- Email: [email protected]
New differential equation model for hydrodynamics with memory effects
L-01. 46 (2000)
NEW
REPORTS
DIFFERENTIAL
ON
MATHEMATICAL
No. l/2
PHYSICS
EQUATION MODEL FOR WITH MEMORY EFFECTS
HYDRODYNAMICS
A. S. MAKARENKO National Technical University of Ukraine (KPI), Department of Mathematical of System Analysis, Pobedy Ave. 37, 03056 Kiev. Ukraine (e-mail: [email protected]) (Received September 1, 1999 -
Revised
February
24,
Methods
2000)
We consider memory effects in hydrodynamic equations. Some new modeling systems of differential equations are proposed, represented by partial differential equations as well as by (finite-) infinite-dimensional systems of ODE. We discuss such topics as blow-up solutions (or collapses), the dimensions of attractors and types of chaos.
1.
Introduction There
are many
model
Kortevegm-de Vries’s, valid merely
equations
Navier-Stokes
for the description
used
in fluid and
equations).
But
of slow movements
script,ion can be based upon model hydrodynamic memory
and nonlocality
effects.
There
gas dynamics
(e.g.
such hydrodynamic
are
More general
de-
which take into account
the
of simple
equations
are many methods
Burger’s,
equations
fluids.
of dynamical
system
research
based either on the analytical and functional calculus or upon the modelling. But only the synthesis of different approaches can give sufficiently deep insight into the complicated system
behaviour.
such synthesis examples, cesses delay.
Investigation
[15,5].
of att,ractors
Memory
let us mention
is a common
Markovian
of dynamical property
chains with memory
[16: 8,111. Infinite-dimensional attractors The so-called anticipatory systems that
are also worth mentioning
system
gives an example
of real systems. or strongly
of
To give only a few nonequilibrium
pro-
are widely encountered in systems wit,h have been widely investigated recently,
here.
So in this paper we briefly
present
the results
of investigations
in dynamical systems. The main goals are to present solit,ons, blow-ups and chaotic behaviour. [I831
entirely
of the role of memory
new model
equ&ions
for
184 2. 2.1.
A. S. MAKARENKO
Model partial differential equations with memory effects Model partial differential equations
The basic equations commonly used for heat and mass transfer are parabolic-type heat equation and Navier-Stokes systems. But such equations lose their applicability in extended media when parameters change during correlation time and inside correlation length when the relaxation or nonlocal effects take place [16,8,11]. Such refined modelling equations have be employed for the investigations of heat conduction and thermoelasticity by Cattaneo, Vernotte, Joseph [4,8]. Taking into account the effects of memory one is also able to obtain new models in hydrodynamical problems. Let us remind some concepts of heat and mass transfer process descriptions that have got recently a deep theoretical background (for review see [8,10,11]). It is well known that there are many hierarchical levels of description of the same phenomena. Studying fundamental works by Zwanzig, Mary, Picirelly and Zubarev one is able to get convinced that the transition from lower to upper levels in hierarchy leads to the presence of memory effects in higher level description. Corresponding equations have integral form with special kernels in integrals. These kernels represent the memory and nonlocality effects. They usually depend on some parameters: relaxation times and characteristic length. Relaxation time has a physical interpretation as the time of essential influence of past states of system on its actual state. Similar equations were derived within thermodynamics with memory effects [16,4]. Relaxation time may have wide range of values from 10 s in gases to thousands of years in geotectonic flows [ll]. The mathematical object on hydrodynamic level is a system of partial differential equations for velocity, pressure and temperature. Such system includes equations of motion as well as constitutive equations which give the expressions for the deformations versus strains dependence [8,16]. Instant connections between displacements and strains lead in the case of the incompressible fluid to NavierStokes equations. The memory effects take on the form of integral constitutive equations. The simplest exponential kernels lead to constitutive equations introduced formerly for visco-elastic media. In such case we obtain the simplest modification of Navier-Stokes equations - the so-called equations for Maxwell media [12,8,9,10,11]). Thus we have the hydrodynamics with relaxation:
d2.i, auk a6
&+dt~fVk
g)
-yV2G-
(l+Tg)gradP+P=O,
(1)
divG = 0, where r is the relaxation time, u viscosity, VI, velocity components, P pressure, F external forces. When 7 = 0 (no memory effects) this system coincides with the Navier-Stokes system. The system (1) is still a very complex mathematical object. So attemping to understand its properties one may resort to the investigation of simpler model equations. It is well known that such model equation for a Navier-Stokes system is a one-dimensional nonlinear Burgers equation [5,8,11]. This equation was introduced in 1948 empirically and later derived within the asymptotic approach. It has been proposed recently [13] to include in the Burgers equation second-order derivative with respect to time. The
NEW
DIFFERENTIAL
EQUATION
MODEL
FOR HYDRODYNAMICS
WITH
MEMORY
185
main goal of such modification was to take into account memory effects. Then the one-dimensional model equation is the so-called hyperbolic modification of the Burgers equation
Its numerical and analytical studies reveal many curious properties. Maybe the most interesting one is the existence of blow-up (collapse) solutions [ll]. But not long ago t,he two- and t,hree-dimensional model equations for system (1) did not exist. Following the lines of deriving the Burgers equation we put forward a multi-dimensional model equations. Analysis of the full system (1) and physical properties of systems with memory (fast heat processes, plasma, turbulence, statistical mechanics, visco-elasticity) allows t,o pick out some necessary properties that should be accounted for by model equations. Such properties are viscosity, mass transfer by convection and finite speed of disturbances propagation. Besides, the properties of such model equations should resemble in some specific cases the behaviour of (1) and, what is more important, the behaviour of real objects (for example the stability properties). Our analysis leads to some new model equations with such properties. The simplest two- and three-dimensional equations have the form (k: = 1.2 or Ic = 1,2,3)
Also the nonlocality 2.2.
