Skiadas C.H., Dimotikalis I. (editors) Chaotic Systems: Theory and Applications
Подождите немного. Документ загружается.
208 A.
B.
Mikishev, A. A. Nepomnyashchy, and
B.
L.
Smorodill
Here
A is
the
heat
conductivity
of
the
liquid;
Qa
and
[2
are
the
amplitude
and
the
frequency
of
the
heat
flux, respectively. A
set
of
two-dimensional
governing
equations
in
the
plane
(x,
z)
is given
by
V·v
=
0,
Vt
+
(v·
V)v
=
_p-1Vp
+ vV
2
v -
ge
z
,
T
t
+
V·
VT
=
XV
2
T.
(2)
(3)
(4)
Here v =
(u,
w), T
and
p
are
fields
of
the
fluid velocity,
temperature
and
pressure,
respectively, v, p, X are, respectively,
kinematic
viscosity,
density
and
thermal
diffusivity, e
z
is
the
unit
vector
in
the
z-direction,
V = (ox, oz).
In
addition
to
the
harmonically
changing
heat
flux
on
the
layer
bottom,
(1), we
assume
the
no-slip,
no-penetration
condition
for
the
velocity
z =
0:
v =
O.
(5)
At
the
free
surface
z =
h(x,
t),
the
boundary
conditions
are,
respectively,
the
kinematic
boundary
condition,
heat
insulation
condition
and
the
balance
of
normal
and
tangential
interfacial
stresses
given as
h
t
+
uh
x
= w,
VT·
n = 0,
2/l-
[ 2 hxx
-p
+
N2
ux(h
x
-
1) -
hx(u
z
+ wx)] =
CJ
N3
'
~[2(wz
-
ux)h
x
+ (u
z
+ w
x
)(I-
h;)] =
osCJ.
(6)
(7)
(8)
(9)
Here
n is
the
unit
vector
normal
to
the
interface
and
directed
outward,
N =
(1
+
h~)1/2,
/l-
=
vp
is a fluid viscosity
and
s
is
arc
length
along
the
interface. To
account
the
Marangoni
effect assume,
that
the
surface
tension
varies
linearly
with
the
temperature
CJ
=
CJa
-
CJTT,
(10)
where
CJa
is
he
reference value
of
surface
tension.
Let
us choose scales
for
coordinates,
time,
velocity,
temperature
and
pressure,
respectively.
Then
the
governing
equations
in
the
nondimensional
form
are
V·v
=
0,
p-l
[Vt + (v . V)v] = - Vp + V
2
v - Ge
z
,
T
t
+ v .
VT
= V
2
T,
(11
)
(12)
(13)
Parametric Excitation
of
a Longwave Marangoni Convection 209
where
P = v
Ix
is
the
Prandtl
number
and
G =
gH3
I (vX) is
the
Galileo
number.
The
boundary
conditions
are
rewritten
as
z = 0 : v = 0, OtT = cos(2wt),
z =
1:
h
t
+
Uh'E
= w,
VT·
n = 0,
2 [ 2 hxx
-P
+ N2 u3;(hx -
1)
-
hx(u
z
+
Wx)]
= E
N3
'
1 2
N
[2(w
z
-
ux)h
x
+ (u
z
+ w
x
)(1 - h
x
)]
=
=
M(T
x
+ hxTz).
Here
the
following dimensionless
parameters
appear:
(14)
(15)
(16)
(17)
(18)
E =
(T
HI
(fiX) -
the
inverse
capillary
number
and
M = (TTQH
2
I
(A/-lX)
-
the
Marangoni
number,
W =
f!H
2
/chi
is
the
non-dimensional
frequency
of
heat
flux
modulation.
For
the
steady
equilibrium
state
the
upper
surface
is
planar
and
the
liquid
is motionless, i.e. v
=
O.
The
pressure
and
temperature
fields are:
Po
=
-G(z
- 1),
r,-, _ - cosh[ex(1 -
z)]
2iwt
.
10-
2 .
l()
e
+c.c.,
exsm 1
ex
(19)
(20)
where
'c.c.'
denotes
the
complex
conjugate.
Here
ex
=
J2iW.
Later
on, we
use
notations
(To)z =
ozT
o
and
(To)zz = ozzTo.
