- Email: [email protected]
Self-synchronization in coupled salt-water oscillators
PHYSlCA® ELS EV 1ER
Physica D 115 (1998) 313 320
Self-synchronization in coupled salt-water oscillators Satoshi Nakata a,,, Takahiro Miyata a, Nozomi Ojima a, Kenichi Yoshikawa b a Department ofChemistr3'. Nara Universi~' of Education. Takabatake-cho, Nara 630, Japan b Graduate School of Human h!lbrmatics, Nagoya Universit3; Nagoya 464-01. Japan
Received 9 July 1997; received in revised form I September 1997: accepted 26 September 1997 Communicated by Y. Kuramoto
Abstract The coupling of nonlinear oscillators was studied experimentally and theoretically. In the experiment, the electrochemical potentials in the salt water in cups dipped in a larger vessel filled with pure water were measured, where both the liquids were made to contact through a small orifice on the bottom of the individual cup. The mode of oscillations was found to depend on the surface area of the pure water. Another experiment was made to observe a coupling between the flows through separate orifices on the bottom of a single cup. The mode of flows depended on the difference in height between the two orifices. These experimental findings are predicted in a semi-quantitative manner by numerical simulations by the use of coupled ordinary differential equations. Copyright © 1998 Elsevier Science B.V. Kevwords: Nonlinear oscillation; Entrainment; Bifurcation; Convection-diffusion; Spatio-temporal sell-organization
1. Introduction Coupling between nonlinear oscillators is useful for understanding the mechanism of self-synchronization in living systems such as in the beating heart, neural networks, circadian rhythm, the organized rhythm in the flashing of swarms of fireflies, and so on [ 1]. There have been many experimental [2-9] and theoretical studies [10-14] on self-synchronization in coupled oscillators. Among the various kinds of oscillators, the coupling of two or more continuous stirred tank reactors (CSTRs) in the Belousov-Zhabotinsky (BZ) reaction has been extensively studied [2-9], since this system is much simpler than those in living systems.
*Corresponding author. Tel.: +81-742-27-9191; fax: +81742-27-9190; e-mail: [email protected].
As a very simple and easily reproducible experiment, in the present paper we will describe coupling between salt-water oscillators. Salt-water oscillators exhibit various nonlinear characteristics such as a limit cycle, bifurcation, quasi-periodicity, period multiplying, and synchronization [15-22]. It has been reported that (1) in the coupling of two salt-water oscillators, various modes of synchronization, i.e., anti-phase, 1 : 2, 1 : 3, and 2 : 3 synchronization, can be observed by changing the diameter of the orifice in the cup and the concentration of NaCI aqueous solution, and (2) in the coupling of three equivalent salt-water oscillators, a bistable tri-phasic mode of synchronization can be observed I 16,17]. In this paper, we report that the mode of synchronization in two coupled salt-water oscillators changes depending on the surface area of the outer phase of pure water in a characteristic manner. The mode of
0167-2789/98/$19.00 Copyright © 1998 Elsevier Science B.V. All rights reserved PII S0167-2789(97)00240-6
314
S. Nakata et al./Physica D 115 (1998) 31.?-320
(b-l)
Laser displacement meter
Salt-water oscillator
L_J (a)
Personal computer
Jack
circular stationary flow (bistable)
(b-2) Plastic board tfixed)
o
/\, I
2
2
i \
oscillatory flow (synchronization)
",, "1\
// s Mode IIl
Ag/AgC1 elcctrode Fig. 1. (a) Experimental apparatus for measuring of the electrical potential m two coupled salt-water oscillators, as an experimental index of the direction of flow. (bl) Experimental apparatus tor observing the interaction between the two fluid flows through the two orifices in a single cup. (b2) Schematic representation of two fluid flows through the two orifices in a single cup.
this synchronization, e.g. anti-phase or quasi-periodic one, is analyzed by a return map of the phase difference between the two oscillators. As tor coupled flow through two orifices in a single cup, either circular flow or oscillatory flow was seen depending on the difference in height between the two orifices. These experimental trends were reproduced by a numerical simulation based on coupled nonlinear differential equations.
2. Experimental The salt-water oscillator was constructed as in previous studies [16,17]. An outer glass vessel (volume: 2000 ml) and a cylindrical acrylic cup contained pure water and 3 M NaCI aqueous solution (60ml),
respectively; inner diameter of cylinder: 4 6 m m , height of cylinder: 60 ram, diameter of the orifice in the bottom of the cylinder: 1.0 mm, and thickness of orifice: 0.5 mm. The level of the pure water in the outer vessel was made almost equal to that of the NaCI aqueous solution in the plastic cup in terms of the difference in hydrostatic pressure at the position of the orifice, by adjusting the initial volume of pure water in the outer vessel. In the experiment with coupled oscillators, two equivalent cups with the same diameter and orifices of the same size were placed in a single outer vessel (Fig. l(a)). The direction of the oscillatory flow in the salt-water oscillators was monitored as an electrochemical potential by using A g / A g C I electrodes and a mV meter with an internal impedance higher than 1012~. In our previous experiment on
S. Nakata et al./Phvsica D 115 (1998) 313 320 the simultaneous measurement of the surface level and electrical potential in a salt-water oscillator, we confirmed that high and low potentials in the oscillation correspond to the outflow of salt-water and the inflow of pure water, respectively [16,17]. Since the maximum change in the water level is of the order of 0.1 mm, we chose to measure the electrical potential rather than the water level in the cup, the latter of which would have required expensive and delicate equipment. To change the surface area of the outer vessel, a plastic plate ithickness: 10mm) of the desired size was fixed at the water level of the outer vessel, as illustrated in Fig. l(a). For the experiment on coupled flow through two orifices in a single cup, we used a cylindrical cup with two 1 mm orifices in the bottom with a fixed distance (3 ram) between their centers. To produce a difference in height between the two orifices, a plate connected to the salt-water cell was tilted with a jack, as shown in Fig. l(b). The difl'erence in the heights of the two orifices was monitored with a laser displacement meter (Keyence. LPB-02, Japan).