may be incorporated
Collapse (blow-up)
in the model equations.
solutions
The common spaces for the solutions of partial differential equations are Sobolev spaces. But there are solutions which are out of the above mentioned classes. For example. see t,he vortices, defects, monopolies etc. One of the classes of such solutions are blow-up solutions or solutions of peaking regimes if to follow the terminology of A. Samarsky’s group. In that case the solutions gain infinite values in finite time [14]. In this paper we remind the results on the singular solutions of the hyperbolic equations of second order. First class consists of t,he so-called boundary blow-up regimes, going to infinity in finite time when the solutions on the boundary get the infinite values in finit,e Cme. The properties of such regimes have been considered, and the statement,s on the peculiarities of these solutions, existence of second boundary layer have been proved. The second class consists of the blow-up solutions of equations with increasing nonlinear source in the RHS, and charact,erizing solutions gaining infinite values inside a domain. Some theorems on the nonexistence of solutions were proved in this case. The third class of blow-up solutions arises in the hydrodynamic-type problems without, explicit increasing source. There were considered the questions of collapses arising in the hyperbolic modification of the Burgers equation. Preliminary computer and analytic investigations show some very unexpected results. We found strong dependence of solution bchaviour on the flow velocity. In fast flows the solutions blow-up. Especially int,eresting is the behaviour of the vortices. In fast, flows vortices are strengthened with reducing
186
A. S. MAKARENKO
thickness of vortices filament. Also we considered the break-up of initial vortex on many small vortices. We suppose that just the same mechanism exists in real turbulence. Such phenomenon suggest prospective investigations of the model equations above by methods used for distributed hydrodynamic and active chemical media as well as methods of the theory of superconductivity and quantum field theory (see [17,3,6]). The blow-up solutions are good candidates for the investigation of different types of noncorrectness of equations. They can also serve as source of measure-valued solutions. 3.
Finite-dimensional
systems
for hydrodynamics
with
memory
effects
One of the approaches to the investigation of hydrodynamics equations is the Galerkin method. Using this method it is easy to construct a low-dimensional dynamical system. In particular, for the hydrodynamical equations one of such systems is the well-known Lorenz system. But, as was described, more accurate is the hydrodynamics with memory effects. In 1979, Boldrighini and Franceschini investigated five-dimensional systems for plane flows [ 11. Using the projection methods we derived from (1) a low-dimensional system of ODE. With T = 0 (no memory effects) this system coincides with the system from [l], while with 0 < r < 1 a singular perturbation of Boldrighini-Franceschini systems arises. Also, in [9] there are described systems of ODE for generalised hydrodynamics with memory effects for three-dimensional flows with slip boundary conditions. We present some properties of such systems and the results of computer simulations. 3.1.
Finite-dimensional
systems
for three-dimensional
flows
The first class of dynamical models was derived for three-dimensional flows with the stick boundary condition. Such boundary conditions are imposed as follows. Let the rigid boundary of the fluid volume be marked as da. Then the stick boundary condition means that the velocity 3 is vanishing on the rigid boundary (nonslip boundary conditions). Physically this means that fluid doesn’t move near the boundary i&o = 0 (see [15], pp. 103-104). In the Galerkin method the solutions are looked for as series expansions (see [2] for the NavierStokes system)
where i&(t), k = 1,2, . . . . are unknown coefficients and {&}, k = 1,2,. system of orthogonal eigenfunctions for linear eigenvalue problem
VA&= -I”k& + grad pk,
diV&
= 0,
{~k}h2
=
. form the full
0.