3
Long-wavelength
linear
stability
analysis
Introducing
perturbations
of
the
dependent
variables
of
the
problem,
like
v = vo + v(x,
z,
t), T =
To
+
B(x,
z,
t), P =
Po
+ p(x,
z,
t), we
obtain
the
set
of
the
equations
and
boundary
conditions
in
terms
of
the
perturbation
functions (tildes will
be
omitted).
Eliminating
the
horizontal
component
of
velocity,
pressure
disturbances
and
omitting
nonlinear
terms
in
the
system
we
formulate
the
linearized
problem
around
the
steady
state
in
terms
of
normal
velocity
component,
w,
temperature
distubances,
B,
and
deviation,
(,
from
the
unperturbated
flat
surface
z =
1:
P-
1
(W
zz
t +
WX.Tt)
= W
zzzz
+ 2w
xxzz
+ W
xxxx
,
B
t
+ w(To)z =
Bxx
+ B
zz
,
The
boundary
conditions
become
z = 0 : W = W
z
= 0,
Bz
= 0,
z =
1:
c't
=
W,
(21 )
(22)
(23)
(24)
210 A.
B.
Mikishev, A. A. Nepomnyashchy, and
B.
L.
Smorodin
8
z
+ ((To)
zz
= 0, (25)
Wxx
- W
zz
= M[8
xx
+ (xx (To)z] ,
(26)
-p-1Wzt
+ W
zzz
+ 3w
zxx
+ G(xx -
IX
xxxx =
O.
(27)
For linear analysis of
the
system
we
introduce
pe
rturbations
(
W)
(W(Z
'
t))
8 = iJ(z,t) ex
p(ikx+,t)
,
( ((t)
(28)
where k
and,
are, respectively,
the
dim
ensionless wave
number
and
growth
rate
of
the
disturbances
, functions
W,
iJ
and
(
are
periodic in
time
with
the
period 7r
/w
For
the
amplitudes
we
obt
ain
the
following
system
(hats
are
omitted):
p-l[V/'
+
,w"
- k
2
1V
-,k
2
w]
=
=
wIV
_
2k
2
w"
+ k
4
w,
e
+,8
+ w(To)z =
8"
- k
2
8,
and
boundary
conditions
z =
0:
w =
w'
= 8' = 0,
z =
1:
( + ,( =
w,
8' + ((To)zz =
0,
w"
+ k
2
w = Mk2[8 + ((To)z],
p-l(W'
+ , w') -
w"'
+
3k
2
w'
+ (G +
Ek
2
)k
2
(
= o.
(29)
(30)
(31)
(32)
(33)
(34)
(35)
Here
the
prime
denotes
the
coordinate
derivative
and
th
e
dot
denot
es
the
derivative
with
respect
to
time. To analyze
the
system
(29)-(35)in
th
e long-
wavelength
limit
we
assum
e
that
all dimensionless
parameters,
namely
the
Prandtl,
Mar
angoni, Galileo
and
capillary
numbers,
are
of
order
unit
y. For
asymptotic
analysis we employ
the
stand
a
rd
scaling for
the
long-scale insta-
bility: k = EK, where E is a small scaling
parameter
serving
as
a meas
ure
of
smallness
of
the
disturbance
wave
number
.
The
growth
rate
of
th
e distur-
bances is
expanded
as
2 ( 2 4 )
, = E ,0 + E
,2
+ E ,4 +
...
a
nd
appears
as
a function of
the
physical
and
modulation
parameters.
The
a
mplitudes
of
the
perturbation
functions
are
expanded
in
the
form
2 ( 2 4 )
W = E
Wo
+ E
w2
+ E
W4
+ ... ,
8 = 8
0
+ E
2
8
2
+ E
4
8
4
+
...
,
( = (0 +
E2(2
+
E4(4
+ .. .
Parametric Excitation
of
a Longwave Marangoni Convection
211
and
substituted
into
(29)-(35).
At
zero
order
the
solution
is given by
(0 =
Co,
8
0
= 8
0
+
8t(z)e
2wit
+
8o(z)e
-
2wit,
Wo
= wot(z) +
wt(z)e
2wit
+
wo(z)e
-
2wit
(36)
(37)
(38)
Here we
denote
by "
+"
thc
pulsating
part
of
the
variable
, in
this
case
altering
with
2w
frequ
ency
and
by
" - " -
its
complex
conjugated,
latter
in
the
sam
e
manner
we
denote
parts
of
(To)z
and
(To)
zz.