3. Results and discussion 3. I. Experiments on the synchronization ~1 salt-water oscillators 3.1.1. Mode qf synchronization of two coupled salt-water oscillators depending on the surface area qf the outer vessel Fig. 2 shows the time-course of the electrical oscillations of two salt-water oscillators for outer vessels with different surface areas: (a) 20.7 cm 2 and (b) 32.7 cm 2. The phase difference, q~n, between the oscillators is calculated based on the time-trace of the electrical potential of the two oscillators, where 4',, is defined as 49,, = 2zrrn/T,, (r,, is the time difference is) between the oscillators and T,, is the period is) of the oscillations in oscillator 1). It is apparent that the mode of the oscillation in Fig. 2(a) is locked as antiphase. In contrast, the time-course of the phase difference in Fig. 2(b) is not locked, and changes gradually with time, i.e., quasi-periodic mode. The modes are
315
diagrammed in Fig. 3(a) as a function of the relative surface area, 79, (79 = so~S, so is the inner surface area (39.3 cm 2) with salt water and S is the outer surface area with pure water); i.e., 79 is inversely proportional to the outer surface area S. The modes of synchronization can be characterized by using a return map of the phase difference. 4~,,. as shown in Fig. 3(b). A - E in Fig. 3(b) correspond to those in Fig. 3(a), and the data in Fig. 3(d)E and (b)C correspond to those in Figs. 2(a) and (b), respectively. It is noted that coupling between the two oscillators is mediated by a change in the water level in the outer vessel, i.e., the coupling strength is inversely proportional to the outer surface area, S [17]. When the coupling parameter, 79, is smaller than 0.2, the two salt-water oscillators become almost independent, since the water level of the outer surface with a significant large surface area remains almost constant independent of the rhythmic flow in the individual oscillators, where the volume change during one oscillation in a cup is ca. 0.1 ml. Self-synchronization in the anti-phase mode was observed when 79 > 1.3 or S < 30cm 2. It is expected that the change in the height of the surface in the outer vessel is minimized in this anti-phase mode l 17]. Thus. a bifurcation point lies at 79 ~ 1.25 between the quasi-periodic and antiphase modes (Fig. 3(b)C and D).
3.1.2. Stationar3,flow and rhythmic.flow through two orifices in a single cup When a single cup with two orifices in its base was used, either circular flow or oscillatory flow (see Fig. lib)) was generated. The flow mode depended on the initial difference in height (Ah = h2 - h i ) between the two orifices isee Fig. l(b2)). In this experiment, the distance between the centers of two orifices was fixed at 3 mm and the cup contained 3.0 M NaCI in aqueous solution. When we start the experiment by removing small plugs from the orifices under the condition that IAhl is smaller than a critical value, ht0, a rhythmic change in the direction of flow is generated in the in-phase mode (Mode III in Fig. l(b2)), as in Fig. 4(a2). The value hto was found to be around 0.5 mm. Then we
S. Nakata et al./Physica D 115 (1998) 313-320
316
(a)
(a)
D
Independent
Tn
",
0,2
0.4 I
0.6 I
(=s0/S, s0=39-3 cm2)
0.8 I
1.0 I
L2 I
1.4 I
1.6 i
1.8 t
1.9
Oscillator 1 i'tl
~ :
.
i :
•
i :
•
:
,
Anti-phase
Quasi-periodicity
::'~n : :
:
•
:
!' l
Oscillator 2
i
I
I
i
AB
(b)
I
C D
A
E
C
B
2x
2-t,
(b) ,~-o
I
~o 0 0
2~
0
2~
o ~ o',~° 0
2x
Oscillator 1 E
D 2n
2~
Oscillator 2
I
I
0
2~
0
2x
50 sec Fig. 2. Time-course of electrical potential for two coupled salt-water oscillators. The surface area of the outer vessel is (a) 20.7cm 2 and (b) 32.7cm 2. The symbol q~ in each figure indicates the phase difference between the two oscillators; fb,, = 2zr r,, / T,, .
have examined the effect of the gradual change of At on the stability of Mode III. The oscillatory mode bifurcates into the circular flow (Mode I or II) when IAhl is increased above hto. When orifice 1 is the below 2 under the condition of Ah > hi0 as seen in Fig. l(b2), Mode 1 is generated, due to the higher water pressure at orifice 1 than at orifice 2. After Mode I begins, this mode persists even if Ah is gradually decreased to a negative value, indicating a memory effect for the flow direction. When Ah is decreased until Ah = - h t , , Mode I1 appears instead of Mode I. In other words, at the height differences of - h t , and ht,, the bifurcation between Mode I and Mode II is induced, as a Fig. 4(al). Fig. 4(b) shows the dependence of h,, on the viscosity of the aqueous solution in the inner cup. The aqueous solution in the inner cup was maintained at a constant density (1.12 ± 0.01 g / m l at 293 K) by dis-
Fig. 3. (a) Phase diagram of the mode of synchronization of two coupled salt-water oscillators depending on the coupling strength 79(= so/S ), where s0 (=39.3 cm 2) and S are the surface areas of the cup and the outer vessel. (b) Return map for the phase difference ~b, vs. 4~,+1 in the two coupled salt-water oscillators. Here A-E in correspond to those in (a). solving adequate concentrations of NaCI and glycerin in the salt water. It is clear that hr,. (or the hysteresis region for circular flow) decreases with an increase in the viscosity. This result is attributable to the damping effect of a high-viscosity solution on oscillatory flow. For the in-phase synchronized mode (Mode IlI), there exists a critical value of Ah; hro = 30 ~m. When the cup is tilted beyond this critical value, the flow mode switches into either Mode I or Mode II, depending on the direction of the tilting.
