Structural-phase transitions and state function in unstable convective medium
Convection in a thin layer of liquid (gas) with temperature independent viscosity between poorly heat conducting boundaries is studied within framework of the Proctor-Sivashinsky model. We have shown by numerical simulation of the Proctor-Sivashinsky model that the state with certain topology can be...
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irk-123456789-1122222017-01-19T03:02:36Z Structural-phase transitions and state function in unstable convective medium Gushchin, I.V. Kirichok, A.V. Kuklin, V.M. Нелинейные процессы в плазменных средах Convection in a thin layer of liquid (gas) with temperature independent viscosity between poorly heat conducting boundaries is studied within framework of the Proctor-Sivashinsky model. We have shown by numerical simulation of the Proctor-Sivashinsky model that the state with certain topology can be described by the state function, which is the sum of squared mode amplitudes of spatial temperature spectrum. The transitions between these states are characterized by splashes in time-derivative of this function and different meta-stable structures, corresponding to different values of the state function have different visually distinguishable topologies. Конвекцію в тонкому шарі рідини (газу) між поверхнями, що недостатньо добре проводять тепло, розглянуто в умовах придатності моделі Проктора-Сівашинського за відсутністю залежності в’язкості від температури. З використанням числового моделювання показано, що кожен стан може бути представлений за допомогою функції стану, що дорівнює сумі квадратів мод просторового спектра температури на поверхні. Перехід між станами характеризується значенням похідної за часом від функції станів. Різниця між метастабільними станами, які відрізняються топологією, визначається різними значеннями функції стану. Конвекция в тонком слое жидкости (газа) между плохо проводящими тепло поверхностями рассмотрена в условиях применимости модели Проктора-Сивашинского при отсутствие зависимости вязкости от температуры. С помощью численного анализа показано, что каждое состояние может быть описано с помощью функции состояния, которая равна сумме квадратов мод спектра пространственного распределения температуры на поверхности. Переход между состояниями характеризуется изменением производной по времени от этой функции. Различие между метастабильными состояниями, которые отличаются топологией, определяется разными значениями функции состояния. 2015 Article Structural-phase transitions and state function in unstable convective medium / I.V. Gushchin, A.V. Kirichok, V.M. Kuklin // Вопросы атомной науки и техники. — 2015. — № 4. — С. 252-254. — Бібліогр.: 13 назв. — англ. 1562-6016 PACS: 47.20.-k http://dspace.nbuv.gov.ua/handle/123456789/112222 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Нелинейные процессы в плазменных средах Нелинейные процессы в плазменных средах |
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Нелинейные процессы в плазменных средах Нелинейные процессы в плазменных средах Gushchin, I.V. Kirichok, A.V. Kuklin, V.M. Structural-phase transitions and state function in unstable convective medium Вопросы атомной науки и техники |
| description |
Convection in a thin layer of liquid (gas) with temperature independent viscosity between poorly heat conducting boundaries is studied within framework of the Proctor-Sivashinsky model. We have shown by numerical simulation of the Proctor-Sivashinsky model that the state with certain topology can be described by the state function, which is the sum of squared mode amplitudes of spatial temperature spectrum. The transitions between these states are characterized by splashes in time-derivative of this function and different meta-stable structures, corresponding to different values of the state function have different visually distinguishable topologies. |
| format |
Article |
| author |
Gushchin, I.V. Kirichok, A.V. Kuklin, V.M. |
| author_facet |
Gushchin, I.V. Kirichok, A.V. Kuklin, V.M. |
| author_sort |
Gushchin, I.V. |
| title |
Structural-phase transitions and state function in unstable convective medium |
| title_short |
Structural-phase transitions and state function in unstable convective medium |
| title_full |
Structural-phase transitions and state function in unstable convective medium |
| title_fullStr |
Structural-phase transitions and state function in unstable convective medium |
| title_full_unstemmed |
Structural-phase transitions and state function in unstable convective medium |
| title_sort |
structural-phase transitions and state function in unstable convective medium |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2015 |
| topic_facet |
Нелинейные процессы в плазменных средах |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/112222 |
| citation_txt |
Structural-phase transitions and state function in unstable convective medium / I.V. Gushchin, A.V. Kirichok, V.M. Kuklin // Вопросы атомной науки и техники. — 2015. — № 4. — С. 252-254. — Бібліогр.: 13 назв. — англ. |
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Вопросы атомной науки и техники |
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2025-07-08T03:33:39Z |
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| fulltext |
ISSN 1562-6016. ВАНТ. 2015. №4(98) 252
STRUCTURAL-PHASE TRANSITIONS AND STATE FUNCTION
IN UNSTABLE CONVECTIVE MEDIUM
I.V. Gushchin, A.V. Kirichok, V.M. Kuklin
Y.N. Karazin Kharkiv National University, Institute for High Technologies, Kharkov, Ukraine
E-mail: sandyrcs@gmail.com
Convection in a thin layer of liquid (gas) with temperature independent viscosity between poorly heat conduct-
ing boundaries is studied within framework of the Proctor-Sivashinsky model. We have shown by numerical simula-
tion of the Proctor-Sivashinsky model that the state with certain topology can be described by the state function,
which is the sum of squared mode amplitudes of spatial temperature spectrum. The transitions between these states
are characterized by splashes in time-derivative of this function and different meta-stable structures, corresponding
to different values of the state function have different visually distinguishable topologies.