The
term
with"
st"
corresponds
to
the
part
independent
on
timc.
Co
and
8
0
are
constants
yet
to
be
det
erm
ined.
At
this
order
of
approximation
the
coefficients
are
8
+ ( ) = (0 cosh(
az)
o z 2 '
2
sinh
(a)
st()
K
2
(M8
0
- G(0)Z2 GK2(0 3
Wo
Z = 2 +
--6-
z ,
MPK
2
(ocoth(a)sinh
2
(
7P)
w+(z) = 2
o a
2
sinh(a)
cosh(
Jp)
We
can
find
Co
and
8
0
from solvability conditions. For
this
purpos
e we
need
the
equations
at
the
second
order
IV
p-1·"
P - 1K2 . +
w
2
-
W2
= -
Wo
+(P-
1
,O
+
2K2)WO",
8
2
"
-
8.
2
=
(,0
+
K2)eO
+ wo(To)z
with
boundary
conditions
z = 0
W2
=
w/
= 8
2
'
= 0
Z = 1
(2
+
,0(0
=
wo,
82' +
(2(T
o
)
zz
= 0,
W2"
+ K2wo = MK2[8
2
+ (2 (To)z],
p -
l[W·2'
+ 10WO']-
W2
111
+ 3K2wo'
+GK
2
(2
+
EK4(0
=
o.
(39)
(
40)
( 41)
( 42)
(43)
(44)
( 45)
We
integrat
e over
the
time
the
condition
(42).
Taking
into
account
the
time
periodicity
of functions, we find
the
first solvability
condition
which
can
be
written
as
( 46)
Obtain
the
second
solvability
condition
we
can
if we
integrate
(40) over
the
interval
0
:::::
z
:::::
1
and
re
sulting
integral
for
the
stationary
part
of
8
2
is
(8~t")
= (,0 +
K2)8
0
+ (wt(To-)z + wo(To+)z)),
(47)
212 A.
B.
Mikishev,
A.
A. Nepomnyashchy, and
B.
L.
Smorodin
where
(1) =
Jo
1
Jdz. Using
the
boundary
conditions
(41)
and
(43)
and
the
relation
( 48)
we
obtain
the
second
solvability
equation
in
the
form
( 49)
where
S(w, P) is a
function
depending
only
on
the
frequency
and
the
Prandtl
number
Now
using
these
two
solvability
conditions,
we find
that
/0
:
/o±
= -
(G
+6
3
)K
2
±
~2
)(G
_
3)2
+ 18M2S(w, P).
(50)
The
lines
of
the
constant
value
of
the
growth/decay
rate
/0
on
the
plane
(M.K
o
)
are
determined
by
the
relation
M(/o,K)
= ±
2(/0
+
K2)(3/
0
+
GK2)
3K4S(W,
P)
(51)
To find
the
neutral
curve
of
the
long
wavelength
convection
we
take
/0
= 0
in
(51),
then
we recover
the
same
expression
as
obtained
in
the
paper
of
Or
and
Kelly (2002)
[2].
It
is
interesting
to
compare
this
result
with
the
similar
one
which
has
been
obtained
in
the
case
when
the
heat
flux
modulation
is
applied
onto
the
upper
free layer, see
Smorodin
et
al. (2009)[3].
In
latter
case,
the
minimum
of
neutral
curve
lies lower ( 14.01
at
w = 2.65),
than
in
the
case
which
is
considered
here,
where
the
minimum
equals
to
105.26
at
w = 11.29, see Fig.
1.
The
instability
domains
are
situated
above
the
neutral
curves.
The
crucial
point
is
whether
the
critical
Marangoni
number
M(O, 0) cor-
responds
to
a
minimum
or
a
maximum
of
the
neutral
curve.
To
answer
that
question,
- we
need
to
consider
the
next
order
in
the
expansion
with
respect
to
the
parameter
E.
For
the
deviation
of
the
interface
from
the
flat surface,
in
the
second
order,
we
obtain:
(52)
The
temperature
and
velocity
disturbances
in
this
order
are
sought
as
e
2
=
fh
+
e~t(z)
+ et,2(z)e
2wit
+ et,4(z)e
4wit
+
c.c.