3.2. Numerical simulation of the mode of coupled nonlinear oscillators 3.2.1. Effect qf the coupling strength Based on a semi-quantitative analysis using the Navier-Stokes equation for a salt-water oscillator, it
317
S. Nakata et al./Phvsica D 115 (1998) 313 320
(a-l)
has become clear that the essential features of a single salt-water oscillator can be described with an ordinary
Mode I
differential equation, as indicated in the following i
i
Mode II
equation [ 17-20]: .~: + c~,f3 - #.f + ~o2x = O,
-.-ile-
I
o! i
io!5
0
-htc
htc
Ah (mm)
0-2)
( 1)
where x (m) is the height of the salt-water, c~ ( s m 2) is a coefficient lbr the degree of the pressure-loss induced when the liquid flows through the orifice in the cup, # ( = #0 - v) (s ~) is a coefficient lbr the acceleration of the fluid flow (/30) due to the buoyancy force, which is partly diminished by the viscosity of the fluid (t,),
Mode I l~i Mode It
Oscillation
L ~" I -50
i!
and oo (s - I ) is a term mainly reflected the effect of the
....
hydrostatic pressure being proportional to the density gradient between the salt water and the pure water.
i "==' "° .....I ="-= .........-i
A -hto
A hto
6h
50
The interaction between two oscillators can be described with the following equations: +
-
+
(b)
= -y~(xt
+ x2) - rl(~l + ~72),
(2)
-)
0.45 =
~
o.4
- y ~ D ( x l + x2) - r/(x~ + .t2),
(3)
where "/9 is a parameter regarding the coupling strength (7) = so~S; so (m 2) is the surface area of the salt-water
cup and S (m 2) is the surface area of the outer vessel), 0.35
r~ (s I ) is a parameter to interpret the dissipation due to I
I
I
I
2
3
4
5
Relative viscosity Fig. 4. Experimental results for a single cup with two orifices ( 1 mm in diameter) on the bottoms. (a) Diagrams of the regions of circular (al) and oscillatory flow la2) depending on the difference in height (Ah = h2 - h i ) between the two orifices, when the cup is tilted and recovered to horizontal at a rather slow rate (ca. 0.008 tad/rain). The distance between the centers of the two orifices was fixed at 3 mm and the cup contained 3.0M NaCI in aqueous solution. The directions of flows in orifices 1 and 2 are marked as in Fig. I(b2). In (al), the directions of outflow (or down-flow) and inflow (or up-flow) in orifice 1 were inverted at -ht,. and ht,, respectively. In (a2), the oscillatory mode in orifice 1 changed to the outflow a t ht o, or the inflow at -hro- (b) Dependence of ht,. o n the viscosity of the NaCI aqueous solution with glycerin.
the effect of viscous d a m p i n g on the rate of the motion of the level of water in the outer vessel, and y (s - 2 ) is a coefficient for the adjustment of the d i m e n s i o n in the equations. It has been known that the coupled system is bistable with in-phase and anti-phase modes when ~1 = 0 [20]. With the introduction of the viscous term, the anti-phase mode becomes more stable than the in-phase mode, which corresponds to the actual experiment. Fig. 5 shows (a) a phase map and (b) a return map of the phase difference based on a numerical simulation for interaction of two salt-water oscillators, where the intrinsic oscillatory periods of oscillators 1 and 2 are slightly different (o~t = 1.0 and co2 = 1.1). A E in Fig. 5(b) correspond to those in Fig. 5(a). Time course of synchronization between the two oscillators
S. Nakata et al./Physica D 115 (1998) 313-320
318
(a) Coupling strength: 4D 0.05
(c-1)
0.10
0.15
,
I
lndependent
II 2
Quasi-periodicity
B
A
Anti-phase
CD
~"
E -2 ,
(b)
-3 0 A
B
2(1
.
.
40
2~ C
,
60
80
100
time o,°
(c-2) 3 2
2~t
%+1
t'q
D
PD 1
F
E
II
0
-2 0
2Jr
%
-3
I
I
I
I
20
40
60
80
100
time
Fig. 5. Numerical simulation of the effect of the surface area of the outer vessel ( D = so~S) on the m a n n e r of synchronization between two oscillators I and 2. (a) Phase diagram of the mode of synchronization between two oscillators. (b) Return map for the phase difference q~, vs. q~,+l. Xl and x2 are the heights of the salt water in two oscillators 1 and 2. The two oscillators 1 and 2 are slightly different from one another (~1 = ~2 = 0.04, /41 = /42 = 0.1, co I = 1, co2 = I.I. r/ = 0.01. and }: = 1.79 = 0.01 (A), 79 = 0.1 (B), 79 = 0.12 (C), 79 = 0.13 (D), 79 = 0.15 (E), and 79 = 0.5 (F). Here A - E in Fig. 5(b) correspond to those in (a). ~b was obtained from the m a x i m u m value of each oscillation. (c) Time-course of synchronization between the two oscillators, Xl (solid line) and x2 (dotted line). The data in ( c l ) and (c2) correspond to D and E in (a) and (b).
is shown in Fig. 5(c). The data in (c l) and (c2) correspond to D and E, respectively, in Figs. 5(a) and (b). The characteristics of the synchronization between the two salt-water oscillators change depending on the coupling strength for the relative surface area, 79(= so~S). When 79 is significantly large, the mode of oscillation is locked as anti-phase (Fig. 5(b)F). With a decrease in 79, the phase difference between the two oscillators deviates from Jr (Fig. 5E), and synchronization bifurcates to the quasi-periodic mode at 79 = 0.1 4 (Figs. 5B, C and D). The periods and phases of the two oscillators become independent at a significantly small 79 (Fig. 6A). Thus, the characteristic modes of synchronization in the actual experiments (Fig. 3) can be reproduced by these theoretical simulations as
a function of the coupling strength for the surface area.
