NEW RESULTS REGARDING THE MOTION OF NEWTONIAN AND NON-NEWTONIAN FLUIDS

DUE TO A SHEAR STRESS ON THE BOUNDARY

Name : Mehwish Rana

Year of Admission : 2009

Registration No. : …-GCU-PHD-SMS-09

Abdus Salam School of Mathematical Sciences

GC University Lahore, Pakistan

i

NEW RESULTS REGARDING THE MOTION OF NEWTONIAN AND NON-NEWTONIAN FLUIDS

DUE TO A SHEAR STRESS ON THE BOUNDARY

Submitted to

Abdus Salam School of Mathematical Sciences

GC University Lahore, Pakistan

in the partial fulfillment of the requirements for the award of degree of

Doctor of Philosophy

in

Mathematics

By

Name : Mehwish Rana

Year of Admission : 2009

Registration No. : 113-GCU-PHD-SMS-09

Abdus Salam School of Mathematical Sciences

GC University Lahore, Pakistan

ii

DECLARATION

I, Mehwish Rana Registration No. 113-GCU-PHD-SMS-09 student at Abdus

Salam School of Mathematical Sciences GC University in the subject of

Mathematics year of admission 2009, hereby declare that the matter printed in

this thesis titled

“NEW RESULTS REGARDING THE MOTION OF

NEWTONIAN AND NON-NEWTONIAN FLUIDS DUE TO A SHEAR

STRESS ON THE BOUNDARY”

is my own work and that

(i) I am not registered for the similar degree elsewhere contemporaneously.

(ii) No direct major work had already been done by me or anybody else on

this topic; I worked on, for the Ph. D. degree.

(iii) The work, I am submitting for the Ph. D. degree has not already been

submitted elsewhere and shall not in future be submitted by me for

obtaining similar degree from any other institution.

Dated: ------------------------- ------------------------------------

Signature

iii

RESEARCH COMPLETION CERTIFICATE

Certified that the research work contained in this thesis titled

“NEW RESULTS REGARDING THE MOTION OF NEWTONIAN AND

NON-NEWTONIAN FLUIDS DUE TO A SHEAR STRESS ON THE

BOUNDARY”

has been carried out and completed by Mehwish Rana Registration No.

113-GCU-PHD-SMS-09 under my supervision.

----------------------------- -------------------------------

Date Constatin Fetecau

Supervisor

Submitted Through

Prof. Dr. A. D. Raza Choudary --------------------------------

Director General Controller of Examination

Abdus Salam School of Mathematical Sciences GC University Lahore

GC University Lahore Pakistan.

Pakistan.

iv

To

My Beloved Parents, Husband

Brothers and Sisters

and

Professors

Whose support, encouragement

and fruitful prayer

made this

.....

Possible

Contents

Contents v

Abstract viii

Acknowledgements x

Preface 1

1 Preliminaries 6

1.1 Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Categorization of Fluids . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Fundamental Laws of Dynamics . . . . . . . . . . . . . . . . . . . . . 9

1.4 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5.1 Inverse Laplace Transform . . . . . . . . . . . . . . . . . . . . 13

1.6 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6.1 The Inverse Fourier Transform . . . . . . . . . . . . . . . . . . 13

1.6.2 Fourier Cosine and Sine Transforms . . . . . . . . . . . . . . . 13

2 Radiative and Porous Effects on Free Convection Flow near a Ver-

tical Plate that Applies a Shear Stress to the Fluid 15

2.1 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Dimensionless Analytical Solutions . . . . . . . . . . . . . . . . . . . 17

v

2.3 Limiting Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 Solution in the Absence of Thermal Radiation (Nr → 0) . . . 21

2.3.2 Solution in the Absence of Mechanical Effects . . . . . . . . . 21

2.3.3 Solution in the Absence of Porous Effects(Kp → 0) . . . . . . 21

2.4 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.1 Case f(t) = fH(t) . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.2 Case f(t) = fta (a > 0) . . . . . . . . . . . . . . . . . . . . . 23

2.4.3 Case f(t) = f sin(ωt) . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . 24

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 General Solutions for the Unsteady Flow of Second Grade Fluids

over an Infinite Plate that Applies Arbitrary Shear to the Fluid 31

3.1 Flow Between Side Walls Perpendicular to a Plate . . . . . . . . . . . 32

3.1.1 Case: f(t) = fta (a > 0) (the Plate Applies an Accelerated

Shear to the Fluid) . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.2 Flow due to an Oscillating Shear Stress . . . . . . . . . . . . 36

3.1.3 Case: f(t) = fH(t) (Flow due to a Plate that Applies a Con-

stant Shear to the Fluid) . . . . . . . . . . . . . . . . . . . . . 36

3.2 Limiting Case h → ∞ (Flow over an Infinite Plate) . . . . . . . . . . 38

3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 The Influence of Deborah Number on Some Couette Flows of a

Maxwell Fluid 42

4.1 Problem Formulation and Calculation of the Velocity Field . . . . . . 42

4.2 Calculation of the Shear Stress . . . . . . . . . . . . . . . . . . . . . . 46

4.3 Some Particular Cases of the Motion . . . . . . . . . . . . . . . . . . 47

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

vi

5 Exact Solution for Motion of an Oldroyd-B Fluid over an Infinite

Flat Plate that Applies an Oscillating Shear Stress to the Fluid 55

5.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.2 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . 56

5.3 Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.3.1 Calculation of the Velocity Field . . . . . . . . . . . . . . . . . 57

5.3.2 Calculation of Shear Stress . . . . . . . . . . . . . . . . . . . . 60

5.4 Particular Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.4.1 λr = 0 (Maxwell Fluid) . . . . . . . . . . . . . . . . . . . . . 63

5.4.2 λ → 0, λr → 0 (Newtonian Fluid) . . . . . . . . . . . . . . . 64

5.5 Numerical Results and Conclusions . . . . . . . . . . . . . . . . . . . 65

Bibliography 73

Appendix 81

vii

Abstract

The present PhD dissertation has completed research on results related to

flow behavior due to shear stress on the boundary of some Newtonian and non-

Newtonian fluids under different circumstances. Firstly, we have discussed some

concepts related to Newtonian and non-Newtonian fluids, constitutive equations,

equations of motion and integral transforms. Secondly, we have presented the exact

solutions of velocity, temperature and shear stress fields corresponding to some flows

of Newtonian, second grade, Maxwell and Oldroyd-B fluids.

We have established general solutions for the unsteady free convection flow of

an incompressible viscous fluid due to an infinite vertical plate that applies a shear

stress f(t) to the fluid, when thermal radiation and porous effects are considered.

These general solutions may generate a large class of exact solutions corresponding

to different motions with technical relevance. Some special cases are investigated

under the effects of pertinent parameters on the fluid motion.

Unsteady motion of second grade fluids induced by an infinite plate that applies

a time-dependent shear stress f(t) to the fluid is also studied. General solutions

may be reduced to new solutions of Newtonian fluids or they may be used to obtain

known solutions from the literature. Furthermore, in view of an important remark,

general solutions for the flow due to a moving plate may be developed.

We have also studied the Couette flows of a Maxwell fluid caused by the bottom

plate applying shear stress on the fluid. Exact expressions for velocity and shear

stress corresponding to the fluid motion are determined using the Laplace transform.

Two particular cases with constant shear stress on the bottom plate or sinusoidal

oscillations of the wall shear stress are further discussed. Some important charac-

teristics of fluid motion are highlighted through graphs.

The unsteady motion of an Oldroyd-B fluid over an infinite flat plate is studied

by means of the Laplace and Fourier transforms. After time t = 0, the plate applies

cosine/sine oscillating shear stress to the fluid. The solutions obtained are presented

viii

as a sum of steady-state and transient solutions, which may easily be reduced to

the similar solutions corresponding to Newtonian or Maxwell fluids. A central issue

namely, obtaining the time for which the steady-state is reached is address by means

of numerical calculations and graphical illustrations.

The influence of oscillations frequency or of material parameters on this time

corresponding to the steady-state is also analyzed. It is lower for cosine oscillations

in comparison to sine oscillations of the shear, decreases with respect to ω and λ

and increases with regard to λr.

ix

Acknowledgements

Firstly, I would like to express my deepest gratitude to Almighty Allah who

has bestowed his blessing upon me in order to strong enough to complete my present

thesis.

Secondly, I wish to convey my sincere gratitude to my thesis advisor Prof. Con-

stantin Fetecau for his invaluable support, permanent encouragement and guidance.

I am deeply grateful to Prof. Vieru Dumitru for his continuous help and support

throughout my research period. I would like to extend my gratitude to the entire

ASSMS faculty, especially to those Professors whom I have learnt from. I acknowl-

edge Director General of ASSMS, Prof. Dr. A. D. Raza Choudary for providing

us an opportunity to learn. Furtheron, I would also thank the administrative staff

of Abdus Salam School of Mathematical Sciences specially to the Assistant Admin

Officer Mr. Awais Naaeem for his help and support.

Lastly, I could not overlook and constant support provided by my beloved hus-

band Captain Hafiz Adnan Rana, my beloved parents, my brothers and sisters, my

friend Faira Kanwal Janjua, my class fellows and my research fellows, whose encour-

agements proved vital in my professional development.

Lahore, Pakistan Mehwish Rana

December, 2013

x

Preface

Fluid dynamics is defined as the study of physics of continuous materials,

which take the shape of their containers. Fluid comes from French meaning “that

which flows” and mechanics corresponds to “applied mathematics.” The history be-

hind fluid mechanics is quite extensive, running back to ancient civilizations. The

little knowledge of fluid dynamics understood at the time was used to solve flow

problems for irrigation. It was used in the development of oars for powering ships

as well. There is a long list of people who contributed to the aforementioned field,

but some stand out more than others. The first person to provide humanity with

information on fluid dynamics was Archimedes of Syracuse. He discovered the law

of buoyancy which was named The Archimedes Principle. This formula states that

the buoyant force is equal to the weight of the displaced liquid. This became quite

important as it soon introduced the concept of density. As a next step, density was

then able to explain the reason why objects such as ships float on the water surface.

Information regarding fluid dynamics only slowly improved until the Renaissance

and Leonardo da Vinci. He was able to derive the equation of conservation of mass

for one-dimensional steady flow. His interest in fluid dynamics also led him to design

a submarine.

The next person to delve into the research of the fluid dynamics was Sir Isaac

Newton. He introduced the laws of motion, which affect not only solids but fluids

as well. However his main contribution to fluid dynamics was the law of viscosity

for linear fluids. It states that the viscosity of a fluid is proportional to the velocity

that the parts of the fluid separate. This law only applied to substances known as

Newtonian fluids, which are defined as fluids that continue to flow despite the forces

that are acting on them.

Newton’s discovery sparked the rise of many followers to follow his line of re-

search. One person to use this information was Leonhard Euler. He was able to

derive equations that related the velocity and the pressure of a liquid to its den-

1

sity. These equations proved quite complicated, yet they were later simplified by

the Navier-Stokes equations and by Ludwig Prandtl’s writings. Euler’s close friend,

Daniel Bernoulli, contributed to the field of fluid mechanics, by inventing the equa-

tion for incompressible flow. This equation is part of what later became to be known

as Bernoulli’s Principle and it was based on the law of Conservation of Energy: the

velocity of a fluid increases as the pressure decreases.

Osborne Reynolds was another important person in the history of fluid dynamics.

He was closely involved in the investigation of forces that act on fluids. His research

consisted of various pipe-flow experiments which ultimately allowed him to formu-

late his famous equation. This equation compares two forces on a fluid, namely the

inertial and viscous forces. The end result is a ratio of the two forces that ends in a

dimensionless number. If this number is greater than 4000 then the fluid is defined

as turbulent, which means that it flows very chaotically. If the number is less than

2100 then the fluid is laminar (fluids flow in parallel layers). Fluids have certain

well established characteristics: pressure, velocity, density, viscosity and body force.

The knowledge of these characteristics has greatly helped us in being able to define

a fluid in its environment.

Radiative convective flows of an incompressible viscous fluid past a vertical plate

have applications in many industrial processes [10]. Radiative heat transfer plays

an important role in manufacturing industries, filtration processes, drying of porous

materials in the textile industry, solar energy collectors, satellites and space vehicles,

etc. Unsteady convective radiative flows have important applications in geophysics,

geothermics, chemical and ceramics processing. Many studies analyzing effects of

thermal radiation in convection flows through porous media have recently appeared.

A short presentation of the main results up to 2007 is given by Ghosh and Beg

[32] who studied the convective radiative heat transfer past a hot vertical surface in

porous media.

Over the last years, problems of free convection and heat transfer through porous

media have attracted the attention of many researchers. Flows through porous me-

2

dia have numerous engineering and geophysical applications in chemical engineering

for filtration and purification processes, agriculture engineering to study the under-

ground water resources and petroleum technology, etc. Some of the most recent

and interesting results in the present field are the works of Chaudary et al. [14],

Narahary [45], Rajesh [52], Toki [69], Chandrakala and Bhaskar [12], Narahary and

Ishak [46], Samiulhaq et al. [68] and therein references. However, it is noteworthy

to highlight the fact that all these papers have a common specific feature: they solve

problems in which the velocity is given on the boundary.

Generally speaking, there are three types of boundary value problems in fluid

mechanics: a) velocity is given on the boundary; b) shear stress is given on the

boundary; c) mixed boundary value problems. From a theoretical and from a prac-

tical point of view, all three types of boundary conditions are equally important due

to fact that in some problems the most important detail is the specific force applied

on the boundary. It is also important to bear in mind that the ’no slip’ boundary

condition may not be necessarily applicable to flows of polymeric fluids that may slip

or slide on the boundary. Thus, the shear stress boundary condition is particularly

meaningful. To the best of our knowledge, the first exact solutions for motions of

non-Newtonian fluids in which the shear stress is given on the boundary are those of

Waters and King [73] and Bandelli et al. [8]. Recently, numerous similar solutions

have been established by different authors [29, 75]. However, neither of these papers

considers the radiative or the porous effects. The purpose of our work in Chapter 2

is to provide exact solutions for the unsteady free convection flow of an incompress-

ible viscous fluid over an infinite vertical plate that applies a time-dependent shear

stress f(t) to the fluid. The obtained results have been published in Z. Naturforsch

68a (2013) 130-138 (see the reference [27]).

