AN ABSTRACT OF THE THESIS OF
Nicholas C. Vlessis
Mathematics
for the degree of
Master of Science
in
presented on March 8, 1984.
Title: Nonlinear Stability of Couette
Flows of Micropolar Fluids
Abstract approved:
Redacted for Privacy
Professor M. N. L. Narasimhan
After deriving the closed-form solution
for steady, laminar plane
Couette and rotational Couette flows of a micropolar
fluid, these two
basic flows are altered by a finite two-dimensional
and a finite axisymmetric disturbance, respectively.
from which the disturbance
Disturbance equations are derived,
energy integrals are found.
Then, utilizing
the solutions of the linearized disturbance
equations, the amplitude
equations are derived, in accordance with
the procedures of the Stuart
energy method.
An expression for the marginal stability
surface is
formulated, and expressions for the critical numbers
R
c
,
R
gc
,
and R
kc
are given.
An elucidation of the flow mechanisms, induced
to deal with the
energies imparted by the disturbance
on the basic flow, is given.
The
new concepts of swirl, microenergy of rotation,
and mean couple stress,
are explained during the physical interpretation of
the disturbance
energy equations.
Also, inequalities describing flow stability (or
instability) are presented.
The numerical procedures, needed to quantitatively sustantiate
the qualitative non-linear stability analysis of this thesis, are
outlined.
Nonlinear Stability of Couette Flows
of Micropolar Fluids
by
Nicholas C. Vlessis
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Master of Science
Completed March 8, 1984
Commencement June 1984
APPROVED:
Redacted for Privacy
Professor of Mathematics in charge of major
Redacted for Privacy
Head of Department of Mathematics
Redacted for Privacy
Dean of Graduate Sc
of
0
Date thesis is presented
Typed by and for
March 8, 1984
Nicholas C. Vlessis
ACKNOWLEDGEMENTS
I wish to acknowledge the patient guidance of my mentor, Prof.
M. N. L. Narasimhan.
The excellent courses, he taught, on classical
continuum mechanics, and on micropolar and non-local continuum physics,
inspired me to do my Master's thesis.
The countless hours of enlight-
ening discussion with Prof. Narasimhan are represented by this thesis.
The financial support provided by Oregon State University, in the
form of a graduate teaching assistantship, was gieatly appreciated.
Special thanks are extended to Prof. P. M. Anselone, mathematics department chairman, and to Prof. B. E. Petersen, graduate committee
chairman, for their support.
I, also, appreciated having access to
the essential materials in Kerr Library.
I dedicate this thesis to my parents, Christopher and Rosina, to
my brothers, Angelo and Damon, and to my sister, Lisa.
TABLE OF CONTENTS
I.
II.
III.
IV.
V.
INTRODUCTION
1
I.1
Why Micropolar Theory
6
1.2
Balance Laws of Micropolar Theory
7
1.3
Field Equations -- Micropolar Fluid Dynamics
10
1.4
Basic Assumptions and the Surmised Field Equations
11
BASIC PLANE COUETTE FLOW
12
II.1
Geometry of Plane Couette Flow
12
11.2
Field Equations -- Rectangular Coordinates
15
11.3
Laminar Plane Couette Flow
16
BASIC ROTATIONAL COUETTE FLOW
20
III.1
Geometry of Rotational Couette Flow
20
111.2
Field Equations -- Cylindrical Coordinates
21
111.3
Laminar Rotational Couette Flow
23
STABILITY OF A BASIC PLANE COUETTE FLOW
27
IV.1
Linear Stability Analysis
28
1V.2
Imposition of Finite Disturbances on a Basic Flow
36
IV.3
Disturbance Energy Equations
42
1V.4
Physical Interpretation of the MOS-Energy Equations
44
1V.5
Amplitude Equations
48
IV.6
Criticality
56
IV.7
Determination of the Constant, A
58
STABILITY OF A BASIC ROTATIONAL COUETTE FLOW
62
V.1
Linear Stability Analysis
63
V.2
Imposition of Finite Disturbances on a Basic Flow
69
V.3
Disturbance Energy Equations
75
V.4
Physical Interpretation of the MOS-Energy Equations
78
V.5
Amplitude Equations
82
V.6
Criticality
91
VI.
NUMERICAL PROCEDURES -- PLANE COUETTE FLOW
VI.1
Algorithm to Determine X
95
VI.2
Ratios of Nondimensional Numbers
96
VI.3
Graphs of the Laminar Velocity and Microgyration
97
Fields
VII.
94
VI.4
Determining the Functions
VI.5
Values for the Coefficients of the Amplitude Equations
SUMMARY AND CONCLUSIONS
(Or,
(Pi,
(Pr, and (Pi
99
100
102
VII.1
Discussion of the Results
102
VII.2
Scope of Further Work
104
ENDNOTES
106
BIBLIOGRAPHY
109
APPENDICES
Appendix A
112
Appendix B
116
LIST OF FIGURES
Page
Figure
6.1
Steady, laminar velocity for a micropolar fluid
97
6.2
Steady, laminar microgyration for a micropolar fluid
98
NONLINEAR STABILITY OF COUETTE FLOWS
OF MICROPOLAR FLUIDS
I.
INTRODUCTION
The immediate objective of the theory of hydrodynamic
stability
is to understand the mechanisms of instability in
laminar flow and to
obtain criteria for its occurence.
A more fundamental objective is to
understand how, and under what circumstances, turbulence
may arise from
laminar instability.
All the possible transitions of a flow profile,
from the placid patterns of simple laminar flows,
to the chaotic com-
plexity of highly turbulent flows, should be elucidated.
Thus, because
of the inherent non-linearity of the equations of motion
governing a
hydrodynamical system, this ambitious pursuit should
employ nonlinear
stability analysis.
1
The mathematical problem of nonlinear hydrodynamic
stability can
be formulated, by taking a given steady-state solution of
the equations
of motion, and superimposing a disturbance of a suitable kind; this
results in a set of nonlinear 'disturbance' equations which
govern the
behavior of the disturbance.
If the solution of the equations shows
that any disturbance ultimately decays to
zero, the flow is said to be
temporally stable; whereas if the disturbance can be permanently different from zero, the flow is unstable.
Note that instability of a
laminar flow does not always imply turbulent motion, but very often
results in another (possibly more complex) form of laminar motion.
2
Some preliminary insight is gained when infinitesimal disturbances
are considered.
For these disturbances of small amplitude, the solution
of the disturbance differential equations is simplified (in fact, the
governing equations are linearized).
However, the initial growth of
the disturbance only can be determined in most problems.
On the basis
of this linearized theory, it is possible to consider disturbances which
contain an exponential time factor of the form
exp(kt), t being the
The boundary conditions on the disturbance equations require the
time.
vanishing on the boundaries of all disturbance quantities like disturbance velocity components and disturbance microgyration components,
relative to the boundaries.
Consequently, the boundary conditions are
homogeneous, and there arises an eigenvalue problem for the determination of possible eigenvalues, k.
In this (linear) case, if k has a
positive real part, the flow is unstable; otherwise, the flow is stable.
To comprehend more than just the initial growth of the disturbance,
requires that the disturbance be of finite amplitude, in that, a finite
disturbance.
Features of the nonlinear terms of the equations of motion
can now be studied.
Furthermore, a clarification of the connection be-
tween linear and nonlinear theories can be assessed.
In cases of instability of fluid flow, the disturbance is usually
periodic in at least one spatial direction.
Thus, it is convenient to
take averages with respect to one of the spatial dimensions, and also
to separate the flow into a mean part and a disturbance part
(with zero
mean).
Now, consider a flow, with local non-dimensional parameters
(classically, a Reynolds number) that do not vary, as in the case of
flows between parallel plates or coaxial, rotating cylinders.
A
3
synopsis of nonlinear stability then reports as follows.
Initially, a
disturbance, superimposed on a given laminar flow,
grows exponentially
with time according to the linear theory; but
eventually it reaches
such a size that the transport of momentum and
microinertia by the
finite flucuations become appreciable and the associated
mean stress
(Reynolds stress) and the associated mean couple
stress then have a
significant effect on the mean flow.
This distortion of the mean flow
modifies the rate of transfer of energies from
the mean flow to the
disturbance, and since this energy transfer is the cause of
the growth
of the disturbance, there is a modification of
the rate of growth of
the disturbance.
An equilibrium state may be possible, in which the
rate of transfer of energies from the distorted mean flow to the
disturbance, balances
precisely the rate of viscous dissipations of the energy of the
disturbance.
In such an equilibrium state, the disturbance will have
a definite finite amplitude and the mean flow will be distorted
from
its original laminar form.
An example of an equilibrium state of this
kind occurs between coaxial, rotating cylinders, when the
instability
is observed to take the form of cellular, toroidal
vortices (Taylor
vortices) spaced regularly along the axes of the cylinders
(Coles,
1965).
Elucidation of this curious phenomenon of Taylor vortices will
be a prize for nonlinear hydrodynamic stability theory.
The philosophy of purpose for this thesis is encoded in
the following quotation.
"Nonlinear hydrodynamic stability theory is really concerned, ultimately, with phenomena such as transition to turbulence.
In practice, however, that phenomenon is so complex
as to defy rational understanding at the present time. A
4
more limited objective is that of gaining some understanding
of nonlinear processes in fluid mechanics, perhaps with reference to the early, relatively-simple stages of the evolution
of laminar flow to turbulence. Even then, the mathematical
(Stuart,l977)
problems posed are challenging enough."
This thesis employs micropolar fluid dynamics to the problems of
flows of micropolar fluids between two parallel plates and between two
coaxial, rotating cylinders.
Closed-form solutions to these two pro-
blems are obtained for laminar flow, and are presented in chapters II
and III.
An understanding of nonlinear processes in fluid mechanics
is gained, with reference to the early, relatively-simple stages of the
evolution of laminar flow to turbulence.
Chapters IV and V deal, respectively, with two-dimensional disturbance plane Couette flow and axisymmetric disturbance Couette flow.
These chapters study the basic flow of an incompressible viscous fluid
that is altered by a finite disturbance flow.
The resulting flow must
satisfy the equations of motion and the same boundary conditions as the
basic flow, but the disturbance flow is otherwise arbitrary.
The anal-
ysis employs the procedures of the Stuart energy method to study the
time-rate of change of the disturbance energies.
By determining what
becomes of the energy imparted to the basic flow by the disturbance
flow, we begin to unravel such complex phenomena as transition to turbulence.
The linear theory of micropolar fluid dynamics,for plane Couette
and rotational Couette flows, is briefly pursued, so that the micropolar analog of the Orr-Sommerfeld energy equations can be derived.
The solution to these equations is assumed to be the spatial form (shape)
of the superimposed nonlinear disturbance.
5
Returning to nonlinear stability analysis, the
disturbance energy
equations are derived; and a physical interpretation of
the mechanisms
induced to deal with the energies of the disturbance
flow are discussed.
The new concepts of swirl, microenergy of rotation, and
mean couple
stress are introduced, because of micropolar theory,
into the discussion.
Amplitude equations are next derived from the corresponding
nonlinear disturbance energy equations.
nonlinear stability
Theoretical predictions concerning
2
are finally presented, which include implicit
values for the critical numbers, and marginal stability
surfaces.
Con-
sequently, we establish the threshold of nonlinear stability
of Couette
flows of micropolar fluids.
In the sequel, equations are labelled as
(f.g.h), where f corre-
sponds to the chapter, g to the section of the f:th chapter,
and h to
the h:th equation in the g:th section. (Terms in the
text, with a
superscript number a, will indicate that further
elucidations can be
found in the endnote with that number a.)
The covariant derivative of a contravariant vector is given by
k
k
1
v
= 31.7 /x
+ vs
;1
sk 1
k
s
is a Christoffel symbol of the second kind relative to the spatial
curvilinear coordinate system
x
1
.
Similarly, the partial derivative of a contravariant vector is
denoted by
v
k
1
=
k
1
Dv /bc
,
and so forth for higher order partial derivatives.
The material time derivative of a spatial vector
f
m
m
= Df
Dt
(?5,t)
=
m
+
vk fm
;k
fm(x,t)
is defined as
6
1.1
Why Micropolar Theory
The point is that, it is not a point.
Non-polar continuum
theories embellish this approximation that the
constituent objects
being mathematically modelled are "material points".
These zero-
dimensional points, comprising a theoretical material, enable
a non-
polar theory to conveniently ignore the existence of
body couples and
couple stresses
3
.
Such ignorance waned, when A. C. Eringen published
a microcontinuum theory (Eringen,1964).
Eringen derived the basic equations of microcontinuum
theory, with
deformable vectors, now assigned to each material point.
he called micromorphic theory.
This theory,
Thus, in the micromorphic theory, each
material point can translate, independently
rotate, and/or deform,
In 1966, Eringen elucidated a special case of the micromorphic
theory, called the micropolar theory (Eringen,1966).
The micropolar
theory allows each material point the freedom to translate,
independently rotate, but not to deform.
In this theory the 'material points'
are considered to be 'geometrical points' that possess properties
similar to rigid particles.
Moreover, this polar theory can recognize
the existence of body couples and couple stresses.
A large class of real materials of great physical importance
are
known to be composed of a substructure with tiny aggregates of
molecules which can be considered very nearly rigid.
Some examples of such
micropolar materials include fibrous and granular media like wood and
wood composites, solid rocket propellant grains, colloidal
suspensions,
7
animal blood, liquid crystals, and polymeric fluids.
We begin our study by stating the balance laws and constitutive
equations of the micropolar theory, and then deriving the governing
for micropolar fluid dynamics.
field equations
1.2
Balance Laws of Micropolar Theory
The balance laws of micropolar media (Eringen,1976) are given
locally as follows.
Conservation of mass for the body is stated as
Conservation of Mass:
usual by
p/at.
+
t is time,
where
=
(pvk)
(1.2.1)
0,
p is mass density per unit volume, and v
k
is the
velocity vector.
Conservation of Micromoment of Inertia (Microinertia):
Conservation
of microinertia is an entirely new balance law which is stated as
k1
3
m
.k1
/at + 3
v
1
.km
v
- 3
.m1
-
v
k
= 0,
(1.2.2)
;In
jkl is the microinertia tensor, and
where
Conservation of Linear Momentum:
kl
v
is the gyration tensor.
The time-rate of change of the momen-
tum of a material body is equal to the total force acting on the body.
Mathematically, this is expressed as
lk
+
t
where
mass.
t
lk
p(f
k
k
- v )
(1.2.3)
= 0,
is the stress tensor and
k
f
is the body force per unit
8
The time-rate of change of moment
Conservation of Moment of Momentum:
of momentum of the body is equal to the total torque acting on the body.
This is expressed as
m
rk
m
where
k
2,
is
kl
+ e
klm
t
;r
rk
+ p(2,
lm
k
- a )
k
klm
s the couple stress tensor, e
is
the body couple per unit mass,
(1.2.4)
= 0,
and
a
k
is the alternating tensor,
is the inertial spin vector.
is the inertial spin tensor.)
(a
The time-rate of change of the total energy
Conservation of Energy:
of the body is equal to the rate of working of the external loads and
the heat energy.
pc = t
kl
(v
Mathematically,
+ v
1;k
kl
v
+ m
)
kl
k
+ ph,
(1.2.5)
;k
c is the internal energy density per unit mass, v 1 is the ang-
where
is the heat vector directed out of the body,
ular velocity vector, qk
h
and
+ q
1;k
is the heat source per unit mass.
We have listed above only the five balance laws of micropolar theory,
with which we are concerned.
From the linear theory of isotropic micropolar fluids, the
following constitutive equations are derived (Eringen,1976):
t
D
kl
m
kl
= A vm
v
= ae
;m
klm
8,
g
kl
m
+ (211.. + Kv)dkl
m
+ avv
kl
1;m
qk
Ke'k
Xv, Pv, Kw, Ur, 3.,
g
eklmv
Ku(v1;k
+ a ,
v
+
k;1
v
(1.2.6)
)
(1.2.7)
1;k
(1.2.8)
eklmv
K, a, and R are the viscosity coefficients.
9
is the deformation rate tensor, and
8 is absolute temperature, d_
kl
is the metric tensor.
functional.
where
The carat
g
symbolizes a constitutive response
For example, the dissipative stress tensor
t =
I is the identity tensor, t is the stress tensor, and
+ t
it is the
thermodynamic pressure.
Eringen also derives that the viscosity coefficients must obey
the following inequalities:
34-
+ 211,
+ Kir
> 0,
21.11
+
KU.
> 0,
Ky.
> 0,
(1.2.9)
3av
Yr
> 0,
Yv- + 13.v.
> 0,
(a
V0)2
Because of the classical limit, we also assume
(Eringen,1966).
kl
< 2K(yv
u rr
>0
130/e.
10
1.3
Field Equations -- Micropolar Fluid Dynamics
Inserting the constitutive equations (1.2.6) - (1.2.8) into
the balance laws yields the nine field equations of Micropolar
Fluid Dynamics (Eringen, 1976) for constant viscosity coefficients,
err, p,, Ku, au, (3u, yv, K, a, and R.
+ V-(pv)
3p/Bt
jkl
.k1
ajkl
= 0
v
m
+ (e
kmr
(av
p88 D2T/82
kl
v1;k
+ M
T = e
Here
On
lmr
.k
3
)v
m
r
= 0
+
(1.3.3)
- yfr.VXVX.X.,),
KvVXV
(1.3.4)
= 0
+ p(2, - 2)
- 2Kuv
+ e
= 0
p(f - v)
yr)V(71?)
ar
1
- (pv + Ku)VxVxy
+ (Xv. + 2pv + Ku)V(V-y)
K VXV
.
+1;k1(v
- 837/98 7v
- ph
- 0728
- eklm vm)
1;k
+
(1.3.5)
= 0
represents the free energy.
is the entropy
(n
density.)
Note that
1(
v
.1(
a
k
= 3v /at
.k1
= 3
+ v
k
v
1
(1.3.6)
and
;1
(3v1/t. + v
1:m
vm)
kmr .1
3
- e
m
v
v
r
.
(1.3.7)
1
Also of importance are the following basic relations of micropolar
theory.
.
Symmetry of the microinertia tensor:
.1k
k1
=
7
11
mk
Skew-symmetry of gyration tensor:
6k k
Spin momentum:
1.4
.
=
k1
3
v
km
v
-v
e
kmr
vr .
from which equation (1.3.7) is derived.
1
Basic Assumptions and the Surmised Field Equations
We now introduce the following simplifying assumptions
into the
theory:
(i)
The fluid is isothermal, which implies that
(ii)
0,k = O.
The fluid is incompressible, homogeneous, and isotropic.
result of incompressibility, the thermodynamic
pressure
by the hydrostatic pressure
(iii)
kl
Tr
As a
is replaced
p.
The fluid is assumed to be microisotropic; that is,
.k1
= j g
(iv)
where
is a scalar which is taken as a constant here.
j
There are no body forces and body couples; that is, fk
= 0 = tk.
These assumptions are not unrealistic since there exist
a wide
class of fluids, as listed previously, for which they
are known to be
valid.
Furthermore, these assumptions are found to simplify the field
equations to a more tractable form.
As a consequence of the above assumptions, the field equations
(1.3.1)
(1.3.5) take the form:
0v =
0
-Op
(11 + K) V2v
(a +
+
)0(0v)
+
yV2v
laxv
+
-
KVXV
pv
-
=
2Kv
0
-
pa
=
0
It is these seven surmised field equations that we will be using
for investigating the nonlinear stability of Couette flows of
micropolar fluids.
12
II.
II.1
BASIC PLANE COUETTE FLOW
Geometry of Plane Couette Flow
Plane Couette flow is defined to mean any flow, occurring between
two parallel plates, that is caused by the two plates moving relative
to each other.
We will use a rectangular Cartesian coordinate system (x,y,z),
where
x
denotes the distance parallel to the plates, and
denotes
z
the distance normal to the plates as measured from the channel center.
The total distance between the plates always equals
2h.
The plates are assumed to be of infinite extent in the
xy-plane.
For simplicity, the relative motion of the two plates will be chosen
so
that the upper plate is moving at constant velocity
x-direction), and the lower plate is at rest.
