hydrodynamic stability of newtonian and non-newtonian fluids · hydrodynamic stability of newtonian...

Hydrodynamic Stability of Newtonian and

Non-Newtonian Fluids

Julian Mak

Supervisor: Dr. Miguel A. Moyers-Gonzalez

University of Durham

Department of Mathematical Sciences

April 2009

i

AbstractThis report aims to provide an introduction to the study of linear hydrodynamic stability in both

Newtonian and non-Newtonian fluids. We first give some motivations as to why one studies

stability of fluid flows in the first place. In Chapter 2 we specify the mathematical tools required

for our study. In Chapters 3 and 4 we look at the linear stability of Newtonian and non-Newtonian

fluids and compare the relevant results. For our study of non-Newtonian fluids we will consider

shear-thinning fluids where viscosities decrease with increasing shear. Our principal hypothesis

may be given as follows: the presence of a shear-thinning viscosity stabilises the flow.

The main emphasis of this report is linear stability for parallel shear flows. In Chapter 5 we

evaluate the methods used and discuss the limitations of the methods. We are mainly interested in

the mathematics involved, but for completeness we will mention the physics where relevant. The

computer code used is given in the appendix.

ii

AcknowledgementsPhotographs in Chapter 1 are taken from D.V. Boger and K. Walters, “Rheological Phenomena in

Focus”, courtesy of Elsevier Science Pub Co. The program used for computations are MATLAB

and MAPLE 11 (under the Durham University license). All are edited for cosmetic purposes in

GIMP 2 and Macromedia Fireworks MX.

The author would like to thank: Dr. Djoko Wirosoetisno for introducing the author to field of fluid

mechanics and the associated research within the field during a summer research placement in the

academic year 2007/2008; to Dr. Miguel Moyers-Gonzalez for his patience in teaching coding and

research skills to a complete novice; to the various 4H colleagues who proof read the numerous

drafts of this report. Their respective contributions, layout suggestions and grammar corrections

have been invaluable and are much appreciated.

iii

NotationVectors and tensors (including matrices) will be given in boldface type. We have given vectors

and tensors both in direct form (for example, v) and component form (for example, vij). Multiple

meanings of symbols which have more than one meaning are divided by a semi-colon, and symbols

with more than one commonly known name are given first, with other names given in brackets.

b body force

F total force; deformation gradient tensor

I identity tensor

n outward pointing normal vector to a surface

u velocity field; the disturbance velocity field

U , U the baseflow

p pressure

x position vector x = xiei

γ Strain tensor (right relative Cauchy-Green strain tensor)

γ, γij rate of strain tensor

γ second invariant of the rate of strain tensor

δij Kronecker delta

εijk Levi-Civita symbol (permutation symbol)

η normal vorticity

µ dynamic viscosity; shear viscosity

µt tangent viscosity

ν kinematic viscosity ν = µ/ρ

ρ fluid density

σ, σij stress tensor

σn stress vector (traction vector)

τ anisotropic part of the stress tensor (extra stress tensor) σ = −pδij + τ

D d/dy

I2n

∫|v(n)|2 dy

Re Reynolds number

O big O notation

∇ gradient operator ∂/∂xjD/Dt material derivative ∂/∂t+ u ·∇u

iv

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

1 Introduction 1

1.1 Turbulence and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Variable viscosity and stress effects . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Mathematical Setup 6

2.1 Preliminary assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 The Continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.3 Stress vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.4 Reynolds transport theorem . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.5 Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.6 Conservation of linear momentum . . . . . . . . . . . . . . . . . . . . . 12

2.1.7 Conservation of angular momentum . . . . . . . . . . . . . . . . . . . . 13

2.2 Stress, strain and viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.2 Deformation and Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.3 Further properties of the stress tensor . . . . . . . . . . . . . . . . . . . 17

2.2.4 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

CONTENTS v

3 Stability of Newtonian fluids 23

3.1 The equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1 Navier-Stokes equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.2 Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1.3 Linearised momentum equation . . . . . . . . . . . . . . . . . . . . . . 27

3.1.4 Orr-Sommerfeld and Squire’s equation . . . . . . . . . . . . . . . . . . 29

3.2 Analytical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.1 Squire’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.2 Eigenvalue bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.1 Spectral method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 Stability of non-Newtonian fluids 48

4.1 The equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.1.1 Baseflow profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.1.2 Linearised momentum equation and tangent viscosity . . . . . . . . . . . 50

4.1.3 Modified Orr-Sommerfeld and Squire equations . . . . . . . . . . . . . . 52

4.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.1 Finding the baseflow U(y) . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.2 Computing the viscosity functions . . . . . . . . . . . . . . . . . . . . . 57

4.2.3 The spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.4 Doing the general case . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3 Analytical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3.1 Bounds for the Squire modes . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3.2 Bounds for the modified Orr-Sommerfeld equation . . . . . . . . . . . . 60

5 Conclusion and Discussion 63

5.1 Numerical problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.1.1 Discretisation error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

CONTENTS vi

5.1.2 Spectrum sensitivity and pseudospectra . . . . . . . . . . . . . . . . . . 63

5.2 Some further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2.1 Rescaling and the Reynolds number . . . . . . . . . . . . . . . . . . . . 64

5.2.2 Energy method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.3 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Appendix 69

A MATLAB codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Bibliography 78

1

Chapter 1

Introduction

Fluids are all around us. If one glances around, it becomes obvious that a good proportion of the

universe is in a fluid state, mostly in gas form. A study of fluids is highly relevant in our quest

to understand more about nature. From a practical point of view, understanding fluid behaviour

is indispensable: aircraft and ships travel through fluids, lubricants for mechanical devices are

fluids, the atmosphere and the weather are governed by fluid dynamics... the list goes on. From a

mathematical point of view, it is extremely interesting, with many open questions that remain to be

answered. For example, the existence of a classical solution to the nonlinear and coupled Navier-

Stokes equation subject to suitable initial data is very much an open question; simulation of fluid

flow requires extra care, because solutions often become ill-defined in finite time; the analysis of

the equations rapidly become complicated due to the nature of the problem. Understanding the

behaviour of fluid models has been and will continue to challenge researchers in a wide variety

of fields, such as physics, engineering, geography, oceanography, astrophysics, and of course,

mathematics.

The area of fluid mechanics we shall be investigating here is the topic of hydrodynamic stability.

This is concerned with when and how laminar flows break down, their subsequent development,

and their eventual transition to turbulence. By laminar, we mean a fluid flowing in parallel layers,

with no disruption between the layers. In nonscientific terms, a laminar flow is “smooth”, whereas

a turbulent flow is “rough”. We shall not limit ourselves to hydrodynamics, for we will not be

limiting our investigation solely to water. The term rheology (generally understood to be the

science of of deformation and flow of materials - from the Greek word rheos, meaning flow) might

be more appropriate in our case.

1.1 Turbulence and stability

Turbulent flows occur in everyday events. Common examples include cigarette smoke breaking

up shortly after it leaves the stub, or the jet of water breaking up when the flow is suitably fast.

Chapter 1. Introduction 2

Turbulent flow usually carries a lot of energy and it is of engineering interest to reduce this

presence of energy. For example, in aircraft design where this may cause structural damage to

the body of the aircraft, or in the prototype fusion power plants where turbulent flow in the plasma

causes difficulty in plasma containment. Trying to understand turbulence has been an important

engineering problem as well as a mathematical one.

In his 1883 paper, Osborne Reynolds found that when a fluid is driven sufficiently hard, eddies

and vortices began to appear and we have transition from a laminar flow to turbulent flow (Figure

1.1). He proposed a dimensionless parameter encompassing the relevant physical data of a flow,

such as the velocity and the fluid’s viscosity, and carried out experiments with varying values

of this parameter. We now call this parameter the Reynolds number in honour of him. For our

purposes we are only going to focus on the mathematical development, but for completeness we

will mention the physics behind the phenomena where relevant.

Figure 1.1: (a) Laminar flow in a pipe. (b) Transition to turbulent flow in a pipe. (c) Transition to turbulent

flow as seen when illuminated by a spark. Taken from Reynolds (1883).

1.2 Variable viscosity and stress effects

The term fluid is a rather loose one - for our purposes here we shall take a fluid to be a continuous

medium which deforms smoothly under stress. We will make precise this concept in the next

chapter. Intuitively, we can think of a fluid as anything that flows.

Most fluids are “sticky” to some extent, and we term the degree of stickiness viscosity.

Mathematically, this is defined as the ratio between the stress and the strain. The viscosity clearly

depends on the fluid under consideration, but a natural question to ask is whether the fluid’s

viscosity depends on other factors such as time, temperature, and the force applied to the fluid.


Some fluids such as water have viscosity which is independent of stress or time; this was the

assumption Newton made when he first proposed a model for fluid studies. We call such fluids

Newtonian, where the relation between the stress and the strain is linear, i.e., the viscosity is

constant. It soon became clear from physical observations that some fluids exhibit variable

viscosity. In other words, the stress-strain relation is nonlinear, and we call these non-Newtonian

fluids. Some examples of these are:

1. Shear-thinning fluids. Substances and mixtures such as oil-based paint, toothpaste,

mayonnaise and quicksand are fairly viscous at first glance, almost gel-like. Under stress,

they seem to flow more easily, as observed by stirring a pot of paint, or stepping onto

quicksand. Fluids of this type have decreasing viscosity under increasing stress. Sometimes

they are called pseudo-plastics, and in fact the majority of non-Newtonian fluids are shear-

thinning.

2. Shear-thickening fluids. Custard and corn starch mixture hardens under stress, in effect

becoming more viscous. One well known experiment involves people running across pools

of custard without sinking into it, thanks to the fluid possessing a more solid like structure

under stress. Fluids of this type which have increasing viscosity under increasing stress are

also known as dilatant fluids.

3. Food products, such as mayonnaise, and biological fluids, such as synovial fluid, are

thixotropic: they decrease in viscosity the longer they are exposed to stress. Mud exhibits

this property, and in earthquake zones liquefaction occurs, causing landslides. This

structural weakening is due to the forces acting on the substance.

4. The opposite to thixotropy is rheopexy, but this is a much rarer property. Here, the longer

a fluid is exposed to stress, the higher its viscosity. This type of fluid is heavily researched

since there are obvious benefits in body-armour and automobile design. Printing ink

possesses this sort of property.

5. Bingham plastics. These are characterised by the presence of a yield stress: it remains a

solid until a certain magnitude of stress is exceeded, then it flows like a fluid. This concept

is used in engineering as a approximation to things like mayonnaise, drilling muds and

toothpaste where a yield stress exists. In this respect, the material is more like a solid than

a liquid. Its associated viscosity is usually known as the plastic viscosity.

6. Viscoelastic fluids. Materials of this class experience viscosity like a fluid, but also elasticity

like a solid. Extremely curious phenomena are observed, for example, the Weissenberg

effect (Figure 1.3), self-syphoning (Figure 1.4), and the Barus effect (Figure 1.5). Examples

include eggs, magma, glass, oils and blood. Most materials may be considered as

viscoelastic, because they are viscous and elastic up to a certain point, due to the molecular

interactions that occur in any sort of matter.


We define a purely viscous fluid to be one whose viscosity is time independent. A graphical

representation of the stress-strain relation is given below:

Figure 1.2: A graphical representation of the shear-dependent types shear dependent viscosity. We can

represent the strain on a body of fluid by the shear rate when we are considering shear motions.

From the thixotropy and rheopexy property displayed by fluids, we see the viscosity may depend

on what is happening to them at the present time as well as what has happened to them in the past.

The stress and deformation of the body thus depend on time. We might expect that the longer

the interval from the present to the past time, the smaller the contribution to the current stress

resulting from a given strain; this is the principle of fading memory. Viscoelastic materials also

exhibit memory since they experience both viscosity and elasticity, so there is a time-dependent

deformation associated with them.

In this report we shall only consider purely viscous fluids. We shall now specify mathematically

the concepts introduced.


Figure 1.3: The Weissenberg or rod-climbing effect. The fluid is high-molecular weight PIB in low-

molecular weight polyisobutylene. Compare with water, where the swirling effect causes a depression

rather than a climbing effect. From Boger & Walters (1993).

Figure 1.4: The self-syphoning phenomenon. When a 0.75% aqueous solution of Polyox WSR 301 is

spilled over the side, the liquid pours itself out. The pictures are taken after the initial spilling occurs. From

Boger & Walters (1993).

Figure 1.5: When a fluid of 2.0% aqueous solution of polyacrylamide exits the capillary, it exhibits a swell

known as the Barus effect. This is also observed in dyes as well as the Polyox solution in Figure 1.4. From

Boger & Walters (1993).

6

Chapter 2

Mathematical Setup

We assume the reader is reasonably fluent in vector calculus, linear algebra, complex analysis, as

well as a little tensor calculus and basic mechanics. Einstein summation is used unless otherwise

stated, and we will use both index notation and vector notation depending on which one is more

convenient.

Most of the material are taken from Tanner (1985) and Batchelor (2000). The section on tensor

invariants is taken from Tanner (1985) and the section on stability from Drazin & Reid (1981)

and Chandrasekhar (1981). These are further supplemented with lecture notes from a course on

non-Newtonian fluid mechanics given by C. Nouar.

2.1 Preliminary assumptions

2.1.1 The Continuum hypothesis

All materials possess a microstructure at the molecular level. Often, we are only interested in the

macroscopic behaviour of the material, and because of the sheer number of microstructures present

it would be unrealistic to work out their behaviour by computing the numerous equations of motion

from Newton’s formulation for example. Instead, we assume that a material is continuous, that is,

the macroscopic properties such as volume, density, velocity, pressure and temperature are well-

defined at an infinitesimal point. These properties are also assumed to vary continuously from

one point to another, and the fact that microstructures exist is ignored. Thus the application of

differential calculus is justified.

How accurate this description is depends on the material in question. Indeed, some polymeric

fluids are better described by statistical mechanics, but we will assume the continuum hypothesis

holds for the fluids we are dealing with, and we will derive the relevant equations in the continuum

mechanics framework.

Chapter 2. Mathematical Setup 7

Figure 2.1: Motion of a body from time t to t’

2.1.2 Motion

We first consider a fixed Eulerian frame of reference. We can consider a particle P in a body of

fluid at present time t located at position x with respect to the origin O (see Figure 2.1). At some

later time t′, it will be at a position x′. We can consider x as fixed, and thus the particles trace out

a path line. The actual trajectory of P is given by

x′(t′) = x.

The velocity u of P is then given by

u(t′) =dx′

dt′,

and similarly, we can define the velocity field u (assumed to be smooth) so that for each position

x at time t,

u = u(x, t).

The acceleration of P at x is given by

a =d

dtu(x, t) =

∂u

∂t+ u · ∇u ≡ Du

Dt

where we have used the chain rule. The operator

D

Dt:=

∂

∂t+ u · ∇ (2.1)


is called the material derivative, which we see is is a directional derivative following the particle.

∇ = ∂/∂xj is the gradient operator according to the orthogonal coordinate system we use. The

operator goes by several other names but we will call it the material derivative here.

The rate of change of the velocity component ui with respect to the coordinate xj is given by

∂ui/∂xj . For the total change at a fixed time, we have

dui =∂ui∂xj

dxj = Lijdxj ,

where L is called the velocity gradient tensor and describes the rate of change of the velocity with

respect to the fixed Eulerian coordinate system. Note that L is not necessary a symmetric second

order tensor. Since (∇u)ij = ∂uj/∂xi by the definition of the gradient operator, we write the

velocity gradient tensor asLT = ∇u

Lji =∂uj∂xi

.(2.2)

It is known that all second order tensors such as L may be split into a symmetric and an anti-

symmetric part as follows:

Lij =12

(Lij + Lji) +12

(Lij − Lji).

This fact is easily verified by an interchange of the index. In light of this, we define the rate of

strain tensor to be the symmetric part of the velocity gradient tensor, i.e.,

γ =12

[∇u+ (∇u)T ]

γij =12

(∂ui∂xj

+∂uj∂xi

).

(2.3)

It will be seen later on why γ is called the rate of strain tensor. In common literature, the rate of

strain tensor (or the rate of deformation tensor) is often labelled by D; we will not use that here

because we reserve D for denoting the derivative. For completeness, the antisymmetric part of the

velocity gradient tensor is called the vorticity tensor. This deals with the rotational aspect of the

fluid but we will not require it here.

2.1.3 Stress vector

Stress is how one specifies the mechanical interaction between one part of the material body and

another. It is usually defined as the force acting on a body per unit area. This is in contrast to body

forces such as gravity and magnetic forces which act from a distance.

Suppose we have an ensemble of infinitesimal fluid elements occupying a volume V (t) with a

closed boundary surface S(t) and an outward pointing normal n as given in Figure 2.2. If we


Figure 2.2: diagram of the setup

consider an infinitesimal surface area elementA of area dS, at a position vector x in the Euclidean

frame, we define the stress vector σn(x, t) by requiring that the force F exerted on A at time t by

the fluid in the direction of n to be σn dS, i.e.,

dF = σn dS.

So the total force over S is

F =∫S(t)

σn · dS,

with dS = n dS. The fluid in V (t) will, in general, also experience body forces as well as those

acting just on S(t). If ρ(x, t) denotes the fluid density in V (t) and b the body force per unit mass,

then the total force experienced by the fluid in V (t) is given by

F =∫S(t)

σn · dS +∫V (t)

ρb dV. (2.4)

2.1.4 Reynolds transport theorem

In fluid mechanics, we work in a control volume by construction. The relevant physical

conservation laws from classical mechanics or thermodynamics are given in terms of the system of

particles and not the control volume. To derive our equations of motion, we will need to recast the

conservation laws by relating the system equations to the volume equations. This link is provided

by the Reynolds transport theorem:


Theorem 2.1.1 Suppose we have a control volume V (t) moving with the fluid flow. Consider a

quantity f(x, t) which is some fluid quantity per unit volume, then the total amount of f in V (t)

is given by

F (t) =∫V (t)

f(x, t) dV.

Then we havedFdt

=∫V (t)

[∂f

∂t+∇ · (uf)

]dV. (2.5)

We require this theorem because our domain of integration is a function of time, and thus the

time derivative cannot be taken inside the integral directly. In the discrete case where we are

“integrating” over the contributions of each individual particles, taking the time derivative inside

is legitimate since the “integral” is then just a sum and the derivative operator is linear.

Before we begin the proof, we give a lemma which concerns the material derivative of the

Jacobian.

