on the stability of the non-newtonian boundary-layer flow over … · 2019. 11. 14. · hawa milad...

On The Stability of The Non-NewtonianBoundary-layer Flow Over a Rotating

Sphere

Thesis submitted for the degree of

Doctor of Philosophy

at the University of Leicester

by

Hawa Milad Ibraheem Egfayer

Department of Mathematics

University of Leicester

May 2019

On the stability of the non-Newtonian boundary-layer

flow over a rotating sphere

Hawa Milad Ibraheem Egfayer

Abstract

The majority of this work is concerned with the local-stability of incompressible non-

Newtonian boundary-layer flows over rotating spheres. Convective stability given by the

power-law fluids is considered for fluids that adhere to a non-Newtonian governing vis-

cosity relationship. The velocity distribution of the base flow is described by ordinary

differential equations with the power-law flow index as the only parameter. The lami-

nar boundary layer flow is studied by extending the perturbation method suggested by

Howarth and used by Banks. The laminar mean velocity profiles are obtained by solv-

ing the resulting ordinary differential equations assuming that the flow is axisymmetric

and time independent. Having the base flow solutions, we investigate the convective

instabilities associated with flows in the limit of large Reynolds number inside the three-

dimensional boundary-layer. The local rotating sphere analyses are conducted at various

latitudes from the axis of rotation (θ). A linear stability analysis is conducted using the

Chebyshev collocation method. Extensive computation results are given for the flow in-

dex 0.6 ≤ n ≤ 1.0. Akin to previous Newtonian studies at sphere latitudes θ = 10−70

in ten degrees increments, it is found that there exists two primary modes of instability; the

upper-branch Type I modes (cross-flow) and the lower-branch Type II modes (streamline-

curvature). The results of the convective instabilities are presented in terms of neutral

curves and growth rates. The predictions of Reynolds number, vortex angle and vortex

ii

speed at the onset of convective instability when the power-law index n = 1 are consistent

with previous Newtonian studies. Our predictions and neutral curves for convective insta-

bility of the power-law boundary layer on rotating spheres approach those of the rotating

disk as we approach the pole. Roughly speaking, our findings reveal that shear-thinning

power-law fluids over rotating spheres have a universal stabilising effect.

This thesis is dedicated to my loving parents, my brother Dr. Salem Egfayer and my

beloved husband Salah

Acknowledgements

Firstly, I would like to express my sincere gratitude to my supervisor Prof. Stephen Garrett

for the excellent supervision and the continuous support I received during my Ph.D study,

for his patience, motivation, and immense knowledge. His guidance helped me for all the

time of my research and writing of this thesis. Additionally, I would like to acknowledge

Dr. Paul Griffiths for his help during my research. I also wish to acknowledge financial

support from Libyan government, Ministry of Higher Education. I am extremely grateful

for the continued support of my brother Dr. Salem Egfayer who supports me spiritually

throughout stages of my studies and my life in general. Last but not the least, I am grateful

to all my family and friends for their support and encouragement.

Contents

Contents v

List of Figures vii

List of Tables xii

Nomenclature xvii

1 Introduction 1

1.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Contributions and Outline of thesis . . . . . . . . . . . . . . . . . . . . . 5

2 Formulation of the steady equations 7

2.1 The power-law non-Newtonian fluid (Ostwald-de Waele Model) . . . . . 7

2.2 The governing boundary-layer equations for Power-law fluids . . . . . . . 9

2.3 The dimensionless boundary-layer equations of Power-law fluids over a

rotating sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Solutions of the steady-flow equations 19

3.1 The series expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Matlab solver for the steady mean flow . . . . . . . . . . . . . . . . . . . 33

3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Derivation of the perturbation equations 42

Contents vi

4.1 The perturbation equations . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2.1 Fundamental background of the spectral methods . . . . . . . . . 50

4.2.2 Implementation of Chebyshev collocation method . . . . . . . . 51

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 Convective instability analysis varying the latitude 61

5.1 The convective instability analysis . . . . . . . . . . . . . . . . . . . . . 61

5.2 Neutral curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.2.1 Case 1 (The power-law index n = 1) . . . . . . . . . . . . . . . . 62

5.2.2 Case 2 (The power-law index n = 0.9) . . . . . . . . . . . . . . . 66




5.3 Growth rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6 Convective instability analysis varying the power-law index 80

6.1 Neutral curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.2 Growth rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7 Conclusion 94

7.1 Completed work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

7.2 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

A Derivation of the steady-flow and disturbance viscosity functions 98

B Neutral Curves 101

References 103

List of Figures

1.1 Transition on a rotating sphere. Photograph by Kobayashi et al. (1983) . . 3

2.1 Sketch of the coordinate system . . . . . . . . . . . . . . . . . . . . . . 10

3.1 Plots of F1; G1; H1 and µ1 versus η , the η-axis has been truncated at

η = 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Plots of F3; G3; H3 and µ3 versus η for n values. The η-axis has been

truncated at η = 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Plots of F5; G5; H5 and µ5 versus η . The η-axis has been truncated at

η = 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4 Plots of F7; G7; H7 and µ7 versus η . The η-axis has been truncated at

η = 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5 Plots of U ; V and W versus η at latitudes of θ = 10−80 in ten degree

increments for n = 1. The η-axis has been truncated at η = 20. . . . . . . 30

3.6 Plots of U ; V ; W and µ versus η at latitudes of θ = 10 − 80 in ten

degree increments for n = 0.9. The η-axis has been truncated at η = 20. . 30


degree increments for n = 0.8. The η-axis has been truncated at η = 20. 31





List of Figures viii

3.10 Mean velocity profiles U ; V and W at latitudes of θ = 10− 80 in ten

degree increments when n = 1. (Each figure is normalised independently

by its maximum value.) . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.1 The two spatial branches for the case when n = 1 at θ = 10 showing

Type I instability from branch 1 only at R = 2400 and Type II instabilities

from the modified branch 1 at R = 2500. . . . . . . . . . . . . . . . . . 63

5.2 The two spatial branches for the case when n = 1 at θ = 70 showing a

kink in branch 2 and a region of instability caused by branch 1. . . . . . 64

5.3 The neutral curves of convective instability for stationary vortices at latit-

udes of θ = 10−70 (right to left) when n = 1. . . . . . . . . . . . . . 64

5.4 The neutral curves of convective instability in the(RS, n)− plane for sta-

tionary vortices of θ = 10−70 (right to left) when n = 1. . . . . . . . . 65

5.5 A comparison of the critical RLvalues for convective instability at each

latitude when n = 1 with those of the Griffiths et al. (2014a) for the rotat-

ing disk (horizontal lines). . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.6 The two spatial branches for the case when when n = 0.9 at θ = 10

showing Type I instability from branch 1 only at R = 2760 and Type II

instabilities from the modified branch 1 at R = 3159.90. . . . . . . . . . 66

5.7 The two spatial branches for the case when when n = 0.9 at θ = 70

showing a kink in branch 2 and a region of instability caused by branch 1. 67


udes of θ = 10−70 (right to left) when n = 0.9 . The R−axis has been

truncated at R = 3000. . . . . . . . . . . . . . . . . . . . . . . . . . . . 67


latitude when n = 0.9 with those of the Griffiths et al. (2014a) for the

rotating disk (horizontal lines). . . . . . . . . . . . . . . . . . . . . . . . 69

List of Figures ix

5.10 The two spatial branches for the case when n = 0.8 at θ = 10 showing

Type I instability from branch 1 only at R = 3251.75 and Type II instabil-

ities from the modified branch 1 at R = 4051.40. . . . . . . . . . . . . . 69

5.11 The two spatial branches for the case when n = 0.8 at θ = 70 showing a




truncated at R = 3000. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70









truncated at R = 3000. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72







5.18 The two spatial branches for the case when n = 0.6 at θ = 70 showing a




truncated at R = 3000. . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

List of Figures x





udes of θ = 10 for decreasing values of n. . . . . . . . . . . . . . . . . 77

5.22 Growth rates for Type I mode for shear-thinning power-law fluids for

0.6 ≤ n ≤ 1.0 at latitudes of θ = 10−60. . . . . . . . . . . . . . . . . 78


ude θ = 10 when 0.6 ≤ n ≤ 1. . . . . . . . . . . . . . . . . . . . . . . 81


ude θ = 20 when 0.6 ≤ n ≤ 1. . . . . . . . . . . . . . . . . . . . . . . 82


ude θ = 30 when 0.6 ≤ n ≤ 1. . . . . . . . . . . . . . . . . . . . . . . 83


ude θ = 40 when 0.6 ≤ n ≤ 1. . . . . . . . . . . . . . . . . . . . . . . 85


ude θ = 50 when 0.6 ≤ n ≤ 1. . . . . . . . . . . . . . . . . . . . . . . 86


ude θ = 60 when 0.6 ≤ n ≤ 1. . . . . . . . . . . . . . . . . . . . . . . 88


ude θ = 70 when 0.6 ≤ n ≤ 1. . . . . . . . . . . . . . . . . . . . . . . 89

6.8 Growth rates for Type I mode for shear-thinning power-law fluids for de-

creasing values of n at latitudes of θ = 10−60. . . . . . . . . . . . . . 91

6.9 Growth rates for Type II mode for shear-thinning power-law fluids for

decreasing values of n at latitudes of θ = 50−70. . . . . . . . . . . . 92

B.1 The neutral curves of convective instability in the(RS, β

)−plane for sta-

tionary vortices of θ = 10−70 (right to left) when n= 0.9, 0.8, 0.7 and 0.6.

101

List of Figures xi

B.2 The neutral curves of convective instability in the(RS, β

)−plane for sta-

tionary vortices of θ = 10−70 (right to left). . . . . . . . . . . . . . . 102

List of Tables

3.1 Numerical values of the mean velocity flow parameters F ′1 (0); G′

1 (0) and

H1 (η∞) for n values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24


3 (0) and

H3 (η∞). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25


5 (0) and

H5 (η∞). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27


7 (0) and

H7 (η∞)for n values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.1 The values of the critical Reynolds number R, wavenumbers α , azimuthal

wave-number, β and the angle of spiral vortices φ corresponding to the

value of n = 1 on the upper-branch for stationary vortices of θ = 10−

60. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65


wave-number, β and the angle of spiral vortices φ corresponding to value

of n = 1 on the lower-branch for stationary vortices of θ = 10−70. . . 65



of n = 0.9 on the upper-branch for stationary vortices of θ = 10−60. . 68



of n = 0.9 on the lower-branch for stationary vortices of θ = 10−70. . 68

List of Tables xiii



of n = 0.8 on the upper-branch for stationary vortices of θ = 10−60 . 70


wave-number, β and the angle of spiral vortices φ corresponding to values















wave-number, β and the angle of spiral vortices φ for power-law fluids at

latitude θ = 10 on the both modes Type I and Type II. . . . . . . . . . . 81


wave-number, β and the angle of spiral vortices φ corresponding to de-

creasing values of n on the upper-branch for stationary vortices at latitude

θ = 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

List of Tables xiv



creasing values of n on the lower-branch for stationary vortices of θ =

20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83




θ = 30. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84



creasing values of n on the lower-branch for stationary vortices of θ =

30. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84




θ = 40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85



creasing values of n on the lower-branch for stationary vortices of θ = 40. 86




θ = 50. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87




List of Tables xv




θ = 60. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88






creasing values of n on the lower-branch for stationary vortices at latitude

θ = 70. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Nomenclature

Roman Symbols

u, v, w, p spectral representations of the perturbation velocities and pressure, re-

spectively

a∗ radius of the sphere

n power-law index

p, P perturbation and mean pressure, respectively

R Reynolds number

r radial coordinate

Rc critical Reynolds number at the onset of the Type I mode

t time

u, U perturbation and mean azimuthal velocity, respectively

v, V perturbation and mean axial velocity, respectively

w, W perturbation and mean radial velocity, respectively

Greek Symbols

φ the angle of spiral vortices on the sphere surface

ρ density

α the wavenumber in the θ−direction

β The number of spiral vortices on the sphere surface

β azimuthal wave-number

γ rate-of-strain tensor

η dimensionless similarity coordinate

κ modified wavenumber

Nomenclature xvii

µ shear-rate viscosity

ν kinematic viscosity

ω frequency

σi j anti-symmetric viscous stress terms

τ shear stress tensor

δ ∗ non-dimensional length scale

Ω∗ system rotation rate

Subscripts

∗ dimensional quantity

Chapter 1

Introduction

1.1 Literature review

The laminar boundary layer flow of non-Newtonian fluids over rotating disks were first

studied by Mitschka (1964) who extended the von Kármán (1921) solution to fluids that

adhere to the power-law relationship and applied a boundary layer approximation since

the basic flow is not an exact solution of the Navier–Stokes equations. Mitschka and

Ulbrecht (1965) and Andersson et al. (2001) investigated the basic flow for both shear

thickening and shear-thinning fluids numerically. For the more common shear-thinning

fluids, Denier and Hewitt (2004) presented corrected similarity solutions of the boundary-

layer equations and revealed that beyond a critical level of shear-thinning, the basic flow

solution grows in the far field, so it cannot be matched to an external flow. Griffiths et al.

(2014b) extended the inviscid stability analysis used to describe the upper-branch station-

ary neutral modes of the von Kármán flow (for R ≫ 1) to incorporate the rheology of a

power-law fluid and considered the stability of the flow due to a rotating disk for shear-

thinning fluids that satisfy the power-law (Ostwald-de Waele) relationship. Specifically,

they presented an asymptotic analysis in the limit of large Reynolds number and extended

the previous works concerning convective instability of Newtonian flows to include the

additional viscous effects of a power-law fluid. Following the approach of Hall (1986),

they investigate the so called stationary “inviscid instabilities” with vortices occurring

1.1 Literature review 2

due to the location of an inflection point of the effective velocity profile. The prediction

of Griffiths et al. (2014b) for the wavenumber and wave angle of the disturbances sug-

gest that shear-thinning fluids may have a stabilising effect on the flow. The angle of the

spiral vortices resulting from the instability for the case when n = 1 (the Newtonian limit)

agreed well with existing numerical and experimental Newtonian results.

Griffiths et al. (2014a) investigated the convective instabilities associated with the boundary-

layer flow due to a rotating disk considering shear-thinning fluids that adhere to the power-

law relationship numerically. This work was an extension to their prior work Griffiths

et al. (2014b)and complete agreement was found with their prior asymptotic analysis.

These two papers can be considered as the non-Newtonian generalisations of Hall (1986)

and Malik (1986), respectively . The results of Griffiths et al. (2014a) indicate that, in

terms of the critical Reynolds number and growth rate, an increase in shear-thinning has

a stabilising effect on both the Type I and II modes. Moreover, the results presented in

Griffiths et al. (2014a) help to confirm previous suggestions of Griffiths et al. (2014b) that

shear thinning fluids have a stabilising effect on the flow.

The three-dimensional flow engendered by a sphere rotating in Newtonian still fluid was

first investigated theoretically by Howarth (1951). Subsequent theoretical papers have

concentrated on the laminar mean flow, for instance Nigam (1954), Stewartson (1957)

and Banks (1965). Howarth (1951) made boundary-layer approximations to the steady

Navier-Stokes equations and used a series solution to calculate the mean flow, showing

that near the poles the flow behaved like that due to a rotating disk. He suggested that the

assumptions on which the boundary layer theory is based were invalid near the equator

and consequently the motion there could not be described by the boundary layer equa-

tions. Nigam (1954) claimed that the boundary layer assumptions would be valid over

the whole flow field and it was possible to find a solution at the equator by assuming

special forms for the series expansions. Stewartson (1957) shows that a collision of the

boundary layers is unavoidable hence the flow claimed by Nigam (1954) is not possible.

Banks (1965) used the series expansion proposed by Howarth (1951) to investigate the

laminar boundary on a rotating sphere numerically. Banks (1965) reported that Nigam’s


model for the flow field is not correct and his special forms for the series approach raised

a divergent series. More accurate finite difference approximation techniques had been

used by Manohar (1967) and Banks (1976). The experimental papers for the Newtonian

fluids over rotating spheres by Sawatzki (1970) and Kobayashi et al. (1983) stated that

the flow remains laminar around the pole, at higher latitude, co-rotating spiral vortices

characteristic of crossflow instabilities appear as it is shown in Figure 1.1.

Figure 1.1: Transition on a rotating sphere. Photograph by Kobayashi et al. (1983)

The flow undergoes transition and becomes fully turbulent at a higher latitude, Garrett

(2002). The theoretical paper on the stability of the rotating sphere boundary layer by

Taniguchi et al. (1998) uses local-linear stability theory to predict the onset of convective

instability and the appearance of the spiral vortices on a sphere rotating in otherwise

undisturbed fluid. The approximate mean-flow profiles of Banks (1965) are used in their

paper to solve the perturbation equations at each latitude. Taniguchi et al. (1998) show

that the instability dominated by crossflow effects near to the pole and the streamline-

curvature instability dominates near to the equator. When below 70, the number of

vortices at the onset of instability are seen to decrease as the latitude increased. The

discrepancy between the critical Reynolds numbers for the onset of spiral vortices and

those reported in experimental investigations increases slightly but are still in reasonably

good agreement. On the other hand, a sharp divergence between the predicted critical


Reynolds numbers and the experimental values is seen for latitudes above 70. Moreover,

the results of Taniguchi et al. (1998) do not expect the stability characteristics of the

rotating-sphere boundary layer to tend towards those of the rotating disk as the analysis

moves towards the pole. Garrett and Peake (2002) conducted local convective instability

analyses on the rotating-sphere boundary layer using a different formulation to Taniguchi

et al. (1998). The approach was taken by Taniguchi et al. (1998) is that for a particular

θ , the perturbation equations are solved for neutral stability, ωi = 0, at each Reynolds

number then the real part of complex frequency, ωr, and the wavenumber in the θ−

direction, α are calculated. Garrett (2002) takes another approach where he insists that

the vortices rotate at some fixed multiple of the sphere surface velocity, thereby fixing the

ratio ωr/β , and then α and β are calculated using a spatial analysis. The approach taken

by Taniguchi et al. (1998) fails to observe the streamline-curvature instability lobe in their

neutral curves for latitudes lower than θ = 40 as they fix the vortex angle at 14. This is

in contrast to the approach taken by Garrett (2002) where the vortex angle is calculated

as an out put. In contrast to Taniguchi et al. (1998), the neutral curves of Garrett (2002)

for both convective and absolute instability of the boundary layer on the rotating sphere

approach those of the rotating disk as we approach the pole. However, the results of

Garrett (2002) show similar behaviour to that found by Taniguchi et al. (1998) where a

discrepancy between the experimental critical values and those predicted for stationary

vortices is seen and it quickly increases beyond a latitude of θ = 60. Garrett (2002)

demonstrates that the discrepancy is due to the stationary vortex assumption being invalid

at higher latitudes. More recently, Segalini and Garrett (2017) considered the boundary-

layer flow over the rotating sphere, both in terms of the computation of the steady flow

and also its linear stability properties. Their steady-flow solution has been improved

significantly from the original formulation proposed by Howarth (1951) and solved by

Banks (1965). The linear stability of the flow was considered as a function of Reynolds

number using a weakly non-parallel analysis. Type I and Type II were found to dictate the

local properties of the flow. Parallel and non-parallel eigenmodes of both mode types have

been computed for the first time. The spatial evolution of these suggests that, while the

1.2 Contributions and Outline of thesis 5

Type I mode is important at all latitudes, the Type II mode is potentially important at only

larger latitudes, at which point unstable Type I modes would already dominate the flow.

Segalini and Garrett (2017) extend those originally presented by Garrett and Peake (2002)

and Garrett (2010), performed using the parallel-flow approximation and Banks steady-

flow solutions. The non-parallel correction was found to be significant, particularly for

low Reynolds numbers and small mode numbers.

The flow and the convective instabilities associated with the boundary-layer flow due to

axisymmetric surfaces considering shear-thinning fluids that adhere to the power-law re-

lationship have not been studied theoretically or experimentally. This thesis is concerned

with the stability of incompressible power-law non-Newtonian fluid over rotating spheres.

The work was prompted by the investigations by Garrett (2002) of incompressible New-

tonian flows over rotating sphere and those of incompressible power-law non-Newtonian

flows over rotating disks done by Griffiths et al. (2014a). The analyses through this thesis

are restricted to a local analysis due to using the linear stability theory where a parallel-

flow approximation is made. This is discussed in Chapter 4.

1.2 Contributions and Outline of thesis

The intention of this work is to improve upon the current understanding of non-Newtonian

rotating boundary-layer flows.The main aim is to generalize the linear study of Newto-

nian boundary-layer flows over rotating spheres of Garrett and Peake (2002). Following

the approach of Banks (1965), we extend the perturbation method of Howarth (1951) to

study the laminar boundary layer flow of rotating sphere. Having the basic flow pro-

files, we compute curves of neutral stability that can then be directly compared to the

predictions of Garrett and Peake (2002) for the power-law index n = 1 and to those of nu-

merical investigation of convective instabilities associated with the boundary-layer flow

due to a rotating disk of Griffiths et al. (2014a) as the analysis moves towards the pole.