PACS: 47.20.-k
Nonlinear systems with many degrees of freedom
can undergo non equilibrium phase transitions charac-
terized by a large variety of spatial or spatiotemporal
patterns. Transitions and competition among these pat-
terns of different symmetries are fundamental problems,
which have attracted considerable interest in recent dec-
ades. Convection in a horizontal fluid layer subject to a
vertical temperature gradient is very convenient for their
study [1 - 5] due to its relative simplicity and great vari-
ety of observed patterns.
The Proctor-Sivashinsky model [6, 7] is found to be
very attractive for studying the processes of pattern
formation in systems, which possess a preferred charac-
teristic spatial scale of interaction between the elements
of future structure. This model was developed for de-
scription of convection in a thin layer of liquid between
poorly conducting horizontal boundaries. Authors of [8]
have found the stationary solutions with a small number
of the spatial modes, one of which (convective cells)
was steady and the second one turned out to be unstable
(convective rolls). A particular future of the model is
that it forces a preferred spatial scale of interaction,
leaving the system a chance of selecting the symmetry
during evolution. It was found, that the type of sym-
metry and hence the characteristics of the structure are
determined by the minima of the potential of interaction
between modes lying on a circle in k-space. Even within
the Proctor-Sivashinsky model not all processes and the
phenomena were studied. The detailed analysis of insta-
bility leading to the formation of a metastable structure
(convective rolls) will be presented below. Earlier, it
was found that at first stage of the instability evolution
the metastable long-lived state (the curved quasi-one-
dimensional convective rolls) arises. And later, after a
lapse of time (which is considerably greater than the
reverse linear increment of the process), the system
transforms to the steady state (square convective cells)
[9, 10]. The detailed treatment of the Proctor-
Sivashinsky model presented below shows that this
structural transition demonstrates all the characteristics
of the second order phase transition (the continuity of
the sum of squared mode amplitudes over the spectrum
that the same, the continuity of density of this value and
discontinuity of its time derivative. The existence of
preferred scale (the distance between the regular spatial
perturbations) and the possibility to select the type of
symmetry (the regular spatial configuration) motivate
the interest to this physical model, particularly for de-
scription of processes in solid state physics, where the
characteristic distance between elements of spatial
structures (atoms, molecules) in their condensed state is
almost invariable. The objective of this work is investi-
gation of the mechanisms of pattern formation and
mode competition in convective medium. The nature
and evolution of structural phase transitions between
patterns of different topology are considered.
THE PROCTOR-SIVASHINSKY MODEL
When the Rayleigh number Ra exceeds the critical
value corresponding to the onset of convective flow, the
three-dimensional convection begins in a thin layer of
liquid between poorly conducting horizontal plates
heated from below [2], which can be described by the
Proctor-Sivashinsky equation [6, 7]. This equation de-
termines the dynamics of temperature field in the hori-
zontal plane (x,y):
( )
2 2 2
2 2
( ) (1 )
1 ,
3
f
ε γ
ε
Φ = Φ + ⋅∇ Φ∇Φ − −∇ Φ +
+ ∇ ∇Φ Φ +
(1)
where f is the random function describing the external
noise, and the quantity ε determines the convection
threshold overriding, which is assumed to be sufficient-
ly small ( 0 1ε< < ). The term ( )γ∇ Φ∇Φ describes the
temperature dependence of viscosity. Further, we as-
sume 0γ = for simplicity. In this case we shall find the
solution in the form
exp( )j j
j
a ik rεΦ = ∑
(2)
with | | 1jk =
. Renormalizing the time units 2ε∝ , we
obtain the evolution equation for slow amplitudes aj:
2
1
| |
N
j j mj m j
m
a a V a a
=
= −∑ , (3)
where interaction coefficients are determined as follows
=1,jjV (4)
( )( ) ( )2 2= (2 3) 1 2 (2 3) 1 2cosij i jV k k ϑ− = +
. (5)
Here ϑ is the angle between vectors ik
and jk
.