Lemma 2.1.2 Consider a transformation from the spatial representation to the material

representation i.e., a frame moving with that particle, given by

x = G(X, t), (2.6)

where we assume G is an appropriate diffeomorphism. Then

DJ

Dt= J ∇ · u,

where J is the Jacobian of the transform G in (2.6) given by

J = det(∂xi∂Xj

)ij

= εijk∂xi∂X1

∂xj∂X2

∂xk∂X3

. (2.7)

Here, εijk is the Levi-Civita symbol. Since we assume G is a diffeomorphism, J is bounded above

by infinity and is non zero.

ProofSince D/Dt is differentiation with X constant, we can interchange the order of differentiation:

D

Dt

(∂xi∂Xj

)=

∂

∂Xj

DxiDt

=∂ui∂Xj

.

The chain rule gives∂ui∂Xj

=∂ui∂xm

∂xm∂Xj

.


We will only show the operations acting on the i-th term; the operations for the j-th and the k-th

terms are similar. Applying the above to the Jacobian, we have, by the product rule,

DJ

Dt= εijk

D

Dt

(∂xi∂X1

)∂xj∂X2

∂xk∂X3

+ · · ·

= εijk∂ui∂xm

∂xm∂X1

∂xj∂X2

∂xk∂X3

+ · · ·

= εijk∂u1

∂x1

∂xi∂X1

∂xj∂X2

∂xk∂X3

+ · · ·

=∂u1

∂x1J +

∂u2

∂x2J +

∂u2

∂x2J

= J (∇ · u)

where in the third line we have used the fact that the determinant of a matrix with repeated rows

is identically zero. 2

We now give the proof for the Reynolds transport theorem (Theorem 2.5).

ProofWe transform all the relevant objects from a spatial representation to material representation using

the transformation given in (2.6). We see the differential volume element at time t is related to the

volume at time t = 0 by

dV = J dV0.

By expressing f in the new reference coordinates as f(X, t), we have

dF

dt=

d

dt

∫V0

f(X, t) J dV0.

Now V0 is independent of time, and the time derivative may be taken inside the integral. By the

product rule,dF

dt=∫V0

[df(X, t)

dt

∣∣∣∣X

J +dJ

dt

∣∣∣∣X

f(X, t)]dV0,

but since we are in material coordinates, X being held fixed corresponds to taking the material

derivative. Using the definition of the material derivative in (2.1), this gives

df(X, t)dt

∣∣∣∣X

=Df

Dt=∂f

∂t

∣∣∣∣x

+ u · ∇f,

dJ

dt

∣∣∣∣X

=DJ

Dt= J ∇ · u,

Combining all of these results, we have

dF

dt=∫V0

[∂f

∂t+ u · ∇f +∇ · u

]J dV0.


Transforming back into spatial representation and invoking the appropriate vector identities, we

have, at time t,d

dt

∫V (t)

f(x, t) dV =∫V (t)

[∂f

∂t+∇ · (uf)

]dV

as required. This completes the proof. 2

Using the divergence theorem on the second term of (2.5), we have

d

dt

∫V (t)

f(x, t) dV =∫V (t)

∂f

∂tdV +

∫S(t)

fu · dS.

This says the rate of change of F is equal to the sum of the time change of f in the control

volume and the net rate of flux of f through the boundary surface. This form of the result may be

physically more intuitive.

We are now in a position to recast the conservation laws in the continuum model.

2.1.5 Conservation of mass

The mass of the fluid elements occupying V (t) is∫V (t) ρ dV , and so the conservation of mass

requires thatddt

∫V (t)

ρ dV = 0.

By Reynolds transport theorem, we have∫V (t)

(Dρ

Dt+ ρ∇ · u

)dV = 0. (2.8)

Since V (t) is arbitrary, we haveDρ

Dt+ ρ∇ · u = 0,

and this is called the continuity equation. A fluid is said to be incompressible if Dρ/Dt ≡ 0.

Since we assume ρ > 0, an equivalent condition for incompressibility is

∇ · u = 0. (2.9)

2.1.6 Conservation of linear momentum

From the definition of momentum, it can easily be seen that the momentum of the fluid occupying

the volume V (t) is ∫V (t)

ρu dV,


so that from the principle of linear momentum conservation and the definition of the force (2.4),

we haveddt

∫V (t)

ρu dV = F =∫S(t)

σn · dS +∫V (t)

ρb dV. (2.10)

By Reynolds transport theorem and using the continuity equation (2.1.5),∫V (t)

ρDu

DtdV =

∫S(t)

σn · dS +∫V (t)

ρb dV. (2.11)

2.1.7 Conservation of angular momentum

Assuming that the net torque experienced by the fluid is due only to the stress and body force, the

conservation of angular momentum states that

ddt

∫V (t)

(x× ρu) dV = x× F =∫S(t)

(x× σn) · dS +∫V (t)

(x× ρb) dV,

where x is the position vector. Again, using Reynolds transport theorem, the continuity equation

(2.1.5) and the divergence theorem,∫V (t)

(x× ρDu

Dt

)dV =

∫V (t)

[∇ · (x× σn) + x× ρb] dV. (2.12)

2.2 Stress, strain and viscosity

2.2.1 Stress tensor

We shall now construct an object which we will need later in the equations of motion.

Proposition 2.2.1 Let D be some bounded region in R3 and let σn(x, t) be the stress vector

defined throughout D. Then there exists a second order tensor σ(x, t) called the stress tensor

such that:

1. σn = σn

2. ρ(Du/Dt) = ∇ · σ + ρb (also known as Cauchy’s deformation equation)

3. σ is a symmetric tensor.

Note the gradient operator acts on the tensor to give a vector (which is a first order tensor), and

the tensor transforms the vector into another vector.

Proof


Figure 2.3: The tetrahedron Tε with outward pointing normal n

1. We fix a time t at a point p in the fluid, and assuming we are working in an orthogonal basis

{ei| i = 1, 2, 3}, we can write n = niei, and we assume n1n2n3 6= 0.

Let the infinite family of tetrahedra {Tε}ε>0 be such that for any ε > 0, Tε is the region in

R3 bounded by the three co-ordinate planes through p and the plane given by x ·n = ε, the

plane which is a distance ε away from p in the direction n (see Figure 2.3)

We label the faces of Tε having outward unit normal −ei by pi with corresponding areas

Ai. We label the plane x · n = ε by pn with area An. From trigonometry, we have, for

i, j, k = {1, 2, 3}, i, j, k all distinct,

Ai =ε2

2njnk, An =

ε2

2ninjnk, . (2.13)

So we have

Ai = niAn. (2.14)

Let V (t) = Tε, and from the conservation of linear momentum (2.11),∫TερDu

DtdV =

3∑k=1

(∫pk

σ−ek · dS)

+∫pn

σn · dS +∫

Tερb dV. (2.15)

Let c be a positive constant chosen such that for 0 < E < ∞ and 0 < ε < E, the uniform

bound below holds for all x within the domain of concern:∣∣∣∣ρDuDt − ρb∣∣∣∣ < c. (2.16)


Using (2.16) on (2.15) and noting (2.13) we have the following inequality:∣∣∣∣∣3∑

k=1

(∫pk

σ−ek · dS)

+∫pn

σn · dS

∣∣∣∣∣ < c · vol(Tε) =c

6n1n2n3ε3 =

c

3Anε. (2.17)

Noting Ai = ε2/2njnk, dividing (2.17) by An gives∣∣∣∣∣3∑

k=1

nkAk

(∫pk

σ−ek · dS)

+1An

∫pn

σn · dS

∣∣∣∣∣ < c

3ε. (2.18)

Taking the limit ε→ 0, we have

−3∑

k=1

nkσ−ek(0, t) = σn(0, t)

since nk/Ak = 1/An. We thus define the stress tensor σ = σij as

σij = (−σ−ej )i,

and indeed we have,

(σn)i = σijnj , or σn = σn. (2.19)

2. Substituting the definition (2.19) into the integral form of the conservation of linear

momentum (2.11), we get∫V (t)

ρDu

DtdV =

∫S(t)

σn dS +∫V (t)

ρb dV. (2.20)

Using the divergence theorem on (2.20) gives∫V (t)

(ρDv

Dt−∇ · σ − ρb

)dV = 0,

and since V (t) was arbitrary, assuming the integrand is continuous, we have Cauchy’s

deformation equation

ρDu

Dt= ∇ · σ + ρb. (2.21)

3. Again, using (2.19) and the integral form of the conservation of angular momentum (2.12),

we have, in index form,∫V (t)

εkjixjρDuiDt

dV =∫S(t)

εkjixjσilnl dS +∫V (t)

εkjixjρbi dV, (2.22)

where εijk is again the Levi-Civita symbol. Applying the divergence theorem to (2.22) we

have, using the product rule,

0 =∫V (t)

εijk

(xjρ

DukDt− ∂

∂xl(xjσkl)− xjρbk

)dV

=∫V (t)

εijkxj

(ρDukDt− ∂

∂xlσkl − ρbk

)dV −

∫V (t)

εijk∂

∂xlxjσkl dV.

(2.23)


We see the first integral vanishes by the Cauchy deformation equation (2.21). Noting that

∂xl/∂xj = δjl, the Kronecker delta, we have∫V (t)

εijkδjlσkl dV =∫V (t)

εijkσkj dV = 0. (2.24)

Again, V (t) was arbitrary, so the integrand in (2.24) must be identically zero. By the

summation convention, we have

ε123σ32 + ε132σ23 + ε1jj(· · · ) = 0

ε213σ31 + ε231σ13 + ε2jj(· · · ) = 0

ε312σ21 + ε321σ12 + ε3jj(· · · ) = 0.

By the property of the Levi-Civita symbol, we conclude that

σij = σji. (2.25)

Hence there exists the stress tensor σ, a second order symmetric tensor which completely

specifies the stress acting on a body of fluid. 2

Using the definition of the stress tensor (2.19) in the conservation of linear momentum (2.11) as

well as using the divergence theorem on the surface term in (2.11), we have∫V (t)

ρDu

DtdV =

∫V (t)

[∇ · σ + ρb] dV.

Since V (t) was arbitrary, we have the momentum equation

ρDu

Dt= ∇ · σ + ρb, (2.26)

and we will need this for deriving the equations of motion later.

2.2.2 Deformation and Strain

In general, the application of stress on a body will result in a displacement of the body from its

original position, as well as changes to the relative distance between the particles of the body. The

former is called rigid-body displacement and the latter is called the deformation. The geometric

measure of deformation is called strain, which is given as the change in the relative distance

between the particles in the deformed state and the undeformed state, and this is measured in units

of length.

Using the set up in Figure 2.1 we define the deformation gradient tensor F (x, t′, t) as the mapping

which relates dx at time t to dx′ at time t′, with the mapping given by

dx′ = F dx, Fij =∂x′i∂xj

. (2.27)


Note that F is not necessarily symmetric. As a measure of the relative magnitude of the

differentials, we consider the following relation:

(dx′)2 = (dx′)T · dx′ = (F dx)T · (F dx) = (dx)T (F T · F ) dx,

where the superscript T denotes the transpose of the object. We thus define the strain tensor as

γ = F T · F . (2.28)

This is sometimes called the right-relative Cauchy-Green strain tensor but we shall not be

employing this term. We now consider taking the derivative of γ with respect to t′, assuming

we have sufficiently smooth motion. Since we have, from (2.28) and (2.27),

γij(t′) = FkiFkj =∂x′k∂xi

∂x′k∂xj

.

By observing that x′k is a function of t′ but xj is not, differentiating with respect to t′ gives

dγijdt′

=∂x′k∂xi

∂uk∂xj

+∂uk∂xi

∂x′k∂xj

. (2.29)

Evaluating this at t = t′, we have x′k = xk and hence

dγijdt′

∣∣∣∣t′=t

= δki∂uk∂xj

+∂uk∂xi

δki =∂ui∂xj

+∂uj∂xi

= 2γij , (2.30)

where γij was defined in (2.3); indeed, this was why γ is denoted the rate of strain tensor.

Sometimes (2.30) is called the first Rivlin-Ericksen tensor, denoted A(1)ij . We can take further time

derivatives and obtain the higher subsequent Rivlin-Ericksen tensors which is often used in some

non-Newtonian fluid related problems; we shall not need them here.

2.2.3 Further properties of the stress tensor

Our main investigation in this report will require us to deal with the stress tensor σij given in (2.19)

and the rate of strain tensor γij given in (2.3). We note that these are both second order symmetric

tensors, and hence may be represented by symmetric matrices.

It will be assumed later that the stress tensor σ is isotropic, i.e., it does not depend on the

orientation). It can be shown by considering the action of the stress tensor on a cube, that elements

on the main diagonal of the tensor act normal to the faces of the cube, and the other elements acts

tangent to the faces (it shears the cube). Thus We may decompose the stress tensor as

σ = −pI + τ (2.31)


where p is the pressure and τ is called the extra-stress tensor or the anisotropic part of the stress

tensor describing the shear stresses. It shall be argued later in Chapter 3 why we have the pressure

appearing as part of the stress tensor. We seek to formulate a relation between the shear stress τ

and the strain γ.

Recall a that real symmetric operator in 3D (which may be represented by a symmetric matrix

with real entries),A, has eigenvalues λ1, λ2, λ3 (see Shores (2007) or Gould (1957) for example).

By expanding the characteristic equation

det |A− λI| = 0,

where I is the identity matrix, we have the following equation:

λ3 −H1λ2 +H2λ−H3 = 0. (2.32)

H1, H2, H3 are called the principal invariants ofA, and are given by

H1 = trA,

H2 = λ1λ2 + λ1λ3 + λ2λ3,

H3 = detA.

(2.33)

These are invariants as we rotate the axes since the eigenvalues are fixed numbers, independent of

whatever axes we measure from. When we have some extra symmetries the eigenvalues may be

degenerate.

By construction, the stress tensor σ and the rate of strain tensor γ are symmetric tensors of second

order. Thus, by the Cayley-Hamilton theorem, (again, see a linear algebra book) we have

σ = ψ0I + ψ1γ + ψ2γ2,

ψi = ψi(Iγ , IIγ , IIIγ),

withIγ = tr γ = H1,

IIγ = tr γ2 = γij · γij = H21 − 2H2,

IIIγ = tr γ3 = 3H3 + I31 − 3H1H2.

(2.34)

These are known as the invariants of γ. It can be shown that the diagonal elements of γ are

associated with the volume of the fluid. Hence for an incompressible fluid the volume element

cannot be changed so γ should be trace free (i.e.,H1 = 0). Experimental evidence does not

support the existence of ψ2 6= 0 so we assume for a purely viscous fluid we have ψ2 = 0.

Importance of H3 is often disputed but taking it to be zero is usually a good approximation for

calculations. Thus we are left with ψ1, which we will denote as µ, the viscosity function. µ should

only depend on the second invariant of the rate of strain tensor and we will denote this by

γ = IIγ =

√(12γ : γ

)(2.35)


where we have a involved the factor of 1/2 in the definition to give us the constitutive relation for

non-Newtonian fluids:

τ = µ(γ)γ. (2.36)

We will be dealing with purely viscous and incompressible fluids, so we require that our viscosity

function should satisfy the relation (2.36).

2.2.4 Viscosity

In Chapter 1 we mentioned that viscosity sometimes depends on stress, strain, and various other

parameters such as time or temperature. As mentioned before, viscosity is defined as the ratio

between stress and strain, as given in (2.36). A Newtonian fluid is one where the relation is linear

(i.e.,µ is just a constant function). We assume all other parameters that may cause a change in a

fluid’s viscosity are held fixed (e.g. temperature). For the Newtonian case, µ is called the dynamic

viscosity.

Fluids that do not satisfy this linear behaviour are called non-Newtonian fluids, or sometimes

generalised Newtonian fluids. Since we are dealing with purely viscous fluids, µ is scalar function

with a dependence on the shear stress; we call µ the shear viscosity. We will mainly be concerned

with shear-thinning fluids and for this there are several models for the viscosity that one can

choose. We are going to use the Carreau model (named after P.J. Carreau (1972) who first derived

it from Lodge’s molecular network theory):

µ− µ∞µ0 − µ∞

= [1 + (λγ)2](n−1)/2. (2.37)

Here, µ0 and µ∞ are respectively the viscosity at zero shear rate and the limiting viscosity at

high shear rate, λ > 0 a time constant, n > 0 the shear thinning exponent, and γ is the second

invariant of the rate of strain tensor. Indeed, this satisfies the relation (2.36), because γ is the

only variable appearing here. The other parameters are constants which are fitted according to the

specific fluid. Generally, for shear-thinning fluids, we have 0.2 ≤ n < 1, O(0.1) ≤ λ < O(100),

and µ∞/µ0 << 1; see Tanner (1985) for some examples of rheological data.

Note that when λ = 0 or n = 1 we recover the Newtonian case, and when λ is very large, by a

redefinition of the parameters we get the power-law model

µ = Kγn−1.

A special case of the Carreau model is the Cross model given by

µ− µ∞µ0 − µ∞

=1

1 + (λγ)m.

There are many other models but we choose the Carreau model because it has a sound theoretical

basis, and stability analysis data is widely available in the literature as it is frequently adopted.


Figure 2.4: Graph of viscosity µ against shear rate γ using the Carreau model. We took some arbitrary

values of λ, n and µ∞/µ0.

The shear-dependent viscosity is plotted in Figure 2.4. We will make use of this viscosity model

when we numerically compute the spectrum in Chapter 4.

2.3 Stability analysis

Suppose we have a system with parameters {Xi| i = 1, 2...} which completely defines the system

(for example, it could be pressure, temperature, velocity, and so on), and suppose we can add

a small disturbance term to perturb the system away from the original set up as defined by the

parameters Xi. Essentially, we seek to determine the reaction of the system to these disturbances.

We call a system unstable if there is one such disturbance which causes the system to depart and

drift away from this original state when following the equations of motion that govern the system.

We call a system stable if there are no such disturbances which causes the system to drift away

from this original state. The case of neutral stability, represented by the marginal states, separates

these two possible cases. The equation of these marginal states can usually be expressed in the

form

f(X1, X2, ...) = 0.

One of the main goals in the study of stability is to find the set of marginal states in the phase

space we are concerned with. Often, it is convenient mathematically and in practical applications

to vary one parameter continuously whilst keeping all the others fixed, and determine the critical

values which give rise to instability. This is known as Lyapunov stability: In this setup, we will

need to specify some positive definite norm, and so stability or instability depends on the choice

of norm.


For our stability analysis, we take a steady flowU as our base solution (the initial state). We take a

steady flow so that the coefficients in the equation of motion may depend on space but not on time,

thus simplifying our analysis. We perturb the system a little by adding a (infinitesimal) disturbance

term u to U . By substituting the initial flow with a perturbation into the full equations of motion,

we linearise by neglecting all terms which contain products or higher powers of the increment u,

since they are small by comparison. By this method we will arrive at the linearised equations of

motion. This is a key assumption: we assume we have an infinitesimally small disturbance (and

the meaning of small depends on which norm we choose). We will deal with linear theories of

stability in the rest of this report, but mention briefly a nonlinear method in Chapter 5.