A Chebyshev polynomial method is used after applying the linear convective instability

to consider the effects of power-law fluids on the Type I and Type II modes. This thesis

1.2 Contributions and Outline of thesis 6

is organised as follows: in Chapter 2, we use dimensional continuity and momentum

equations to apply suitable boundary-layer approximations in order to determine the gov-

erning boundary-layer equations of base flow. In Chapter 3, these mean flow equations are

solved by extending the series expansion suggested by Howarth (1951) and used by Banks

(1965) and Segalini and Garrett (2017). The four sets of ordinary differential equations

are two-point boundary value problems which are solved numerically to determine the

corresponding base flow profiles of power-law fluids on a rotating sphere. The convective

instabilities for the power-law fluids over the outer surface of a rotating spheres are con-

ducted in Chapters 4,5 and 6, respectively. In Chapter 4, the linear perturbation equations

are formulated by applying parallel-flow approximation and then are solved using Cheby-

shev collocation method in order to present the convective neutral curves. In Chapter 5,

these neutral curves and our theoretical predictions of the critical Reynolds number and

vortex angle at the onset of convective instability are presented and confirmed by present-

ing the growth rates at latitudes 10− 70 for n = 1.0,0.9,0.8,0.7 and 0.6 respectively,

then at each latitude for the power-law index 0.6 ≤ n ≤ 1 in Chapter 6. Finally, in Chapter

7, we conclude and summarise our findings and comment on possible extensions of this

work.

Chapter 2

Formulation of the steady equations

This chapter is concerned with the governing equations for power-law non-Newtonian

fluids over a rotating sphere.We will obtain these equations to investigate the laminar-

flow profiles and their stability properties.

The flow of power-law non-Newtonian fluids on axi-symmetric surfaces has received

less attention than the laminar boundary layer of the Newtonian fluid which have been

received constant research, see for example Howarth (1951), Banks (1965), Taniguchi

et al. (1998) and Garrett and Peake (2002).

2.1 The power-law non-Newtonian fluid (Ostwald-de Waele

Model)

Newtonian fluids are fluids which have a constant viscosity at all shear rates. This defini-

tion leads to the following governing relationship

τττ = µnγγγ, (2.1)

where τττ is the shear stress tensor, µn is the constant viscosity and γγγ = ∇∇∇uuu+ (∇∇∇uuu)T is

the rate-of-strain tensor for any arbitrary flow field uuu = uuu(x1,x2,x3, t), in an arbitrary

coordinate system. Thus in the case of Newtonian fluids, shear stress is a linear function

2.1 The power-law non-Newtonian fluid (Ostwald-de Waele Model) 8

of the shear rate.

For a generalised Newtonian fluid, the viscosity is no longer constant and it is a func-

tion of the shear rate such that, µ = µ (γ), hence

τττ = µ (γ) γγγ, (2.2)

where here µ is a function of the invariants of the tensor γ . The invariants of a tensor

are defined such that their values are independent of the choice of coordinate system. For

example, the scalar invariant of an arbitrary vector vvv = (v1,v2,v3), in three dimensions, is

|vvv| =√

v21 + v2

2 + v23 =

√∑i v2

i . This scalar invariant of vvv is independent of the choice of

coordinate system.

The invariants of γ are given in Bird et al. (1977) as

I = trγ = ∑i

γii,

II = trγγγ2 = ∑i

∑j

γi jγ ji = ∑i

∑j

γ2i j,

III = trγγγ3 = ∑i

∑j∑k

γi jγ jkγki.

The magnitude of the rate-of-strain tensor is given by |γγγ| = γ =√

(γγγ ::: γγγ)/2 where the

double dot operation is defined by Griffiths (2016) as follows

(γγγ ::: γγγ) =

[(∑

i∑

jδδδ iδδδ jγi j

):

(∑k

∑l

δδδ kδδδ l γkl

)],

= ∑i

∑j∑k

∑l

(δδδ iδδδ j : δδδ kδδδ l

)γi jγkl,

= ∑i

∑j∑k

∑l

(δδδ j·δδδ k

)(δδδ i·δδδ l) γi jγkl = ∑

i∑

j∑k

∑l

δilδ jkγi jγkl,

= ∑i

∑j

γi jγ ji = ∑i

∑j

γ2i j = II.

2.2 The governing boundary-layer equations for Power-law fluids 9

The governing relationship for µ (γ), when considering a power-law fluid is

µ = m(γ)n−1 .

This is also known as the Ostwald-de Waele, where m(Pasn) is the consistency coefficient

and n the dimensionless power-law index, For shear-thickening (dilatant) flows with n >

1, the fluid’s viscosity increases with increasing shear rate. Whilst for shear-thinning

(pseudoplastic) flows with n< 1, the fluid’s viscosity decreases with increasing shear rate.

Substituting n = 1 into the power-law model returns the Newtonian viscosity relationship

where m = µn.

The power-law relationship is only applicable over a finite range of shear rates, this

is because the model predicts an infinite viscosity at rest and a vanishing viscosity as the

shear rate approaches infinity, when shear-thinning fluids are consedired, and vice-versa

for shear-thickening fluids. This unphysical behaviour is described as follows

for n < 1 : limγ→0

µ (γ)→ ∞, limγ→∞

µ (γ)→ 0,

for n < 1 : limγ→0

µ (γ)→ 0, lim˙γ→∞

µ (γ)→ ∞.

2.2 The governing boundary-layer equations for Power-

law fluids

The objective of this section is to introduce and derive the governing equations of the

boundary-layer flow for power-law fluids, considered in spherical polar coordinates fixed

in space with origin located at the centre of the sphere. The radius of the sphere is a∗, r∗

is the distance measured radially from the centre of the sphere, θ is the angle of latitude

measured from the axis of rotation, φ is the angle of azimuth. Figure 2.1 shows the

coordinate system used in this analysis.


Figure 2.1: Sketch of the coordinate system

The sphere rotates at a constant angular frequency Ω ∗ in the φ direction in a power-

law non-Newtonian fluid of infinite extension, ν∗ denoted the coefficient of kinematic

viscosity and ρ∗ denoted the density (asterisks indicate dimensional quantities).

The continuity and momentum equation in spherical coordinates are

1r∗2

∂

(r∗

2W ∗)

∂ r∗+

1r∗ sinθ

∂ (U∗ sinθ)

∂θ+

1r∗ sinθ

∂V ∗

∂φ= 0, (2.3)

∂W ∗

∂ t∗+W ∗∂W ∗

∂ r∗+

U∗

r∗∂W ∗

∂θ+

V ∗

r∗ sinθ

∂W ∗

∂φ− U∗2

+V ∗2

r∗=− 1

ρ∗∂P∗

∂ r∗

+1

ρ∗

[1

r∗2

∂

(r∗

2τ∗r∗r∗

)∂ r∗

+1

r∗ sinθ

∂(τ∗r∗θ

sinθ)

∂θ+

1r∗ sinθ

∂τ∗r∗φ

∂φ−

τ∗θθ

+ τ∗φφ

r∗

], (2.4a)

∂U∗

∂ t∗+W ∗∂U∗

∂ r∗+

U∗

r∗∂U∗

∂θ+

V ∗

r∗ sinθ

∂U∗

∂φ+

W ∗U∗

r∗− V ∗2

cotθ

r∗=− 1

r∗ρ∗∂P∗

∂θ

+1

ρ∗

1r∗3

∂

(r∗

3τ∗r∗θ

)∂ r∗

+1

r∗ sinθ

∂(τ∗

θθsinθ

)∂θ

+1

r∗ sinθ

∂τ∗θφ

∂φ−

τ∗φφ

cotθ

r∗

, (2.4b)


∂V ∗

∂ t∗+W ∗∂V ∗

∂ r∗+

U∗

r∗∂V ∗

∂θ+

V ∗

r∗ sinθ

∂V ∗

∂φ+

W ∗V ∗

r∗− V ∗U∗ cotθ

r∗=− 1

r∗ρ∗ sinθ

∂P∗

∂φ

+1

ρ∗

1r∗3

∂

(r∗

3τ∗r∗φ

)∂ r∗

+1

r∗ sinθ

∂

(τ∗

θφsinθ

)∂θ

+1

r∗ sinθ

∂τ∗φφ

∂φ−

τ∗θφ

cotθ

r∗

. (2.4c)

The components of the stress tensor in Equations (2.4) are expressed as

τ∗r∗r∗ = 2µ

∗(

∂W ∗

∂ r∗

), (2.5a)

τ∗θθ = 2µ

∗(

1r∗

∂U∗

∂θ+

W ∗

r∗

), (2.5b)

τ∗φφ = 2µ

∗(

1r∗ sinθ

∂V ∗

∂φ+

W ∗+U∗ cotθ

r∗

), (2.5c)

τ∗r∗θ = µ

∗(

r∗∂

∂ r∗

(U∗

r∗

)+

1r∗

∂W ∗

∂θ

)= τ

∗θr∗ , (2.5d)

τ∗r∗φ = µ

∗(

1r∗ sinθ

∂W ∗

∂φ+ r∗

∂

∂ r∗

(V ∗

r∗

))= τ

∗φr∗ , (2.5e)

τ∗θφ = µ

∗(

sinθ

r∗∂

∂θ

(V ∗

sinθ

)+

1r∗ sinθ

∂U∗

∂φ

)= τ

∗φθ . (2.5f)

The rate-of-strain tensor γ∗ in spherical coordinates can be written as the following

γ∗ =

√II2=

2[(

∂W ∗

∂ r∗

)2+(

1r∗

∂U∗

∂θ+ W ∗

r∗

)2+(

1r∗ sinθ

∂V ∗

∂φ+ W ∗

r∗ + U∗ cotθ

r∗

)2]

+[r∗ ∂

∂ r∗

(U∗

r∗

)+ 1

r∗∂W ∗

∂θ

]2+[

sinθ

r∗∂

∂θ

(V ∗

sinθ

)+ 1

r∗ sinθ

∂U∗

∂φ

]2

+[

1r∗ sinθ

∂W ∗

∂φ+ r∗ ∂

∂ r∗

(V ∗

r∗

)]2

1/2

,

(2.6)

where W ∗, U∗ and V ∗ are the dimensional velocities in the r∗, θ and φ directions respect-

ively and P∗ is the fluid pressure. In the radial equation, we find P∗ = P∗ (θ) . Since in this

chapter the sphere is rotating in otherwise still fluid P∗ = constant. The boundary-layer

thickness is δ ∗ =(

ν∗/Ω∗(2−n))1/2

, and by assuming that δ ∗/a∗ ≪ 1, we can replace the

r∗ multiplying terms in Equations (2.3) and (2.4) by a∗, which represents a parallel-flow


assumption. Furthermore,

∂W ∗

∂ t∗= 0 and

∂W ∗

∂φ= 0 where W ∗W ∗W ∗ = (W ∗,U∗,V ∗) , (2.7)

due to the time in-dependency (the steady motion) and the symmetry about the axis of the

rotation respectively. Under these three assumptions, Equations (2.3) and (2.4) become

∂W ∗

∂ r∗+

1a∗

∂U∗

∂θ+

U∗

a∗cotθ = 0, (2.8)

W ∗∂W ∗

∂ r∗+

U∗

a∗∂W ∗

∂θ− U∗2

+V ∗2

r∗=− 1

ρ∗∂P∗

∂ r∗

+1

ρ∗

[∂τ∗r∗r∗

∂ r∗+

1a∗

∂τ∗r∗θ

∂θ+

τ∗r∗φ

a∗cotθ −

τ∗θθ

+ τ∗φφ

a∗

], (2.9a)

W ∗∂U∗

∂ r∗+

U∗

a∗∂U∗

∂θ+

W ∗U∗

a∗− V ∗2

cotθ

a∗=− 1

a∗ρ∗∂P∗

∂θ

+1

ρ∗

[∂τ∗r∗θ

∂ r∗+

1a∗

∂τ∗θθ

∂θ+

τ∗θθ

a∗cotθ −

τ∗φφ

cotθ

a∗

], (2.9b)

W ∗∂V ∗

∂ r∗+

U∗

a∗∂V ∗

∂θ+

W ∗V ∗

a∗− U∗V ∗ cotθ

a∗=

+1

ρ∗

[∂τ∗r∗φ

∂ r∗+

1a∗

∂τ∗θφ

∂θ+

τ∗φφ

a∗cotθ −

τ∗θφ

cotθ

a∗

]. (2.9c)

In the fixed frame of reference, Equations (2.8) and (2.9) are subjects to the boundary

conditions

W ∗ = U∗ =V ∗−a∗Ω∗2−n

sinθ = 0 on r∗ = a∗, (2.10a)

U∗ = V ∗ = 0 as r∗ → ∞. (2.10b)

The first equation in (2.10) represents the no-slip condition on the sphere surface while

2.3 The dimensionless boundary-layer equations of Power-law fluids over a rotating sphere13

the second represents the quiescent fluid condition away from the sphere at the edge of

the boundary layer.

The components of the stress tensor under the parallel-flow and (2.7) assumptions are

τ∗r∗r∗ = 2µ

∗(

∂W ∗

∂ r∗

), (2.11a)

τ∗θθ = 2µ

∗(

1a∗

∂U∗

∂θ+

W ∗

a∗

), (2.11b)

τ∗φφ = 2µ

∗(

W ∗+U∗ cotθ

a∗

), (2.11c)

τ∗r∗θ = µ

∗(

∂U∗

∂ r∗+

1a∗

∂W ∗

∂θ

)= τ

∗θr∗, (2.11d)

τ∗r∗φ = µ

∗(

∂V ∗

∂ r∗

)= τ

∗φr∗, (2.11e)

τ∗θφ = µ

∗(

1a∗

(∂V ∗

∂θ

)− 1

a∗V ∗ cotθ

)= τ

∗φθ . (2.11f)

The rate-of-strain tensor γ∗ is expressed as

γ∗ =

√II2=

2[(

∂W ∗

∂ r∗

)2+(

1a∗

∂U∗

∂θ+ W ∗

a∗

)2+(

W ∗

a∗ + U∗ cotθ

a∗

)2]

+[

∂U∗

∂ r∗ + 1a∗

∂W ∗

∂θ

]2+[

1a∗

∂V ∗

∂θ− V ∗

a∗ cotθ

]2

+[

∂V ∗

∂ r∗

]2

1/2

. (2.12)

2.3 The dimensionless boundary-layer equations of Power-

law fluids over a rotating sphere

The non-dimensional mean-flow variables are defined as

W (η1,θ) =A∗

Ω∗W ∗ where A∗ =

[a∗

1−nΩ∗2−n

ν∗

]1/n+1

, and η1 = A∗ (r∗−a∗) ,

U =U∗

a∗Ω∗ , V =V ∗

a∗Ω∗ , µ =[a∗A∗Ω∗]1−n

m∗ µ∗, P =

P∗

ρ∗ (a∗Ω∗)2 . (2.13)


Equation (2.13) is consistent with non-dimensional variables of Garrett and Peake (2002),

however modifications have been made to accommodate the power-law fluid such that all

equations reduce to the Newtonian equations when n = 1.

Substituting (2.13) into (2.8) and (2.9) leads to the following dimensionless continu-

ity and momentum equations

∂W∂η1

+∂U∂θ

+U cotθ = 0, (2.14)

1a∗A∗W

∂W∂η1

+1

a∗A∗U∂W∂θ

−(U2 +V 2)=− 1

a∗A∗∂P∂η1

+

(δ ∗

a∗

)2

(a∗A∗)n

[

∂

∂η1

(2µ

∂W∂η1

)]+[

∂

∂θ

(µ

∂U∂η1

)]+[cotθ

(µ

∂U∂η1

)]

−2(

δ ∗

a∗

)2

(a∗A∗)(n−1)

[µ

∂U∂θ

]+[µU cotθ ]

−2(

δ ∗

a∗

)2

(a∗A∗)(n−2)

[µW ]+ [µU cotθ ]

, (2.15a)

W∂U∂η1

+U∂U∂θ

+1

a∗A∗WU −V 2 cotθ =−∂P∂θ

+∂

∂η1

(µ

∂U∂η1

)

+

(δ ∗

a∗

)2

(a∗A∗)(n−1)

[∂

∂η1

(µ

∂W∂θ

)]+[

∂

∂θ

(2µ

∂U∂θ

)][

∂

∂θ(2µW )

]+[(

2µ cotθ∂U∂θ

)]+[2µU cot2 θ

]

+

(δ ∗

a∗

)2

(a∗A∗)(n−2)

2µU cot2 θ

, (2.15b)


W∂V∂η1

+U∂V∂θ

+UV cotθ +1

a∗A∗WV =

[∂

∂η1

(µ

∂V∂η1

)](

δ ∗

a∗

)2

(a∗A∗)(n−1)

∂

∂θ

[µ

(∂V∂θ

−V cotθ

)]+2cotθ

[µ

(∂V∂θ

−V cotθ

)] . (2.15c)

The rate-of-strain tensor becomes

γ∗2= Ω

∗

2[(

∂W∂η1

)2+(

∂U∂θ

+ 1a∗A∗W

)2+( 1

a∗A∗W +U cotθ)2]

+[(a∗A∗)

(∂U∂η1

)+ 1

a∗A∗

(∂W∂θ

)]2+[(

∂V∂θ

)−V cotθ

]2

+[

1a∗A∗

(∂V

∂η1‘

)]2

1/2

. (2.16)

Now, the Reynolds number for power-law fluids is defined in the following form

Re =

[a∗

2Ω∗2−n

ν∗

]1/(n+1)

. (2.17)

By setting n = 1, the form of the Reynolds number for Newtonian fluid flow in Garrett

and Peake (2002) is recovered. Notice that

a∗A∗ = Re and Re =[

a∗

δ ∗

]2/(n+1)

. (2.18)

Considering n = 1 in (2.17), the latter relationship in (2.18) will give the Newtonian

formula Re = a∗/δ ∗ as appears in Garrett (2002).

Therefore, by substituting (2.18) into (2.15), the scaled governing equations for power-

law fluids over a rotating sphere are obtained as following

∂W∂η1

+∂U∂θ

+U cotθ = 0, (2.19)


Re−2W∂W∂η1

+Re−2U∂W∂θ

−Re−1 (U2 +V 2)=− ∂P∂η1

+Re−2

[

∂

∂η1

(2µ

∂W∂η1

)]+[

∂

∂θ

(µ

∂U∂η1

)]+[cotθ

(µ

∂U∂η1

)]

−2Re−3

[µ

∂U∂θ

]+[µU cotθ ]

−2Re−4

[µW ]+ [µU cotθ ]

, (2.20a)

W∂U∂η1

+U∂U∂θ

+Re−1WU −V 2 cotθ =−∂P∂θ

+∂

∂η1

(µ

∂U∂η1

)

+Re−2

[∂

∂η1

(µ

∂W∂θ

)]+[

∂

∂θ

(2µ

∂U∂θ

)][

∂

∂θ(2µW )

]+[(

2µ cotθ∂U∂θ

)]+[2µU cot2 θ

]

+Re−3

2µU cot2 θ

, (2.20b)

W∂V∂η1

+U∂V∂θ

+UV cotθ +Re−1WV =

[∂

∂η1

(µ

∂V∂η1

)]

+Re−2

∂

∂θ

[µ

(∂V∂θ

−V cotθ

)]+2cotθ

[µ

(∂V∂θ

−V cotθ

)] . (2.20c)


The rate-of-strain tensor becomes

γ2 =

(∂U∂η1

)2

+

(∂V∂η1

)2

+Re−2

2(

∂W∂η1

)2+2(

∂U∂θ

)2+2U2 cot2 θ +2

(∂U∂η1

)(∂W∂θ

)+(

∂V∂θ

)2−2(

∂V∂θ

)V cotθ +V 2 cot2 θ

+Re−3

[4W(

∂U∂θ

)+4WU cotθ

]+Re−4

[4W 2 +

(∂W∂θ

)2],

(2.21)

where

µ =

[(∂U∂η1

)2

+

(∂V∂η1

)2

+O(Re−2)](n−1)/2

, (2.22)

is the dimensionless viscosity function defined by

µ =[ReΩ∗]1−n

m∗ µ∗.

We now proceed by making a boundary-layer approximation by eliminating terms in-

volving inverse powers of the Reynolds number by assuming that Re ≫ 1, the continuity

equation remains unchanged. Since we assumed that inside the boundary-layer the pres-

sure is a function of θ only, we have that P ≡ 0. The non-dimensional pressure P is not

required to conduct the instability analysis discussed later. Thus, only the equations of

W , U and V are presented here, and are completely defined by the system (2.23). This

approach is entirely consistent with Garrett and Peake (2002).

