′′ ′′′ 4 −1/2
1/2
Rez Cf = f (0)−W e[(3−4λγ)f (0)−λf (0)] and Rez N u = − 1+ Rd θ′ (0),
3
(3.12)
cz 2
where the dimensionless local Reynolds number is represented by Rez = νl .

3.2 Method For Solution

Comprehending Walters-B fluid behavior is critical for creating efficient systems in
sectors where precise fluid flow control is necessary. Mathematically, the governing
equation of motion for Walters-B fluid is of higher order compared to those for New-
tonian and other non-Newtonian fluids. To solve the modelled problem, we employed
the optimal homotopy analysis technique (OHAM), by utilizing the MATHEMATICA
program BVPh 2.0, and obtained a series solution. This technique provides various
benefits over numerical and approximation analytical approaches, such as perturbation
techniques and few others. It gives analytical answers defined in terms of recognized
functions, which promotes a better grasp of the behavior of system. Unlike numerical
approaches, which involve dividing up the domain and may cause mistakes and infor-
mation loss, OHAM bypasses this action. Additionally, the control of the convergence
rate and solution precision is made possible by the addition of a new parameter in
OHAM. The convergence and accuracy of the OHAM solution rely on the choice of ini-
tial assumptions and linear operators, which may often be tricky to pick effectively. In
certain circumstances, OHAM’s convergence properties may limit its ability to gener-
ate solutions to very small values of the embedded parameters. It has been shown that
the convergence of series solutions based on homotopy methods can be significantly
enhanced through the use of optimized auxiliary parameters. The following formulas

37
are used to express the unknown functions f and θ in the first phase of the procedure:
∞
X ∞
X
f (η) = fk (η), θ(η) = θk (η). (3.13)
k=0 k=0

Initial guesses for f and θ denoted by f0 and θ0 are defined below:

f0 (η) = λ + 1 − e−η , θ0 (η) = e−η . (3.14)

Moreover, the auxiliary linear operators are defined as:

d3 f df d2 θ
Lf ≡ − , Lθ ≡ − θ. (3.15)
dη 3 dη dη 2

The series solutions include a parameter ℏ that is crucial for the convergence of the
solutions. The total squared residual ET,k across the interval [a∗ , b∗ ] will be defined as
follows:
ET,k = Ef,k + Eθ,k , (3.16)

0.005
0.005
Rd = 0.01, We = 0.00, γ = 0.10, Rd = 0.01, We = 0.10, γ = 0.10,
M = 0.10, λ = 0.05, Pr = 3.00 M = 0.10, λ = 0.10, Pr = 3.00
0.001 0.001
ET ,k

ET ,k

5.×10-4 5.×10-4

1.×10-4
1.×10-4
5.×10-5
2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16
k k
Figure 3.2: Fluctuation in ET,k Figure 3.3: Fluctuation in ET,k
corresponds to order of corresponds to order of
approximations k when W e = 0.00 approximations k when W e = 0.10

38
 0.005 0.005
Rd = 0.01, We = 0.15, γ = 0.10, Rd = 0.01, We = 0.20, γ = 0.10,
M = 0.10, λ = 0.10, Pr = 3.00 M = 0.10, λ = 0.10, Pr = 3.00
0.001
0.001
ET ,k

ET ,k
5.×10-4
5.×10-4

1.×10-4
1.×10-4
5.×10-5
2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16
k k
Figure 3.4: Fluctuation in ET,k Figure 3.5: Fluctuation in ET,k
corresponds to order of corresponds to order of
approximations k when W e = 0.15 approximations k when W e = 0.20

where the average squared residuals are Ef,k and Eθ,k , as determined by Liao [30]:
Z b∗
1  2
Ef,k = ∗ Nf∗ [fk (η)] dη, (3.17)
b − a∗ a∗
Z b∗
1  2
Nθ∗ fk (η), θk (η) (3.18)

Eθ,k = ∗ dη,
b − a∗ a∗
where Nf∗ and Nθ∗ represent the corresponding non-linear differential operators.

3.2.1 Convergence and Error Analysis for OHAM Solution

The graphical outputs for several choices of controlling parameters are obtained using
the above-described analytical procedure. The total squared residual ET,k is plotted
in Figures (2)-(5) as a function of k, the order of OHAM iterations/approximations,
to confirm the accuracy of the analytical method. It is observed that, by varying
Weissenberg number W e, ET,k decreases as k increases. With certain precise selections
of other parameters(Rd = 0.01, γ = 0.10, M = 0.10, λ = 0.10, P r = 3.00), it may be
deduced that 16th-order iterations/approximations are enough to obtain fairly accurate
outputs for 0.0 < W e < 0.2.