Stability of a system means stability with respect to all possible (infinitesimal) disturbances. For an

investigation of stability to be complete it is necessary to examine all possible disturbances. This

is usually done by expressing an arbitrary disturbance as a superposition of some basic modes and

examining the stability of the system with respect to each basic mode. This normal mode analysis

is valid if we can show these basic modes are complete. We will be dealing with bounded plane

geometry and it can be shown we do have completeness because of the boundedness property. The

disturbances may be expanded in wave-like Fourier modes

u(x, t) = u(y) exp[i(αx+ βz − ωt)]. (2.38)

Here, u′ is the disturbance, u is the amplitude of the disturbance which is a scalar function, α

and β are the wave numbers, and ω is the wave frequency. It is understood that the real parts

of these expressions are taken to obtain physical quantities. Since the perturbation equations are

linear by construction, the disturbance can be determined if we know the reaction of the system to

all the basic disturbances of the assigned wave numbers and use the principle of superposition to

construct the end response. In particular, the stability of the system will depend on its stability to

disturbances of all wave number; instability will follow from its instability to disturbances to just

one wave number. This is called a normal mode analysis.

We consider the so called temporal analysis: In (2.38), we take the wave numbers α and β to be

real and positive, but ω = cα (c the wave speed) may be complex. Thus we can write ω = ωr+iωior c = cr + ici. Abstractly, this is equivalent to taking Fourier transforms in space and a Laplace

transform in time (with the Bromwich contour taken so we can invert the Laplace transform if we

need to).

For finite dimensional systems the stability problem is easier to analyse because we have a finite

number of degrees of freedom, so we end up dealing with a system of ODEs. This is not the

case for fluid systems, for we are dealing with infinite dimensional systems and hence systems of

PDEs. The technical difficulties associated with PDEs (in our case the Navier-Stokes equation) are

overcome only for a few classes of flows where some sort of symmetry is present. Here, because

we have taken plane geometry we can expand the solution in Fourier modes which satisfy the

superposition principle. Thus we have made the analysis more accessible: instead of dealing with


PDEs, we have a system of ODEs with an associated eigenvalue relation. Our eigenvalue will be

ω (or c), and our eigenfunction will be u. It is this eigenvalue problem we will be investigating.

Going back to (2.38), our solution grows or decays in time like exp(ωit), so our classification

becomes:

1. A disturbance is stable if ωi ≤ 0.

2. A disturbance is unstable if ωi > 0.

3. A disturbance is marginally stable if ωi = 0.

Together with the boundary conditions, our eigenvalue relation will be of the form

F(α, β, ω) = 0.

We will tend to take c as the eigenvalue for analytical work and take ω as the eigenvalue for

numerical problems.

The role of viscosity

The analysis of stability for fluids was formulated in Reynold’s time. In his 1883 paper he

considered the role of viscosity by comparing flows of a viscous fluid with that of an inviscid

fluid (fluid with zero viscosity), assuming both have the same basic velocity profile. From this, he

was led to formulate two hypotheses:

1. The inviscid fluid may be unstable and the viscous fluid stable; the effect of viscosity is then

purely stabilising.

2. The inviscid fluid may be stable and the viscous fluid unstable; viscosity would then be the

cause of instability.

The first hypothesis is intuitive; viscosity dissipates energy (usually from kinetic energy to thermal

energy), and a flow with less energy is more stable as it is less excited. The second hypothesis

seems to disagree with our physical intuition, and indeed Reynolds was unable to suggest a

physical mechanism by which viscosity causes instability, but nevertheless refused to exclude

this case. Reynolds was right in thinking that, because there are flow profiles for inviscid fluids

which are absolutely stable according to the inviscid theory, but the same flow profile for viscous

fluids can be unstable. We give an example of this in Chapter 3.

Intuitively, we expect the more viscous fluids are more stable. Over the next two chapters we will

see to what extent this is true. We will give physical explanations to how viscosity affects the flow

stability (or instability) where appropriate.

23

Chapter 3

Stability of Newtonian fluids

As defined in Chapter 2, a Newtonian fluid is one where the viscosity is a constant function. We

will first derive the equations of motion, and consider parallel shear flows between two rigid plates

with no slip conditions, i.e., we take the basic steady flowU = U(x, y, z) in Euclidean space (see

Figure 3.1) withU = U(y)ex (y1 ≤ y ≤ y2),

U(y1) = U(y2) = 0.(3.1)

For simplicity we will take the upper plate to be at y = 1 and the lower plate at y = −1. For the

rest of this report we will take y ∈ [−1, 1]; we can always rescale the domain if required.

One way to go about studying the stability problem is to solve the linearised equations of motion

as an initial value problem for some initial conditions. The problem with this is that, even for very

simple initial conditions, the analysis of it rapidly becomes complicated. We will be trying to find

criteria for instability so a normal mode analysis will suffice. We assume the modes are complete

and consider a temporal analysis of it.

For parallel shear flows, the PDE reduces down to a system of fourth order ODEs for the amplitude

of the disturbance; if this amplitude grows in time, the system is unstable, and if it decays in

time, the system is stable. Our eigenvalue relation is then a fourth order ODE problem, and we

investigate the set of eigenvalues associated with it.

3.1 The equations

3.1.1 Navier-Stokes equation

Following Stokes’ formulation, we assume stress and rate of strain are related linearly as well

being time independent. In index notation, our constitutive model is

σij = ωij + qijklγkl, (3.2)

Chapter 3. Stability of Newtonian fluids 24

Figure 3.1: Geometry and coordinate system for a channel flow

where ωij and qijkl are second and fourth order tensor respectively. For a fluid at rest, there is no

deformation or shear stress, and hence all the terms of the stress tensor not on the main-diagonal

are zero. Our model (3.2) reduces to

σij = ωij = −pδij . (3.3)

We assume the pressure p acts equally in all directions, and we took the minus sign for directional

considerations (this is what Euler assumed in his theory of inviscid flow). To take this assumption

further, we assume the fluid is isotropic, i.e., uniformity of properties in all directions and

orientations. As a result, the tensors themselves are isotropic with entries independent of the

co-ordinate system. Hence we can write our arbitrary tensors as

ωij = αδij ,

qijkl = λδijδkl + µδikδjl + βδilδjk,(3.4)

where α, β, λ, µ are scalars; see the book by Jeffreys & Jeffreys (1946) for a full derivation. We

see we have already fixed α to be −p from the static consideration (3.3). The stress tensor σij was

derived to be symmetric, so we have qijkl = qjikl, which implies β = µ in (3.4). This gives

σij = −pδij + λδij γkk + 2µγij . (3.5)

Since p was defined to be the isotropic pressure when there is no strain,

σii = −3p = −3p+ 2µγii + 3λγkk,

because δii = 3. Hence we deduce λ = −2µ/3, and so

σij = −pδij + 2µγij −2µ3δij γkk. (3.6)


We define µ to be the dynamic viscosity: it could be a linear scalar function depending on the

position and other parameters, but not on stress. We take µ to be a constant. Substituting (3.6) into

the linear momentum equation (2.26), and using the definition of the rate of deformation tensor

(2.3), we have

ρDuiDt

= ρbi −∂

∂xip+

∂

∂xj

[µ

(∂

∂xjui +

∂

∂xiuj

)− 2µ

3∂

∂xkuk

]. (3.7)

This is the compressible form of the Navier-Stokes equation in index notation. Assuming

incompressibility (and hence the density ρ could also be treated as a constant), we have ∂kuk =

∇ · u = 0, and the incompressible Navier-Stokes equation in compact form is given by

ρDu

Dt= ρb−∇p+ µ∆u (3.8)

with ∆ is the Laplacian operator. Along with appropriate boundary conditions and the continuity

equation, this set of equations specifies the dynamics of the fluid in the domain of definition. As

yet there are no closed methods in obtaining the solution to this set of second-order PDEs, mainly

due to the fact that this vector equation is nonlinear and coupled. This can be seen explicitly by

recallingD/Dt = ∂/∂t+u ·∇u and the vector identity u ·∇u = (∇|u|2)/2+u×(∇×u). The

mathematical difficulties presented by the complete equation are such that most existing solutions

are ones which reduce the equation down to a linear one. Among these are parallel shear flows

mentioned in the beginning of this chapter, and we see with a quick computation that parallel shear

flows reduces the inertial term u · ∇u to zero. We assume the flow is steady and unforced, so the

Navier-Stokes equation (3.8) reduces to

dp

dx= µ

d2U

dy2. (3.9)

Two important flows which are feasible solutions to (3.9) are:

1. Plane Couette flow. This is given by U(y) = y. This corresponds to p = const, and the

flow is driven by a moving plate, which we can take to be the upper plate at y = 1; see

Figure 3.2. This was first considered by Couette for a fluid trapped between two concentric

cylinders moving relative to each other.

2. Plane Poiseuille flow. This is given by U(y) = 1 − y2. This corresponds to p = p(x) and

the motion of the flow is driven by the pressure gradient. This gives the familiar parabolic

profile; see Figure 3.2. It was named after Poiseuille who was a physician, and he was

interested in how blood flows in the blood vessels (modelled as a non-porous rigid pipe).

The pipe Poiseuille flow is sometimes known as the Hagen-Poiseuille flow.


Figure 3.2: a) Plane Couette flow, upper plate moving at velocity V relative to the lower plate. b) Plane

Poiseuille flow.

3.1.2 Reynolds number

The equations of motion now depend on the two parameters ρ and µ which are assumed to be

constants and uniform throughout the fluid. We wish to consider the effects of the flow upon

changing these parameters, and a way to do this is to write the equations of motion in terms of

dimensionless variables. To do this, we introduce a characteristic scale which reduces the equation

into non-dimensional form. Consider the change of variables:

t′ =t

T, x′ =

x

L, m′ =

m

M, (3.10)

where T is a time scale, L is a length scale, m is the mass and M is a mass scale. The new

variables are now dimensionless parameters. We define the velocity scale V and density scale ρ as

V =L

T, ρ =

M

L3.

We now consider the following change of variables:

u′ =u

V, p′ = p

ML

T 2=p− p0

ρV 2, (3.11)

where p0 is some representative value of the modified pressure in the fluid and is a constant. We

see all the quantities are dimensionless from the basic principles of mechanics. By the chain rule,

we have∂

∂t′= T

∂

∂t=L

V

∂

∂t, ∇′ = L∇. (3.12)

Using (3.11) and (3.12) on the Navier-Stokes equation (3.8), and recalling the definition of the

material derivative D/Dt = ∂/∂t+ u · ∇, we have

Du′

Dt′= −∇′p′ + µ

ρLV∇′2u′. (3.13)


We thus define the Reynolds number as

Re =ρLV

µ, (3.14)

and the (incompressible, unforced) non-dimensional Navier-Stokes equation is given by

Du

Dt= −∇p+

1Re∇2u, ∇ · u = 0, (3.15)

where we have removed the unnecessary dashes for ease of reading. We see now the equations

contain only the dimensionless parameter Re, and the solution for the dependent variables u

and p satisfying appropriate boundary conditions can only depend on x, t, Re and the scales we

used. These scales usually depend on the geometry of the boundary and initial conditions. We

see that, for a given geometry of the boundary and initial conditions, there is no more than one

singly-infinite family of solutions in dimensionless form, with different members of the family

corresponding to different values of Re. In the flows we will be interested in, the characteristic

scales used are usually:

1. Plane Couette flow: V = velocity of the upper plate, L = half the channel width,

2. Plane Poiseuille flow: V = maximum velocity at the centre of the channel, L = half the

channel width,

but of course these are non-unique and are only chosen for convenience.

The Reynolds number may be regarded as providing an estimate of the relative importance of the

inviscid and viscous forces acting on a unit volume of the fluid. The lower the Reynolds number,

the more important the viscous force on the fluid. It was Reynolds who, in his 1883 paper, gave

descriptions of experiments he did where he varied the parameters to give different values of Re.

He found that when a certain critical value Rec of a flow was exceeded, the laminar flow starts

to break down and becomes turbulent (see Figure. 1.1). This suggests the Reynolds number is a

parameter that plays a role in the stability of a fluid and requires further investigation.

Intuitively, we expect the role of viscosity to be stabilising. Noting the definition of the Reynolds

number, a lower viscosity (or a faster flow) corresponds to a higher Reynolds number, so we expect

that as the Reynolds number increases, the flow becomes less stable.

3.1.3 Linearised momentum equation

Assuming then our initial flow takes the form given by (3.1). We denote the velocity field by

u = (u, v, w), where u, v, w are the x, y and z components of the velocity field respectively, and


are scalar functions depending on (x, t). We consider adding a small disturbance to the initial flow

u and to the pressure p. We have, for U the baseflow and P its associated pressure,

u(x, t) = U(x, t) + u′(x, t) = U(y)ex + u′(x, t),

p(x, t) = P + p′(x, t).(3.16)

We substitute (3.16) into our momentum equation, which in this case is the (non-dimensional)

Navier-Stokes equation given by (3.15). In doing so, we assume terms involving the disturbance

terms of quadratic order or higher is small and thus negligible (namely the inertial term u′ · ∇u′).We also note we get−dP/dx+d2U/dy2 which reduces to zero because U is our solution of (3.9)

with associated pressure P . We end up with an equation where only U(y) and disturbance terms

appear, so we omit the dashes for clarity purposes. The linearised momentum equation is thus

∂u

∂t+ U

∂u

∂x+ v(DU)ex = −∇p+

1Re

∆u, (3.17)

∇ · u = 0, (3.18)

where D = d/dy and everything is expressed in terms of the disturbance field and pressure.

Taking the divergence of (3.17), we note that the divergence operator commutes with the time and

spatial derivatives. The incompressibility condition means the first and second term on the left

hand side and the Laplacian term on the right hand side reduce to zero. The remaining terms are

∆p = −2(DU)∂v

∂x. (3.19)

Now, the y-component of (3.17) is given by

∂v

∂t+ U

∂v

∂x= −∂p

∂y+

1Re

∆u. (3.20)

By taking the laplacian of (3.20), we can eliminate p from (3.20) using (3.19):[(∂

∂t+ U

∂

∂x

)∆− (D2U)

∂

∂x− 1Re

∆2

]v = 0. (3.21)

We require a second equation to complete our three-dimensional description of the flow field. We

adopt the velocity-vorticity formulation by considering the normal vorticity η:

η =∂u

∂z− ∂w

∂x. (3.22)

Taking ∂/∂z of the x-component of (3.17) and ∂/∂x of the z-component of (3.17), we note that

the two operators commute and hence we can eliminate p from our equation. Our equation for the

normal vorticity η is then given by(∂

∂t+ U

∂

∂x− 1Re

∆)η = −(DU)

∂v

∂z. (3.23)

The boundary conditions for (3.21) and (3.23) are

v = Dv = η = 0, −1 ≤ y ≤ 1. (3.24)


3.1.4 Orr-Sommerfeld and Squire’s equation

We expand the disturbance as normal modes given by

v(x, t) = v(y) exp[i(αx+ βz − ωt)],

η(x, t) = η(y) exp[i(αx+ βz − ωt)].(3.25)

We consider a temporal analysis: we take the wave numbers α and β to be real and the frequency

ω (and hence wave speed c, where ω = cα) to be complex. Substituting the modal solutions (3.25)

into (3.21) and (3.23) accordingly (this is equivalent to taking the Fourier transform of (3.21) and

(3.23) in the horizontal directions), we have the Orr-Sommerfeld equation for the disturbance

amplitude v(y) (Orr, 1907; Sommerfeld, 1908):[(−ω + αU)(D2 + k2)− α(D2U) +

i

Re(D2 − k2)2

]v = 0, (3.26)

as well as Squire’s equation for the amplitude of the normal vorticity η(y) (Squire, 1933):[(αU) +

i

Re(D2 − k2)

]η + β(DU)v = ωη, (3.27)

where we have defined k2 = α2 + β2.

The Orr-Sommerfeld equation and Squire’s equation, together with the boundary conditions, can

be regarded as an eigenvalue problem. For our study of stability we are interested in trying to

find the values of ω (in particular ωi). The eigenvalue depends on the wave numbers, Reynolds

number, and the shape of our initial flow. Since ω is complex, the eigenfunctions v and η are

in general complex. Note that the Orr-Sommerfeld equation (3.26) is homogeneous, and the fact

that the solutions to Squire’s equation (3.27) are forced by the solutions of the Orr-Sommerfeld

equation unless β or v are identically zero.

We divide the eigensolutions into the following two classes:

1. Orr-Sommerfeld modes (OS modes)

{vn, ηpn, ωn}Nn=1 (3.28)

This is where we find vn and ωn by solving the Orr-Sommerfeld equation (3.26), and ηpn is

found by solving the inhomogeneous Squire equation (3.27) with vn on the right hand side.

We use the symbol ηpn to emphasise that it is equivalent to a particular solution of the driven

Squire equation.

2. Squire modes (SQ modes)

{v = 0, ηm, ωm}Mm=1 (3.29)

In this case the solution to the Orr-Sommerfeld equation (3.26) is identically zero. Thus

Squire’s equation (3.27) is homogeneous and we can form an eigenvalue problem for ηmand ωm.


In general, the set of Orr-Sommerfeld eigenvalues {ωn}Nn=1 are different from the set of Squire

eigenvalues {ωm}Mm=1.

Considering this as one system, we start by introducing the vector quantity(v

η

).

We can now write the Orr-Sommerfeld equation (3.26) and Squire’s equation (3.27) in matrix

form: (L 0

β(DU) S

)(v

η

)= ω

(D2 − k2 0

0 1

)(v

η

), (3.30)

where the Orr-Sommerfeld operator and the Squire operator are given by

L = αU(D2 − k2)− α(D2U) +i

Re(D2 − k2)2, (3.31)

S = αU +i

iRe(D2 − k2). (3.32)

This can be regarded as a generalised eigenvalue problem: if we let

v =

(v

η

), A =

(D2 − k2 0

0 1

), B =

(LOS 0

βDU LSQ

),

then our eigenvalue problem (3.30) becomes

Av = ωBv.

We see the off-diagonal coupling term β(DU) in the matrix implies that the Squire equation is

driven by solutions of the Orr-Sommerfeld equation, unless v or β are zero. The solution to this

system with the boundary conditions (3.24) gives the eigenmodes (3.28) and (3.29) as discussed

earlier. We emphasise that, formally, they are all eigenfunctions of the same system. It is only due

to the properties of the coupling term that we consider these modes as two distinct families.

3.2 Analytical results

We now use dashes to denote derivatives with respect to the argument; we will change back to

using D at the beginning of the next chapter. For simplicity, we drop the hats on v and η; in the

rest of the report when we write v or η we will mean the amplitude of the disturbances. Every time

we employ the integration sign without the integration limits we will mean the definite integral

over y ∈ [−1, 1], unless stated otherwise.