U (in the positive
Also, the fluid flows
to be considered are assumed to be under no external pressure gradients.
The velocity field will be
v = (u, v, w).
The microgyration field will be
1) = (C, v, Ti).
We will be non-dimensionalizing all equations, choosing as reference length, h, which represents one-half the distance between the
parallel plates, and as reference velocity, U, which represents the
constant velocity of the upper plate.
along with the constant density
These reference parameters,
p, are used to provide a reference
time, t = ht/U; a reference pressure gradient,
reference microinertia,
v = Uv/h.
Vp = pU 24 p/h; a
j = h25; and a reference microgyration,
The over-bar denotes a dimensionless variable.
13
Four nondimensional numbers, for the micropolar theory, are
defined as follows:
R = phU;
Rk = phU;
Rg = ph3U;
and
Rb = ph3U
a+8
(2.1.1)
.
We define
1/M
=
1/R
+ 1/Rk
.
In accordance with the constitutive theory, given by equations
(1.2.6) - (1.2.8), we present the following nomenclature for the
viscosity coefficients:
cosity;
p = dynamic viscosity;
K = gyrational vis-
y = gyrational-gradient viscosity (right);
gradient viscosity (left); and
Also, do not confuse
8 = gyrational-
a = dilational-gyration viscosity.
R with the classical Reynolds number (Re), even
though,their definitions are very similar, with the only difference
being that the
Physically,
p
listed here is from micropolar theory.
R
represents the ratio of inertial force to viscous
force,(inertial force being
Hence,
ph2U2
and viscous force being
phU).
R plays the same role as the classical Reynolds number, in
that, the
p-viscosity is created by relatively translating volume
elements.
Physically,
R
k
represents the ratio of inertial force to
gyrational-viscous force,(gyrational-viscous force being
KhU).
Since
the micropolar theory has the added feature of geometrical points
rotating, the adjective 'gyrational' is inserted to emphasize that the
K-viscosity is created by relatively rotating volume elements.
Physically,
R
represents the ratio of inertial couple to
14
gyrational-gradient couple, (gyrational-gradient couple being
yU).
The adjective 'gyrational-gradient' is inserted to emphasize that the
y-viscosity is created by relatively rotating and translating volume
elements.
Physically,
Rb
represents the ratio of inertial couple to
dilational-gyration couple, (dilational-gyration couple being (a+)U).
The adjective 'dilational-gyration' is inserted to emphasize that the
(a+(3)-viscosity is created by relatively rotating, translating, and
dilating volume elements.
For reference purposes, the surmised field equations are presented
in rectangular Cartesian coordinates.
15
11.2
Field Equations -- Rectangular Coordinates
(1.4.3) that
Listed below are the seven field equations (1.4.1)
These non-dimensionalized field equations,
were deduced in section 1.4.
in rectangular Cartesian coordinates, are:
u,
-A .
+ w,
+ v,
x
Y
x
+ Cu,
= u,
Y
= v,
-p,
z
XX
XX
+ uw,
t
+ w,
(c,
yx
+ v,
yy
= 2v/Rk
+ (n,
) /M
+ wu,
Y
+ v,
yy
zz
)/M
Y
+ w,
yy
zz
)/M
Y
=
- n, )/Rk
x
z
(2.2.3)
z
=
C, )/Ric
x
y
(2.2.4)
z
+ ("xx 1- "yy + "zz)/Rg
+ (w'y
v'z)/Rk
(2.2.5)
+ j(c,t + uc,x + vc,y + wC,z)
+ n,
yz
)/R
+ (v,
xx
+ v,
+ j(v,t + uv,x + vv,
yy
+ v,
+ wv,
Y
(C'zx + v, zy + n' zz )/Rb + (n' XX
= 2n/Rk
=
z
(2.2.2)
+ (v.
+ ww,
+ vw,
x
v, )/Rk
y
z
+ (c,
+ wv,
+ vv,
("xx + "9,xy + n'xz)/Bb
= 2C/Rk
zz
+ vu,
x
+ w,
(2.2.1)
yy
x
+ v,
+ uv,
t
+ (w,
= w,
+ v,
+ uu,
t
+ (v,
-p,
xx
= 0
z
+ un,
+ j(n,
t
x
+ n,
yy
Y
)/R
g
+ (u,
- w, )/R
z
x
k
(2.2.6)
)
z
+ n,
+ wn,
+ vn,
zz
)
zz
- u,y)/Rk =
)/R + (v,
g
x
(2.2.7)
z
The over-bars on nondimensional quantities have been omitted for convenience, and a concise notation for partial derivatives employed.
16
Note that assumption (iii) stated in section 1.4 and the expression
for spin momentum, allow the inertial spin vector to be expressed as:
k
Dv
Dt
a
11.3
k
k
j(Bv /3t + vk
(2.2.8)
vm ).
Laminar Plane Couette Flow
The plane Couette flow problem for steady, laminar micropolar
fluid flows in the xz-plane is now presented and solved.
The laminar flow is assumed to be in the x- direction.
For steady, laminar plane Couette flow, we prescribe the velocity and
microgyration fields, respectively, to be:
v = {u(z), 0, 0}
and
v = {0, v(z), 0}.
(2.3.1)
The field equations (2.2.1) - (2.2.7), in accord with the prescription (2.3.1) and pertinent nondimensional numbers (2.1.1),
reduce to:
-
dp
dx
+
d2v
1
FT
g dz
3p/By
2
1 d2u
M
2
dz
R.
du
Rk dz
1
= 0 =
dv
dz
1
2v
Rk
Bp/Dz.
= 0,
(2.3.2)
(2.3.3)
(2.3.4)
17
Enforcing the strict adherence boundary conditions 4
gives,
non-dimensionally,
u(-1) = 0 = v(±1)
and
u(1) = 1.
(2.3.5)
We now solve equations (2.3.2-3) for
v(z)
and
u(z).
From
equation (2.3.2), we obtain
dp
dx
dv
1
1
2
2
- ridu/dz
(2.3.6)
Note that since the right-hand side of (2.3.6) is
a function of z,
and the left-hand side is a function of x, it follows that
Integrating equation (2.3.6) with respect to
Az + B
where
B
1
=
1
1 du
M dz
-
z
A = constant.
gives
(2.3.7)
'
is a constant of integration.
Substituting for
du/dz
from (2.3.7) into (2.3.3) yields
d2v/dz2 -
A2v =
MRg(Az + B1)/Rk
(2.3.8)
where
R (R + 2Rk)
A2
h2K(211 + K)
g
Y(11
Rk(R "k)
Of importance is the fact that
(2.3.9)
K)
A2 > 0
via the inequalities (1.2.9).
The general solution of (2.3.8) is
v(z)
=
B e
Az
2
where
B
2
and
+
B e
3
B
3
-Az
RRk
-
R + 2R
(Az + B ),
1
are additional constants of integration.
With this expression for
v(z), equation (2.3.7) reveals
(2.3.10)
18
u(z) = -214 B eXz
2
XRk
(Az2 + 2B z)
k)
M(R 2R Rk)
B e-Xz )
-
R +
3
1
+
B
4
(2.3.11)
where
B
is another integration constant.
4
Assuming the experimentally achievable assumption that
- dp/dx
A =
= 0, and with the boundary conditions (2.3.5), the
integration constants
B
2'
for equations (2.3.10-11) are
B
B
B
3'
4
found to be:
B1 = X(R + 2Rk)
B2 = B3 = XRRksech(X)U2, and
,
B4 = 1/2,
(2.3.12)
where
2RM tanh(X)
=
4ARRk.
Therefore, the microgyration is
v(z)
=
(2.3.13)
11.
XRRkVcosh(Az)/cosh(X)
And the velocity is
u(z)
=
RM
sinh(Xz)/cosh(X)
2ARRkCz
+ 1/2
.
(2.3.14)
Equations (2.3.13) and (2.3.14) completely determine the flow
profiles for steady, laminar plane Couette flow of a micropolar fluid
for the given boundary conditions (2.3.5) and
For the corresponding unsteady,
laminar plane Couette flow pro-
blem, the velocity and microgyration fields
respectively, reduce to
u(z)
and
A = 0.
v(z).
u(z,t)
That is,
and
v(z,t),
19
u(z,t) = u(z)
and
v(z,t) = v(z).
(2.3.15)
Especially note that solving equation (2.3.8) when
(i.e.
Rk
00), we derive
v(z) = 0
and
A2 = 0
u(z) = (z + 1)/2
.
As
expected, the fields revert back to those derived
for classical
incompressible viscous flows.
20
III.
III.1
BASIC ROTATIONAL COUETTE FLOW
Geometry of Rotational Couette Flow
The rotational Couette flow is defined to mean any flow, occurring
in the annulus between two coaxial cylinders, rotating relative to each
other about their common axis.
These cylinders have
We will use cylindrical coordinates (r,e,z).
r denotes the radial direction, and
the z-axis as the center axle.
e
The inner cylinder is of radius R1,
denotes the azimuthal direction.
and the outer cylinder has radius R2, with
0 < R1 < R2
Each cylinder is assumed to be of infinite length.
always.
The relative
motion of the two cylinders will be chosen so that the inner cylinder
is rotating at the constant angular velocity Qv and the outer cylinder
is rotating at the constant angular velocity 02.
Also, the fluid flows
to be considered are assumed to be under no external pressure gradients.
v = (u, v, w).
The velocity field will be
v = (c, n, v).
The microgyration field will be
We will be non-dimensionalizing all equations, choosing as reference
length,
d = R2 - R1, which represents the gap-width between the cylin5
ders, and as reference velocity
velocity of the inner cylinder.
the constant density
t = .Ed/0 R
,
1
R
which represents the constant
These reference parameters, along with
p, are used to provide a reference time,
a reference pressure gradient, Vp =
1 17
microinertia,
ns-g1 R2 Op /d;
1
'-'
j = 5d2; and a reference microgyration,
The over-bar denotes a dimensionless variable.
v =
a reference
21
Four non-dimensional numbers, for the micropolar theory, are
defined as follows:
pdO R
R =
R
pdS2
=
1 1;
R
1;
1
=
pd3Q R
1
Rb =
1; and
pd g 1 R1.
a+f3
(3.1.1)
As before, we define
1/M
I/R
=
1/Rk
+
.
For reference purposes, the surmised field equations are next
presented in cylindrical coordinates.
Field Equations -- Cylindrical Coordinates
111.2
(1.4.3) that
Listed below are the seven field equations (1.4.1)
were deduced in section 1.4.
These non-dimensionalized field equations,
in cylindrical coordinates, are:
1
+
r(ru),,r
p,
r
1
+
+ uu,
t
1
+
6
= v,t + uv,
r
u,
00
+ ru,
zz
+ u,
- v/r 2 + wu,
+ T u,
r
0
1
,
cry,
1-
(3.2.1)
= 0
+ w,
v,
r
+ r1(ru,
rr
M
= u,
-
1
+ 177v,
rr
+
v,
00
+ rv,
zz
u
r
r
-
2
r
v,
)
+
8
1
1
e- v,
Rk r
- n, z) =
(3.2.2)
z
+ v,
+ uv
+ wv,z
-
-
r
v
2
+ T v,
)
+
1
(z
v,
r
)
(3.2.3)
=
22
= w,
r
1
Rio(rC,
+ uw,
t
+
r
+
rr
-n,
0
r
r
r
(3.2.4)
1
+ n,
+ rv,
re
rz
+ j(C,t
(c,
+
Or
r
C,
0
1
+
--(rn, rr
rR
g
+ j(n,
, +
R
zr
+
r
1
rR
vn
+ wC,
z
r
e
+ r-n, 00 + v, ez
+1r n, 08
+ rn,
+
+
+ un,
r
r-II, 0
c,
(rv,
z
rr
+
n,
r
+
Rk
1
-w,
r
e
-r)
=
zz
vc
1
r
ze
v,
+ v,
zz
+ rv,
00
)
+
Rk
+
(3.2.5)
+ i : (u,
+ n,
)
z
)
1
)
- v,
2c
11,0
r
- w,
z
+2r
r
r
)
+
2n
c,
2-
r
8
=
)
rRk
+ wn, 2 )
(3.2.6)
, +
1
1
1
1
(c
t
r,
r
1
1
1
+
11C1r
+
)
2
1
1
) =
r
z
1
C
+ C,
+ ww,
w,
r
+ C,
+ rc,
+
C.
+ ---(rC,
r
zz
60
r
rR
rr
rR
1
n
Rkrr
1
1
1
+ w, ) + --ln,
+ rw,
+ --w,
+ --(rw,
r
zz
08
r
rr
z
rM
,
- p,
+
Rk
(v
+ v,
zz
r
)
r
=
v
1
r
r
2v
R
u,
)
+
0
+
g
+ j(v,t + uv,r +
-'717 v,e + wv,z)
(3.2.7)
The over-bars on nondimensional quantities have been omitted for
convenience, and a concise notation for partial derivatives was
employed.
23
Laminar Rotational Couette Flow
111.3
The rotational Couette flow problem for steady, laminar micropolar
fluid flows in the re-plane is now presented and solved.
For steady,
The laminar flow is assumed to be in the 0- direction.
laminar rotational Couette flow, we prescribe the velocity and microgyration fields, respectively, to be:
v = {0, v(r), o}
and
(3.3.1)
v = {0, 0, v(r)}.
(3.2.7), in accord with the
The field equations (3.2.1)
prescription (3.3.1), reduce to:
(d2v /dr2
1
+ 1r (v/r)
d
1 dv
r dr
d2
(v/dr2
1
=
dv
(3.3.2)
Rk dr
1
2v
111.I
Rk Cdr
r /
(3.3.3)
'
g
dp
dr
v
T2
(3.3.4)
3p/3z = 0
because of axial symmetry, and
3p/30 = 0
Note that
since there is no axial motion.
4
Enforcing the strict adherence boundary conditions
gives,
non-dimensionally,
v(R
where
R
)
1
= 0 = v(R
= R /d
1
v(R
),
2
1
and
)
= 1, and
1
= R /d.
R
2
v(R
)
2
R /Q R
= SI
2
2
,
(3.3.5)
1 1
The over-bars on these nondimensional
2
quantities are omitted for convenience.
24
We now solve equations (3.3.2-3) for
Integrating equation (3.3.2) with respect to
1 (dv
dr
where
B
v
v(r).
yields
r
v
r
(3.3.6)
1
Rk
dv/dr + v/r
Substituting for
is a constant of integration.
1
and
v(r)
from (3.3.6) into equation (3.3.3) yields
1 dv
r dr
+
d2v/dr2
=
X2v
MR B /R
(3.3.7)
k
g 1
again (see expressions (2.3.9)) defining
R (R + 2R,)
=
2
K
g
ROR
R)
The general solution of (3.3.7) is well-known, in terms of the
modified zero-order Bessel functions
I
0
K
and
0
of the first and
second kind respectively, to be:
RR B
v(r)
where
B
2
=
B I
and
expression for
0
2
(Xr)
+
B K (Xr)
3
0
-It
Rk
R + 21
(3.3.8)
'
are additional integration constants.
B
With this
3
v(r), equation (3.3.6) reveals
RRkE31 r
v(r) =
{B I
2
(Xr)
1
-
B K (Xr)].
3
R + 2R
1
+
B /r
,
4
(3.3.9)
where
I
1
and
K
are modified first-order Bessel functions of the
1
first and second kind respectively, and
constant.
B4
is another integration
25
With boundary conditions (3.3.5), the integration constants
B
,
1
B2, B3, and B4
for equations (3.3.8-9) are found to be:
B1 = -(R + 2Rk)gB3/(ARRk),
B2 = -CB3/A,
B
= R R {S' + gR
B
1 2
4
- R )A/G,
= (SIR
3
2
1
(3.3.10)
- Q(S + gR2) } /G.
2
Here
A = I (XR
0
I
)
0
1
C = K (XR
(XR ),
0
2
S = MIK0(XR2)I1(XR1)
0
G = g(1,2
-
)
0
1
12322)
+
2
- K (XR
0
Ki(XR1)I0(XR2)
+
S' = -M{K0(XR1)I1(XR2)
g = I (XR )K (XR
)
1
+
I
0
1
_
R2S'
1/(XR1)}/(XRk),
-
1/(XR2)}/(XRk),
K1(XR2)I0(X121)
(XR )K
0
(XR ),
),
2
Q = Q R /(Q R
2
2
),
1 1
2
R1S.
Therefore, the microgyration is
- R
(2R
v(r)
)
1
2
=
{AK0(Xr)
G
-
CI
o
(Xr)
+
(3.3.11)
g}.
And the velocity is
M(R
v(r)
=
1
AR
- QR
G
g(2R
)
2
{AK (Xr) + CI (Xr)}
1
+
R
1
)
1
2
G
r
+
B /r
4
(3.3.12)
Equations (3.3.11) and (3.3.12) completely determine the flow
profiles for steady, laminar rotational Couette flow of a micropolar
fluid for the given boundary conditions (3.3.5).
For the corresponding unsteady, laminar rotational Couette flow
problem, the velocity and microgyration fields
v(r,t)
and
v(r,t),
.
26
reduce to
and
v(r)
v(r).
and
v(r,t) = v(r)
That is,
Especially note that when
v(r) = 0
(3.3.13)
v(r,t) = v(r).
X2 = 0 (i.e.
Rk - 00), one derives
and
1
v(r) =
R2 - R2
2
r(Q R2 - Q R2)
1 1
2 2
-
R2 R2
1 2
r
02
2
Q )).
1
1
As expected, the fields revert back to those derived for classical
incompressible viscous flows.
27
IV.
STABILITY OF A BASIC PLANE COUETTE FLOW
The laminar plane Couette flow elucidated in chapter
II, will now
be disturbed by the imposition of a disturbance wave.
The stability
analysis of this chapter follows the procedures of the
Stuart energy
method (Stuart,1958).
Note that this procedure will use the solutions
of the linearized theory, which are derived in section
IV.l.
The sol-
utions sought, in this (linear) case, satisfy the micropolar analog
to
the Orr-Sommerfeld (MOS-) energy equations, which are also derived.
To study its stability, the basic flow is superimposed with
a twodimensional, finite disturbance.
The imposed disturbances, having zero
mean, provide homogeneous boundary conditions for the nonlinear
equations of motion governing the disturbance flow.
(See section IV.2.)
The disturbance energy equations are derived from the disturbance
equations in section IV.3.
The energy equations invite a physical in-
terpretation of the possible mechanisms involved in the transition from
stable to unstable flow.
(See section IV.4.)
These nonlinear energy
equations (hence, the nomenclature of energy method) are then
assumed
to be solved by wave forms of the same spatial form as the
'marginal'
disturbances of the linearized theory, but with unknown amplitude.
In
fact, the solution to the nonlinear disturbance energy equations are
assumed to be separable into a spatial part, which is known from the
linearized theory, and a temporal (time) part, which defines the amplitude of the imposed disturbances (at least, near marginal or critical
stability).
28
Since the spatial part of the disturbance is known, ordinary
differential equations, describing the disturbance amplitudes, are
found from the disturbance energy equations.
Such equations are called
amplitude equations when micropolar theory is involved, or else Landau
equations when classical theory is used.
The possible growth, decay,
or equilibrium states of these disturbance amplitudes can then provide
For instance, in sections IV.6 and IV.7, we
the stability criteria.
derive the marginal stability surface, and extract a theoretical prediction for the critical non-dimensional numbers,
R , R
c
gc
, and Rk
c
involved in the stability of plane Couette flows.
In essence, we are re-working the stability problem for plane
Couette (parallel) flows, with the enhanced insight permitted by the
micropolar theory of fluid dynamics.
IV.1
Linear Stability Analysis
Employment of the Stuart energy method will require the shape of
the marginal disturbances of the linearized theory, so that numerical
calculations for theoretical predictions can be performed.
Thus, the
main goal of this section is to derive the MOS-energy equations.
The
solution to these coupled equations yields the 'shape' of the disturbances that we will be utilizing in later calculations.