The flow of a second grade fluid over an infinite plate, with suitable boundary

and initial conditions, has been investigated by many authors. It can be realized

if the plate is moving in its plane or applies a tangential shear stress to the fluid.

In the second case, unlike the usual no slip condition, a boundary condition on the

3

shear stress is used. This is very important as in some problems, what is specified is

the force applied on the boundary. Meanwhile, exact solutions for different motions

of viscous and second grade fluids have been established [18, 31, 34, 53]. Chapter

3 provides general solutions for the unsteady motion of a second grade fluid induced

by an infinite plate that applies a shear stress f(t) to the fluid. The results obtained

in this chapter have been published in Z. Naturforsch 66a (2011) 753-759 (see the

reference [29]).

The Maxwell fluid model has become a subject of study for many researchers be-

cause of its simple and convenient approach with regards to determining analytical

solutions for various fluid motion problems. In time, the motion of Maxwell fluids

[7, 23, 36, 54, 55, 56, 58, 59] has been studied in various circumstances e.g. the fluid

owing its motion to the motion of boundary, the application of a body force, the

imposition of pressure gradient or the application of tangential shear. The first exact

solution of Rayleigh Stokes’ problem for Maxwell fluids seems to be that given by

Tanner [70]. Some other interesting solutions of Stokes’ first and second problems

corresponding to non-Newtonian fluids have been determined in [40, 41], [48, 50].

Christov [9] has proposed convincing results corresponding to Stokes’ first problem

for Oldroyd-B fluids.

The flow of a fluid is called Couette flow if the fluid is bounded by two parallel

walls such that they are in relative motion [38, 39]. The flow between two parallel

plates such that one plate is at rest and the other one is moving in its plane with

a constant speed, is called the simple Couette flow. The flow between two plates

produced by a constant pressure gradient in the direction of the flow is termed as

Poiseuille flow. The generalized Couette flow is a superposition of the simple Cou-

ette flow over Poiseuille flow [62]. Some practical applications of this type of flows

have been presented in the reference [21]. Recently, considerable amount of research

has been completed regarding Couette flow problem. Siddiqui et al. [67] considered

the problem of steady plane Couette flows between two parallel plates sliding with

respect to each other. Asghar et al. [5] studied the behavior of unsteady Couette

4

flow for second grade fluids. Jha [37] analyzed natural convection effects on fluid in

an unsteady MHD Couette flow. Marques et al. [44] brought into light the effects

of fluid slip at the boundary for Couette flow under steady state conditions. Khalid

and Vafai [42] have studied the effect of slip condition on Couette flows due to an

oscillating wall. Denn and Porteous [17] have presented interesting results regarding

unsteady Couette flow of a Maxwell fluid between two infinite parallel plates while

similar solution corresponding to second grade fluids was established by Jordan [38].

Some interesting results regarding Couette or Stokes’ flows of non-Newtonian fluids

can be found in the references [16], [33] and [35]. In the Chapter 4, we have dealt

with Couette flows of a Maxwell fluid caused by the bottom plate that applies a

shear rate ∂u(0,t)∂y

= τ0µf(t). The results corresponding to this chapter have been pub-

lished in International Journal of Mechanics March 17(2013) 1-12 (see the reference

[64]).

Over the past few decades, the unsteady flows of viscoelastic fluids caused by the

oscillations of the boundary have become of considerable interest. Rajagopal [51]

found steady-state solutions for some oscillating motions of second grade fluids and

Erdogan [20] provided two starting solutions for the motion of a viscous fluid due

to cosine and sine oscillations of a flat plate. Flows of the fluids due to oscillating

boundary for different constitutive models may be found in [26, 49, 72, 76]. Recently,

some solutions have been obtained for the motion of Newtonian fluids induced by

an infinite plate that applies oscillating shear stresses to the fluid in [30]. The aim

of this Chapter 5 is to determine starting solutions for the unsteady motion of

an incompressible Oldroyd-B fluid due to an infinite plate that applies an oscillat-

ing shear to the fluid. The results regarding this chapter have been published in

Boundary Value Problems, (2012) 2012:48 [66].

5

Chapter 1

Preliminaries

1.1 Fluids

A fluid cannot oppose shear stress by a static deflection and it moves and deforms

continuously as long as the shear stress is applied. Fluid mechanics is the study of

fluids either in motion (fluid dynamics) or at rest (fluid statics). Both, liquids and

gases are classified as fluids.

Since the scope of fluid dynamics is vast and has several applications in engi-

neering and human activities. Examples are medical studies of breathing and blood

flow, oceanography, hydrology, energy generation. Other engineering applications

include: fans, turbines, pumps, missiles and airplanes to name a few. Viscosity is

an internal property of a fluid that offers resistance to flow. Viscosity increases the

difficulty of the basic equations. It also has a destabilizing effect and gives rise to

disorderly, random phenomena called turbulence. The phenomenon of response to

normal stresses (pressure) acting on fluid elements further subdivides the fluids into

compressible and incompressible fluids.

When a fluid element responses to varying changes in pressure by adjusting its

volume and consequently its density, the fluid is regarded as ’compressible’. When

no volume or density changes occur with pressure or temperature, the fluid is consid-

ered as incompressible. Liquids having very small expansion coefficients as compared

to gases which have much larger expansion coefficients, fulfill the condition of in-

6

compressibility with precision.

There are two classes of fluids:

Liquids: are composed of relatively closepacked molecules with strong cohesive

forces. Liquids have constant volume (almost incompressible) and will form a free

surface in a gravitational field.

Gases: molecules are widely spaced with negligible cohesive forces. A gas is free to

expand until it encounters confining walls. A gas has no definite volume and it forms

an atmosphere when it is not confined. Gravitational effects are rarely concerned.

Liquids and gases can coexist in two-phase mixtures such as steam-water mixtures.

Eulerian and Lagrangian Point of View:

There are two different points of view in analyzing problems in mechanics.

In the Eulerian point of view, the dynamic behavior of the fluid is studied from a

fixed point in space. Therefore, fluid properties and parameters are computed as

filed functions, e.g. p(x, y, z, t). Most measurement devices work based on Eulerian

method.

The system concept represents a Lagrangian point of view where the dynamic

behavior of a fluid particle is considered. To stimulate a Lagrangian measurement,

the probe would have to move downstream at the fluid particle speed.

Fluid velocity field:

Velocity: the rate of change of fluid position at a point in a flow field. Velocity in

general is a vector function of position and time, thus it has three components u, v

and w each a scalar field in itself:

V(x, y, z, t) = u(x, y, z, t)i + v(x, y, z, t)j + w(x, y, z, t)k,

where i, j and k are unit vectors.

Velocity is used to specify flow filed characteristics, flow rate, momentum, and vis-

cous effects for a fluid in motion. Furthermore, velocity field must be known to solve

heat and mass transfer problems.

Thermodynamic properties of a fluid:

Thermodynamic properties describe the state of a system. Any characteristic of

7

a system is called a property. In this work, the fluid is assumed to be a continuum,

homogenous matter with no microscopic holes. This assumption holds as long as

the volumes, and length scales are large with respect to the intermolecular spacing.

System is defined as a collection of matter of fixed identity that interacts with

its surroundings. For a single-phase substance such as water or oxygen, two basic

(independent) properties such as pressure and temperature can identify the state of

a system; and thus the value of all other properties.

Temperature:

Temperature is a measure of the internal energy, it is also a pointer for the

direction of energy transfer as heat.

1.2 Categorization of Fluids

Generally, the fluids are categorized into two groups

a) Newtonian fluids

b) Non-Newtonian fluids

a) Newtonian fluids:

Fluids in which the shear stress is directly proportional to the rate of defor-

mation are called Newtonian fluids. Mathematically, for one-dimensional Newtonian

flow, this fact is presented as [25]

τyx = µdu

dy, (1.2.1)

where τ is the shear stress, u is the velocity of fluid and µ is the dynamic viscosity.

Water, air, gasoline, etc. are few examples of Newtonian fluids.

b) Non-Newtonian fluids:

Fluids in which shear stress is not linearly related to the rate of deformation

are considered to be non-Newtonian fluids. These fluids have a threshold or yield

stress below which they behave like solids. Polymer solutions, mud flows, sludge,

blood, paints, facial products, oils, edibles, etc. are examples of non-Newtonian

8

fluids [6]. Non-Newtonian fluids exhibit time-dependent or time-independent be-

havior. For example, the time-independent one-dimensional non-Newtonian fluid

can be described in the context of Newtons’s law of viscosity [25] i.e.

τyx = k

(du

dy

)n

, (1.2.2)

where ”k” is consistency index and ”n” represents flow behavior index. The non-

Newtonian behavior of fluids is also described in reference to ’apparent viscosity’

which further helps to classify the fluids into shear thickening and shear thinning

fluids. Most of the non-Newtonian fluids are shear thinning fluids in which apparent

viscosity decreases with increasing deformation rate. Polymer solutions, colloidal

suspensions and paper pulp, etc. are few examples of shear thinning fluids.

1.3 Fundamental Laws of Dynamics

The motion of a fluid is governed by two fundamental laws of dynamics

a) Mass Conservation

b) Newton’s second law of motion

a) Mass conservation (Continuity equation):

To agree with mass conservation in fluids flow, a ’closed fluid system’ can

always be found i.e a system whose total mass M is constant. This fact is apparent

in the case of fluid mass which is stored in a container. For all other fluids flows,

’control volumes’ approach is taken into account within which the system’s total

mass can be stated as constant. Mathematical description of mass conservation (in

fluid flow) leads to an equation known as continuity equation i.e.

∂ρ

∂t+ ∇(ρv) = 0, (compressiblefluids)

where ”ρ” is the density of fluid and v is the velocity vector.

Also,

∇.v = 0. (incompressiblefluids)

9

b) Newton’s Second Law (Equation of motion):

Newton’s second law, in the context of fluids flows, is stated as:

The time derivation of momentum in a particular direction is equal to the sum of

the external forces acting in that direction on the fluid element, plus the temporal

change of momentum caused by molecular movement input. The external forces

comprise mass forces caused by gravitational forces and electromagnetic forces as

well as surface forces caused by pressure.

Fluid elements behave like rigid bodies as they don’t intend to change their state

of motion (momentum) unless mass or surface forces act on them and molecular-

dependent momentum input is present. Mass forces acting on fluid elements can be

expressed in terms of acceleration and, in the context of the definition of fluids, the

surface forces are those imposed by the molecular pressure. Mathematical descrip-

tion of Newton’s second law leads to the equation of motion (momentum equation).

The differential form of the equation of motion is

divT + ρb = ρa,

where T is the Cauchy stress tensor. b and a represent the body force and acceler-

ation, respectively.

1.4 Constitutive Equations

Complete description of fluid flow problem requires constitutive equation along

with the continuity equation and the equation of motion. Constitutive equation

corresponding to various fluids represents the relation between the shear stress and

the rate of deformation.

10

a) Newtonian fluids model:

Constitutive equation for Newtonian fluids is mathematically expressed as

T = −pI + µA, (1.4.1)

where I is spherical unit tensor, A = L+LT is the first Rivlin-Ericksen tensor, L is

velocity gradient, p is the hydrostatic pressure, µ is the dynamic viscosity and the

superposed ’T’ represents the transpose.

For Newtonian fluids model, owing to the continuity equation, the equation of

motion and the constitutive equation, are expressed in the form of Navier-Stokes

equations. These equations in Cartesian coordinates directions are expressed as

ρ

(∂ui

∂t+

3∑j=1

uj∂ui

∂xj

)= ρgi −

∂p

∂xi

+ µ3∑

j=1

∂2ui

∂xj2 ; j = 1, 2, 3. (1.4.2)

b) Non-Newtonian fluids models:

The non-Newtonian fluids have been classified as differential, rate and integral

type fluids. To describe the behavior of non-Newtonian fluids, various models have

been introduced in the course of history considering different characteristics of these

fluids. Among them the mostly used models are as following:

Second grade fluids model:

The second grade fluids are a subclass of non-Newtonian fluids and, these

fluids are considered to form the simplest subclass of differential type fluids. This

model is preferred due to its relatively simple structure. The constitutive equation

corresponding to second grade incompressible fluids is expressed as

T = −pI + S = −pI + µA1 + α1A2 + α2A12, (1.4.3)

where α1 and α2 are material constants, A1 = A and A2 is the second Rivlin-

Ericksen tensor.

In the construction of second grade fluids flow problems, it is usually assumed [18, 19]

µ ≥ 0, α1 ≥ 0, α1 + α2 = 0.

11

Maxwell fluids model:

Some characteristics of non-Newtonian fluids are better described by rate

type fluid models. Maxwell fluid model is a rate type fluid model. The constitutive

equations corresponding to the Maxwell fluids are defined as

T = −pI + S , S + λ(S − LS − SLT ) = µA, (1.4.4)

where the superposed dot denotes the material time derivative.

Oldroyd-B fluids model:

Other characteristics of non-Newtonian fluids like stress-relaxation, creep,

normal stress differences occurring during the motion of the fluid, etc. are better

described by the Oldroyd-B fluids model. This model has turned out to be the

most successful model to analyze the response of a subclass of polymeric liquids.

Apparently, an Oldroyd-B fluid arises from the mixture of two viscous fluids. This

model is also considered as a rate type fluid model.

The constitutive equations corresponding to Oldroyd-B fluids are given by

T = −pI + S , S + λ(S − LS − SLT ) = µ[A + λr(A − LA − ALT )], (1.4.5)

where λ and λr are relaxation and retardation times, respectively.

1.5 Laplace Transform

If f(t) is defined for all values of t > 0, then the Laplace transform of f(t) is denoted

by F (s) or £[f(t)](s) and is defined by the integral

F (s) = £[f(t)](s) =

∫ ∞

0

e−stf(t)dt, (s ∈ C) (1.5.1)

where s is a positive real number or a complex number with a positive real part so

that the integral is convergent.

12

1.5.1 Inverse Laplace Transform

The inverse Laplace transform which is denoted by £−1[F (s)] = f(t) is defined as

£−1[F (s)](t) =1

2πi

∫ γ+i∞

γ−i∞estF (s)ds, (1.5.2)

where γ > 0.