39
3.3 Findings and Analysis
Analytical results are offered in order to improve comprehension of the physical prob-
lem using graphical illustrations. Figures 6-12 present analytical results, illustrating
the impact of numerous non-dimensional parameters, namely, Prandtl number (P r),
Weissenberg number (W e), curvature parameter (γ), suction/injection parameter (λ),
Hartmann number (M ), and Radiation parameter (Rd) on similarity, flow and tem-
perature of the fluid (as a baseline for the results, P r = 3.0 is fixed).

1.0

Solid (λ = 0.3)
Dashed (λ = 0.0)
0.5
Dotted (λ = -0.3)
f

We = 0.00, 0.05, 0.10

0.0

Rd = 0.1, γ = 0.1, M = 0.1, Pr = 3.0

0 2 4 6 8
η
Figure 3.6: Fluctuation in f (η) with various values of W e and λ

40
 1.0 1.0
Rd = 0.1, γ = 0.1, We = 0.1, M = 0.1, Pr = 3.0 Rd = 0.1, We = 0.1, M = 0.1, Pr = 3.0
0.8 0.8

Solid (λ = 0.3)
0.6 0.6 Dashed (λ = -0.3)
f′

f′
0.4 λ = -0.4, -0.2, 0.0, 0.2, 0.4 0.4 γ = 0.00, 0.05, 0.10

0.2 0.2

0.0 0.0
0 2 4 6 8 10 12 0 2 4 6 8 10 12
η η
Figure 3.7: Fluctuation in f ′ (η) with Figure 3.8: Fluctuation in f ′ (η) with
various values of both suction and various values of γ for both suction
injection and injection

1.0 1.0
Rd = 0.1, We = 0.1, γ = 0.1, Pr = 3.0 Rd = 0.1, γ = 0.1, M = 0.1, Pr = 3.0
0.8 0.8

Solid (λ = 0.3) Solid (λ = 0.3)

0.6 0.6
Dashed (λ = -0.3) Dashed (λ = -0.3)
f′

f′

0.4 0.4
M = 0.0, 0.4, 0.8 We = 0.0, 0.1, 0.2

0.2 0.2

0.0 0.0
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12
η η
Figure 3.9: Fluctuation in f ′ (η) with Figure 3.10: Fluctuation in f ′ (η) with
various values of M for both suction various values of W e for both suction
and injection and injection

41
 1.0 1.0
Rd = 0.1, M = 0.1, We = 0.1, γ = 0.1, Pr = 3.0 M = 0.1, We = 0.1, γ = 0.1, Pr = 3.0
0.8 0.8

Solid (λ = 0.3)
0.6 0.6
Dashed (λ = -0.3)

θ
θ

0.4 0.4
λ = -0.8, -0.4, 0.0, 0.4, 0.8 Rd = 0.0, 1.0, 2.0, 3.0

0.2 0.2

0.0 0.0
0 2 4 6 8 10 0 5 10 15 20
η η
Figure 3.11: Fluctuation in θ(η) with Figure 3.12: Fluctuation in θ(η) with
various values of both suction and various values of and Rd for both
injection suction and injection

3.3.1 Radial component of velocity f (η) vs associated parame-

ters
The radial component of velocity f (η) for different values of W e for three scenarios,
for suction, injection, and rigid surface (means no suction/injection) are shown in
Figure 6. It is evident from boundary conditions 3.10, that f (η), which begins from
the value of λ increases until it reaches its final predicted value f (∞) which is at a
considerable distance from the cylinder’s surface. Therefore, as λ increases, f (η) also
increases. Additionally, it is anticipated that with the increase in Weissenberg number,
f (η) decreases because a higher Weissenberg number indicates stronger elastic effects
in the fluid, which can disrupt the flow pattern and reduce f (η).

3.3.2 Axial component of velocity f ′ (η) vs associated parame-

ters
From the momentum equation 3.8, it is evident that the radiation parameter and
Prandtl number do not influence the velocity profiles, as they are uncoupled from the
energy equation 3.9. Figure 7 illustrates the velocity profiles for various values of the
suction/injection parameter λ. The figure demonstrates that, in all scenarios (suction,