Enroute, we
will also prove, why only considering two-dimensional infinitesimal
disturbances, is sufficient to obtain the minimum critical non-dimensional numbers
R
,
c
R
,
kc
and R
gc
.
This proof suggests the feasibility
29
of the two-dimensional nonlinear stability analysis that
we will be
undertaking in later sections.
From section 1.4, the surmised field equations (1.4.1)
(1.4.3),
in non-dimensional form, are:
Vv = 0,
1
1, 2
- Vp + -N -v + --Nxv
M
Rk
+ vVv,
1
1
1
--V(V-v) + --V2v + --vxv
= 2v +
Rb
where
R
Rk
v = (u, v, w) and
.3v
at
Rk
v = (t, v, n).
+ jvVv,
(4.1.3)
We are still using a rec-
tangular Cartesian coordinate system, in that,
x = (x,y,z).
In section 11.4, the basic flow was derived to be of the form
v = u(z)i
and
v = v(z)j, where
i
and
j
are respectively, unit
vectors along the x- and y-axes of the rectangular Cartesian coordinate
system.
To study the stability of this flow, we now superimpose a
disturbance on the basic flow as follows:
-
-
v(x,t) = u(z)i + v'(x,t);
,..
-...:
..,
_
v(x,t) = v(z)j + v'(x,t);
...,
p(x,t) = constant + p'(x,t).
where
v-
is the disturbance velocity,
gyration, and
p"
(4.1.4)
v'
is the disturbance micro-
is the disturbance pressure.
On substituting these
expressions into equations (4.1.1) -(4.1.3), we obtain the equations
of motion governing the disturbed flow.
By utilizing the fact that the basic flow already satisfies the
equations of motion, we have
30
Vv" = 0,
1
,
1
- D
Vxv',
V-v +
i +N 17"Vv"
= - Vp" +
-47' + w'd1.1
____,
.Rk
M
Dx dz
,)
a
+ u
(Dt
and
D
J
(
E + Ju
5
1
a
3x
+
v2v,
,
+
.. + Jw-dv j + jv .vv- . Rb v(vv-)
-
2
Rk
dz
1
(4.1.7)
Rk
Rg
v"Vy
By neglecting the quadratic terms
and v"Vv'
(or
equivalently, assuming the disturbances are infinitesimal), we obtain
the linearized equations of motion governing the disturbed flow.
the coefficients of
v"
and
x", in the linearized equations, depend
only on z, the equations admit solutions which depend on
exponentially.
Since
x, y, and t
Consider therefore solutions of the form
v'(x,t) = v(z) exp(i(bx + ay - bct))
;
v-(x,t) = v(z) exp(i(bx + ay - bct))
;
p"(x,t) = p(z) exp(i(bx + ay
.
bct))
(4.1.8)
The real parts of the expressions are to be taken to obtain physical
quantities.
Requiring that the solutions remain bounded as
implies that the wavenumbers
b
and
a
must be real.
cmaybecmplex,i.e.c=cr+ic..The expressions
x,y
±00
The wave speed
thus represent
waves which travel in the direction (b,a,0), with wave speed
bc /(b2 + a2)12, and which grow or decay like
exp(bc.t).
Note that a
r
wave is said to be (asymptotically) stable if
bc. > 0, and neutrally stable if
bc. = 0.
bc.
< 0, unstable if
Marginal stability occurs
31
if
bc. = 0
for critical values of the parameters (e.g. R, Rk, and Rg)
on which the 'eigenvalue'
depends, but
c
bci > 0
for some neigh-
boring values of the parameters.
for marginal stab-
R, R , and R
The ratios of the parameters
k
ility are found in section IV.6 and IV.7, which can thus give some
criteria for stability.
The critical relationship between the para-
meters, when discovered, yields the marginal stability surface.
that neutral stability is not necessarily marginal stability.
parison, note that on a neutral atability surface, bc. = 0, but
Note
For combe,
is not necessarily positive for any neighboring values of the parameters.
The minimum values of R, Rk, and Rg on the marginal stability surface
are called the critical numbers
instability for any
R > Rc, Rk
R
,
R,
c
KC
, and R
Rkc' or
R
gc
;
hence, there is flow
> P.
Forcompleteness,wementionthatifbcr+Oasbc.approaches
zero from above for a disturbance, oscillatory instability sets in.
This is sometimes called overstability.
Also, if
be = 0
at marginal
stability, i.e. bci = 0 = be , then there is said to be an 'exchange
r
of stabilities', whereby instability sets in as a steady secondary
flow, such as in the case of the convection cells that arise when a
fluid is heated from below (Perez-Garcia & Rubi,l982).
If we now let
D = d/dz, then on substituting the expressions
(4.1.8) into the linearized equations
(4.1.5) - (4.1.7), we obtain
the following (coupled) system of ordinary differential equations:
32
(4.1.9)
i(bu + av) + Dw = 0,
(D2 - (b2 + a2) - ibM(u - c))
u = Mu'w + (Dv - ian)M/Rk + ibMp,
(4.1.10)
(i)2
(b2
a2)
- ibM(u -
(D2
- (b2
a2)
- ibM(u - c)) w = MDp
(b2 + a2)
- ibjR (u - c) - 2R g/
(D2 -
c))
v = iaMp - (Dc - ibn)M/Rk,
(4.1.11)
(ibv - iac)M/Rk,
(4.1.12)
= (1021
+ bav - ibDn)R /R_
g b
(4.1.13)
+ (Dv - iaw)R /R
g
K
A
A
(?2
- (b2 + a2) - ibjR (u - c) - 2R /R)v = (bac +
g k
-
A
b
g
A
+ jR v'w - (Du - ibw)R
(D2 -
- iaDn)R /R +
g
(4.1.14)
/R_ ,
k
(b2 + a2) - ibjR (u - c) - 2R g/R.K )n =
- i(r)00
+ av +
Dr1))
R /R
g
b
(4.1.15)
(iau - ibv) R /R. .
g k
Here primes denote differentiation with respect to z.
The strict adherence boundary conditions, applied to the
disturbance flow, imply
u =v=w= 0 = c =v=1-1
at z= ±1.
(4.1.16)
The three-dimensional problem defined by equations (4.1.9) (4.1.16) can be reduced to an (almost) equivalent two-dimensional
problem by the use of the micropolar analog of the Squire transformation.
33
Let
B = (b2
a2)11,
W = w,
(as used in classical viscous theory).
BM = bM
C = c, and
P/B = p/b,
BV = bu + av,
(4.1.17)
Also, 'let
%
by
= BN,
ac
ti
(as is now needed
BRg = bRg
and
BRk = bRk,
for micropolar theory).
By introducing the relations (4.1.17) into equations (4.1.9)
(4.1.16), and after some simple manipulations, we obtain
iBV + DW = 0,
%
(D2
rt,
BM(11 - C):)V = Mu'W + MDN/Rk + iBMP,
B2 - iBM(u
B
(D 2
- B2
(D2
B2
IBM(u - C)) W = MDP - (iBN)M/Rk,
ti
(Ti
- C) - 222 /ON
k
g
-
'ij2
g
(DV -
iBW)Ak +
;'W,
(4.1.21)
with boundary conditions
V = W = 0 = N
at
(4.1.22)
z = ±1.
Equations (4.1.18)
(4.1.22) have the same mathematical structure
as equations (4.1.9) - (4.1.16) with
= n, and
a = v = 0 =
define the equivalent two-dimensional linear problem.
they thus
Note, however,
that transforms for Rb and n were not required in the derivation, and
hence are still arbitrary.
trivial equality
(Moreover, equation (4.1.15) implies the
0 = 0.)
%
,A,
Since
B > b, it immediately follows that
M < M,
,t,
R
< R .
Since by definition
(Recall that the tilda "A,
1/M = 1/R + 1/Rk,
R < R
Rk < Rk, and
also follows.
terms stem from the two-dimensional problem.)
34
We have thus proved, for infinitesimal disturbances, the following
Theorem.
and R
,
To obtain minimum critical nondimensional numbers
R
,
c
R
,
kc
it is sufficient to consider only two-dimensional disturbances.
gc
(For a rendition of Squire's theorem for classical viscous theory,
refer to Drazin and Reid, Hydrodynamic Stability, p. 155.)
Restricting now to two-dimensional disturbances in the xz-plane,
we introduce the disturbance stream function
tr(x,z,t).=
(0(z) exp(ib(x
ct))
.
Thus, the two components of the disturbance velocity will be
u" = 8t' /8z
and
w' = - W/9x.
Furthermore,
u = d4 /dz
Notice that
(4.1.23)
and w = - ibcP.
4(z) is a complex-valued function, in that,
(0(z) = (f)r(z) + icyz)
where
is the imaginary part of
(Pr
is the real part of cp(z)
and
(1)i
(1)(z).
From equations (4.1.10) - (4.1.16) with
a = v = 0 =
= n,
we now have
(D2 - b2 - ibM(u - c))
u = Mu'w + MDv/Rk + ibMp,
(D2 -
b2 - ibM(u - c)) w = MD; - M(ib;)/R
k'
(D2 -
b2-ibjR (72 - c) - 2R /R.); = - R (Du - ibw)/Rk + jR
g
;17,
1
k
(4.1.26)
35
with boundary conditions
u = w = 0 = v
at
z = ±1.
(4.1.27)
Remember, in these equations, primes denote differentiation with
respect to z,
i = 47T, and the scalar
is the microinertia.
j
By inserting (4.1.23) into the three equations (4.1.24-26), we
reduce the unknowns to three:
{D2
ibjR
b2
c)
(171
cl),
v, and p.
The resulting equations are:
2R /R }v = ibjRg
g k
- R (q)"
b24))/Rk
,
(4.1.28)
{D2
{D2
b2
ibM(u - c)}(-ib(0) = MDp - M(ibV)/Rk
(4.1.29)
I
b2 - ibM(u - c)} (p' = MU'(-ibcp) + MDv/Rk + ibMp.
Now, eliminating p
(4.1.30)
from the above equations, and effecting some
rearrangement, gives the micropolar analog to the Orr-Sommerfeld (MOS-)
energy equations.
The result is
(D2 _ b2)2
(I) = (u
ibM
c) (D2
102)(1)
.11,,(1)
(D2
b2)v
ibRk
(4.1.31)
and
(D2 - 132)v
'
ibR
+
(D2 - b2)(1)
g
ibR
j(u - c)v +
2v
.-
i bRk + 3v1(1)
(4.1.32)
with boundary conditions
= bcp = 0 = v
at z = ±1.
(4.1.33)
36
The unknowns
(f)
and
v
governed by (4.1.31-32) will be required
in later calculations, as disturbance shapes.
and
u
v
Also, in the calculations,
are taken as (2.3.14) and (2.3.13), respectively.
Note,
however, that in the nonlinear theory, the mean velocity u and the
vim
mean micogyration v are different from those of laminar flow (as taken
above) beacause of the interactions between the mean flow and the disturbances.
Next, we study, with the nonlinear theory, the stability of a
basic flow, which is disturbed by a two-dimensional, finite disturbance.
Imposition of Finite Disturbances on a Basic Flow
IV.2
Two-dimensional plane Couette flow refers to a plane Couette flow
that is theoretically restricted to two spatial variables.
We will be
analyzing the stability of a basic plane Couette flow, that is disturbed
by a two-dimensional, finite disturbance in the xz-plane.
Moreover,
since the channel is assumed to extend to infinity in both the positive
and negative y-directions, the velocity and microgyration fields will
be independent of the y-coordinate.
Restricting to two-dimensional flows, the continuity equation
au/ax
+ Bw/az = 0
defines a stream function for the fluid flow.
This
stream function is decomposed into the sum of two functions, one representing the mean flow, the other representing the finite disturbance
flow.
The analysis of the nonlinear field equations governing the
finite disturbance flow employs procedures embodied in the Stuart energy
method, with which we begin.
37
The energy method to be used here, was established by J. T. Stuart
The energy
in his fundamental paper published in 1958 (Stuart,1958).
method of Stuart is an approximate method which assumes that the spatial
form (shape) of the nonlinear disturbances is the same as the shape of
marginal disturbances of the linearized theory, but with unknown amprefer to D. D.
(For additional references and criticism,
litude.
Joseph's book, Stability of Fluid Motions I (Joseph, 1976).)
energy method was further developed and applied by Stuart
This
(Stuart,1960)
Also, of special note are the elucida-
and by J. Watson (Watson,1960).
tions of the Stuart energy method made by A. Davey (Davey,1962).
For flow under no pressure gradient between two parallel plates
in constant relative motion (plane Couette flow), we impose a disturbance travelling in the direction of the basic flow, which has the form
11)(x,z,t) = Ki1(z)
exp(ib(x
- ct))
+
(z)
exp(ib(x
- ct))
1
(4.2.1)
K
where
(c
r
is an arbitrary complex scalar, wave speed
> 0), and the tilda
ti
c = c
r
+ ic.
denotes a complex conjugate.
Now, let the stream function for the flow be represented by the
Fourier series expansion in x (Watson,1960), as
flx,z,t)
=
(13,
{(pn(z,t) enibx +
= (00(2,t) +
n=1
n
(z,t)
e-nibx}.
(4.2.2)
6
38
In the linearized theory, (I) represents the steady stream function, and
cr reduces to (4.2.1)
(or equivalently, the function 4/"(x,z,t) used in
For the nonlinear theory, the sum on the right repre-
section IV.1).
sents the finite disturbance cr, while (I) is the mean stream function,
where the mean (average) is taken with respect to x over a disturbance
wavelength of
27r/b.
b is the period
7
of the disturbance wave.
With the stream function IP, the Fourier series expansion gives
u = 911)/3z
+ u" =
=
co
= u(z,t) +
X
{u'(z,t) e
nibx
+
;
n=1
w = -DTP/x
-nibx
1;
= w' =
co
=
(z,t) e
1 {w-(z,t) e
n=1
nibx
+ Wn(z,t) e
-nibx
};
v = v + v' =
co
= v(z,t) +
n=1
{v"(z,t) e
n
nibx
+
n
(z,t) e
-nibx
(4.2.3)
}.
The over-bar now denotes a mean value, and the primed terms denote
disturbance flow variables.
Defined for equations (4.2.3) are
u(z,t) = Bo/z;
u"(z,t) = B(I)
/3z; and w"(z,t) =
n
n
(n > 1)
(4.2.4)
For the resulting two-dimensional flow, mathematically, we
prescribe that the velocity and microgyration vector fields are,
39
respectively,
and
v = {u(x,z,t), 0, w(x,z,t)1
v = {0, v(x,2,t), 0}.
(4.2.5)
Applying the prescription (4.2.5) to the field equations (2.2.1) (2.2.7) yields:
3w
Du
= 0,
+
3x
Dz
/ a2u
(4.2.6)
a2u
1
3x + M ax2 + az2
R
ap
3v
32
au
32
au
3x
au
at
(4.2.7)
-.(
1 D2w
3p
az
( axe
32w
---a2
2
ax2
az2
Rk
3v
c
au
1
R
1
3w
5r;
3w
Dw
(4.2.8)
+ w 3w
11
2v
Rk +
av
(3v
u
av
w TD.
(4.2.9)
The conditions to be applied throughout this procedure are:
(C1)
that the mean velocity u and the mean microgyration v assume the
same values on the plates as do the undisturbed velocity and the undisturbed microgyration;
(C2)
that the disturbance velocities 11' and w', and the disturbance
microgyration v- vanish on the plates; and
(C3)
that there is a suitable condition on the mean pressure gradient
in the flow direction or, equivalently on the mean velocity and the
mean microgyration.
Consequently, for disturbed plane Couette flow, acceptable nondimensional boundary conditions are:
40
-
-
-
u = 0
z = -1,
at
at
at
z = ±1.
(1)11/3z = 0 = (1)11 = v'
v = 0
z = 1,
u = 1
z = +1;
at
(4.2.10)
(n = 1,2,3,...)
In order that the pressure gradients shall balance the remaining
terms, the pressure must be of the form
p = xp*(t) + p**(z,t) + p'(x,z,t) =
co
nibx
X {pi'(z,t)e
= xp*(t) + p**(z,t) +
,I
+
n=1
p,
where the part
p**(z,t)
'
n
(z,t)e
-nibx
1
(4.2.11)
is a purely time-dependent mean pressure term,
p*(t)
is the mean pressure independent of x, and
p'(x,z,t)
is
the disturbance pressure.
Substitute equations (4.2.3) and (4.2.11) into equations (4.2.7) (4.2.9), and equate the Fourier components.
The equations arising from
equating the terms independent of x are equivalently found by taking the
mean of equations (4.2.7) - (4.2.9).
u,t + u'u;
u'w",
x
+ w'u;
x
+ w'w;
u,- z/k
+ v,
= - p**,
z
2v/
+ v'zz/Rg
/R = - p* + u,
z
z
Rk
The result is:
K
ZZ
/M
(4.2.13)
z
+ j(v,t + u'v;x + w'v;z)
172, ;-72, u'w', v'2, u'v', and w'v'
Note that the quantities
(4.2.12)
(4.2.14)
are
Also, note that the continuity equation (4.2.6)
independent of x.
implies that
u;
x
=
w;
(4.2.15)
z
The equations governing the disturbance field quantities are now
found on subtracting (4.2.12) from (4.2.7), (4.2.13) from (4.2.8), and
41
(4.2.14) from (4.2.9), to be:
311'
at
3U
- Du'
+ x
+ w'
+ u
aw'
ax
- aw+ x
u
2
-
= 1
3x
22:
1
+ u
Dx2
w
az2
ax2
+ w
1
aZ2
Rk az
(4.2.16)
2
a2w'
3v"
3v'
92u'
3
m
R
az
1_.(Dw' - au')
+ 3
az
ax
+ 1
A
+ x3
(4.2.17)
ay'
1
Rk 3x
1 (32v"
32v-
ax2
az2
2v'
Rk
(4.2.18)
where
x
1
= u
.au'
_au'
+ w
az
ax
3
Aaw-
a
X2
u ax
x3
u
_Dv
3x
w 3z
+ w
-ay
3z
--(u w ),
az
-72,
I'
azw
3
3z
,
A
,,
v ).
These nonlinear equations will be referred to as the disturbance
equations.
B
As a preview, the terms
Reynolds stress terms.
term.
---(u w )
az
The term
a.
y
and
---(w
az
)
are the familiar
(wAvA) is a mean couple stress
These concepts are discussed in section IV.4.
42
Disturbance Energy Equations
IV.3
In this section we will derive the disturbance energy balance
equations for two-dimensional, finite disturbances.
By properly pre-
paring the disturbance equations (4.2.16) - (4.2.18), we derive the
two-dimensional form of the micropolar analog to the Reynolds-Orr
(MRO-) energy equations.
the x-direction,
Note that since the channel is unbounded in
assumed the disturbances
we have
u"
and
to
v'
be (spatially) periodic in x, and thus, the following integrations
with respect to x can be taken over exactly one wavelength.
u' and (4.2.17) by
Begin by multiplying (4.2.16) by
w".
Add the
equations; then utilize simplifications similar to those demonstrated
in Appendix A, while integrating
dxdz.
Thus, the first Disturbance
Energy Equation is derived without approximation to be:
at
rr
.01(u'
.1
+ w'2) dxdz
rr rvr
M ff
;112
=
ff
dxdz
(-u'ur.)21-1
3z
-
Rk
dxdz
ff
- -11f)v" dxdz
(4.3.1)
where the integrals are evaluated over a volume bounded by the plates
z = ±1, and by one wavelength
x = 0, 27r/b.
Note that in the first integral on the right-hand side of (4.3.1),
u'w" was replaced by
u'1,7"
because only the mean part contributes to
the integral (Stuart,1956).
(4.2.12),
The mean velocity occurring in (4.3.1), is derived from
43
and is given by
"
T; (uw) + Rk Dz
Du
= _ DP
3
at
"2;
+
(4.3.2)
M
av
v', simplify,
Next, multiply the remaining equation (4.2.18) by
then integrate; thus deriving the companion Disturbance Energy Equation
without approximation to be:
a
at
dxdz
ff
ff (Jw-v-)
=
dxdz
_1
((av-)
R II
"
+
\3x
(av-)2)
dxdz
g
r (9wII
1
dxdz
v
Tz-
-
2
(4.3.3)
ff v-2 dxdz
where again the integrals are evaluated over a volume bounded by the
plates
z = ±1, and by one wavelength
was replaced by
v'w'
,
x = 0, 27/b.
v'w'
Note that
as was similarly done in (4.3.1).