The functional space is the closure of the set of smooth divergence free functions in the La metrics. The usual projection procedure of Galerkin’s method gives the equation
NEW
DIFFERENTIAL
EQUATION
In the case of F(z,t)
MODEL
WITH
hlEhlOR\I
187
= F( 2 ) we obtain the infinite system of ODE’s CklmZkZ,
k=l
1 = 1.2.3.
FOR HYDRODYNAMICS
+
“PlZl
+ ‘i-
m=l
. ., where f’ = k
i?m$dr.
i’;)
When r = 0 then the system (7) coincides with the coresponding system of ODE‘s obtained from Navier-Stokes equations. For 7 < 1 we have the singularly perturbed system. Thus especially interesting are the problems of singularly perturbed chaos. The simple low-dimensional systems as usually are received by reduction of infinite series. Brushlinskaya [2] received the three-dimensional system: it = -v,Xi
i, j, k = 1.2,3,
f AiXjXk $ I$,
i#j#k.
(8,
The counterpart to this system, with memory effects taken into account is the following six-dimensional system: i,
=
r -l(-51
-
25Xfi
&
=
7 -1(-x2
+
25&j
j_,
=
7 -‘(-x3 id
=
+%5x4
Xl,
25
-
vxq
=
+
KE5 v56 x2.
Fl) +
+
-
F2) F3)
(23x5
2&).
2(X156 + 2Qj).
+ -
+
(x2x:,
L-i& =
+xtjx2).
(!))
x3.
Here in (9) the variables 24, z5,26 correspond to the first three variables zl ,22; -?:Iin (7). introduction of such variables is sometimes useful for the reduction the ODE’s syst,em of second-order to the first-order system in time. 3.2.
Finite-dimensional condition
systems
for a-dimensional
flows with a periodicity
We considered two-dimensional
flows of fluids with memory on a plane region T” = The geometry of flow and boundary conditions were imposed just like in [l]. In such case we consider the flows with the velocity component v, = 0. We assume that the flow is space-periodic in the (.c, ;u)plane with the periods 2x. This implies that the flow and flow derivatives are periodic with the period 27r. Moreover, we also assume the mean flow averages over the region T2 = [0, 2n] x [0, 27r] (for details see [I] and [15], pp. 103~mlO4): [0, 27r] x [0, 27~1with periodical boundary conditions.
s T2
+dx.
In our case t)he solution is considered as the series on harmonics exp(&i), co-ordinates and & are wave vectors with integer components, 5(x, t) = 2 k=l
rk(t) exp(&z)
where i are
(10)
188
A. S. MAKARENKO
After long calculations r-
d2Tk+
we obtained the system of equations for the coefficients ok,
2 = -i c
y//f&k; + k;)
k’+k”+k
1
(II)
This system has infinite dimension. After reduction to a low-dimensional choice of a small number of wave vectors in [l] the system was obtained: ?l =
27’1
+
4y2ys
4y4ys,
+
$4 = -574
$2 = -
-9y2
+
+3 = -573
?5 = -75
YlY5,
The memory effects lead to the ten-dimensional
-
-
7ylyz
+ T,
3YlY4.
counterpart
(12)
of (12):
r&
=
(-51
- 2Zs + 4S7Zs + 4 X9X10) + ~T(XZX~ + X7X3) + 47(24210 + x9x5),
722
=
(--lc2
-
9x7 +
3x&r&) + s(xixs
rLi3
=
(-X3
-
52s - 720x7)
-
7(X1X7
724
=
(-X4
-
5x9
X6510)
-
(21x10
725
=
(-x5
-
X10 -
3xsxg)
-
3(x129
i,
4.
3yiys,
system by a
= Xl,
-
lc7 =x2,
Analysis of low-dimensional
i,
+ X623)7, + X622)7
+ R,
+ X625)7,
(13)
+ X624)7, = x3,
$9 = x4,
iTI
= x5.