3.2.1 Squire’s Theorem

For deriving conditions for stability and instability, we investigate how the amplitude of the modes

behave in time. If it grows then we have instability, and if it damps we have stability. Squire’s

theorem states that for a parallel shear flow, the flow first becomes unstable to 2D perturbations.

Hence for finding conditions for instability, we only need to consider the 2D case of (3.30), with β

set to zero. This decouples the Orr-Sommerfeld and Squire equation. To prove Squire’s theorem,

first we need to show that, to each 3D OS mode, there corresponds a 2D OS mode at a lower

Reynolds number. We then show that the SQ modes are always damped, hence they do not

contribute to the flow instability. This implies we only need to consider the Orr-Sommerfeld

equation in our stability analysis.

Theorem 3.2.1 (Squire’s theorem) Given ReL as the critical Reynolds number for the onset of

linear instability for some given α, β, the Reynolds number ReC below which no exponential

instabilities exist for any wave numbers satisfies

ReC ≡ minα,β

ReL(α, β) = minαReL(α, 0). (3.33)

ProofWe first make use of an alternative form of the Orr-Sommerfeld equation (3.26): since ω is just

the frequency, we make use of the identity

ω = αc

with α the wavenumber (or the wavelength) and c the wave speed. ω is complex and α is real,

so c is complex and we can write it as c = cr + ici. An alternative form of the Orr-Sommerfeld

equation is given as

(U − c)(v′′ − k2v)− U ′′v +i

αRe(viv − 2k2v′′ + k4v) = 0. (3.34)

Consider then the 2D case of the Orr-Sommerfeld equation, where we artificially set β = 0:

(U − c)(v′′ − α2v)− U ′′v +i

αRe(viv − 2α2v′′ + α4v) = 0. (3.35)

Comparing (3.34) and (3.35), these have identical solutions v if the following two conditions hold:

α = k =√α2 + β2, (3.36)

αRe = αRe. (3.37)

This is known as Squire’s transform. (3.37) implies that if β 6= 0,

Re = Reα

k< Re. (3.38)


So, to every 3D OS mode, we have a corresponding 2D OS mode at a lower Reynolds number.

We now show the solutions to Squire’s equation are always damped, i.e., ci < 0 ∀ α, β,Re.

Since the equations are now decoupled, we multiply the homogeneous Squire equation (3.27) by

the complex conjugate η and integrate across the y-domain. This gives

c

∫|η|2 dy =

∫U |η|2 dy +

i

αRe

∫η(η′′ − k2η) dy. (3.39)

Integrating the second term on the right hand side by parts and noting all the boundary terms

vanish due to the boundary conditions, we get∫η(η′′ − k2η) dy =

∫ [−|η′|2 − k2|η|2

]dy. (3.40)

Taking the imaginary part of (3.39) and noting (3.40), we have

ci

∫|η|2 dy = − 1

αRe

∫ [|η′|2 + k2|η|2

]dy < 0, (3.41)

because integrand is positive definite if we assume we have the non-trivial case, i.e., η 6= 0. So ciis strictly negative and hence the modes are always damped. This concludes the proof of Squire’s

theorem. 2

We shall now focus on extracting information from the Orr-Sommerfeld equation. We can solve

the Orr-Sommerfeld equation numerically, but first we give some analytical results which provide

stability conditions and eigenvalue bounds.

3.2.2 Eigenvalue bounds

The main result of this section is the eigenvalue bounds first given by Synge (1938), later

generalised by Pai (1954), and further improved by Joseph (1968; 1969). Before we start on

that, we will give some theorems and lemmas for which we will either give a quick sketch of the

proof, or omit the proof entirely; we refer the reader to the relevant literature that are available.

Theorem 3.2.2 (Second integral mean value theorem) If φ and ψ are continuous real functions

on a closed interval [a, b], and, additionally, ψ ≥ 0 on the open interval (a, b), then there exists

some ζ ∈ (a, b) such that ∫ b

aφ(y)ψ(y) dy = φ(ζ)

∫ b

aψ(y) dy.

This is a basic result from analysis which uses the intermediate value theorem in the proof. The

reader can find a proof of this readily on the internet or in an elementary calculus or real analysis

book.


Lemma 3.2.3 We denote∫ 1−1 |φ

(n)|2 dy by I2n. For φ ∈ H whereH is a Hilbert space containing

the complex-valued elements

{φ(y) | φ(±1) = φ′(±1) = 0 ,−1 ≤ y ≤ 1}

appearing as limit points of sequences of four times continuously differentiable functions in the

completion under the norm√I2

2 (the L2 norm), we have the following inequalities:

I21 ≥ λ2

1I20 , I2

2 ≥ λ22I

21 , I2

2 ≥ λ23I

20

λ21 =

π2

4, λ2

2 = π2, λ23 = (2.365)4.

We will give a quick sketch of the techniques used and derive the inequalities without providing a

proof for the method. The reader is referred to the book by Gould (1957) for a more comprehensive

and rigorous discussion.

First of all recall that functions defined on the interval y ∈ [−1, 1] can be viewed as elements in a

vector space. Call this the function space H; note this is an infinite-dimensional space. The inner

product (which induces the metric) is given by

(φ, ψ) =∫ 1

−1φ · ψ dy.

One thing to note is that this space is not complete with respect to the metric; there are Cauchy

sequences that do not converge inH. For us to apply variational techniques we require a complete

space, hence we consider a completion of the space. One way to do it is to consider adding the

appropriate limit points of the Cauchy sequences into H, and the way described above gives us a

functional completion with the desired properties. We denote the completion of the function space

by H. Thus we have completeness of the space; this happens to be a separable Hilbert space

because of the choice of norm we use. This is a technical detail we require if we are to apply

variational techniques to the function.

Now let H be some linear operator which acts of elements on this Hilbert space H. hen we note

that (Hφ, φ) and (φ, φ) are linear functionals. Now suppose we seek λ such that

λ = min(Hφ, φ)(φ, φ)

.

To proceed, we consider a variation of the functional and set it zero to find the extremum (in our

case we want the minimum). It can be shown that this variation is zero when φ is an eigenfunction

of H , and associated with the variational problem is the Euler-Lagrange equation given by

Hφ− λφ = 0.

This is an eigenvalue equation with the appropriate boundary conditions we impose, as well as

any additional natural boundary conditions that may arise if the boundary conditions we impose


are not sufficient to completely determine the solution. We solve it and find the minimum of

the eigenvalues. Now we have the technical details sorted out, we provide a derivation of the

inequalities.

ProofFor our first inequality, we take H = −d2/dy2 which gives

λ1 = min−∫φ′′φ dy∫|φ|2 dy

= minI2

1

I20

where we have used integration by parts, noting that we have homogeneous boundary conditions

for φ ∈ H, and made use of our definitions. It can be shown that−d2/dy2 is a self-adjoint positive

definite operator so all its eigenvalues are real and positive with limit point at positive infinity (see

the book by Gould on a derivation). The associated Euler-Lagrange equation is then

φ′′ + λ1φ = 0

with homogeneous boundary conditions; in this case we don’t have any natural boundary

conditions. This is the equation for simple harmonic motion. Imposing the boundary conditions

and solving this we get two equations with two constants to be determined; writing this as a linear

system gives (sinλ1 cosλ1

− sinλ1 cosλ1

)(A

B

)=

(0

0

).

For this to give us a non-trivial solution we require the determinant of the matrix to vanish. This

gives 2 cosλ1 sinλ1 = sin(2λ1). Since all the eigenvalues are real, positive and bounded below,

we take the smallest positive eigenvalue to give us

I0 ≤π

2I1,

and squaring this gives us the inequality we desire.

For the second inequality, again we have homogeneous boundary conditions, no natural boundary

conditions, and the Euler-Lagrange equation is given by

φiv + λ2φ′′ = 0.

The determinant equation we obtain is sinλ(λ cosλ − sinλ) = 0 and this gives sinλ = 0 or

λ = tanλ. The smallest non-zero positive root is given by λ2 = π. Again, the operator is

self-adjoint and positive definite.

For the third inequality, the boundary conditions are the same but we take H = d4/dy4, which

gives the Euler-Lagrange equations as

φiv − λ3φ = 0.


This is in fact the equation for a vibrating rod (clamped at y = ±1 if we employ homogeneous

boundary conditions). The eigenvalues of this was first computed by Rayleigh (1894) using the

Rayleigh-Ritz method. The relevant determinant equation is tan2(λ1/43 ) = tanh2(λ1/4

3 ), and

solving this by the method of bisection, we obtain λ3 = 2.365... as the smallest non-zero positive

root. Again, the operator −d4/dy4 can be shown to be positive definite and self-adjoint. We have

thus derived our desired inequalities. 2

The first of these inequalities is sometimes known as the Poincare inequality for the 1D case.

Theorem 3.2.4 (Cauchy-Schwartz inequality) Let φ(y) and ψ(y) be two functions in our Hilbert

spaceH with the L2 norm, then∣∣∣∣∫ φ(y)ψ(y) dy∣∣∣∣ ≤ ∫ |φ(y)| dy ·

∫|ψ(y)| dy, (3.42)

with equality if and only if ψ is linearly dependent on φ. Note the integrals are definite integrals

taken over y ∈ [−1, 1].

The Cauchy-Schwartz inequality is a result arising from the study of vector spaces. We use the

L2 norm because this is the norm that gives us a Hilbert space; a more general result is the Holder

inequality for the Lebesque Lp spaces (which are Banach spaces), but we will not require it here.

The proof is readily available on the internet or in an elementary book about vector spaces or linear

algebra.

We now come to the main result of this section. This result and the idea of the proof is due to

Joseph. The proof for parallel shear flows is given in his 1968 paper, but he gives a sharper result

in his 1969 paper. Joseph also takes the domain to be 0 ≤ y ≤ 1. We will give the sharper

results for parallel shear flows between −1 ≤ y ≤ 1 here; see also Drazin & Reid (1981)1.

The isoperimetric inequalities given in Lemma 3.2.3 are used to derive rigorous estimates of the

amplification rates ci and wave speeds cr over the entire set of solutions. The proof has been

elaborated slightly for clarity purposes.

Theorem 3.2.5 Let c(α,Re) be any eigenvalue of the Orr-Sommerfeld equation

(U − c)(v′′ − α2v)− U ′′v = − i

αRe(viv − 2α2v′′ + α4v),

−1 ≤ y ≤ 1, v(±1) = v′(±1) = 0,(3.43)

where we assume U is a real and twice continuously differentiable function, and v ∈ H is a

four times continuously differentiable function (H the Hilbert space as defined before). Let q =

max{|U ′(y)| : −1 ≤ y ≤ 1}, then we have the following results:1However we think their second isoperimetric inequality is numerically incorrect, with an extra factor of 1/4 on

it; we suspect it is a misprint since the results are contradictory using their inequalities, and in Joseph’s 1969 paper it

makes it clear that the first and second inequality should be different.


1.

ci ≤q

2α− 1Re

(π2(π2 + α2)π2 + 4α2

+ α2

). (3.44)

2. No amplified disturbances (modes with ci > 0) of (3.43) exist if

qαRe < f(α) ≡ max[M1,M2],

M1 = λ3π + 23/2α3

M2 = λ3π + πα2,

(3.45)

where λ3 = (2.356)2 is the least eigenvalue of a vibrating rod clamped at y = ±1.

3.

U ′′min ≤ 0 : Umin < cr < Umax +2U ′′maxπ2 + 4α2

U ′′min ≤ 0 ≤ Umax : Umin +2U ′′minπ2 + 4α2

< cr < Umax +2U ′′maxπ2 + 4α2

U ′′max ≤ 0 : Umin +2U ′′minπ2 + 4α2

< cr < Umax.

(3.46)

Here, the subscript max and min denote the maximum and minimum values on the range of

U(y) and U ′′(y) for y ∈ [−1, 1].

ProofWe multiply the Orr-Sommerfeld equation (3.43) by the conjugate eigenfunction v and integrate

across the y-domain. Before we proceed, we make a few observations:

• We impose homogeneous boundary conditions on our solutions. As a consequence, v

satisfies the same boundary conditions as v, and when we integrate by parts accordingly

all the boundary terms will vanish.

• We will end up considering the real and imaginary part of the resulting equation separately.

When we integrate by parts we aim to manipulate the integrand in the form∫|v(n)|dy.

These are all real.

• We employ our isoperimetric inequalities to give numerical bounds.

Using these observations, we note the right hand side of (3.43) will be purely imaginary, and using

integration by parts we get the following expression:

−∫

i

αRe(viv − 2α2v′′ + α4v)v dy = − i

αRe(I2

2 + 2α2I21 + α4I2

0 ). (3.47)

The left hand side of (3.43) gives∫[U(v′′ − α2v)− U ′′v]v dy =

∫[U |v′|2 + (α2U + U ′′)|v|2] dy +

∫U ′v′v dy. (3.48)


The first term of the right hand side here is completely real, but the second is ambiguous. From

complex analysis, we have the identity

z =12

(z + z) +12

(z − z),

where the first part is purely real and the second part is purely imaginary. Since U is real, we have∫U ′v′v dy =

12

∫U ′(v′v + v′v) dy +

12

∫U ′(v′v − v′v) dy

= −12

∫U ′′|v′|2 + i(Q−Q)

(3.49)

where Q = (i/2)∫U ′v′v dy (multiplying by i means we need to swap the signs). So the first

term is purely real and the second is purely imaginary. Now, for the remaining term involving

c = cr + ici, we integrate by parts as well and we get

−(cr + ci)∫

(v′′ − α2v)v dy = (cr + ici)(I21 + α2I2

0 ). (3.50)

Putting all the above identities together and considering the real and imaginary part separately, we

have

ci =(Q−Q)− (αRe)−1(I2

2 + 2α2I21 + α4I2

0 )I2

1 + α2I20

(3.51)

cr =∫

[U |v′|2 + (α2 + U ′′/2)|v|2] dyI2

1 + α2I20

. (3.52)

1. We consider first the imaginary component (3.51). We have the following inequality:

|Q−Q| = 12

∣∣∣∣∫ U ′(v′v + v′v) dy∣∣∣∣ ≤ 1

2· 2∫|U ′| · |v′| · |v| dy ≤ qI0I1 (3.53)

where we have used the Cauchy-Schwartz inequality, the second integral mean value

theorem on the last inequality and employed our definitions. Making use of our

isoperimetric inequalities given in Lemma (3.2.3), we note the following:

0 ≤ (I21 − αI2

0 ) = I21 − 2αI1I0 + α2I2

0 ⇒ 2αI1I0 ≤ I21α

2I20 , (3.54)

I22 + 2α2I2

1 + α4I20

I21 + α2I2

0

=I2

2/I20 + 2α2I2

1/I20 + α4

I21/I

20 + α2

≥ π2I21/I

20 + 2α2I2

1/I20 + α4

I21/I

20 + α2

≥ π2(π2/4) + 2α2(π2/4) + α4

(π2/4) + α2.

(3.55)

Combining all of these with (3.51), we get

ci ≤qI1I0

2αI1I0− 1αRe

π2(π2 + α2) + (π2 + 4α2)α2

π2 + 4α2(3.56)

and we establish our first result (3.44).


2. For our second result, we go back to the expression for the imaginary component of the

Orr-Sommerfeld equation (3.51). Using (3.53), this gives

ci ≤qI1I0 − (αRe)−1(I2

2 + 2α2I21 + α4I2

0 )I2

1 + α2I20

. (3.57)

For no amplified disturbances to exist, we require ci < 0. This means the nominator of

(3.57) has to be strictly negative, i.e.,

qαRe <I2

2 + 2α2I21 + α4I2

0

I1I0. (3.58)

We can bound the right hand side of (3.58): the argument is that either we can bound it

using our isoperimetric inequalities, and when we can’t bound some of the inequalities (e.g.

I0/I1), we throw them away as long as they are positive. We have the following:

I22 + 2α2I2

1 + α4I20

I1I0≥ I2

I1

I2

I0+

2α2

I1I0

(I2

1 +α2I2

0

2

)=I2

I1

I2

I0+

2α2

I1I0

[(I1 −

αI0√2

)2

+2α2

√2I1I0

]

≥ I2

I1

I2

I0+

2α2

I1I0

2α2

√2I1I0 ≥ λ3π + 23/2α3,

(3.59)

I22 + 2α2I2

1 + α4I20

I1I0≥ I2

I1

I2

I0+

2α2I21

I1I0≥ λ3π + 2α2π

2(3.60)

and these correspond to our M1 and M2 respectively. We require two different bounds

because for small values of α, α2 > α3 (and vice versa), and for the most relaxed bound

(with numbers given explicitly) we take the maximum of these; a plot of this is given in

Figure 3.3. This establishes the second result (3.45).

3. For our last result, we take the real part of the Orr-Sommerfeld equation (3.52) and apply

the second integral mean value theorem. This gives

cr = U(y1)I2

1 + α2I20 + U ′′(y0)/2

I21/I0

2 + α2(3.61)

where y1 and y2 are some mean values in [−1, 1]. Then for the subscript max and min

denoting the maximum and minimum value of the function U and U ′′ over y ∈ [−1, 1], we

have the following relation:

2U ′′minπ2 + 4α2

<U ′′(y0)/2I2

1/I20 + α2

<2U ′′maxπ2 + 4α2

. (3.62)

Accordingly, for our specified cases, to get the bound with the biggest range, we have:

U ′′min ≤ 0 : 0 <U ′′(y0)/2I2

1/I20 + α2

<2U ′′maxπ2 + 4α2

U ′′min ≤ 0 ≤ Umax :2U ′′minπ2 + 4α2

<U ′′(y0)/2I2

1/I20 + α2

<2U ′′maxπ2 + 4α2

U ′′max ≤ 0 :2U ′′minπ2 + 4α2

<U ′′(y0)/2I2

1/I20 + α2

< 0.

(3.63)


Figure 3.3: Linear stability bounds for the Orr-Sommerfeld equation. Here, we have plotted f(α)/α. The

region of certain linear stability lies to the left of the curve. This gives minf(α)/α ≈ 18.1, with α ≈ 1.46.