The boundary-layer equations for power-law fluid on a rotating sphere are

∂W∂η1

+∂U∂θ

+U cotθ = 0, (2.23a)

W∂U∂η1

+U∂U∂θ

−V 2 cotθ =∂

∂η1

(µ

∂U∂η1

), (2.23b)

2.4 Summary 18

W∂V∂η1

+U∂V∂θ

−UV cotθ =∂

∂η1

(µ

∂V∂η1

), (2.23c)

where

µ =

[(∂U∂η1

)2

+

(∂V∂η1

)2](n−1)/2

, (2.24)

is the viscosity function.

The system of Equations (2.23) is subject to the following boundary conditions

W = U =V − sinθ = 0 on η1 = 0, (2.25a)

U = V = 0 as η1 → ∞, (2.25b)

which again represent the non-slip and quiescent flow conditions respectively.

2.4 Summary

In this chapter we gave a brief introduction to describe the power-law fluids and the phe-

nomenological relationship between the stress tensor and the rate of deformation tensor

of the fluid. The boundary-layer equations have been derived, for the first time, in order

to describe the boundary-layer flow. assuming the time independency and the symmetry

about the axis of rotation, then applying the boundary-layer approximation, the governing

equations are formulated. These governing boundary-layer equations are the main re-

quirement to obtain the steady mean flow solutions and subsequently study the transition

from laminar to turbulence via the stability properties of the flow.

Chapter 3

Solutions of the steady-flow equations

The objective of this chapter is to solve the boundary-layer equations of the mean flow

as determined in Chapter 2 to determine and describe the steady mean flow profiles and

obtain the steady mean flow solutions. These will be used to study the transition from

laminar to turbulence later in this thesis.

In this chapter, the flow of laminar boundary-layer fluids on a rotating sphere are

studied theoretically by the extension of Banks’ method.

3.1 The series expansions

Starting from the governing boundary-layer equations (2.23) and (2.24) along with the

boundary conditions (2.25), we introduce asymptotic series expansions for the velocity

components and viscosity modification as follows

U (η1,θ) = F1θ +F3θ3 +F5θ

5 +F7θ7 + · · · , (3.1a)

V (η1,θ) = G1θ +G3θ3 +G5θ

5 +G7θ7 + · · · , (3.1b)

W (η1,θ) = θ(n−1)/(n+1)

[H1 +H3θ

2 +H5θ4 +H7θ

6 + · · ·], (3.1c)

µ (η1,θ) = µ1 +µ3θ2 +µ5θ

4 +µ7θ6 + · · · . (3.1d)

3.1 The series expansions 20

Here the velocity functions Fi, Gi, Hi for i = 1, 3, 5, . . . are dimensionless and are func-

tions of an appropriate coordinate η where

η = θ(1−n)/(n+1)

η1. (3.2)

Notice that:

sinθ = θ − θ 3

6+

θ 5

120− θ 7

5040+ · · · , (3.3a)

cotθ =1θ− θ

3− θ 3

45− 2θ 5

945− θ 7

4725−·· · . (3.3b)

Starting from the viscosity function

µ =

[(∂U∂η1

)2

+

(∂V∂η1

)2](n−1)/2

= θ2(n−1)/(n+1)

[(F

′1

)2+(

G′1

)2]+[2F

′1F

′3 +2G

′1G

′3

]θ 2

+

[(F

′3

)2+(

G′3

)2+2(

F′1F

′5 +G

′1G

′5

)]θ 4

+[2(

F′1F

′7 +F

′3F

′5 +G

′1G

′7 +G

′3G

′5

)]θ 6 + · · ·

(n−1)/2

. (3.4)

Using the binomial theorem then comparing of like powers of θ gives

µ1 =

[(F

′1

)2+(

G′1

)2](n−1)/2

, (3.5)

µ3 = (n−1)(F ′

1F ′3 +G′

1G′3)[(

F ′1)2

+(G′

1)2](n−3)/2

, (3.6)

µ5 =(n−1)(n−3)

2(F ′

1F ′3 +G′

1G′3)2[(

F ′1)2

+(G′

1)2](n−5)/2

+(n−1)

2

(F ′

3)2

+(G′

3)2

+2(F ′

1F ′5 +G′

1G′5)[(

F ′1)2

+(G′

1)3](n−3)/2

,

µ7 =(n−1)(n−3)(n−5)

6(F ′

1F ′3 +G′

1G′3)3[(

F ′1)2

+(G′

1)2](n−7)/2

(3.7)

+(n−1)(n−3)

2

[(

F ′3)2

+(G′

3)2

+2(F ′

1F ′5 +G′

1G′5)]

(F ′

1F ′3 +G′

1G′3)[

(F ′1)

2 +(G′1)

2](n−5)/2

+ (n−1)

(F ′

1F7 +G′1G′

7 +F ′3F ′

5 +G′3G′

5)[(

F ′1)2

+(G′

1)2](n−3)/2

. (3.8)


Now substituting (3.1), (3.2) and (3.3) into system (2.23) and its boundary conditions

(2.25) gives

(H

′1 +H

′3θ

2 +H′5θ

4 +H′7θ

6 + · · ·)+(

F1 +3F3θ2 +5F5θ

4 +7F7θ6 + · · ·

)+

[(1−n1+n

)η1F

′1 +

(1−n1+n

)η1F

′3θ

2 +

(1−n1+n

)η1F

′5θ

4 +

(1−n1+n

)η1F

′7θ

6 + · · ·]

+

[F1 +

(F3 −

F1

3

)θ

2 +

(F5 −

F3

3− F1

45

)θ

4 +

(F7 −

F5

3− F3

45− 2F1

945

)θ

6 + · · ·]= 0,

(3.9)

H1F′1θ +

(H1F

′3 +H3F

′1

)θ 3 +

(H1F

′5 +H3F

′3 +H5F

′1

)θ 5

+(

H1F′7 +H3F

′5 +H5F

′3 +H7F

′1

)θ 7 + · · ·

(3.10)

+[F2

1 θ +4F1F3θ3 +(6F1F5 +3F2

3)

θ5 +(8F1F7 +8F3F5)θ

7 + · · ·]

+

(1−n

1+n

)η1F1F

′1θ +

(1−n1+n

)η1

(F1F

′3 +F3F

′1

)θ 3

+(1−n

1+n

)η1

(F1F

′5 +F3F

′3 +F5F

′1

)θ 5

+(1−n

1+n

)η1

(F7F

′1 +F5F

′3 +F3F

′5 +F1F

′7

)θ 7 + · · ·

+

G21θ +

(2G1G3 −

G21

3

)θ 3

+(

G23 +2G1G5 − 2G1G3

3 − G21

45

)θ 5

+

2G1G7 +2G3G5 − 2G1G345

−(G23+2G1G5)

3 − 2G1945

θ 7 + · · ·

=

∂

∂η

µ1F ′1θ +

(µ1F ′

3 +µ3F ′1)

θ 3 +(µ1F ′

5 +µ3F ′3 +µ5F ′

1)

θ 5

+(µ1F ′

7 +µ3F ′5 +µ5F ′

3 +µ7F ′1)

θ 7 + · · ·

,


H1G′1θ +

(H1G

′3 +H3G

′1

)θ 3 +

(H1G

′5 +H3G

′3 +H5G

′1

)θ 5

+(

H1G′7 +H3G

′5 +H5G

′3 +H7G

′1

)θ 7 + · · ·

(3.11)

F1G1θ +(3F1G3 +F3G1)θ 3 +(5F1G5 +3F3G3 +F1G1)θ 5

+(F7G1 +3F5G3 +5F3G5 +7F1G7)θ 7 + · · ·

(1−n1+n

)η1F1G

′1θ +

(1−n1+n

)η1

(F1G

′3 +F3G

′1

)θ 3

+(1−n

1+n

)η1

(F1G

′5 +F3G

′3 +F5G

′1

)θ 5

+(1−n

1+n

)η1

(F1G

′7 +F3G

′5 +F5G

′3 +F7G

′1

)θ 7 + · · ·

F1G1θ +(

F1G3 +F3G1 − F1G13

)θ 3

+(

F1G5 +F3G3 +F5G1 − (F1G3+F3G1)3 − F1G1

45

)θ 5 F1G7 +F3G5 +F5G3 +F7G1 − 2F1G1

945

− (F1G5+F3G3+F5G1)3 − (F1G3+F3G1)

45

θ 7 + · · ·

=

∂

∂η1

µ1G′1θ +

(µ1G

′3 +µ3G

′1

)θ 3 +

(µ1G

′5 +µ3G

′3 +µ5G

′1

)θ 5

+(

µ1G′7 +µ3G

′5 +µ5G

′3 +µ7G

′1

)θ 7 + · · ·

,where the boundary conditions when η = 0 are

F1 (0)θ +F3 (0)θ3 +F5 (0)θ

5 +F7 (0)θ7 + · · ·= 0, (3.12)

G1 (0)θ +G3 (0)θ3 +G5 (0)θ

5 +G7 (0)θ7 + · · ·= θ − θ 3

6+

θ 5

120− θ 7

5040+ · · · ,

(3.13)

H1 (0)+H3 (0)θ2 +H5 (0)θ

4 +H7 (0)θ6 + · · ·= 0, (3.14)

and at η → ∞ are


F1 (∞)θ +F3 (∞)θ3 +F5 (∞)θ

5 +F7 (∞)θ7 + · · ·= 0, (3.15)

G1 (∞)θ +G3 (∞)θ3 +G5 (∞)θ

5 +G7 (∞)θ7 + · · ·= 0. (3.16)

We now proceed by dividing Equations (3.10) and (3.11) by θ then comparing like power

of θ gives the following set of ordinary differential equations for the velocity functions.

At each order, the first, third, fifth and seventh order Equations (3.17), (3.19), (3.21) and

(3.23) are solved and the unknowns values of F ′i (0) and G′

i (0) are determined using the

MATLAB function bvp4c. The numerical method of this function is based on a finite

difference code implementing the three stage Lobatto IIIa formula, that can be viewed as

an implicit Runge-Kutta formula with a continuous interpolant, Kierzenka and Shampine

(2001). The Lobatto IIIa method is a collocation method that provides a C1 continuous

solution that is fourth-order accurate uniformly in a finite interval [a,b].

First-order functions are given by

H′1 +2F1 +

(1−n1+n

)ηF

′1 = 0, (3.17a)[

H1 +

(1−n1+n

)ηF1

]F ′

1 +F21 −G2

1 =d

dη

(µ1F ′

1), (3.17b)[

H1 +

(1−n1+n

)ηF1

]G′

1 +2F1G1 =d

dη

(µ1G′

1), (3.17c)

0 2 4 6 8 10 12 14 16 18 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2n=1n=0.9n=0.8n=0.7n=0.6

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1n=1n=0.9n=0.8n=0.7n=0.6

0 2 4 6 8 10 12 14 16 18 20-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0n=1n=0.9n=0.8n=0.7n=0.6

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20n=1n=0.9n=0.8n=0.7n=0.6

Figure 3.1: Plots of F1; G1; H1 and µ1 versus η , the η-axis has been truncated at η = 20.


with boundary conditions

H1 (0) = 0, G1 (0) = 1, F1 (0) = 0, (3.18a)

F1 (∞) = G1 (∞) = 0. (3.18b)

The initial values of mean velocity flow parameters F ′1 (0), G′

1 (0) and H ′1 (η∞) are stated

in Table 3.1

n F ′1 (0) −G′

1 (0) −H1 (η∞)

1 0.5102 0.6159 0.8845

0.9 0.5069 0.6243 0.9698

0.8 0.5039 0.6362 1.0957

0.7 0.5017 0.6532 1.3051

0.6 0.5005 0.6778 1.7329

Table 3.1: Numerical values of the mean velocity flow parameters F ′1 (0); G′

1 (0) andH1 (η∞) for n values.

Third-order functions are as following

H′3 +4F3 +

(1−n1+n

)ηF

′3 −

F1

3= 0, (3.19a)[

H3 +

(1−n1+n

)ηF3

]F ′

1 +

[H1 +

(1−n1+n

)ηF1

]F ′

3

+4F1F3 −2G1G3 +G2

13

=d

dη

(µ1F ′

3 +µ3F ′1),

(3.19b)[H3 +

(1−n1+n

)ηF3

]G′

1 +

[H1 +

(1−n1+n

)ηF1

]G′

3

+2F3G1 +4F1G3 −F1G1

3=

ddη

(µ1G′

3 +µ3G′1),

(3.19c)


with the boundary conditions

F3 (0) = 0, G3 (0) =−16, H3 (0) = 0, (3.20a)

F3 (∞) = G3 (∞) = 0. (3.20b)

The initial values of mean velocity flow parameters F ′3 (0), G′

3 (0) and H ′3 (η∞) are stated

in Table 3.2.

0 2 4 6 8 10 12 14 16 18 20-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

n=1n=0.9n=0.8n=0.7n=0.6

0 2 4 6 8 10 12 14 16 18 20-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

n=1n=0.9n=0.8n=0.7n=0.6

0 2 4 6 8 10 12 14 16 18 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4n=1n=0.9n=0.8n=0.7n=0.6

0 2 4 6 8 10 12 14 16 18 20-2.5

-2

-1.5

-1

-0.5

0

0.5

n=1n=0.9n=0.8n=0.7n=0.6

Figure 3.2: Plots of F3; G3; H3 and µ3 versus η for n values. The η-axis has beentruncated at η = 20.

n −F ′3 (0) G′

3 (0) H1 (η∞)

1 0.2213 0.2476 0.1607

0.9 0.2314 0.2647 0.1539

0.8 0.2429 0.2853 0.1466

0.7 0.2562 0.3109 0.1411

0.6 0.2717 0.3437 0.1458


3 (0) andH3 (η∞).


Fifth-order functions are as following

H′5 +6F5 +

(1−n1+n

)ηF

′5 −

F3

3− F1

45= 0, (3.21a)[

H5 +

(1−n1+n

)ηF5

]F ′

1 +

[H3 +

(1−n1+n

)ηF3

]F ′

3 +

[H1 +

(1−n1+n

)ηF1

]F ′

5

+3F23 +6F1F5 −G2

3 −2G1G5 +2G1G3

3+

G21

45=

ddη

(µ1F ′

5 +µ3F ′3 +µ5F ′

1),

(3.21b)

[H5 +

(1−n1+n

)ηF5

]G

′1 +

[H3 +

(1−n1+n

)ηF3

]G

′3 +

[H1 +

(1−n1+n

)ηF1

]G

′5

+4F3G3 +6F1G5 +2F5G1 −(F1G3 +F3G1)

3− F1G1

45=

ddη

(µ1G′

5 +µ3G′3 +µ5G′

1).

(3.21c)

0 2 4 6 8 10 12 14 16 18 20-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

n=1n=0.9n=0.8n=0.7n=0.6

0 2 4 6 8 10 12 14 16 18 20

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

n=1n=0.9n=0.8n=0.7n=0.6

0 2 4 6 8 10 12 14 16 18 20-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3n=1n=0.9n=0.8n=0.7n=0.6

0 2 4 6 8 10 12 14 16 18 20

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8n=1n=0.9n=0.8n=0.7n=0.6

Figure 3.3: Plots of F5; G5; H5 and µ5 versus η . The η-axis has been truncated at η = 20.


The boundary conditions are

H5 (0) = 0, G5 (0) =1

120, F5 (0) = 0, (3.22a)

F5 (∞) = G5 (∞) = 0. (3.22b)

The velocity functions profiles of fifth order are shown in Figure 3.3. The initial values

of mean velocity flow parameters F ′5 (0), G′

5 (0) and H ′5 (η∞) are stated in Table 3.3.

n F ′5 (0) G′

5 (0) H5 (η∞)

1 0.0207 0.0257 0.0008

0.9 0.0246 0.0436 0.0611

0.8 0.0321 0.0725 0.1410

0.7 0.0467 0.1205 0.2303

0.6 0.0747 0.2038 0.2928


5 (0) andH5 (η∞).

Seventh-order functions are as following

H′7 +8F7 +

(1−n1+n

)ηF

′7 −

F5

3− F3

45− 2F1

945= 0, (3.23a)

+

[H7 +

(1−n1+n

)ηF7

]F ′

1 +

[H5 +

(1−n1+n

)ηF5

]F ′

3 +

[H3 +

(1−n1+n

)ηF3

]F ′

5

+

[H1 +

(1−n1+n

)ηF1

]F ′

7 +

(G2

3 +2G1G5)

3−2(G1G7 +G3G5)+

2G1G3

45+

2G21

945

+8(F1F7 +F3F5)+

(G2

3 +G1G5)

3

=d

dη

(µ1F ′

7 +µ3F ′5 +µ5F ′

3 +µ7F ′1),

(3.23b)


[H7 +

(1−n1+n

)ηF7

]G′

1 +

[H5 +

(1−n1+n

)ηF5

]G′

3 +

[H3 +

(1−n1+n

)ηF3

]G′

5

+

[H1 +

(1−n1+n

)ηF1

]G′

7 +6F3G5 +4F5G3 +8F1G7 +4F5G3 +2F7G1 −2F1G1

945

−(F1G3 +F3G1)

45− (F1G5 +F3G3 +F5G1)

3

=d

dη

(µ1G′

7 +µ3G′5 +µ5G′

3 +µ7G′1).

(3.23c)

n F ′7 (0) G′

7 (0) H7 (η∞)

1 -0.0019 0.0018 0.0008

0.9 0.00311 -0.0249 -0.2742

0.8 0.0162 -0.0738 -0.5484

0.7 0.0450 -0.1613 -0.7114

0.6 0.1014 -0.3159 -0.5552


7 (0) andH7 (η∞)for n values.

0 2 4 6 8 10 12 14 16 18 20-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01n=1n=0.9n=0.8n=0.7n=0.6

0 2 4 6 8 10 12 14 16 18 20-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

n=1n=0.9n=0.8n=0.7n=0.6

0 2 4 6 8 10 12 14 16 18 20-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

n=1n=0.9n=0.8n=0.7n=0.6

0 2 4 6 8 10 12 14 16 18 20-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

n=1n=0.9n=0.8n=0.7n=0.6

Figure 3.4: Plots of F7; G7; H7 and µ7 versus η . The η-axis has been truncated at η = 20.


The boundary conditions are

H7 (0) = 0, G7 (0) =− 15040

, F7 (0) = 0,

F7 (∞) = G7 (∞) = 0. (3.24)

The velocity functions profiles of Seventh order are shown in Figure 3.4. The initial

values of mean velocity flow parameters F ′7 (0), G′

7 (0) and H ′7 (η∞) are stated in Table

3.4.

The MATLAB function solver and the numerical method applied to solve the system

of Equations (3.17), (3.19), (3.21) and (3.23) are explained later in §3.2. When n = 1,

Equations (3.17), (3.19), (3.21) and (3.23) are reduced to the results given by Banks

(1965). The flow index n is the only parameter of the velocity functions Fi, Gi and Hi..

The first-order velocity functions F1, G1 and H1 which are corresponding to the case of

rotating disk Griffiths et al. (2014a) are the limiting solutions at the poles of the rotating

sphere. The four sets of ordinary differential equations given by (3.17), (3.19), (3.21)

and (3.23) are two-point boundary value problems, which are solved numerically in the

next section. Except for the radial velocity in the vicinity of the equator, it was found that

the series expansion is everywhere convergent so the four sets of Fi, Gi and Hi, where

i = 1,3,5,7, and for all the power-law index n values are sufficient to discuss the flow

over the whole surface of the sphere from the pole to the equator, excluding the eruption

region near the equator where the fluid erupts to form a swirling radial jet, Banks (1965).

Figures 3.5, 3.6, 3.7, 3.8 and 3.9 show the three velocity components U , V and W

as well as the viscosity function µ for n = 0.9− 0.6 in increments of 0.1 at latitudes

of θ = 10− 80 in ten degree increments. The latitudinal velocity U is inflectional at

all latitudes and the fluid is entrained radially into the boundary layer at all the latitudes

shown. However, the non-convergence of the radial velocity is not so drastic for large η ,

where it indicates, as found by Howarth (1951), that inflow occurs right up to the equator.


0 2 4 6 8 10 12 14 16 18 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18=10=20=30=40=50=60=70=80

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1=10=20=30=40=50=60=70=80

0 2 4 6 8 10 12 14 16 18 20-1

-0.8

-0.6

-0.4

-0.2

0

0.2=10=20=30=40=50=60=70=80

Figure 3.5: Plots of U ; V and W versus η at latitudes of θ = 10− 80 in ten degreeincrements for n = 1. The η-axis has been truncated at η = 20.

0 2 4 6 8 10 12 14 16 18 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18=10=20=30=40=50=60=70=80

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1=10=20=30=40=50=60=70=80

0 2 4 6 8 10 12 14 16 18 20-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5=10=20=30=40=50=60=70=80

0 2 4 6 8 10 12 14 16 18 20

-14

-12

-10

-8

-6

-4

-2

0

2

4

=10=20=30=40=50=60=70=80

Figure 3.6: Plots of U ; V ; W and µ versus η at latitudes of θ = 10−80 in ten degreeincrements for n = 0.9. The η-axis has been truncated at η = 20.