1.6 Fourier Transform

If u(x, t) is a continuous, piecewise smooth, and absolutely integrable function, then

the Fourier transform of u(x, t) with respect to x ∈ R is denoted by U(k, t) and is

defined as [15]

Fu(x, t) = U(k, t) =1√2π

∫ +∞

−∞eikxu(x, t)dx, (1.6.1)

where k is called the Fourier transform variable and e−ikx is called the kernel of the

transform.

1.6.1 The Inverse Fourier Transform

For all x ∈ R, the inverse Fourier transform of U(k, t) is defined by [15]

F−1U(k, t) = u(x, t) =1√2π

∫ +∞

−∞e−iqxU(k, t)dk, (1.6.2)

where F−1 is called inverse Fourier transform operator.

1.6.2 Fourier Cosine and Sine Transforms

The Fourier cosine and the Fourier sine transforms of a function f(x), x ∈ (0,∞)

are defined as [15]

Fcf(x) = fc(k) =

√2

π

∫ ∞

0

cos(kx)f(x)dx, (1.6.3)

13

Fsf(x) = fs(k) =

√2

π

∫ ∞

0

sin(kx)f(x)dx, (1.6.4)

respectively. Also, the corresponding inverse Fourier cosine and sine transforms are

given by the formulas [15]

F−1c fc(k) = f(x) =

√2

π

∫ ∞

0

cos(kx)fc(k)dk, (1.6.5)

F−1s fs(k) = f(x) =

√2

π

∫ ∞

0

sin(kx)fs(k)dk, (1.6.6)

respectively.

14

Chapter 2

Radiative and Porous Effects on

Free Convection Flow near a

Vertical Plate that Applies a

Shear Stress to the Fluid

The idea behind this chapter is to provide exact solutions for the unsteady

free convection flow of an incompressible viscous fluid over an infinite vertical plate

that applies a time-dependent shear stress f(t) to the fluid. The viscous dissipa-

tion is neglected therefore, radiative and porous effects are taken into consideration.

General solutions that have been obtained satisfy all given initial and boundary

conditions and are not common in the literature. They generate a large class of

exact solutions for different motion problems that are similar to fluid motions in

which velocity is given on the boundary. To illustrate their theoretical and practical

importance, three special cases are considered and the effects of appropriate param-

eters on the dimensionless velocity and temperature are graphically underlined.

15

2.1 Mathematical Formulation

Lets consider the unsteady flow of an incompressible viscous radiating fluid over

an infinite hot vertical plate embedded in a porous medium. The x-axis of the

Cartesian coordinate system is taken along the plate in the vertical direction and

the y-axis is normal to the plate. Initially, the plate and the fluid are at the same

temperature T∞ in a stationary condition. After time t = 0+, we applies a time

dependent shear stress f(t) to the fluid along the x-axis. In the same time the

temperature of the plate is raised to Tw. The radiative heat flux is considered to

be negligible in the x-direction in comparison with the y-direction. The fluid is

gray absorbing-emitting radiation but no scattering medium. Assuming that the

viscous dissipation is negligible and using the usual Boussinesq’s approximation,

the unsteady flow is governed by the following equations [32]

∂u(y, t)

∂t= ν

∂2u(y, t)

∂y2+ gβ[T (y, t) − T∞] − ν

Ku(y, t); y, t > 0, (2.1.1)

ρCp∂T (y, t)

∂t= k

∂2T (y, t)

∂y2− ∂qr(y, t)

∂y; y, t > 0, (2.1.2)

where u,T ,ν,g,β, ρ, K, Cp,k and qr are the velocity of the fluid, its temperature,

the kinematic viscosity of the fluid, the gravitational acceleration, the coefficient of

thermal expansion, the constant density of the fluid, the permeability of the porous

medium, the specific heat at constant pressure, the thermal conductivity of the fluid

and the radiative heat flux respectively.

Assuming that no slip appears between the plate and fluid, the suitable initial

and boundary conditions are:

T (y, 0) = T∞, u(y, 0) = 0 for y ≥ 0,

T (0, t) = Tw, ∂u(y,t)∂y

|y=0=f(t)µ

for t > 0,

T (y, t) → T∞, u(y, t) → 0 as y → ∞,

(2.1.3)

where µ = ρν is coefficient of viscosity and function f(t) satisfies the condition

f(0) = 0.

16

In the following we adopt the Rosseland approximation for the radiative flux qr [32],

[46], [63],[65] and [68], namely

qr = − 4σ

3kR

∂T 4

∂y, (2.1.4)

where kR is the mean spectral absorption coefficient or the Rosseland mean atten-

uation coefficient and σ is the Stefan-Boltzmann constant[47]. Assuming that the

temperature difference between the fluid temperature T and the free stream tem-

perature T∞ is sufficiently small, now expands T 4 in a Taylor series about T∞ and

ignore higher order terms, we find that

T 4 ≈ 4T 3∞T − 3T 4

∞. (2.1.5)

It is worth pointing out that Eq. (2.1.5) is widely used in computational fluid

dynamics involving absorption problems [11]. Introducing Eq. (2.1.5) into Eq.

(2.1.4) and by using the result in the governing equation (2.1.2), we find that

Pr∂T (y, t)

∂t= ν(1 + Nr)

∂2T (y, t)

∂y2; y, t > 0, (2.1.6)

where the Prandtl number and the radiation-conduction parameter are defined by

[32],

Pr =µCp

krespectively Nr =

16σT 3∞

3kkR

. (2.1.7)

Furthermore, the dimensionless solutions of coupled partial differential equations

(2.1.1) and (2.1.6) with the initial and boundary conditions (2.1.3) will be computed

by means of Laplace transforms.

2.2 Dimensionless Analytical Solutions

In order to obtain non-dimensional forms of governing equations (2.1.1) and

(2.1.6) and to reduce the number of essential parameters, let us introduce the fol-

17

lowing dimensionless quantities

u∗ = uU, T ∗ = T−T∞

Tw−T∞, y∗ = U

νy,

t∗ = U2

νt, f ∗(t∗) = 1

ρU2 f

(ν

U2 t∗)

and Kp = ν2

U21K

,(2.2.1)

where Kp is the inverse permeability parameter for the porous medium. In order to

reduce essential parameters, we choose reference velocity U = 3√

gβν(Tw − T∞).

Indroducing Eqs. (2.2.1) into Eqs. (2.1.1), (2.1.6) and dropping out the star nota-

tion, we find non-dimensional governing equations in the suitable forms

∂u(y, t)

∂t=

∂2u(y, t)

∂y2+ T (y, t) − Kpu(y, t); y, t > 0, (2.2.2)

Preff∂T (y, t)

∂t=

∂2T (y, t)

∂y2; y, t > 0, (2.2.3)

where Preff = Pr1+Nr

is the effective Prandtl number [[43], Eq. (10)]. The corre-

sponding boundary conditions are

T (y, 0) = 0, u(y, 0) = 0 for y ≥ 0,

T (0, t) = 1, ∂u(y,t)∂y

|y=0= f(t) for t > 0,

T (y, t) → 0, u(y, t) → 0 as y → ∞,

(2.2.4)

The dimensionless temperature and the surface heat transfer rate, as it results

from [[32], Eqs. (13) and (15)] are given by

T (y, t) = erfc

(y

2

√Preff

t

),∂T (y, t)

∂y|y=0= −

√Preff

πt, (2.2.5)

where erfc(.) is complementary error function of Guass.

By using the Laplace transform to Eq. (2.2.2) and bearing in mind the corre-

sponding initial and boundary conditions for u(y, t), we find that

∂2u(y, q)

∂y2− (q + Kp)u(y, q) = −1

qexp(−y

√Preffq), (2.2.6)

where the Laplace transform u(y, q) of u(y, t) has to satisfy the following conditions

∂u(y, q)

∂y|y=0= F (q); u(y, q) → 0 as y → ∞. (2.2.7)

18

Here, F (q) is the Laplace transform of f(t). The solution of Eq. (2.2.6) with the

conditions (2.2.7) is given by

u(y, q) =exp(−y

√Preffq)

q[q(1 − Preff ) + Kp]−

√Preffqexp(−y

√q + Kp)

q[q(1 − Preff ) + Kp]√

q + Kp

(2.2.8)

− F (q)exp(−y

√q + Kp)√

q + Kp

.

In order to obtain the (y, t)-domain solution for velocity, we rewrite u(y, q) in

the equivalent but suitable form

u(y, q) =1

Kp

[exp(−y√

Preffq)

q−

exp(−y√

Preffq)

q − b

](2.2.9)

+b√

Preff

Kp

1√

q(q − b)

exp(−y√

q + Kp)√q + Kp

− F (q)exp(−y

√q + Kp)√

q + Kp

,

where b = Kp/(Preff − 1) if Preff 6= 1. Applying the inverse Laplace transform to

Eq. (2.2.9), the velocity u(y, t) can be written as a sum, namely

u(y, t) = ut(y, t) + um(y, t) for Preff 6= 1 and Kp 6= 0, (2.2.10)

where

ut(y, t) =1

Kp

erfc

(y

2

√Preff

t

)(2.2.11)

− ebt

2Kp

[exp(−y

√bPreff )erfc

(y√

Preff

2√

t−√

bt

)

+ exp(y√

bPreff )erfc

(y√

Preff

2√

t+√

bt

)]

+b√

Preff

2Kp

√π(b + Kp)

∫ t

0

ebt

√t − s

[exp(−y

√b + Kp)

× erfc( y

2√

s−

√(b + Kp)s

)− exp(−y

√b + Kp)erfc

( y

2√

s+

√(b + Kp)s

)]ds,

19

corresponds to the thermal effects and

um(y, t) = − 1√π

∫ t

0

f(t − s)√s

exp

(− y2

4s− Kps

)ds, (2.2.12)

becomes zero if the shear stress on the boundary vanishes.

By straightforward computations show that T (y, t) and u(y, t) given by Eqs.

(2.2.5) and (2.2.10) satisfy all enforced initial and boundary conditions. In order to

show that u(y, t) satisfies the boundary condition (2.2.4)3, for instance, let us firstly

observe that direct computations imply

∂ut(y, t)

∂y|y=0= 0, and

∂um(y, t)

∂y=

y

2√

π

∫ t

0

f(t − s)

s√

sexp

(− y2

4s− Kps

)ds.

(2.2.13)

The last equality is equivalent to

∂um(y, t)

∂y=

2√π

∫ ∞

y/2√

t

f(t − y2

4s2

)exp

(− s2 − Kp

y2

4s2

)ds, (2.2.14)

that clearly implies ∂um(y,t)∂y

|y=0= f(t).

Finally, let us observe that T (y, t) given by Eq. (2.2.5) is valid for all positive

values of Preff while the component ut(y, t) of the solution for velocity is not valid

for Preff = 1.

Consequently, in this case ut(y, t) has to be rederived starting again from Eq. (2.2.8).

By making Preff = 1 in the first two terms of Eq. (2.2.8) and applying again the

inverse Laplace transform, we find that

u(y, t) =1

Kp

erfc( y

2√

t

)− 1

πKp

∫ t

0

1√s(t − s)

exp

(− y2

4s− Kps

)ds (2.2.15)

− 1√π

∫ t

0

f(t − s)√s

exp

(− y2

4s− Kps

)ds for Kp 6= 0.

2.3 Limiting Cases

In the following, for completion, let us consider some limiting cases of general

solutions.

20

2.3.1 Solution in the Absence of Thermal Radiation (Nr →

0)

In the absence of thermal radiation, namely in the pure convection, the corre-

sponding solutions can directly be obtained from general solutions by substituting

Preff (effective Prandtl number) by Pr (Prandtl number). The dimensionless tem-

perature T (y, t) and the surface heat transfer rate, for instance, take the simplified

forms

T (y, t) = erfc

(y

2

√Pr

t

),∂T (y, t)

∂y|y=0= −

√Pr

πt. (2.3.1)

2.3.2 Solution in the Absence of Mechanical Effects

Let us now assume that the infinite plate is kept at rest all the time. In this

case, the function f(t) is zero for each real value of t and the component um(y, t) of

velocity is identically zero. Consequently, the velocity of the fluid u(y, t) reduces to

the thermal component ut(y, t) given by Eq. (2.2.11). Its temperature, as well as

the surface heat transfer rate is given by the same equality (2.2.5).

2.3.3 Solution in the Absence of Porous Effects(Kp → 0)

The temperature distribution in the fluid mass, as it results from Eq. (2.2.5) is

not affected by the porosity of medium and the velocity corresponding to the purely

fluid regime, i.e. infinite permeability, cannot be obtained from general solution

(2.2.9) by making Kp → 0. So,

we must start again from Eq. (2.2.8). For Kp = 0 this equality becomes

u(y, q) =1

1 − Preff

[exp(−y√

Preffq)

q2−

√Preff

exp(−y√

q)

q2

](2.3.2)

− F (q)exp(−y

√q)

√q

,

21

and the velocity of the fluid is

u(y, t) =1

1 − Preff

[(t +

y2

2Preff

)erfc

(y

2

√Preff

t

)(2.3.3)

− y

√Preff t

πexp

(− y2

4tPreff

)]−

√Preff

1 − Preff

[(t +

y2

2

)erfc

( y

2√

t

)− y

√t

πexp

(− y2

4t

)]− 1√

π

∫ t

0

f(t − s)√s

exp(− y2

4s

)ds if Preff 6= 1.

Furthermore if Preff = 1 then

u(y, q) =1

2q

(1

q+

y√

q

)exp(−y

√q) − F (q)

exp(−y√

q)√

q, (2.3.4)

and the corresponding velocity is

u(y, t) =1

2

[(t − y2

2

)erfc

( y

2√

t

)+ y

√t

πexp

(− y2

4t

)](2.3.5)

− 1√π

∫ t

0

f(t − s)√s

exp(− y2

4s

)ds.

2.4 Special Cases

In order to underline the theoretical value of the general solution (2.2.10) for

velocity, as well as to gain physical insight of the flow regime, we consider some

special cases whose technical relevance is well known in the literature.