3.2.2. Interference between two flows within the single cup The interaction between the flows within a single cup may be interpreted with the following equations: 2j = - ~ j x ~ + fl].tl - yD(x] +x2) x~ -7(21 + 2 2 ) + 6 ~ - s g n ( 2 2 ) ,
(4)
-r2 = - - ~ 2 2 ~ + fi2X2 -- g 7 9 ( X l + X2) X ~
-7(2] +22)+6~-21sgn(21)+KAH(t),
(5)
S. Nakata et al./Phvsica D 115 (1998) 313-320
319
(a) -1"
~
:...j.-~:-~ . . . . . . . . . . . . . . . .
0
¥ t-
-1
. x~ -2
I
I
I
20
40
60
time (b-l) 2
!'~ , .~ ,-.......
T
i V:
(c) 1.8
o
1.6
";7, -1 . x=
1.4
-2
10
20
30
40
50
time
1.2
(b-2) 2
1
"-r-
0.8 o
0.6
7, -t -2
0
10
20
30
40
50
time
0.4 -0.2
I
I
I
I
I
0
0.2
0.4
0.6
0.8
V
Fig. 6. N u m e r i c a l s i m u l a t i o n o f s t a t i o n a r y flow ((a) M o d e s 1 a n d ll) a n d o s c i l l a t o r y flow ((b) M o d e IlI) with a c h a n g e in the d i f f e r e n c e in height ( A H = H2 - H I ) b e t w e e n the t w o orilices, oe I = ct 2 = 0.4. 2D = 1.0. 1 = 2.0, q = 0.1. y = 1. a n d ,v = I. (a) A H is g r a d u a l l y c h a n g e d until t = 5 0 a n d then kept c o n s t a n t . T h e intersection o f the c u r v e s for ,t I a n d -/:2 c o r r e s p o n d s to the switch f r o m M o d e II to M o d e 1. /31 = / 3 2 = 0.5, H I = 0 . 0 5 , H 0 = - 1 , ,~ = 0.2. A H = Hit + 14o at 0 < t < 50, A H = 1.5 at t > 50. (b) T h e i n - p h a s e m o d e ( M o d e III) c o n t i n u e s even if A H is c h a n g e d linearly with time ( b l ) or is kept c o n s t a n t a f t e r a linear i n c r e a s e (b2). /31 = / 3 2 = 1.0, 5 = 0.2. ( b l ) M o d e 11I s w i t c h e s to M o d e I over A H = 0 . 6 7 5 at t = 45. H I = 0 . 0 1 5 a n d H 0 = 0. (b2) T h e in-phase m o d e ( M o d e 111) c o n t i n u e s even if H I = 0 . 0 1 5 , H 0 = 0. A H = Hit + Ho at 0 < t < 25. a n d A H = 0 . 3 7 5 at t > 25. (c) N u m e r i c a l results for Hr, d e p e n d i n g on v(v = / 3 0 - / 3 , , /30 = 1.0, n = I or 2). All o t h e r p a r a m e t e r s are the s a m e as those in (a).
where A H (m) is the difference between the heights of the two orifices relative to orifice 1, 79 is a coefficient of the coupling strength for the level of the salt water, which changes with the amount of fluid
is the distance between the two orifices, and x (s 2) is a coefficient for the adjustment of the dimension. Since the angle of tilting is small in our experiment, the distance, ~//2 - A H 2, between the orifices is ap-
flow through the two orifices, 6 (m) is a coefficient for the effect of flow through the other orifice. / (m)
proximated to be constant as I. A H varies linearly as
Ho + Hit (140 and Hi are constant). The filth term in
320
S. Nakata et al./Physica D 115 (1998) 313-320
these equations is due to the negative pressure, or the
Acknowledgements
head loss, produced by the neighboring flow, which is proportional to the square of the flow velocity and to
We thank Prof. Y. Taniguchi (Nara University
the square of 1/l [23]. Fig. 6 shows the numerical results of the change
of Education, Japan), Messrs. Y. Hayashi (Nagoya University, Japan) and M. Yagi (Nara University of
in .t for (a) the circular flow and (b) the oscillatory
Education) for their technical assistance. This work
flow as a function of A H . Here, the positive and neg-
was supported in part by a Grant-in-Aid for Science
ative value of 5: mean the up-flow (or inflow) and the
Research from the Ministry of Education, Culture and
down-flow (or outflow), respectively. The switching
Science, the Shimadzu Science Foundation, and the
between stationary flow (Modes I or II) in Fig. 4(al)
Nestl6 Science Promotion Committee.
and the oscillatory flow (Mode II1) in Fig. 4(a2) was reproduced in the simulation, as shown in Figs. 6(a) and (b). The transition point of the direction of fluid flow, lit,, changes depending on the parameter v ( = ~0 - / ~ , , ( n = I or 2)), which is related to the viscosity of fluid through the orifice, as shown in Fig. 6(c). The dependence of ht, on the viscosity of salt-water phase in the experiment (Fig. 4(b)) may correspond to that on parameter v theoretically (Fig. 6(c)); i.e., the hysteresis region depends on the degree of the autocatalytic process of oscillation. Oscillatory flow was maintained even if A H was slightly changed from zero, but was bifurcated over A H = 0.675 (t = 45) in Fig. 6(bl). Thus, the stationary flow (Mode I or II) and oscillatory flow (Mode Ill) were also quantitatively reproduced by the theoretical simulation.
4. Conclusion In two coupled salt-water oscillators, the mode of synchronization changes depending on the surface area of outer vessel. Various modes of synchronization can be categorized using the return map of the phase difference between the two oscillators. The manner in which synchronization depended on the surface area is reproduced by a theoretical simulation based on nonlinear differential equations. In experiments with two orifices in a single cup, either circular flow or oscillatory flow was observed as a function of the difference in height between the two orifices, and the region of circular flow changed depending on the viscosity of the salt-water phase. These phenomena for the two orifices in a single cup are also reproduced by the theoretical simulation.