0 2 4 6 8 10 12 14 16 18 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18=10=20=30=40=50=60=70=80

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1=10=20=30=40=50=60=70=80

0 2 4 6 8 10 12 14 16 18 20-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

=10=20=30=40=50=60=70=80

0 2 4 6 8 10 12 14 16 18 20-12

-10

-8

-6

-4

-2

0

2

4

6

8

=10=20=30=40=50=60=70=80


0 2 4 6 8 10 12 14 16 18 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18=10=20=30=40=50=60=70

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1=10=20=30=40=50=60=70

0 2 4 6 8 10 12 14 16 18 20-2.5

-2

-1.5

-1

-0.5

0

0.5=10=20=30=40=50=60=70

0 2 4 6 8 10 12 14 16 18 201

2

3

4

5

6

7

8

9

10

11=10=20=30=40=50=60=70



0 2 4 6 8 10 12 14 16 18 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18=10=20=30=40=50=60

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1=10=20=30=40=50=60

0 2 4 6 8 10 12 14 16 18 20-2.5

-2

-1.5

-1

-0.5

0

0.5=10=20=30=40=50=60

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20=10=20=30=40=50=60


Figure 3.10: Mean velocity profiles U ; V and W at latitudes of θ = 10− 80 in tendegree increments when n = 1. (Each figure is normalised independently by its maximumvalue.)

Garrett (2002) used the NAG routine D03PEF to find the mean flow at each latitude.

This NAG routine is a general PDE solver that reduces the system of PDEs to a system of

ODEs in η . Figure 3.10 shows the three velocity components at latitudes of θ = 10−80

in ten degree increments when the power-law index n = 1. The resulting profiles have

3.2 Matlab solver for the steady mean flow 33

compared to the NAG routine results of Garrett (2002) and complete agreement is found

up to the equator.

3.2 Matlab solver for the steady mean flow

The first, third, fifth and seventh order Equations (3.17), (3.19), (3.21) and (3.23) solved

using the MATLAB function bvp4c. The numerical method of this function is based

on a finite difference code implementing the three stage Lobatto IIIa formula, that can

be viewed as an implicit Runge-Kutta formula with a continuous interpolant, Kierzenka

and Shampine (2001). The Lobatto IIIa method is a collocation method that provides a

C1-continuous solution that is fourth-order accurate uniformly in a finite interval [a,b].

The error estimation and mesh selection of the method are based on the residual of the

continuous solution.

In order to obtain the numerical solutions of the mean flow profiles of Fi, Gi and Hi

and their derivatives F′i , G

′i and H

′i where (i = 1, · · · ,20), the governing non-dimensional

mean flow Equations (3.17), (3.19), (3.21) and (3.23) have to be expressed as a system

of first order ordinary differential equations. These systems of equations is written as four

sets as following.

The system (3.17) is written as a five coupled first order equations in terms of the new

five dependent transformation variables φn (n = 1,2, · · · ,5) where

φ1 = F1, φ2 = F ′1, φ3 = G1, φ4 = G′

1 and φ5 = H1. (3.25)

The transformed first order ODE system is stated in (3.26) with no-slip boundary condi-

tions for power-law fluids given by (3.26c)


φ′1 = φ2, (3.26a)

φ′2 =

(φ 2

2 +nφ 24)

K +(1−n)φ2φ4Lnµ1

(φ 2

2 +φ 24) ,

φ′3 = φ4, (3.26b)

φ′4 =

(nφ 2

2 +φ 24)

L+(1−n)φ2φ4Knµ1

(φ 2

2 +φ 24) ,

φ′5 =−2φ1 −

(1−n1+n

)ηφ2,

φ1 (0) = 0,φ3 (0) = 1 and φ5 (0) = 0, (3.26c)

φ1 (η → ∞) = 0,φ3 (η → ∞) = 0, (3.26d)

where

K =

[φ5 +

(1−n1+n

)ηφ1

]φ2 +φ

21 −φ

23 ,

L =

[φ5 +

(1−n1+n

)ηφ1

]φ4 +2φ1φ3,

µ1 =[φ

22 +φ

24](n−1)/2

.

Using the solutions of φ ′2 and φ ′

4 we could introduce the third order Equations (3.19)

as other new five coupled first order equations system in terms of the new five dependent

transformation variables φn (n = 6,7, · · · ,10), where

φ6 = F3, φ7 = F ′3, φ8 = G3, φ9 = G′

3 and φ10 = H3, (3.27)

The transformed first order ODE system is stated in (3.28a) with no-slip boundary con-

ditions for power-law fluids given by (3.28c)


φ′6 = φ7, (3.28a)

φ′7 =

(φ 2

2 +nφ 24)

M+(1−n)φ2φ4Nnµ1

(φ 2

2 +φ 24) ,

φ′8 = φ9, (3.28b)

φ′9 =

(nφ 2

2 +φ 24)

N +(1−n)φ2φ4Mnµ1

(φ 2

2 +φ 24) ,

φ′10 =

φ1

3−4φ6 −

(1−n1+n

)ηφ7.

φ6 (0) = 0,φ8 (0) =−16,φ10 (0) = 0, (3.28c)

φ6 (η → ∞) = 0,φ8 (η → ∞) = 0, (3.28d)

where

M =

[φ10 +

(1−n1+n

)ηφ6

]φ2 +

[φ5 +

(1−n1+n

)ηφ1

]φ7 +4φ1φ6 −2φ3φ8 +

φ 233

−(n−1)[φ

22 +φ

24]n−3/2(

φ2φ′2 +φ4φ

′4)

φ7 +(φ2φ7 +φ4φ9)φ′2 +(φ7φ

′2 +φ9φ

′4)

φ2

−(n−1)(n−3)[φ

22 +φ

24]n−5/2

(φ2φ7 +φ4φ9)(φ2φ

′2 +φ4φ

′4)

φ2,

N =

[φ10 +

(1−n1+n

)ηφ6

]φ4 +

[φ5 +

(1−n1+n

)ηφ1

]φ9 +2(φ1φ8 +φ3φ6)−

φ1φ3

2

−(n−1)[φ

22 +φ

24]n−3/2(

φ2φ′2 +φ4φ

′4)

φ9 +(φ2φ7 +φ4φ9)φ′4 +(φ7φ

′2 +φ9φ

′4)

φ4

−(n−1)(n−3)[φ

22 +φ

24]n−5/2

(φ2φ7 +φ4φ9)(φ2φ

′2 +φ4φ

′4)

φ4.

Proceeding in similar way, the new five dependent variables φn (n = 11,12, · · · ,15) are

φ11 = F5, φ12 = F ′5, φ13 = G5, φ14 = G′

5 and φ15 = H5, (3.29)

regarding to the system (3.2), the transformed first order ODE system with no-slip bound-


ary conditions are given by

φ′11 = φ12,

φ′12 =

(φ 2

2 +nφ 24)

P+(1−n)φ2φ4Qnµ1

(φ 2

2 +φ 24) ,

φ′13 = φ14, (3.30a)

φ′14 =

(nφ 2

2 +φ 24)

Q+(1−n)φ2φ4Pnµ1

(φ 2

2 +φ 24) ,

φ′15 =

φ6

3+

φ1

45−6φ11 −

(1−n1+n

)ηφ12.

φ11 (0) = 0,φ13 (0) =1

120,φ15 (0) = 0,

φ11 (η → ∞) = 0,φ13 (η → ∞) = 0, (3.30b)

where

P =

[φ15 +

(1−n1+n

)ηφ11

]φ2 +

[φ10 +

(1−n1+n

)ηφ6

]φ7 +

[φ5 +

(1−n1+n

)ηφ1

]φ12 +3φ

26

+6φ1φ11 −2φ3φ13 −φ28 +

2φ3φ8

3+

φ 23

45

−(n−1)(n−3)(n−5)[φ

22 +φ

24]n−7/2 (

φ2φ′2 +φ4φ

′4)(φ2φ7 +φ4φ9)

2φ2

−(n−1)(n−3)[φ

22 +φ

24]n−5/2

(φ2φ7 +φ4φ9)(φ2φ ′

2 +φ4φ ′4)φ7 +(φ2φ7 +φ4φ9)

2φ ′

2

+12

[2(φ2φ12 +φ4φ14)+φ 2

7 +φ 29](φ2φ ′

2 +φ4φ ′4)φ2

+(φ2φ7 +φ4φ9)(φ2φ ′

7 +φ7φ ′2 +φ4φ ′

9 +φ9φ ′4)

φ2

−(n−1)[φ

22 +φ

24]n−3/2

(φ2φ ′2 +φ4φ ′

4)φ12 +(φ2φ7 +φ4φ9)φ ′7

+12

[2(φ2φ12 +φ4φ14)φ 2

7 +φ 29]

φ ′2

+(φ2φ ′

7 +φ7φ ′2 +φ4φ ′

9 +φ9φ ′4)

φ7

+(φ12φ ′

2 +φ7φ ′7φ14φ ′

4 +φ9φ ′9)

φ2

,


Q =

[φ15 +

(1−n1+n

)ηφ11

]φ4 +

[φ10 +

(1−n1+n

)ηφ6

]φ9 +

[φ5 +

(1−n1+n

)ηφ1

]φ14 +4φ6φ8

+6φ1φ13 +2φ3φ11 −φ28 −

(φ1φ8 +φ6φ3)

3− φ1φ3

45

−(n−1)(n−3)(n−5)[φ

22 +φ

24]n−7/2 (

φ2φ′2 +φ4φ

′4)(φ2φ7 +φ4φ9)

2φ4

−(n−1)(n−3)[φ

22 +φ

24]n−5/2

(φ2φ7 +φ4φ9)(φ2φ ′

2 +φ4φ ′4)φ9 +(φ2φ7 +φ4φ9)

2φ ′

4

+12

[2(φ2φ12 +φ4φ14)+φ 2

7 +φ 29](φ2φ ′

2 +φ4φ ′4)φ4

+(φ2φ7 +φ4φ9)(φ2φ ′

7 +φ7φ ′2 +φ4φ ′

9 +φ9φ ′4)

φ4

−(n−1)[φ

22 +φ

24]n−3/2

(φ2φ ′2 +φ4φ ′

4)φ14 +(φ2φ7 +φ4φ9)φ ′9

+12

[2(φ2φ12 +φ4φ14)φ 2

7 +φ 29]

φ ′4

+(φ2φ ′

7 +φ7φ ′2 +φ4φ ′

9 +φ9φ ′4)

φ9

+(φ12φ ′

2 +φ7φ ′7φ14φ ′

4 +φ9φ ′9)

φ4

.

The system (3.23) is written as a five coupled first order equations in terms of the new

five dependent transformation variables φn (n = 16,17, · · · ,20) where

φ16 = F7, φ17 = F ′7, φ18 = G7, φ19 = G′

7 and φ20 = H20. (3.31)

The transformed first order ODE system with no-slip boundary conditions are given by

φ′16 = φ17,

φ′17 =

(φ 2

2 +nφ 24)

R+(1−n)φ2φ4Snµ1

(φ 2

2 +φ 24) ,

φ′18 = φ19, (3.32a)

φ′19 =

(nφ 2

2 +φ 24)

S+(1−n)φ2φ4Rnµ1

(φ 2

2 +φ 24) ,

φ′20 =

φ11

3+

φ6

45+

2φ1

945−8φ16 −

(1−n1+n

)ηφ17,


where

R =

[φ20 +

(1−n1+n

)ηφ16

]φ2 +

[φ15 +

(1−n1+n

)ηφ11

]φ7 +

[φ10 +

(1−n1+n

)φ6

]φ12

+

[φ5 +

(1−n1+n

)ηφ1

]φ17 +

(φ 2

8 +2φ3φ13)

3−2(φ3φ18 +φ8φ13)+8(φ1φ16 +φ6φ11)

+2φ3φ8

45+

2φ 23

945

−(n−1)(n−3)(n−5)(n−7)[φ

22 +φ

24]n−9/2 (

φ2φ′2 +φ4φ

′4)(φ2φ7 +φ4φ9)

3φ2

−(n−1)(n−3)(n−5)[φ

22 +φ

24]n−7/2

(φ2φ7 +φ4φ9)2 (φ2φ ′

2 +φ4φ ′4)φ7 +(φ2φ7 +φ4φ9)

3φ ′

2

+[2(φ2φ12 +φ4φ14)+φ 2

7 +φ 29](φ2φ7 +φ4φ9)(φ2φ ′

2 +φ4φ ′4)φ2

+(φ2φ ′

7 +φ7φ ′2 +φ4φ ′

9 +φ9φ ′4)(φ2φ7 +φ4φ9)

2φ2

−(n−1)(n−3)

[φ

22 +φ

24]n−5/2

(φ2φ7 +φ4φ9)(φ2φ ′2 +φ4φ ′

4)φ12 +(φ2φ7 +φ4φ9)2

φ ′7

+12

[2(φ2φ12+φ4φ14)+φ 2

7 +φ 29](φ2φ ′

2 +φ4φ ′4)φ7

+[2(φ2φ12 +φ4φ14)+φ 2

7 +φ 29](φ2φ7 +φ4φ9)φ ′

2

+(φ2φ17 +φ7φ12 +φ4φ19 +φ9φ14)(φ2φ ′2 +φ4φ ′

4)φ2

+(φ2φ ′

7 +φ7φ ′2 +φ4φ ′

9 +φ9φ ′4)(φ2φ7 +φ4φ9)φ7

+12

[2(φ2φ12 +φ4φ14)+φ 2

7 +φ 29](

φ2φ ′7 +φ7φ ′

2 +φ4φ ′9 +φ9φ ′

4)

φ2

+(φ2φ ′

12 +φ12φ ′2 +φ7φ ′

7 +φ4φ ′14 +φ14φ ′

4 +φ9φ ′9)(φ2φ7 +φ4φ9)φ2

−(n−1)

[φ

22 +φ

24]n−3/2

(φ2φ ′2 +φ4φ ′

4)φ17 +(φ2φ7 +φ4φ9)φ ′12

+12

[2(φ2φ12 +φ4φ14)+φ 2

7 +φ 29]

φ ′7

+(φ2φ ′

17 +φ7φ ′12 +φ4φ ′

19 +φ9φ ′14)

φ ′2

+(φ2φ ′

7 +φ7φ ′2 +φ4φ ′

9 +φ9φ ′4)

φ12

+(φ2φ ′

12 +φ12φ ′2 +φ7φ ′

7 +φ4φ ′14 +φ14φ ′

4 +φ9φ ′9)

φ7(φ17φ ′

2 +φ7φ ′12 +φ12φ ′

7 +φ19φ ′4 +φ9φ ′

14 +φ14φ ′9)

φ2

,


S =

[φ20 +

(1−n1+n

)ηφ16

]φ4 +

[φ15 +

(1−n1+n

)ηφ11

]φ9 +

[φ10 +

(1−n1+n

)φ6

]φ14

+

[φ5 +

(1−n1+n

)ηφ1

]φ19 −

(φ1φ8 +φ3φ6)

45− (φ1φ13 +φ6φ8 +φ3φ11)

3

+6φ6φ13 +4φ8φ11 +8φ1φ18 +2φ3φ16 −2φ1φ3

945

−(n−1)(n−3)(n−5)(n−7)[φ

22 +φ

24]n−9/2 (

φ2φ′2 +φ4φ

′4)(φ2φ7 +φ4φ9)

3φ4

−(n−1)(n−3)(n−5)[φ

22 +φ

24]n−7/2

(φ2φ7 +φ4φ9)2 (φ2φ ′

2 +φ4φ ′4)φ9 +(φ2φ7 +φ4φ9)

3φ ′

4

+[2(φ2φ12 +φ4φ14)+φ 2

7 +φ 29](φ2φ7 +φ4φ9)(φ2φ ′

2 +φ4φ ′4)φ4

+(φ2φ ′

7 +φ7φ ′2 +φ4φ ′

9 +φ9φ ′4)(φ2φ7 +φ4φ9)

2φ4

−(n−1)(n−3)

[φ

22 +φ

24]n−5/2

(φ2φ7 +φ4φ9)(φ2φ ′2 +φ4φ ′

4)φ14 +(φ2φ7 +φ4φ9)2

φ ′9

+12

[2(φ2φ12+φ4φ14)+φ 2

7 +φ 29](φ2φ ′

2 +φ4φ ′4)φ9

+[2(φ2φ12 +φ4φ14)+φ 2

7 +φ 29](φ2φ7 +φ4φ9)φ ′

4

+(φ2φ17 +φ7φ12 +φ4φ19 +φ9φ14)(φ2φ ′2 +φ4φ ′

4)φ4

+(φ2φ ′

7 +φ7φ ′2 +φ4φ ′

9 +φ9φ ′4)(φ2φ7 +φ4φ9)φ9

+12

[2(φ2φ12 +φ4φ14)+φ 2

7 +φ 29](

φ2φ ′7 +φ7φ ′

2 +φ4φ ′9 +φ9φ ′

4)

φ4

+(φ2φ ′

12 +φ12φ ′2 +φ7φ ′

7 +φ4φ ′14 +φ14φ ′

4 +φ9φ ′9)(φ2φ7 +φ4φ9)φ4

−(n−1)

[φ

22 +φ

24]n−3/2

(φ2φ ′2 +φ4φ ′

4)φ19 +(φ2φ7 +φ4φ9)φ ′14

+12

[2(φ2φ12 +φ4φ14)+φ 2

7 +φ 29]

φ ′9

+(φ2φ ′

17 +φ7φ ′12 +φ4φ ′

19 +φ9φ ′14)

φ ′4

+(φ2φ ′

7 +φ7φ ′2 +φ4φ ′

9 +φ9φ ′4)

φ14

+(φ2φ ′

12 +φ12φ ′2 +φ7φ ′

7 +φ4φ ′14 +φ14φ ′

4 +φ9φ ′9)

φ9(φ17φ ′

2 +φ7φ ′12 +φ12φ ′

7 +φ19φ ′4 +φ9φ ′

14 +φ14φ ′9)

φ4

,

3.3 Summary 40

with the boundary conditions

φ16 (0) = 0,φ18 (0) =− 15040

,φ20 (0) = 0,

φ16 (η → ∞) = 0,φ18 (η → ∞) = 0. (3.33a)

The strategy of using the bvp4c function to evaluate the series solutions can be sum-

marised as follows

1. The MATLAB function bvpinit is used over an initial finite interval to obtain a

solution guess for the first boundary value problem solver bvp4c.

2. The MATLAB bvp4c solver is used to evaluate the solution of first order boundary

value problem in this interval.

3. By continuation over larger intervals (domain), the solution is then extended so the

solution of former interval is used as a solution guess for the next.

4. Repeating the above process until the desired domain size is achieved.

5. The first order solutions are then used to obtain the third order solutions and so on

until the seventh order solutions are evaluated.

3.3 Summary

An appropriate coordinate transformation and a series expansion in terms of the angle

measured from the axis of rotation are used to describe the velocity functions of the coef-

ficients of the velocity components by ordinary differential equations of first, third, fifth

and seventh order, the coefficients of the asymptotic series are described by second-order

ordinary differential equations with the flow index n as the only parameter. Four terms are

found to quite well describe the velocity distributions. The first-order velocity functions

which are the limiting solutions at the poles of the rotating sphere are corresponding to

the case of the rotating disk presented by Griffiths et al. (2014a). When the power-law

3.3 Summary 41

index n = 1, our series solutions are reduced to the series solution of the boundary-layer

equations for the rotating sphere proposed by Howarth (1951) and developed by Banks

(1965).

Chapter 4

Derivation of the perturbation

equations

The convective and absolute instabilities in the boundary layer flow over the outer surface

of a sphere rotating in an otherwise still fluid have been conducted by Taniguchi et al.

(1998) and Garrett and Peake (2002). This chapter presents the linear perturbation equa-

tions applicable to power-law fluids over the rotating sphere. These perturbation equations

are formulated with a view to studying the occurrence of linear convective instabilities.

The stability analysis conducted at a particular latitude involves imposing infinitesimal

small perturbations on the steady mean flow at that latitude, for the still fluid. The lin-

ear perturbation equations are solved by applying the Chebyshev collocation method de-

scribed in §4.2. Appelquist (2014) used this numerical method to compute the neutral

curves for the convective instability of Newtonian fluids on a rotating disk.