The mean microgyration occurring in (4.3.3), is derived from
(4.2.14), and is given by
1
311
RIc 7z--
1
B2)
2;
/7;
3z2
Rk
.(3.;
3
,
.,1
(4.3.4)
Dz °4") ))
Equations (4.3.1) and (4.3.3) comprise the two-dimensional
MRO-energy equations.
It can be shown, so far as two-dimensional dis-
turbances are concerned, that the energy integrals of the MOS-energy
equations are equivalent to the above two-dimensional form of the MR0energy equations.
-
(To see this, let
Reaboib(x-ct),1
and
u" = Re{(D(p)
v" = Refv(z) e
eib
ib(x-ct)1.
(x-ct)1,
Then integrating
44
over one wavelength and dividing by the wavelength (i.e. averaging
with respect to x), one derives the imaginary part of the energy inteThis is to be expected, since in
gral of the MOS-energy equations.)
the derivation of equations (4.3.1) and (4.3.3), the nonlinear terms,
)(1,
)(.2, and x3, in the disturbance equations disappear in the process
of integration.
Following a physical interpretation of the flow mechanisms suggested by equations (4.3.1) and (4.3.3), we will derive the amplitude
equations with the Stuart energy method.
IV.4
Physical Interpretation of the MOS-Energy Equations
The goal of this section is to identify and physically interpret
the terms appearing in the disturbance energy equations (4.3.1) and
(4.3.3).
To enrich the physical interpretation, equations (4.3.1) and
(4.3.3) are, respectively, re-written in dimensional form and a briefer
notation is introduced.
2--ff2-(u-2 + w-2) dxdz
2
3t
=
31.1- 2
(P+Off(
Du
ff(-pu ,
-
dxdz
-
dxdz
Kff(
-5"
-
Du')
3z i v
dxdz.
Briefly,
3E
at
=
- (p+K)I
(4.4.1)
+ K1
2
3.
45
TTff 12pjv-2 dxdz
- K ifr
=
Dw"
(-
.w
103 v
ff
3u")
3x
(
V
dxdz
3v
dxdz
az
)
dxdz -
J
3av
z
- 2K ff v-2 dxdz.
Briefly,
3e
at
H1
YH2
(4.4.2)
KI3 - 2KH3.
In equation (4.4.1), the term on the left-hand side gives the rate
of growth of the disturbance kinetic energy within the volume considered.
On the right-hand side of (4.4.1), the term
I1
is the integral of the
product of the Reynolds stress and the mean velocity gradient, and represents the "translational" rate of transfer of kinetic energy from
The term
the mean flow to the disturbance.
so
represents the rate of
-(11-110I
(u+K)I
is always positive;
2
(0-0-viscous dissipation of the
2
kinetic energy of the disturbance due to translational and rotational
effects of the macro-volume elements in the volume considered.
The term
is the common link between equations (4.4.1) and
KI
3
(4.4.2).
The term
is the integral of the dot product of the curl v
I
3
and the microgyration,
and physically
Mathematically,
by the disturbance.
0
0
I
3
= v"(Vxv") =
-
3/3x
u'
3/ay
0
3/3z
,(3w'
3u'
3z
w"
Notice that the scalar triple product
in two other forms.
represents the Swirl created
v"(Vxv")
can also be written
In that,
v".(Vxv") = v".(Vxv-) = v.(v-xv-)
The third variation is the divergence of the Coriolis acceleration.
46
The Coriolis acceleration, common to the mechanics of moving
2y-xv-.
coordinate systems, equals
ular velocity.
Note that microgyration is ang-
For a fluid motion described by micropolar theory, the
Coriolis acceleration, 2y"xv", represents the resultant from the interaction of the rotation of moving micro-volume elements and the present
motion of the ambient macro-volume elements for the existing flow in
the volume considered.
Thus, a nonzero swirl, i.e.
= V(v-xv")
t 0, acts as a source, if
- -
y"(Vxv") =
> 0 (or a sink, if I
I
3
< 0),
3
for spreading (or gathering) the energy necessary to create a turbulent
flow.
Most of all, the swirl is the coupling mechanism between the
micro- and macro-continuum volume elements.
In equation (4.4.2), the term on the left-hand side gives the rate
of growth of the disturbance Microenergy of Rotation
considered.
8
within the volume
On the right-hand side of (4.4.2), the term
H
is the
1
integral of the product of the mean couple stress and the mean microgyration gradient, and represents the "rotational" rate of transfer
of microenergy of rotation from the mean (micro)flow to the disturbThe term
ance.
rate of
yH
is always positive; so
-yH
2
represents the
2
y-viscous dissipation of the microenergy of rotation of the
disturbance due to the translational and rotational effects of the
micro-volume elements in the volume considered.
always positive; so
represents the
-2KH
The term
2KH
is
3
K-viscous dissipation of
3
the microenergy of rotation of the disturbance due to the rotational
effects of the micro-volume elements in the volume considered.
In equation (4.4.1), the nonlinear Reynolds stress term
has the units of force per area.
mean couple stress term
pjw"v"
pu'w'
In equation (4.4.2), the nonlinear
has the units of force times distance
47
With microspin, Q = jv, the mean couple stress term can also
per area.
be written as
One of the main advantages that microcontinuum
pw"c'.
mechanics has over classical continuum mechanics, is its recognition of
couple stresses (and body couples).
In summary, fluid flow stabilizers are the terms
and 2KH
which represent viscous dissipation mechanisms.
Fluid flow
3
E, e, I
destabilizers are the terms
and H
(p+K)I2, yH2,
and H1.
,
Note that listing
I
1
1
as destabilizers, presumes these terms to be positive (which
1
The intermediary between stability and in-
may not always be true).
stability is the swirl term
KI3.
Combining equations (4.4.1) and (4.4.2), reveals
a(E + e)
+ Hi - (p+K)I
=
3t
+ 2KI
- yH2 - 2KH
2
.
(4.4.3)
3
3
Suppose
I
1
+ H
+ 2KI
1
> 0.
3
If
I
1
+ H
+ 2KI
1
>
(p+K)I
3
+ yH
2
2
+ 2KH
3
then
a(E + e)/at > 0.
This means the disturbance energies are growing, and the disturbances
are increasing in amplitude (i.e. the flow is becoming unstable).
Conversely, if
+ 2KI
+ H
I
<
+ 2KH
+ yH
(4+K)I
2
3
1
2
3
then
a(E + e)/at < O.
This means the disturbance energies are decaying, and the disturbances
are decreasing in amplitude (i.e. the flow is becoming stable).
Ideally, if
I
1
+ H
1
+ 2KI
3
< 0
then
(E + e)/a t < 0,
48
in that, the flow is becoming stable.
Finally, equations (4.3.2) and
(4.3.3) show how the distribution of mean velocity and mean microgyra9
tion are affected by the viscous stresses , pressure gradients, Reynolds
stress, and mean couple stress, due to the disturbance.
An equilibrium
flow is possible if u and v can be so distorted, by the Reynolds stress
and the mean couple stress, that
+ H1 + 2KI3 = (p+K)I2 + 'H2 + 2KH3,
which implies
am + e)/at
in that, equilibrium.
= 0,
The equilibrium state will play a crucial role
in the analysis presented in the next sections.
IV.5
Amplitude Equations
Recall, in the Stuart energy method, the stream function for the
disturbed flow, 4), represents a mean flow together with a periodic
disturbance consisting of the fundamental harmonic, 4)0, with wavelength
27r/b, and higher harmonics,
4)
p2, ..., having wavenumbers nb (n > 1),
l'
indepenbut the same (real) wave velocity, cr, which is assumed to be
dent of time.
The amplification or damping of the finite disturbance,
microgyand the consequent changes in the mean velocity u and the mean
ration T), are accounted for by the dependence of all the 4-functions
on time t.
We assume that the higher harmonics 4)2,
403,
... are zero.
Further-
conditions
more, we assume that the disturbances are under 'supercritical'
meaning that the non-dimensional numbers R, Rk, R
,
and Rb are above the
value which is critical for the linearized instability theory.
(For
49
motivation, see the proof of the micropolar analog to Squire's theorem,
Moreover, a disturbance under supercritical
given in section IV.1.)
A suitable initial condition,
conditions amplifies for small amplitudes.
therefore, is that the function 0 (z,t) shall be an exponentially in1
t 4- -co; in fact, 01 has to be
creasing function of time in the limit as
the appropriate function, 0(z)exp(bcit), where
ci > 0, of the linearized
12
instability theory (Stuart, 1960).
are similar in 'shape' to the
u', w', and v'
Assume disturbances
solution given by an amplitude factor, a(t) or, A(t) in the case of v'.
That is
01(z,t) = a(t)0(z)
and
v'(z,t) = A(t)v(z).
For an equilibrium state, we presume that
a ; /at
= 0 = DU/3t.
With this presumption, and assuming constant mean pressure p, we have
equations (4.3.2) and (4.3.4) yielding
d13
k
dz
d2u /dz2
-
-a-z-(u'w-)
(4.5.1)
and
du
dz
2%)
+
Rki
-
R
(4.5.2)
g
Integrating (4.5.1), and using (4.5.2), reveals
d2v /dz2 - X2; = f(z) = R
where
f(z)
.d
g
- MR (u'w")/Rk - MRgK 1 /Rk 2
(4.5.3)
represents the non-homogeneous part of (4.5.3), K1 is an
50
integration constant, and
is as given in (2.3.9)1.
A2
The homogeneous equation from (4.5.3) is solved by
V
h
(z) = K
2
K
+
e
-Az
(4.5.4)
3
are additional integration constants.
K
and
K
where
Az
e
2
3
dv/dz
Remember, we are seeking
and
du/dz for the disturbance
energy equations (4.3.1) and (4.3.3), so that the amplitude equations
can be derived.
Using variation of parameters, we find a particular solution for
equation (4.5.3) of the form
f
v (z) =
f(s) ds.
sinh(A(z - s))
j
(4.5.5)
1
So, in integral form, the general solution to equation (4.5.3) for
the mean microgyration is found to be
v- (z) =
K
e
Az
+ K
e
-Az
z
+
3
2
1
f
sinh(A(z - s)) f(s) ds.
(4.5.6)
1
The mean microgyration gradient is
dv
=
dz
A(K e
2
Az
- K e
3
z
-Az
)
cosh(A(z-4 f(s) ds.
+ f
(4.5.7)
1
having
Next, integrating equation (4.5.2) with respect to z, after
incorporated (4.5.6), gives the mean velocity to be:
51
R
u(z) =
+ R)
x
Az
(K e
Rk
K
-Az
)
+
2
rz
rr
1
1
2
+ Rkj(w'y') -
where
- K3e
R
sinh(A(z-s))f(s) dsdr +
z
g
(4.5.8)
+ K
cosh(A(z-s),)f(s) ds
f
4
1
is another integration constant.
4
The mean velocity is
du
dz
R + Rk
z
(K e
Az
+ K e
2
-Az
)
+
3
(RR+ R )
sinh(A(z-s),)f(s) ds
f
r
f cosh(A(z-s))f(s) dsdr + Mu'w' + MK /R
+ 2 f
(4.5.9)
.
g
1
1
+
1
1
The strict adherence boundary conditions (4.2.10) are:
As an integrating aid for
in the mean
functions.
function.
U"
f(z), it is reasonable to suppose that
is an odd function, and that
Hence, u'w'
(4.5.10)
u(1) = 1.
and
v(±1) = 0 = v'(±1) = u'(±1) = w'(±1) = u(-1)
and
c7.7"
7)-
is an even
w"v"
is an odd function, and
are even
Also, facts like integrating an even function gives an odd
function are used when determining the integration constants.
The integration constants
K , K
1
,
2
K
,
3
and K
for equations
4
(4.5.6) and (4.5.8) are found to be:
R + 2Rk
K
=
1
R(cosh(2X) -
C
1)(B
A
(e
- e
-3A
)
I*
+
X
,
K2 =
C
B
e
-2A
-A
K3 =
where
,
and
K
4
= 1 +
2RCe
AB(R +
(4.5.11)
'
52
(R
)
(cosh(2A) - 1)(
1 sinh(2X)
C =
Rk
(R + 2Rk)
B =
2(cosh(2X) - 1)
X(R+2Rk)sinh(2X)
k)
tX(R+2Re
r
,
-
X
X,kcosh(2X) + 1) + R(Cosh(2X)
-1
+
{R jXcosh(Xs) w'v-
I* = cosh(X) f
I*
tanh(X) I*
MR
g sinh(Xs) u-w-
1)}.
ds.
Rk
1
Incorporating the expressions (4.5.11), equations (4.5.7) and (4.5.9)
become
dv
dz
XC
X(z-2)
{e
B
.
+ e
-Xz,
}
z
MR
g
-
K
{cosh(X(z -1))
1
MR
d
ds
--av (u"w")} ds,
-k
cosh(X(z-s)) {R j ---(w-v")
+ f
g
1
- 1} +
(4.5.12)
and
du
dz
RC
B(R+Rk)
MK,
{e -Xz - e X(z-2) } +
R+2Rk
z
X
(R+R )
cosh(X(z-s)){R
f
{cosh(X(z-1)) - 1} +
MR
d
g
j
g
-(w v-)
,71;
+ Mu-w-
MR K
g 1
(u w-)J ds +
-k
1
MK
(z - 1) cosh(X(z-1)) +
1
R
A.Rk2
MR
Z r
(u-w")} dsdr.
+ 2f f cosh(X (z -s)) {R j 4--(w-v") 1 1
g
ds
(4.5.13)
-1(
If this were classical viscous theory, the amplitude equations,
that we derive in this section, would be referred to as a Landau
equation.
We would then comment on the Landau equation as an appro-
priate description of the nonlinear self-interaction of the most unstable mode (stemming from normal mode analysis) when slightly super-
53
We assume that this single, weakly unstable mode and its
critical.
lower harmonics (e.g. (00 and cp
)
dominate the flow.
1
The derivation takes the MOS-energy equations, and substitutes a
solution with the same spatial form as the solution of the linearized
problem.
Thus, we make what is called the 'shape assumption', namely
that the
finite
disturbances (e.g. u', w", and v") have the same
spatial structure as the linear ones, although their amplitudes (e.g.
a(t)
and
A(t)
)
may differ.
This approximation serves to give a
simplified derivation of the amplitude equations by neglect of the
It is
harmonics and neglect of the distortion of the fundamental (1)0.
a good approximation only if the total nonlinear effect is nearly the
same as that due only to the distortion of the mean flow.
Similar computations, to those needed in deriving equations
(4.5.15) and (4.5.16), are presented in Appendix B.
The amplitude equation for equation (4.3.1), incorporating (4.5.13),
is found to be
y
da2
= - y2 a2 - y3 a3A
1 dt
y4 a4 - y5 a3A - y6 a4 -
a2
8 aA,
(4.5.14)
where
1
1
f
1
for
11).12
dz;
b21412
-1
RC
F (z) =
B(R+Rk)
1
,
ie
y2 = 2bf F (z)(4);(0i
1
-1
ea (z
-Xz
MK
-2)
} +
{cosh (X(z-1))
R+21 Rk
MK
MR K
g 1
1)
(z
cosh(X(z-1)) +
z1 ,
g
XRk2
1
y3 = 4b2
f
(4)'<i)
1
ri
ir
(1)(1)
)
(c1)
fl G ids
ri J
1
(Prql dz,
4). )
ri
dsdz,
1}
54
RR
for
G1 =
j
X(R+R
1
cosh(X(z-s))
k
;
)
1
y
4
Wy5.
= 4b2 f
(P!(P
ir
ri
)
1
G
XRk(R+Rk)
2
(Vq5.
cosh(X(z-s))
1
= 4b2 f
5
for
01(1).
T1
1
G
- (0(15.
1 r
)
ir
4,!(t,
)
dsdz,
z r
f f
G33
1 1
= 2R j cosh(X(z-s))
;
dsri
-
(I)
cp.)
ri
dsdrdz,
;
3
1
y
2r1
RMR
for
y
G
f
- 1
= 4b2 f
(cP',1).
ri
6
- Cc!)
)
z r
f f
114
G4 ri
ir
(0!(1)
)
dsdrdz,
MR
for
cosh(X(z-s))
= -
G
;
4
1
Y
7
{14)"12 + 2b2W 12
= 2f
b4102} dz;
- 1
1
y
s
= 2f {b2($
1
rr
+ (1).4).)
(01)"(1)
r
r
+ (1)!'(D.)1 dz.
1
1
Primes indicate differentiation with respect to z.
We recall that, by solving equations (4.1.31-32), the unknowns
(I)
and
v
are determined.
In the above,
(1)1 = ept. + iii =
(I)
and v =
(1)=(Dr+ic1).1 .Also, we applied the shape assumption when we utilized
the expressions
d(1)
u"(z,t) = a(t)
dz
,
fl
w"(z,t) = - ib4(z)
a(t),
v"(z,t) = A(t) ()(z).
Continuing, the amplitude equation corresponding to equation (4.5.3),
incorporating (4.5.12), is found to be
55
dA2
6
- 6 aA - 6 a2A2 -
1 TE=
6
4
3
2
a 3A -
A2
5
R
aA -
-
66A2,
(4.5.15)
k
where
1
1
(Pry dz,
62 = 2bj f F2(z)f(Pr4i
-1
MR K
X(z-2)
g 1
-Az}
{cosh(X(z-1))
+ e
61 = j f 1012 dz;
- 1
for
F2(z) = -
11;
XRk2
z
1
6
3
= 4b2 f
j(cP 0. -
G
f
)
r(Pi
1
- 1
5 ds
(cp
0
ri
)
r(Pi
dsdz,
for
G5 = Rgj cosh(X(z-s));
1
6
4
= 4b2 f
j(cp 0
ri
1
-
ri
)
G
f
1
6i
- (pi cp
((pr'cp
dsdz,
MR
for
G
6
cosh(X(z-s));
= -
1
1
= 2 f
6
)
r
14012 + b21.1)12
} dz; and
5
1
= 2 f 1012 dz
6
6
= 261 /j.
-1
We have thus derived the amplitude equations (4.5.14-15) for
finite disturbances imposed on a basic plane Couette flow between two
parallel plates.
Next, we examine the disturbance amplitudes at the threshold
between stability and instability.
56
IV.6
Criticality
The threshold between stability and instability is criticality.
For the amplitude equations, criticality implies that the magnitude of
all the disturbance amplitudes are not changing as time changes.
Math-
ematically, such a state of equilibrium implies that
da
dt
0 =
dA
dt
(4.6.1)
.
a4 = 0 = a3A = a2A2; in that,
Additionally, we assume that
A2
an
a 2 and
a4, a3A, and a2A2.
are much greater than
Hence, at critical stability (criticality), the amplitude
equations (4.5.14) and (4.5.15) yield
,
0 = y
2c
a2 +
Yo
a 2 + ;a
(4.6.2)
aA,
'skc
and
8
0 = 8
2c
aA +
R
8
A2
+ 2
6
A
2
+
Rkc
gc
Y8
(4.6.3)
aA.
Rkc
The 'c' affixed to nondimensional numbers indicates a 'critical value'.
13
Note that y
2c
and
S
2c
contain critical numbers.
Integrating relation (4.6.1) suggests that
a = mA, in that,
these two disturbance amplitudes are multiples of each other at criticality.
For instance,
m = a(0)/A(0).
Remember that initial condi-
tions are plausible since disturbances are under supercritical conditions.
In particular, if we select
m = 1, then the two disturbance
amplitudes are initially of equal magnitude.
and (4.6.3) become, respectively,
Then equations (4.6.2)
57
°
i2c
°
62c
(4.6.4)
Y8/Rkc'
Y7/Mc
and
1-
Y8/kc
6 5 IRgc
(4.6.5)
266IRkc.