2.4.1 Case f(t) = fH(t)

Let us firstly consider f(t) = fH(t) where f is a dimensionless constant and

H(.) is the Heaviside unit step function. In this case, after the time t = 0, the

infinite plate applies a constant shear stress to the fluid. The thermal component of

velocity ut(y, t) remain unchanged,

while um(y, t) takes the simplified form

um0(y, t) = − f√π

∫ t

0

1√s

exp

(− y2

4s− Kps

)ds, (2.4.1)

22

or equivalently

um0(y, t) = − f

Kp

exp(−Kpy) +2f√π

∫ ∞

√t

exp(− y2

4s2− Kps

2)ds; Kp 6= 0. (2.4.2)

In the case Kp = 0 Eq. (2.4.1) takes the simplified form (in agreement with [[29],

Eq. (23)])

um0(y, t) = − f√π

∫ t

0

1√s

exp

(− y2

4s

)ds, (2.4.3)

or evaluating the integral

um0(y, t) = fyerfc

(y

2√

t

)− 2f

√t

πexp

(− y2

4t

). (2.4.4)

2.4.2 Case f(t) = fta (a > 0)

Introducing f(t) = fta into Eq. (2.2.12) we get

uma(y, t) = − f√π

∫ t

0

(t − s)a

√s

exp(− y2

4s− Kps

)ds. (2.4.5)

Expression of the mechanical component of velocity corresponding to Kp = 0,

namely

uma(y, t) = − f√π

∫ t

0

(t − s)a

√s

exp(− y2

4s

)ds, (2.4.6)

is equivalent to Eq. (4.1) from [71] with α = 0. This motion, unlike those corre-

sponding to the cases 2.4.1 and 2.4.3, is unsteady and remain unsteady all the time.

Of interest is the case a = 1 when the plate applies a constantly accelerating shear

stress to the fluid. The corresponding expression of the mechanical component

um1(y, t), resulting from Eq. (2.4.5) is

um1(y, t) = − f√π

∫ t

0

(t − s)√s

exp(− y2

4s− Kps

)ds =

∫ t

0

um0(y, s)ds. (2.4.7)

23

2.4.3 Case f(t) = f sin(ωt)

By now letting f(t) = f sin(ωt) in the general expression (2.2.12) of um(y, t), it

results that

um(y, t) = − f√π

∫ t

0

sin[ω(t − s)]√s

exp(− y2

4s− Kps

)ds. (2.4.8)

This is the mechanical component of the fluid velocity in the motion induced

by an infinite plate that applies an oscillating shear stress to the fluid. It can be

written as a sum between steady-state and transient solutions

ums(y, t) = − f√π

∫ ∞

0

sin[ω(t − s)]√s

exp(− y2

4s− Kps

)ds, (2.4.9)

umt(y, t) =f√π

∫ ∞

t

sin[ω(t − s)]√s

exp(− y2

4s− Kps

)ds. (2.4.10)

Into above relations ω is the dimensionless frequency of the shear stress. In the

absence of porosity, the steady-state component

ums(y, t) = − f√π

∫ ∞

0

sin[ω(t − s)]√s

exp(− y2

4s

)ds, (2.4.11)

can be written in the simplified form,

ums(y, t) =f√ω

exp(− y

√ω

2

)cos

(ωt − y

√ω

2+

π

4

). (2.4.12)

For a check of results, let us determine the steady shear stress component corre-

sponding to the steady-state velocity (2.4.12), namely [[30], Eq. (24)]

τms(y, t) = f exp(− y

√ω

2

)sin

(ωt − y

√ω

2

). (2.4.13)

As expected, it is in accordance with the dimensional form resulting from [[29], Eq.

(30)].

2.5 Numerical Results and Discussion

To study the behavior of dimensionless velocity and temperature fields and to

get some physical insight of the obtained results, a series of numerical calculations

24

was carried out for different values of relevant parameters that describe the flow

characteristics. All graphs correspond to the case when the plate applies a constant

shear stress to the fluid. Fig. 1 exhibits the dimensionless velocity profiles at differ-

ent times and fixed values of the material parameters Preff and Kp and the shear

f on the boundary. As expected, the fluid velocity increases in time and smoothly

decreases to zero for y going to infinity. Fig. 2 depicts the influence of the effective

Prandtl number on velocity. The fluid velocity is a decreasing function with respect

to Preff . This result agrees well with that resulting from [[32], Fig. 3] because

Preff decreases if the radiation-conduction parameter Nr increases.

The effects of permeability parameter Kp on the spatial distribution of the di-

mensionless velocity are presented in Fig. 3. The inverse permeability parameter

Kp, as defined by Eq. (2.2.1) is inverse proportional to the permeability of the

medium. The resistance of porous medium increases if its permeability decreases.

Consequently, the velocity of the fluid decreases with respect to Kp. However, this

change of velocity is maximum near the plate decreases with respect to y and finally

approach to zero. The profiles of velocity monotonically decay for all values of Kp

and the boundary layer thickness decreases when Kp increases. The spatial variation

of the dimensionless velocity u(y, t) with the shear stress f induced by the boundary

plate is plotted against y in Fig. 4. As expected, the velocity of the fluid decreases

for increasing values of f (by negative values) and this result is in accordance with

that of Erdogan [[18], Fig. 3]. The influence of thermal effects on the fluid motion is

shown by Fig. 5 where the dimensionless velocity u(y, t) against y is compared with

its thermal component ut(y, t) . As expected, the mechanical effects are stronger

but the thermal influence on velocity is also significant.

Expressions of the dimensionless temperature and surface heat transfer rate, as

we previously specified, are identical to those from [[32], Eqs. (13) and (15)]. Con-

sequently, there is no reason to present again their variations with respect to time

and Prandtl number or radiation-conduction parameter. Of interest seems to be

here their variations against y for different values of the effective Prandtl number.

25

Such a variation for temperature is presented in Fig. 6 for t = 1. It is observed that

an increase of the effective Prandtl number Preff implies a significant decrease of

the temperature throughout the fluid. The temperature of fluid, for different values

of Preff smoothly decreases from a maximum value at the boundary to a minimum

value for large values of y. Further, the values of T (y, t) at any distance y from the

plate are always higher for Preff = 0.175 than those for Preff = 0.233 or 0.350.

The thermal boundary layer thickness also decreases for increasing Preff .

2.6 Conclusions

Heat transfer and the motion of a viscous fluid over a heated infinite plate that

applies an arbitrary shear stress f(t) to the fluid are analytically studied. Radiative

and porous effects are taken into consideration and exact solutions for the dimension-

less velocity and temperature are obtained by means of the Laplace transforms.

These solutions are presented in simple forms in terms of the complementary error

function of Gauss and they satisfy both governing equations and all imposed initial

and boundary conditions. The dimensionless temperature depends only on Preff

and the fluid velocity is presented as a sum of thermal and mechanical components.

All results regarding velocity are new. What is more, its mechanical component is

reduced to the already- known forms in literature in absence of porous effects.

Some significant limiting cases, excepting those corresponding to Preff = 1 and

Kp = 0, whose solutions are separately established, are easy obtained from general

solutions. In all cases, the temperature of the fluid does neither depend on porosity

nor on shear stress on the boundary. This is possible as the viscous dissipation is

not taken into consideration. Further on, as we have expected, both components

of velocity are affected by the porosity of the medium and the number of essential

parameters is reduced by a suitable selection of the reference velocity U .

Finally, in order to underline some physical insight of the present results, three

special cases of technical relevance motions are considered. The first case corre-

26

sponds to the fluid motion due to an infinite plate that applies a constant shear to

the fluid. Figs. 1-4 are prepared to bring to light the effects of pertinent parame-

ters on the velocity field. A comparison of the dimensionless velocity ut(y, t) with

its thermal component ut(y, t) is presented in Fig. 5. We may conclude that the

thermal effects, as well as the mechanical ones have a significant influence on the

fluid motion. Therefore, our main conclusions are as follows:

1. Dimensionless temperature and the surface heat transfer rate are not influenced

by the porosity of medium and by the shear stress on the boundary. They depend

only on the effective Prandtl number Preff .

2. Dimensionless velocity is presented as a sum of thermal and mechanical compo-

nents. The influence of thermal effects on velocity is also significant.

3. The fluid velocity is a decreasing function with respect to Preff , Kp and f .

4. Boundary layer thickness as well as the thermal boundary layer thickness de-

creases when the effective Prandtl number increases.

27

Fig. 1. Non-dimensional velocity profiles for

.

Preff = 0.35 (N P K

t fr r p= 1, = 0.7), = 1, and

different values of when the plate applies a constant shear stress = -2 to the fluid

Fig. 2.

.

Non-dimensional velocity profiles for t = 1, = 1 and different values

of when the plate applies a constant shear stress = -2 to the fluid

K

Pr fp

eff

0 1 2 3 4

0

0.5

1

1.5

2

u1 y( )

u2 y( )

u3 y( )

y

t = 0.2

t = 0.4

t = 0.6

0 1 2 3 40

1

2

u1 y( )

u2 y( )

u3 y( )

y

Preff = 0.140

Preff = 0.233

Preff = 0.467

28

Fig. 3. Non-dimensional velocity profiles for t = 1, = 0.35 and different

values of when the plate applies a constant shear stress = -2 to the fluid.

Pr

K feff

p

Fig. 4. Non-dimensional velocity profiles for t = 1, = 0.35,

= 1 and different values of the constant shear stress

Pr

K feff

p .

Kp = 1

Kp = 2

Kp = 3

0 1 2 3 40

1

2

u1 y( )

u2 y( )

u3 y( )

y

0 1 2 3 40

2

4

6

8

10

u1 y( )

u2 y( )

u3 y( )

y

f = -1

f = -5

f = -10

29

Fig. 5. Comparison between the dimensionless velocity ( , ) and its thermalcomponent ( , ) for = 0.35, = 1, f = -2 and = 0.5 and 0.8 .

u y tu y t Pr K tt eff p

u y( ) for t = 0.5

0 1 2 3 4

0

1

2

u1 y( )

u2 y( )

u3 y( )

u4 y( )

y

u y( ) for t = 0.8

u yt( ) for t = 0.5

u yt( ) for t = 0.8

Fig. 6. Dimensionless temperature profiles for = 1 and different values oft Preff .

0 1 2 3 40

0.2

0.4

0.6

0.8

1

T1 y( )

T2 y( )

T3 y( )

y

Preff = 0.233

Preff = 0.350

Preff = 0.175

30

Chapter 3

General Solutions for the

Unsteady Flow of Second Grade

Fluids over an Infinite Plate that

Applies Arbitrary Shear to the

Fluid

In this chapter we provide general solutions for the unsteady motion of a

second grade fluid induced by an infinite plate that applies a shear stress f(t) to the

fluid. In addition to being a study of a general time-dependent problem, it leads to

exact solutions. Such solutions are uncommon in the literature and they provide a

very important check for numerical methods that are used to study flows of such

fluids in a complex domain. For generality, the solutions are firstly established for

the motion between two parallel walls perpendicular to the plate. These solutions,

in the absence of the side walls, reduce to the similar solutions over an infinite plate.

In order to illustrate their importance, some special cases are considered and known

solutions from the literature are recovered. Finally, relying on an immediate con-

31

sequence of the governing equations, an important relation with the motion over a

moving plate is brought to light.

3.1 Flow Between Side Walls Perpendicular to a

Plate

Lets consider an incompressible second grade fluid at rest occupying the space

above an infinite plate perpendicular to the y-axis and between two side walls situ-

ated in the planes z = 0 and z = d of a fixed Cartesian coordinate system x, y and

z. When time t = 0+ the plate is pulled with the time-dependent shear stress f(t)

along the x-axis and f(0) = 0. Due to the shear the fluid is gradually moved and

its velocity is of the form

v = v(y, z, t) = u(y, z, t)i, (3.1.1)

where i is the unit vector along the x-direction. For such a flow the constraint of

incompressibility is satisfied while the governing equation is given by [29], [75]

∂u(y, z, t)

∂t=

(ν + α

∂

∂t

)(∂2

∂y2+

∂2

∂z2

)u(y, z, t); y, t > 0 and z ∈ (0, d),

(3.1.2)

where α = α1/ρ (α1 is a material constant and ρ is the fluid’s density). The suitable

initial and boundary conditions areu(y, z, 0) = 0 for y > 0 and z ∈ [0, d],

τ(0, z, t) =

(µ + α1

∂∂t

)∂u(y,z,t)

∂y|y=0= f(t) for ; z ∈ (0, d) and ; t > 0,

u(y, 0, t) = u(y, d, t) = 0 ; for y, t > 0; u(y, z, t) → 0 as y → ∞.

(3.1.3)

In Eq. (3.1.3) µ = ρν is the dynamic viscosity of the fluid and τ(y, z, t) = Sxy(y, z, t)

is one of the non-trivial shear stresses.

32

In order to solve this initial and boundary value problem, we use the Fourier

transforms [60], [61]. Consequently, multiplying (3.1.2) by√

2/π cos(yξ) sin(λnz)

integrating the result with respect to y from 0 to ∞ and z from 0 to d, respectively,

and taking into account the conditions (3.1.3), we obtain

∂ucn(ξ, t)

∂t+

ν(ξ2 + λ2n)

1 + α(ξ2 + λ2n)

ucn(ξ, t) = −√

2

π

f(t)

ρλn

1 − (−1)n

1 + α(ξ2 + λ2n)

, ξ, t > 0,

n = 1, 2, 3, ...

(3.1.4)

where λn = nπ/d and the double Fourier sine and cosine transform ucn(ξ, t) of

u(y, z, t) must satisfy the initial condition

ucn(ξ, 0) = 0 for ξ, t > 0, n = 1, 2, 3, ... (3.1.5)

Integrating Eq. (3.1.4) with the initial (3.1.5), inverting the result by using the

Fourier inversion formulae [60], [61], setting d = 2h and changing the origin of the

coordinate system to the middle of channel, we can write the velocity field u(y, z, t)

in the suitable form

u(y, z, t) =4

ρπh

∞∑n=1

(−1)n cos(µmz)

µm

×∫ ∞

0

cos(yξ)

1 + α(ξ2 + µ2m)

∫ t

0

f(s) exp

[− ν(ξ2 + µ2

m)(t − s)

1 + α(ξ2 + µ2m)

]dsdξ,

(3.1.6)

where µm = (2n − 1)π/(2h), m = 2n − 1.