The instability interval in k-space represents a ring
with average radius equal to unit and the width is order
of relative above-threshold parameter ε, i.e. much less
than unity. During the development of the instability,
the effective growth rate of modes that are localized
mailto:sandyrcs@gmail.com
ISSN 1562-6016. ВАНТ. 2015. №4(98) 253
outside of the very small neighborhood near the unit
circle will decrease due to the growth of the nonlinear
terms and can change sign which will lead to a narrow-
ing of the spectrum to the unit circle in the k-space.
Since the purpose of further research will be the study
of stability of spatial structures with characteristic size
of order 2 / 2kπ π∝ and the important characteristic
for visualization of simulation results will be evidence
of these structures, so we restrict ourselves by consider-
ing some idealized model of the phenomenon, assuming
that the oscillation spectrum is already located on the
unit circle in the k-space.
SIMULATION RESULTS
It was shown in [9, 11] that in the absence of tem-
perature dependence of viscosity and when the number
of modes is sufficiently large, the system delayed the
development while remaining in a dynamic equilibrium.
Development of perturbations in the system, as shown
by the numerical analysis will be as follows [9]. Starting
from initial fluctuations, the modes over a wide range of
ϑ begin grow. The value of the quadratic form of the
spectrum 2
jj
I a= ∑ can be estimated to obtain as result
a value close to 0.75. It was shown that in the absence of
temperature dependence of viscosity and when the num-
ber of modes is sufficiently large, the system delayed the
development while remaining in a dynamic equilibrium.
For further development − "crystallization", one of the
modes must get a portion of the energy, which excesses
some threshold value. That is, in these case, it is neces-
sary a certain level of noise (fluctuations).
a
b
Fig. 1. Convective structures: rolls (a); square cells (b)
If one of the modes gets the proper amount of ener-
gy, then the process of formation of a simplest convec-
tive structure – rolls begins (Fig. 1,a). Note that in the
nature, the thin clouds also can form the roll structure.
The value of I in this case tends to unity ( 1I → ). How-
ever, this state is not stable and then we can see the next
structural transition: convective rolls are modulated
along the axis of fluid rotation, and the typical size of
this modulation phases down. In this transition state, the
system stays for a sufficiently long time (which slightly
increases within some limits with increase in the num-
ber of modes), and the value 1.07I ≈ remains constant
during this time. After a rather long time, ten times
more than the inverse linear growth rate of the initial
instability only the one mode “survives” from newly
formed “side” spectrum, which amplitude is comparable
with the amplitude of the primary leading mode. In the
end, the stable convective structure – square cells is
generated (Fig. 1,b), and the quadratic form I reaches
the value of 1.2I = .
Further researches of this process have found the fol-
lowing dynamics of quadratic form 2
jj
I a= ∑
with time
(Fig. 2). Exactly after the first peak of the derivative, the
metastable structure – a system of convective rolls is
formed, and up to the moment when the second burst
have appeared with value of 1I ≈ it remains un-
changed. The next burst indicates the onset of a second-
ary metastable structure with a new value of 1.07I ≈ .
After the second burst of the quadratic form deriva-
tive, a stable structure of squared convective cells is
started to build up. Such behavior proves the existence
of structural-phase transitions in the system.
Fig. 2. The evolution of the derivative dI dt (in relative
units) of the integral quadratic form 2
jj
I a= ∑
Generally speaking, the characteristic times of relax-
ation processes during evolution of the system to more
equilibrium state are determined as usual by the differ-
ence of the state function values before the transition
and after it. The greater this difference, the faster the
transition from one state to another. It is important to
keep in mind that the sequence of state transitions is
determined by the characteristic times of instabilities
(which play the role of relaxation processes) that pro-
vide a cascade evolution of the system to the most equi-
librium state. Initially, the fastest relaxation processes
take place that associated with large difference of the
state function values corresponding to different equilib-
rium states.