From this, we can easily establish the result (3.46) by demanding that we have the inequality with

the smaller range. This concludes our proof. 2

Before we leave the analytical portion of this chapter and do some numerical calculations on

the Orr-Sommerfeld equation, a few comments should be made regarding Joseph’s result. These

conditions are sufficient conditions for linear stability. The imaginary part of the Orr-Sommerfeld

equation involving ci may be regarded as the 2D energy equation of the disturbance. Noting the

result (3.45) (or graphically, Figure 3.3), we see that for each fixed Re, there is some minimum

wavenumber αmin below which there is linear stability; this corresponds to the case where we

know for certain that the dissipation of energy in the system due to viscosity is higher than the

transfer of energy from baseflow to disturbance, hence we have stability. By the same observation,

for each fixed Re there is some maximum wavenumber αmax above which there is also linear

stability. This may seem counter intuitive, but it corresponds to the case where the disturbance

vorticity is so great that the energy dissipation again overcomes the production of energy. This is

easily seen if we consider the limit α→∞ in (3.44) and (3.45): ci behaves like −α, and f(α)/α

tends to infinity. One final remark regarding (3.45) is that it suggests, for every finite α, there

exists an Re(α) above which linear stability cannot be deduced.


(3.46) shows the wave speed of the disturbance wave for all parallel motions is restricted to an

interval only slightly larger than the range of U . Also, it suggests we can have negative wave

speeds occurring.

In the case where Re → ∞ (the zero viscosity or inviscid limit), (3.44) reduces to the inviscid

result of Høiland (1953). Here, the presence of the viscous term (the negative term) suggests that

viscosity is stabilising, in that it provides a sharper bound than the inviscid case.

A similar bound exists for Squire modes (Davis & Reid, 1977):

Umin < cr < Umax, ci < −π2/4 + k2

αRe(3.64)

and the method used to derive this is similar to that in deriving Joseph’s bounds.

We aim to use a similar approach in Chapter 4 when we are dealing with non-Newtonian fluids

to try and derive some bounds for the analogue of the Orr-Sommerfeld equation and Squire’s

equation.

3.3 Numerical results

Giving a full account of numerical analysis is clearly out of place in this report. Instead, we

mention a few ideas without developing it rigorously, and refer the reader to the excellent book

by Iserles (1996) for an introduction to numerical analysis, and the book by Trefethen (2000) on

spectral methods. For the reader who is familiar with spectral Chebyshev collocation methods,

please skip to the end of chapter for the numerically computed eigenvalue spectrum of the Orr-

Sommerfeld equation. For those who are unfamiliar and those who would like a quick reminder,

please read on.

3.3.1 Spectral method

With the increasing computational power of computers, numerical tools are more accessible now

than ever before. Ultimately, we seek a method which converges to the solution (convergent), does

not ‘misbehave’ (stable), and converges to the solution fast enough (the rate of convergence is high

enough). For the Orr-Sommerfeld eigenvalue problem, we choose to use the spectral method,

specifically using a Chebyshev collocation method. This is not the only method available; indeed,

before the development of the spectral method in the 70’s people used a variety of methods such as

shooting methods or finite-elements method to solve fluid mechanics problems. The appropriate

method to use depends on the context of the problem.

Differentiation matrices

To numerically take the derivative of some function f(y), we could proceed as follows:


1. Discretise our domain into {y1, ..., yN}, and sample our function f(y) on these grid points.

Call f(yj) := fj .

2. Interpolate these sampled points with some known function p(y), with p(yj) := pj = fj ∀j.

3. Take the derivative of p and call this the approximation to f ′(y), and let w(yj) := wj = p′j .

4. Invoke any extra conditions as necessary.

Then it is intuitive to think that the finer we discretise our domain, i.e., the more grid points and

samples we take, the better our approximation becomes; this is generally true (sometimes we have

stability issues where the errors do not die away when we take a finer spacing, but this is usually

due to the numerical scheme). The interpolant used is usually a polynomial (and there is always

a unique polynomial of degree N through all fj , a result proved by Lagrange), but it could also

be a function, such as the trigonometric functions; again, the appropriate interpolant depends on

the context of the problem. If we write w and f as column vectors, we can represent the relation

between w and f as a matrix-vector product

w = Df ,

where D is some N by N matrix with the entries depending on the interpolant; D is known as

the differentiation matrix. If we take a polynomial of degree less than N , then A will be suitably

sparse. The idea of spectral methods is we take an interpolant that gives us a dense differentiation

matrix.

Advantages of Spectral methods

We use spectral methods because of the following reasons:

• High accuracy. The error of an approximation to the solution of a typical numerical method

tend to decrease like O(N−m) for some positive constant m that depends on the numerical

scheme and the smoothness of the solution, N being the number of grid points we take. For

spectral methods, convergence at O(N−m) for every m is achieved provided the solution is

C∞ (infinitely differentiable), and O(cN ) for 0 < c < 1 if the solution is suitably analytic.

We expect our solution to the Orr-Sommerfeld equation to be smooth enough, hence we try

and exploit the spectral accuracy accordingly.

• Efficiency. Due to spectral accuracy, we do not need to take N to be too large because we

have guaranteed convergence rates. This is also of importance because dealing with large

and dense matrices is computationally not desirable. This is still an advantage: for the same

accuracy, dealing with a dense 150 by 150 dense matrix may still be faster than dealing with

a suitably sparse 5,000 by 5,000 matrix.


• Suitability. It is known that the method of finite elements is much more flexible than spectral

methods with regards to the geometry of the domain. That is certainly true, but we don’t

need this flexibility here because our domain is simple enough for the spectral method to be

easily applied.

Interpolants and Gauss-Lobatto points

The choice on interpolants largely depends on the problem at hand. If we have periodic domains

then it is natural to consider trigonometric functions as interpolants. Indeed, there are many tools

associated with it, namely that of Fourier transforms. It was the discovery of the Fast Fourier

Transform (FFT) that led to the surge of interest in the 1970s for spectral methods (FFT works at

O(N logN) operations which is substantially less than the conventional O(N2) of the Discrete

Fourier Transform).

Here we do not have the periodic domains so instead we use polynomial interpolants. It seems

intuitive to take equally spaced grid points, but this is in fact an extremely bad move. With

increasing N , the error of polynomial interpolation on equispaced points increase exponentially;

this is known as the Runge phenomenon. The correct move is actually to take unevenly spaced

points. There are a variety of ways to do this, and the scheme we shall use are the so called

Chebyshev points, or Gauss-Lobatto points. These are given by

yj = cos(jπ

N

), j = 0, 1, ..., N.

Geometrically, we can visualise this as the projection of equally spaced points on the upper-half

unit circle on to the axis (see Figure 3.4). It will be seen shortly that the Gauss-Lobatto points are

exactly the roots of a degree N Chebyshev polynomial TN (y), so it is natural to take TN (y) as our

interpolant. This is by no means the only way to take unevenly spaced points; for example, the

famed Gauss quadrature uses Gauss-Legendre points instead (the roots of a degree N Legendre

polynomial).

Chebyshev polynomials

Chebyshev polynomials (of the first kind) are solutions to the differential equation

d

dy

(√1− y2

d

dyTn(y)

)+

n2√1− y2

Tn(y) = 0 (3.65)

for y ∈ [−1, 1]. It may be noted that the solutions TN (y) = cos(N arccos(y)) are solutions

of the above singular Sturm-Liouville eigenvalue problem (in self-adjoint form). Because we

have a solution to the Sturm-Liouville problem then the eigenfunctions have nice properties: the

eigenvalues are all real, to each eigenvalue there is a unique eigenfunction (up to a normalisation


Figure 3.4: Gauss-Lobatto points. Note that y1 = 1 and yN = −1, so we count from the right to the left.

factor), and the eigenfunctions are complete and form a basis (and may be normalised accordingly

to get an orthonormal basis). Compare this is to a Fourier expansion of a function in terms of

trigonometric functions on a symmetric domain on the real line. The same can be done with

Chebyshev polynomials: we may approximate a function f(y) on [−1, 1] by

f(y) =N∑n=0

anTn(y) (3.66)

and evaluate (3.66) on the Gauss-Lobatto points yj = cos(jπ/N). This discretisation gives us a

set of numbers: writing then fj = a1T1(yj) + ...+ aNTN (yj), we havef1

...

fN

=

T1(y1) T2(y1) · · · TN (y1)

T1(y2) · · · · · · TN (y2)...

. . ....

T1(yN ) · · · TN (yN )

a1

...

aN

. (3.67)

Here, the column vector a is the unknown. We can readily find fj , and evaluate the elements of

the matrix via a direct formula for Chebyshev polynomials, or by the Rodrigues’ formula. For

computational purposes, we use the recurrence relation given by

T0(y) = 1, T1 = y, Tn+1(y) = 2yTn(y)− Tn−1(y), n = 2, 3, 4... . (3.68)

If we need to find the column vector a we solve the system of linear equations. We will need to if

we need to take the derivative of f . It is clear that a does not depend on y so when we approximate

the derivative we just take

f (k)(y) =N∑n=0

anT(k)n (y). (3.69)


A similar recurrence relation exists for the derivatives of Chebyshev polynomials, given by

T(k)0 (y) = 0, T (k)

1 (y) = T(k−1)0 (y), T (k)

2 (y) = 4T (k−1)1 (y),

T (k)n (y) = 2nT (k−1)

n−1 (y) +n

n− 1T

(k)n−1(y), n = 3, 4, ...

(3.70)

So if we need to approximate f (k), we would first solve the system of linear equations given

by (3.67), find a, find the n-th differentiation matrix using the recurrence relations, construct the

matrix as in (3.69), and compute the matrix vector product to find the column vector f (k)j . We will

need to do this in the non-Newtonian case: in the Newtonian case we have an analytic expression

for the baseflow U(y) (either Couette or Poiseuille), but for the non-Newtonian case we will only

have a numerical solution for the baseflow, so we need to take numerical derivatives of it when we

discretise the eigenvalue equation before we can compute the spectrum.

Clenshaw-Curtis Quadrature

To numerically integrate, the idea is essentially the same: to integrate a function g(y), we

interpolate it with an interpolant, and integrate this interpolant. Of course here we will need to

assume g is suitably smooth and free of singularities otherwise the result will be nonsense. This

is the case here because we expect the baseflow to be smooth. The Clenshaw-Curtis quadrature

uses Chebyshev polynomials as the interpolant, evaluates the approximation at the Gauss-Lobatto

points, and integrates it over -1 and 1.

Consider the integration of g over [−1, 1] as a linear functional I(g), then its numerical

approximation is another linear functional IN (g). By the Riesz representation theorem, we can

represent this linear functional IN (g) as an inner product of g = (g1, ..., gN ) with some weight

function W (y), i.e.,

IN (g) = (W , g).

Taking the Euclidean inner product, we have∫ 1

−1g(y) dy =

N∑j=0

g(yj)W (yj),

with W (yj) the Chebyshev integration weight function, and these fourmlas ensure we have

spectral accuracy. We are not going to derive the weight function here, but we refer the reader

to Trefethen (2000) (Chapter 12) or Schmid & Henningson (2001) (Appendix A).

Other remarks

There is a correspondence between FFT and our Chebyshev method and indeed we could combine

the two together to speed up the methods we are going to use. We are not going to do that here


because we are not dealing with N large enough for this difference in computation time to be

remotely noticeable (although we do make use of it in our quadrature program when we need to

numerically integrate the baseflow in Chapter 4). This is one of the reasons why spectral methods

became more popular. Please see Trefethen (2000) for more details.

We could use [a, b] instead of [−1, 1]; we can always find some mapping (with suitable conditions)

of the interval [a, b] to [−1, 1] and invert the result we get from our spectral method. We make use

of this in our quadrature program, but otherwise we do not require this mapping.

This method is known as a collocation method, where we choose a finite-dimensional space of

candidate solutions and a number of points in the domain (called collocation points), and to

select that solution which satisfies the given equation at the these points. In our case we choose

Chebyshev polynomials as candidate solutions and Gauss-Lobatto points as collocation points.

There are different formulations and there is considerable debate about which one is more superior

or applicable. We use this because of the properties of the equations we are dealing with; the Orr-

Sommerfeld operator is a highly non-normal operator and the eigenvalue spectrum is extremely

sensitive to perturbations, and it is known that collocation methods give better results. We will

give a brief discussion of this in Chapter 5.

3.3.2 Results

We will now compute the spectrum of the Orr-Sommerfeld equation. We choose to use the

program MATLAB to impliment the spectral collocation method mentioned in the previous

section. The MATLAB codes are given in Appendix A. The spectrum is given in Figure 3.5.

The eigenvalues of the Poiseuille flow are located on three branches which are labelledA (cr → 0),

P (cr → 2/3) and S (cr ≈ 2/3) by L. M. Mack. We used the so called critical Reynolds number

here, as computed by Orszag (1971) (the values he got are in fact Re ≈ 5772.22, with α ≈ 1.02).

We see there is one mode on the A branch which is on the borderline of being unstable; if we use

Re = 5773, we get max{Im(ω)} > 0 and we do have instability. This unstable mode is called

a Tollmien-Schlichting wave (Tollmien, 1935; Schlichting, 1933) in honour of W. Tollmien and

H. Schlichting who first showed that the Orr-Sommerfeld equation has unstable modes for flows

without inflexion points (more next paragraph). Thus Poiseuille flow is linearly unstable when

the critical Reynolds number is exceeded. When we increase the viscosity (in effect lowering

the Reynolds number), the least stable eigenvalue has a smaller imaginary part, thus we conclude

viscosity is stabilising.

Now compare this briefly with the inviscid case (where there is an absence of viscosity). A result

due to Lord Rayleigh (1880) known as Rayleigh’s inflexion point theorem states that a necessary

condition for an inviscid parallel plane flow to be unstable is that the flow profile should have

an inflexion point within the domain. The point here is Poiseuille does not have an inflexion

point, so the inviscid Poiseuille cannot be unstable. But Poiseuille flow can be unstable in the


Figure 3.5: Spectrum for a) Poiseuille flow with N = 150, Re = 5772, α = 1 and β = 0. b) Couette

flow with N = 100, Re = 1000, α = β = 1. For a), we took N = 150 for comparison purposes with the

non-Newtonian spectrum later in Chapter 4. N = 100 is actually sufficient for discretisation errors to not

occur in the range of ωi and ωr specified here. For an example where this is not the case, see b) of Fig.4.4.

Newtonian case. We must then conclude viscosity acts in a dual manner, both stabilising and

destabilising, even though we don’t seem to have a physical explanation on why viscosity should

be destabilising.

Th spectrum for plane Couette flow does not have a P branch, but instead has two A branches.

The S branch corresponds to a phase speed equal to the average speed of the mean flow (in this

case zero). Numerical experiments suggest that plane Couette flow is linearly stable for all values

of Re, and an analytic proof of this was supplied by Romanov (1973) for parallel shear flows.

There is as yet no analytical proof for the stability of a pipe Couette flow, although numerical

results suggests it is always linearly stable. It is also known that the Hagen-Poiseuille flow (pipe

Poiseuille flow) is linearly stable for all Reynolds number; compare this to the plane Poiseuille

flow.

When considering disturbances with α = 0, the spectrum is different to when we consider c as the

eigenvalue. The spectrum in this case consists of just one branch, and using the same technique we

used to prove that all the Squire modes are damped, we can show that the modes corresponding

to α = 0 are always damped. We can do some analytical work in this special case: the Orr-

Sommerfeld equation (3.26) and Squire equation (3.27) reduce to a constant coefficient differential

equation, so the dispersion relation and the eigenfunctions may be determined analytically. We

refer the reader to Chapter 3 of Schmid & Henningson (2001).

We have now given some analytical and numerical results for the Newtonian case. We aim to do

exactly the same thing for the non-Newtonian case: we derive the equations of motion, consider

modal perturbations, derive its associated stability equation, numerically compute its spectrum


and derive some eigenvalue bounds. The added complication now is that viscosity is no longer

just a constant, but a function that varies across the domain. Also, the viscosity function affects

what baseflows we have. There are no analytical expressions for the baseflow (apart from when

we choose parameters such that it reduces to the Newtonian case) so we also have to numerically

compute the baseflow. It is obvious that the spectrum will depend on the baseflow, hence we

expect different results emerging. Will a non-Newtonian fluid be more stable or more unstable?

48

Chapter 4

Stability of non-Newtonian fluids

Again we take parallel shear flows between two boundary plates at y = ±1. We perturb the

baseflow, substitute it into the momentum equation, linearise, and study its stability via the

eigenvalue relation. Most of the method and techniques used parallels that of the Newtonian case

we have discussed already. We concentrate on shear-thinning fluids and use the Carreau model to

provide us with a viscosity function.

Our principle hypothesis of this chapter is that the presence of a shear thinning viscosity stabilises

the flow, up to a certain point, and when this point is breached it has a destabilising effect. A

stronger hypothesis is given and discussed in Chapter 5.

We first derive an analogue of the Orr-Sommerfeld equation and the Squire equation, but we

compute the spectrum before we provide some analytical results. The reason for this is that

our principle hypothesis may be demonstrated by the numerical results, but it is not as less

obvious trying to demonstrate this using the analytical results. Also, most of the literature on

non-Newtonian fluids is based on a numerical investigation, so there is considerably more data

that we can refer to in our discussion.

4.1 The equations

Now that viscosity is a function rather than a constant, we do not have an equivalent of the

Navier-Stokes equation. Recalling the linear momentum equation (2.26), and how we decomposed

the stress tensor σ into the pressure term and the anisotropic part τ in (2.31), we have as our

momentum equation

ρDu

Dt= −∇p+∇ · τ , (4.1)

along with the incompressibility condition ∇ · u = 0. Note that the gradient operator acts on the

tensor to give a vector. Our constitutive relation (2.36) is given by

τ = µ(γ)γ,

Chapter 4. Stability of non-Newtonian fluids 49

with γ = (∇u) + (∇u)T as the rate of strain tensor. Again, if we consider parallel shear flows of

the form U = U(y)ex, then the general momentum equation reduces to

dp

dx=

∂

∂y

(µ(γ)

dU

dy

). (4.2)

If we now scale as appropriate and non-dimensionalise everything, recall the viscosity function

for the Carreau model (2.37), we have (as before but removing all the dashes)

Du

Dt= −∇p+∇ · τ , ∇ · u = 0, τ =

1Re

µ(γ)γ (4.3)

µ =µ∞µ0

+[1− µ∞

µ0

][1 + (λγ)2](n−1)/2 (4.4)

Re =ρLV

µ0, (4.5)

and all quantities here are non-dimensional, normalised by the half channel thickness L and the

centre line velocity V . We have redefined the Reynolds number and rescaled the viscosity by the

viscosity at the zero-shear rate, and this is consistent with the Newtonian case where µ ≡ µ0.

Recall that γ is the second invariant of the rate of strain tensor, µ∞/µ0 << 1, n, λ are time

constants, and when n = 1 or λ = 0 we recover the Newtonian case.