In order to discuss the shear stress in planes parallel to the bottom wall, as well

as the shear stress on the side walls, the expressions of the non-trivial shear stresses

are needed. The first of these, for instance, has the form

τ(y, z, t) = − 2f(t)

h

∞∑n=1

(−1)n cos(µmz)

µm

e−µmy − 2

π

∫ ∞

0

ξ sin(yξ)

(ξ2 + µ2m)[1 + α(ξ2 + µ2

m)]dξ

− 4ν

πh

∞∑n=1

(−1)n cos(µmz)

µm

∫ ∞

0

ξ sin(yξ)

[1 + α(ξ2 + µ2m)]2

∫ t

0

f(s)

× exp

[− ν(ξ2 + µ2

m)(t − s)

1 + α(ξ2 + µ2m)

]dsdξ. (3.1.7)

33

Taking α → 0 into above relations, the similar solutions

uN(y, z, t) =4

ρπh

∞∑n=1

(−1)n cos(µmz)

µm

∫ ∞

0

cos(yξ)

∫ t

0

f(s)e−ν(ξ2+µ2m)(t−s)dsdξ,

(3.1.8)

τN(y, z, t) = − 4ν

πh

∞∑n=1

(−1)n cos(µmz)

µm

∫ ∞

0

ξ sin(yξ)

∫ t

0

f(s)e−ν(ξ2+µ2m)(t−s)dsdξ,

(3.1.9)

corresponding to a Newtonian fluid performing the same motion are obtained. In

view of the entry 5 of Table 4 from [61] and its immediate consequence∫ ∞

0

ξ sin(yξ)e−νtξ2

dξ =y

4νt

√π

νtexp

(− y2

4νt

),

the solutions (3.1.8) and (3.1.9) can be written under the simplified forms

uN(y, z, t) =2

ρh√

νπ

∞∑n=1

(−1)n cos(µmz)

µm

∫ t

0

f(t − s)√s

exp

(− y2

4νs− νµ2

ms

)ds,

(3.1.10)

τN(y, z, t) = − y

h√

νπ

∞∑n=1

(−1)n cos(µmz)

µm

∫ t

0

f(t − s)

s√

sexp

(− y2

4νs− νµ2

ms

)ds.

(3.1.11)

Integrating by parts the last integrals from Eqs. (3.1.8) and (3.1.9) and using

the entries 6 and 7 of Tables 4 and 5 from [61], the Newtonian solutions can also be

written in the equivalent forms

uN(y, z, t) =2f(t)

µh

∞∑n=1

(−1)n cos(µmz)

µ2m

e−µmy − 1

µh

∞∑n=1

(−1)n cos(µmz)

µ2m

×∫ t

0

f ′(t − s)

e−µmyErfc

(µm

√νs − y

2√

νs

)+ eµmyErfc

(µm

√νs +

y

2√

νs

)ds;

(3.1.12)

µm 6= 0,

34

τN(y, z, t) = −2f(t)

h

∞∑n=1

(−1)n cos(µmz)

µm

e−µmy +1

h

∞∑n=1

(−1)n cos(µmz)

µm

×∫ t

0

f ′(t − s)

e−µmyErfc

(µm

√νs − y

2√

νs

)− eµmyErfc

(µm

√νs +

y

2√

νs

)ds,

(3.1.13)

in terms of the error complementary function of Gauss Erfc(·) which can be ob-

tained from tables [1].

To the best of our knowledge, the general solutions (3.1.6) and (3.1.7) for second

grade fluids, as well as the solutions (3.1.8)-(3.1.13) for Newtonian fluids are new in

the literature and their value for theory and practice can be significant. They can

provide exact solutions for different motions with physical relevance of these fluids.

In order to bring to the light the theoretical importance of these general solutions,

some known solutions from the literature will be recovered as limiting cases.

3.1.1 Case: f(t) = fta (a > 0) (the Plate Applies an Acceler-

ated Shear to the Fluid)

Putting f(t) = fta into Eqs. (3.1.6) and (3.1.7), the corresponding solutions

(3.12) and (3.14) from [71] are recovered. The solutions corresponding to a =

2, 3, ..., n, as it was proved in [71], can be written as simple or multiple integrals of

u1(y, z, t) and τ1(y, z, t). The similar solutions for Newtonian fluids are immediately

obtained from any one of Eqs. (3.1.8) and (3.1.9), (3.1.10) and (3.1.11) or (3.1.12)

and (3.1.13). By making f(t) = ft in Eq. (3.1.13), for instance, we obtain the shear

stress

τ1N(y, z, t) = −2ft

h

∞∑n=1

(−1)n cos(µmz)

µm

e−µmy +f

h

∞∑n=1

(−1)n cos(µmz)

µm

×∫ t

0

e−µmyErfc

(µm

√νs − y

2√

νs

)− eµmyErfc

(µm

√νs +

y

2√

νs

)ds.

(3.1.14)

Further, unlike the next two cases, this motion is unsteady and remains unsteady.

35

3.1.2 Flow due to an Oscillating Shear Stress

By now letting f(t) = f sin(ωt) into Eqs. (3.1.6)-(3.1.9), the corresponding

solutions obtained in [30] and [31] are recovered. The velocity field for second grade

fluids

us(y, z, t) =4f

µπh

∞∑n=1

(−1)n cos(µmz)

µm

sin(ωt)

∫ ∞

0

(ξ2 + µ2m) cos(yξ)

(ξ2 + µ2m)2 + (ω/ν)2[1 + α(ξ2 + µ2

m)]2dξ

− ω

νcos(ωt)

∫ ∞

0

[1 + α(ξ2 + µ2m)] cos(yξ)

(ξ2 + µ2m)2 + (ω/ν)2[1 + α(ξ2 + µ2

m)]2dξ

+ω

ν

∫ ∞

0

[1 + α(ξ2 + µ2m)] cos(yξ)

(ξ2 + µ2m)2 + (ω/ν)2[1 + α(ξ2 + µ2

m)]2exp

[− ν(ξ2 + µ2

m)t

1 + α(ξ2 + µ2m)

]dξ,

(3.1.15)

is identical to that given by [[31], Eq. (23)]. It is presented as a sum of steady-state

and transient solutions and describes the motion of the fluid some time after its

initiation. After this time, when the transients disappear, it tends to the steady-state

solution that is periodic in time and independent of the initial condition. However, it

satisfies the boundary conditions and the governing equation. An important problem

regarding the technical relevance of starting solutions is to find the approximate time

after which the fluid is moving according to the steady-state solutions. More exactly,

in practice, it is necessary to find the required time to reach the steady-state.

3.1.3 Case: f(t) = fH(t) (Flow due to a Plate that Applies

a Constant Shear to the Fluid)

In this case, as well as for f(t) = fH(t) cos(ωt), where f is a constant and H(.)

is the Heaviside unit step function, the solution is obtained following the same way

as in [13]. However, it is worth noticing that the corresponding solutions can also

be obtained from general solutions (3.1.6) and (3.1.7). Taking f(t) = fH(t) into

Eq. (3.1.6), for instance, the corresponding velocity u0(y, z, t) takes the simplified

36

form [[71], Eq. (3.16)]

u0(y, z, t) =2f

µh

∞∑n=1

(−1)n cos(µmz)

µm

e−µmy

µm

− 2

π

∫ ∞

0

cos(yξ)

(ξ2 + µ2m)

exp

[− ν(ξ2 + µ2

m)t

1 + α(ξ2 + µ2m)

]dξ

,

(3.1.16)

which is equivalent to the result obtained by Yao and Lin [[75], Sect. 4]. By now

letting α = 0 in Eq. (3.1.16), the solution (16) from [19] is recovered. Of course,

this last solution is equivalent to the velocity field

u0N(y, z, t) =2f

µh

∞∑n=1

(−1)n cos(µmz)

µ2m

e−µmy

− f

µh

∞∑n=1

(−1)n cos(µmz)

µ2m

[e−µmyErfc

(µm

√νt − y

2√

νt

)+ eµmyErfc

(µm

√νt +

y

2√

νt

)],

(3.1.17)

resulting from Eq. (3.1.12) for f ′(t) = fH ′(t) = fδ(t), where δ(.) is the Dirac delta

function. The corresponding shear stress, namely

τ0N(y, z, t) = −2f

h

∞∑n=1

(−1)n cos(µmz)

µm

e−µmy

+f

h

∞∑n=1

(−1)n cos(µmz)

µm

e−µmyErfc

(µm

√νt − y

2√

νt

)− eµmyErfc

(µm

√νt +

y

2√

νt

),

(3.1.18)

is immediately obtained from Eq. (3.1.13). It is clearly seen from Eqs. (3.1.16),

(3.1.17) and (3.1.18) that for large times the last terms tend to zero. Consequently,

this flow also becomes steady and the steady solutions are the same for both types of

fluids (Newtonian and second grade). Furthermore, as it immediately results from

Eqs. (3.1.14) and (3.1.18),

τ1N(y, z, t) =

∫ t

0

τ0N(y, z, s)ds.

37

3.2 Limiting Case h → ∞ (Flow over an Infinite

Plate)

In the absence of the side walls, namely when h → ∞, the general solutions

(3.1.6)-(3.1.9) take the simplified forms

u(y, t) = − 2

ρπ

∫ ∞

0

cos(yξ)

1 + αξ2

∫ t

0

f(s) exp

[−νξ2(t − s)

1 + αξ2

]dsdξ, (3.2.1)

τ(y, t) = f(t) − 2

πf(t)

∫ ∞

0

sin(yξ)

ξ(1 + αξ2)dξ +

2ν

π

∫ ∞

0

ξ sin(yξ)

(1 + αξ2)2

∫ t

0

f(s) exp

[−νξ2(t − s)

1 + αξ2

]dsdξ,

(3.2.2)

uN(y, t) = − 2

ρπ

∫ ∞

0

cos(yξ)

∫ t

0

f(s)e−νξ2(t−s)dsdξ, (3.2.3)

τN(y, t) =2ν

π

∫ ∞

0

ξ sin(yξ)

∫ t

0

f(s)e−νξ2(t−s)dsdξ, (3.2.4)

correspond to the motion over an infinite plate that applies a shear stress f(t) to the

fluid. The Newtonian solutions, as it results from Eqs. (3.1.10), (3.1.11), (3.1.13),

(3.2.3) and the identity∫ ∞

0

1 − e−νξ2t

ξ2cos(yξ)dξ =

√νπt exp

(− y2

4νt

)− πy

2Erfc

(y

2√

νt

),

can also be written in the equivalent forms

uN(y, t) = − 1

ρ√

νπ

∫ t

0

f(t − s)√s

exp

(− y2

4νs

)ds, (3.2.5)

τN(y, t) =y

2√

νπ

∫ t

0

f(t − s)

s√

sexp

(− y2

4νs

)ds, (3.2.6)

uN(y, t) =y

µ

∫ t

0

f ′(t − s)Erfc

(y

2√

νs

)ds − 2

µ

√ν

π

∫ ∞

0

√sf ′(t − s) exp

(− y2

4νs

)ds,

(3.2.7)

38

τN(y, t) =

∫ t

0

f ′(t − s)Erfc

(y

2√

νs

)ds. (3.2.8)

If f(t) is a periodic function, all general solutions that have previously been

developed can be written as a sum of steady-state and transient solutions. The

Newtonian shear stress (3.2.6), for example, can be written as

τN(y, t) = τNs(y, t) + τNt(y, t), (3.2.9)

where

τNs(y, t) =y

2√

νπ

∫ ∞

0

f(t − s)

s√

sexp

(− y2

4νs

)ds,

τNt(y, t) = − y

2√

νπ

∫ ∞

t

f(t − s)

s√

sexp

(− y2

4νs

)ds. (3.2.10)

Choosing f(t) = f sin(ωt) into last relations, we find that

τN(y, t) =fy

2√

νπ

∫ ∞

0

sin[ω(t − s)]

s√

sexp

(− y2

4νs

)ds

− fy

2√

νπ

∫ ∞

t

sin[ω(t − s)]

s√

sexp

(− y2

4νs

)ds. (3.2.11)

Under this form, the corresponding boundary condition τN(0, t) = f sin(ωt)

seems not to be satisfied. In order to do away with this inconvenience we shall

present the steady-state solution (3.2.10)1 in a more appropriate form. Indeed, mak-

ing the change of variable s = 1/σ and using the fact that cos x = cosh(ix), sin x =

−i sinh(ix) and the known result∫ ∞

0

exp[−a2s − (b2/4s)]

s√

sds =

√π

2ae−ab,

we find, after lengthy but straightforward computations

τNs(y, t) = f exp

(−y

√ω

2ν

)sin

(ωt − y

√ω

2ν

). (3.2.12)

Finally, taking the function f(t) to be f(t) = fH(t) or f(t) = ft in Eq. (3.2.8),

we obtain for the shear stress the simple but elegant expressions

τ0N(y, t) = fErfc

(y

2√

νt

)and τ1N(y, t) = f

∫ t

0

Erfc

(y

2√

νs

)ds, (3.2.13)

which are identical as form to v0n(y, t) and v1n(y, t) corresponding to the flow due to

a flat plate that moves in its plane with the velocities V H(t) and V t, respectively.

39

3.3 Conclusions

The motion of a second grade fluid due to an infinite plate that applies a time-

dependent shear f(t) to the fluid is studied by means of integral Fourier transforms.

General solutions are obtained for the motion between two infinite parallel walls

perpendicular to the plate. These solutions may easily be used to recover different

known solutions from the literature or to develop new similar solutions for suitable

selections of the function f(t). Similar solutions for Newtonian fluids performing

the same motion are obtained as special cases of general solutions. They are also

written in simpler forms, Eqs. (3.1.10)-(3.1.13), in terms of the elementary function

exp() and of the error complementry function Erfc().

In the absence of the side walls, when the distance between walls tends to infinity,

the general solutions take simplified forms like those given by Eqs. (3.2.1)-(3.2.8)

and they correspond to the motion over an infinite plate. If the plate applies an

oscillating shear to the fluid, the corresponding solutions can be presented as a sum

of steady-state and transient solutions. These solutions depict the motion of the

fluid some time after its initiation. After a while, when the transients disappear,

they tend to the steady-state solutions that are periodic in time and independent of

the initial conditions. Some of the present results can be extended to fluid motions

in cylindrical domains [74].