We have discovered, with some approximation, the critical
relationship between the parameters
R, R
and Rk
,
This critical
!
relationship is defined by equations (4.6.4-5), and thus yields the
marginal stability surface,
value' c
S
= Sm(b,R,R
m
gk
-,c)
where the 'eigen-
is the wavespeed with restrictions imposed on it by the
assumed supercritical conditions
10
.
The marginal stability surface is
Sm
62c
Y2c
The graph of
of parameters
(4.6.6)
+ 65/Rgc + 2d6/Rkc - y7/Mc = 0.
Sm would indicate, at a glance, the combination
that lead to a stable flow, an unstable
R, R , and Rk
flow, or a flow in equilibrium.
surface, only traces of
Before graphing
Sm
Since the graph of
Sm
is a
hyper-
can be plotted.
Sm, we should decompose
Y
'2c
and
6
2c'
with the
intention of recovering the critical numbers that these relations contain.
The major difficulty is liberating
X
from the exponential and
hyperbolic functions, while maintaining the existing integrity of the
integrations.
First, we could empirically estimate the
We have three options.
probable expressions for
y
2c
and
6
and then try to write a costly
2c
algorithm to generate the traces of S.
Second, we could linearize
58
and
F (z)
1
F (z), thereby extracting
from the exponential and
A
2
hyperbolic functions.
The third option is to find another option,
like the one we will pursue in the next section.
The second option is known as the narrow gap approximation.
Math-
ematically, this approximation means that z (for us, Az) is assumed
small.
Employing this approximation at this stage of the analysis is
burdened by the difficulty of knowing how to express such constants,
as
cosh(X), linearly.
To ease this burden, the narrow gap approxima-
tion should first be utilized when equation (4.5.4) is invoked into
the "nonlinear" analysis; that is, linearize
v
and
u.
Determination of the Constant, A
IV.7
In section 11.3, equation (2.3.14) describing the velocity field
for steady, laminar plane Couette flow was derived, in accordance with
the assumptions of section 1.4 and boundary conditions (2.3.5), to be:
u(z) =
sinh(Az)/cosh A - 2XAz
2tanh A - 4XA
1
(4.7.1)
2
where
A
=
Rk/M
=
1 + Rk/R
(4.7.2)
> 1.
Similarly, equation (2.3.13) describing the microgyration field for
steady, laminar plane Couette flow was derived to be:
AA {cosh(Az) /cosh A
v(z) =
We notice that
2tanh A
-
4XA
1}
(4.7.3)
59
Rg (R + 2Rk)
A
2
=
=
Rk(R
r(2A - 1)
(4.7.4)
A
Rk)
where
r
So,
(4.7.5)
Rg/Rk > o.
=
X = x(r,A)
is a function of the two ratios, r and A.
If any two
of the triple, r, A, and X, is known, the other can be determined.
We rely, as one ultimately must, on experimental data to dictate
the value of
X
for the fluid flowing between the experimenter's
parallel plates.
To illustrate the selection of X, and to demonstrate the range of
velocity and microgyration, as X varies, for fixed A and z, we present
Table 4.1.
Input values for the calculations are z, A, and X.
Output
values for u(z) and v(z) are calculated from equations (4.7.1) and
(4.7.3), respectively.
Also, (4.7.4)
allows us to determine r from
2
the input values.
As the tabulations in Table 4.1 indicates, the values of u(z) and
v(z) vary slightly for X > 10 (at a fixed z).
Furthermore, increasing
A tends to promote a more rapid convergence to velocity (and microgyration) values that we would expect from classical viscous theory.
Notice, also, the lower values, at
z = 0.99, for the microgyration as
it complies with the boundary condition of
Once X is determined,
numbers can be calculated.
X2
g
useful ratios of the nondimensional
From the expression
- M)/R
R (2R
=
k
v(1) = 0.
(4.7.6)
we get
r
=
R
g
/Rk
=
X2/(2 - M/Rk).
(4.7.7)
60
Table 4.1.
Velocity and microgyration for various X.
FIX1u(.25)1
v(.25)! u(.50) J v(.50) ! u(.75)
1
v(.75) 1
v(.99) 1
*** A = 2 ***
7 E-5 0.01 0.62500
7 E-3 0.10 0.62506
0.667 1
0.62912
16.67 5
0.63101
66.67 10 0.62820
267
20 0.62658
30
600
0.62605
1667
50 0.62563
4267
80 0.62539
8067 110 0.62528
15000 150 0.62521
0.00002
0.00155
0.10239
0.25646
0.25627
0.25316
0.25210
0.25126
0.25078
0.25057
0.25042
0.75000
0.75010
0.75665
0.76101
0.75632
0.75316
0.75210
0.75126
0.75078
0.75057
0.75042
0.00001
0.00124
0.08314
0.24141
0.25468
0.25315
0.25210
0.25126
0.25078
0.25057
0.25042
0.87500
0.87509
0.88091
0.88720
0.88356
0.87970
0.87815
0.87688
0.87618
0.87585
0.87563
0.00001
0.00073
0.04971
0.18772
0.23536
0.25146
0.25196
0.25126
0.25078
0.25057
0.25042
3 E-7
0.00003
0.00234
0.01283
0.02440
0.04589
0.06534
0.09886
0.13810
0.16716
0.19454
*** A
5 E-5
5 E-3
0.526
13.2
52.6
211
474
1316
3368
6368
11842
= 10 ***
0.01 0.62500
0.10 0.62501
1
0.62569
5
0.62615
10 0.62563
20 0.62531
30
0.62521
50 0.62513
80
0.62508
110 0.62506
150 0.62504
0.00001
0.00123
0.08618
0.24610
0.25112
0.25063
0.25042
0.25025
0.25016
0.25011
0.25008
0.75000
0.75002
0.75112
0.75211
0.75124
0.75063
0.75041
0.75025
0.75016
0.75011
0.75008
0.00001
0.00098
0.06997
0.23166
0.24956
0.25062
0.25042
0.25025
0.25016
0.25011
0.25008
0.87500
0.87501
0.87600
0.87837
0.87668
0.87584
0.87563
0.87538
0.87523
0.87517
0.87513
0.00001
0.00057
0.04184
0.18014
0.23063
0.24894
0.25028
0.25025
0.25016
0.25011
0.25008
3 E-7
0.00003
0.00197
0.01231
0.02391
0.04543
0.06490
0.09847
0.13775
0.16686
0.19428
*** A = 100 ***
5 E-5 0.01 0.62500
0.005 0.10 0.62500
0.62507
0.503 1
12.56 5
0.62511
50.25 10 0.62506
201.0 20 0.62503
452.2 30 0.62502
50 0.62501
1256
80 0.62501
3216
110
0.62501
6080
11307 150 0.62500
0.00001
0.00117
0.08321
0.24388
0.24999
0.25006
0.25004
0.25003
0.25002
0.25001
0.25001
0.75000
0.75000
0.75011
0.75021
0.75012
0.75006
0.75004
0.75003
0.75002
0.75001
0.75001
0.00001
0.00094
0.06757
0.22957
0.24844
0.25005
0.25004
0.25003
0.25002
0.25001
0.25001
0.87500
0.87500
0.87510
0.87523
0.87517
0.87509
0.87506
0.87504
0.87502
0.87502
0.87501
0.00001
0.00055
0.04040
0.17852
0.22959
0.24838
0.24990
0.25002
0.25002
0.25001
0.25001
3 E-7
0.00002
0.00190
0.01220
0.02380
0.04533
0.06481
0.09838
0.13768
0.16679
0.19422
v(.25)
u(.50)
v(.50)
u(.75)
v(.75)
v(.99)
1'
x
u(.25)
61
Then
Rg/R
=
(Rg/Rk)(Rk/M) - Rg/Rk
(4.7.8)
R/Rk
=
(Rg/Rk)(R/Rg).
(4.7.9)
and
The numbers, that are the ratios (4.7.7-9), still apply at criticality.
The values of the critical non-dimensional numbers
R
kc
,
,
c
R
,
gc
and
involved in the stability of plane Couette flows, can now be
From the marginal stability surface (4.6.6),
theoretically predicted.
the relation
Rkc
R
P.
S
xc m
= 0
{-(kc/Rgc)65
yields
266
(kc/Mc)Y71/(62c
12c).
(Rgc/Mc)Y71/(62c
12c"
(4.7.10)
Similarly,
Rgc = {-,55 - 2(Rgc/Rkc) 66
Mc
f-(Mc/Rgc)(55
Rc
McRkc/(Rkc
2(Mc/Rkc)66
Y71/(62c
(4.7.11)
(4.7.12)
Y2c"
(4.7.13)
Mc).
Recall that the difference, S2c
numbers.
y
2c
, contains additional critical
Consequently, the relations (4.7.10-13) only implicitly
establish values for R , R
c
,
gc
and R
kc
.
Or so it seems.
With the
ratios of the nondimensional numbers (4.7.7-9), S 2c - y 2c
(Refer to section VI.5.)
be shown to be a constant.
though, requires that
A
,
can indeed
This result,
is known.
Numerical procedures for this plane Couette flow problem are
postponed until chapter VI.
62
V.
STABILITY OF A BASIC ROTATIONAL COUETTE FLOW
The laminar rotational Couette flow elucidated in chapter III, will
now be disturbed by the imposition of a disturbance wave.
The stability
analysis of this chapter follows the procedures of the Stuart energy
method (Stuart,1958).
Note that this procedure will use the solutions
of the linearized theory, which are pursued in section V.1.
The solu-
tions sought, in this (linear) case, satisfy the micropolar analog to
the Orr-Sommerfeld (MOS-) energy equations, which are also derived.
To study its stability, the basic flow is superimposed with an axisymmetric, finite disturbance.
(See section V.2.)
The imposed distur-
bance, having zero mean, provides homogeneous boundary conditions for
the nonlinear equations of motion governing the disturbance flow.
The disturbance energy equations are derived from the disturbance
equations in section V.3.
The energy equations suggest a physical in-
terpretation of the possible mechanisms involved in the transition from
stable to unstable flow.
(See section V.4.)
These nonlinear energy
equations (hence, the nomenclature of energy method) are then assumed
to be solved by wave forms of the same spatial form as the 'marginal'
disturbances of the linearized theory, but with unknown amplitude.
In
fact, the solution to the nonlinear disturbance energy equations are
assumed to be separable into a spatial part, which is known from the
linearized theory, and a temporal (time) part, which defines the amplitude of the imposed disturbances (at least, near marginal or critical
stability).
63
Since the spatial part of the disturbance is known, ordinary
differential equations, describing the disturbance amplitudes, are
Such equations are called
found from the disturbance energy equations.
amplitude equations when micropolar theory is involved, or else Landau
equations when classical theory is used.
The possible growth, decay,
or equilibrium states of these disturbance amplitudes can then provide
the stability criteria.
For instance, in section V.6, we derive
marginal stability surfaces, and extract a theoretical prediction for
the critical nondimensional numbers, Rc
,
R
,
gc
and R
kc
,
involved in the
stability of rotational Couette flows.
In essence, we are re-working the stability problem for Couette
flows between coaxial, rotating cylinders, with the enhanced insight
permitted by the micropolar theory of fluid dynamics.
V.1
Linear Stability Analysis
Employment of the Stuart energy method, will require the shape of
the marginal disturbances of the linearized theory, so that numerical
calculations for theoretical predictions can be performed.
Thus, the
goal of this section is to derive the MOS-energy equations pertaining
to the linearized rotational Couette disturbed flow problem.
The sol-
ution to these coupled equations is the shape of the marginal disturbances that we will be utilizing in later calculations.
From section 1.4, the surmised field equations
in nondimensional form, are:
(1.4.1) - (1.4.3),
= 0,
- Vp +
1
M
2
V v +
V(Vv) +
k
1
1
vxv
Rk
2
V v +
R
v = (u,
where now
v = (c,
and
v, w)
+ j~vv,
= 2Vk + j3)2/3t
Vxv
Rk
+ vVv,
3v/3t
=
(5.1.3)
We will maintain a
n, v).
x = (r,0,z).
cylindrical coordinate system, in that, the point
In section 111.3, the basic flow was derived to be of the form
.
.
v = v(r) e
v = v(r) e
0, -
,
.
,
and p = p(r), where
and
e
8
z
e
z
are re-
spectively unit vectors along the 0- and z-axes of the cylindrical
coordinate system.
To study the stability of this flow, let
v(x,t) = v(r) e
v(x,t) = v(r) enu + v'(x,t),
z
+v"(x,t),
(5.1.4)
p(x,t) = iS(r) + p'(x,t),
v'
,
where
is the disturbance velocity, v'
gyration, and
is the disturbance microOn substituting these
p' is the disturbance pressure.
expressions into equations (5.1.1) - (5.1.3), we obtain the equations
of motion governing the disturbed flow.
By utilizing the fact that the basic flow already satisfies the
equations of motion, we have
(5.1.5)
V17' = 0,
- Vp
1
,
+
M
V-D 17" +
.-
Rk
3
Vxv' =
Dt
,--
+
,dv
v D
+ v"-Vv,
e
v' + u
,
dr 0
r D8
(5.1.6)
1
.v) +
Rb
+
1
m2 ,
v v +
Rk
vm
xv
)
= (.D
33t + 3r 30
+
Rk
v
+ 3u
dr
+
e
z
(5.1.7)
65
By neglecting the quadratic terms
y"Vy'
and y"Vv'
(or equi-
valently, assuming the disturbances are infinitesimal), we obtain the
Since the
linearized equations of motion governing the disturbed flow.
coefficients of
v'
and
v', in the linearized equations, depend only
on r, the equations admit solutions which depend on z and t exponentially.
To maintain a physically realistic wave, the effects of axisym-
metry (i.e. solutions independent of 0) are induced.
Consider therefore solutions of the form
x'(1c,t) = v(r) exp{ib(z - ct)};
v'(x,t) = v(r) exp {ib(z - ct)1;
(5.1.8)
p'(x,t) = p(r) exp{ib(z - ct)}.
The real parts of the expressions must be taken to obtain physical
quantities.
Requiring that the solutions remain bounded as
implies that the wavenumber b must be real.
The wave speed
+03
z -4-
may
c
becorwlex,inthat,c=cr+ic..The expressions thus represent
waves which travel in the direction (0,0,b), with wave speed
which grow or decay in time like exp(bcit).
to be (asymptotically) stable if
neutrally stable if
bc. = 0.
bc.
c
r
,
and
Note that a wave is said
< 0, unstable if
bc. > 0, and
Marginal stability occurs if
for critical values of the parameters (e.g. R, Rk, R
,
bc. = 0
and Rb) on which
for some neighboring values
the'eigenvalue'cdepends,butbc.>0
of the parameters.
The ratios of the parameters R, Rk, R , and Rb for marginal stabg
ility are found in section V.7, which can thus give some criteria for
stability.
The critical relationship between the parameters, when dis-
covered, yields the marginal stability surface.
Note that neutral
66
For comparison, note
necesarily marginal stability.
stability is not
bci
that on a neutral stablility surface, bci = 0, but
is not neces-
sarily positive for any neighboring values of the parameters.
The
minimum values of R, Rk, R , and Rho on all the marginal stability surfaces are called the critical numbers
there is flow instability for any
R
,
c
R > RC,
R.
xc
,
R , and P.
gc
be
Rk >
R.
xc
,
hence,
;
> R
R
g
gc
,
and
> Rbc.
Forcompleteness,wementionthatifbcr+Oasbc.approaches
zero from above for a disturbance, oscillatory instability sets in.
This is sometimes called overstability.
Also, if
be = 0
at marginal
stability (i.e. bc. = 0 = bc ), then there is said to be an 'exchange
r
of stabilities', whereby instability sets in as a steady secondary flow,
such as in the case of the convection cells that arise when a fluid is
heated from below (Perez-Garcia & Rubi, 1892).
If we now let
D = d/dr
and
D* = d/dr + l/r, then on substituting
the expressions (5.1.8) into the linearized equations (5.1.5-7), we obtain, after some rearrangement, the following (coupled) system of ordinary differential equations:
D*u
+
bw
= 0,
(DD*
b2 + ibMc)u = MDp + ibMn/Rk,
(D*D
b2
_ l/r
+ ibMc)v = M(ibc - Dv)/Rk + Mv'u,
(DID - b2 + ibMc)w = ibMp - mip*n/Rk,
(DD* - b2 + ibcjRg - 2R /Rk )c = ibRgv/Rk - Rg(DD* + ibD)v/Rb, (5.1.13)
g
(D*D
b2 - 1/r2 + ibcjRg - 2Rg/Rk)n = Rg(ibu
Dw)/Pk,
(5.1.14)
67
R D*v/Rk +
2R /Rk )v = ibR (D*c - ibv)/Rb
(D*D - b2 + ibcjRg
g
(5.1.15)
+ jR ;'u.
g
Here primes denote differentiation with respect to r.
Notice that
DD* t D*D.
The strict adherence boundary conditions, applied to the disturbance flow, imply
v
w-O c-n-v
at r= R 1 ,R 2
(5.1.16)
.
Having restricted to axisymmetric disturbances, we introduce the
disturbance stream function
V(r,z,t) = (D(r) exp {ib(z - ct)}.
Thus, two of the disturbance velocity components will be
u
= -
1 3T
r 3z
and
w- = 1 3Y
r 3r
Furthermore,
and
u = -ib(D/r
r
Notice that
1 d(1)
w = T
1
dr
= D
*
Wr).
is a complex-valued function.
By inserting (5.1.17) into the six equations
eliminating p
(5.1.17)
(5.1.10-15), and
from these equations, we reduce the unknowns to five:
v, c, n, and v.
Finally, effecting some rearrangement gives the
micropolar analog to the Orr-Sommerfeld (MOS-) energy equations.
result is
The
68
b2)21A.
(DID*
ibM
r
(D *D - b2
1/r2)v
(DD*
(DD
b2)"
ibRg
+
DD v
*
ibRb
b2 - 1/r2)
ibRg
cv
+
ibM
(DD
b2)^
ibRk n
(2E
Y
Dv
Rb
+
-
=
(5.1.18)
r
ibc - Dv
ibRk
(5.1.19)
2c
2n
ibRk
ibRk
.
D v
b2)"
+ jv 0/r
v + cjv +
ibRk
ibR
.
=
(5.1.20)
ibRk
Rk
b2)10\
\r/
(DD*
+
-
cjC
+
c
*
v'O/r
+
)"
2(7
b
2v
ibRk
+
b2 v
D C
*- +
ibRb
Rb
(5.1.21)
(5.1.22)
g
The boundary conditions are
A
A
A
b0 =v= D0 =0=C=n=v
A
The unknowns
A
at r= R 1 ,R 2
.
A
0, v, c, n, and v
governed by (5.1.18-22) will be
required in later calculations, as disturbance shapes.
calculations, v
spectively.
and
v
(5.1.23)
Also, in the
are taken as in (3.3.12) and (3.3.11), re-
Note, however, that in the nonlinear theory, the mean
velocity v and the mean microgyration v are different from those of
laminar flow (as taken above) because of the interactions between the
mean flow and the disturbances.
Next, we study, with the nonlinear theory, the stability of the
basic flow, which is disturbed by an axisymmetric, finite disturbance.
69
Imposition of Finite Disturbances on a Basic Flow
V.2
Axisymmetric Couette flow refers to a Couette flow regime that is
(theoretically) independent of the azimuthal coordinate 0.
We will be
analyzing the stability of a basic rotational Couette flow, that is
disturbed by an axisymmetric, finite disturbance in the rz-plane.
More-
over, since the cylinders are assumed to extend to infinity in both the
positive and negative z-directions, the velocity and microgyration
fields will be independent of the 0-coordinate.
Restricting to axisymmetric flows, the continuity equation
1
-(ru) +
r ar
2
0
3z
defines a Stokes stream function for the fluid
This stream function is decomposed into the sum of two functions,
flow.
one representing the mean flow, the other representing the finite disturbance flow.
The analysis of the nonlinear field equations governing
the finite disturbance flow employs procedures embodied in the Stuart
energy method, with which we begin.