systems
The systems (8), (9), (12) and (13) were investigated numerically and analytically. First of all we describe some results of computer modelling. For the six-dimensional system (9) we found the emergence of periodical solutions. Such solutions were found also without external forces (F = 0) with nonzero initial conditions. We also found in some cases phenomena similar to “intermittency” (bursts in solutions). The ten-dimensional system (13) was investigated for some parameters. We changed the values of relaxation time and initial conditions. In the case with no memory (r = 0) the projections of the phase portrait on two-dimensional planes have a “butterfly” type of attractor similar to the pictures for Lorenz chaotic attractor [5,1]. In the case with r # 0 there is complex behaviour of a new type. Some examples of numerical solution are desribed in [9,4]. Visually trajectory fills densely a bounded volume (named “container”). Trajectory projections on the planes have broken form in many points. Locally the projection of phase portrait remembers involved ball of thread or else “patience”. Visually the behaviour is similar to two-dimensional mappings with homoclinic tangency and quasi-attractors which was described in [7]. Similar pictures were found in systems with the so-called “fat-fractals” with larger fractal dimension then Cantor-type set. In
NEW
DIFFERENTIAL
EQUATION
MODEL
FOR HYDRODYNAMICS
WITH
MEMORY
189
many cases the results of numerical calculations remember the projections of the motion on torus. The first Liapunov characteristic exponent stayed positive for a long time (but with diminishing value). Numerical investigation of (13) with different values of R displayed the stiff excitation of complex behaviour. Standard one-dimensional bifurcation diagram is entirely different from the usual period doubling scenario of transit,ion t,o chaos. We also made some numerical investigations of bifurcation for the lo-dimensional system. We found that in case of vanishing external forces (I? = 0) there was a unique stationary point with zero co-ordinates. The Jacobian of the right part of (13) had a pair of pure imaginary eigenvalues. Some further bifurcation was received by increasing the number of stationary points to ten under increasing the values of R. At this processes t,he pairs of complex conjugate eigenvalues cross the imaginary axis from left to right. Further investigations of the systems above look very promissing in the cast 7 < 1. This is the singular perturbation of the usual systems of ODE’s with chaos. The evaluation of bounds for the attractor dimension in the case r + 0 is interesting (especially in the limit N or/and R tending to infinity). Remark that (7) consist of ODE’s of sccond order in time. Hence such a system reminds the collection of oscillators. Each 3k corresponds to a wave number of harmonics k. So, we may anticipate the propert,ies like transmission of energy on the spectrum of harmonics, the existence of harmonics clust,ers, resonances and so on. Also some ideas of memory effects in turbulence may be reconsidered. There arc many space and time scales in fluid flows. This implies the existence of many types of chaotic behaviour in fluids. The account of memory leads presumably to a new type of chaos similar to the chaos in media constructed from oscillators. Let us remark also that the above complex behaviour may serve as a prototype of new possible type of chaos in media with finite speed of propagation and with gauge symmetry. 5.
Conclusions
In this paper the short report on the investigation of new model equations is One is able to conclude from these results that the memory effects present hydrodynamic equations in many cases change the qualitative and quantitative of their solutions in an essential way. The main conclusion is the possibility t,ype of chaos and vorticity behaviour.
presented. in model brhaviour of a new
Acknowledgements The author is grateful to Dr. J. Burzlaff, Dr. H. True for their interest and comments about the results presented above and to Dr. V. Vladimirov for many useful discussions and help in preparing this paper. REFERENCES Boldrighini and N. Franceschini: Commun. Math. Phys. 64, 159-170 (1979) [2] N. Brushlinskaya: Doklady AN USSR, 162, 731-734 (1964). [l]
C.
A. S. MAKARENKO
190
[3] J. Burzlaff and W. Zakrzewski: Nonlinearity, 11, 1311-1320 (1998). [4] V. Danilenko, V. Korolevich, A. Makarenko and V. Christenuk: Selforganization in Strongly Nonequilibria Media. Collapses and Structures, Inst. of Geophys., Kiev 1992. [5] U. Frish: Turbulence, Cambridge Univ. Press, Cambridge 1996. [6] [7] [8] [9] [lo] [ll] [12]
W. F’nshchich and R. Zhdanov: Phys. Rept. 172, 123-174 (1989). S. Gonchenko, D. Turaev and L. Silnikov: Physica D 62, l-14 (1993). D. Joseph and L. Preziosi: Rev. Mod. Phys. 61, No. 2E, 47-73 (1989). A. S. Makarenko: Doklady Akademii Nauk of Ukraine 2, 85-87 (1994). A. S. Makarenko: Control and Cybernetics 25, 621-630 (1996). A. Makarenko, M. Moskalkov and S. Levkov: Phys. Lett. A 235, 391-397 (1997). A. Oskolkov: Proc. Math. Inst. AN USSR CLIX, 103-131 (1983).
[13] V. Rudiak and S. Smagulov: Dokladi Akademii Nauk USSR 255, 801-804 (1980). [14] A. Samarsky, B. Galaktionov, S. Kurdiumov and A. Mickhailov: Blow-up Solutions in Problems for Quasilinear Parabolic Equations, Nauka, Moscow 1987. [15] R. Temam: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, N. Y. 1988. [16] C. Truesdell: First Course in Rational Continuum Mechanics, The Johns Hopkins Univ., Baltimore 1972. [17] A. Winfree:
The Geometry of Biological Time, Springer, Berlin 1980.