Finally, taking f(t) = ft, f sin(ωt) or fH(t) in Eq. (3.2.2) we obtain the shear

stresses

τ(y, t) = ft − 2f

νπ

∫ ∞

0

1 − exp

(− νξ2t

1 + αξ2

)sin(yξ)

ξ3dξ, (3.3.1)

τ(y, t) = f sin(ωt) − 2f

π

ω

νcos(ωt)

∫ ∞

0

ξ sin(yξ)

ξ4 + (ω/ν)2(1 + αξ2)2dξ

− 2f

π

(ω

ν

)2

sin(ωt)

∫ ∞

0

(1 + αξ2) sin(yξ)

ξ[ξ4 + (ω/ν)2(1 + αξ2)2]dξ

+2f

π

ω

ν

∫ ∞

0

ξ sin(yξ)

ξ4 + (ω/ν)2(1 + αξ2)2exp

(− νξ2t

1 + αξ2

)dξ, (3.3.2)

40

τ(y, t) = fH(t)

[1 − 2

π

∫ ∞

0

sin(yξ)

ξ(1 + αξ2)exp

(− νξ2t

1 + αξ2

)dξ

], (3.3.3)

As form, these expressions are identical to those of the velocity field v(y, t) (see

[[28], Eq.(23)], [[22], Eq.(3.9)] and [[13], Eq.(3)]) corresponding to the motion in-

duced by a plate that moves in its plane with the velocities V t, V sin(ωt) or V H(t),

respectively. This is not a surprise because a simple analysis shows that the shear

stress τ(y, t) in such motions of second grade fluids satisfies the governing equation

∂τ(y, t)

∂t=

(ν + α

∂

∂t

)∂2τ(y, t)

∂y2, (3.3.4)

which is identical to that for the velocity v(y, t) [[8], Eq. (2.12)]. Consequently,

the velocity field v(y, t) corresponding to the unsteady motion of a second grade

or Newtonian fluid due to an infinite plate that slides in its plane with a velocity

V (t)H(t) is given by anyone of the relations (3.2.2), (3.2.4), (3.2.6), (3.2.8) or (3.2.9)

with V (t) instead of f(t).

41

Chapter 4

The Influence of Deborah Numberon Some Couette Flows of aMaxwell Fluid

In this chapter, we have dealt with Couette flows of a Maxwell fluid caused by

the bottom plate which applies on the fluid a shear rate of the form ∂u(0,t)∂y

= τ0µf(t).

Similar solutions of the same generality have been recently obtained by [29] for sec-

ond grade fluids. Laplace transform has been used to determine exact expressions

for shear stress and velocity corresponding to the fluid motion. In particular, the

cases of constant shear rate on the bottom plate and sinusoidal oscillations of the

wall shear rate are studied. Some relevant properties of velocity and shear stress are

brought to light through graphical illustrations.

4.1 Problem Formulation and Calculation of the

Velocity Field

Let us consider an incompressible, homogeneous Maxwell fluid between two flat,

infinite solid plates situated in the planes y = 0 and y = h of a Cartesian coordinate

system Oxyz with the positive y-axis in the upward direction, Fig. 1.

42

Initially, both the fluid and the plates are considered to be at rest. At the

moment t = 0+ the motion of the fluid is caused by the bottom plate that applies

a shear stress of the form τ0µf(t) to the fluid. Here f(t) is a piecewise continuous

function defined on [0,∞) and f(0) = 0. We also assume that the Laplace transform

of function f(t) exists.

For the present fluid motion problem, the velocity vector has the form [40], [41]

V = u(y, t)i. (4.1.1)

while the constitutive and governing equations imply(1 + λ

∂

∂t

)τ(y, t) = µ

∂u(y, t)

∂y, (4.1.2)

(1 + λ

∂

∂t

)∂u(y, t)

∂t= ν

∂2u(y, t)

∂y2, (y, t) ∈ (0, h) × (0,∞), (4.1.3)

where τ(y, t) is the tangential shear stress.

43

Also, the initial and boundary conditions are given by

u(y, 0) = 0,∂u(y, 0)

∂t= 0, τ(y, 0) = 0, y ∈ [0, h], (4.1.4)

∂u(0, t)

∂y=

τ0

µf(t), u(h, t) = 0. (4.1.5)

By using the following dimensionless variables and functions

y∗ =y

h, t∗ =

νt

h2, τ ∗ =

τ

τ0

, u∗ =u

hτ0/µ, g(t∗) = f

(h2t∗

ν

), (4.1.6)

we obtain the next non dimensional initial boundary value problem (dropping the

star notation)(1 + D

∂

∂t

)τ(y, t) =

∂u(y, t)

∂y, (y, t) ∈ (0, 1) × (0,∞), (4.1.7)

(1 + D

∂

∂t

)∂u(y, t)

∂t=

∂2u(y, t)

∂y2, (y, t) ∈ (0, 1) × (0,∞), (4.1.8)

∂u(0, t)

∂y= g(t), u(1, t) = 0, t ≥ 0, (4.1.9)

u(y, 0) = 0,∂u(y, 0)

∂t= 0, τ(y, 0) = 0, y ∈ [0, 1], (4.1.10)

where D = λ(h2/ν

) is the Deborah number.

By applying the temporal Laplace transform L [15], to Eqs. (4.1.7)-(4.1.9) and

employing the initial conditions (4.1.10), we obtain the problem

(1 + Dq)τ(y, q) =∂u(y, q)

∂y, y ∈ (0, 1), Req > 0, (4.1.11)

∂2u(y, q)

∂y2= (Dq2 + q)u(y, q), y ∈ (0, 1), Req > 0, (4.1.12)

∂u(0, q)

∂y= G(q), u(1, q) = 0, (4.1.13)

44

where τ(y, q) = Lτ(y, t), u(y, q) = Lu(y, t), G(q) = Lg(t) are the Laplace

transforms of the functions τ(y, t), u(y, t) and g(t), respectively.

The transform domain solution of Eq. (4.1.12) with the boundary conditions

(4.1.13) is given by

u(y, q) = G(q)G1(y, q), (4.1.14)

where

G1(y, q) =sh[(y − 1)

√Dq2 + q]√

Dq2 + qch(√

Dq2 + q). (4.1.15)

In order to find the inverse Laplace transform of the right part of Eq. (4.1.14), we

consider the auxiliary function

F1(y, q) =sh[(y − 1)

√q]

√qch(

√q)

, (4.1.16)

which is the image of the function

f1(y, t) = −2∞∑

n=0

cos(αny) exp(−αn2t), (4.1.17)

with αn = (2n+1)π2

, n=0,1,2,...

Since G1(y, q) = (F1ow)(q) = F1(y, w(q)),

with w(q) = Dq2 + q = D

(q + 1

2D

)2

− 14D

,

then its inverse Laplace transform is

g1(y, t) = L−1G1(y, q) =

∫ ∞

0

f1(y, z)p(z, t)dz, (4.1.18)

where

p(z, t) = L−1e−zw(q) = L−1

e

z4D .e

−zD

(q+ 1

2D

)2. (4.1.19)

By using Eq. (A.1) from Appendix, we obtain

p(z, t) =t

2e

z−2t4D

∞∑k=0

(−Dz)k

(k + 1)!(2k + 1)!

∫ ∞

0

J2(2√

xt)dx, (4.1.20)

45

where Jν(.) is the Bessel function of first kind and order ν.

Replacing (4.1.17) and (4.1.20) into (4.1.18) we find that

g1(y, t) = −te−t

2D

∑∞n=0 cos(αny)

∑∞k=0

(−Dz)k

(k+1)!(2k+1)!

×∫ ∞0

x2k+1J2(2√

xt)dx∫ ∞

0zke

−

(αn

2− 14D

)z

dz,

= −te−t

2D

∞∑n=0

cos(αny)

∫ ∞

0

J2(2√

xt)∞∑

k=0

(−D)kΓ(k + 1)x2k+1

(k + 1)!(2k + 1)!bk+1n

dx, (4.1.21)

where bn = αn2 − 1

4D> 0 and Γ is the Gamma function.

By using Eq. (A.2) from Appendix, we obtain the following simpler form of the

function g1(y, t):

g1(y, t) = −2t

De−

t2D

∞∑n=0

cos(αny)

∫ ∞

0

J2(2√

xt)1

x

[1 − cos

(x

√D

bn

)]dx. (4.1.22)

Now, using the properties of the Bessel functions ([1],[2]), we obtain

g1(y, t) = −2e−t

2D

∞∑n=0

cos(αny)√bnD

sin

(t

√bn

D

). (4.1.23)

Finally, using Eqs. (4.1.14), (4.1.23) and the convolution theorem we obtain the

expression of the velocity given by

u(y, t) = (g ∗ g1)(t) = −2∞∑

n=0

cos(αny)√bnD

∫ t

0

g(t − s)e−s

2D sin

(s

√bn

D

)ds. (4.1.24)

4.2 Calculation of the Shear Stress

To determine the shear stress τ(y, t), we use Eqs. (4.1.11), (4.1.14) and (4.1.23).

Introducing Eq. (4.1.14) into Eq. (4.1.11) we find that

τ(y, q) =1

1 + DqG(q)

∂G1(y, q)

∂y=

G(q)

D

1

q + 1D

∂G1(y, q)

∂y=

1

DG(q)G2(y, q),(4.2.1)

where the function

G2(y, q) =1

q + 1/D

∂G1(y, q)

∂y(4.2.2)

46

has the inverse Laplace transform

g2(y, t) =

∫ t

0

e−(t−s)

D∂g1(y, s)

∂yds = 2e−

tD

∞∑n=0

αn sin(αny)√bnD

∫ t

0

es

2D sin

(s

√bn

D

)ds.

By evaluating the last integral, it results

g2(y, t) = e−tD + 2e−

t2D

∞∑n=0

sin(αny)

αn

[1

2√

bnDsin

(t

√bn

D

)− cos

(t

√bn

D

)].(4.2.3)

Consequently, the shear stress may be written in the simpler form

τ(y, t) =1

D(g ∗ g2)(t) =

1

D

∫ t

0

g(t − s)g2(y, s)ds, (4.2.4)

where g2(y, s) is given by the above relation.

4.3 Some Particular Cases of the Motion

In this section we consider the following two expressions of the function g(t)

• g(t) = H(t), H(t) =

0, t ≤ 0

1, t > 0being the Heaviside unit step function;

• g(t) = sin(Ωt), Ω > 0 is the constant frequency of the oscillations.

In the first case, replacing g(t − s) = 1 into Eq. (4.1.24) we obtain

u(y, t) = −2∞∑

n=0

cos(αny)√bnD

∫ t

0

e−s

2D sin

(s

√bn

D

)ds

= −2∞∑

n=0

cos(αny)

αn2

+ 2e−t

2D

∞∑n=0

cos(αny)

αn2

[1

2√

bnDsin

(t

√bn

D

)+ cos

(t

√bn

D

)].

Applying Eq. (A.3) from Appendix in order to get the velocity in the following form

u(y, t) = y − 1 + 2e−t

2D

∞∑n=0

cos(αny)

αn2

[1

2√

bnDsin

(t

√bn

D

)+ cos

(t

√bn

D

)].(4.3.1)

The velocity given by Eq. (4.3.1) has the following temporal limits:

limt→0+

u(y, t) = 0, limt→∞

u(y, t) = y − 1. (4.3.2)

47

From Eq. (4.3.2) it results that the velocity feild u(y, t) does not exhibit a jump

of discontinuity at t=0 and, for t → ∞ it reduces to the ”permanent solution” (or

steady solution) us = y − 1. Furthermore, all initial and boundary conditions are

clearly satisfied.

By using the diagrams generated from the Mathcad software, we may discuss

some physical aspects of the flow. In all figures we used ν = 0.1655 m2/s λ =

0.062951s, ρ = 840 kg/m3. In Fig. 2, we have plotted the profiles of the velocity

u(y, t) given by Eq. (4.3.1), versus y ∈ [0, 1], t ∈ 1, 1.5, 3 and for different values of

the Deborah number D. Its obvoius that the absolute values of the velocity decrease

if the Deborah number decreases.

For large values of the time t the diagrams of the velocity tend to the diagram

of the ”permanent velocity” up = y − 1. Figure 3 contains diagrams of velocity

u(y, t) , versus t, for y ∈ 0.1, 0.4, 0.6 and different values of Deborah number D.

For smaller values of the time t the influence of the Deborah number on the velocity

is insignificant. In the interval t ∈ [1, 4] the influence of the Deborah number on the

velocity is significant and the velocity increases if the Deborah number decreases.

For t≥ 4 the velocity tends to the permanent velocity.

To determine the velocity field corresponding to the oscillating shear rate g(t) =

sin(Ωt) we use Eq. (4.1.24) with g(t − s) = sin Ω(t − s) and have

u(y, t) = up(y, t) + ut(y, t) (4.3.3)

where

up(y, t) = 2Ω cos(Ωt)∞∑

n=0

cos(αny)

(αn2 − DΩ2)2 + Ω2

−2 sin(Ωt)∞∑

n=0

(αn2 − DΩ2) cos(αny)

(αn2 − DΩ2)2 + Ω2

, (4.3.4)

ut(y, t) = −Ωe−t

2D

∞∑n=0

cos(αny)√bnD[(αn

2 − DΩ2)2 + Ω2]

×[2√

bnD cos

(t

√bn

D

)+ [1 − 2D(αn

2 − DΩ2)] sin

(t

√bn

D

)]. (4.3.5)

48

The permanent solution (4.3.4) can also be written in the simpler form

up(y, t) =1

(A2 + B2)(sh2A + cosh2B)

[AM2(y) − BM1(y)] cos(Ωt)

+[AM1(y) + BM2(y)] sin(Ωt)

,

(4.3.6)

where

2A2 = Ω√

D2Ω2 + 1 − DΩ2; 2B2 = Ω√

D2Ω2 + 1 + DΩ2, (4.3.7)

M1(y) = chA cos Bsh[A(y − 1)] cos[B(y − 1)]

+shA sin Bch[A(y − 1)] sin[B(y − 1)], (4.3.8)

M2(y) = chA cos Bch[A(y − 1)] sin[B(y − 1)]

−shA sin Bsh[A(y − 1)] cos[B(y − 1)]. (4.3.9)

Physical aspects of the flow in the case of sinusoidal shear rate on the bottom plate

are illustrated by means of the figures 4 and 5.