42
injection, and rigid surface), the velocity as well as the momentum boundary layer
thickness diminishes. This is due to the fact that the fluid is either being pulled in or
pushed out more aggressively, compressing the flow near the surface. Figure 8 depicts
the impact of the curvature parameter γ on the axial component of velocity f ′ (η) for
both suction and injection. It is observed that an increase in the curvature of the
cylinder results in higher velocity profiles within the boundary layer. This effect oc-
curs because an increased curvature, characterized by a higher γ, lowers the cylinder’s
surface area, thereby accelerating the flow of fluid close to the surface. Because of
the increased flow rate, the surface velocity gradient becomes steeper, causing the mo-
mentum boundary layer to thin out. Figure 9 displays the fluctuation in the velocity
profile for various values of the magnetic parameter M for both suction and injec-
tion. The graph demonstrates that when the magnetic parameter M rises, f ′ (η) drops.
This phenomenon may be traced to the introduction of a transverse magnetic field to
an electrically conducting fluid, which causes a Lorentz force. This force serves as a
resistive force, therefore slowing the fluid’s velocity. This reduction in velocity leads
to a thicker momentum boundary layer because the fluid takes longer to reach the
free-stream velocity, causing the transition from the surface velocity to the free-stream
velocity to occur over a larger distance. Figure 10 elucidates the impact of the Weis-
senberg number W e on f ′ (η) for both suction and injection. It is noted that an increase
in the Weissenberg number W e leads to a drop in the f ′ (η). The Weissenberg number
measures the ratio of elastic to viscous forces. As the Weissenberg number climbs, the
elastic effects become more evident. In viscoelastic fluids, this greater elasticity leads
in stronger resistance to deformation and slower fluid flow toward the surface, which
appears as a flatter or lowered velocity profile. As a result, the momentum boundary
layer thickness increases. The increased elasticity slows down the velocity near the
surface, which makes the transition from the wall to the free-stream velocity occur
over a larger region, thus thickening the boundary layer.

3.3.3 Fluid’s temperature θ(η) vs associated parameters

Figure 11 indicates the fluctuation in the curves of temperature for distinct numer-
ics of the suction/injection parameter λ, for all three scenarios (suction, injection and
rigid surface). As the parameter rises, the temperature profile drops. This happens
because suction removes warmer fluid from the surface and replaces it with compar-
atively cooler fluid from the surroundings, while injection adds comparatively colder
fluid, boosting heat transfer and subsequently reducing the temperature at the surface.
Consequently, the thickness of the thermal boundary layer reduces. Since the temper-
ature gradient near the surface steepens due to the decreased temperature near the
surface, the thermal boundary layer, where temperature changes from the surface to
the free-stream temperature, becomes narrower. Figure 12 demonstrates the influence
of radiation parameter on the temperature profile. Temperature profile rises with an

43
increase in the radiation parameter. This may be ascribed to numerous things. Firstly,
a greater radiation parameter implies increased radiative heat transfer inside the fluid.
This extra mechanism of heat transmission increases the total energy intake, elevating
the fluid temperature. Secondly, an increased radiation parameter suggests that the
fluid absorbs more thermal energy from radiative sources, which raises its internal en-
ergy and temperature. Thirdly, more radiation may lead to a thicker thermal boundary
layer, which holds more heat, resulting in a higher temperature profile.

3.3.4 Physical quantities vs associated parameters

The fluctuation in skin friction coefficient and local Nusselt number for numerous values
of associated parameters are displayed numerically in Table 1. The fluid’s shear stress
on the cylinder’s surface is indicated by the skin friction coefficient. With the increase
in suction/injection parameter, the expression for skin friction coefficient decreases.
This is because the fluid in the boundary layer close to the surface is removed by
suction (λ > 0). Consequently, the boundary layer reduces, which lowers the surface
velocity gradient. Reduced shear stress results from a decreased velocity gradient,
which lowers the skin friction coefficient. It is also noticed that higher the value of
Weissenberg number, so is the expression of skin friction coefficient. The fluid’s elastic
to viscous force ratio is indicated by the Weissenberg number. Stronger elastic effects
are indicated by a higher Weissenberg number. These elastic effects typically result
in increased resistance to deformation, which manifests as higher shear stress at the
cylinder’s surface, thereby increasing the skin friction coefficient. With the increase
in curvature parameter, the expression for skin friction coefficient decreases. This is
because a larger curvature parameter indicates a more noticeable cylinder curvature.
The formation of the boundary layer is impacted by this sharp curvature since it thins
the layer. Skin friction coefficients drop when the boundary layer gets thinner because
it lessens the velocity gradient close to the surface. Because of the increasing curvature,
centripetal pressures act more strongly on the fluid, which directly causes the boundary
layer thickness to decrease. It is observed that by raising magnetic parameter M , the
expression for skin friction coefficient falls. The decrease in skin friction coefficient is
because the shear stress at the surface and the velocity gradient are reduced due to
the magnetic damping effect and the decreased fluid velocity. Results in this table also
shows that with the increase in radiation parameter Rd, there is no significant effect
on the expression of skin friction coefficient. This is due to the fact that the radiation
parameter typically affects the thermal boundary layer by altering the temperature
distribution but does not directly influence the velocity field or the shear stress at the
surface.
The convective heat transfer at the cylinder’s surface is measured by the local
Nusselt number. As the suction/injection parameter λ rises, so is the expression for
the local Nusselt number. This is because suction (λ > 0) makes the thermal boundary