For flow between two coaxial cylinders in constant relative
rotation (rotational Couette flow), we impose a disturbance travelling
in the direction of the basic flow which has the form
(5.2.1)
'Y(r,z,t) = KT (r) exp{ib(z - ct)} + K T (r) exp{-ib(z - ct)}
1
where
(c
K
1
is an arbitrary complex constant, wave speed
c = c
r
+ ic.
> 0), and the tilda ti denotes a complex conjugate.
r =---
Now, let the stream function for the flow be represented by the
Fourier series expansion in z, as
70
T(r,z,t) =
+
=
=
+
{4 )m(r,t) e
X
mibz
-mibz,
1(0,
+
(5.2.2)
y.
m(r,t) e
m=1
In the linearized theory,
and
reduces to (5.2.1)
(0'
represents the steady stream function,
j
(or equivalently, the function
T'(r,z,t)
For the nonlinear theory, the sum on the right
used in section V.1).
represents the finite disturbance (0' , while
(0
is
the mean stream
i
function, where the mean (average) is taken with respect to
7
z
over a
of the disturbance
disturbance wavelength of 21T/b, b being the period
wave.
'Y, the Fourier series expansion gives
With the stream function
CO
u' =
u = -
mibz
X {u'(r,t) e
m=1
v = v + v' = v(r,t) +
X
+
{v"(r,t) e
-(r,t) e
-mibz
m
mibz
+ v (r,t) e
},
-mibz
m=1
w =
1
a = w' =
{w"(r,t) e
mibz
+ ie(r,t) e
-mibz
1,
m=1
= v(r,t) +
v = v + v
X
{v"(r,t) e
mibz
+
-mibz,
M(r,t) e
m=1
-mibz
mibz
m=1
co
mibz
n = n" =
X
m=1
{nr'n(r,t) e
+
m
(r,t) e
-mibz,
f-
(5.2.3)
71
The over-bars now denote a mean value, and the primed terms denote
disturbance flow variables.
Defined for equations (5.2.3) are
w'(r,t) =
u"(r,t) = - mib0 /r,
m
1
r
DO /Dr
(m > 1).
m
(5.2.4)
For the resulting axisymmetric flow, mathematically, we prescribe
that the velocity and microgyration vector fields are, respectively,
v = (u(r,z,t), v(r,z,t), w(r,z,t)}, v = {(r,z,t), n(r,z,t), v(r,z,t)}.
(5.2.5)
Applying the prescription (5.2.5) to the field equations (3.2.1) (3.2.7) yields:
r ar
3r
+
(ru)
+
:1-:
M
az
{32u/art
3u
au
= TT
+ u TT
(5.2.6)
= 0,
a2u/3z2}
2(u/r)
+
v2
-r-
+ w
au
(5.2.7)
{92v/3r2 + 1-( v/r) + a2v/z2}
=
_
av
+ u
1 1.,2
P
1-d
=
3w
3t
Dv
3v
uv
+ w
+
3z
ar
u
3w
3r
r
w
1
Rk
(
az
(5.2.8)
w/3z21 + 1
k
3w
3z
,
Br
'
2
h'
w/ar2 +
=
9T1
Rk 3z
Dr
3n
n
-r
=
ar + r
(5.2.9)
72
02c/are
R
3raz
3r
Rb
a
{32u3r2
__.(c/r)
B
a2c/Bz2}
=
r
g
1
3v
2C
Rk
Rk
3C
at
u
vn
3r
r
+ jOn/3t
Rb Dr3z
r Bz
+ u 3n/3r
a2,/z2)
+ vc/r
--Br
3u )
2
3z
Rk r
ri
+ w 3n/3z1
02,/are
1
(5.2.16)
(3w
Rk
3r
Rg
BC\
w 3z)
32,/3 21
2._(n/r)
2 n/Br2
DC
(5.2.11)
D2v/Dz21
av
r Br
R
g
-
1 ( Dv
Rk
Dr
v )
r
3v
Ti..)
2v
Rk
j
;t
(5.2.12)
Dr
The conditions to be applied throughout this procedure are:
(C1)
that the mean velocity v and the mean microgyration ; assume
the same values on the cylinders as do the undisturbed velocity and
microgyration;
(C2)
that the disturbance velocities
microgyrations
(C3)
c", n', v'
u', v', w'
and the disturbance
vanish on the cylinders; and
that just enough external power is supplied to maintain the
angular speeds of the cylinders at constant values, in accordance with
the variation with time of the mean skin friction on the cylinders.
Consequently, for disturbed rotational Couette flow, acceptable nondimensional boundary conditions are:
v = 1
and
at r = R
1,
v = Q2 R2 /S2 R
1 1
at r = R
u' = v' = w' = 0 = c' = n' = y'
,
2
at r = R ,R
v = 0
at r = R ,R
1
1
.
2
,
2
(5.2.13)
73
In order that the pressure gradients will also balance the
remaining terms, the pressure must be of the form
p = rp*(t) + p**(r,t) + p'(r,z,t) =
= rp*(t) + p**(r,t) +
-mibz
mibz
+ Pjm(r,t) e
{pm(r,t) e
}
X
m=1
(5.2.14)
where asterisks denote labels for the terms in the decomposition of
Substitute equations
the mean pressure, as used in equation (4.2.11).
into equations (5.2.7) - (5.2.12),
(5.2.3) and (5.2.14)
The equations arising from equating
and equate the Fourier components.
are equivalently found by taking the mean
the terms independent of
z
of equations (5.2.7-12).
The result is:
1
a
P* = r 3r (ru
1
1 02/3r2
1 3
o =
.
o=j
=
(1 a
)
(1
31-T_ 02/3r2
(5.2.15)
+ v'2),
1 3v
Rk3r
1 1r2 v
r 3r
P**(r,t)
2
(v
)
3v
1
at
T.2 3r
3
(5.2.16)
(r 2u v'),
(5.2.17)
(ru"w") = 0,
(5.2.18)
- v"n7r
,--
(ru n') + v-c-/r I
(5.2.19)
,
1- 3
1
(
1
r 3r
Rk\ 3r
r
g
v +
23
.
+ 3
317)
1 3
+ .i77(TI(ru"v')
.
(5.2.20)
Note that quanities involving the mean disturbance variables, like
u'v"
and
u"v", are independent of z.
Also, note that the continuity
74
1
aw
D
-7,--(ru') = -
equation (5.2.6) implies that
(5.2.21)
r or
The equations governing the disturbance field quantities are now
found on subtracting (5.2.15) from (5.2.7), (5.2.16) from (5.2.8),
,
and (5.2.20) from (5.2.12), to be:
BP-4. 1(22:
M
Br
(2:4 11.1:
Rk az
RV
1
(v-)
a
-51-r7/
1 a2w-+ 1 aw-4. 92w'
2
r ar
az2
Br
m
az
lt2c."
Rb ar2
fc-)
32v-
BrAr )
r.az
(C
2c'
B
4- ID
j
( B2n'
1
R
g
3r
at
in')
arr/
2
in'
r
av'
(ac-._ av-
+ Rk az
az2
-a7-2-
ap'
(5.2.22)
X1
r
4
az 2
ar 2
ila2v-
nv v
auat
an'
1
Brkr/
-5--r
v
(a;
7--dr
+ X2'
(5.2.23)
--r
\
1
41.-+
Rk
0
32c-
R
,
+ u
J.-
3r2
g
Bw
at
(5.2.24)
3
9
av"
Rk 3z
1
3r r
3z2
(5.2.25)
1-
X4)
92n1
'
au')
1
Rk
az 2
Car
2
n'
+ Rk r
az
(3n'
(5.2.26)
X5)
3
1
Rb
( 32C'
araz
+
32v1
1 BC'
r 3z
j
Rk
t
+ u
1
Rg
3z2
(v-*
2v'
'
-3;
Br
a
1
_
Br2
r Br
2
(
+ X6)
ay-
32v3z2
= _
1
Dv'
v'
)
Rk Car
(5.2.27)
75
where
u
x1
1
...312-
3r
-
.av-
v-2
311'
+
3z
-
1 D
(ru--)
+
1 -7.2
v
,
1 3
w..31.7"
u' v'
X2 = u
-9,
,
r 3r
3z
.23r(r21-1-v)
'
1 3
...3w"
,3wkru-w-)
+ w
r ar
3z
3r
,
x3 = u
+ w, 2 : -
X4 = u 3r
X5 = u
x
6
= w
vn
v
1
,an'
...3v"
9z
+
v'c'
+ w
r ar
v'c'
,3n"
3z
1 3
-Dv'
A As
- ----kru v )
r Dr
Dr
,
As
t
)
----ru C
az
-
1 D
(ru'r1")
,
,
,
+ u
.
These nonlinear equations will be referred to as the disturbance
equations.
V.3
Disturbance Energy Equations
In this section we will derive the disturbance energy balance
equations for axisymmetric, finite disturbances.
By properly preparing
the disturbance equations (5.2.22) - (5.2.27), we derive the integral
form of the MOS-energy equations.
Note that since the cylinders are
unbounded (in the z-direction), we have assumed the disturbances
1.1'
and
v-
to be (spatially) periodic in
z, and thus, the following
integrations with respect to z can be taken over exactly one wavelength.
76
Begin by multiplying (5.2.22) by
u-, (5.2.23) by
v', and
w'. Add these equations; then utilize the simplifications
(5.2.24) by
and integrations demonstrated in Appendix A.
Thus, the first
Disturbance Energy Equation is derived without approximation to be:
a
f r
M
1
Rk
r
(
Dw'
317-12
ff (u-v-)
=
ff (u'2 + v-2 + 14'2) rdrdz
9u" 12
+
)
_ r
laar,rv-))
) rdrdz
rdrdz
k3r.
3z /
ri(r-9vaz
v-9
-;Tr(ry )
9u-1
T; )
ar
n
rdrdz
(5.3.1)
where the integrals are evaluated over a volume bounded by the
cylinders
r = R1,R2
The mean velocity
and by one wavelength
v
z = 0,27/b.
occurring in equation (5.3.1), is derived
from (5.2.16), and is given by
{32/ar2
9
r 9r
_ 1/r2};
Next, multiply (5.2.25) by
by
Bv
1
IT;
C.,
9v
1
3
-r729r(r2u-v-).
(5.2.26) by
(5.3.2)
n-, and (5.2.27)
v"; simplify; then integrate; thus deriving the companion
Disturbance Energy Equation without approximation to be:
77
rr
TE- JJ1/23(CA2
.
-
2
n'2 + vA2) rdrdz
.2
r
n2
+
ff(-juAvA)
=
+ vA2) rdrdz
3v
rdrdz
-
-
k
1
az
Rk
(
1
+ 2
r 3r
(
1
(
/-
vA3
ar.ry
(
2
rdrdz
-
-
\ 24. / an -
/
3z )
+
ar
+
)
al2) rdrdz
av'aC
3r 3z
(r1 3
)2
a
r 3r
+
3z
2
a
Rb
3u"1
nAl3w'
kar
(c,31.7"
(5.3.3)
)rdrdz.
3z
az
where again the integrals are evaluated over a volume bounded by the
r = R ,R
cylinders
1
and by one wavelength
z = 0,27/b.
2
The mean microgyration
v
occurring in equation (5.3.3), is
derived from (5.2.20), and is given by
132
1 3 )-
R Tr2
T 3r 1)
2)
Rk
1
Rk ar
1V177.
.
'
3
+ 1 3 (ru-v")
3t
r 3r
(5.3.4)
g
Equations (5.3.1) and (5.3.3) comprise the integral form of the
micropolar analog to the Orr-Sommerfeld (MOS-) energy equations.
Following a physical interpretation of the flow mechanisms suggested by equations (5.3.1) and (5.3.3), we will derive the amplitude
equations,
with the Stuart energy method.
78
V.4
Physical Interpretation of the MOS-Energy Equations
The goal of this section is to identify and physically interpret
the terms appearing in the disturbance energy equations (5.3.1) and (5.3.3).
(Interpretation is similar to that given in section IV.4.)
To enrich
the physical interpretation, equations (5.3.1) and (5.3.3) are, respectively, re-written in dimensional form and a briefer notation introduced.
12p(u-2 + v'2 + w'2) rdrdz
9--at If
taw"
av-I2
ff(-pu'v , )
=
,(aw'
3v'
Or
3E
= II
at
a
If 1/2p7(c...2
3u')
v
az
r 3r
- (u+K)I2
12
rdrdz
=
ff(-Pju'v')
r Br
fl,12
az )
r
- Kij (c ,317" +
az
ar
rdrdz
av-i2
+ (1r
ar
(rri'))
+
)
+
)
/
rdrdz
4Aaz )
rg
Dr
-
-
-vD
(rv )
r 3r
)
3z
-
(5.4.1)
+ KI2.
V-2 + n'2) rdrdz
2---(rc))
rdrdz -
rdrdz.
\ 2
- y 11(
Br- v
T
3z
Briefly,
v
3v
/1 a
au- 12
(p+K)ff(-
Kfr(c,
(
rdrdz
-
79
,-2
-
2Kff(C2 +
+ v"2) rdrdz -
r
1 3
(a+f3)ff
(
= H
-33e
E
+ 2
2r '2z:
-; Y:
(
Briefly,
2
1(re))
- yH
1
- 2KH
+ KI
2
3
rdrdz.
+ (113z 12
- (a+a)H
3
(5.4.2)
4.
In equation (5.4.1), the term on the left-hand side gives the rate
of growth of the disturbance kinetic energy within the volume considered.
On the right-hand side of (5.4.1), the term
II
is the integral of the
(3;/3r - 177/r), and
product of the Reynolds stress and the flow shear
represents the "translational" rate of transfer of kinetic energy from
the mean flow to the disturbance.
so
-(p+K)I
The term
represents the rate of
(p+K)I
is always positive;
2
(p+K)-viscous dissipation of the
2
kinetic energy of the disturbance due to translational and rotational
effects of the macro-volume elements in the volume considered.
The term KI 3
(5.4.2).
The term
is the common link between equations (5.4.1) and
is the integral of the dot product of the micro-
I
3
gyration with the curl v, and geometrically represents the Swirl created
by the disturbance.
= v"(Vxv') =
I
Mathematically,
1-
3/3r
rri"
v"
13/30
3/3z
3
rv"
Notice that the scalar triple product
in two other forms.
In that,
v"(Vxv")
can also be written
80
v'.(Vxv') = v".(Vxv-) = V-(v'xy').
The third variation is the divergence of the Coriolis acceleration.
The Coriolis acceleration, common to the mechanics of moving
coordinate systems, equals
Note that microgyration is angular
2y-xv-.
For a fluid motion described by micropolar theory, the
velocity.
Coriolis acceleration, 2v'xv",
represents the resultant from the interaction of the rotation of moving micro-volume elements and the present
motion of the ambient macro-volume elements for the existing flow in
the volume considered.
v'(Vxv') =
Thus, a nonzero swirl, i.e.
= V(v-xy') t 0, acts as a source, if
> 0 (or a sink, if
I
3
< 0),
1
3
for spreading (or gathering) energy necessary to create a turbulent
flow.
Most of all, the swirl is the coupling mechanism between the
micro- and macro-continuum volume elements.
In equation (5.4.2), the term on the left-hand side gives the rate
8
of growth of the disturbance Microenergy of Rotation
within the volume
On the right-hand side of (5.4.2), the term
considered.
H1
is the
integral of the product of the mean couple stress and the mean micro-
gyration gradient, and represents the "rotational" rate of transfer of
microenergy of rotation from the mean (micro)flow to the disturbance.
yH
The term
is always positive; so
2
-yH
represents the rate of
2
y-viscous dissipation of the microenergy of rotation of the disturbance
due to the translational and rotational effects of the micro-volume
elements in the volume considered.
so
The term
is always positive;
2KH
3
represents the K-viscous dissipation of the microenergy of
-2KH
3
rotation of the disturbance due to the rotational effects of the micro-
81
volume elements in the volume considered.
(a+)H4
The term
always be positive depending on the sign and magnitude of 2
if
H
-(a+a)H
then
> 0
4
21)-111';
ar az
(a+a)-viscous
represents the rate of
4
may
dissipation of the microenergy of rotation of the disturbance due to
the rotational, translational, and dilational effects of the microvolume elements in the volume considered.
(11+0'2, 1H2,
In summary, fluid flow stabilizers are the terms
which represent viscous dissipation mechanisms.
2KH3, and (a+a)H4
listing
and
I
1
positive.
H
Note that
E, e, Il, H1.
Fluid flow destabilizers are the terms
as destabilizers, presumes that these terms are
1
The intermediary between stability and instability is the
swirl term
KI3.
Combining equations (5.4.1) and (5.4.2), reveals
(E + e)
(Note that
=
- (p+K)I
+ H
I
1
1
-
- 2KH
2
2
3
- (a+a)H
+ 2KI
4
.
3
(5.4.3)
now possesses the factor of 2, in agreement with the
2KI
3
Suppose that
accepted definition of the Coriolis acceleration.)
I
1
+ H
then
1
+ 2KI
If
> 0.
3
a(E + e)/at > 0.
I
1
+ H
> (u+K)I
+ 2KI
1
3
2
+ yH
2
+ 2KH
3
+ (a+a)H 4
This means the disturbance energies are growing,
and the disturbances are increasing in amplitude (i.e. the flow is becoming unstable).
2KH
3
+ (a+a)H
4
Conversely, if
then
11 + H1 + 2KI3 < (11+K)I2 + yH2 +
a(E + e)/at < 0.
This means the disturbance
energies are decaying, and the disturbances are decreasing in amplitude
(i.e. the flow is becoming stable).
then
Ideally, if
+ 2KI
+ H
I
1
a(E + e)/Bt < 0, i.e. the flow is becoming stable.
1
< 0
3
82
Finally, equations (5.3.2) and (5.3.4) show how the distribution
of mean velocity and mean microgyration are affected by the viscous
stresses, pressure gradients, Reynolds stress, and mean couple stress,
due to the disturbance.
and
v
An equilibrium flow is possible if
v
can be so distorted, by the Reynolds stress and the mean couple stress,
I1 + H1 + 2K13 = (u +K)I2 + yH2 + 2KH3 + (a+OH4, which implies
that
then that
3(E + e)/3t = 0; in that, equilibrium.
The equilibrium
state will play a crucial role in the analysis presented in the next
sections.
V.5
Amplitude Equations
Recall, in the Stuart energy method, the stream function for the
disturbed flow, 'P, represents a mean flow together with a periodic
disturbance consisting of the fundamental harmonic, (1)0, with wavelength
2n/b, and higher harmonic components, (1)/,
wavenumbers
4)2,
,
having
nb (n > 1), but the same (real) wave velocity, cr, which
is assumed to be independent of time.
The amplification or damping of
the finite disturbance, and the consequent changes in the mean velocity
v
and the mean microgyration
v
are accounted for by the dependence
of all the (1)-functions on time t.
We assume that the higher harmonics
,
(1)
(f)
2'
are zero.
3
Furthermore, we assume that the disturbances are under 'supercritical'
conditions meaning that the nondimensional numbers
R, R,, R
,
and R
are above the value which is critical for the linearized instability
theory.
b
83
Moreover, a disturbance under supercritical conditions amplifies for
small amplitudes.
the function
(I)
1
A suitable initial condition, therefore, is that
shall be an exponentially increasing function
(r,t)
-00; in fact, 01 has to be the appropriate
of time t in the limit as t
function(D(r)exp(bcit),wherec.>0, of the linearized instability
theory (Stuart, 1960)12
u', v', w', c", n", and v'
Assume disturbances
are similar in
'shape' to the solution given by the linear theory, but that the solu-
a(t) in the case of v', and
tion is multiplied by an amplitude factor:
A(t) in the case of v'.
(1);.(r,t) = a(t)(1)(r),
I.e.
vi:(r,t) = a(t)v(r),
v'(r,t) = A(t)v(r).
and
1
For an equilibrium state, we presume that
aT)/9t = 0 = al7/at.