4.1.1 Baseflow profile

First we will illustrate how the viscosity affects the shape of the baseflow. We take our y-domain

to be [0,1] (from the wall to the centre line) and extend our results by a symmetry argument. Since

we are taking unidirectional flowsU = U(y)ex as potential base solutions, it can be shown that γ

reduces to |dU/dy|. We expect our solution to be almost like a parabola similar to the Poiseuille

flow, and hence in the region y ∈ [0, 1], γ = −dU/dy. Integrating both sides of (4.2) with respect

to y, we have

ydp

dxRe+ µ(γ)γ = 0. (4.6)

µ here is given by (4.4). We thus need to solve this non-linear equation for γ and then integrate it

with with respect to y to find the baseflow U .

We can’t solve for γ analytically unless we have certain well chosen values of the parameters.

Instead, we solve this numerically by a bisection method: we choose some a and b in the y-

domain, compute the midpoint c, and compute the function values at these points. If f(a)f(c) < 0

then because γ should be continuous by the intermediate value theorem it crosses the root of the

equation, so we take c to be our new b and iterate. Otherwise, if f(b)f(c) < 0, we take b to be our

new a and iterate. If neither case is true then there is something wrong and we need to consider

other starting values of a and b.


Once we have found all the roots of the equation to (4.6), we will integrate it to find the baseflow.

We do this by noting for some x ∈ [0, 1),∫ 1

xγ dy =

∫ 1

x−dUdy

dy = −U(1) + U(x) = 0 + U(x) (4.7)

because of the no-slip boundary conditions we impose. We make use of this relation and

numerically integrate γ to get the baseflow U(y). We will not be using these computed values

as our baseflow in the numerics later when we are computing the spectrum; to do the general case

for arbitrary parameters it will be argued later that it is computationally expensive and not very

revealing. Computing the baseflow here is purely for illustrative purposes on how the viscosity

affects the shape of the baseflow, and instead of integrating γ by a spectral method, we are going to

take a uniform grid on the y-domain and use the trapezium rule built into the MATLAB programs.

The resulting diagrams are given in Figure 4.1.

We have also varied the parameter Re(dp/dx) accordingly, so that the baseflow profiles are such

that U(0) = 1. We do this by using the bisection method again but this time treating Re(dp/dx)

as the variable. We take the starting values a and b to be some different numbers. For these two

different numbers we compute the respective flow profiles, test the values U(0) of each flow and

iterate accordingly to get U(0) = 1.

The reason why we use the bisection method is that it guarantees convergence and stability,

as long as the function we are dealing with is continuous and has the appropriate roots. γ is

clearly continuous by construction, since we require U to be at least twice differentiable, and the

fundamental theorem of algebra guarantees that we have roots to the equation (one trivial case is

when y = 0 since this is the turning point of U by symmetry arguments). Something like the

Newton-Rhapson method may not converge to the root because we are likely to have problems

cropping up when we take the derivative of the appropriate function. Although the bisection

method is very slow, it is very robust, and, due to our imposed conditions, we can be sure that we

always have convergence.

We observe from the shapes of the baseflows that with increasing deviation from the Newtonian

viscosity, we get a ‘flattening’ of the baseflow. This is one reason why shear thinning fluids are

sometimes called pseudoplastics. Compare this with how a tube of toothpaste would flow when

squeezed.

4.1.2 Linearised momentum equation and tangent viscosity

Coming back to our stability analysis, we now consider adding a small perturbation term to our

baseflowU (with associated pressure P ) which is a solution of (4.3). If we denote our perturbation

by u′ and p′, then we have

τ ′ =1Re

µ′(γ′)γ ′ (4.8)


Figure 4.1: Baseflow of Carreau model for varying values of the time constants n, λ computed using a

bisection method and the trapezium rule. a) fixed n. b) fixed λ. Notice how the profiles flatten out as we

have increasing deviation from a Newtonian fluid.

where γ′ is the second invariant of the perturbed rate of strain tensor:

γ ′ =[∇(U + u′)

]+[∇(U + u′)

]T =[∇U + (∇U)T

]+[∇u′ + (∇u′)T

]. (4.9)

The second invariant and the viscosity is now a function of (U + u′). Considering a Taylor

expansion of the viscosity function in powers of u′ about U , we have

µ(γ(U + u′)) = µ(γ(U)) + γij(u′) ·∂µ

∂γij(U) +O(|u′|2). (4.10)

Putting (4.9) and (4.10) together, we have an expression for the perturbed anisotropic part of the

stress tensor:

Re · τ ′ =[µ(U) + γij

∂µ

∂γij(U) +O(|u′|2)

] [γ(U) + γ(u′)

]= µ(U)γ(U) + µ(U)γ(u′) + γij(u′)

∂µ

∂γij(U)γ(U) +O(|u′|2).

(4.11)

We have abused the notation somewhat: when we write γ(U) or γ(U), we mean the function

or the tensor with respect to the argument U (or u′). When we linearise, we drop the O(|u′|2)

terms, and the rest of the terms in the general momentum equation is as it was in the Newtonian

case. When putting everything together, µ(U)γ(U) along with ∇P cancels out because U is a

solution of the equation (4.2). By dropping all the dashes, the linearised momentum equation for

the disturbance is∂u

∂t+ (u · ∇)U + (U · ∇)u = −∇p+∇τ, (4.12)

Re · τ = µ(U)γ(u) + γij(u)∂µ

∂γij(U)γ(U) = µ(U)γ(u) + µγ(U). (4.13)

µ is called the viscosity perturbation. We also have the incompressibility condition as before.


We argue now because we are dealing with a unidirectional shear flow, γ(U) = γ(U(y)ex) is

only non-zero on the off diagonal entries of the anisotropic stress tensor. We are not concerned

with any terms of τ which have a z-component because we are dealing with a flow which depends

on y, flowing in the x direction. By this argument, we have

τij =

{µ(U)γij(u) for ij 6= xy, yx

µt(U)γij(u) for ij = xy, yx,(4.14)

where

µt = µ(U) +dµ

dγxy(U)γxy(U) (4.15)

is termed the tangent viscosity. Also note that γ is symmetric so we may swap the index. For a

unidirectional shear flow, the tangent viscosity is defined by µt = dτxy/dγxy; in experiments, the

value of viscosity usually measured is the effective viscosity µeff = τxy/γxy. For shear thinning

fluids we have (by analysing the viscosity models such as the Carreau model) µt < µ everywhere

in the y-domain except at the centre of the channel, where the two quantities are equal. For an

illustration of this, please see Figure 4.3. This is an important observation which we shall make

use of later.

4.1.3 Modified Orr-Sommerfeld and Squire equations

If we now proceed as in the Newtonian case, we write the linearised momentum equation for the

disturbance in components. We take the divergence, make use of the incompressibility condition

and find an expression for p which we can use to eliminate p in the equations. We consider the

velocity-vorticity as before, taking η = ∂u/∂z − ∂w/∂x, where u = (u, v, w). Taking modal

solutions of the formv(x, t) = v(y) exp[i(αx+ βz − ωt)],

η(x, t) = η(y) exp[i(αx+ βz − ωt)],

and dropping all the hats knowing we have an equation for the amplitude of the disturbance and

normal vorticity, the modified Orr-Sommerfeld and modified Squire’s equation are given by(L C1

C2 S

)(v

η

)=

(∆v

η

), (4.16)

L =α[U∆− (D2U)] +i

Re[µ∆2 + 2(Dµ)D3 + (D2U)D2 − 2k2(Dµ)D + k2(D2µ)]

+iα2

Re k2(D2 + k2)[(µt − µ)(D2 + k2)],

C1 =− iαβ

Re k2(D2 + k2)[(µt − µ)D],

C2 =β(DU)− iαβ

Re k2D[(µt − µ)(D2 + k2)],

S =αU +i

Reµ∆ +

i

Re(Dµ)D +

i

Re

β2

k2D[(µt − µ)D],

(4.17)


with k2 = α2 + β2, D = d/dy, ∆ = D2 − k2. Note these are operators; when we expand these

out we need to be careful about the order of operation. Also note that there are cases when we will

need to apply the identity

Dnφ =n∑r=0

(n

r

)φ(n−r)Dr (4.18)

where in this instance φ may be η, v, µ or µt. A bracket in this case always has priority. From

this set up we note that if µ is a constant function, µt − µ is zero because dµ/dγxy is zero. All

derivatives of µ are zero and in this reduction we recover the Newtonian case, namely that of the

Orr-Sommerfeld and Squire operators in (3.30), with the coupling operator C2 reducing to β(DU)

and C1 reducing to zero as required.

Together with the boundary conditions v = Dv = η = 0 at y = ±1, this forms a generalised

eigenvalue problem, and we can compute its spectrum as before.

4.2 Numerical results

In the Newtonian case, we derived the analytical results first before computing the spectrum. In

particular, in deriving Squire’s theorem, we showed that to obtain a criterion for instability it is

enough to consider the 2D linearised and decoupled equations for the amplitude of the disturbance,

namely that of the Orr-Sommerfeld equation. As far as we know there is no Squire’s theorem

equivalent for a parallel shear flow in the non-Newtonian case. The difficulty it seems is that

when we take a spanwise-homogeneous disturbance (β = 0), some terms in the Orr-Sommerfeld-

like operator L and Squire-like S are artificially forced to zero, whilst the coupling operators C1

and C2 are identically zero. In the Newtonian case, when β was forced to zero, the coupling term

β(DU) was the only one that is identically zero, and the corresponding 2D equations had a similar

structure to those in the full 3D case. It is not immediately obvious how we would find a Squire

transform equivalent to deal with this.

Numerical tests have been carried out and the common conclusion seems to be that the lowest

critical Reynolds number is obtained with spanwise-homogeneous disturbances. Authors therefore

generally take the stability of shear flows to be governed by the modified Orr-Sommerfeld equation

(for example, in Govindarajan et al. 2001, 2003; Nouar et al. 2007). Of course, numerical testing

does not constitute a proof, but we argue that there is no reason putting off a field of study just

because there is a lack of rigourous proof available, especially when the numerical results clearly

suggests the result is valid. Without turning this into a philosophical debate, we will take the

numerical results to be true, and take β to be zero for our stability analysis since we seek the

most unstable modes. With this in mind, we observe that the equations decouple; it can be shown

analytically that the decoupled Squire modes are always damped (Theorem 4.3.1) by considering

the homogeneous modified Squire equation, i.e., v forced to zero. Thus we only need to consider


the modified Orr-Sommerfeld problem given by

Lv = ω(D2 − α2)v, v(±1) = (Dv)(±1) = 0.

We compute the spectrum with a Chebyshev spectral collocation method as before, taking into

account our new form of the Orr-Sommerfeld operator L, as well as computing the viscosity

function accordingly. We will give the results first (Figure 4.2, 4.3, 4.4), then explain and comment

how we got them.

We take n = 0.5 in our computation. In Figure 4.4, the thing to note is that the maximum

imaginary part of the eigenvalue given in Figure 4.4 is negative when λ = 1 but when λ = 4 this is

positive (Reynolds number Re = 5772 and α = 1 as before). So comparing this to the Newtonian

spectrum for Poiseuille flow (Figure 3.5), we conclude that the presence of a shear thinning fluid

stabilises the flow, but too much deviation from the Newtonian fluid causes a destabilisation of

the flow. This agrees with the results give by Nouar et al. (2007). We have computed the spectra

for varying values of λ, from which we find that we have stabilisation until λ ≈ 3, and this also

agrees with the reported results.

In their paper, Nouar et al. argue that the presence of a shear thinning viscosity should consistently

stabilise the flow. Indeed, this is a well known result: a small viscosity contrast greatly stabilises

the flow from experimental data, and analysis of the flow of these fluids show that one can

raise the critical Reynolds number by up to fifteen times (Govindarajan et al., 2001, 2003). The

physical mechanism for this stabilisation appears to be the drop in the production of energy by the

disturbance. This stabilisation does not seem to appear in our case for all values of λ, but Nouar

et al. claims we can show that we have consistent stabilisation by considering a different scaling.

We will discuss this briefly in Chapter 5.


Figure 4.2: Baseflow of Carreau model with n=0.5. a) λ = 1. b) λ = 4. We have a ‘flattening’ of the flow

for a shear thinning fluid driven by the pressure.

Figure 4.3: Viscosity functions of Carreau model with n=0.5. a) λ = 1. b) λ = 4. Note how µt is always

less than or equal to µ for shear thinning fluids.

Figure 4.4: Spectrum of Carreau model with n=0.5. a)λ = 1. b) λ = 4. Note that for λ = 1 we have

stabilisation, where as for λ = 4 we have an unstable mode. Also note the ‘splitting’ that occurs near the

bottom of b).


4.2.1 Finding the baseflow U(y)

We take y ∈ [−1, 1] again. Before, we argued that γ = −dU/dy, but here we argue that we can

re-insert our original definition back in, namely that of γ = |dU/dy|. We take n = 0.5 in the

Carreau model and assume the ratio µ∞/µ0 is small and may thus be neglected. The choice of

this particular value of n gives us shear-thinning behaviour, and the general form of (4.6) given by

y(dp/dx) = µ(γ) · (±γ) takes on the simpler form

ydp

dxRe =

±γ[1 + (λγ)2]1/4

. (4.19)

Raising everything to the fourth power removes the troublesome plus or minus sign. Tidying this

up, we get

γ4 − λ2

(ydp

dxRe

)4

γ2 −(ydp

dxRe

)4

= 0, (4.20)

and this is a quadratic equation for γ2 which we can solve. After solving for γ2, we then take the

square root of it to get an expression for γ. It is clear that we will get two plus or minus signs with

this approach: one inside a square root and one outside the square root. The positive sign should

be taken for the one inside the square root because γ should be real. The one outside depends: if

we use our identity (4.7) ∫ 1

xγ dy = U(x)

then we take the minus sign to give us a baseflow that we want (if we took the plus sign then we

can then we have a flow in essence going the other way; all the numbers are right, we just have

the wrong sign). This seems a more conventional way of doing it, going from the lower limit

to the upper limit, but the problem we are faced with is that if we want to do this numerically

using the Clenshaw-Curtis quadrature, we have to take into account the Gauss-Lobatto points are

defined backwards: we start from y = 1 as our first point and go backwards to y = −1. Thus for

numerical purposes, we take the following approach:∫ x

1γ dy = −U(x),

hence we take the plus sign. To summarise, the value inside the square root should be positive, and

for our purposes we are going to take the outside sign to be a plus. This generalises: for arbitrary

n, we can find some numerical method to compute the γ, then use a quadrature appropriately

to numerically compute the baseflow. A scale factor needs to be found so that U(0) = 1 for

comparison purposes. This resulting baseflow profile is given in Figure 4.2.

Of course we could always try integrating it analytically. The expressions to integrate are the

most simplified when we take n = 0.5, and even then it requires numerous substitutions and

invoking various identities. The outcome is extremely complicated and not easy to generalise.

The observant reader may notice that if we obtain a γ like this, it will not be strictly positive over


the y-domain, and this contradicts what we originally imposed. This is indeed a valid question: we

argue that it doesn’t matter for finding the baseflow because of the baseflow possessing symmetry

about the centre line. It does matter when we are trying to find the viscosity.

4.2.2 Computing the viscosity functions

Recall that for our purposes, we take the Carreau model to be

µ =1

[1 + (λγ)2]1/4. (4.21)

Now that we can in theory obtain an expression for γ, we insert this in to above to find µ as a

function of y, and thus we can find the appropriate derivatives with respect to y. To find µt, we

invoke the definition (4.15) and take the appropriate y and γ derivatives.

The only problem now arises from the definition of µt = µ + (dµ/dγ)γ. We have no problems

taking the γ derivative, but when we take the y derivative we have to take dγ/dy. The modulus

function is involved so this has the danger of not being differentiable at y = 0. Clearly the

viscosity function is a smooth function of γ so we argue that by substituting the expression

for γ into the Carreau model of the viscosity, the operations smooth out the potential problem.

The resulting analytical expressions are difficult to manipulate, so instead we will interpolate the

viscosity functions and take the derivatives numerically. As a result, this places extra demand

in the computational step because we need to solve the system of linear equations, but the extra

demands on computation time is not overly significant in our case.

The computed viscosity functions are given in Figure 4.3.

4.2.3 The spectrum

The spectrum is given in Figure 4.4. The procedure is the same although we need to be a bit

careful when we are expanding the operators. As with Poiseuille flow we have the characteristic

Y -shape spectrum with the three branches. The results are given in the caption of the figure and

have been discussed already, so we will comment on the numerics.

Notice that there is a splitting of the S branch for the spectrum when λ = 4. This is the

discretisation error, and is due to not taking N big enough for this particular combination of

the parameters. This occurs also with the Newtonian case, but our choice of focus of the domain

hides the fact that they exist. The choice of N = 150 was chosen out of hindsight: initially,

we had N = 100, but this was not quite enough for the non-Newtonian case and the splitting

occurred very near the point where the three branches meet. As already mentioned N = 100 was

sufficient for the Newtonian case but for comparison purposes with the non-Newtonian case we

chose N = 150.


4.2.4 Doing the general case

The main reason why we have only done the specific case of n = 0.5 is because the code for

doing arbitrary parameters will be complicated and computationally demanding. In general,

the algorithm will involve finding the roots to γ with the appropriate values of the parameters,

numerically integrate to find the baseflow, scale accordingly to get U(0) = 1 for comparison

purposes, numerically differentiate the viscosity functions, put everything together, and then

compute the spectrum. We also have the added complication of the discretisation error: when

the fluid deviates from the Newtonian case (n small or λ large) we need to take large values of N .

As an example, in MATLAB for the case n = 0.5 and λ = 7, we need N = 500 or so; it is clear

how the magnitude of the computation time will scale with this increase of N . When λ gets to

about ten, even N = 1000 doesn’t seem to remove the error to a visually acceptable level.

To get our scaling, we apply another bisection method on two different and well chosen initial

values of the scaling factor on top of using it to find γ. This means we need to numerically

integrate γ every time in an iteration, and because the bisection program runs on linear time (it

converges to the solution at almost the slowest rate possible), we need to do a lot of iterations,

and the effect of N being large kicks in. We could always set some tolerance condition to vary N

appropriately, but again this is extra complication in the programming.

It is certainly possible to do the general case numerically, but there are certainly other more

efficient and elegant analytical ways to come to the same conclusions as ours. We shall now

progress on to the analytical portion of this chapter, knowing we already have the result we wanted.

4.3 Analytical results

We aim to emulate the results we had earlier for Newtonian fluids, that is, to provide some bounds

for the eigenvalues and give some criteria for a flow to be stable. We follow Joseph’s approach

and provide some numerical bounds by making use of the isometric inequalities. The results will

appear very similar to the Newtonian case, and the proofs are similar. Because of this, we make

frequent references to the tools we gave in Chapter 3, and the steps given here will be less detailed

compared to before.