4.1 The perturbation equations

In this section we formulate the stability problem. The perturbation equations are de-

rived using the dimensional continuity and motion Equations (2.3−2.4) and the Reynolds

numbers that will be used in this investigation are discussed. We impose infinitesimally

small perturbations on the steady mean flow in the boundary layer at a particular latitude

4.1 The perturbation equations 43

on the rotating-sphere. The dimensional velocity and pressure of the perturbed flow are

formed from a dimensional basic flow component (denoted by an upper-case quantity)

and a perturbing quantity (denoted by a lower-case hatted quantity). The expansions are

as follows

(U∗

,V ∗,W ∗

,P∗)=(U∗+ u∗,V ∗+ v∗,W ∗+ w∗,P∗+ p∗

). (4.1)

Throughout this thesis, we assume that the imposed disturbances are sufficiently small so

that the transition process is controlled by the primary linear stability of the mean flow

instead of any secondary instability that can occur if the perturbations are large enough

to deform the mean flow profiles. Physically, secondary instabilities can occur if the

perturbations are large enough to significantly distort the mean velocity profiles, Garrett

(2002). Furthermore, a linear analysis is conducted so the non-linear terms arising from

products of the small perturbation quantities are also sufficiently small to be ignored in

the equations. Note that bypass transition, where the non-linear effects dominate from

the start due of being the initial perturbations sufficiently large, are not considered. The

dimensional perturbation variables (denoted by lower case hatted quantities) are assumed

to have the normal-mode form

(u∗, v∗, w∗, p∗

)= (u∗ (r∗) ,v∗ (r∗) ,w∗ (r∗) , p∗ (r∗))ei(α∗a∗θ+β ∗a∗φ sinθ−ω∗t∗). (4.2)

The distance measured over the surface of the sphere from the pole to the latitude un-

der consideration is a∗θ , and α∗ is the dimensional wavenumber of disturbance in this

direction. The distance measured along a circular section of the sphere by a plane perpen-

dicular to the axis of rotation is a∗φ sinθ , and β ∗ is the dimensional wavenumber in this

direction.

In order to obtain the dimensional perturbation equations, the perturbed flow compon-

ents (4.1) are firstly substituted in Equations (2.3−2.4) then by applying the differenti-

ations and linearisation with respect to perturbation quantities and subtracting the mean


flow components, we find the first-order perturbation equations to be

∂ w∗

∂ r∗+

2r∗

w∗+1r∗

cotθ u∗+∂ u∗

∂ r∗∂ r∗

∂θ+

∂ u∗

∂θ+

1r∗ sinθ

∂ v∗

∂φ= 0, (4.3)

∂ w∗

∂ t∗+W ∗∂ w∗

∂ r∗+ w∗∂W ∗

∂ r∗+

U∗

r∗∂ w∗

∂θ+

u∗

r∗∂W ∗

∂θ+

V ∗

r∗ sinθ

∂ w∗

∂φ

2(U∗u∗+V ∗v

)r∗

=1

ρ∗∂ p∗

∂ r∗+

1ρ∗

[1

r∗2∂

∂ r∗

(2r∗

2µ∗

∂W ∗

∂ r∗ + ∂ w∗∂ r∗

)]+[

1r∗ sinθ

∂

∂θ

(sinθ µ∗r∗ ∂

∂ r∗

(U∗+u∗

r∗

))]+[

1r∗ sinθ

∂

∂θ

(sinθ µ∗ 1

r∗

∂W ∗

∂θ+ ∂ w∗

∂θ

)]+ 1

r∗ sinθ

∂

∂φ

[µ∗

1r∗ sinθ

∂ w∗∂φ

+ r∗ ∂

∂ r∗

(v∗r∗

)]

,

(4.4a)

∂ u∗

∂ t∗+W ∗∂ u∗

∂ r∗+ w∗∂U∗

∂ r∗+

U∗

r∗∂ u∗

∂θ+

u∗

r∗∂U∗

∂θ+

V ∗

r∗ sinθ

∂ u∗

∂φ− 2V ∗v∗ cotθ

r∗

+

(W ∗u∗+U∗w∗

)r∗

=− 1ρ∗r∗

∂ p∗

∂θ+

1ρ∗

1r∗2

∂

∂ r∗

[r∗

2µ∗(

r∗ ∂

∂ r∗

(U∗

r∗

)+ 1

r∗∂W ∗

∂θ

)]+ 1

r∗2∂

∂ r∗

[r∗

2µ∗(

r∗ ∂

∂ r∗

(u∗r∗

)+ 1

r∗∂ w∗∂θ

)]+ 1

r∗ sinθ

∂

∂θ

[2sinθ µ∗

(1r∗

∂U∗

∂θ+ W ∗

r∗

)]+ 1

r∗ sinθ

∂

∂θ

[2sinθ µ∗

(1r∗

∂ u∗∂θ

+ w∗r∗

)]+ 1

r∗ sinθ

[∂ µ∗

∂φ

(sinθ

r∗∂

∂θ

(V ∗

sinθ

))]+ ∂

r∗ sinθ∂φ

[µ∗(

sinθ∂

r∗∂θ

(v∗

sinθ

)+ 1

r∗ sinθ

∂ u∗∂φ

)]−cotθ

r∗[2µ∗ ( 1

r∗ (W∗+U∗ cotθ)

)]−cotθ

r∗

[2µ∗

(1

r∗ sinθ

∂ v∗∂φ

+ w∗+u∗ cotθ

r∗

)]

,

(4.4b)


∂ v∗

∂ t∗+W ∗∂ v∗

∂ r∗+ w∗∂V ∗

∂ r∗+

U∗

r∗∂ v∗

∂θ+

u∗

r∗∂V ∗

∂θ+

V ∗

r∗ sinθ

∂ v∗

∂φ− V ∗u∗ cotθ

r∗

+U∗v∗ cotθ

r∗+

(W ∗v∗+V ∗w∗

)r∗

=− 1ρ∗r∗ sinθ

∂ p∗

∂φ

+1

ρ∗

1r∗2

∂

∂ r∗

[r∗

2µ∗(

r∗ ∂

∂ r∗

(V ∗

r∗

))]+ 1

r∗2∂

∂ r∗

[r∗

2µ∗(

r∗ ∂

∂ r∗

(v∗r∗

)+ 1

r∗ sinθ

∂ w∗∂φ

)]+ 1

r∗ sinθ

∂

∂θ

[sinθ µ∗

(sinθ

r∗∂

∂θ

(V ∗

sinθ

))]+ 1

r∗ sinθ

∂

∂θ

[sinθ µ∗

(sinθ

r∗∂

∂θ

(v∗

sinθ

)+ 1

r∗ sinθ

∂ u∗∂φ

)]+ 1

r∗ sinθ

∂

∂φ

[2µ∗

(W ∗+U∗ cotθ

r∗

)]+ 1

r∗ sinθ

∂

∂φ

[2µ∗

(1

r∗ sinθ

∂ v∗∂φ

+(w∗+u∗ cotθ)

r∗

)]+cotθ

r∗

[µ∗(

sinθ

r∗∂

∂θ

(V ∗

sinθ

))]+cotθ

r∗

[µ∗(

sinθ

r∗∂

∂θ

(v∗

sinθ

)+ 1

r∗ sinθ

∂ u∗∂φ

)]

.

(4.4c)

We non-dimensionalize (4.3) and (4.4) using the local similarity variables (2.13) for the

mean flow variables where the non-dimensional perturbing quantities are written as

u = u∗/U∗m, v = v∗/U∗

m, w = w∗/U∗m, t = A∗U∗

mt∗,

α = α∗/A∗, β = β

∗/A∗, ω = ω∗/(A∗U∗

m) , p = p∗/ρ∗ (U∗

m)2 ,

δ1 = 1/(a∗A∗) = 1/R, ℓ= 1/(

1+δ1ηθ(n−1)/(n+1)

). (4.5)

When n = 1, Equation (4.5) reduced to the dimensionless perturbation quantities ap-

pearing in Garrett (2002), particularly 1/A∗ is reduced to the boundary-layer thickness

δ ∗ =√

ν∗/Ω∗, which is used as the length scale . The quantity U∗m = a∗Ω∗ is the max-

imum rotation speed and it used as the velocity scale. This is consistent with the non-

dimensionlization of the mean flow variables in §??.

The non-dimensional perturbation equations are given by

∂w∂η

+2ℓw+ ℓcotθu+ ℓ∂u∂η

∂η

∂θ+ ℓ

∂u∂θ

+ℓ

sinθ

∂v∂φ

= 0, (4.6)


∂w∂ t

+δ1W∂w∂η

+δ1w∂W∂η

+δ1ℓU[(

1−n1+n

)η

θ

∂w∂η

+∂w∂θ

]+δ1ℓ

Vsinθ

∂w∂φ

−2δ1ℓ(Uu+V v) =∂ p∂η

+1R

[∂ µmean

∂η

∂w∂η

+µmean∂ 2w∂η2

]

+1

R2

ℓµ

(∂U∂η

∂ 2u∂η∂θ

+ ∂V∂η

∂ 2v∂η∂θ

)∂U∂η

+ ℓµ

sinθ

(∂U∂η

∂ 2u∂η∂θ

+ ∂V∂η

∂ 2v∂η∂θ

)∂V∂η

1

R3

[ℓ2

µmean∂ 2w∂θ 2 +

ℓ2

sin2θ

∂ 2w∂φ 2

],

(4.7a)

∂u∂ t

+δ1W∂u∂η

+w∂U∂η

+δ1ℓU[(

1−n1+n

)η

θ

∂u∂η

+∂u∂θ

]+δ1ℓu

∂U∂θ

+δ1ℓUw

+δ1ℓV

sinθ

∂u∂φ

−2δ1ℓcotθV v =− 1R

[(1−n1+n

)η

θ

∂ p∂η

+∂ p∂θ

]

+1R

µmean∂ 2u∂η2 +

∂ µmean∂η

∂u∂η

+

µ

(∂U∂η

∂u∂η

+ ∂V∂η

∂v∂η

)∂ 2U∂η2

+

µ ′(

∂U∂η

∂u∂η

+ ∂V∂η

∂v∂η

)∂U∂η

+

µ

(∂ 2U∂η2

∂u∂η

+ ∂U∂η

∂ 2u∂η2 +

∂ 2V∂η2

∂v∂η

+ ∂V∂η

∂ 2v∂η2

)∂U∂η

+

1R2

∂ µmean

∂η

∂w∂θ

+1

R3ℓ2

sin2θ

∂ 2u∂φ 2 ,

(4.7b)

∂v∂ t

+δ1W∂v∂η

+w∂V∂η

+δ1ℓU[(

1−n1+n

)η

θ

∂v∂η

+∂v∂θ

]+

δ1ℓVsinθ

∂v∂φ

+δ1ℓwV

+δ1ℓu[(

1−n1+n

)η

θ

∂V∂η

+∂V∂θ

]+δ1ℓcotθ (Uv+Vu) =

1R

ℓ

sinθ

∂ p∂φ

+1R

µmean∂ 2v∂η2 +

∂ µmean∂η

∂v∂η

+

µ

(∂U∂η

∂u∂η

+ ∂V∂η

∂v∂η

)∂ 2V∂η2

+

µ ′(

∂U∂η

∂u∂η

+ ∂V∂η

∂v∂η

)∂V∂η

+

µ

(∂ 2U∂η2

∂u∂η

+ ∂U∂η

∂ 2u∂η2 +

∂ 2V∂η2

∂v∂η

+ ∂V∂η

∂ 2v∂η

)∂V∂η

+

1R2

ℓ

sinθ

∂ µmean

∂η

∂w∂φ

+1

R3 ℓ2µmean

∂ 2v∂θ 2 +

1R3

ℓ2

sin2θ

µmean∂ 2v∂φ 2 .

(4.7c)


The additional viscous terms µ

(U ′ ∂ 2u

∂η∂θ+V ′ ∂ 2v

∂η∂θ

)U ′, µ

(U ′ ∂ 2u

∂η∂θ+V ′ ∂ 2v

∂η∂θ

)V ′,

µ

(U ′ ∂u

∂η+V ′ ∂v

∂η

)U ′′, µ

(U ′ ∂u

∂η+V ′ ∂v

∂η

)V ′′, µ ′

(U ′ ∂u

∂η+V ′ ∂v

∂η

)U ′, µ ′

(U ′ ∂u

∂η+V ′ ∂v

∂η

)V ′,

µ

(U ′′ ∂u

∂η+U ′ ∂ 2u

∂η2 +V ′ ∂v∂η

+V ′ ∂ 2v∂η2

)U ′ and µ

(U ′′ ∂u

∂η+U ′ ∂ 2u

∂η2 +V ′′ ∂v∂η

+V ′ ∂ 2v∂η

)V ′ appear-

ing in the perturbation Equations (4.7) are due to the first-order terms of the crossproduct

associated with the generalised binomial expansion of the perturbed viscosity function.

These terms do not appear in Garrett (2002) as they are reduced to unity in the case of

Newtonian fluids. the derivation of the perturbed viscosity function is given in Appendix

A.

To conduct a normal-mode analysis, the disturbances have the dimensionless normal

mode form

u = u(η ;α,β ,γ;R,n)ei(αRθ+βRφ sinθ−ωt),

v = v(η ;α,β ,γ;R,n)ei(αRθ+βRφ sinθ−ωt),

w = w(η ;α,β ,γ;R,n)ei(αRθ+βRφ sinθ−ωt),

p = p(η ;α,β ,γ;R,n)ei(αRθ+βRφ sinθ−ωt),

where the eigenvalue problem is solved for either α or ω . Here u, v, w and p are the

spectral representations of the perturbation velocities and pressure, respectively. The

wavenumber in the θ−direction, α , and the frequency, ω , are in general complex and

we write these quantities as α = αr + iαi and ω = ωr + iωi. In contrast, the azimuthal

wave number, β , is real. The integer number of complete cycles of the disturbance round

the azimuth is

β = βRsinθ . (4.8)

The angle that the phase fronts make with a circle parallel to the equator is denoted φ , and

is found from

φ = tan−1 (β/αr) . (4.9)

As presented by Garrett (2002), β and φ are identified as the angle and number of spiral

vortices on the sphere surface respectively. After differentiations and neglecting O(R−2)


terms, the non-dimensional perturbation equations are found to be

(δ1W +δ1ℓηU −δ1µ

′−δ1F ′UU)

u′+[

i(αU +βV )ℓ−ω+δ1ℓ∂U∂θ

+δ1ℓηU ′]

u

−2V δ1ℓcotθ v+(U ′+δ1ℓU

)w =−iαℓp+

1Rℓη p′+

1R

(µ +FUU) u′′

−ℓ2µ(α2 +β 2) u

−ℓµ ′w+(

v′FUV

)′ ,(4.10a)(

δ1W +δ1ℓηU −δ1µ′−δ1F ′

VV)

v′+[i(αU +βV )ℓ−ω+δ1ℓU cotθ ] v

+

(∂V∂θ

+V cotθ + ηV ′)

δ1ℓu+(V ′+δ1ℓV − iδ1ℓβ µ

′) w = iβℓp

+1R

[(µ +FVV ) v′′− ℓ2

µ(α

2 +β2) v+

(u′FUV

)′],

(4.10b)(δ1W +δ1ℓηU −2δ1µ

′) w′+[i(αU +βV )ℓ−ω+δ1W ′] w−2Uδ1ℓu−2V δ1ℓv

=−p′+1R

µw′′− ℓ2µ(α2 +β 2) w

+iβ(

u′FUV + v′FVV

)+ iα

(u′FUU + v′FUV

) ,(4.10c)

w′+2δ1ℓw =−ℓ(iα +δ1 cotθ) u+δ1η u′+ iβ v

, (4.11)

where

η =

(1−n1+n

)η

θ,

FUU = µ(U ′)2

,

FUV = µU ′V ′,

FVV = µ(V ′)2

,

F ′UU = µ ′

(U ′)2

+2µU ′U ′′,

F ′UV = µ ′U ′V ′+ µ

(U ′V ′′+V ′U ′′) ,

F ′VV = µ ′

(V ′)2

+2µV ′V ′′.

4.2 Numerical methods 49

In order to ensure that the disturbances are contained within the boundary layer, all

perturbations must decay as η → ∞ , hence all perturbation quantities are naturally set to

zero at the far end of the physical domain so the boundary conditions are

u(η) = v(η) = w(η) = w′ (η) = p(η) = 0 at η = 0,

u(η)→ 0, v(η)→ 0, w(η)→ 0 and p(η)→ 0 as η → ∞. (4.12)

Factors ℓ= 1/(

1+δ1ηθn−1/n+1

)appear multiplying terms in the perturbation Equations (4.10)−

(4.11). These factors are set to unity in an approximation that is similar to the parallel-

flow approximations made in many other boundary-layer investigations, Garrett (2002).

Terms multiplied by 1/R and δ1 are the terms arising from viscosity and streamline curvature

respectively. The frequency ω is set to be ω = β sinθ in order to study the occurrence

of stationary instabilities relative to the sphere surface. It is noted that the perturbation

equations when n = 1 are entirely consistent with the Newtonian set of transformed per-

turbation Newtonian equations of Garrett and Peake (2002).

The perturbation Equations (4.10)− (4.11) are written as the eigenvalue problem

(4.31) and solved by using a Galerkin projection in terms of Chebyshev polynomials

described in §4.2. The stability analysis of the perturbation equations of Newtonian fluid

on the rotating disk have been done, either using transformed variables and a shooting

method (Griffiths et al. (2014a); Lingwood (1997); Lingwood and Garrett (2011)), or

primitive variables, when considering a rotating sphere, as in this thesis. An advantage

of the spectral method by Chebyshev polynomials over the shooting method is that the

perturbation equations are not transformed to a new system, and also for this method all

eigenvalues are found at the same time instead of looking for one by one by using a qual-

ified guess. A disadvantage is that only eigenvalues of α are found, Appelquist (2014).

4.2 Numerical methods

In this section we present an overview of the spectral Chebyshev collocation technique

used to solve the perturbation Equations (4.12)− (5.1) in order to study the occurrence


of convective instabilities. The spectral methods are reviewed briefly in §4.2.1, in §4.2.2

the implementation of the Chebyshev collocation method and the eigenvalue problems for

power-law are described.

4.2.1 Fundamental background of the spectral methods

The full theoretical background of the spectral methods used is given by Peyret (2013).

Spectral methods are considered as a general class of weighted residual methods where

the approximation solutions are defined as a truncated series expansion. The error or

residual of the approximations should be set approximately to zero, Finlayson (2013).

This is satised through the following process

The truncated series expansion of a function u(x) defined on the interval [a,b] for

given orthogonal basis functions ϕk (x) is stated in (4.13). In spectral analysis, the or-

thogonal basis functions for periodic problems are usually chosen to be the trigonometric

functions eikx while Chebyshev Tk (x) or Legendre Lk (x) polynomials are usually used for

non-periodic problems.

uN (x) =N

∑k=0

ckϕk (x) , a ≤ x ≤ b, (4.13)

where the expansion coefficients denoted by ck are the unknowns of the approximation.

The residual RN (x) is defined as

RN (x) =Lun − f , (4.14)

where uN (x) is the approximated solution of the differential equation

Lu− f = 0, (4.15)

where L is a partial differential operator subject to the appropriate boundary conditions.

f is assumed to be a continuous function. The residual then is forced to be zero by setting


the following scalar product to zero,

(RN ,ψi)w∗ =∫ b

aRNψiw∗dx = 0, i ∈ IN . (4.16)

Here ψi (x) are the weighting functions, w is the weight and IN is the discrete set where

its dimension is the number of the collocation points xi. The choice of the weighting

functions and the weight determines the distinct formulation types of the spectral meth-

ods. The weighting functions for Galerkin and tau formulations are the same as the basis

functions and the weight is the same weight associated with orthogonality of the basis

functions. For the Chebyshev collocation method they are chosen as

ψi (x) = δ (x− xi) and w∗ = 1, (4.17)

where δ is the Dirac delta-function and xi are selected collocation points in [a,b].

It is now clear from (4.16) and (4.17) that

RN (xi) = 0, (4.18)

which also implies, from denition of the residual, that

uN (xi) = u(xi) , i = 0, . . . ,N. (4.19)

Equation (4.19) raises an algebraic system of N +1 coefficients ck defined as follows

N

∑k=0

ckϕk (xi) = u(xi) , i = 0, . . . ,N. (4.20)

4.2.2 Implementation of Chebyshev collocation method

An advantage of analysing the linear stability by applying the spectral method using

Chebyshev polynomials over the shooting method (first used by Malik (1986) for an in-

stability analysis of the von Karman flow) also been used is that the former method allows


all instability modes to be obtained simultaneously by computing the entire spectrum of

eigenvalues in one calculation. That is, all eigenvalues are found at the same time instead

of looking for one by one by using a qualified guess. Another advantage over the shoot-

ing method, is that the approach uses the primitive forms of the governing equations so

the perturbation equations are not transformed to a new system. On the other hand, by

applying the spectral method using Chebyshev polynomials, only eigenvalues of α are

found, which can be counted as a disadvantage of this method. The approach should not

in principle give different results to Garrett (2002) results which he obtained using the

shooting method .