With this presumption, we have equations (5.3.2) and (5.3.4) yielding
4d2;
d (T21
M
drIr
d
2
d;
d
1
Rk dr
9
(5.5.1)
-172 dr (r-
and
2v +
Iv
r
.
1 d
Rk
k(d2v
(ru'v')
Rg
+
1
d;)
(5.5.2)
a 17
dr2
Integrating (5.5.1) and using (5.5.2) leads to
d2v
+
dr2
r dr
MR
-k
where
f(r)
-
X2;
=
f (r)
MR
g
(u v )
MR
r
1-
f
-
Rk
1 d
(ru'v') + --g- C
u'v' ds + jRg
Rk
1
dr
R
1
represents the non-homogeneous part of
(5.5.3)
(5.5.3), C1
is an
84
integration constant, and again
X2
is as given in (2.3.9)1.
The homogeneous equation from (5.5.3) is solved by
v (r)
=
I
and
h
where
0
C
I
2
K
0
+ C
(Xr)
3
(5.5.4)
K (Xr)
0
are modified zero-order Bessel functions of the
0
are additional
C2
and
C3
ci/dr
and
dT7/dr - ;/r, for the
first and second kind, respectively; and
integration constants.
Our immediate purpose is to find
disturbance energy equations (5.3.1) and (5.3.3), so that the amplitude
equations can be derived.
Using variation of parameters, we find a particular solution for
equation (5.5.3) of the form
v (r)
=
I
o
(Xr)
sK0(Xs) f(s) ds
f
sI
- K0(Xr) f
(Xs) f(s) ds.
o
1
1
(5.5.5)
So, in integral form, the general solution to equation (5.5.3)
is the mean microgyration, given by
=
v- (r)
v
h
The mean microgyration gradient,
dv
dr
in the radial direction, is
sK (Xs)f(s) ds +
C K (Xr)) + XI (Xr) f
1
3 1
R
X(C I (Xr)
1
2
(5.5.6)
(r) + vp (r).
0
1
r
+ XK
(Xr)
sI
f
1
1
(Xs)f(s) ds.
o
(5.5.7)
85
Next, integrating equation (5.5.2) with respect to r, after having
incorporated (5.5.6), leads to the mean velocity as follows:
Let
d;/dr + v/r
where
C
=
=
v(r)
Then
represent the right-hand side of (5.5.2).
g(r)
1
has solution
g(r)
r
isg(s) ds
(5.5.8)
+ C4)
is another integration constant.
4
The mean velocity is
dTr
dr
1
r
+ C4
jsg(s) ds
T2
=
(5.5.9)
+ g(r).
Therefore, the flow shear, required in (5.3.1), is given by
-
dv
dr
-
v
=
-
F(s)
=
r
r
2 (isg(s)
ds
Let
(5.5.10)
+ g(r).
+ C4
MR C
g 1
f(s)
Rk
Then
MR
f sg(s) ds
=
R r
(C 2 I 1 (Xr) - C 3 K 1 (Xr)
_.
A(R+ Rk)
R
ARk
Rg
r2
C
+
1
sI
R
0
(Xs)F(s) dsdE +
1
1
+ Rkj(ru'v') -
-
sK (Xs)F(s) dsdE - 2fEK0 (XE)f
0
+ 2f EI (XE) f
0
)
sKo(Xs)F(s) ds + K1(Ar)
Xr(Ii(Xr)f
R
R
1
sio(xs)F(s)ds).
f
1
(5.5.11)
86
For brevity in the calculations that follow, we write
fsg(s) ds
=
MR
R r
A(R+
C I
2
(Ar) - C K (Ar)
1
I
C1 r2 +
ARk
1
3
G(r),
1
(5.5.12)
where
represents the last four terms on the right-hand side of
G(r)
contains all the nonlinear mean stress and mean couple
G(r)
(5.5.11).
sg(s).
stress terms for this integral of
Utilizing the boundary conditions (5.2.13), the integration con-
C
H
MR2
g
1
(H
(3:2
0
= I (AR
0
1
(R -R
2
=
AR
0
2
)
2
R
1
1 - AR {I (AR )K (AR 1
2
0
2
sI
o
(As)F(s) ds,
1
)
- K (AR )I (AR )1
0
1
,
1
1
2
2
H' = 1 - QR
- R2
)
2
+
AH
2
2 R
H
2
+ G(R )/R
1
2
2
= I (AR )K (AR
0
0
r
)
1
(R 2
0
(5.5.13)
K (AR
sK (As)F(s) ds
)
R
0
R
2
2
= H'I (XR )/H,
1
3
2
R
H
g
1
H R2
1 1
AH
x147
(R+Rk)
C
- H1K (AR )/H,
2
g R2
O 1 )
'
Here
2
C
+ g0H' /H),
1
H
C4 = 1 4-
H
for equations (5.5.6) and (5.5.8) are:
Cl, C2, C3, and C4
stants
1
0
2
)
- I
0
(AR )K (AR
2
0
1
2
,
=
2
R
1 1
'
)
g
H =
AR {K (AR )I (AR 2) + Io(AR1 )K 1(X112)}- 1 ) + MHO
0
2
1
0
1
2
X- (RRk) (
2
2
(R2
-
Incorporating these integration constants (5.5.13), equations
(5.5.7) and (5.5.10) become:
2
87
X(r
XH'
dv
dr
H
{Ko(XR1)Ii(Xr) + Io(XR1)Ki(Xr)1 +
HH
- K1(Xr)I1(XR1)}{HH1 + g0H'} + XI1(Xr) f
R
K (Xr) f
+
1
sI
R2
o
R
1
)
{I
1
(Xr)K (AR
)
1
1
-
2
+
sK0(Xs)F(s) ds
2
(5.5.14)
(Xs)F(s) ds,
and
d;
;
HH
(X-2)
X-+ I
0
HH
2RH'
+
X(R+
4
r
fK (AR )I
0
1
1
(Xr) +
RH'
fI (XR )K (Xr) - K (AR )I (Xr) } +
0
1
0
0
1
0
)H
(R+
)
1
XH r
XM
2 c
r
1
1
2
2
(AR )K (Xr)} +
2(r - R
+
1 + gOH'
{HH
+ g H' }(Xr{I (Xr)K1 (XR1) + K0 (Xr)
C (r-R ){K (Xr)I
1
1
1
I
0
0
1
(XR
1
)
- I
1
1
1
(XR )1 - 1
(5.5.15)
(Xr)K (XR )} + x(r),
1
J +
1
1
where
X(r)
4
-2 /CI 0 (A) f sK o (Xs)F(s) dscg
r
R
=
+
R
--(ru'v')
+
r dr
1
4
+ -2jEK (XE) f sI (As)F(s) dsdE
r
0
r
-
o
Ri
2
-R j(u'v')
r k
R+RRk
{I (Xr) f sK (Xs)F(s) ds
0
Ri
- r
R
{I
-
o
-
1
r
(Xr) f sK0(Xs)F(s) ds
1
+
K0(Xr) f sI (Xs)F(s) ds}
o
R
r
X lc.
F(r)
r
r
+
+
+
K (Xr) f sI (Xs)F(s) ds}.
1
o
g
R1
Ri
The symbol
0(amp)n
will be used to denote an n:th order amplitude
that is formed by any product of
a(t)
and/or
A(t).
The expression
88
O(amp)4
x(r), as expresses above leads to
terms in the amplitude
equations.
The derivation takes the MOS-energy equations, and substitutes a
solution with the same spatial form as the solution of the linearized
Thus, we make what is called the 'shape assumption', namely
problem.
that the finite disturbances (e.g. v' and v')have the same spatial
structure as the linear ones, although their amplitudes (e.g. a(t) and
) may differ.
A(t)
This approximation serves to give a simplified
derivation of the amplitude equations by neglect of the harmonics, as well as
neglect of the distortion of the fundamental cp
It is a good approxi0.
mation only if the total nonlinear effect is nearly the same as that
due only to the distortion of the mean flow.
Representative computations and important relations are presented
in Appendix B.
The amplitude equation for equation (5.3.1), incorporating (5.5.15),
is
2
da2
Y30(amp)4 - y
12a2
Y1(717-
--a-
4 M
-
aA
15 Rk
(5.5.16)
where
Y
1
=
f z ID
1v12
4)12
1
14'12} rdr;
I
R
1
Y2 = 2b
rR2
Q(r){4.v r
4
r
v.} dr,
Q(r)
a;
dr
7
.1-
X(r)
R1
(see (5.5.15)
R
3
= 2b f
R
2
1
x(r){4.lv
r
- 4 v.} dr;
r
);
89
R
{2b1
Y4 = f
4
12
IW12 + 2 4112 +
+
r
R1
I
r
I
2b2
4
p(ciT4); + (IT(Di) + -.p(O;.Or +
5
2
r
=
1
1,
+
+
+ 21v'12
2
(7 v, + v.v!)} rdr;
r r r
+
y
+ WO.) 413(WO
r r
r
1341(02 -
,
- C v.) +
t2b(C.v
ri
1
1
+ nisi')
1
r rr
2(nr vr
-
11.4)!)
1
)2(n
2
+ v.v.) + (v v, + v.v!)} rdr.
r r
-(%) v
+1-1.(E)
rr
r
r
r
Primes indicate differentiation with respect to r.
that, by solving equations (5.1.24-28), the unknowns
Werecall
A
(1),
A
A
A
are determined.
v, C, 11, and v
v = v = vr + iv.,
In the above,
qh
1
=
=
r
+ iO
,
i
C = C = Cr + iC.,
v=v=vr+iv..Also, we applied the shape assumption when we
w"(r,t) = a(t) dO/dr,
O(r)
u'(r,t) = -ib
utilized the expressions:
C(r,t) = AWC(r),
a(t),
v'(r,t) = a(t)v(r),
n'(r,t) = A(t)n(r), and
All disturbances are approximated up to the first
v'(r,t) = A(t)v(r).
harmonic, so that, for example,
c'=
The amplitude equation for equation (5.3.3), incorporating (5.5.14), is
dA 2
d
1 a
=
2
aA
6
3
0(amp)
A2
4
d
4 R
aA
y 5
g
where
R
=
6
1
I 2 {l
R
1
12
InI2
1v121
rdr;
A2
2A2
5 R
(S
6
(5.5.17)
90
R,
6
2
= 2bj f
2
Q
(r){(1).v
1
R1
Q (r) = -
+
{K (AR )I
H
1
v.} dr,
r 1
-
r
0
X(r-R
1
HH
)
1
fI
1
(Ar) + I (AR )K (Ar)}
1
0
1
(Ar)K (AR
1
)
1
- K (Ar)I
1
1
1
(AR )1{HH
1
1
+ g H1};
o
2
R,
6
3
= 2bj f
R
d.)
Q
2
(r){(1).v
r
-
r
v.} dr,
Q1 (r)
Q2(r)
1
(see (5.5.14)
=
R,
j2 {(1
2
rb2)(k12
In12
1v12)
ric112
);
rIn.12
riv,12
R1
+
V
r r
+
11
+
n'
TT
+ T1TI!1 } dr;
R2
6
5
= 2 f
trk12 + Ini2 + riv121 dr;
R1
2
2 f R2 {.i..2k12
6
6
lc,12
1v112
1(crc;
+2b(vir -v;.1 rdr.
R1
We have thus derived the amplitude equations (5.5.16) and (5.5.17) for
finite disturbances imposed on a basic Couette flow between rotating,
coaxial cylinders.
Next, we examine the disturbance amplitudes at the threshold
between stability and instability.
91
V.6
Criticality
The threshold between stability and instability is criticality.
For the amplitude equations, criticality implies that the magnitude of
all the disturbance amplitudes are not changing as time changes.
Math-
ematically, such a state of equilibrium proposes that
dA
dt
da
dt
(5.6.1)
Additionally, we assume that
0(amp)n = 0
O(amp)2 is much greater than
0(amp)4.
for
n > 3; specifically,
Hence, at critical stability (criticality), the amplitude
equations (5.5.16) and (5.5.17) produce
0 = y2ca 2 +
a2
y
Y44 Mc
aA
Kc
aA
A2
°
62caA
(5.6.2)
,
5 R
64 IT
gc
Y5
)c
c
2A 2
65 1c1--(6
A2
(5.6.3)
Rbe
The 'c' affixed to nondimensional numbers indicates a 'critical value'.
Note that
y
2c
and
6
2c
contain critical numbers.
Integrating relation (5.6.1) suggests that
a = mA, in that, these
two disturbance amplitudes are multiples of each other at criticality.
For instance, m = a(0)/A(0).
Remember that initial conditions are
plausible since disturbances are under supercritical conditions.
particular, if we select
m = 1, then the two disturbance amplitudes
are initially of equal magnitude.
become, respectively,
In
Then equations (5.6.2) and (5.6.3)
92
0 = y
2c
Y5/Rkc
Y4/Mc
62c "4/Rgc
°
Y5/Rkc
265/Rkc "6/Rbc.
We have discovered, with some approximation, the critical
relationship between the parameters
R, Rk, R
and Rb
,
This
!
critical relationship is defined by equations (5.6.4-5), and thus
= Sm(b,R,Rk,R ,P ,c) where
yields the marginal stability surface, S
is the wave speed with restrictions imposed on it
the 'eigenvalue' c
by the assumed supercritical conditions
10
.
The marginal stability surface is
Sm
62c
Y2c "4/Rgc
The graph of
of parameters
,
Y4/Mc
66/Rbc
(5.6.6)
= O.
would indicate, at a glance, the combination
Sm
R, R
+ 265/Rkc
Rb, and Rk
that lead to a stable flow, an un-
stable flow, or a flow in equilibrium.
Since the graph of
S
Sm
hypersurface, only (two- or three-dimensional) traces of
is a
m
can be
plotted.
Before graphing
S
m
, we should decompose
y
2c
and
6
2c
with the
intention of recovering the critical numbers that these relations conThe major difficulty is liberating
tain.
X
from the Bessel functions,
while maintaining the existing integrity of the integrations.
One option is to make the narrow gap approximation.
this approximation means that the gap-width
Mathematically,
d = R2 - R1 << 1.
Em-
ploying this approximation at this stage of the analysis is burdened
by the difficulty of knowing how to express such constants, as
and
I
0
(XR ), linearly
2
11
.
K (XR
0
)
1
To ease this burden, the narrow gap approxi-
93
mation should first be utilized when equation (5.5.4) is invoked into
the "nonlinear" analysis; that is, linearize
v
and
v.
The option we would pursue, involves the utilization of known
experimental data, and is similar to the procedure of section VI.1.
The ratios (4.7.7) - (4.7.9) are still applicable here, once
determined (for fixed A).
The
A
is
determined for plane Couette flow
A
could be used here, if the same fluid is involved, and vice versa.
in section IV.7, the values of the critical numbers, Rc
,
As
R , and P.
-kc'
gc
involved in the stability of rotational Couette flows, can now be theoNote, however, that we must assume
retically predicted.
From the adjusted marginal stability surface (5.6.6), the re-
a = -13).
lation
Rkc
Rb = co (i.e.
R,
S
KC M
yields
= 0
265 + (Rkc/Mc)Y41/(62c
Y2c).
)5 + (Rgc/Mc) 141/(62c
2(Rgc/Rkc(S
Y2c "
{-(Rkc/Rgc)64
(5.6.7)
Similarly,
R
gc
= {-64
Mc
{-(Mc/Rgc)64
Rc
McRkc/(Rk c
2(Mc/Rkc)65
Y41/(62c
Y2c"
(5.6.8)
(5.6.9)
(5.6.10)
Mc).
Utilizing the ratios of the nondimensional numbers (4.7.7-9), the
constant, S
2c
- y
2c
,
can be evaluated, once
A
is known.
Finally,
to determine Rb, also, requires more detailed analysis of S m (for instance, its graph).
Numerical procedures, for the rotational Couette flow problem,
would be similar to the numerical procedures outlined for the plane
Couette flow problem in chapter VI.
94
PLANE COUETTE FLOW
NUMERICAL PROCEDURES
VI.
In chapter IV, a qualitative analysis of the stability of a basic
plane Couette flow was presented.
In this chapter, we will outline the
numerical procedures that will quantitatively substantiate the nonlinear
analysis for a basic plane Couette flow.
We begin by listing the sequence of steps necessary to graph the
marginal stability surface Sm, and to calculate the theoretically predicted critical numbers R ,
R,
C
,
KC
and R
Step 1:
Algorithm to determine X;
Step 2:
Determine the ratios of the nondimensional numbers;
Step 3 (optional):
(4.7.3)) for
Step 4:
(a).
Plot
I:I
and
v
(equations (4.7.1) and
-1 < z < 1;
Determine the unknowns
(b)(optional).
cpr,
vr, and v.;
Plot these functions for
-1 < z < 1;
Step 5:
Determine the coefficients of the amplitude equations;
Step 6:
Calculate the values of the critical numbers; and
Step 7:
Plot traces of
S
m
= S (b,R,R ,R ,c)
k g
m
To demonstrate these steps, we choose
was made because
A = 5
A = 2
and
for fixed b and c.
A = 5.
This choice
seems to represent fairly, a typical fluid
with moderate micropolar properties.
95
Algorithm to Determine A
VI.1
If experimental velocity data is available for steady, laminar
plane Couette flow, we can then determine
by solving equation (4.7.1) for X.
of
A, for any predetermined
A,
From (4.7.1), we obtain a function
only, such that
A
sinh(Xz)
F(A) = G(z)sinh A
+
2XA(z
G(z))cosh A
= 0
(6.1.1)
where
(6.1.2)
G(z) = 2u(z) - 1.
F(A),
A = 0 (which represents the classical case) is a root of
Since
we divide through by A (now requiring
F(X)
P(X) =
G(z)sinh A
-
sinh(Xz)
0 ), to get
A
+
2A(z - G(z))cosh A
=
0.
(6.1.3)
Using Newton's method, we obtain the following iterative formula:
X
an
n+1
-
P(X )/P'(X )
n
n
(n = 0,1,2,3,...)
(6.1.4)
where
P'(X) = {G(z)cosh A
-
z cosh(Az) }(A
1)/A2
+
2A(z - G(z))sinh A.
(6.1.5)
We can predict an initial approximation, A0, from Table 4.1; or we can
use a method such as linear interpolation on P(A).
Warning:
If
G(z) = z, at least to within the accuracy of the data,
then no other roots will be found for P(A).
expect
that
G(z) = z, because classically
G(z) = z.
Note that we will always
u(z) = (z + 1)/2, which implies
96
To reduce the choice of A, from being predetermined, to being
'experimentally' determined, in conjunction with the above algorithm
(6.1.4), requires an additional constraint, such as that provided by
Of course, we would then require accurate experimental
equation (4.7.3).
microgyration data for steady, laminar plane Couette flow.
A source of experimental data for velocity (although given graphically) is Reichardt (1956).
Other papers of interest, that present
numerical insight into classical plane Couette flow are Ellingsen,
(Refer to the
Gjevik, and Palm (1970), and Orszag and Kells (1980).
bibliography for the journal reference.)
Ratios of Nondimensional Numbers
VI.2
Formulas for useful ratios between the parameters, R, R , and Rk,
were derived in section IV.7.
and
With our demonstration values of
X = 5
A = 2, we compute, from equations (4.7.7-9), that
=
Rg /Rk
=
50/3
(6.2.1)
= 16.6,
R /R
=
50/3,
(6.2.2)
R/Rk
=
1.
(6.2.3)
These ratios, (6.2.1-3), are required in the execution of the remaining
steps.
Also, according to our foregoing choices of
note from (6.2.3) that
R
coincides with
Rk (i.e.
X
and
p = K).
A, we
97
VI.3
Graphs of the Laminar Velocity and Microgyration Fields
Figure 6.1 illustrates the velocity given by (2.3.14), with
and
A = 2.
x = 5
Recall that all quantities (e.g. u and z) are nondimensional.
The dashed line in figure 6.1 represents the
(Refer to section II.1.)
classical velocity field, u(z) = (z + 1)/2.
1.00
u
,
0.75--*
0.50
0.25
C.25
.00 -0.75 -0.50 -0.25
U. 5U
U.
/5
1.
00
Z
-0.25
-0.50
-0.75
-1.00
Figure 6.1.