Let us verify that in this case all the phenomena oc-
cur in the same order and within the framework of the
foregoing scenario. The numerical analysis of the model
allows confirming these considerations.
It can be seen that the times of state formation τn are
inversely proportional to 2
i
i
I A= ∑ the difference be-
tween the values ( ) 2 ( )( )
in n
i
I A+ += ∑ after n-th structural
phase transition ( ) 2 ( )( )
in n
i
I A+ += ∑ and before it
( ) 2 ( )( ) ,
in n
i
I A− −= ∑
2 ( ) 2 ( ) 1 1{( ) ) } .
i in n n n
i i
A A Iτ + − − −− = ∆∑ ∑
(6)
It follows from this that
3 2 2 3/ /I Iτ τ ≈ ∆ ∆ . (7)
ISSN 1562-6016. ВАНТ. 2015. №4(98) 254
Thus, we have shown by numerical simulation of the
Proctor-Sivashinsky model that the state with certain to-
pology can be described by the state function, which is the
sum of squared mode amplitudes. The transitions between
these states are characterized by splashes in time-derivative
of this function and different meta-stable structures, corre-
sponding to different values of the state function have dif-
ferent visually distinguishable topologies.
The fact that the metastable states are characterized
by specific values of the state function was highlighted
in our earlier works [11 - 13]. The numerical study, pre-
sented in this paper, confirmed two observations:
1) the difference between the values of the state
function before and after the structural phase transition
is inversely proportional to the characteristic time of the
corresponding structural-phase transition;
2) the evolution of the planar convective structure
under consideration demonstrates all the features of a
relaxation process, i.e. the fast structural-phase transi-
tion is succeeded by more slow ones. Thus, a fuller pic-
ture of the process becomes clear.
CONCLUSIONS
The special feature of the Proctor-Sivashinsky mod-
el with temperature independent viscosity is the exist-
ence of three possible metastable states, which corre-
spond to patterns of different symmetries. The times of
structural transitions between these metastable states are
much less than the times of their existence. Each state
has a definite topology and can be characterized by def-
inite steady value of the state function. The metastable
states are destroyed with time for the instabilities, the
growth rate of which can be evaluated from the ampli-
tude of splashes of time-derivative of the state function.
It is shown, that the characteristic times of the instabili-
ties, which destroy the previous state and form a new
one are inversely proportional to the difference between
the values of the state function before and after the
structural phase transition. In addition, we show that the
faster relaxation processes, i.e. structural phase transi-
tions take priority over more slow ones.
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11. E.V. Belkin, I.V. Gushchin, A.V. Kirichok, V.M. Kuklin.
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Article received 12.05.2015
СТРУКТУРНО-ФАЗОВЫЕ ПЕРЕХОДЫ И ФУНКЦИЯ СОСТОЯНИЯ В НЕСТАБИЛЬНОЙ КОНВЕКТИВНОЙ
СРЕДЕ
И.В. Гущин, А.В. Киричок, В.М. Куклин
Конвекция в тонком слое жидкости (газа) между плохо проводящими тепло поверхностями рассмотрена
в условиях применимости модели Проктора-Сивашинского при отсутствие зависимости вязкости от темпе-
ратуры. С помощью численного анализа показано, что каждое состояние может быть описано с помощью
функции состояния, которая равна сумме квадратов мод спектра пространственного распределения темпера-
туры на поверхности. Переход между состояниями характеризуется изменением производной по времени от
этой функции. Различие между метастабильными состояниями, которые отличаются топологией, определя-
ется разными значениями функции состояния.
СТРУКТУРНО-ФАЗОВІ ПЕРЕХОДИ ТА ФУНКЦІЇ СТАНУ В НЕСТАБІЛЬНОМУ КОНВЕКТИВНОМУ
СЕРЕДОВИЩІ
І.В. Гущин, О.В. Киричок, В.М. Куклін
Конвекцію в тонкому шарі рідини (газу) між поверхнями, що недостатньо добре проводять тепло, розг-
лянуто в умовах придатності моделі Проктора-Сівашинського за відсутністю залежності в’язкості від тем-
ператури. З використанням числового моделювання показано, що кожен стан може бути представлений за
допомогою функції стану, що дорівнює сумі квадратів мод просторового спектра температури на поверхні.
Перехід між станами характеризується значенням похідної за часом від функції станів. Різниця між метаста-
більними станами, які відрізняються топологією, визначається різними значеннями функції стану.
|