The Chebyshev collocation method is used by Appelquist (2014) to solve linear gov-

erning perturbation equations for the Newtonian fluids on a rotating disk. We use this

method to solve the governing equations for power-law on a rotating sphere. The Cheby-

shev collocation method is based on the Chebyshev polynomials defined recursively on

the interval y ∈ [−1,1] as follows

T0 (y) = 1,

T1 (y) = y, (4.21)

Tk+1 (y) = 2yTk (y)−Tk−1 (y) .

Because the governing perturbation equations (4.5) and (5.5) for power-law flow over

a rotating sphere involve second order ODEs, Only the first and second derivatives of the

Chebyshev polynomials are needed and they can be defined in a recurrence relation as

T ′0 (y) = 0, T ′′

0 (y) = 0,

T ′1 (y) = 1, T ′′

1 (y) = 0,

T ′2 (y) = 4T ′

1 (y) , T ′′2 (y) = 4T ′′

1 (y) , (4.22)

T ′k (y) = 2Tk−1 (y)+2yiT ′

k−1 (y)−T ′k−2 (y) ,

T ′′k (y) = 4T ′

k−1 (y)+2yiT ′′k−1 (y)−T ′′

k−2 (y) ,


for k = 3,4, . . . ,N. Superscripts ′ and ′′ denote the first and second derivatives with respect

to y.

With the purpose of solving the eigenvalue problem and obtaining the eigenvalues of

the radial wavenumber and the corresponding eigenfunctions of the perturbation quant-

ities (u, v, w, p), the Chebyshev expansions of these quantities should be introduced at a

number of points in the physical domain of the power-law flow, called collocation points.

The determination of these collocation points is based on a transformation of the

Gauss-Lobatto collocation points yiyj , defined as

yi = cos(iπ/N) , i = 0,1, . . . ,N, (4.23)

at N + 1 number of points in the interval [−1,1] into the physical domain [0,20], [0,60],

[0,160], [0,370] and [0,1680] when n = 1, 0.9, 0.8, 0.7 and 0.6 respectively.

In this thesis we use an exponential mapping function described as

η = −4log(

y−AB

),

A = −1−B, (4.24)

B = 2/(

e−ηmax

4 −1),

to distribute 50 collocation points between η = 0 and the top of the domain, ηmax =

20, 60, 160, 370, and 1680 when n = 1, 0.9, 0.8, 0.7 and 0.6 respectively. The expo-

nential mapping transformation distributes the collocation points mainly into the bound-

ary layer. This is necessary in a boundary layer flow because the discrepancies of the

quantities are high and more calculations should be performed near the lower surface to

ensure a higher degree of accuracy.

In the physical space of the rotating sphere flow, the Chebyshev polynomials and their


derivatives are constructed using the chain rule as

Sk (η) = Tk (y) ,

S′k (η) =dTk (y)

dη= T ′

k (y)dydη

, (4.25)

S′′k (η) =d2Tk (y)

dη2 = T ′′k (y)

(dydη

)2

+T ′k (y)

d2ydη

.

The superscripts ′ and ′′ denote the first and second derivativesof Sk (η) and Tk (y) with re-

spect to η and y respectively. The truncated series of the perturbation quantities (u, v, w, p)

and of their derivatives at collocation points η j are given as the sum of the contributions

of all the transformed Chebyshev polynomials. They are described as

u(ηi) =N

∑k=0

cukSk (ηi) , v(ηi) =

N

∑k=0

cvkSk (ηi) ,

w(ηi) =N

∑k=0

cwk Sk (ηi) , p(ηi) =

N

∑k=0

c pk Sk (ηi) , (4.26)

u′ (ηi) =N

∑k=0

cukS′k (ηi) , v′ (ηi) =

N

∑k=0

cvkS′k (ηi) ,

w′ (ηi) =N

∑k=0

cwk S′k (ηi) , p′ (ηi) =

N

∑k=0

c pk S′k (ηi) , (4.27)

u′′ (ηi) =N

∑k=0

cukS′′k (ηi) , v′′ (ηi) =

N

∑k=0

cvkS′′k (ηi) ,

w′′ (ηi) =N

∑k=0

cwk S′′k (ηi) , p′′ (ηi) =

N

∑k=0

c pk S′′k (ηi) . (4.28)

For the boundary conditions, notice that from (4.12), all perturbations should decay when

η → ∞ . At the wall, velocity perturbations must be zero and from Equation (4.11) we

found that the first derivative of w should also be zero. The boundary conditions at top


and bottom of the domain are

u(η0) =N

∑k=0

cukSk (η0) = 0, v(η0) =

N

∑k=0

cvkSk (η0) = 0,

w(η0) =N

∑k=0

cwk Sk (η0) = 0, p(η0) =

N

∑k=0

c pk Sk (η0) = 0, (4.29)

u(ηN) =N

∑k=0

cukSk (ηN) = 0, v(ηN) =

N

∑k=0

cvkSk (ηN) = 0,

w(ηN) =N

∑k=0

cwk Sk (ηN) = 0, p(ηN) =

N

∑k=0

cpk Sk (ηN) = 0. (4.30)

Now inserting the Chebyshev expansions of the perturbation quantities, along with the

boundary conditions (4.29) into Equations (4.10)-(4.11) gives the eigenvalue problem

for the wavenumber α for power-law fluids formed as

(A2α

2 +A1α +A0)V = 0, (4.31)

where V is the matrix of the eigenfunctions and A2, A1 and A0 are matrices of size

4(N +1) " 4(N +1), where 4 is the number of unknown quantities. The form of the

matrices are stated in (4.32)-(4.34).

4.2N

umericalm

ethods56

A2 =

SN (η0) 0 0 0 . . .

0 SN (η0) 0 0 . . .

0 0 SN (η0) 0 . . .

0 0 0 SN (η0) . . .(µℓ2/R

)S0 (η1) 0 0 0 . . .

0(

µℓ2/R)

S0 (η1) 0 0 · · ·

0 0(

µℓ2/R)

S0 (η1) 0 · · ·

0 0 0 0 · · ·...

......

... · · ·

SN (ηN) 0 0 0 . . .

0 SN (ηN) 0 0 . . .

0 0 SN (ηN) 0 . . .

0 0 S′N (ηN) 0 . . .

,

(4.32)

4.2N

umericalm

ethods57

A1 =

εSN (η0) 0 0 0 . . .

0 εSN (η0) 0 0 . . .

0 0 εSN (η0) 0 . . .

0 0 0 εSN (η0) . . .

iℓUS0 (η1) 0 0 iℓS0 (η1) . . .

0 iℓUS0 (η1) 0 0 · · ·

−(iFUU/R)S′0 (η1) −(iFUV/R)S′0 (η1) iℓUS0 (η1) 0 · · ·

iℓS0 (η1) 0 0 0 · · ·...

......

... · · ·

εSN (ηN) 0 0 0 . . .

0 εSN (ηN) 0 0 . . .

0 0 εSN (ηN) 0 . . .

0 0 εS′N (ηN) 0 . . .

, (4.33)

4.2N

umericalm

ethods58

A0 =

εSN (η0) 0 0 0 . . .

0 εSN (η0) 0 0 . . .

0 0 εSN (η0) 0 . . .

0 0 0 εSN (η0) . . .

A51 A52 A53 A54 . . .

A61 A62 A63 A64 · · ·

A71 A72 A73 A74 · · ·

A81 A82 A83 A84 · · ·...

......

... · · ·

εSN (ηN) 0 0 0 . . .

0 εSN (ηN) 0 0 . . .

0 0 εSN (ηN) 0 . . .

0 0 εS′N (ηN) 0 . . .

. (4.34)


Where

A51 =

[(iβℓV

Rsinθ− iω +

ℓ2µβ 2

R3 sin2θ

)+

ℓ

R

(∂U∂θ

)+

ℓηU ′

R

]S0 (η1)

+

(WR+

ℓηUR

− µ ′

R−

F ′UUR

)S′0 (η1)−

(µ +FUU)

RS′′0 (η1) ,

A52 =

(−2ℓcotθV

R

)S0 (η1)−

F ′UVR

S′0 (η1)−FUV

RS′′0 (η1) ,

A53 =

(U ′+

ℓUR

+ℓµ

R

)S0 (η1) ,

A54 = −ℓη

RS′0 (η1) ,

A61 =

[(ℓ

R

)(∂V∂θ

+V cotθ + ηV ′)]

S0 (η1)−F ′

UVR

S′0 (η1)−FUV

RS′′0 (η1) ,

A62 =

[(iβℓV

Rsinθ− iω +

ℓ2µβ 2

R3 sin2θ

)+

ℓU cotθ

R

]S0 (η1)

+

(WR+

ℓηUR

− µ ′

R−

F ′VVR

)S′0 (η1)− (µ +FVV )S′′0 (η1) ,

A63 =

(V ′+

ℓVR

− iℓβ µ ′

R2 sinθ

)S0 (η1) ,

A64 =iβℓ

RsinθS0 (η1) ,

A71 = −2ℓUR

S0 (η1)−iβFUV

R2 sinθS′0 (η1) ,

A72 = −2ℓVR

S0 (η1)−iβFVV

R2 sinθS′0 (η1) ,

A73 =

[(iℓβV

Rsinθ− iω +

ℓ2µβ 2

R3 sin2θ

)+

W ′

R

]S0 (η1)

+

(WR+

ℓηUR

− 2µ ′

R

)S′0 (η1)−

(µ

R

)S′′0 (η1) ,

A74 = S′0 (η1) ,

A81 =

(ℓcotθ

R

)S0 (η1)+

(η

R

)S′0 (η1) ,

A82 =

(iℓβ

Rsinθ

)S0 (η1) ,

A83 =

(2ℓR

)S0 (η1)+S′0 (η1) ,

The complex parameter ε in the matrices (4.33) and (4.34) indicates a complex values

set to 30i, 60i 160i 370i and 1680i for n = 1, 0.9, 0.8, 0.7 and 0.6 respectively, where i is√−1. The eigenvalue problem is solved by using MATLAB solver function (polyeig) in

4.3 Summary 60

the spectral code to compute the solutions and the eigenvalues α for fixed values of R and

various values of β iteratively. In this code the solutions and eigenvalues are computed

for fixed values of R and iteratively changed values of β , the branch point is selected to

be the mode with smallest imaginary part Im(αi) then the iteration runs until a neutral

point on the neutral curve with zero imaginary part is found. In order to construct the

entire neutral curve of convective instability, this procedure is repeated iteratively for a

wide range of R.

4.3 Summary

In this chapter the perturbation equations for power-law fluids over a rotating sphere are

derived where infinitesimal small perturbations on the steady mean flow are imposed at

a particular latitude. In the case when the power-law index n = 1, the perturbation equa-

tions of Newtonian fluid on rotating sphere presented by Garrett (2002) are recovered.

A normal-mode analysis is made using the dimensionless normal mode form of the per-

turbation variables. The eigenvalue problem for power-law fluids is formed in order to

construct the entire neutral curve of convective instability. The perturbation equations are

written as an eigenvalue problem for α and solved by using spectral method by Chebyshev

polynomials.

Chapter 5

Convective instability analysis varying

the latitude

In this chapter the convective instability analyses and the neutral stability curves are

presented. Our theoretical predictions are compared with results of Garrett and Peake

(2002) when the power-law index n = 1, and with those for the rotating disk conducted

by Griffiths et al. (2014a) as the analyses approach toward the pole.

5.1 The convective instability analysis

In this chapter we analyse the characteristics of convective instability in terms of neut-

ral curves in §5.2 and growth rates in §5.3. The set of Equations (4.10)− (4.11) with

the boundary conditions (4.12) leads to a dispersion relation for the rotating sphere at a

latitude θ

D(η ;α,β ,ω;R,n) = 0.

This dispersion relation allows an unknown parameter (α, β , or ω) to be calculated given

the others at each R and n. The Chebyshev polynomial method used in this thesis solves

the eigenvalue problems of the perturbation Equations (4.10)− (4.11) using Chebyshev

discretization of the wall-normal coordinate. A spatial stability analysis will be performed

as we are interested in the disturbance generated at a fixed position in space and the growth

5.2 Neutral curves 62

from the source. In this analysis, the eigenvalue of the problem is the complex radial

wavenumber, α , for a fixed real frequency, ωr. In the Briggs-Bers procedure we reduce

the imaginary part of the frequency down to zero since the flow is supposed not to be

absolutely unstable in the first instance, so that ωi = 0. β is assumed to be real and O (1).

In much the same way as the approach presented by Garrett (2002) where the vortices are

insisted to rotate at some fixed multiple of the sphere surface velocity, we fixed the ratio

ωi/β and calculate α and β using a spatial analysis. Equating the relevant multiple of

the non-dimensional speed of the surface of the sphere, sinθ , with the disturbance phase

velocity ωi/β in the same direction, gives

ωr = cβ sinθ . (5.1)

If the vortices are to rotate with the sphere, the relationship (5.1) must be satisfied with

c = 1.0 and c = 0.76 if the vortices are those reported by Kobayashi and Arai (1990) at

high latitudes, Garrett (2002). There is no evidence that non-Newtonian flows give rise to

travelling vortices and here we consider c = 1.0 only. The amplification of a normal mode

in (4.10)−(4.11) is therefore given by −Im(α)> 0. We solve this eigenvalue problem in

MATLAB with sets of conditions, R, β and mean velocity profiles for the rotating-sphere

flow.

5.2 Neutral curves

5.2.1 Case 1 (The power-law index n = 1)

Considering vortices that rotate with the surface of the sphere, i.e. fixing c = 1.0, we have

found that at each latitude, there are two spatial branches in the complex α − plane that

determine the convective instability characteristics of the system. A branch with a region

lying below the αr − axis indicates convective instability.


0 0.1 0.2 0.3 0.4 0.5 0.6-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0 0.1 0.2 0.3 0.4 0.5 0.6-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Figure 5.1: The two spatial branches for the case when n = 1 at θ = 10 showing TypeI instability from branch 1 only at R = 2400 and Type II instabilities from the modifiedbranch 1 at R = 2500.

Figure (5.1) shows these spatial branches in the complex α − plane for θ = 10 and

R = 2400. The effects of crossflow-instability mode cause branch 1, branch 2 is due to

the streamline curvature. From Figure 5.1 at R = 2500 we observe that an exchange of

modes has occurred between the two branches, the modified branch 1 now determines the

region of convective instability. Increasing the value of R causes the peak between the

two minima on branch 1 to move downwards and the points where the branch crosses the

line αi = 0 move apart thereby widening the regions of instability and mapping out two

lobes on the neutral curve. Above a certain value of R, the peak moves below the line

αi = 0 and further increases in R change the region of instability, producing the upper and

lower branches of the neutral curve.

For all latitudes below θ = 66, this branches behaviour is typical. Above θ = 66 and

as presented by Garrett (2002) , the two branches only ever appear like those described

when the peak in the modified branch 1 has moved below the line αi = 0 after the branch

exchange, therefore the neutral curves for latitudes above θ = 66 do not have the two

lobed structure. The branches when n = 1 at θ = 70 and R = 100 are shown in Figure

5.2


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.02

0

0.02

0.04

0.06

0.08

0.1

Figure 5.2: The two spatial branches for the case when n = 1 at θ = 70 showing a kinkin branch 2 and a region of instability caused by branch 1.

Figure 5.3 shows the neutral curves for convective instability when n = 1 at latitudes

of θ = 10−70. Each curve encloses a region that is connectively unstable.

0 500 1000 1500 2000 2500 30000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1 =10=20=30=40=50=60=70

0 500 1000 1500 2000 2500 30000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22 =10=20=30=40=50=60=70

0 500 1000 1500 2000 2500 30001012141618202224262830323436 =10

=20=30=40=50=60=70

0 500 1000 1500 2000 2500 30000

50

100

150

200

250

300=10=20=30=40=50=60=70

Figure 5.3: The neutral curves of convective instability for stationary vortices at latitudesof θ = 10−70 (right to left) when n = 1.

Figure 5.4 shows our results in the(RS, β

)−plane. Here RS = R2 is the spin Reynolds

number and β is the number of vortices from (4.8). For n = 1, the Figure 5.4 shows a

comparison of our neutral curves with the theoretical neutral curves of Taniguchi et al.

(1998). The value of the critical Reynolds number on both the upper and lower lobes is

shown in Tables 5.1 and 5.2 respectively.


104 105 106 1070

50

100

150

200

250

300=10=20=30=40=50=60=70

Figure 5.4: The neutral curves of convective instability in the(RS, n)−plane for stationaryvortices of θ = 10−70 (right to left) when n = 1.

θ R α β β φ ()

10 1601.62 0.3847 0.0779 21.65 11.44

20 777.30 0.3806 0.0771 20.50 11.45

30 498.02 0.3711 0.0751 18.69 11.44

40 353.40 0.3583 0.0721 16.37 11.37

50 260.29 0.3404 0.0679 13.53 11.27

60 188.72 0.3151 0.0622 10.17 11.17

Table 5.1: The values of the critical Reynolds number R, wavenumbers α , azimuthalwave-number, β and the angle of spiral vortices φ corresponding to the value of n = 1 onthe upper-branch for stationary vortices of θ = 10−60.

θ R α β β φ ()

10 2476.27 0.1338 0.04691 20.17 19.32

20 1172.06 0.1358 0.0469 18.80 19.05

30 727.47 0.1364 0.0463 16.82 18.73

40 494.25 0.1368 0.0452 14.34 18.26

50 340.41 0.1385 0.0438 11.42 17.54

60 216.68 0.1466 0.0429 8.05 16.32

70 91.29 0.1984 0.0489 4.20 13.85

Table 5.2: The values of the critical Reynolds number R, wavenumbers α , azimuthalwave-number, β and the angle of spiral vortices φ corresponding to value of n = 1 on thelower-branch for stationary vortices of θ = 10−70.


The rotating disk investigations, Malik (1986), Lingwood (1995) and Griffiths et al.

(2014a) use a Reynolds number based on the local disk velocity at the radius under invest-

igation and the local boundary layer thickness. For our investigation on rotating sphere,

the equivalent Reynolds number is written as RL = Rsinθ . Using this Reynolds number,

when n = 1 a comparison between our results and those of Griffiths et al. (2014a) for the

rotating disk is made.

θ

10 15 20 25 30 35 40 45 50 55 60

R L=Rsin

θ

150

200

250

300

350

400

450

500

Streamline curvature mode on rotating disk

Crossflow mode on rotating disk

Crossflow modeStreamline curvature mode

Figure 5.5: A comparison of the critical RLvalues for convective instability at each latitudewhen n= 1 with those of the Griffiths et al. (2014a) for the rotating disk (horizontal lines).

Figure 5.5 shows that the critical Reynolds numbers of the rotating sphere boundary-

layer approach the disk as we approach the pole, i.e. as θ → 0. Plots of the neutral curves

at θ = 10 when n = 1 are shown in Figure 6.1.

5.2.2 Case 2 (The power-law index n = 0.9)

When n = 0.9, Figure 5.6 shows the structure of the two branches for R = 2760 and

R = 31590.

0 0.1 0.2 0.3 0.4 0.5 0.6-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0 0.1 0.2 0.3 0.4 0.5 0.6-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Figure 5.6: The two spatial branches for the case when when n = 0.9 at θ = 10 show-ing Type I instability from branch 1 only at R = 2760 and Type II instabilities from themodified branch 1 at R = 3159.90.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.02

0

0.02

0.04

0.06

0.08

0.1

Figure 5.7: The two spatial branches for the case when when n = 0.9 at θ = 70 showinga kink in branch 2 and a region of instability caused by branch 1.

The branches when n = 0.9 at θ = 70 and R = 150 are shown in Figure 5.7.

0 500 1000 1500 2000 2500 30000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1 =10=20=30=40=50=60=70

0 500 1000 1500 2000 2500 30000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22 =10=20=30=40=50=60=70

0 500 1000 1500 2000 2500 30001012141618202224262830323436 =10

=20=30=40=50=60=70

0 500 1000 1500 2000 2500 30000

50

100

150

200

250

300=10=20=30=40=50=60=70

Figure 5.8: The neutral curves of convective instability for stationary vortices at latitudesof θ = 10−70 (right to left) when n= 0.9 . The R−axis has been truncated at R= 3000.


θ R α β β φ ()

10 1959.91 0.3601 0.0758 25.80 11.89

20 949.49 0.3559 0.07481 24.29 11.87

30 607.35 0.3467 0.0725 22.03 11.86

40 430.09 0.3334 0.0691 19.09 11.70

50 317.22 0.3154 0.0642 15.44 11.19

60 228.98 0.2940 0.0589 11.68 11.33

Table 5.3: The values of the critical Reynolds number R, wavenumbers α , azimuthalwave-number, β and the angle of spiral vortices φ corresponding to value of n = 0.9 onthe upper-branch for stationary vortices of θ = 10−60.