References [ I [ A.T. Winfree, The Geometry of Biological Time, Springer, New York, 1980. [2] R.J. Field, E. K6r6s, R.M. Noyes, J. Am. Chem. Soc. 94 (1972) 8649. [3] R.J. Field, R.M. Noyes. J. Chem. Phys. 60 (1974) 1877. [4] M. Marek, 1. Stuchl, Biophys. Chem. 3 (1975) 241. [5] K. Nakajima, Y. Sawada, J. Chem. Phys. 72 (1980) 2231. [6] M.E Crowley, R. Epstein, J. Phys. Chem. 93 (1989) 2496. [7] J. Laplante, T. Erneux, J. Phys. Chem. 96 (1992) 4931. [8] N. Nishiyama. K. Eto. J. Chem. Phys. 100 (1994) 6977. [9] M. Yoshimoto. K. Yoshikawa. Y. Mori, Phys. Rev. E 47 (1993) 864. [10] J.J. Tyson, S. Kauffman, J. Math. Bio. 1 (1975) 289. 111] Y. Kuramoto, Prog. Theor. Phys. Suppl. 79 (1984) 223. 112[ Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer, New York, 1984. [13] H. Daido. Rev. Phys. Lett. 61 (1988) 231. [141 S. Omata. Y. Yamaguchi. H. Shimizu, Physica D 31 (1988) 397. 115] S. Martin, Geophys. Fluid Dyn. I (1970) 143. [16] K. Yoshikawa, S. Nakata, M. Yamanaka,T. Waki. J. Chem. Edu. 66 (1989) 205. 7] K. Yoshikawa, N. Oyama, M. Shoji, S. Nakata, Am. J. Phys. 59 (1991) 137. 8] K. Yoshikawa, K. Fukunaga, H. Kawakami. Chem. Phys. Len. 174 11990) 203. 1191 K. Yoshikawa. T. Yoshinaga, H. Kawakami. S. Nakata, Physica A 188 (1992) 243. 120] K. Yoshikawa, H. Kawakami, Appl. Math. (Japanese) 4 (1994) 238. [21] A.K. Das, R.C. Srivastava. J. Chem. Faraday Trans. 89 (1993) 905. [221 L. Adamciko'vfi. React. Kinet. Catal. Lett. 48 (1992) 649. [231 L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Course of Theoretical Physics, vol. 6, 2nd ed., Pergamon Press, London, 1987.
Physica D 115 (1998) 313 320
Self-synchronization in coupled salt-water oscillators Satoshi Nakata a,,, Takahiro Miyata a, Nozomi Ojima a, Kenichi Yoshikawa b a Department ofChemistr3'. Nara Universi~' of Education. Takabatake-cho, Nara 630, Japan b Graduate School of Human h!lbrmatics, Nagoya Universit3; Nagoya 464-01. Japan
Received 9 July 1997; received in revised form I September 1997: accepted 26 September 1997 Communicated by Y. Kuramoto
Abstract The coupling of nonlinear oscillators was studied experimentally and theoretically. In the experiment, the electrochemical potentials in the salt water in cups dipped in a larger vessel filled with pure water were measured, where both the liquids were made to contact through a small orifice on the bottom of the individual cup. The mode of oscillations was found to depend on the surface area of the pure water. Another experiment was made to observe a coupling between the flows through separate orifices on the bottom of a single cup. The mode of flows depended on the difference in height between the two orifices. These experimental findings are predicted in a semi-quantitative manner by numerical simulations by the use of coupled ordinary differential equations. Copyright © 1998 Elsevier Science B.V. Kevwords: Nonlinear oscillation; Entrainment; Bifurcation; Convection-diffusion; Spatio-temporal sell-organization
1. Introduction Coupling between nonlinear oscillators is useful for understanding the mechanism of self-synchronization in living systems such as in the beating heart, neural networks, circadian rhythm, the organized rhythm in the flashing of swarms of fireflies, and so on [ 1]. There have been many experimental [2-9] and theoretical studies [10-14] on self-synchronization in coupled oscillators. Among the various kinds of oscillators, the coupling of two or more continuous stirred tank reactors (CSTRs) in the Belousov-Zhabotinsky (BZ) reaction has been extensively studied [2-9], since this system is much simpler than those in living systems.
*Corresponding author. Tel.: +81-742-27-9191; fax: +81742-27-9190; e-mail: [email protected].
As a very simple and easily reproducible experiment, in the present paper we will describe coupling between salt-water oscillators. Salt-water oscillators exhibit various nonlinear characteristics such as a limit cycle, bifurcation, quasi-periodicity, period multiplying, and synchronization [15-22]. It has been reported that (1) in the coupling of two salt-water oscillators, various modes of synchronization, i.e., anti-phase, 1 : 2, 1 : 3, and 2 : 3 synchronization, can be observed by changing the diameter of the orifice in the cup and the concentration of NaCI aqueous solution, and (2) in the coupling of three equivalent salt-water oscillators, a bistable tri-phasic mode of synchronization can be observed I 16,17]. In this paper, we report that the mode of synchronization in two coupled salt-water oscillators changes depending on the surface area of the outer phase of pure water in a characteristic manner. The mode of
0167-2789/98/$19.00 Copyright © 1998 Elsevier Science B.V. All rights reserved PII S0167-2789(97)00240-6
314
S. Nakata et al./Physica D 115 (1998) 31.?-320
(b-l)
Laser displacement meter
Salt-water oscillator
L_J (a)
Personal computer
Jack
circular stationary flow (bistable)
(b-2) Plastic board tfixed)
o
/\, I
2
2
i \
oscillatory flow (synchronization)
",, "1\
// s Mode IIl
Ag/AgC1 elcctrode Fig. 1. (a) Experimental apparatus for measuring of the electrical potential m two coupled salt-water oscillators, as an experimental index of the direction of flow. (bl) Experimental apparatus tor observing the interaction between the two fluid flows through the two orifices in a single cup. (b2) Schematic representation of two fluid flows through the two orifices in a single cup.
this synchronization, e.g. anti-phase or quasi-periodic one, is analyzed by a return map of the phase difference between the two oscillators. As tor coupled flow through two orifices in a single cup, either circular flow or oscillatory flow was seen depending on the difference in height between the two orifices. These experimental trends were reproduced by a numerical simulation based on coupled nonlinear differential equations.