(53)
W2
=
w~t(z)
+
wt(z)e
2wit
+
C.C
(54)
Parametric Excitation
of
a Longwave Marangoni Convection 213
M{2G13)-1/2
104.--rr---r--,r-~r---r---r---r---r---r---,,-,
5
10
15
ro
20
25
Fig.
1.
The
first
long-wavelength
regions
of
Marangoni
instability.
1 -
heating
from
the
surface, 2 -
heating
from
the
bottom.
and
the
explicit expressions for
these
functions are
very
cumbersome.
In
these
expressions indices like
{2,2}
indicate
that
this
terms
correspond
to
the
perturbation
harmonic
with
frequency
2w
of
the
temperature
or velocity
disturbance
of
the
second order. Here C
2
and
8
2
are
constants
yet
to
be
determined.
To
obtain
how
/2
depends
on
Marangoni
number
we
need
to
apply
the
solvability
criteria
to
the
fourth
order
system, which is
wiv
-
P-
1
W4"
=
_p-1K2W2
+
(p-l/O
+ 2K2)W2/'
+p-l/2
W
O/l
-
K2(K2
+ p-l/O)WO,
8
4
/1 - 8
4
=
ho
+
K2)8
2
+
/280
+
w2(T
o
)z
with
boundary
conditions
z = 0
W4
=
W4'
= 8
4
'
= 0
z = 1
(4
+/0(2
+/2(0
=
W2,
8
4
'
+
(4(T
o
)zz =
0,
W4/1
+ K
2
w2
=
MK2[8
4
+ (4 (To)z],
p-l
[W4'
+
/OW2'
+
/2WO']
-
W4/1'
+3K2W2' +
GK2(4
+
EK4(2
=
o.
(55)
(56)
(57)
(58)
(59)
(60)
(61)
The
solvability conditions here
are
constructed
in
the
same
way as in
the
pre-
vious
order
they
are
based
on
(58)
and
(56)
and
include functions
obtained
for
the
second order.
Substituting
the
expressions for
the
second
order
func-
tions
into
the
solvability conditions we
obtain
a linear
system
for C
2
and
8
2
.
214 A.
B.
Mikishev,
A.
A. Nepomnyashchy, and
B.
L.
Smorodin
Its
solu
tion
gives us a cumbersome expression for "/2.
The
va
lue
"/2
in
the
point
M = M
o
,
corresponding
to
"/0
= 0, is
of
the
ma
jor
importanc
e.
This
expression depends
on
G,
P,
E
and
w. We calculate
this
expression numeri-
ca
lly for diffe
rent
ranges
of
param
eters.
Figure
2 shows
the
chang
ing
of
"/2
depe
nding
on
the
frequency w for
the
parameter
va
lues P = 7, G = 1400
and
E = 1580.
y2
LO
5
Y2
W
50
10
0
150
20
250
- 5
n
-10
Fig.
2.
The be
havior
of 12 for different w
at
P = 7, G = 1400, E = 1580.
Th
e long-wave nonlinear e
quation
that
governs
the
spatiotemporal
dy-
namics of
the
thin
layer is derived using
the
results
of
the
lin
ear
analysis.
Acknowl
e
dments
A.M.
and
A.N. acknowledge
partial
support
from
the
I
srae
l
Ministry
of
Science
through
Grant
No. 3-5799. B.S. acknowledges
a
partial
support
by
the
Russian Basic Research
Found
a
tion
(projects Nos.
06-01-72031
and
06-08-00289).
The
research was
partially
supported
by
the
Europ
e
an
Union
via
the
FP7
Mar
ie
Curie
scheme [PITN-GA-2008-214919
(MULTIFLOW)].
References
1.8. H. Davis.
Thermoc
ap
illary instabilities.
Annu.
Rev
. Fluid Mechanics, 19:403,
1987.
2.A.C.
Or
and
R.E
. Kell
y.
The
effects of
thermal
modulation
upon
the
onset
of
marangoni-benard
convection. Journal Fluid Mechanics, 456:161, 2002.
3.B.L.
8morodin
, A.B. Mikishev, A.A. Nepomnyashchy,
and
B.I. Myznikova.