In Fig. 4 we plotted the velocity u(y, t) given by Eq. (4.3.3), versus y, for

Ω = 2, t ∈ 5, 10, 15 and different values of the Deborah number D. As shown

in these diagrams, for a fixed time t, the influence of the Deborah number on the

velocity can be different. For example, for t ∈ 5, 15 the velocity increases if the

Deborah number decreases, and for t = 10 the velocity decreases if the Deborah

number decreases.

Fig. 5 contains the profiles of the starting velocity u(y,t) given by Eq. (4.3.3) and

the ”permanent solution” given by Eq. (4.3.4).These diagrams were plotted versus

t, for y = 0.5, Ω ∈ 0.5, 1.2 and for different values of the Deborah number D. An

important practical aspect of this type of flow is the achievement of ”steady-state”

flow. In this case the flow is in accordance with the permanent solution and it is

achieved after a time t from which the transient solution can be neglected. It is

clear that, for given values of the frequency of oscillations of the shear rate, the time

to reach the steady-state is decreasing if the Deborah number decreases. Also, this

time decreases if the frequency Ω increases.

49

4.4 Conclusions

The aim of this chapter is to find exact solutions for Couette flows of a Maxwell

fluid generated by a time-dependent shear rate given on the plate. Expressions for

velocity and shear stress are obtained for the general case ∂u(y,t)∂y

∣∣∣∣y=0

= τ0µf(t). Two

particular cases corresponding to a constant shear rate and sinusoidal oscillations of

the shear rate are analyzed. The influence of Deborah number on the fluid motion

was studied by means of numerical and graphical results generated with the software

Mathcad. The time to reach the steady-state flow can also be obtained by graphical

illustrations. The dependence of this time of the Deborah number has been studied

as well.

In the case of constant shear rate, influence of Deborah number on the velocity

field is insignificant for several values of time t. for large values of time t, the influence

of Deborah number on velocity becomes significant and velocity increases if the

Deborah number decreases. In the case of sinusoidal oscillations of the shear rate,

we shown that the Deborah number play an important role on the flow behavior.

The time required to reach the movement described by the ”permanent solution”

(or steady-state solution) decreases with decreasing Deborah number.

50

51

52

53

54

Chapter 5

Exact Solution for Motion of anOldroyd-B Fluid over an InfiniteFlat Plate that Applies anOscillating Shear Stress to theFluid

In this chapter, we discussed the starting solutions for the unsteady motion of

an incompressible Oldroyd-B fluid due to an infinite plate that applies an oscillating

shear to the fluid. Such exact solutions provide an important check for numerical

methods that are used to study flows of these fluids in a complex domain. They are

given as a sum of steady-state and transient solutions and satisfy both the governing

equations and all given initial and boundary conditions. Furtheron, the similar solu-

tions for Maxwell and Newtonian fluids can be obtained as limiting cases of general

solutions. Finally, the influence of material parameters on the fluid motion and the

required time to reach the steady-state are determined by graphical illustrations.

This time is lower for the cosine oscillations in comparison with the sine oscillations

of the shear, decreases with respect to the relaxation time λ and the frequency ω of

the shear and increases with respect to the retardation time λr.

55

5.1 Governing Equations

An incompressible Oldroyd-B fluid is characterized by the constitutive equations

([3], [4], [26], [57]) given by Eq. (1.4.5)

In the following analysis, we will consider a unidirectional flow whose velocity is

given by

V = V(y, t) = u(y, t)i, (5.1.1)

where i denotes the unit vector along the x-direction of the Cartesian coordinate

system x, y and z. For the velocity field (5.1.1) the continuity equation is satisfied.

We also assume that the extra-stress tensor S and the velocity V depends only on

y and t. In the absence of a pressure gradient in the flow direction and ignoring the

body forces, the governing equation is given by [4]

∂u(y, t)

∂t+ λ

∂2u(y, t)

∂t2= ν(1 + λr

∂

∂t)∂2u(y, t)

∂y2; y, t > 0, (5.1.2)

where ν = µ/ρ is the kinematic viscosity and ρ is the fluid’s constant density. The

non-trivial shear stress τ(y, t) = Sxy(y, t) satisfies the partial differential equation

[[28], Eq. (4)] (1 + λ

∂

∂t

)τ(y, t) = µ

(1 + λr

∂

∂t

)∂u(y, t)

∂y; y, t > 0. (5.1.3)

5.2 Formulation of the Problem

Lets suppose an incompressible Oldroyd-B fluid at rest over an infinite plate. At

time t = 0+ the plate applies an oscillating shear to the fluid (f sin ωt or f cos ωt

where f and ω are constants). Owing to the shear, the fluid is gradually moved. Its

velocity has the form of Eq. (5.1.1), the governing equation is given by Eq. (5.1.2)

and the suitable initial and boundary conditions are

u(y, 0) = 0,∂u(y, 0)

∂t= 0, τ(y, 0) = 0, y > 0, (5.2.1)

56

(1 + λ

∂

∂t

)τ(y, t) |y=0= µ

(1 + λr

∂

∂t

)∂u(y, t)

∂y|y=0= f sin ωt or f cos ωt t > 0.

(5.2.2)

Furtheron, the natural condition

u(y, t) → 0 as y → ∞, (5.2.3)

also has to be satisfied.

5.3 Exact Solutions

In the following, lets denote by us(y, t), τs(y, t) and uc(y, t), τc(y, t), the solutions

corresponding to the two problems and by

V (y, t) = uc(y, t) + ius(y, t), T (y, t) = τc(y, t) + iτs(y, t), (5.3.1)

the complex velocity and the complex tension, respectively. In view of the above

equations, the functions V (y, t) and T (y, t) have to be solutions of the next initial

and boundary values problems

∂V (y, t)

∂t+ λ

∂2V (y, t)

∂t2= ν

(1 + λr

∂

∂t

)∂2V (y, t)

∂y2y, t > 0, (5.3.2)

(1 + λ

∂

∂t

)T (y, t) = µ

(1 + λr

∂

∂t

)∂V (y, t)

∂yy, t > 0, (5.3.3)

V (y, 0) = 0,∂V (y, 0)

∂t= 0, T (y, 0) = 0 y > 0, (5.3.4)(

1 + λ∂

∂t

)T (y, t) |y=0= µ

(1 + λr

∂

∂t

)∂V (y, t)

∂y|y=0= feiωt t > 0, (5.3.5)

V (y, t) → 0 as y → ∞ t > 0. (5.3.6)

5.3.1 Calculation of the Velocity Field

To determine the solution of initial-boundary values problem (5.3.2),

(5.3.4)1,2, (5.3.5)2 and (5.3.6), we firstly take the Laplace transform [15] of Eq.

57

(5.3.2) and obtain

qV (y, q) + λq2V (y, q) = ν (1 + λrq)∂2V (y, q)

∂y2, (5.3.7)

where the Laplace transform V (y, q) of function V (y, t) has to satisfy the conditions

∂V (y, q)

∂y|y=0=

f

µ(q − iω)(1 + λrq), (5.3.8)

V (y, q) → 0 as y → ∞. (5.3.9)

Multiplying Eq. (5.3.7) by√

2π

cos(yξ), integrating the result with respect to y from

0 to infinity and using Eqs. (5.3.8) and (5.3.9), we obtain

Vc(ξ, q) = −√

2

π

f

ρ

1

(q − iω)[λq2 + (1 + λrνξ2)q + νξ2], (5.3.10)

where

Vc(ξ, q) =

√2

π

∞∫0

V (y, q) cos(yξ)dy, (5.3.11)

denotes the Fourier cosine transform [60] of function V (y, q). Eq. (5.3.10) can be

written as

Vc(ξ, q) = Vc1(ξ, q) + Vc2(ξ, q), (5.3.12)

where

Vc1(ξ, q) = −√

2

π

f

ρ

(νξ2 − λω2) − iω(1 + λrνξ2)

(νξ2 − λω2)2 + ω2(1 + λrνξ2)2

1

q − iω, (5.3.13)

Vc2(ξ, q) =

√2

π

f

ρ

(νξ2 − λω2) − iω(1 + λrνξ2)

(νξ2 − λω2)2 + ω2(1 + λrνξ2)2

λq + [iλω + (1 + λrνξ2)]

λq2 + (1 + λrνξ2)q + νξ2.

(5.3.14)

Now Apply the inverse Laplace transform and then the inverse Fourier cosine trans-

form to Eq. (5.3.13), we get the following

V1(y, t) = − 2

π

feiωt

ρν2α

∞∫0

(νξ2 − λω2) − iω(1 + λrνξ2)

(ξ2 − β2)2 + γ2cos(yξ)dξ, (5.3.15)

where

α = 1 + λr2ω2, β2 =

ω2(λ − λr)

ν(1 + λ2rω

2), γ =

ω(1 + λλrω2)

ν(1 + λ2rω

2).

58

Using (A.4) and (A.5) from Appendix, we get the following simplified expression

V1(y, t) =f

µ

√ν

ω

e−yB

4√

(1 + λ2rω

2)(1 + λ2ω2)

[cos(ωt − yA + ϕ +

π

2) + i sin(ωt − yA + ϕ +

π

2)],

(5.3.16)

where

2A2 =√

β4 + γ2 + β2, 2B2 =√

β4 + γ2 − β2 and

tan ϕ =

√1 + λ2ω2 − λω

√1 + λ2

rω2√

1 + λ2rω

2 + λrω√

1 + λ2ω2.

Now, for Eq. (5.3.14), we introduce the function

F (ξ, q) =λq + [iλω + (1 + λrνξ2)]

λq2 + (1 + λrνξ2)q + νξ2, (5.3.17)

which can be written in the following equivalent form

F (ξ, q) =q + b(ξ)

2λ

(q + b(ξ)2λ

)2 − ( c(ξ)2λ

)2+

b(ξ) + i2λω

c(ξ)

c(ξ)2λ

(q + b(ξ)2λ

)2 − ( c(ξ)2λ

)2, (5.3.18)

where

b(ξ) = 1 + λrνξ2, c(ξ) =√

(1 + λrνξ2)2 − 4νλξ2.

Applying the inverse Laplace transform and then the inverse Fourier transform to

Eq. (5.3.14) and using Eq. (5.3.18), we obtain the following expression

V2(y, t) =2

π

f

νµα

∞∫0

cos(yξ)

(ξ2 − β2)2 + γ2

[(νξ2 − λω2)ch(

c(ξ)t

2λ) +

(νξ2 + λω2)b(ξ)

c(ξ)sh(

c(ξ)t

2λ)

]e−

b(ξ)t2λ dξ

−i2

π

fω

νµα

∞∫0

cos(yξ)

(ξ2 − β2)2 + γ2

[b(ξ)ch(

c(ξ)t

2λ) +

b2(ξ) − 2λ(νξ2 − λω2)

c(ξ)sh(

c(ξ)t

2λ)

]e−

b(ξ)t2λ dξ.

(5.3.19)

Finally, the velocity associates with the cosine oscillations of the shear is given by

uc(y, t) =f

µ

√ν

ω

e−yB

4√

(1 + λ2rω

2)(1 + λ2ω2)cos(ωt − yA + ϕ +

π

2) +

2

π

f

νµα

×∞∫0

cos(yξ)

(ξ2 − β2)2 + γ2

[(νξ2 − λω2)ch(

c(ξ)t

2λ) +

(νξ2 + λω2)b(ξ)

c(ξ)sh(

c(ξ)t

2λ)

]e−

b(ξ)t2λ dξ,

(5.3.20)

59

while that related to sine oscillations has the form

us(y, t) =f

µ

√ν

ω

e−yB

4√

(1 + λ2rω

2)(1 + λ2ω2)sin(ωt − yA + ϕ +

π

2) − 2

π

fω

νµα

×∞∫0

cos(yξ)

(ξ2 − β2)2 + γ2

[b(ξ)ch(

c(ξ)t

2λ) +

b2(ξ) − 2λ(νξ2 − λω2)

c(ξ)sh(

c(ξ)t

2λ)

]e−

b(ξ)t2λ dξ.

(5.3.21)

The starting solutions (5.3.20) and (5.3.21) corresponding to cosine and sine

oscillations of the shear on the boundary, are established as a sum between the

steady-state and transient solutions. They depict the motion of the fluid some time

after its initiation. After that time, in which the transients disappear, the starting

solutions tend to the steady-state solutions

ucs(y, t) =f

µ

√ν

ω

e−yB

4√

(1 + λ2rω

2)(1 + λ2ω2)cos(ωt − yA + ϕ +

π

2)

=f

µ

√ν

ω

e−yB

4√

(1 + λ2rω

2)(1 + λ2ω2)sin(ωt − yA + ϕ + π), (5.3.22)

respectively,

uss(y, t) =f

µ

√ν

ω

e−yB

4√

(1 + λ2rω

2)(1 + λ2ω2)sin(ωt − yA + ϕ +

π

2). (5.3.23)

which are periodic in time and independent of the initial conditions. As expected,

they differ with a phase shift.

5.3.2 Calculation of Shear Stress

In order to get the corresponding shear stresses, we apply the Laplace transform

to Eq. (5.3.3) and the inverse Fourier cosine transform to Eq. (5.3.10). Combining

the results, we get the following expression for the Laplace transform of the complex

tension T (y, t):

T (y, q) =2

π

fµ

ρ

∫ ∞

0

ξ sin(yξ)1

1 + λq

1

q − iω

λrq + 1

λq2 + (1 + λrνξ2)q + νξ2dξ. (5.3.24)

60

Eq. (5.3.24) can be written as

T (y, q) = T 1(y, q) + T 2(y, q) + T 3(y, q), (5.3.25)

where

T 1(y, q) =−2f

π(1 + λ2ω2)

1 − iλω

q + 1λ

∫ ∞

0

sin(yξ)

ξdξ, (5.3.26)

T 2(y, q) =2f

π(1 + λ2ω2)

1

q − iω

∫ ∞

0

ξ sin(yξ)

[ξ2 − β2 − λωγ

(ξ2 − β2)2 + γ2− i

λωξ2 − λωβ2 + γ

(ξ2 − β2)2 + γ2

]dξ,

(5.3.27)

T 3(y, q) =2

π

fµ

ρ

∞∫0

ξ sin(yξ)M(ξ)q + N(ξ)

λq2 + (1 + λrνξ2)q + νξ2dξ, (5.3.28)

with

M(ξ) =λ

ν(1 + λ2ω2)

ξ2(λωγ − β2) + (γ2 + β4) + i[ξ2(γ + λωβ2) − λω(γ2 + β4)]

ξ2((ξ2 − β2)2 + γ2)

,

N(ξ) =1

ω(1 + λ2ω2)

−ξ2(γ + λωβ2) + λω(γ2 + β4) + i[ξ2(λωγ − β2) + (γ2 + β4)]

(ξ2 − β2)2 + γ2)

.