NEW
REPORTS
DIFFERENTIAL
ON
MATHEMATICAL
No. l/2
PHYSICS
EQUATION MODEL FOR WITH MEMORY EFFECTS
HYDRODYNAMICS
A. S. MAKARENKO National Technical University of Ukraine (KPI), Department of Mathematical of System Analysis, Pobedy Ave. 37, 03056 Kiev. Ukraine (e-mail: [email protected]) (Received September 1, 1999 -
Revised
February
24,
Methods
2000)
We consider memory effects in hydrodynamic equations. Some new modeling systems of differential equations are proposed, represented by partial differential equations as well as by (finite-) infinite-dimensional systems of ODE. We discuss such topics as blow-up solutions (or collapses), the dimensions of attractors and types of chaos.
1.
Introduction There
are many
model
Kortevegm-de Vries’s, valid merely
equations
Navier-Stokes
for the description
used
in fluid and
equations).
But
of slow movements
script,ion can be based upon model hydrodynamic memory
and nonlocality
effects.
There
gas dynamics
(e.g.
such hydrodynamic
are
More general
de-
which take into account
the
of simple
equations
are many methods
Burger’s,
equations
fluids.
of dynamical
system
research
based either on the analytical and functional calculus or upon the modelling. But only the synthesis of different approaches can give sufficiently deep insight into the complicated system
behaviour.
such synthesis examples, cesses delay.
Investigation
[15,5].
of att,ractors
Memory
let us mention
is a common
Markovian
of dynamical property
chains with memory
[16: 8,111. Infinite-dimensional attractors The so-called anticipatory systems that
are also worth mentioning
system
gives an example
of real systems. or strongly
of
To give only a few nonequilibrium
pro-
are widely encountered in systems wit,h have been widely investigated recently,
here.
So in this paper we briefly
present
the results
of investigations
in dynamical systems. The main goals are to present solit,ons, blow-ups and chaotic behaviour. [I831
entirely
of the role of memory
new model
equ&ions
for
184 2. 2.1.
A. S. MAKARENKO
Model partial differential equations with memory effects Model partial differential equations
The basic equations commonly used for heat and mass transfer are parabolic-type heat equation and Navier-Stokes systems. But such equations lose their applicability in extended media when parameters change during correlation time and inside correlation length when the relaxation or nonlocal effects take place [16,8,11]. Such refined modelling equations have be employed for the investigations of heat conduction and thermoelasticity by Cattaneo, Vernotte, Joseph [4,8]. Taking into account the effects of memory one is also able to obtain new models in hydrodynamical problems. Let us remind some concepts of heat and mass transfer process descriptions that have got recently a deep theoretical background (for review see [8,10,11]). It is well known that there are many hierarchical levels of description of the same phenomena. Studying fundamental works by Zwanzig, Mary, Picirelly and Zubarev one is able to get convinced that the transition from lower to upper levels in hierarchy leads to the presence of memory effects in higher level description. Corresponding equations have integral form with special kernels in integrals. These kernels represent the memory and nonlocality effects. They usually depend on some parameters: relaxation times and characteristic length. Relaxation time has a physical interpretation as the time of essential influence of past states of system on its actual state. Similar equations were derived within thermodynamics with memory effects [16,4]. Relaxation time may have wide range of values from 10 s in gases to thousands of years in geotectonic flows [ll]. The mathematical object on hydrodynamic level is a system of partial differential equations for velocity, pressure and temperature. Such system includes equations of motion as well as constitutive equations which give the expressions for the deformations versus strains dependence [8,16]. Instant connections between displacements and strains lead in the case of the incompressible fluid to NavierStokes equations. The memory effects take on the form of integral constitutive equations. The simplest exponential kernels lead to constitutive equations introduced formerly for visco-elastic media. In such case we obtain the simplest modification of Navier-Stokes equations - the so-called equations for Maxwell media [12,8,9,10,11]). Thus we have the hydrodynamics with relaxation:
d2.i, auk a6
&+dt~fVk
g)
-yV2G-
(l+Tg)gradP+P=O,
(1)
divG = 0, where r is the relaxation time, u viscosity, VI, velocity components, P pressure, F external forces. When 7 = 0 (no memory effects) this system coincides with the Navier-Stokes system. The system (1) is still a very complex mathematical object. So attemping to understand its properties one may resort to the investigation of simpler model equations. It is well known that such model equation for a Navier-Stokes system is a one-dimensional nonlinear Burgers equation [5,8,11]. This equation was introduced in 1948 empirically and later derived within the asymptotic approach. It has been proposed recently [13] to include in the Burgers equation second-order derivative with respect to time. The
NEW
DIFFERENTIAL
EQUATION
MODEL
FOR HYDRODYNAMICS
WITH
MEMORY
185
main goal of such modification was to take into account memory effects. Then the one-dimensional model equation is the so-called hyperbolic modification of the Burgers equation
Its numerical and analytical studies reveal many curious properties. Maybe the most interesting one is the existence of blow-up (collapse) solutions [ll]. But not long ago t,he two- and t,hree-dimensional model equations for system (1) did not exist. Following the lines of deriving the Burgers equation we put forward a multi-dimensional model equations. Analysis of the full system (1) and physical properties of systems with memory (fast heat processes, plasma, turbulence, statistical mechanics, visco-elasticity) allows t,o pick out some necessary properties that should be accounted for by model equations. Such properties are viscosity, mass transfer by convection and finite speed of disturbances propagation. Besides, the properties of such model equations should resemble in some specific cases the behaviour of (1) and, what is more important, the behaviour of real objects (for example the stability properties). Our analysis leads to some new model equations with such properties. The simplest two- and three-dimensional equations have the form (k: = 1.2 or Ic = 1,2,3)
Also the nonlocality 2.2.