2. Experimental The salt-water oscillator was constructed as in previous studies [16,17]. An outer glass vessel (volume: 2000 ml) and a cylindrical acrylic cup contained pure water and 3 M NaCI aqueous solution (60ml),
respectively; inner diameter of cylinder: 4 6 m m , height of cylinder: 60 ram, diameter of the orifice in the bottom of the cylinder: 1.0 mm, and thickness of orifice: 0.5 mm. The level of the pure water in the outer vessel was made almost equal to that of the NaCI aqueous solution in the plastic cup in terms of the difference in hydrostatic pressure at the position of the orifice, by adjusting the initial volume of pure water in the outer vessel. In the experiment with coupled oscillators, two equivalent cups with the same diameter and orifices of the same size were placed in a single outer vessel (Fig. l(a)). The direction of the oscillatory flow in the salt-water oscillators was monitored as an electrochemical potential by using A g / A g C I electrodes and a mV meter with an internal impedance higher than 1012~. In our previous experiment on
S. Nakata et al./Phvsica D 115 (1998) 313 320 the simultaneous measurement of the surface level and electrical potential in a salt-water oscillator, we confirmed that high and low potentials in the oscillation correspond to the outflow of salt-water and the inflow of pure water, respectively [16,17]. Since the maximum change in the water level is of the order of 0.1 mm, we chose to measure the electrical potential rather than the water level in the cup, the latter of which would have required expensive and delicate equipment. To change the surface area of the outer vessel, a plastic plate ithickness: 10mm) of the desired size was fixed at the water level of the outer vessel, as illustrated in Fig. l(a). For the experiment on coupled flow through two orifices in a single cup, we used a cylindrical cup with two 1 mm orifices in the bottom with a fixed distance (3 ram) between their centers. To produce a difference in height between the two orifices, a plate connected to the salt-water cell was tilted with a jack, as shown in Fig. l(b). The difl'erence in the heights of the two orifices was monitored with a laser displacement meter (Keyence. LPB-02, Japan).
3. Results and discussion 3. I. Experiments on the synchronization ~1 salt-water oscillators 3.1.1. Mode qf synchronization of two coupled salt-water oscillators depending on the surface area qf the outer vessel Fig. 2 shows the time-course of the electrical oscillations of two salt-water oscillators for outer vessels with different surface areas: (a) 20.7 cm 2 and (b) 32.7 cm 2. The phase difference, q~n, between the oscillators is calculated based on the time-trace of the electrical potential of the two oscillators, where 4',, is defined as 49,, = 2zrrn/T,, (r,, is the time difference is) between the oscillators and T,, is the period is) of the oscillations in oscillator 1). It is apparent that the mode of the oscillation in Fig. 2(a) is locked as antiphase. In contrast, the time-course of the phase difference in Fig. 2(b) is not locked, and changes gradually with time, i.e., quasi-periodic mode. The modes are
315
diagrammed in Fig. 3(a) as a function of the relative surface area, 79, (79 = so~S, so is the inner surface area (39.3 cm 2) with salt water and S is the outer surface area with pure water); i.e., 79 is inversely proportional to the outer surface area S. The modes of synchronization can be characterized by using a return map of the phase difference. 4~,,. as shown in Fig. 3(b). A - E in Fig. 3(b) correspond to those in Fig. 3(a), and the data in Fig. 3(d)E and (b)C correspond to those in Figs. 2(a) and (b), respectively. It is noted that coupling between the two oscillators is mediated by a change in the water level in the outer vessel, i.e., the coupling strength is inversely proportional to the outer surface area, S [17]. When the coupling parameter, 79, is smaller than 0.2, the two salt-water oscillators become almost independent, since the water level of the outer surface with a significant large surface area remains almost constant independent of the rhythmic flow in the individual oscillators, where the volume change during one oscillation in a cup is ca. 0.1 ml. Self-synchronization in the anti-phase mode was observed when 79 > 1.3 or S < 30cm 2. It is expected that the change in the height of the surface in the outer vessel is minimized in this anti-phase mode l 17]. Thus. a bifurcation point lies at 79 ~ 1.25 between the quasi-periodic and antiphase modes (Fig. 3(b)C and D).
3.1.2. Stationar3,flow and rhythmic.flow through two orifices in a single cup When a single cup with two orifices in its base was used, either circular flow or oscillatory flow (see Fig. lib)) was generated. The flow mode depended on the initial difference in height (Ah = h2 - h i ) between the two orifices isee Fig. l(b2)). In this experiment, the distance between the centers of two orifices was fixed at 3 mm and the cup contained 3.0 M NaCI in aqueous solution. When we start the experiment by removing small plugs from the orifices under the condition that IAhl is smaller than a critical value, ht0, a rhythmic change in the direction of flow is generated in the in-phase mode (Mode III in Fig. l(b2)), as in Fig. 4(a2). The value hto was found to be around 0.5 mm. Then we
S. Nakata et al./Physica D 115 (1998) 313-320
316
(a)
(a)
D
Independent
Tn
",
0,2
0.4 I
0.6 I
(=s0/S, s0=39-3 cm2)
0.8 I
1.0 I
L2 I
1.4 I
1.6 i
1.8 t
1.9
Oscillator 1 i'tl
~ :
.
i :
•
i :
•
:
,
Anti-phase
Quasi-periodicity
::'~n : :
:
•
:
!' l
Oscillator 2
i
I
I
i
AB
(b)
I
C D
A
E
C
B
2x
2-t,
(b) ,~-o
I
~o 0 0
2~
0
2~
o ~ o',~° 0
2x
Oscillator 1 E
D 2n
2~
Oscillator 2
I
I
0
2~
0
2x
50 sec Fig. 2. Time-course of electrical potential for two coupled salt-water oscillators. The surface area of the outer vessel is (a) 20.7cm 2 and (b) 32.7cm 2. The symbol q~ in each figure indicates the phase difference between the two oscillators; fb,, = 2zr r,, / T,, .