Th
er-
mocapill
ary
instability
of
a li
quid
layer u
nder
heat
flux
mod
ulation. submitted
to Physics
of
Fluids, 2009.
Symmetry
Broken
in
Low
Dimensional
N-Body
Chaos
David
C.
Nil
and
Chou
Hsin
Chin
2
1
Direxion
Technology
9F,
177-1
Ho-Ping
East
Road
Section
1
Taipei,
Taiwan,
R.O.C.
(e-mail:
davidcni@yahoo.com)
2
Department
of
Electro-Physics
National
Chiao
Tung
University
Hsin
Chu,
Taiwan,
R.O.C.
(e-mail:
jchchin@cc.nctu.edu.tw)
215
Abstract:
In
this
paper,
we
explore
domains
in
conjunction
with
the
defined
para-
metric
space
of
function
f = h(z)
Il{exp(gi(z))
[(ai
-
z)/(l
- aiz)]},
which
repre-
sents
a
N-body
system
in
complex
form.
Interested
observations
on
the
computed
results
reveal
limited
levels
of
Herman
ring
fractals,
symmetry
broken,
and
transi-
tion
to
chaos
based
on
phase
parameter
exp(gi (z))
and
ai.
Keywords:
N-body
chaos,
Herman
ring,
fractals,
meromophic,
symmetry
broken.
1
Introduction
Herman
Rings
represent
a class of
fractals
with
hierarchical
structure
in mero-
morphic
dynamical
systems.
Due
to
the
richness
of
mathematical
elaboration
and
computing
sophistication,
we have
observed
limited
efforts
on
simple
ra-
tional
functions
[1-4].
In
the
previous
efforts, we
explored
complex
function:
f = h(z) I14{exp(gi(Z))
[(ai
-
z)/(l
- (liZ)]),
and
revealed some
interested
characteristics
of
this
function
[5].
We
extended
our
observations
to
the
ap-
plications in
the
antenna
areas
[6,7]
and
elaborated
the
mathematical
foun-
dations
[8].
We have
then
classified
the
function
f = h(z) I1i{ exp(gi (z))
[(
ai
-
z)
/
(1
-
(liZ)]),
and
revealed
the
characteristics
of
this
function
[9]:
1.
There
are
at
most
five levels
of
partial
and
localized
Herman
Ring
do-
mains
at
different
complex
number
z scales.
2.
For
the
orders, which
are
equal
and
higher
than
fourth
orders,
there
exist
five
and
only
five level domains.
3.
The
normalized
phase
and
energy
level
parameters
can
induce
symmetry
broken.
In
this
paper,
we
further
examine
in
details
about
the
symmetry
broken
and
transition
from
stable
fractal
domains
to
chaotic
domains
based
on
two
216 D.
C.
Ni
and
C.
H.
Chin
parameters:
exp(gi (z))
and
ai
in
the
paramtric
space
as
listed
above.
These
two
parameters
define
thestable
and
chaotic
domains
of
initial
condition
of
functions.
2
FUnction
and
Parametric
Space
2.1
Function
The
function
set
is
re-write
as f =
zq
IIi
C
i
with
h(z) =
zq,
and
may
have
the
following
third-order
form:
(1)
Where
z is a
complex
variable, q is
an
integer
and
C
i
has
following form:
C
i
= exp(gi(z))[(ai -
z)/(l
-
(liZ)]
(2)
Here
(li
is
the
complex
conjugate
of
complex
number
ai.
We
propose
this
form
based
on
the
following form
known
in
the
theory
of
special
relativity
by
A.
Einstein:
(3)
The
zq
term
represents
the
interaction
of
C
i
.
Here, we
set
q =
-1
rep-
resenting
potential
energy
of
interaction
is
propotional
to
the
reverse
of
dis-
tance.
The
term
exp(gi(z))
represents
the
phase,
where
gi(Z) is a
complex
function. A given
domain
can
be
a
domain
of
complex
numbers,
z = x +
yi
,
with
(x
2
+ y2)1/2
::;
R. Here, R is a
real
number.
We also use a
simpler
domain,
(x
2
+ y2)1/2 =
R,
to
observe
directly
a
mapping
or
a
transformation
from
original
domain
to
the
set
of
x +
yi
after
function
iteration.