θ R α β β φ ()

10 3140.52 0.1200 0.0429 23.37 19.65

20 1501.72 0.1202 0.0425 21.81 19.45

30 940.03 0.1195 0.0415 19.52 19.17

40 644.67 0.1185 0.0401 16.61 18.68

50 448.81 0.1165 0.0379 13.98 18.04

60 268.28 0.1244 0.0372 8.63 16.64

70 91.50 0.1978 0.0480 4.12 13.63

Table 5.4: The values of the critical Reynolds number R, wavenumbers α , azimuthalwave-number, β and the angle of spiral vortices φ corresponding to value of n = 0.9 onthe lower-branch for stationary vortices of θ = 10−70.

Figure 5.8 shows the neutral curves for convective instability when n = 0.9 at latitudes of

θ = 10− 70. The value of the critical Reynolds number on both the upper and lower

lobes is shown in Tables 5.3 and 5.4 respectively. A comparison between our results

and those of Griffiths et al. (2014a) for the rotating disk when n = 0.9 is made using

the equivalent Reynolds number RL = Rsinθ . Figure 5.9 shows that, when n = 0.9, the

critical Reynolds numbers of the rotating sphere boundary-layer approach the disk as we

approach the pole, i.e. as θ → 0.


θ

10 15 20 25 30 35 40 45 50 55 60

R L=Rsin

θ

150

200

250

300

350

400

450

500

550

600 Streamline curvature mode on rotating disk



Figure 5.9: A comparison of the critical RLvalues for convective instability at each latitudewhen n = 0.9 with those of the Griffiths et al. (2014a) for the rotating disk (horizontallines).


The structure of the two branches when n = 0.8 are illustrated by Figure 5.10 for R =

3251.75 and R = 4051.40.

0 0.1 0.2 0.3 0.4 0.5 0.6-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0 0.1 0.2 0.3 0.4 0.5 0.6-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Figure 5.10: The two spatial branches for the case when n = 0.8 at θ = 10 showing TypeI instability from branch 1 only at R = 3251.75 and Type II instabilities from the modifiedbranch 1 at R = 4051.40.

The branches when n = 0.8 at θ = 70 and R = 170 are shown in Figure 5.11.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.02

0

0.02

0.04

0.06

0.08

0.1

Figure 5.11: The two spatial branches for the case when n = 0.8 at θ = 70 showing akink in branch 2 and a region of instability caused by branch 1.

Figure 5.12 shows the neutral curves for convective instability when n = 0.8 at lat-


itudes of θ = 10− 70. The value of the critical Reynolds number on both the upper

and lower lobes is shown in Tables 5.5 and 5.6 respectively. A comparison between our

results and those of Griffiths et al. (2014a) for the rotating disk when n = 0.8 is made

using the equivalent Reynolds number RL = Rsinθ .

0 500 1000 1500 2000 2500 30000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1 =10=20=30=40=50=60=70

0 500 1000 1500 2000 2500 30000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22 =10=20=30=40=50=60=70

0 500 1000 1500 2000 2500 30001012141618202224262830323436 =10

=20=30=40=50=60=70

0 500 1000 1500 2000 2500 30000

50

100

150

200

250

300=10=20=30=40=50=60=70


θ R α β β φ ()

10 2451.75 0.3380 0.0741 31.54 12.36

20 1185.83 0.3329 0.0727 29.47 12.31

30 757.94 0.3234 0.0701 26.55 12.22

40 536.45 0.3078 0.0656 22.63 12.04

50 387.14 0.2866 0.0594 17.62 11.71

60 279.70 0.2670 0.0527 12.77 11.17

Table 5.5: The values of the critical Reynolds number R, wavenumbers α , azimuthalwave-number, β and the angle of spiral vortices φ corresponding to value of n = 0.8 onthe upper-branch for stationary vortices of θ = 10−60 .


θ R α β β φ ()

10 4085.36 0.1078 0.0390 27.69 19.90

20 1974.92 0.1068 0.0383 25.90 19.75

30 1248.19 0.1047 0.0371 23.15 19.51

40 862.66 0.1017 0.0351 19.46 19.03

50 578.51 0.1022 0.0329 14.56 17.82

60 318.92 0.1054 0.0312 8.63 16.51

70 77.58 0.2100 0.0470 3.42 12.61

Table 5.6: The values of the critical Reynolds number R, wavenumbers α , azimuthalwave-number, β and the angle of spiral vortices φ corresponding to values of n = 0.8 onthe lower-branch for stationary vortices of θ = 10−70.

θ

10 15 20 25 30 35 40 45 50 55 60

R L=Rsin

θ

200

300

400

500

600

700

800Streamline curvature mode on rotating disk



Figure 5.13: A comparison of the critical RLvalues for convective instability at each latit-ude when n = 0.8 with those of the Griffiths et al. (2014a) for the rotating disk (horizontallines).

Figure 5.13 shows that, when n= 0.8, the critical Reynolds numbers of the rotating sphere

boundary-layer approach disk as we approach the pole, i.e. as θ → 0.


Figure 5.14 shows the structure of the two branches when n = 0.7 for R = 3944.20 and

R = 5444.20.


0 0.1 0.2 0.3 0.4 0.5 0.6-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0 0.1 0.2 0.3 0.4 0.5 0.6-0.06

-0.04

-0.02

0

0.02

0.04

0.06


0 500 1000 1500 2000 2500 30000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1 =20=30=40=50=60

0 500 1000 1500 2000 2500 30000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22 =20=30=40=50=60

0 500 1000 1500 2000 2500 30001012141618202224262830323436 =20

=30=40=50=60

0 500 1000 1500 2000 2500 30000

50

100

150

200

250

300=20=30=40=50=60


Figure 5.15 shows the neutral curves for convective instability when n = 0.7 at lat-

itudes of θ = 10− 60. The value of the critical Reynolds number on both the upper

and lower lobes is shown in Tables 5.7 and 5.8 respectively. A comparison between our

results and those of Griffiths et al. (2014a) for the rotating disk when n = 0.7 is made

using the equivalent Reynolds number RL = Rsinθ .


θ R α β β φ ()

10 3144.20 0.3177 0.0724 39.53 12.84

20 1518.85 0.3124 0.0708 36.76 12.76

30 971.51 0.3021 0.0677 32.87 12.63

40 689.78 0.2838 0.0621 27.54 12.35

50 494.18 0.2577 0.0539 20.39 11.80

60 346.08 0.2397 0.0453 13.58 10.70


θ R α β β φ ()

10 5469.25 0.0967 0.0354 33.58 20.09

20 2674.99 0.0947 0.0344 31.50 19.97

30 1710.82 0.0911 0.0328 28.04 19.79

40 1198.38 0.0869 0.0303 23.32 19.21

50 749.64 0.0876 0.0277 15.90 17.54

60 349.51 0.0918 0.0260 7.86 15.80


θ

10 15 20 25 30 35 40 45 50 55 60

R L=Rsin

θ

200

300

400

500

600

700

800

900

1000

1100Streamline curvature mode on rotating disk






boundary-layer approach disk as we approach the pole, i.e. as θ → 0.


Figure 5.17 shows the structure of these two branches when n = 0.6 for R = 4947.56 and

R = 7347.60. The branches when n = 0.6 at θ = 70 and R = 200 are shown in Figure

5.18

0 0.1 0.2 0.3 0.4 0.5 0.6-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0 0.1 0.2 0.3 0.4 0.5 0.6-0.06

-0.04

-0.02

0

0.02

0.04

0.06


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.02

0

0.02

0.04

0.06

0.08

0.1

Figure 5.18: The two spatial branches for the case when n = 0.6 at θ = 70 showing akink in branch 2 and a region of instability caused by branch 1.

Figure 5.19 shows the neutral curves for convective instability when n = 0.6 at latit-

udes of θ = 10−70. The value of the critical Reynolds number on both the upper and

lower lobes is shown in Tables 5.9 and 5.10 respectively.


0 500 1000 1500 2000 2500 30000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1 =20=30=40=50=60=70

0 500 1000 1500 2000 2500 30000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22 =20=30=40=50=60=70

0 500 1000 1500 2000 2500 30001012141618202224262830323436 =20

=30=40=50

0 500 1000 1500 2000 2500 30000

50

100

150

200

250

300=20=30=40=50=60=70


θ R α β β φ ()

10 4147.56 0.3005 0.0709 51.09 13.28

20 2002.45 0.2947 0.0690 47.26 13.18

30 1285.16 0.2820 0.0651 41.83 13.00

40 921.73 0.2597 0.0580 34.35 12.59

50 662.26 0.2289 0.0476 24.14 11.74

60 516.92 0.2169 0.0386 17.29 10.10



θ R α β β φ ()

10 7575.43 0.0870 0.0319 42.00 20.14

20 3752.99 0.0841 0.0307 39.43 20.07

30 2439.25 0.0797 0.0288 35.10 19.87

40 1745.18 0.0734 0.0255 28.63 19.16

50 921.18 0.0782 0.0235 16.56 16.70

60 341.11 0.0974 0.0224 6.61 12.93

70 97.60 0.1460 0.0243 2.22 9.44


A comparison between our results and those of Griffiths et al. (2014a) for the rotating

disk when n = 0.6 is made using the equivalent Reynolds number RL = Rsinθ .

θ

10 15 20 25 30 35 40 45 50 55 60

R L=Rsin

θ

200

400

600

800

1000

1200

1400

1600

Streamline curvature mode on rotating disk





boundary-layer approach disk as we approach the pole, i.e. as θ → 0. The neutral curves

in Figures 5.3, 5.8, 5.12, 5.15 and 5.19 show that the rotating-sphere boundary layer is

increasingly stable as we decrease the latitude from the equator towards the pole, which,

for the Newtonian case when n = 1, is consistent with the Garrett (2002) results and the

experimental results of Sawatzki (1970) and Kobayashi et al. (1983). For all n values, and

at all latitude in figures except θ = 70, a two-lobed structure is seen. Plots of the neutral

curves at θ = 10when 1 6 n 6 0.6 are shown in Figure 5.21.

5.3 Growth rates 77

0 200 400 600 800 1000 1200 14000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1n=1n=0.9n=0.8n=0.7n=0.6

0 200 400 600 800 1000 1200 14000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2n=1n=0.9n=0.8n=0.7n=0.6

0 200 400 600 800 1000 1200 14000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1n=1n=0.9n=0.8n=0.7n=0.6

0 200 400 600 800 1000 1200 140010

12

14

16

18

20

22

24

26

28

30n=1n=0.9n=0.8n=0.7n=0.6

Figure 5.21: The neutral curves of convective instability for stationary vortices at latitudesof θ = 10 for decreasing values of n.

Our theoretical predications and neutral curves in Figure 5.21 are in good agreement

with those due to the rotating disk conducted by Griffiths et al. (2014a). We note here that

κ is the neutral wavenumber where

κ =√

α2 +β 2. (5.2)

5.3 Growth rates

This section presents the growth rates of the Type I instability mode of boundary layer

flows for power-law fluids over rotating spheres. The growth rate of the instability mode is

measured as the absolute value of the negative imaginary part of the latitudinal wavenum-

ber, |αi|, at the particular values of number of spiral vortices β . We only present here the

growth rates of the Type I instability mode since the growth rates of the secondary Type II

mode has very small values in comparison with the dominant Type I mode, furthermore,

the power-law fluids have only a slight effect on the Type II mode. Figure 5.22 shows the

convective growth rates of the dominant Type I at R = Rc+25, at a fixed distance into the

connectively unstable region for a variety of flow, as a function of β . for various value of

5.3 Growth rates 78

n at latitudes of θ = 10−60. Here Rc denotes the critical Reynolds number at the onset

of the Type I mode presented in Tables 5.1, 5.3, 5.5, 5.7 and 5.9 for the particular n and

latitude θ , β is the number of spiral vortices around the sphere surface given in Equation

(4.8).

4 6 8 10 12 14 16 18 20 22 240

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018=10=20=30=40=50=60

(a) n=1.0

8 10 12 14 16 18 20 22 24 26 28 300

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018=10=20=30=40=50=60

(b) n=0.9

10 15 20 25 30 350

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018=10=20=30=40=50=60

(c) n=0.8

10 15 20 25 30 35 40 450

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018=10=20=30=40=50=60

(d) n=0.7

20 30 40 50 600

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018=10=20=30=40=50

(e) n=0.6

Figure 5.22: Growth rates for Type I mode for shear-thinning power-law fluids for 0.6 ≤n ≤ 1.0 at latitudes of θ = 10−60.

Figure 5.22 reveals a stabilising effect on the growth rates of the Type I for power-law

flow as the value of the maximum growth rate indicated by a red dot decreases when

we decrease the latitude from the equator towards the pole. It is interesting to note that

the maximum growth rate is pushed to higher β as the power-law index decreases. The

stabilising results seen here are consistent with the critical values for each power-law

5.4 Conclusion 79

index value reported in §5.1.

5.4 Conclusion

In this chapter we have investigated the convective instability of stationary disturbances

of power-law fluids on a rotating sphere The investigation conducted in Chapter 3 shows

that the steady laminar flow profiles of power-law fluid of rotating sphere boundary layer

reduce to those of the rotating disk near to the pole. In this chapter we have conducted a

linear stability analyses and revealed that the neutral curves for the convective instability

of the boundary layer of power-law fluid on the rotating sphere approach those of the

rotating disk as we approach the pole. Our results and the neutral curves for the con-

vective instability when the power-law index n = 1 are in excellent agreement with those

presented by Garrett (2002) for the Newtonian case. Similar to the results revealed by

Garrett and Peake (2002) for the Newtonian fluid. The convective instability analysis

when n = 0.9,0.8,0.7 and 0.6 shows that crossflow instabilities dominate below θ = 66,

whilst streamline-curvature instabilities dominate above this latitude due to a region of

reverse flow in the radial component of the mean flow. The number of spiral vortices at

the onset of instability are predicted to decrease with increased latitude. As the analysis

moves towards the pole, this number approaches the theoretical prediction for the rotat-

ing disk of Griffiths et al. (2014a). They found being β ≈ 22,26,32,40 and 51 when

n = 1,0.9,0.8,0.7 and 0.6 respectively. Roughly speaking, at the onset of instability

the stationary vortices at each latitude were predicted to have the same vortex angles

for each mode. At the onset of crossflow instabilities, these values are found to be

φ = 11.4,11.8,12.4,12.8 and 13.2when n = 1,0.9,0.8,0.7 and 0.6 respectively and

at streamline-curvature instabilities, φ = 19.3,19.6,19.9, 20.1 and 20.2 when n =

1,0.9,0.8,0.7 and 0.6 respectively. The boundary layer is known to erupt at the equator

causing the boundary-layer assumptions to invalid there Garrett (2002), θ = 80 may

be close enough to be affected by this, giving an explanation for being the convective

instability results show a large discrepancy there.

Chapter 6

Convective instability analysis varying

the power-law index

In this chapter, we present the results of Chapter 5 but in terms of fixed the latitude θ

and varying the power-law index n. This is intended to help us understand how the non-

Newtonian power-law effects the stability at each latitude.

6.1 Neutral curves

Case 1 (θ = 10)

Figure 6.1 presents the neutral curves at θ = 10 for values of the power-law index ranging

from n = 0.6− 1 in increments of 0.1 in the (R,αr), (R,β ) ,(R, β

)and (R,φ) planes.

These curves enclose a region in which the boundary-layer is convectively unstable. The

neutral curves show that decreasing the power-law index has a stabilising effect on the

boundary-layer flow. The value of the critical Reynolds number is increased on both the

upper and lower lobes as n decreases, as shown in Table 6.1.


0 1000 2000 3000 4000 5000 6000 7000 8000 90000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1n=1n=0.9n=0.8n=0.7n=0.6

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2n=1n=0.9n=0.8n=0.7n=0.6

0 1000 2000 3000 4000 5000 6000 7000 8000 900010

12

14

16

18

20

22

24

26

28

30n=1n=0.9n=0.8n=0.7n=0.6

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

50

100

150

200

250

300n=1n=0.9n=0.8n=0.7n=0.6

Figure 6.1: The neutral curves of convective instability for stationary vortices at latitudeθ = 10 when 0.6 ≤ n ≤ 1.

n R α β β φ ()

1.0 1601.62 0.3847 0.0779 21.65 11.44

0.9 1959.91 0.3601 0.0758 25.80 11.89

0.8 2451.75 0.3380 0.0741 31.54 12.36

0.7 3144.20 0.3177 0.0724 39.53 12.84

0.6 4147.56 0.3005 0.0709 51.09 13.28

n R α β β φ ()

1.0 2476.27 0.1338 0.04691 20.17 19.32

0.9 3140.52 0.1200 0.0429 23.37 19.65

0.8 4085.36 0.1078 0.0390 27.69 19.90

0.7 5469.25 0.0967 0.0354 33.58 20.09

0.6 7575.43 0.0870 0.0319 42.00 20.14

Table 6.1: The values of the critical Reynolds number R, wavenumbers α , azimuthalwave-number, β and the angle of spiral vortices φ for power-law fluids at latitude θ = 10

on the both modes Type I and Type II.


Case 2 (θ = 20)

The neutral curves at θ = 20 for values of the power-law index n = 0.6− 1 presents in

Figure 6.2. These curves enclose a region in which the boundary-layer is convectively

unstable.

0 1000 2000 3000 4000 50000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1n=1n=0.9n=0.8n=0.7n=0.6

0 1000 2000 3000 4000 50000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2n=1n=0.9n=0.8n=0.7n=0.6

0 1000 2000 3000 4000 500010

12

14

16

18

20

22

24

26

28

30n=1n=0.9n=0.8n=0.7n=0.6

0 1000 2000 3000 4000 50000

50

100

150

200

250

300n=1n=0.9n=0.8n=0.7n=0.6


A stabilising effect on the boundary-layer flow is shown as the power-law index is de-

creasing. The value of the critical Reynolds number is increased on both the upper and

lower lobes as n decreases, as shown in Tables 6.2 and 6.3, respectively.

n R α β β φ ()

1.0 777.30 0.3806 0.0771 20.50 11.45

0.9 949.49 0.3559 0.07481 24.29 11.87

0.8 1185.83 0.3329 0.0727 29.47 12.31

0.7 1518.85 0.3124 0.0708 36.76 12.76

0.6 2002.45 0.2947 0.0690 47.26 13.18

Table 6.2: The values of the critical Reynolds number R, wavenumbers α , azimuthalwave-number, β and the angle of spiral vortices φ corresponding to decreasing values ofn on the upper-branch for stationary vortices at latitude θ = 20.


n R α β β φ ()

1.0 1172.06 0.1358 0.0469 18.80 19.06

0.9 1501.72 0.1202 0.0425 21.81 19.45

0.8 1974.92 0.1068 0.0383 25.90 19.75

0.7 2674.99 0.0947 0.0344 31.50 19.97

0.6 3752.99 0.0841 0.0307 39.43 20.07

Table 6.3: The values of the critical Reynolds number R, wavenumbers α , azimuthalwave-number, β and the angle of spiral vortices φ corresponding to decreasing values ofn on the lower-branch for stationary vortices of θ = 20.

Case 3 (θ = 30)

Figure 6.3 presents the neutral curves at θ = 30 for values of the power-law index ran-

ging from n = 0.6− 1 in increments of 0.1. These curves enclose a region in which the

boundary-layer is convectively unstable.

0 500 1000 1500 2000 2500 30000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1n=1n=0.9n=0.8n=0.7n=0.6

0 500 1000 1500 2000 2500 30000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2n=1n=0.9n=0.8n=0.7n=0.6

0 500 1000 1500 2000 2500 300010

12

14

16

18

20

22

24

26

28

30n=1n=0.9n=0.8n=0.7n=0.6

0 500 1000 1500 2000 2500 30000

50

100

150

200

250

300n=1n=0.9n=0.8n=0.7n=0.6


The neutral curves show that decreasing the power-law index has a stabilising effect on

the boundary-layer flow. The value of the critical Reynolds number is increased on both

the upper and lower lobes as n decreases, as shown in Tables 6.4 and 6.5, respectively.


n R α β β φ ()

1.0 498.02 0.3711 0.0751 18.69 11.44

0.9 607.35 0.3467 0.0725 22.03 11.86

0.8 757.94 0.3234 0.0701 26.55 12.22

0.7 971.51 0.3021 0.0677 32.87 12.63

0.6 1285.16 0.2820 0.0651 41.83 13.00


n R α β β φ ()

1.0 727.47 0.1364 0.0463 16.82 18.73

0.9 940.03 0.1195 0.0415 19.52 19.17

0.8 1248.19 0.1047 0.0371 23.15 19.51

0.7 1710.82 0.0911 0.0328 28.04 19.79

0.6 2439.25 0.0797 0.0288 35.10 19.87


Case 4 (θ = 40)

At θ = 40, Figure 6.4 presents the neutral curves for values of the power-law index

0.6 ≤ n ≤ 1.0. These curves enclose a region in which the boundary-layer is convectively

unstable.