may be incorporated
Collapse (blow-up)
in the model equations.
solutions
The common spaces for the solutions of partial differential equations are Sobolev spaces. But there are solutions which are out of the above mentioned classes. For example. see t,he vortices, defects, monopolies etc. One of the classes of such solutions are blow-up solutions or solutions of peaking regimes if to follow the terminology of A. Samarsky’s group. In that case the solutions gain infinite values in finite time [14]. In this paper we remind the results on the singular solutions of the hyperbolic equations of second order. First class consists of t,he so-called boundary blow-up regimes, going to infinity in finite time when the solutions on the boundary get the infinite values in finit,e Cme. The properties of such regimes have been considered, and the statement,s on the peculiarities of these solutions, existence of second boundary layer have been proved. The second class consists of the blow-up solutions of equations with increasing nonlinear source in the RHS, and charact,erizing solutions gaining infinite values inside a domain. Some theorems on the nonexistence of solutions were proved in this case. The third class of blow-up solutions arises in the hydrodynamic-type problems without, explicit increasing source. There were considered the questions of collapses arising in the hyperbolic modification of the Burgers equation. Preliminary computer and analytic investigations show some very unexpected results. We found strong dependence of solution bchaviour on the flow velocity. In fast flows the solutions blow-up. Especially int,eresting is the behaviour of the vortices. In fast, flows vortices are strengthened with reducing
186
A. S. MAKARENKO
thickness of vortices filament. Also we considered the break-up of initial vortex on many small vortices. We suppose that just the same mechanism exists in real turbulence. Such phenomenon suggest prospective investigations of the model equations above by methods used for distributed hydrodynamic and active chemical media as well as methods of the theory of superconductivity and quantum field theory (see [17,3,6]). The blow-up solutions are good candidates for the investigation of different types of noncorrectness of equations. They can also serve as source of measure-valued solutions. 3.
Finite-dimensional
systems
for hydrodynamics
with
memory
effects
One of the approaches to the investigation of hydrodynamics equations is the Galerkin method. Using this method it is easy to construct a low-dimensional dynamical system. In particular, for the hydrodynamical equations one of such systems is the well-known Lorenz system. But, as was described, more accurate is the hydrodynamics with memory effects. In 1979, Boldrighini and Franceschini investigated five-dimensional systems for plane flows [ 11. Using the projection methods we derived from (1) a low-dimensional system of ODE. With T = 0 (no memory effects) this system coincides with the system from [l], while with 0 < r < 1 a singular perturbation of Boldrighini-Franceschini systems arises. Also, in [9] there are described systems of ODE for generalised hydrodynamics with memory effects for three-dimensional flows with slip boundary conditions. We present some properties of such systems and the results of computer simulations. 3.1.
Finite-dimensional
systems
for three-dimensional
flows
The first class of dynamical models was derived for three-dimensional flows with the stick boundary condition. Such boundary conditions are imposed as follows. Let the rigid boundary of the fluid volume be marked as da. Then the stick boundary condition means that the velocity 3 is vanishing on the rigid boundary (nonslip boundary conditions). Physically this means that fluid doesn’t move near the boundary i&o = 0 (see [15], pp. 103-104). In the Galerkin method the solutions are looked for as series expansions (see [2] for the NavierStokes system)
where i&(t), k = 1,2, . . . . are unknown coefficients and {&}, k = 1,2,. system of orthogonal eigenfunctions for linear eigenvalue problem
VA&= -I”k& + grad pk,
diV&
= 0,
{~k}h2
=
. form the full
0.