44
layer thin, so it improves heat transfer. The temperature gradient at the surface is more
prominent when the thermal boundary layer is thinner. The local Nusselt number is
directly increased by a steeper temperature gradient since it denotes more effective heat
removal from the surface. It is evident from table that with the increase in Weissenberg
number, the expression for the local Nusselt number decreases. This is because of
the decrease in convective heat transfer efficiency and the thermal boundary layer’s
thickening, leading to a smaller temperature gradient at the boundary. Also, with
the increase in curvature parameter, the expression for the local Nusselt number also
increases. This is because of the thermal boundary layer’s compression and enhanced
convective heat transfer, which leads to a more pronounced temperature gradient at the
surface and enhanced heat transfer efficiency. By increasing the magnetic parameter
M , the expression for the local Nusselt number drops. The decrease in the local
Nusselt number is because of the thickening of the thermal boundary layer and reduced
efficiency of convective heat transfer, which lowers the temperature gradient at the
surface. Lastly, by increasing the radiation parameter Rd, the expression of the local
Nusselt number decreases. The decrease in the local Nusselt number is because of the
increased radiative heat transfer, which lowers the surface temperature gradient and
decreases convective heat transfer efficiency.

45
 Table 3.1. Fluctuation in skin friction coefficient and local Nusselt number with
distinct values of associated parameters

λ We γ M Rd −Rez 1/2 Cf Rez −1/2 N u

-0.2 0.1 0.1 0.1 0.1 0.72004 2.10824
-0.1 - - - - 0.75286 2.23023
0.0 - - - - 0.78661 2.35868
0.1 - - - - 0.82127 2.49361
0.2 - - - - 0.85675 2.63472
0.1 0.0 - - - 1.09237 2.51443
- 0.1 - - - 0.82127 2.49361
- 0.2 - - - 0.49838 2.46335
- 0.3 - - - 0.08807 2.41886
- 0.1 0.05 - - 0.79760 2.47667
- - 0.10 - - 0.82127 2.49361
- - 0.15 - - 0.84502 2.51121
- - 0.20 - - 0.86887 2.52889
- - 0.1 0.0 - 0.81751 2.49501
- - - 0.2 - 0.83239 2.48942
- - - 0.4 - 0.87505 2.47312
- - - 0.6 - 0.94085 2.44754
- - - 0.1 0.0 0.82126 2.67937
- - - - 1.0 - 1.62513
- - - - 2.0 - 1.23012
- - - - 3.0 - 1.01116

3.3.5 Validation of current analysis

Table 2 represents a comparison of numerical values of skin friction coefficient compared
to Pillai et al. [?], Nandeppanawar et al. [?] and Abid et al [26] for distinct values of
W e with γ = M = 0. This table shows that the computed solution is highly accurate
and convergent.

46
Table 3.2. Present results of skin friction coefficient compared to Pillai et al. [53],
Nandeppanawar et al. [54] and Abid et al [40] for different values of W e with γ = M =
0.

We Pillai et al. [53] Nandeppanawar et al. [54] Abid et al [40] Present results
0 1.0 1.0 1.0 1.0
0.0001 1.00005 1.00005 1.00005 1.00005
0.001 1.0050 1.0050 1.0050 1.0051
0.005 - 1.00251 1.00250 1.00253
0.01 1.00504 1.00504 1.00504 1.00503
0.03 - 1.01535 1.01534 1.01534
0.05 - 1.02598 1.02597 1.02596
0.1 1.05409 1.05409 1.05406 1.05407
0.2 1.11803 1.11803 1.11797 1.11797
0.3 1.19523 1.19523 1.19512 1.19513
0.4 1.29099 1.29099 1.29079 1.29080

3.4 Conclusion
The behavior of flow of Walters-B fluid over an elongating cylinder with the effects of
magnetohydrodynamics, suction/injection and thermal radiation has been analyzed.
The optimal homotopy analysis method (OHAM) is employed to tackle with the gov-
erning equations. The findings highlight the significant impact of the parameters on
the velocity, temperature profiles, and essential elements of flow such as the local Nus-
selt number and skin friction coefficient. The results based on the obtained data are
as follows:
• The similarity profile diminishes for distinct values of Weissenberg number for all
three scenarios (suction, injection and rigid surface).
• With the increase in suction/injection parameter, velocity and temperature profile
decreases.
• Velocity profile drops as magnetic parameter and Weissenberg number rise and it
rises with the rise of curvature parameter.
• The increase in radiation parameter tends to increase the temperature profile.
• The skin friction coefficient decreases with increased suction/injection param-
eter, magnetic parameter and curvature parameter, while it increases with a higher
Weissenberg number.
• The local Nusselt number increases with suction/injection and curvature but de-
creases with higher Weissenberg number, radiation parameter and magnetic parameter.

47
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