have examined the effect of the gradual change of At on the stability of Mode III. The oscillatory mode bifurcates into the circular flow (Mode I or II) when IAhl is increased above hto. When orifice 1 is the below 2 under the condition of Ah > hi0 as seen in Fig. l(b2), Mode 1 is generated, due to the higher water pressure at orifice 1 than at orifice 2. After Mode I begins, this mode persists even if Ah is gradually decreased to a negative value, indicating a memory effect for the flow direction. When Ah is decreased until Ah = - h t , , Mode I1 appears instead of Mode I. In other words, at the height differences of - h t , and ht,, the bifurcation between Mode I and Mode II is induced, as a Fig. 4(al). Fig. 4(b) shows the dependence of h,, on the viscosity of the aqueous solution in the inner cup. The aqueous solution in the inner cup was maintained at a constant density (1.12 ± 0.01 g / m l at 293 K) by dis-
Fig. 3. (a) Phase diagram of the mode of synchronization of two coupled salt-water oscillators depending on the coupling strength 79(= so/S ), where s0 (=39.3 cm 2) and S are the surface areas of the cup and the outer vessel. (b) Return map for the phase difference ~b, vs. 4~,+1 in the two coupled salt-water oscillators. Here A-E in correspond to those in (a). solving adequate concentrations of NaCI and glycerin in the salt water. It is clear that hr,. (or the hysteresis region for circular flow) decreases with an increase in the viscosity. This result is attributable to the damping effect of a high-viscosity solution on oscillatory flow. For the in-phase synchronized mode (Mode IlI), there exists a critical value of Ah; hro = 30 ~m. When the cup is tilted beyond this critical value, the flow mode switches into either Mode I or Mode II, depending on the direction of the tilting.
3.2. Numerical simulation of the mode of coupled nonlinear oscillators 3.2.1. Effect qf the coupling strength Based on a semi-quantitative analysis using the Navier-Stokes equation for a salt-water oscillator, it
317
S. Nakata et al./Phvsica D 115 (1998) 313 320
(a-l)
has become clear that the essential features of a single salt-water oscillator can be described with an ordinary
Mode I
differential equation, as indicated in the following i
i
Mode II
equation [ 17-20]: .~: + c~,f3 - #.f + ~o2x = O,
-.-ile-
I
o! i
io!5
0
-htc
htc
Ah (mm)
0-2)
( 1)
where x (m) is the height of the salt-water, c~ ( s m 2) is a coefficient lbr the degree of the pressure-loss induced when the liquid flows through the orifice in the cup, # ( = #0 - v) (s ~) is a coefficient lbr the acceleration of the fluid flow (/30) due to the buoyancy force, which is partly diminished by the viscosity of the fluid (t,),
Mode I l~i Mode It
Oscillation
L ~" I -50
i!
and oo (s - I ) is a term mainly reflected the effect of the
....
hydrostatic pressure being proportional to the density gradient between the salt water and the pure water.
i "==' "° .....I ="-= .........-i
A -hto
A hto
6h
50
The interaction between two oscillators can be described with the following equations: +
-
+
(b)
= -y~(xt
+ x2) - rl(~l + ~72),
(2)
-)
0.45 =
~
o.4
- y ~ D ( x l + x2) - r/(x~ + .t2),
(3)
where "/9 is a parameter regarding the coupling strength (7) = so~S; so (m 2) is the surface area of the salt-water
cup and S (m 2) is the surface area of the outer vessel), 0.35
r~ (s I ) is a parameter to interpret the dissipation due to I
I
I
I
2
3
4
5
Relative viscosity Fig. 4. Experimental results for a single cup with two orifices ( 1 mm in diameter) on the bottoms. (a) Diagrams of the regions of circular (al) and oscillatory flow la2) depending on the difference in height (Ah = h2 - h i ) between the two orifices, when the cup is tilted and recovered to horizontal at a rather slow rate (ca. 0.008 tad/rain). The distance between the centers of the two orifices was fixed at 3 mm and the cup contained 3.0M NaCI in aqueous solution. The directions of flows in orifices 1 and 2 are marked as in Fig. I(b2). In (al), the directions of outflow (or down-flow) and inflow (or up-flow) in orifice 1 were inverted at -ht,. and ht,, respectively. In (a2), the oscillatory mode in orifice 1 changed to the outflow a t ht o, or the inflow at -hro- (b) Dependence of ht,. o n the viscosity of the NaCI aqueous solution with glycerin.
the effect of viscous d a m p i n g on the rate of the motion of the level of water in the outer vessel, and y (s - 2 ) is a coefficient for the adjustment of the d i m e n s i o n in the equations. It has been known that the coupled system is bistable with in-phase and anti-phase modes when ~1 = 0 [20]. With the introduction of the viscous term, the anti-phase mode becomes more stable than the in-phase mode, which corresponds to the actual experiment. Fig. 5 shows (a) a phase map and (b) a return map of the phase difference based on a numerical simulation for interaction of two salt-water oscillators, where the intrinsic oscillatory periods of oscillators 1 and 2 are slightly different (o~t = 1.0 and co2 = 1.1). A E in Fig. 5(b) correspond to those in Fig. 5(a). Time course of synchronization between the two oscillators
S. Nakata et al./Physica D 115 (1998) 313-320
318
(a) Coupling strength: 4D 0.05
(c-1)
0.10
0.15
,
I
lndependent
II 2
Quasi-periodicity
B
A
Anti-phase
CD
~"
E -2 ,
(b)
-3 0 A
B
2(1
.
.