Steady, laminar velocity for a micropolar fluid.
In comparing micropolar with classical velocity profiles, we notice
the subtle effects of the hyperbolic functions (largest deviation from
classical theory is 0.0126 at
z = ±0.7), present in the micropolar
98
solutions of steady, laminar plane Couette flow.
Such subtle effects
should not lead one to expect them in the case of transition to turbulence also.
Figure 6.2 illustrates the microgyration profile given by (2.3.13),
with
X = 5
and
A = 2.
v
0.3
0.2
0.1
.
Figure 6.2.
.
Steady, laminar microgyration for a micropolar fluid.
99
Determining the Functions
VI.4
vr, and vi
(Or,
The MOS-energy equations (4.1.31) and (4.1.32), with (4.1.33) as
the boundary conditions, are decomposed into their real and imaginary
parts, thereby resulting in the following system of coupled differential
equations for the unknown functions
(1)r,
Al
A
D(1)1..
Al D(1).
1
A Dv
3
A
3
r
=
u
=
(u - c
+ A
r
D. + A 2 D(1).1
+
)D,1).
r
1
)131)
u- "
()
=
A
j(u - c
=
A v.
4
1
Dcp
2
(u - c
( P .
'
r
r
4
v.
+
r
DV
2
A
+
r
(1)i, vr, and vi:
Dv.
-
jAU"
+
jAu"
i
r
(6.4.2)
,
1
2
) v
j(u - c )v
(6.4.1)
r
,
(6.4.3)
,
(6.4.4)
r
where
A
= 1/(bR ),
3
u
u'
c
'
=
=
A2 = 1/(bRk),
Al = 1/(bM) + c.,
D = d2/dz2 - b2,
A = Rk/M,
A4 = 2/(bRk) - jci,
sinh(Xz)/cosh X 2tanh X - 4XA
2XAz
+
1
-
c
X2sinh(Xz)/cosh X
4XA
2tanh X
Coincidentally,
when
1/M =
A = 5
and
A = 2.
The boundary conditions (4.1.33) are:
a
(q)
r
+ i(1).1 )
= b(q)
r
+ i(1).1 )
= 0 = v
r
+ iv.
1
at
z = ±1.
(6.4.5)
100
The fact that
v' = AU"
was utilized in equations (6.4.3) and (6.4.4).
(Refer to (2.3.2).)
Prior to numerically solving the system (6.4.1-4), we need to be
given the wave speed
c = c
r
+ ic
(obeying the supercritical conditions
i
that we are assuming), the period
b, and the constant microinertia
(For purposes of demonstration, we could arbitrarily select
j.
j = 1.)
Because only values for the ratios of the nondimensional numbers are
known, we also equate the classical Reynolds number, Re, to the micropolar parameter, R.
Now, the ratios of the nondimesional numbers, as
given in (6.2.1-3), can be used to determine the constants
Al, A2, A3,
and A4.
An exact solution of the Orr-Sommerfeld equation for plane
Remark:
Couette flows of classical viscous fluids was obtained by W. H. Reid
(Reid, 1979).
VI.5
Values for the Coefficients of the Amplitude Equations
Only the coefficients, 52, y2, (S5, (56, and 17, that are required
to complete steps 6 and 7, are calculated.
Determination of the coeffi-
cients is accomplished by numerically integrating the following
equations:
1
y
0"2
= 2 f
7
+ 2b2 0,12 + d.,!2)
Tr
Ti
104(th
Tr
2
Ti
2)} if dz,
r
(6.5.1)
66=2f{v2+v.2} dz,
(6.5.2)
-1
1
-1
101
1
{v'2 + v'2 + b2(v
85 = 2 f
r
-1
1
4. v.2)1 dz
2
(6.5.3)
,
r
1
y2 = 2b I
(pr(pil dz
F1(z){(1);(pi
(6.5.4)
,
-1
where
F (z) =
1
X(z-2)
-Az
+ e
C{e
B(1 + Rk/R)
AK (z - 1)cosh {A(z -1)}
}
1
(1 + 2Rk/R)
X
e-3X){cosh{X(z-1)} - 1}
B(1 + R/Rk){cosh(2X) - 1}
C(e
-
(R /R )K
g 1
1 + Rk/R '
(6.5.5)
with
X(1 + Rk/R)/2
C
B
X(1 + R/R)e
(6.5.6)
-X
{cosh(2X) + 1} + cosh(2A) - 1
and
-3A
C(1 + 2Rk/R) (e
K
1
=
e
)
(6.5.6)
B{cosh(2A) - 1}
1
62 = 21:d f
-1
F2(z)-(vr(15,.
(1)r.(1).1 dz
(6.5.7)
,
where
F
(z) =
-X
{e
2
a(z-2)
+ e
-Az
1
}
{cosh{X(z-1)}
11
(6.5.8)
(1 + 2Rk/R)
Note that primes denote differentiation with respect to z.
Also,
recallthaty our shape assumption.
The remaining steps can now be completed.
For step 6, the values
of the critical numbers can be calculated from expressions (4.7.10-13).
For step 7, traces of the marginal stability surface Sm = Sm(b,R,Rk,R ,c)
(refer to (4.6.6)) can be plotted with the values of
in section VI.4.
b
and
c, as used
102
VII.
SUMMARY AND CONCLUSIONS
Discussion of the Results
VII.1
In this thesis we used micropolar fluid dynamics for the problems
of flows of micropolar fluids between two parallel plates (plane Couette
flow) and between two coaxial, rotating cylinders (rotational Couette
flow).
Closed-form solutions to these two problems were obtained for
steady, laminar flow.
A graph of the laminar velocity profile for a
plane Couette flow of a micropolar fluid showed only a subtle deviation
from the classical plane Couette flow solution predicted from the NavierStokes equations.
Also, a graph of the laminar microgyration profile
for a plane Couette flow of a micropolar fluid was presented, predicting
a nearly constant microgyration value of
0.25
throughout the mid-region
between the plates.
The two basic flows, laminar plane Couette and laminar rotational
Couette, were superimposed by a finite two-dimensional and a finite
axisymmetric disturbance, respectively.
The linear theory of micropolar fluid dynamics, for plane Couette
and rotational Couette disturbance flows, was briefly pursued, and thus,
the micropolar analog of the Orr-Sommerfeld energy equations were derived.
Numerical solution of the MOS-energy equations was not performed.
The
solution to these equations was assumed to be the spatial form (shape
assumption) of the superimposed non-linear disturbances.
103
The nonlinear disturbance equations were derived for the finite
disturbance flow.
Then, the nonlinear disturbance energy equations
The functions in the disturbance energy equations were
were derived.
The
assumed to be separable into a spatial part and a temporal part.
spatial part was known from the linearized theory for the disturbance
flow.
Hence, incorporating these (spatial) solutions into the disThen,
turbance energy equations, resulted in the amplitude equations.
the equations governing the finite amplitudes of the disturbance flow
were known.
Finally, the stationary phase of the amplitude functions
led to the discovery of the marginal stability surface.
Also, using
ratios of the nondimensional numbers, the equation for the marginal
stability surface directly produced expressions for the critical numbers,
R
c'
R
gc
, and R.
.
However, for the disturbed rotational Couette flow
ice
problem, the values of the critical numbers could only be implicitly
implied (due to the presence of Rb) from the marginal stability surface.
Of special importance, was the elucidation of the fluid flow mechanisms induced to deal with the energies of the disturbance flow.
A
physical interpretation of the disturbance energy equations introduced
the concepts of swirl, microenergy of rotation, and mean couple stress,
into the repertoire of fluid dynamics.
Recall, the 'swirl' created
by a disturbance is essentially the divergence of the Coriolis acceleration experienced within the volume elements in the fluid, thereby acting
as a source (or sink) for the energy necessary to create turbulent flow.
Also, the swirl is the coupling mechanism between the micro- and macrocontinuum volume elements, and thus, provides a tangible link for
understanding the transition from laminar to turbulent flow.
Recall,
104
the disturbance 'microenergy of rotation' is the kinetic energy of
rotation for a micro-volume element about the principal axes (rectangular
or cylindrical coordinate axes).
A detailed explanation of the fluid
flow stabilizers and destabilizers was presented, including inequalities
describing flow stability or instability.
All numerical calculations hinge on the establishment of the constant A.
We must emphasize that
A
is constant, only because, we have
assumed that R, Rk, and Rg remain constant for a given fluid.
The necessary numerical procedures are outlined in chapter VI.
VII.2
Scope of Further Work
Completion of the numerical procedures, as we began to illustrate
in chapter VI, is needed to quantitatively substantiate the qualitative
nonlinear stability analysis that was presented for disturbed plane
Couette and rotational Couette flows.
The remaining numerical work is
no small task, but will be straightforward from the algorithm of the
procedures given.
We anticipate many interesting results from the numerical work
for plane Couette and rotational Couette disturbance flows.
Also, we
expect, that for the first time, theoretically predicted critical numbers will be calculated for the stability of plane Couette flow.
We would like to see a more convincing argument for, or against,
105
the term
H
(of section V.4) always being positive.
(Refer to Append-
4
ices A and B.)
The implications of
H4
not always being positive are
not contradictory, but suggest that the instability (probably caused
by vigorous dilating) of the microcontinuum volume elements may, in fact,
exist locally; while globally, the flow is stable.
Furthermore, a more
extensive study of all the flow mechanisms should be made for various
flow situations and fluids.
An attempt should be made at finding an exact solution of the MOSequations for plane Couette flow of micropolar viscous fluids, as was
successfully done in the classical case.
Ultimately, we would like to witness, in our lifetime, a presentation of the "closed-form" solutions to the nonlinear equations of micropolar fluid dynamics.
106
ENDNOTES
1.
Hydrodynamic stability applies the abstract concepts of stability
for differential equations.
The ideas are similar; however, here the
physical decay or growth of disturbance waves (solutions)
is of para-
mount importance.
2.
An enlightening synopsis of nonlinear stability theory for class-
ical viscous fluids is presented in the book, Hydrodynamic Stability,
(Drazin & Reid, 1981, pp. 370-464).
3.
A couple in classical continuum mechanics is pictured as a pair of
parallel forces having equal magnitude and opposite sense with respect
to each other, separated by a moment arm.
The moment arm is allowed
to tend to zero in a volume element since the latter, regarded as an
infinitesimal, approches zero, while the forces are assumed to remain
bounded.
Thus, the couple vanishes.
couple stress also, that it vanishes.
4.
A similar argument is made for
(Eringen,1967)
Strict adherence boundary conditions imply (1) that the microgyra-
tion vector vanishes on the plates; and (2) that the fluid comforms to
the no-slip condition, meaning that the fluid velocity is equal to the
plate velocity when in direct contact with the plate.
5.
Selecting this reference velocity precludes the freedom of letting
the inner cylinder be stationary.
107
It is found that in turbulent plane Couette flow, the mean flow is
6.
is
antisymmetric, so that although a disturbance of the form (4.2.1)
possible, it would not, in general, lead to an antisymmetrical mean
flow; while an infinitesimal disturbance, that is composed of two disturbances travelling in opposite directions, with stream function of
the form
r,
1)
(x,z,t) = (4.2.1) + K
1
(-z) exP{ib(x+ct)} + K
(-z) exp {- ib(x +ct) },
1
does.
7.
f = frequency,
Period b = 27f = 27c/A = k (wavenumber), where
A = wavelength, and
c = 1
by the proper choice of units.
Hence, the
wavenumber and the period are 'equivalent', as shown above.
8.
If
col, w2, w3
and
Qi, 522,
represent the magnitudes of the
523
angular velocities and angular momenta about the principal axes, respectively, then
01 = Ilwi,
R2 = I2w2, and
Il, 12,
The kinetic energy of rota-
are the principal moments of inertia.
I
Q3 = 13w3, where
3
T = (I w2 + I w
tion about the principal axes is given by
Thus, we analogously define
e =
j
v ,2
1
1
2
+ I w )/2.
2
3 3
to be called the 'microenergy
2
of rotation' for the disturbance.
9.
10.
E.g.
(11+K)2U/az2, 2j), and ,0 2 \7)/3z2.
Supercritical conditions impose the possibility of flow instabil-
ities prior to the equilibrium state.
just prior to equilibrium.
This instability means
bci > 0
This situation is marginal stability.
108
(Definition given in section IV.1.)
Yes, it is possible that the flow
situation may technically be neutral stability, since we are using nonlinear theory.
T = 0.5772
Euler's constant
11.
For small values of
Ar,
I0(Ar) = 1
K0(Xr) = -T
I1(Ar) = Xr/2
K
1
(Xr)
ln(Ar /2)
(Tranter,1968)
= 1/(Xr)
For the small gap-width approximation, assuming
Ar = ar d2 << 1 (since
A
is fixed and
R
d << 1
< r < R2);
)
implies that
A and r dimensional.
1
The situation is somewhat different when considering a disturbance
12.
under 'subcritical' conditions.
not amplify, but is damped.
In this case, a small disturbance does
And now, a suitable 'terminal' condition
is applied, namely that the function (pi shall be an exponentially de-
creasing function of time in the limit as
t
+°.
By analogy with the
, where
c.
< 0, of the linearized stability theory.
13.
The number
I*, found in the constant C of F1(z) and F2(z), should
be equated to zero (i.e. assumed negligible) since it implies amplitudes
of order a2 and aA.
Hence, as seen from equations (4.6.2) and (4.6.3),
I* will lead to amplitudes of order a4 and a3A, which we assumed to be
zero.
109
BIBLIOGRAPHY
Cole, D.
(1965)
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APPENDICES
112
APPENDIX A
The derivation of the disturbance energy equations will be, in
(5.2.26)
Multiply equation (5.2.25) by
part, now demonstrated.
by n', and (5.2.27) by N.; then add the resulting equations to get
3u v
+
+ T-1-2 + v'2)
3v
+
T4/Rb
=
+
jT
T2/Rk
+
2T3/Rk =
1
3t
(A.1)
T5/Rg ,
+
where
Tl = r.X4
n'x5
v'x6;
3u' 1
9z
9z
2
-2
T3=
T
4
,2
+ nT
+ v ,2
v,( t:
1!.."");
/
;
+
= C'{a2C/Br2 + 1(r'/r) + a2v'/r9z}
3r
+ y'{a2C'/;raz + '-ck(c"/r) +
T5 =
2v"./az2};
1(C/r) + 32C/3z2}
C{a2C/Br2
+
3r
TI-{a2T1-/3r2
2._(n-/r)
21.1.79z2).
3r.'"
v-ta2v-or2
._(v-/r)
1r
2v-/az21
3
For HT 1 rdrdz, the term
Integrate equation (A.1).
v'x
6
= v'w
_Dv'
9z
.
v'a
----ru
t
v)'.
r 3r
,
+ v'u
3r
,
From this, when integrating by parts, a typical term is like
113
II
V U
(A. 2)
rdrdz =
3r
r=R
= Iry "2u'
r=R
2
dz
-
r
fu"v"2 drdz
-
=
jv --(v"u") drdz
3r
1
= Si - S2 - S3.
Due to condition (C2) of section V.2, S 1 = 0.
Integral S
(and, in
2
turn, S ) equals zero in the mean.
3
Mean value of function f on the interval
Recall:
T
ba
a < x < b
is
f(x) dx.
a
Now, since the disturbances are assumed to be of zero mean (with respect
to spatial variable z), integral (A.2) equals zero, in the mean.
Specifically,
27/b
e
27/b
ibz
0
0
(E.g.
v" =
v e
ib(z-ct)
So, quantites like
u"v"2
= 0.
dz = ib eibzi
7'72
was replaced by
are independent of z.
u"v"2
Note, however, that
because only the mean part contributes
to the integral (Stuart,1956).
The above result is typical for the term
rdrdz = 0.
T
1
Let us now look at term
T
T
.
4
Wav-
Because
32C"/3r3z
c"%)'/ar) = 3z 3r
and
---(vA3C/Dz) = 3r Dz
3r
+ v'
2
C/9raz
Ti.
And hence,
114
imply that
a2v'/araz + y' a2c-73raz =
2(v'W/az)
Br
-
ar az
'
we have
2
T
4
1 9
=
)2
,
)
r 3r
v-ac"
av'ac'
+ ---(v-aC/az) - 2 -- + r az
3r
az
ar
a
(3v3z
Now, since
f--(v-aC/ar) rdr
- fv-ac-/az dr,
=
ar
we finally derive
ifT
4
=
rdrdz
Since the term
2-57.
T
fav "12
rdrdz.
in remaining an energy dissipation mechanism.
4
f3C
t
We anticipated
+
may not always be positive, poses uncertainty
2av'a
ar az
as to the fidelity of
ff( 22:
ar
' C
ava
1
- if {r
ll29v--
ar 3z
rdrdz
to be equivalent to the integral
: 2 rdrdz, but this is false, in general.
+
az
If true, then
T
4
(i.e. H
4
)
would always be positive; and hence,
would always represent the rate of
-(a+a)H 4
(coi-)-viscous dissipation
of the microenergy of rotation of the disturbance due to the rotational,
translational, and dilational effects of the micro-volume elements in
the volume considered.
the integrals
f(av'Pr)2 rdr
be zero, in general.
term
H
.
2
The reason this supposition is false, is that
and
f(3C/3z)2 dz
cannot be shown to
Refer to the presence of these integrals in the
115
Remark:
=
c,
(A.3)
a2c /ar2
'4.-(C-aC"/ar)
+
c2/1..2
-
-2/3r.
r ar
Integration by parts yields, with the influence of condition (C2), that
R,
P--(CaC/ar) rdr
ar
= WW/arl
-
R1
R
=
0 - C2/21
2
=
fBC2/3r dr
=
2
O.
R1
Therefore, integral (A.3) =
=
ff{c-2/r2
(ac-/;r)2} rdrdz
=
-
2- 2r
ff((rc"))2
\r a
rdrdz.
The sample calculations presented above thus demonstrate how the
disturbance energy equations (5.3.3) and (5.3.1) were derived.
116
APPENDIX B
Representative computations and important relations for the amplitude equations (5.5.16&17) are now presented.
We have assumed the disturbance stream function (as given by
), to be
(5.2.2)
)gibz
cp'(r,z,t)
=
1
(r,t)eibz
1
(r,t
With the solutions given by the linearized theory (refer to equations
(5.1.18-22) ), we make the shape assumption, in that
v" = a(t)v(r), and
(pi = a(t)(D(r),
v(r) = v
Notation:
v(r) = v
v" = A(t)v(r).
1
r
(E.g. c = c
+ iv. = v
+ iv. = v, and
r
(D(r) =
1
1
+ iC. =
r
(I)
r
+
denoted disturbance quantities.
Above primes
1
Below, apostrophes
denote differentiation with respect to r.
(D, v, C, fl, and v
'
The
tilda 1, denotes a complex conjugate.
Note:
A double arrow -4-4- signifies a result after integrating over
z = 0
is
v". = ve
w"2 44
ibz
2
Also, the modulus of a complex
14)12 =
b2
11'2 = -
z = 2Tr/b.
to
1
e-2ibz)
2ibz
2
(1)2e
ft, -ibz
+ ve
1V1"
v-2 44 21v12
where
(11.1
E d(D/dr.
2b12
rzi
number (1)
on
117
ib
((I)
=
2b
- Irv.)
((1).v
1
1 r
r
Bv-9
2--- -L = 2ib(v'e
Br Bz
ibz
2ib(v'C
+v'e
avBc
'Bz
Br
+
)(v
i
r
- iv.)
&" Reals.
ibz
-ibz
) (Ce
v'C) = 4b(v1!C
2
Important Note:
r
> 0
r
v!C
lr
e
-
-ibz
)
-v'C ).
ri
> V'C..
rl
Also, of concern in section V.4 is the result that
H4 > 0
2
r
(C
rr
+C.C!) + 4b(v!C r
v'.) > 0.
r
This is valid, to the extent of all the assumptions, that we have made
up through section V.5, especially the shape assumption.
The sample expressions presented above thus demonstrate how the
coefficients for the amplitude equations (5.5.16)
derived.
and (5.5.17) were