0 500 1000 1500 2000 2500 30000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1n=1n=0.9n=0.8n=0.7n=0.6

0 500 1000 1500 2000 2500 30000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2n=1n=0.9n=0.8n=0.7n=0.6

0 500 1000 1500 2000 2500 300010

12

14

16

18

20

22

24

26

28

30n=1n=0.9n=0.8n=0.7n=0.6

0 500 1000 1500 2000 2500 30000

50

100

150

200

250

300n=1n=0.9n=0.8n=0.7n=0.6





n R α β β φ ()

1.0 353.40 0.3583 0.0721 16.37 11.37

0.9 430.09 0.3334 0.0691 19.09 11.70

0.8 536.45 0.3078 0.0656 22.63 12.04

0.7 689.78 0.2838 0.0621 27.54 12.35

0.6 921.73 0.2597 0.0580 34.35 12.59



n R α β β φ ()

1.0 494.25 0.1368 0.0452 14.34 18.26

0.9 644.67 0.1185 0.0401 16.61 18.68

0.8 862.66 0.1017 0.0351 19.46 19.03

0.7 1198.38 0.0869 0.0303 23.32 19.21

0.6 1745.18 0.0734 0.0255 28.63 19.16


Case 5 (θ = 50)


from n = 0.6 − 1 in increments of 0.1 . These curves enclose a region in which the


0 500 1000 1500 2000 2500 30000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1n=1n=0.9n=0.8n=0.7n=0.6

0 500 1000 1500 2000 2500 30000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2n=1n=0.9n=0.8n=0.7n=0.6

0 500 1000 1500 2000 2500 300010

12

14

16

18

20

22

24

26

28

30n=1n=0.9n=0.8n=0.7n=0.6

0 500 1000 1500 2000 2500 30000

50

100

150

200

250

300n=1n=0.9n=0.8n=0.7n=0.6






n R α β β φ ()

1.0 260.29 0.3404 0.0679 13.53 11.27

0.9 317.22 0.3154 0.0642 15.44 11.19

0.8 387.14 0.2866 0.0594 17.62 11.71

0.7 494.18 0.2577 0.0539 20.39 11.80

0.6 662.26 0.2289 0.0476 24.14 11.74


n R α β β φ ()

1.0 340.41 0.1385 0.0438 11.42 17.54

0.9 448.81 0.1165 0.0379 13.98 18.04

0.8 578.51 0.1022 0.0329 14.56 17.82

0.7 749.64 0.0876 0.0277 15.90 17.54

0.6 921.18 0.0782 0.0235 16.56 16.70


Case 6 (θ = 60)


from n = 0.6 − 1 in increments of 0.1 . These curves enclose a region in which the



0 500 1000 1500 2000 2500 30000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1n=1n=0.9n=0.8n=0.7n=0.6

0 500 1000 1500 2000 2500 30000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2n=1n=0.9n=0.8n=0.7n=0.6

0 500 1000 1500 2000 2500 300010

12

14

16

18

20

22

24

26

28

30n=1n=0.9n=0.8n=0.7

0 500 1000 1500 2000 2500 30000

50

100

150

200

250

300n=1n=0.9n=0.8n=0.7n=0.6




the upper and lower lobes as n decreases, as shown in Tables 6.10 and 6.11, respectively

n R α β β φ ()

1.0 188.72 0.3151 0.0622 10.17 11.17

0.9 228.98 0.2940 0.0589 11.68 11.33

0.8 279.70 0.2670 0.0527 12.77 11.17

0.7 346.08 0.2397 0.0453 13.58 10.70

0.6 516.92 0.2169 0.0386 17.29 10.10



n R α β β φ ()

1.0 216.68 0.1466 0.0429 8.05 16.32

0.9 268.28 0.1244 0.0372 8.63 16.64

0.8 318.92 0.1054 0.0312 8.63 16.51

0.7 349.51 0.0918 0.0260 7.86 15.80

0.6 341.11 0.0974 0.0224 6.61 12.93


Case 7 (θ = 70)

Figure 6.7 presents the neutral curves at θ = 70 for values of the power-law index ran-

ging from n = 0.6− 1 in increments of 0.1. These curves enclose a region in which the


0 500 1000 1500 2000 2500 30000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1n=1n=0.9n=0.8n=0.6

0 500 1000 1500 2000 2500 30000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2n=1n=0.9n=0.8n=0.6

0 500 1000 1500 2000 2500 3000

10

12

14

16

18

20

22

24

26

28

30 n=1n=0.9n=0.8n=0.6

0 500 1000 1500 2000 2500 30000

50

100

150

200

250

300n=1n=0.9n=0.8n=0.6



the boundary-layer flow. The value of the critical Reynolds number is increased on the

lower lobes as n decreases, as shown in Tables 6.10.

6.2 Growth rates 90

n R α β β φ ()

1.0 91.29 0.1984 0.0489 4.20 13.85

0.9 91.50 0.1978 0.0480 4.12 13.63

0.8 77.58 0.2100 0.0470 3.42 12.61

0.6 97.60 0.1460 0.0243 2.22 9.44

Table 6.12: The values of the critical Reynolds number R, wavenumbers α , azimuthalwave-number, β and the angle of spiral vortices φ corresponding to decreasing values ofn on the lower-branch for stationary vortices at latitude θ = 70.

6.2 Growth rates

We now consider the growth rates of the instability modes of power-law of boundary layer

flows. The growth rates of the Type I and Type II instability within the power-law fluid

at R = Rc + 25 are presented in Figure 6.8 and Figure 6.9 respectively as a function of

the vortex number β . Here Rc is the critical Reynolds number presented in Tables 6.1,

6.2, 6.4, 6.6, 6.8 and 6.10 for the onset of the crossflow instabilities and those presented

in Tables 6.9, 6.11 and 6.12 for streamline-curvature instabilities. Figure 6.8 and Figure

6.9 clearly reveal a stabilising effect on the Type I mode of latitudes 10, 20, 30, 40,

50 and 60 and type II mode of latitudes 50, 60 and 70 as the value of the maximum

growth rate indicated by a red dot decreases when the power-law index n decreases. In

addition, the location of the maximum growth rate shifts to higher values of β , indicating

an overall effect of an increase in the number of vortices as n decreases at latitudes of

θ = 10 − 60 in ten degree increments. For n = 0.6 at latitude θ = 60, the type I

mode has a significantly delayed onset and so the critical results are presented for Type II

only. For Type II mode, Figure 6.9 shows that the point of maximum amplification shifts

to lower values of β , suggesting a reduction in the number of vortices at θ = 60 and

θ = 70.

6.2 Growth rates 91

20 25 30 35 40 45 50 550

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016n=1n=0.9n=0.8n=0.7n=0.6

(a) θ = 10

15 20 25 30 35 40 45 50 550

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016n=1n=0.9n=0.8n=0.7n=0.6

(b) θ = 20

15 20 25 30 35 40 45 500

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016n=1n=0.9n=0.8n=0.7n=0.6

(c) θ = 30

10 15 20 25 30 35 40 450

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016n=1n=0.9n=0.8n=0.7n=0.6

(d) θ = 40

12 14 16 18 20 22 24 26 28 300

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016n=1n=0.9n=0.8n=0.7n=0.6

(e) θ = 50

8 10 12 14 16 180

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016n=1n=0.9n=0.8n=0.7

(f) θ = 60

Figure 6.8: Growth rates for Type I mode for shear-thinning power-law fluids for decreas-ing values of n at latitudes of θ = 10−60.

6.3 Conclusion 92

10 15 20 25 30 35 40 45 50 550

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045n=1n=0.9n=0.8n=0.7n=0.6

(a) θ = 50

4 6 8 10 12 14 16 18 20 220

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045n=1n=0.9n=0.8n=0.7n=0.6

(b) θ = 60

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045n=1n=0.9n=0.8n=0.6

(c) θ = 70

Figure 6.9: Growth rates for Type II mode for shear-thinning power-law fluids for de-creasing values of n at latitudes of θ = 50−70.

6.3 Conclusion

In this chapter we have investigated the convective instability of stationary vortices relat-

ive to the sphere surface when the power-law index n decreases and the analysis moves

from the pole toward the equator. The linear stability analysis revealed that both crossflow

and streamline curvature instabilities occur in the boundary layer over rotating sphere for

all power-law index values at all latitudes except at θ = 70. The neutral curves show that

decreasing the power-law index has a stabilising effect on the boundary-layer flow. The

value of the critical Reynolds number is increased on both the upper and lower lobes as

n decreases. At the onset of crossflow instabilities, the number of spiral vortices is pre-

dicted to increase with decreased the power-law index n at all latitudes θ = 10−60. An

increasing of spiral vortices were predicted as well at the onset of streamline-curvature in-

stabilities except at θ = 60 and θ = 70 where a reduction in the number of vortices can

be seen. At the onset of instabilities the stationary vortices were predicted to have vortex

6.3 Conclusion 93

angles which increase as the power-law index n decreases at latitudes θ = 10−50 and

θ = 10−40 at the onset of crossflow instabilities and streamline-curvature instabilities

respectively, our results revealed a decreasing in vortex angles at latitudes θ = 60 and

70 and θ = 50 and 60 at at the onset of crossflow instabilities and streamline-curvature

instabilities respectively. Roughly speaking, on both the Type I and II modes the onset

of linear convective instability is delayed as n decreases, the critical Reynolds number

is increased and the linear convective growth rates are significantly reduced, effectively

expanding the region of stable flow.

Chapter 7

Conclusion

This thesis consists of a number of chapters presenting both base flow solutions and lin-

ear stability analyses for power-law non-Newtonian fluids over a rotating sphere. In the

second chapter we investigate and derive the non-dimensionalised governing continu-

ity and Navier-Stokes equations of the laminar boundary layer flow of power-law non-

Newtonian fluids of a rotating sphere. The perturbation method of Howarth and Banks is

extended in Chapter Three to investigate the boundary layer flow engendered by a rotating

sphere numerically. The fourth, fifth and sixth chapters are dealing with the stability ana-

lysis of a three dimensional boundary layer problem of power-law non-Newtonian fluids

over a rotating sphere.

7.1 Completed work

This thesis focuses on disturbances in the incompressible power-law non-Newtonian flow

over the surface of rotating spheres, and we have presented both base flow solutions and

linear stability analyses for non-Newtonian fluid model. Consistent with all previous in-

vestigations of the boundary layers over rotating spheres, in our analysis through this

thesis the sphere is considered to rotate in a fixed frame of reference, which is in contrast

to the rotating-disk investigations of Griffiths et al. (2014a). For the laminar boundary

layer flow, we have non-dimensionalised the governing continuity and momentum equa-

7.1 Completed work 95

tions. After a suitable transformation of coordinate variables, the velocity components

are expressed in series of the angle measured from the axis of rotation.We describe the

velocity functions of the coefficients of the velocity components by ordinary differential

equations with the power-law flow index as the only parameter. The viscosity charac-

teristics of the flow are governed by the power-law index n and the viscosity function

for shear-thinning power-law fluids is unbounded in the far-field of the boundary-layer.

Our shear-thinning power-law solutions are identical to those of Banks (1965) when the

power-law index n = 1. The first-order velocity functions are corresponding to the case

of rotating disk presented by Griffiths et al. (2014a) and they are the limiting solutions

at the poles of the rotating sphere. The general shape of velocity functions profiles are

similar for different values of n and convergence is quite satisfactory. The computed

mean flow profiles when n = 1 are very similar to Garrett’s base flow profiles. Not-

ably, the velocity components extend to larger η for smaller values of power-law index n.

Therefore, the boundary layer thickness increases as the flow index decreases. The local

convective instability properties of the boundary layers of power-law fluids have been

studied at various latitudes along the sphere using linear-stability theory and the Briggs-

Bers criterion. Firstly, we perturbed the basic flow and derived the linear disturbance

equations dependent on the relevant non-Newtonian parameters. Clearly our results in

the Newtonian limit are in excellent agreement with the perturbation equations of Newto-

nian flow on rotating sphere presented by Garrett and Peake (2002) when the power-law

index n = 1. Additional viscous terms appeared due to the generalised Newtonian for-

mulation of the problem. We solved the resulting perturbation equations using a spectral

method by Chebyshev polynomials scheme and present solutions for the onset of instabil-

ities in terms of the wavenumber and wave angle. Convective instability of Newtonian

boundary layers flow over a rotating sphere were studied theoretically and experimentally

by many investigators such that Taniguchi et al. (1998) and Garrett and Peake (2002),

and the corresponding convective instability problem of power-law non-Newtonian fluid

from a rotating disk was also investigated theoretically by Griffiths et al. (2014a). How-

ever, to the author’s knowledge the laminar boundary layer and convective instability of

7.1 Completed work 96

power-law non-Newtonian fluids over rotating spheres have not been studied theoretically

or experimentally. The convective instability analyses conducted in this thesis using an

approach that assumes the disturbances to rotate with some fixed multiple of the sphere

surface speed, so we predicted the critical Reynolds numbers and the number and orient-

ation of the spiral vortices for power-law index n = 1−0.6 in 0.1 decrement at latitudes

θ = 10−70 in ten degree increment. The results obtained by assuming stationary vor-

tex for the power-law non-Newtonian boundary-layer flow with assumed vortex speed of

c = 1 give the existing results of Newtonian flow presented by Garrett and Peake (2002)

when the power-law index n = 1, and to existing results for the power-law non-Newtonian

fluids on rotating disk presented by Griffiths et al. (2014a) for all power-law index values

as the analysis moves towards the pole. We used a parallel-flow approximation in the

analyses of the rotating sphere as we were restricted to the local stability characteristics

of the flow. We investigated the growth rates of the both Type I and II instability modes

to present the effects of power-law fluids which revealed a well agreement between the

results of neutral curves and the growth rates. Note that this prediction has been made for

stationary vortices, ωr = Rsinθ , and therefore is not a complete instability analysis. A

complete investigation of the absolute instability mechanism requires a highly intensive

study beyond the scope of this thesis. Nevertheless, results can give credibility to the idea

that the power-law results indicate boundary-layer stabilising effect on both the Type I for

θ = 10−60, and Type II modes for θ = 10−70 in ten degree increment in terms of the

critical Reynolds number and the region of instability from the upper and lower branches.

However, in the context of the rotating sphere model there are currently no experimental

investigations that would support this claim. In summary, in this thesis we have shown

that it is possible to model the instability mechanisms associated with the boundary-layer

flow due to a rotating sphere when considering power-law non- Newtonian fluids.

7.2 Further work 97

7.2 Further work

There is a great deal of scope for future work with regards to this research. For instance, it

would be an interesting extension of Griffiths (2016) investigation to consider the Carreau

model over a rotating sphere and show the laminar flow profiles. Another possible exten-

sion would be to solve the energy balance equation and establish the underlying effects

of decreasing the power-law index on the boundary layer flows over a rotating sphere. In

addition, there is a considerable amount of work remaining to study the absolute instabil-

ity for power-law fluids over a rotating sphere. Previous studies of absolute instability has

been performed for Newtonian flows by Garrett (2002). Chapters 4 and 5 could be gen-

eralized to observe the transition points associated with the onset of absolute instability

and predict the onset of the latitudinal absolute instability which will give an insight into

the onset of turbulence within the boundary layers. This work clearly motivates the need

for detailed experimental results with which to compare our theoretical analysis. To the

best of the author’s knowledge, there is no such experimental study that could confirm the

main findings of this thesis, currently experiments will be required in order to investigate

the validity of most of the predictions made through this thesis. The convective instability

characteristics of the power-law non-Newtonian fluids are investigated here. The con-

vective instability characteristics of the other type of generalised Newtonian fluid models,

Carreau fluids, could be investigated to compare with shear-thinning power-law results

found in this study. Finally, another possible area of future work would be to investigate

the instability mechanisms within the power-law of non-Newtonian of boundary layers

flow over the outer surface of rotating cones.

Appendix A

Derivation of the steady-flow and

disturbance viscosity functions

In this appendix, the viscosity function in the non-dimensional form and the calculations

for adding small perturbing quantities to the mean flow velocities to generate the disturb-

ance viscosity function are presented. The dimensional viscosity function is

µ∗ = m∗

[(∂U∗

∂ r∗

)2

+

(∂V ∗

∂ r∗

)2](n−1)/2

. (A.1)

Using the non-dimensional mean-flow variables (2.13) presented in §2.3 Equation

A.1 becomes

µ∗ = m∗

[(a∗Ω∗

1/A∗

)2(∂U∂η1

)2

+

(a∗Ω∗

1/A∗

)2(∂V ∗

∂η1

)2](n−1)/2

,

µ∗ = m∗ [a∗Ω

∗A∗](n−1)

[(∂U∂η1

)2

+

(∂V ∗

∂η1

)2](n−1)/2

.

Since

µ =µ∗

m∗ (a∗Ω

∗A∗)1−n ,

99

we have

µ =

[(∂U∂η1

)2

+

(∂V∂η1

)2](n−1)/2

.

Now η is an appropriate coordinate given by Equation (3.2) , we have

µ =

[((θ(1−n)/(n+1)

)∂U∂η

)2

+

((θ(1−n)/(n+1)

)∂V∂η

)2](n−1)/2

,

µ = θ(1−n)(n−1)

n+1

[(∂U∂η

)2

+

(∂V∂η

)2](n−1)/2

. (A.2)

To derive the perturbation viscosity we use the perturbed flow components, each of those

components are decomposed into dimensionless mean flow and perturbation parts

µpert = θ(1−n)(n−1)

n+1

[(∂ (U +u)

∂η

)2

+

(∂ (V + v)

∂η

)2](n−1)/2

. (A.3)

µpert = θ(1−n)(n−1)/(n+1)

[(∂U∂η

)2

+

(∂V∂η

)2

+2(

∂U∂η

∂u∂η

+∂V∂η

∂v∂η

)](n−1)/2

= θ(1−n)(n−1)/(n+1)

[(∂U∂η

)2

+

(∂V∂η

)2]n−1/2

1+2[(

∂U∂η

)2+(

∂V∂η

)2] (∂U

∂η

∂u∂η

+∂V∂η

∂v∂η

)(n−1)/2

. (A.4)

Using the binomial theorem, the perturbation viscosity function (A.4) becomes

µpert = µ

1+(n−1)[

(U ′)2 +(V ′)2] (U ′ ∂u

∂η+V ′ ∂v

∂η

)(n−1)/2

,

µpert =

µ +(n−1)µU ′[(U ′)2 +(V ′)2

] ∂u∂η

+(n−1)µV ′[(U ′)2 +(V ′)2

] ∂v∂η

,

100

µpert =

[µ + µ

∂u∂η

+ µ∂v∂η

].

where

µ =(n−1)µ[

(U ′)2 +(V ′)2] = θ

(1−n)(n−1)n+1 (n−1)

[(U ′)2

+(V ′)2

](n−3)/2.

The derivatives of the viscosity and disturbance viscosity functions are given by

∂ µ

∂η= µ

′,

∂ µ

∂η=

(n−3)µ ′[(U ′)2 +(V ′)2

] ,∂ µ

∂θ=

1−n1+n

η

θµ′,

∂ µ

∂θ=

1−n1+n

η

θ

(n−3)µ ′[(U ′)2

+(V ′)2] .

Appendix B

Neutral Curves

In this appendix, in similar way to Garrett (2002) we present the neutral curves of con-

vective instability in the(RS, β

)−plane. Here RS =R2 is the spin Reynolds number and β

is the number of vortices from (4.8). Figure B.1 shows our results in the(RS, β

)−plane

when n = 0.9, 0.8, 0.7 and 0.6

104 105 106 1070

50

100

150

200

250

300=10=20=30=40=50=60=70

104 105 106 1070

50

100

150

200

250

300=10=20=30=40=50=60=70

105 106 1070

50

100

150

200

250

300=20=30=40=50=60

104 105 106 1070

50

100

150

200

250

300=20=30=40=50=60=70

Figure B.1: The neutral curves of convective instability in the(RS, β

)−plane for station-

ary vortices of θ = 10−70 (right to left) when n = 0.9, 0.8, 0.7 and 0.6.

Figure B.2 shows our results in the(RS, β

)−plane at latitudes of θ = 10−70.

102

1070

50

100

150

200

250

300n=1n=0.9n=0.8n=0.7n=0.6

106 1070

50

100

150

200

250

300n=1n=0.9n=0.8n=0.7n=0.6

106 1070

50

100

150

200

250

300n=1n=0.9n=0.8n=0.7n=0.6

106 1070

50

100

150

200

250

300n=1n=0.9n=0.8n=0.7n=0.6

105 106 1070

50

100

150

200

250

300 n=1n=0.9n=0.8n=0.7n=0.6

105 106 1070

50

100

150

200

250

300n=1n=0.9n=0.8n=0.7n=0.6

104 105 106 1070

50

100

150

200

250

300n=1n=0.9n=0.8n=0.6

Figure B.2: The neutral curves of convective instability in the(RS, β

)−plane for station-

ary vortices of θ = 10−70 (right to left).

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