The functional space is the closure of the set of smooth divergence free functions in the La metrics. The usual projection procedure of Galerkin’s method gives the equation
NEW
DIFFERENTIAL
EQUATION
In the case of F(z,t)
MODEL
WITH
hlEhlOR\I
187
= F( 2 ) we obtain the infinite system of ODE’s CklmZkZ,
k=l
1 = 1.2.3.
FOR HYDRODYNAMICS
+
“PlZl
+ ‘i-
m=l
. ., where f’ = k
i?m$dr.
i’;)
When r = 0 then the system (7) coincides with the coresponding system of ODE‘s obtained from Navier-Stokes equations. For 7 < 1 we have the singularly perturbed system. Thus especially interesting are the problems of singularly perturbed chaos. The simple low-dimensional systems as usually are received by reduction of infinite series. Brushlinskaya [2] received the three-dimensional system: it = -v,Xi
i, j, k = 1.2,3,
f AiXjXk $ I$,
i#j#k.
(8,
The counterpart to this system, with memory effects taken into account is the following six-dimensional system: i,
=
r -l(-51
-
25Xfi
&
=
7 -1(-x2
+
25&j
j_,
=
7 -‘(-x3 id
=
+%5x4
Xl,
25
-
vxq
=
+
KE5 v56 x2.
Fl) +
+
-
F2) F3)
(23x5
2&).
2(X156 + 2Qj).
+ -
+
(x2x:,
L-i& =
+xtjx2).
(!))
x3.
Here in (9) the variables 24, z5,26 correspond to the first three variables zl ,22; -?:Iin (7). introduction of such variables is sometimes useful for the reduction the ODE’s syst,em of second-order to the first-order system in time. 3.2.
Finite-dimensional condition
systems
for a-dimensional
flows with a periodicity
We considered two-dimensional
flows of fluids with memory on a plane region T” = The geometry of flow and boundary conditions were imposed just like in [l]. In such case we consider the flows with the velocity component v, = 0. We assume that the flow is space-periodic in the (.c, ;u)plane with the periods 2x. This implies that the flow and flow derivatives are periodic with the period 27r. Moreover, we also assume the mean flow averages over the region T2 = [0, 2n] x [0, 27r] (for details see [I] and [15], pp. 103~mlO4): [0, 27r] x [0, 27~1with periodical boundary conditions.
s T2
+dx.
In our case t)he solution is considered as the series on harmonics exp(&i), co-ordinates and & are wave vectors with integer components, 5(x, t) = 2 k=l
rk(t) exp(&z)
where i are
(10)
188
A. S. MAKARENKO
After long calculations r-
d2Tk+
we obtained the system of equations for the coefficients ok,
2 = -i c
y//f&k; + k;)
k’+k”+k
1
(II)
This system has infinite dimension. After reduction to a low-dimensional choice of a small number of wave vectors in [l] the system was obtained: ?l =
27’1
+
4y2ys
4y4ys,
+
$4 = -574
$2 = -
-9y2
+
+3 = -573
?5 = -75
YlY5,
The memory effects lead to the ten-dimensional
-
-
7ylyz
+ T,
3YlY4.
counterpart
(12)
of (12):
r&
=
(-51
- 2Zs + 4S7Zs + 4 X9X10) + ~T(XZX~ + X7X3) + 47(24210 + x9x5),
722
=
(--lc2
-
9x7 +
3x&r&) + s(xixs
rLi3
=
(-X3
-
52s - 720x7)
-
7(X1X7
724
=
(-X4
-
5x9
X6510)
-
(21x10
725
=
(-x5
-
X10 -
3xsxg)
-
3(x129
i,
4.
3yiys,
system by a
= Xl,
-
lc7 =x2,
Analysis of low-dimensional
i,
+ X623)7, + X622)7
+ R,
+ X625)7,
(13)
+ X624)7, = x3,
$9 = x4,
iTI
= x5.