There
are
two
sets
of
domains
in
conjunction
with
the
function:
the
converged
set
re-
mained
from
original
domains
after
several
orders
of
function
iteration
and
the
mapped
domains
of
original
domains
after
several
orders
of
functional
iterations.
The
function
f will go
through
iteration
as:
r(z)
= f 0
r-
1
(4)
Here, n is a
positive
integer
indicating
the
order
of
functional
iterations.
2.2
Parametric
Space
In
order
to
obtain
the
remained
domains
and
mapped
domains,
we
have
to
determine
the
criteria
of
convergence, divergence,
and
oscillation
of
func-
tional
values
after
several
iterations.
We
adopted
function
values
of
several
consecutive
iterations
for
this
purpose.
We
define
the
parametric
space
or
normalized
space
including
the
following
parameters:
Symmetry Broken
in
Low Dimensional N -Body Chaos 217
1.
z : original
or
mapped
domains
2.
a :
normalized
energy
level
of
interaction
3.
exp(gi(z))
:
normalized
phase
factor
4.
q :
normalized
distance
factor
of
interaction
5.
Iteration
These
parameters
can
transform
stable
domains
into
chaotic
domains.
3
Original
and
Mapped
Domains
Figure
1 (a)
through
(e) show
the
Z-lCIC2C3C4
original
domains.
Figure
l(a),
(b), (c), (d)
and
(e) show a original
domain
z = x +
yi
,
with
(x
2
+
y2)1/2
:s;
R
I
.
forming a
set
of
5-level
of
co-center
partial
Herman-Ring
fractal
domains
at
different scales from a
set
of
parameters
in
parametric
space.
Figure
l(a)
shows a
partial
Herman-Ring
domain
at
scale
of
R
rv
10-
6
..
Figure
1 (b) shows
domain
a
partial
Herman-Ring
domain
at
scale
of
R
rv
10-
2
.
Figure
l(c)
shows a
partial
Herman-Ring
domain
at
scale of R = 1,
the
unit
circle.
The
domain
in
Figure
1 (b)
can
be
seen
locating
around
z = 0
point
at
Figure
l(c).
Figure
l(d)
shows a
partial
Herman-Ring
domain
at
scale of R
rv
10
2
.
Figure
1 (c)
can
be
seen
locating
around
z = 0
point
at
Figure
1 ( d).
Figure
1 ( e) shows a
partial
Herman-Ring
domain
at
scale of
R
rv
10
6
.
Figure
1 (d) is
locating
around
z = 0
point
at
Figure
1 (e).
The
partial
Herman-Ring
domain
at
scale
of
R
rv
10-
6
is
topologically
similar
to
that
at
R
rv
10
2
and
the
domain
at
scale
of
R
rv
10-
2
is topologically
similar
to
that
at
R
rv
10
6
.
We call
this
as skip-level
symmetry.
For
higher
functional
order
than
the
fourth
order
(i.e., f =
Z-lCIC2C3C4),
the
number
of levels keeps
the
same
as f =
Z-IC
I
C
2
C
3
C
4
,
namely,
only
5 levels
are
observed
at
11th
order
or
20th
order
as examples.
The
skip-level
symmetry
is also
observed
in
the
higher-ordered
domains.
Figure
2 (a)
and
(b) show
mapped
domains
offunction
f =
Z-IC
I
C
2
C
3
C
4
.
Figure
2 (a) shows
the
quasi-perodic
orbit
becoming
chaotic
orbit
as
shown
in
Figure
2 (b)
when
the
parameters
in
parametric
space
change.
4
Symmetry
Broken
and
Chaos
The
parameters
in
the
parametric
space
correlate
to
one
another
when
an
original
domain
becoming
a
chaotic
domain.
We define
the
transition
from a
stable
to
a
chaotic
domain
based
on
the
change
of
domain
boundries
by
in-
creasing
the
order
of
iteration.
For
example,
when
the
iteration
order
changes
from
10
to
1000
iterations
and
the
low-iterated
fractal
bondaries
of
the
origi-
nal
domain
remain
the
same, we
then
claim
that
this
is a
stable
domain.
On
the
opposite,
when
the
fractal
boundaries
change
with
iteration
order,
then
original
domain
becoming
a
chaotic
domain
with
the
given
set
of
parameters.
In
the
following, we will
examine
the
impact
of
normalized
energy
level
and