Applying the inverse Laplace transform to Eq. (5.3.26) and using (A.6) from

Appendix, we get

T1(y, t) =

(−f

1 + λ2ω2+ i

fλω

1 + λ2ω2

)e

−tλ . (5.3.29)

Similarly, using (A.7) and (A.8) from Appendix, from Eq. (5.3.27), we obtain the

following appropriate form for T2(y, t)

T2(y, t) =fe−yB

1 + λ2ω2[cos(ωt − yA) + λω sin(ωt − yA) + i(sin(ωt − yA) − λω cos(ωt − yA))] .

(5.3.30)

A direct computation leads to the following simplified form

T2(y, t) =fe−yB

√1 + λ2ω2

[cos(ωt − yA − ψ) + i sin(ωt − yA − ψ)] , (5.3.31)

61

where tan ψ = λω.

Now, we consider the function

G(ξ, q) =M(ξ)q + N(ξ)

λq2 + (1 + λrνξ2)q + νξ2, (5.3.32)

which can be written in the following equivalent form:

G(ξ, q) = M(ξ)q + b(ξ)

2λ

λ

[(q + b(ξ)

2λ)2 − ( b(ξ)

2λ)2

] +

[−b(ξ)M(ξ)

λc(ξ)+

2N(ξ)

c(ξ)

] c(ξ)2λ[

(q + b(ξ)2λ

)2 − ( c(ξ)2λ

)2

] .

(5.3.33)

Applying the inverse Laplace transform to Eq. (5.3.28) and using Eq. (5.3.33), we

obtain

T3(y, t) =2f

π(1 + λ2ω2)

∞∫0

sin(yξ)

ξ[(ξ2 − β2)2 + (γ)2]

[((ξ2(λωγ − β2) + (γ2 + β4))

× ch(c(ξ)t

2λ) − p(ξ)

ωc(ξ)sh(

c(ξ)t

2λ)

)+ i

([ξ2(γ + λωβ2) − λω(γ2 + β4)]ch(

c(ξ)t

2λ) +

r(ξ)

ωc(ξ)sh(

c(ξ)t

2λ)

)]e−

b(ξ)t2λ dξ,

(5.3.34)

where

p(ξ) = νξ4[λrω(λωγ−β2)+2(γ+λωβ2)]+ω(γ2+β4)+ωξ2[(λωγ−β2)−ν(γ2+β4)(2λ−λr)],

r(ξ) = νξ4[2(λωγ−β2)−λrω(γ+λωβ2)]+ξ2[ν(γ2+β4)(2+λλrω2)−ω(γ+λωβ2)]+λω2(γ2+β4).

Using Eqs. (5.3.25), (5.3.29), (5.3.31) and (5.3.34), the shear stress corresponding

to cosine oscillations of the shear can be written in the form

τc(y, t) =−f

1 + λ2ω2e

−tλ +

fe−yB

√1 + λ2ω2

cos(ωt − yA − ψ) +2f

π(1 + λ2ω2)

∞∫0

sin(yξ)

ξ[(ξ2 − β2)2 + (γ)2]

×[(ξ2(λωγ − β2) + (γ2 + β4))ch(

c(ξ)t

2λ) − p(ξ)

ωc(ξ)sh(

c(ξ)t

2λ)

]e

−b(ξ)t2λ dξ.

(5.3.35)

62

Also, the shear stress corresponds to the sine oscillations is given by

τs(y, t) =fλω

1 + λ2ω2e

−tλ +

fe−yB

√1 + λ2ω2

sin(ωt − yA − ψ) +2f

π(1 + λ2ω2)

∞∫0

sin(yξ)

ξ[(ξ2 − β2)2 + (γ)2]

×[[ξ2(γ + λωβ2) − λω(γ2 + β4)]ch(

c(ξ)t

2λ) +

r(ξ)

ωc(ξ)sh(

c(ξ)t

2λ)

]e

−b(ξ)t2λ dξ.

(5.3.36)

Of course, the shear stresses given by Eqs. (5.3.35) and (5.3.36) are also given as

the sum of steady-state and transient solutions. The steady-state solutions

τcs(y, t) =fe−yB

√1 + λ2ω2

cos(ωt−yA−ψ) and τss(y, t) =fe−yB

√1 + λ2ω2

sin(ωt−yA−ψ),

also differ by a phase shift.

5.4 Particular Cases

5.4.1 λr = 0 (Maxwell Fluid)

Letting λr → 0 into Eqs. (5.3.20) and (5.3.21), we get the velocity fields

ucM(y, t) =f

µ

√ν

ω

e−yB

4√

(1 + λ2ω2)cos(ωt − yA + ϕ +

π

2) +

2f

πµν

∫ ∞

0

cos(yξ)

(ξ2 − λω2

ν)2 + (ω

ν)2

×

[(νξ2 − λω2)ch

(√1 − 4λνξ2

2λt

)+

νξ2 + λω2√1 − 4λνξ2

sh

(√1 − 4λνξ2

2λt

)]e−

t2λ dξ,

(5.4.1)

usM(y, t) =f

µ

√ν

ω

e−yB

4√

(1 + λ2ω2)sin(ωt − yA + ϕ +

π

2) − 2

π

fω

µν

∫ ∞

0

cos(yξ)

(ξ2 − λω2

ν)2 + (ω

ν)2

×

[ch

(√1 − 4λνξ2

2λt

)+

1 − 2λ(νξ2 − λω2)√1 − 4λνξ2

sh

(√1 − 4λνξ2

2λt

)]e−

t2λ dξ,

(5.4.2)

63

corresponding to a Maxwell fluid performing the same motion. Similarly, from Eqs.

(5.3.35) and (5.3.36), we obtain the corresponding shear stresses

τcM(y, t) = − f

1 + λ2ω2e−

tλ +

fe−yB

√1 + λ2ω2

cos(ωt − yA − ψ) +2f

π

∫ ∞

0

sin(yξ)e−t

2λ

ξ [(νξ2 − λω2)2 + ω2]

×

[ω2ch

(√1 − 4λνξ2

2λt

)− ω2 + 2νξ2(νξ2 − λω2)√

1 − 4λνξ2sh

(√1 − 4λνξ2

2λt

)]dξ.

(5.4.3)

τsM(y, t) =fλω

1 + λ2ω2e−

tλ +

fe−yB

√1 + λ2ω2

sin(ωt − yA − ψ) +2fω

π

∫ ∞

0

sin(yξ)e−t

2λ

ξ [(νξ2 − λω2)2 + ω2]

×

[(νξ2 − λω2)ch

(√1 − 4λνξ2

2λt

)+

νξ2 + λω2√1 − 4λνξ2

sh

(√1 − 4λνξ2

2λt

)]dξ.

(5.4.4)

5.4.2 λ → 0, λr → 0 (Newtonian Fluid)

Making λ → 0 and λr → 0 into Eqs. (5.3.20), (5.3.21), (5.3.35) and (5.3.36) or

λ → 0 into Eqs. (5.4.1)-(5.4.4), we recover the solutions ([30], Eqs. (20) - (23))

ucN(y, t) =f

µ

√ν

ωe−y

√ω2ν cos(ωt − y

√ω

2ν+

3π

4) +

2f

µπ

∞∫0

ξ2 cos(yξ)

ξ4 + (ων)2

e−νξ2tdξ,(5.4.5)

usN(y, t) =f

µ

√ν

ωe−y

√ω2ν sin(ωt − y

√ω

2ν+

3π

4) − 2f

µπ

ω

ν

∞∫0

cos(yξ)

ξ4 + (ων)2

e−νξ2tdξ,(5.4.6)

τcN(y, t) = fe−y√

ω2ν cos(ωt − y

√ω

2ν) − 2f

π

∞∫0

ξ3 sin(yξ)

ξ4 + (ων)2

e−νξ2tdξ, (5.4.7)

τsN(y, t) = fe−y√

ω2ν sin(ωt − y

√ω

2ν) +

2f

π

ω

ν

∞∫0

ξ sin(yξ)

ξ4 + (ων)2

e−νξ2tdξ, (5.4.8)

corresponding to the flow of a Newtonian fluid.

64

5.5 Numerical Results and Conclusions

In the present chapter we have applied the integral transforms in order to re-

search the unsteady motion of an incompressible Oldroyd-B fluid over an infinite

plate that applies an oscillating shear stress to the fluid. The starting solutions that

have been obtained for velocity and shear stress are presented as a sum of steady-

state and transient solutions. They describe the motion of the fluid some time after

its initiation. After that moment, when the transients disappear, the starting solu-

tions tend to the steady-state solutions that are periodic in time and independent of

the initial conditions. However, they satisfy the governing equations and boundary

conditions. Furthermore, as it was expected, the steady-state solutions correspond-

ing to the cosine oscillations of the shear differ by a phase shift from those due to the

sine oscillations of the shear. This property is not true for the transient components

of solutions. That is the reason why we separately gave the starting solutions for

both cosine and sine oscillations of the shear stress on the boundary.

By making λr = 0 into general solutions (5.3.20), (5.3.21), (5.3.35) and (5.3.36),

we obtain the similar solutions (5.4.1)-(5.4.4) corresponding to a Maxwell fluid per-

forming the same motion. These solutions can also be particularized (by making

λ → 0) to give the similar solutions (5.4.5)-(5.4.8) for Newtonian fluids. It is worth

pointing out that the expressions of τcN(y, t) and τsN(y, t) are identical, as form, with

those of vcN(y, t) and vsN(y, t) corresponding to a similar motion with the boundary

conditions [[24], Eq. (3.1)]

v(0, t) = V cos(ωt) or v(0, t) = V sin(ωt); t > 0. (5.5.1)

The velocity field (see [24], Eqs. (3.11) and (6.2))

vsN(y, t) = V e−y√

ω2ν sin(ωt − y

√ω

2ν) +

2V

π

ω

ν

∞∫0

ξ sin(yξ)

ξ4 + (ων)2

e−νξ2tdξ, (5.5.2)

for instance, has the same form as τsN(y, t) given by Eq. (5.4.8). This is not a sur-

prise because, for Newtonian fluids, Eq. (5.1.3) together with the balance of linear

65

momentum lead to a governing equation for shear stress of the same form as that

for velocity.

Generally speaking, the starting solutions for unsteady motions of fluids are im-

portant for those who need to eliminate the transients from their rheological mea-

surements. Consequently, an important problem regarding the technical relevance

of these solutions is to find the required time to get the steady-state. More exactly,

in practice it is necessary to know the approximate time after which the fluid is

moving according to the steady-state solutions. For this, the variations of starting

and steady-state velocities with the distance from the plate are depicted in Figs. 1-6

for sine and cosine oscillations of the shear stress on the boundary. At small values

of time, the difference between the starting and steady-state solutions is meaningful.

This difference decreases in time and it is clearly seen from figures that the required

time to reach the steady-state for the sine oscillations is higher in comparison to the

cosine oscillations of the shear. This is obvious, as at t = 0 the shear stress on the

boundary is zero for sine oscillations.

Naturally, the required time to reach the steady-state depends on the material

constants and the frequency ω of the shear. Figs. 1 and 4 show the influence of

ω on the fluid motion. Furthermore, the required time to reach the steady-state

decreases for increasing ω. The influence of the relaxation and retardation time’s λ

and λr on the fluid motion is underlined by Figs. 2, 3, 5 and 6. The two parameters,

as expected, have opposite effects on the motion. The required time to reach the

state-state decreases with respect to λ and increases with regard to λr for both types

of oscillating shears. Consequently, the required time to reach the steady-state for a

Newtonian fluid is higher in comparison to Maxwell fluids. On the other hand, the

steady-state is rather obtained for a Maxwell fluid in comparison to an Oldroyd-B

fluid having the same relaxation time λ.

66

67

68

69

70

71

72

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Appendix

L−1

(q +

1

2D

)2

e−zD

(q+ 1

2D

)2=

e−t2D

2

∫ ∞

0

J0(2√

xt)∞∑

k=0

(−Dz)kx2(k+1)

(k + 1)!(2k + 1)!dx, (A.1)

Also,∫ ∞

0(t − s)J0(2

√xs)ds = t

xJ2(2

√xt).

∞∑k=0

(−1)kx2k+1

(k + 1)(2k + 1)!=

2

x(1 − cos x), (A.2)

−∞∑

n=0

cos(αny)

αn2

= y − 1. (A.3)

∫ ∞

0

x2cos(mx)

(x2 − b2)2 + c2dx =

πe−mB

2c[Acos(mA) − Bsin(mA)] , (A.4)

∫ ∞

0

cos(mx)

(x2 − b2)2 + c2dx =

πe−mB

2c(A2 + B2)[Acos(mA) + Bsin(mA)] , (A.5)

∫ ∞

0

sin(yξ)

ξ=

π

2, (A.6)

∫ ∞

0

xsin(mx)(x2 − b2)

(x2 − b2)2 + c2dx =

π

2e−mBcos(mA), (A.7)

∫ ∞

0

xsin(mx)

x [(x2 − b2)2 + c2]dx =

π

2ce−mBsin(mA), (A.8)

∫ ∞

0

sin(mx)

x [(x2 − b2)2 + c2]dx =

π

2c(b4 + c2)

c +

[b2sin(mA) − ccos(mA)

].exp(−mB)

, (A.9)

where

A2 =√

b4 + c2 + b2, 2B2 =√

b4 + c2 − b2.

81