systems
The systems (8), (9), (12) and (13) were investigated numerically and analytically. First of all we describe some results of computer modelling. For the six-dimensional system (9) we found the emergence of periodical solutions. Such solutions were found also without external forces (F = 0) with nonzero initial conditions. We also found in some cases phenomena similar to “intermittency” (bursts in solutions). The ten-dimensional system (13) was investigated for some parameters. We changed the values of relaxation time and initial conditions. In the case with no memory (r = 0) the projections of the phase portrait on two-dimensional planes have a “butterfly” type of attractor similar to the pictures for Lorenz chaotic attractor [5,1]. In the case with r # 0 there is complex behaviour of a new type. Some examples of numerical solution are desribed in [9,4]. Visually trajectory fills densely a bounded volume (named “container”). Trajectory projections on the planes have broken form in many points. Locally the projection of phase portrait remembers involved ball of thread or else “patience”. Visually the behaviour is similar to two-dimensional mappings with homoclinic tangency and quasi-attractors which was described in [7]. Similar pictures were found in systems with the so-called “fat-fractals” with larger fractal dimension then Cantor-type set. In
NEW
DIFFERENTIAL
EQUATION
MODEL
FOR HYDRODYNAMICS
WITH
MEMORY
189
many cases the results of numerical calculations remember the projections of the motion on torus. The first Liapunov characteristic exponent stayed positive for a long time (but with diminishing value). Numerical investigation of (13) with different values of R displayed the stiff excitation of complex behaviour. Standard one-dimensional bifurcation diagram is entirely different from the usual period doubling scenario of transit,ion t,o chaos. We also made some numerical investigations of bifurcation for the lo-dimensional system. We found that in case of vanishing external forces (I? = 0) there was a unique stationary point with zero co-ordinates. The Jacobian of the right part of (13) had a pair of pure imaginary eigenvalues. Some further bifurcation was received by increasing the number of stationary points to ten under increasing the values of R. At this processes t,he pairs of complex conjugate eigenvalues cross the imaginary axis from left to right. Further investigations of the systems above look very promissing in the cast 7 < 1. This is the singular perturbation of the usual systems of ODE’s with chaos. The evaluation of bounds for the attractor dimension in the case r + 0 is interesting (especially in the limit N or/and R tending to infinity). Remark that (7) consist of ODE’s of sccond order in time. Hence such a system reminds the collection of oscillators. Each 3k corresponds to a wave number of harmonics k. So, we may anticipate the propert,ies like transmission of energy on the spectrum of harmonics, the existence of harmonics clust,ers, resonances and so on. Also some ideas of memory effects in turbulence may be reconsidered. There arc many space and time scales in fluid flows. This implies the existence of many types of chaotic behaviour in fluids. The account of memory leads presumably to a new type of chaos similar to the chaos in media constructed from oscillators. Let us remark also that the above complex behaviour may serve as a prototype of new possible type of chaos in media with finite speed of propagation and with gauge symmetry. 5.
Conclusions
In this paper the short report on the investigation of new model equations is One is able to conclude from these results that the memory effects present hydrodynamic equations in many cases change the qualitative and quantitative of their solutions in an essential way. The main conclusion is the possibility t,ype of chaos and vorticity behaviour.
presented. in model brhaviour of a new
Acknowledgements The author is grateful to Dr. J. Burzlaff, Dr. H. True for their interest and comments about the results presented above and to Dr. V. Vladimirov for many useful discussions and help in preparing this paper. REFERENCES Boldrighini and N. Franceschini: Commun. Math. Phys. 64, 159-170 (1979) [2] N. Brushlinskaya: Doklady AN USSR, 162, 731-734 (1964). [l]
C.
A. S. MAKARENKO
190
[3] J. Burzlaff and W. Zakrzewski: Nonlinearity, 11, 1311-1320 (1998). [4] V. Danilenko, V. Korolevich, A. Makarenko and V. Christenuk: Selforganization in Strongly Nonequilibria Media. Collapses and Structures, Inst. of Geophys., Kiev 1992. [5] U. Frish: Turbulence, Cambridge Univ. Press, Cambridge 1996. [6] [7] [8] [9] [lo] [ll] [12]
W. F’nshchich and R. Zhdanov: Phys. Rept. 172, 123-174 (1989). S. Gonchenko, D. Turaev and L. Silnikov: Physica D 62, l-14 (1993). D. Joseph and L. Preziosi: Rev. Mod. Phys. 61, No. 2E, 47-73 (1989). A. S. Makarenko: Doklady Akademii Nauk of Ukraine 2, 85-87 (1994). A. S. Makarenko: Control and Cybernetics 25, 621-630 (1996). A. Makarenko, M. Moskalkov and S. Levkov: Phys. Lett. A 235, 391-397 (1997). A. Oskolkov: Proc. Math. Inst. AN USSR CLIX, 103-131 (1983).
[13] V. Rudiak and S. Smagulov: Dokladi Akademii Nauk USSR 255, 801-804 (1980). [14] A. Samarsky, B. Galaktionov, S. Kurdiumov and A. Mickhailov: Blow-up Solutions in Problems for Quasilinear Parabolic Equations, Nauka, Moscow 1987. [15] R. Temam: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, N. Y. 1988. [16] C. Truesdell: First Course in Rational Continuum Mechanics, The Johns Hopkins Univ., Baltimore 1972. [17] A. Winfree:
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