As far as we are aware, the results we now give are new. We believe its absence in published

literature is that is largely due to an absence of a Squire’s theorem equivalent, so although we

know numerically that a Squire’s theorem equivalent for a parallel shear flow of a non-Newtonian

holds, there is no rigourous support for this conclusion. We derive these results on the assumption

that a Squire’s theorem equivalent is true and a proof of it will be discovered in due course.

We take c the wave speed rather than ω = αc the frequency to be the eigenvalue. We consider

the decoupled equations, so we set the coupling operators C1 and C2 to zero. We take dashes


to denote derivatives with respect to the argument, integrals without limits to be integrals over

y ∈ [−1, 1], and I2n =

∫|φ(n)|2 dy where φ will either be the amplitude of the normal vorticity

(η) or the amplitude of the disturbance (v). The same isoperimetric inequalities apply to both of

these functions since they belong to our functionally completed Hilbert space H. The idea of the

proof is adapted from Joseph’s result.

4.3.1 Bounds for the Squire modes

Theorem 4.3.1 (Damped Squire modes) The eigensolutions to the homogeneous modified

Squire’s equation1αS = c η, η(±1) = 0,

are all damped (ci < 0 ∀α, β,Re); see (4.17) for the definition of S. In addition, they satisfy the

inequalitiesUmin <cr < Umax

ci < −p(π2/4) + k2

Re

(4.22)

where p = min{µt(y)| − 1 ≤ y ≤ 1}.

ProofThe homogeneous modified Squire equation is obtained by setting v ≡ 0 in (4.16) and (4.17).

This gives

cη =Uη +i

αReµ(η′′ − k2η) +

i

αReµ′η′

+i

αRe

β2

k2[(µ′t − µ′)η′ + (µt − µ)η′′].

(4.23)

Here we have expanded the operator S given in (4.17). Multiplying (4.23) by the conjugate η and

integrating over y ∈ [−1, 1] we get

c

∫|η|2 dy =

∫U |η|2 dy +

i

αRe

∫ [µ(η′′η − k2|η|2) + µ′η′η

]dy

+i

αRe

β2

k2

∫ [(µ′t − µ′)η′η + (µt − µ)η′′η

]dy.

(4.24)

Integrating by parts and invoking the homogeneous boundary condition η(±1) = 0, we get

c

∫|η|2 dy =

∫U |η|2 dy +

i

αRe

∫ [µ(|η′|2 − k2|η|2) +

β2

k2(µt − µ)|η′|2

]dy. (4.25)

We can apply the second integral mean value theorem to each of the terms since the individual

integrands are sign definite. Separating the real and imaginary part of (4.25) we get

crI20 = U(y1)I2

0 , (4.26)

ciI20 =

−1αRe

[(1− β2

k2

)µ(y2)I2

1 + µ(y3)k2I20 + µt(y4)

β2

k2I2

1

], (4.27)


with yi ∈ [−1, 1] to be the mean values. The first bound immediately comes from the fact that

Umin < U(yi) < Umax. Note we cannot conclude yet the Squire modes are all damped because

(4.27) is not negative definite due to the term +(β2/k2)µ(y2) being present. Instead, we note that

for shear thinning fluids, µ(yi) > minµ > minµt > 0. We also have µt(yi) > minµt. Thus, for

p = minµt > 0,

ciI20 < −

1αRe

[p

(1− β2

k2

)I2

1 + pk2I20 + p

β2

k2I2

1

]= − p

αRe

[I2

1 + k2I20

]< 0,

(4.28)

so all the Squire modes are damped since I20 is positive. Dividing through by I2

0 and applying our

isoperimetric inequality I21/I

20 ≥ π2/4 given in Lemma 3.2.3 we get our result as required. 2

4.3.2 Bounds for the modified Orr-Sommerfeld equation

Theorem 4.3.2 Let c(α,Re) be any eigenvalue of the modified Orr-Sommerfeld equation

1αLv = c(D2 − α2)v v(±1) = v′(±1) = 0. (4.29)

See (4.17) for the definition of L. We assume U is a real and twice continuously differentiable

function, and v ∈ H is a four times continuously differentiable function (H the Hilbert space as

defined in Section 3.2). Let q = max{|U ′(y)| : −1 ≤ y ≤ 1} and p = min{µt(y)| − 1 ≤ y ≤ 1},then we have the following results:

1.

ci ≤q

2α− p

Re

(π2(π2 + α2)π2 + 4α2

+ α2

). (4.30)

2. No amplified disturbances (modes with ci > 0) of (4.29) exists if

qαRe < p · f(α) ≡ max[M1,M2],

M1 = λ3π + 23/2α3

M2 = λ3π + πα2,

(4.31)

where λ3 = (2.356)2 is the least eigenvalue of a vibrating rod clamped at y = ±1.

3.

U ′′min ≤ 0 : Umin < cr < Umax +2U ′′maxπ2 + 4α2

U ′′min ≤ 0 ≤ Umax : Umin +2U ′′minπ2 + 4α2

< cr < Umax +2U ′′maxπ2 + 4α2

U ′′max ≤ 0 : Umin +2U ′′minπ2 + 4α2

< cr < Umax.

(4.32)

Again, the subscript max and min denote the maximum and minimum values on the range

of U(y) and U ′′(y) for y ∈ [−1, 1].


ProofWe note the right hand side of (4.29) is the same as the Newtonian case so we will quote the result

later we get when we multiply by the conjugate v and integrate. We aim to reduce everything to

previously known cases and so we can apply the logic used in Theorem 3.2.5.

First of all we split the Orr-Sommerfeld operator into smaller components, namely L = A+B+C,

where

A =α[U(D2 − α2)− U ′′],

B =i

Re[µ(D2 − α2)2 + 2µ′D3 + U ′′D2 − 2k2µ′D + k2µ′′],

C =i

Re(D2 + k2)[(µt − µ)(D2 + k2)]

=1Re

(µt − µ)(D2 + k2)2 + 2(µ′t − µ′)(D3 + α2D1) + (µ′′t − µ′′)(D2 + α2),

(4.33)

and respectively, acting these on v, multiplying by v, integrating over y ∈ [−1, 1] and invoking

the boundary conditions, we get

A :∫ [−U |v′|2 − (α2 + U ′′)|v|2

]dy −

∫U ′v′v dy,

B :i

Re

∫ [µ(|v′′|2 + 2α2|v′|2 + α4|v|2) + α2µ′′|v|2

]dy,

C :i

Re

∫ [(µt − µ)(|v′′|2 − 2α2|v′|2 + α4|v|2) + α2(µ′′t − µ′′)|v|2

]dy.

(4.34)

Putting B and C together and removing the derivatives from the viscosity functions by means

integration by parts, we get (see also Equation 3.1 in Nouar et al. (2007))

B + C :i

Re

∫ [µ(4α2|v′|2 + |v′′ + α2|v|2) + (µt − µ)|v′′ + α2|2

]dy. (4.35)

We note A is as before in the Newtonian case, and that we have cancellation for some of the µ

terms in (4.35). Applying the second integral value theorem to all the terms and letting µ and

µt denote the function evaluated at some mean value in the y-domain, then taking the real and

imaginary part of (4.29) gives

ci(I21 + α2I2

0 ) = (Q−Q)− (αRe)−14α2µI21 − (αRe)−1µt

∫|v′′ + α2v|2 dy, (4.36)

cr(I21 + α2I2

0 ) =∫ [

U |v′|2 + (α2U + U ′′/2)|v|2]dy, (4.37)

with Q = (i/2)∫U ′v′v dy. Now that the viscosity function is outside the integral, we expand the

integrand and integrate by parts appropriately. This gives

ci(I21 + α2I2

0 ) = (Q−Q)− (αRe)−1[4α2µI21 − µt(I2

2 − 2α2I21 − α4I2

0 )]. (4.38)

We will now proceed with the proof.


1. Noting that µ > minµ > minµt, we divide across the (I21 + α2I0) term, apply Cauchy-

Schwartz on the (Q −Q) term, apply the inequality we just stated on the rest of the terms,

and use our definitions for p and q. We thus have

ci <qI1I0

I21 + α2I2

0

− (αRe)−1(4α2pI21 + p(I2

2 − 2α2I21 − α4I2

0 ))I2

1 + α2I20

=qI1I0

I21 + α2I2

0

− (αRe)−1p(I22 + 2α2I2

1 + α4I20 ))

I21 + α2I2

0

.

(4.39)

The proof from now on is just the rest of the proof in Section 3.2, from (3.54) to (3.56). We

apply the isoperimetric inequalities as appropriate.

2. As above, this is the same argument as (3.58) and (3.59), but instead of (3.57) we use (4.39).

3. This is exactly the same as (3.61) to (3.63), because the real part of the modified Orr-

Sommerfeld equation is not affected by the presence of the viscosity terms.

The reason why we need the Hilbert spaceH is given in Lemma 3.2.3. This concludes our proof. 2

Here we have given the form of the result which is the most compact rather than one which is

sharper by taking inequalities for the viscosity functions. As a result of this, we have p = minµtappearing and this term decreases with increasing shear-thinning behaviour of the fluid. Because

of this, we note the bounds for both the modified Squire and Orr-Sommerfeld modes are not as

strong compared to the bounds given by Joseph (1969) or Davis & Reid (1977). Nevertheless the

Squire modes are all damped, so they do not contribute to the instability of a parallel shear flow of

a non-Newtonian fluid. We observe, with increasing shear or with increasing Reynolds number,

we again have (4.30) tending to the result of Høiland (1953), which agrees with the properties of

a shear thinning fluid. As noted already, the numerical results suggests we have stabilisation if the

fluid is mildly shear thinning, otherwise we have destabilisation. The analytical results seem to

confirm the destabilisation part of the conclusion, because the factor p decreases when we have

increasing deviation from a Newtonian fluid, thus reducing the negative contribution we would

like for showing stabilisation. The stabilisation part however does not seem to be reinforced by

the analytical results. Refining this result and making it agree with known results for shear thinning

fluids is one of the things we will be trying to do in due course.

63

Chapter 5

Conclusion and Discussion

We have gone into substantial detail in investigating our hypothesis: a shear thinning viscosity

stabilises the flow in a plane domain (up to a certain point). In this closing chapter, we shall assess

to what extent our investigation has been valid, the limitations of the methods used, and clarify

some of the points mentioned. We also give some further results which will be mentioned but not

elaborated.

5.1 Numerical problems

5.1.1 Discretisation error

Already mentioned in Chapter 4 we need to be careful and takeN big enough to avoid the splitting

of the branches. This is due to the interpolating polynomials not resolving the rapid oscillations

appropriately. For doing the numerics the choice of domain of the numerics hides the fact that

there is a splitting of the S branch for the spectrum far away enough from the junction, but this is

shown in Figure 4.4. We are not concerned with these modes since they are all damped and we

are studying the instability of the flow. This is an issue associated with the numerics rather than

the problem itself.

5.1.2 Spectrum sensitivity and pseudospectra

A more fundamental issue is that the Orr-Sommerfeld operators are non-normal. Recall that for an

appropriate inner product (·, ·), the adjoint of the operator L is defined by the operator L† which

satisfies

(Lφ, ψ) = (φ,L†ψ).

An operator which is non-normal is one where it does not commute with its adjoint. This causes

problems because non-normal operators have eigenfunctions which are not necessarily orthogonal

Chapter 5. Conclusion and Discussion 64

to each other, and, as a result, the associated eigenvalues are highly sensitive to perturbation. This

phenomenon is thus associated with the problem rather than the numerical scheme we employ.

The sensitivity needs to be investigated, because when we discretise we get rounding errors, and

the presence of this rounding error may cause deviations of the computed eigenvalue to the actual

eigenvalue by several orders of magnitude.

This prompted an investigation in to something called the pseudospectra. No less than six people

have independently come up with the idea of pseudospectra, but it was Trefethen (1992) who

applied the idea to a wide variety of interesting problems. He is now usually credited with the

theory, especially within the hydrodynamic stability field. A slightly more intuitive definition of

the ε-pseudospectra of a matrixA is the set of eigenvalues of

A = A+E, ‖E‖ < ε

where the norm ‖·‖ is the matrix norm induced by the appropriate inner product. A more useful

definition is given in terms of the resolvent operator ofA (Reddy et al., 1993).

For the Orr-Sommerfeld equation, it is known that the the eigenvalues near the junction joining

the three branches react the most to a perturbation of the operator. Perturbing the discretised Orr-

Sommerfeld operator by a matrix with norm of O(10−5), Schmid & Henningson (2001) showed

that the relative movement of the eigenvalues near the junction was of O(1), which is several

orders of magnitude higher. We have not done a similar test for the modified Orr-Sommerfeld

operator, but, at a guess, the sensitivity of the spectrum will not be of order less than the Orr-

Sommerfeld operator. For these problems, it is argued that computing the pseudospectra is more

informative than the spectra.

It is known that the spectral collocation method is a more accurate method for dealing with highly

non-normal operators such as the one we are dealing with (Trefethen & Embree, 2005). For further

reading, please see Trefethen et al. (1993), Reddy et al. (1993), Trefethen (1997, 1999).

5.2 Some further results

5.2.1 Rescaling and the Reynolds number

When we derived the Reynolds number, we mentioned the choice of scaling and normalisation

was not unique, and are usually chosen for convenience. From a practical point of view, we

might choose to normalise the equation using flow rate (for example), since it may be easier to

control the flow rate rather than the maximum speed of a flow at the centreline. Using different

scalings gives us a different Reynolds number and in turn the critical Reynolds number will shift

accordingly. This opens up the possibility of getting more informative results by using a definition

of the Reynolds number which is more physically relevant.


One such rescaling is the Reynolds number Re based on the viscosity averaged over the channel

(Wall & Wilson, 1996). This was first used in Newtonian fluids to represent the global decrease of

the viscosity when the channel walls are heated, and was later adapted by Chikkadi et al. (2005)

for the study of Carreau fluids. Using this definition, Nouar et al. (2007) reports that the presence

of a shear thinning viscosity does stabilise the flow by reducing the critical Reynolds number.

However when the viscosity fluctuation (the terms involving the tangent viscosity µt) is included

this appears to reduce this stabilisation.

In Nouar et al. (2007), they propose rescaling the Reynolds number as

Retw =ρV L

µtwµ0, (5.1)

where µtw is the tangential viscosity at the wall, and the rest of the parameters are as before. The

details are left out here, but the result reported is that with this definition of the Reynolds number,

the presence of a shear thinning viscosity is consistently stabilising.

In laboratory experiments, it is customary to use the effective viscosity at the wall instead of the

tangent viscosity at the wall, because they are easier to work out and control. In the same paper, it

is reported that, using this rescaling, the qualitative behaviour is the same, and we have consistent

stabilisation of the flow.

5.2.2 Energy method

Throughout this report we have focused on the linear theory and that eigenvalue relation that arises

as a result. This is all very well when we are looking for instability, because if something is not

stable to infinitesimal perturbations, it will not be stable to perturbations of higher magnitude.

The limitation of this theory is that it is limited in scope and of course not very realistic, since

we rarely have infinitesimal disturbances anyway. For giving a criterion for stability, we require

the nonlinear theory. We have mentioned the word energy a few times throughout the report in

relation to hydrodynamic stability, so we will briefly discuss the energy method. This investigates

the energy exchange between the disturbance and the base flow: if energy transferred from the base

flow to the disturbance is higher than the energy dissipated by the viscosity, then the disturbance

grows and becomes unstable due to the surplus of energy. The opposite is true if the dissipation is

bigger than the ‘production’ of energy.

It can be easily derived that the kinetic energy in the domain of interest is given in index form by

EK =12

∫Vuiui dV. (5.2)

We have assumed we are dealing with incompressible fluids so the fluid density has been

normalised accordingly and does not appear in our definition. We aim to get an equation which

tells us the rate of change of the energy of the disturbance. For the Newtonian case, we go back

to the Navier-Stokes equation, consider a parallel flow, perturb the base solution, but this time we


do not throw away the nonlinear term. Then we take the inner product of the equation with the

disturbance (inner product in the sense of function spaces, i.e., multiply and integrate). Writing as

many terms in divergence form as possible, we apply the divergence theorem as appropriate. The

incompressibility condition reduces a substantial amount of terms to zero, and in the end we get

dEKdt

= −∫Vuiuj

∂Ui∂xj

dV − 1Re

∫V

∂ui∂xj

∂ui∂xj

dV. (5.3)

This is known as the Reynolds-Orr equation, with ui the disturbance and Ui the baseflow. This

equation is clearly nonlinear due to the presence of quadratic terms, a consequence of taking the

inner product earlier on. In symbolic form, this is

dJ1

dt= J2 −

1Re

J3. (5.4)

Here, J1 represents the time variation of the disturbance kinetic energy density, J2 the integral of

the product of the Reynolds stresses (uiuj) with the mean velocity gradient which quantifies the

energy available to the disturbance, and J3/Re is the rate of dissipation of kinetic energy due to

the viscosity.

We now go back to giving a physical explanation for our hypothesis. For the Newtonian case, when

viscosity is large enough, the dissipation dominates over the production rate, hence the kinetic

density decreases and the flow is stabilised by viscosity. It must be noted that this energy equation

is not the same as the energy equation derived earlier in Joseph’s theorem (ci = Q − Q − ...);see (3.51). That equation is linear and is the energy equation for a 2D disturbance arising from a

temporal analysis of the normal modes. The physical argument, however, is the same: if the energy

of the disturbance is reduced via either dissipation or a lack of production, then the disturbance

dies down and the flow is stabilised.

For non-Newtonian fluids, an equivalent of the Reynolds-Orr equation may be derived and it takes

the same symbolic form above. For shear-thinning fluids, it is in fact the decrease in production

of energy that causes the stabilisation of the flow, rather than an increase of the dissipation rate

actually varies very little (Govindarajan et al., 2001, 2003; Nouar et al., 2007). The energy is

mainly produced near the critical layer whilst most of the dissipation seems to occur in the wall

layer.

As we said before, in providing a criterion for nonlinear stability we consider a perturbation of

finite amplitude so we analyse the nonlinear equation for the disturbance. To start a discussion

now would be out of place, so instead we provide this theorem for Newtonian fluids which is due

to Serrin (1959):

Theorem 5.2.1 Let an incompressible viscous fluid occupying some region V (t) which may be

enclosed within a sphere of diameter L. Let u be a solution to the Navier-Stokes equation with

|u| < umax in V (t) for all t. Let u∗ be another solution to the Navier-Stokes equation with the


same boundary conditions as u but with different initial conditions. Then the kinetic energy of the

difference flow v = u− u∗ satisfy

EK ≤ E0 exp[(u2max −

3π2ν2

L2

)tν

]where ν = µρ is the kinematic viscosity, and E0 is the initial value of the kinetic energy. Further

more, if

Re =umaxL

ν=umaxLρ

µ< π√

3

then EK → 0 as t→∞ and hence the flow u is (nonlinearly) stable.

This agrees with our intuition: if viscosity is very high or the flow is very slow, then we don’t

have instability. This theorem is stronger because it guarantees we don’t have instability to finite

perturbations. The proof of this is essentially done using the Reynolds-Orr equation, noticing some

bounds for a vector identity that arises, and taking the seemingly obscure approach of considering

the difference flow v. We refer the reader to the book by Acheson (1990) for a proof of it (Chapter

9).

5.3 Closing remarks

One thing we would like to consider in due course is to improve the analytical bound we found

for the non-Newtonian case. As it stands at the moment the bound does not provide the stabilising

effect we expect for parallel shear flows of shear-thinning fluids. One idea is to consider a rescaling

given above, since it appears that this provides the stabilising effect in the numerical analysis of

the problem. We hope to do this in due course; the details appears to be a bit more intricate

than previously thought. We would also like to derive a similar result for pipe flows, since that is

another type of flow possessing the relevant symmetries for our study, and is of practical interest.

Of course, the lack of Squire’s theorem adds to the complication on how valid this result is.

Very few laminar flows correspond to an analytical solution of the equations we considered. Our

analysis has been forced to be narrow, studying flows with symmetries present that reduce the

problem significantly. This unfortunately obscures the reality, since many physical problems often

lack the symmetry we impose for our mathematical study. We have focused on studying a flow’s

stability through the method of normal modes and investigating the associated eigenvalue problem.

There are of course many other ways to study hydrodynamic stability, but it must be realised that

the topic of hydrodynamic stability makes use of a lot of tools which we would not have been

able to use without a lot of assumed knowledge. For example, one way to go about studying

the stability of the Orr-Sommerfeld equation would be to investigate its solution indirectly via

an asymptotic analysis. This might be using methods such as the method of stationary phase

or the WKBJ approximation, and study the solution at some asymptotic limit without actually

knowing the solution explicitly. It is known that the solutions to the Orr-Sommerfeld equation


may be approximated by the generalised Airy functions which takes a complex argument, and

these functions are well documented. To do such an analysis requires knowledge of (but not

limited to): asymptotics, real and complex analysis, as well as plenty of functional analysis. The

analysis for the Newtonian case through this method is complicated enough as it is; doing it for

the non-Newtonian case seems such a daunting task, it is no surprise there appears to be a distinct

lack of literature available on it. The main focus of this report was to compare the available results

on hydrodynamic stability of Newtonian and non-Newtonian fluids, so due to a lack of results to

compare, we have left out a discussion of said method and its associated results.

There are many other problems that we did not go in to detail about. We have flows in different

geometries; some of these may be of practical interest, whilst other may be of more theoretical

interest. There are complications in studying the initial value problem because of the nature of

the nonlinear equations. A spatial analysis rather than a temporal analysis of the disturbance

is possible, again with some minor complications; these are usually more relevant because it

gives a more realistic representation of the transition of a laminar flow into a turbulent one. We

have only investigated the discrete spectrum rather than the continuous spectrum or its associated

eigenfunctions, which again improves our understanding of stability and are required in the initial

value problem. There are different non-Newtonian fluids which we have not even touched on, such

as viscoelastic fluids with a time dependent viscosity. The list goes on. To put them all in would

simply be too much.

Hydrodynamic stability has important applications in industry for it allows us to predict the onset

of instability and dealing with it accordingly. There is so much scope for further investigation this

field of research; we have barely scratched the surface of it.

69

Appendices

A MATLAB codes

The program of choice in this report is MATLAB; all the numerical computation and almost all the

figures are computed in MATLAB. Here we will explain how we put the codes together to compute

the eigenvalue spectrum using the spectral collocation method with Chebyshev polynomials as the

interpolant.

A few technical details first. MATLAB calculations are done in the the standard IEEE double

precision arithmetic with the error in each basic operation such as addition and multiplication

to be 2−53 = O(10−16). Spectral accuracy ensures that the methods are to a similar degree of

accuracy, with rounding errors cropping in around O(10−14) or so. Our chief source of error is

in the bisection program which for simplicity we have imposed an error tolerance of O(10−8);

clearly this eclipses all of the other errors that may occur by a substantial order of magnitude. We

can say our calculations are accurate to at least seven decimal places, and we can easily improve

this by demanding a stricter tolerance at the cost of computation time.

With regards to the computation time, computing the the Newtonian spectrum takes less than a

second because we have an analytical form of the baseflow. For the non-Newtonian case this takes

about three seconds due to our need to numerically compute, interpolate and take the derivative of

the baseflow. For computing the baseflow with arbitrary values of n and λ, this takes about three

seconds for each baseflow. The main bottle neck in each case is the in the bisection program and

in the interpolation (namely that of solving the linear system of equations to find the coefficients).

MATLAB uses the QZ algorithm for computing the eigenvalues and the command in MATLAB is

given by eig(A,B) for a genrealised eigenvalue problem. The built in trapezium rule is called by

the command trapz(y,f) where y is the discretised domain (represented by a vector) and f is

the function. Programs that are taken from elsewhere are referenced in the programs themselves.

Most of the codes here are taken and modified from the book by Schmid & Henningson (2001),

which are in turn adapted from codes in the MATLAB differentiation matrix suite written by

S. Reddy and J.A.C. Weiderman. The codes for the Newtonian case (Poiseuille and Couette

flow) are obtained by removing a large portion of the codes we give here and modifying pois.m

appropriately; we will not include them here for space considerations.

Appendix 70

The algorithm for computing the spectrum with n = 0.5

1. Write y dpdxRe±µ(γ)γ = 0 and solve for γ analytically using the Carreau model at n = 0.5.

2. Numerically integrate γ to find a baseflow (vel.m and fclencurt.m).

3. Find the right scaling factor dpdxRe so that U(y) = 1 (baseflow.m and bisection.m).

4. Generate the differentiation matrices (Dmat.m).

5. Compute the viscosity functions using the scaling factor in 3) (mu.m).

6. Interpolate and compute the approximation to the derivatives of µ, µt and U . Set up the

Orr-Sommerfeld equation and put all the values in (pois.m).

7. Compute the spectrum with the desired values of the parameters α, β, Re, N (osmat.m).

Note that we have the freedom to choose Re because we can always change the pressure term

dp/dx to fit give our appropriate scaling factor. A more comprehensive explanation of the

algorithm is given in Chapter 4.

Appendix 71

Codes

Driver program

clear all; close all; clc% driver program: osmat.m% program to compute the baseflow, spectrum and the viscosity functions% (this is for the case n=0.5 of a Carreau fluid)

% INPUT%% nosmod = number of OS modes% lambda = time constant% R = Reynolds number% alp = alpha (streamwise wave number) [most sensitive at 1.02]% beta = beta (spanwise wave number)%

zi = sqrt(-1);

% input data

nosmod=input(’Enter N the number of OS modes: ’);lambda=input(’Enter lambda the shear-thinning parameter: ’);R=input(’Enter the Reynolds number: ’);alp=input(’Enter alpha: ’);beta=input(’Enter beta: ’);

% Set up Gauss-Lobatto pointsvec=(0:1:nosmod)’;y=cos(pi*vec/nosmod);my=length(y);

% generate Chebyshev differentiation matrices[D0,D1,D2,D3,D4]=Dmat(nosmod);

% set up OS matrices A and B[A,B]=pois(nosmod,alp,beta,R,D0,D1,D2,D3,D4,lambda);

% generate baseflow and viscosity functions for plotting purposes[Rep,u]=baseflow(y,lambda,nosmod);[x,mu0,mu1,mu2,mut0,mut1,mut2] = mu(y,Rep,lambda,D0,D1,D2);

% compute and plot the eigenvalues (not showing the spurious ones)ee=eig(A,B);

% spectrumfigure(1)plot(ee,’.’,’markersize’,12)grid on, axis([0 1 -1 .2]), axis squaretitle([’N = ’ int2str(nosmod), ’ \lambda= ’ num2str(lambda), ...

’ R = ’ num2str(R),’ \omega_{max} = ’ num2str(max(imag(ee)) ...,’%15.11f’)]), drawnow

% viscosity functionsfigure(2)plot(y,mu0,’r’,y,mut0,’g’)title([’\lambda= ’ num2str(lambda)]), drawnow

% baseflow comparison with Newtonian casefigure(3)plot(y,u,’r’,y,1-y.ˆ2,’g’)

Appendix 72

axis([-1 1 0 1])title([’\lambda= ’ num2str(lambda)]), drawnow

Computing Chebyshev Differentiation Matrices

function [D0,D1,D2,D3,D4]=Dmat(N)

% Function to create differentiation matrices%% INPUT%% N = number of modes (ends up being nosmod)%% OUTPUT%% D0 = zero’th derivative matrix% D1 = first derivative matrix (and so on)%

% initialise

num=round(abs(N));

% create D0

D0=[];vec=(0:1:num)’;for j=0:1:num

D0=[D0 cos(j*pi*vec/num)];end;

% higher order derivative matrices (using the recuurence relations)

lv=length(vec);D1=[zeros(lv,1) D0(:,1) 4*D0(:,2)];D2=[zeros(lv,1) zeros(lv,1) 4*D0(:,1)];D3=[zeros(lv,1) zeros(lv,1) zeros(lv,1)];D4=[zeros(lv,1) zeros(lv,1) zeros(lv,1)];for j=3:num

D1=[D1 2*j*D0(:,j)+j*D1(:,j-1)/(j-2)];D2=[D2 2*j*D1(:,j)+j*D2(:,j-1)/(j-2)];D3=[D3 2*j*D2(:,j)+j*D3(:,j-1)/(j-2)];D4=[D4 2*j*D3(:,j)+j*D4(:,j-1)/(j-2)];

end;

Computing the Orr-Sommerfeld matrix for the baseflow

function [A,B]=pois(nosmod,alp,beta,R,D0,D1,D2,D3,D4,lambda)

% Function to create OS matrices using Chebyshev pseudospectral discretisation%% INPUT%% nosmod = number of modes% alp = streamwise wavenumber% beta = spanwise wavenumber

Appendix 73

% R = Reynolds number% Dn = nth derviative matrix% lambda = time constant%% OUTPUT%% A = Orr-Sommerfeld Matrix% B = The bit multiplying the eigenvalue%

zi=sqrt(-1);

% initialise

% define kˆ2 by ak2ak2=alpˆ2+betaˆ2;

Nos=nosmod+1;vec=(0:1:nosmod)’;

% Gauss-Lobatto pointsy=cos(pi*vec/nosmod);

% Baseflow U[Rep,u]=baseflow(y,lambda,nosmod);

% Viscosity functions using the above value of Rep[x,mu0,mu1,mu2,mut0,mut1,mut2] = mu(y,Rep,lambda,D0,D1,D2);

% interpolate and find second derivative of Ucoeffu = D0\u;u2 = D2*coeffu;

% turn into matrices

u=u*ones(1,length(u));u2=u2*ones(1,length(u2));mu0=mu0*ones(1,length(mu0));mu1=mu1*ones(1,length(mu1));mu2=mu2*ones(1,length(mu2));mut0=mut0*ones(1,length(mut0));mut1=mut1*ones(1,length(mut1));mut2=mut2*ones(1,length(mut2));

% set up OS martix (Modify this to get the non-Newtonian case)

B11=D2-ak2*D0;

% Modified Orr-SommerfeldA11=-(mu0.*(D4-2*ak2*D2+ak2ˆ2*D0)...

+2*mu1.*D3+mu2.*D2-2*ak2ˆ2*mu1.*D1+ak2*mu2.*D0).../(zi*R);

% With tangential viscosityA11=A11-alpˆ2*((mut0-mu0).*D4 ...

+2*(mut1-mu1).*D3...+(mut2-mu2).*D2+ 2*ak2*(mut0-mu0).*D2...+2*ak2*(mut1-mu1).*D1...+ak2ˆ2*(mut0-mu0).*D0 + ak2*(mut2-mu2).*D0)/(zi*R*ak2);

A11=A11+alp*u.*B11-alp*u2.*D0;

% Employ boundary conditions

Appendix 74

er=-200*zi;A11=[er*D0(1,:); er*D1(1,:); A11(3:Nos-2,:); ...

er*D1(Nos,:); er*D0(Nos,:)];B11=[D0(1,:); D1(1,:); B11(3:Nos-2,:); D1(Nos,:); D0(Nos,:)];

% combine all blocksA=A11;B=B11;

% Comment - Boundary conditions are employed on the first, second,% last and second to last row of the matrix by sending multiplying% it by an extremely large imaginary number. Another way to do it% would be to strip the rows accordingly and this is the method used by% Trefethen in his book. We can’t do that here because we require a% square matrix to compute its eigenvalues (he sets up his differntiation% matrix differently, using a different formula).

Computing the Baseflow

function [Rep,u]=baseflow(y,lambda,nosmod)

% Scales the baseflow so that U(0)=1%% INPUT%% y = y-domain (Gauss-Lobatto points)% lambda = time constant% nosmod = number of modes%% OUTPUT%% Rep = the number required to scale flow so U(0)=1% u = the baseflow as data points%% NOTE:% m = ratio between initial viscosity and "infinite" shear viscosity,% which we set to zero here% Repu, Repl = Starting values to carry out bisection% fval = u1 at y=0% error tolerance for bisection is 10e-8

Repu=-0.1;Repl=-10;c = bisection(@vel,Repl,Repu,y,my,10e-8,lambda,nosmod);[fval,u1]=vel(y,my,c,lambda,nosmod);Rep=c;u=u1;

Finding the velocity from γ

function [fval,u]=vel(y,my,Rep,lambda,nosmod)

% Computes the baseflow by integrating gammadot using the Clenshaw-% Curtis equarature%% INPUT

Appendix 75

%% y = y-domain (Gauss-Lobatto points)% my = length of the vector y% Rep = Scaling factor% lambda = time constant% nosmod = number of modes%% OUTPUT%% fval = U(0)% u = the baseflow as data points

% Sort Gauss-Lobatto nodes so we have y from -1 to 1y=sort(y);

% precondition (for speeding things up in MATLAB)u=0*y;

% Apply Clenshaw-Curtis quadrature%(f is gammadot when n=0.5 in the Carreau model)% Do this by iteration: scales the interval [-1,1] to between x and 1,% interpolate and integrate, then do the next point x(i+1), etc.for i=2:length(y)

[x,w]=fclencurt(length(y(1:i)),-1,y(i));f = Rep*x.*2ˆ(-1/2).*(x.ˆ2*Repˆ2*lambdaˆ2+...

(lambdaˆ2*x.ˆ4*Repˆ4+4).ˆ(1/2)).ˆ(1/2);u(i) = w’*f;

end

% Apply boundary conditionsu(my)=0;u(1)=0;

% Find y=0 and U(0).I=nosmod/2 + 1;fval=u(I)-1;

Computing the viscosity functions

function [x,mu0,mu1,mu2,mut0,mut1,mut2] = mu(y,Rep,lambda,D0,D1,D2)%% creates the zeroth, 1st, 2nd derivative of the viscosity function,% and also the tangential tangential viscosities and it’s derivative, using% Chebyshev interpolation.%% INPUT%% y = y-domain (Gauss-Lobatto points)% Rep = Scaling factor% lambda = shear rate time constant% Dn = nth derivative matrix%% OUTPUT%% x = gammadot as a function of y with with n=0.5 in Carreau model% mun = nth derivatives of the viscosity function (Carreau law)% mutn = nth derivative of the tangential viscosity (carreau law)

% initialise (analytical expressions)

Appendix 76

x = -Rep*y.*2ˆ(-1/2).*(y.ˆ2*Repˆ2*lambdaˆ2+...(lambdaˆ2*y.ˆ4*Repˆ4+4).ˆ(1/2)).ˆ(1/2);

mu0 = (1+lambdaˆ2*x.ˆ2).ˆ(-1/4);mudot = (-1/2).*(lambdaˆ2.*x)./((1+lambdaˆ2*x.ˆ2).ˆ(5/4));mut0 = mu0 + x.*mudot;

% interpolate and find coefficients

coeffmu0 = D0\mu0;coeffmut0 = D0\mut0;

% find derivativesmu1 = D1*coeffmu0;mu2 = D2*coeffmu0;mut1 = D1*coeffmut0;mut2 = D2*coeffmut0;

Computing the weights for the Clenshaw-Curtis quadrature

function [x,w]=fclencurt(N1,a,b)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fclencurt.m - Fast Clenshaw Curtis Quadrature%% Compute the N nodes and weights for Clenshaw-Curtis% Quadrature on the interval [a,b]. Unlike Gauss% quadratures, Clenshaw-Curtis is only exact for% polynomials up to order N, however, using the FFT% algorithm, the weights and nodes are computed in linear% time. This script will calculate for N=2ˆ20+1 (1048577% points) in about 5 seconds on a normal laptop computer.%% Written by: Greg von Winckel - 02/12/2005% Contact: gregvw(at)chtm(dot)unm(dot)edu%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

N=N1-1; bma=b-a;c=zeros(N1,2);c(1:2:N1,1)=(2./[1 1-(2:2:N).ˆ2 ])’; c(2,2)=1;f=real(ifft([c(1:N1,:);c(N:-1:2,:)]));w=bma*([f(1,1); 2*f(2:N,1); f(N1,1)])/2;x=0.5*((b+a)+N*bma*f(1:N1,2));

Bisection program

function c = bisection(vel,a,b,y,my,delta,lambda,nosmod)

% Computes the root of a continuous function f in the interval [a,b] by% bisectioning.%% INPUT%% vel = function to bisect

Appendix 77

% a = left endpoint% b = right endpoint% y = y-domain% my = length of the vector y% lambda = shear rate time constant% nosmod = number of modes%% OUTPUT%% c = the root ’ie’ the scaling factor Rep to give U(0)=1%

% Compute the baseflow with the initial values of the scaling factor Rep[fvall,u]=velt(y,my,a,lambda,nosmod);[fvalu,u]=velt(y,my,b,lambda,nosmod);

% Bisectionif fvall*fvalu>0

error(’MATLAB:incorrectValue’,...’function values at endpoints must differ in sign’);

end

% number of iterations required to obtain an error smaller than tolerancemaxIt=round((log(abs(b-a))-log(delta))/log(2));k=0;while k<maxIt && abs(b-a)>delta

k=k+1;c=(a+b)/2;[fvalc,u]=vel(y,my,c,lambda,nosmod);if fvalc==0

a=c;b=c;

elseif fvall*fvalc<0b=c;fvalu=fvalc;

elseif fvalu*fvalc<0a=c;fvall=fvalc;

elseerror(’MATLAB:incorrectValue’,’function f has to be continuous’);

endendc=(a+b)/2;end

78

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