40
2~ C
,
60
80
100
time o,°
(c-2) 3 2
2~t
%+1
t'q
D
PD 1
F
E
II
0
-2 0
2Jr
%
-3
I
I
I
I
20
40
60
80
100
time
Fig. 5. Numerical simulation of the effect of the surface area of the outer vessel ( D = so~S) on the m a n n e r of synchronization between two oscillators I and 2. (a) Phase diagram of the mode of synchronization between two oscillators. (b) Return map for the phase difference q~, vs. q~,+l. Xl and x2 are the heights of the salt water in two oscillators 1 and 2. The two oscillators 1 and 2 are slightly different from one another (~1 = ~2 = 0.04, /41 = /42 = 0.1, co I = 1, co2 = I.I. r/ = 0.01. and }: = 1.79 = 0.01 (A), 79 = 0.1 (B), 79 = 0.12 (C), 79 = 0.13 (D), 79 = 0.15 (E), and 79 = 0.5 (F). Here A - E in Fig. 5(b) correspond to those in (a). ~b was obtained from the m a x i m u m value of each oscillation. (c) Time-course of synchronization between the two oscillators, Xl (solid line) and x2 (dotted line). The data in ( c l ) and (c2) correspond to D and E in (a) and (b).
is shown in Fig. 5(c). The data in (c l) and (c2) correspond to D and E, respectively, in Figs. 5(a) and (b). The characteristics of the synchronization between the two salt-water oscillators change depending on the coupling strength for the relative surface area, 79(= so~S). When 79 is significantly large, the mode of oscillation is locked as anti-phase (Fig. 5(b)F). With a decrease in 79, the phase difference between the two oscillators deviates from Jr (Fig. 5E), and synchronization bifurcates to the quasi-periodic mode at 79 = 0.1 4 (Figs. 5B, C and D). The periods and phases of the two oscillators become independent at a significantly small 79 (Fig. 6A). Thus, the characteristic modes of synchronization in the actual experiments (Fig. 3) can be reproduced by these theoretical simulations as
a function of the coupling strength for the surface area.
3.2.2. Interference between two flows within the single cup The interaction between the flows within a single cup may be interpreted with the following equations: 2j = - ~ j x ~ + fl].tl - yD(x] +x2) x~ -7(21 + 2 2 ) + 6 ~ - s g n ( 2 2 ) ,
(4)
-r2 = - - ~ 2 2 ~ + fi2X2 -- g 7 9 ( X l + X2) X ~
-7(2] +22)+6~-21sgn(21)+KAH(t),
(5)
S. Nakata et al./Phvsica D 115 (1998) 313-320
319
(a) -1"
~
:...j.-~:-~ . . . . . . . . . . . . . . . .
0
¥ t-
-1
. x~ -2
I
I
I
20
40
60
time (b-l) 2
!'~ , .~ ,-.......
T
i V:
(c) 1.8
o
1.6
";7, -1 . x=
1.4
-2
10
20
30
40
50
time
1.2
(b-2) 2
1
"-r-
0.8 o
0.6
7, -t -2
0
10
20
30
40
50
time
0.4 -0.2
I
I
I
I
I
0
0.2
0.4
0.6
0.8
V
Fig. 6. N u m e r i c a l s i m u l a t i o n o f s t a t i o n a r y flow ((a) M o d e s 1 a n d ll) a n d o s c i l l a t o r y flow ((b) M o d e IlI) with a c h a n g e in the d i f f e r e n c e in height ( A H = H2 - H I ) b e t w e e n the t w o orilices, oe I = ct 2 = 0.4. 2D = 1.0. 1 = 2.0, q = 0.1. y = 1. a n d ,v = I. (a) A H is g r a d u a l l y c h a n g e d until t = 5 0 a n d then kept c o n s t a n t . T h e intersection o f the c u r v e s for ,t I a n d -/:2 c o r r e s p o n d s to the switch f r o m M o d e II to M o d e 1. /31 = / 3 2 = 0.5, H I = 0 . 0 5 , H 0 = - 1 , ,~ = 0.2. A H = Hit + 14o at 0 < t < 50, A H = 1.5 at t > 50. (b) T h e i n - p h a s e m o d e ( M o d e III) c o n t i n u e s even if A H is c h a n g e d linearly with time ( b l ) or is kept c o n s t a n t a f t e r a linear i n c r e a s e (b2). /31 = / 3 2 = 1.0, 5 = 0.2. ( b l ) M o d e 11I s w i t c h e s to M o d e I over A H = 0 . 6 7 5 at t = 45. H I = 0 . 0 1 5 a n d H 0 = 0. (b2) T h e in-phase m o d e ( M o d e 111) c o n t i n u e s even if H I = 0 . 0 1 5 , H 0 = 0. A H = Hit + Ho at 0 < t < 25. a n d A H = 0 . 3 7 5 at t > 25. (c) N u m e r i c a l results for Hr, d e p e n d i n g on v(v = / 3 0 - / 3 , , /30 = 1.0, n = I or 2). All o t h e r p a r a m e t e r s are the s a m e as those in (a).
where A H (m) is the difference between the heights of the two orifices relative to orifice 1, 79 is a coefficient of the coupling strength for the level of the salt water, which changes with the amount of fluid
is the distance between the two orifices, and x (s 2) is a coefficient for the adjustment of the dimension. Since the angle of tilting is small in our experiment, the distance, ~//2 - A H 2, between the orifices is ap-
flow through the two orifices, 6 (m) is a coefficient for the effect of flow through the other orifice. / (m)
proximated to be constant as I. A H varies linearly as
Ho + Hit (140 and Hi are constant). The filth term in
320
S. Nakata et al./Physica D 115 (1998) 313-320
these equations is due to the negative pressure, or the
Acknowledgements
head loss, produced by the neighboring flow, which is proportional to the square of the flow velocity and to
We thank Prof. Y. Taniguchi (Nara University
the square of 1/l [23]. Fig. 6 shows the numerical results of the change
of Education, Japan), Messrs. Y. Hayashi (Nagoya University, Japan) and M. Yagi (Nara University of
in .t for (a) the circular flow and (b) the oscillatory
Education) for their technical assistance. This work
flow as a function of A H . Here, the positive and neg-
was supported in part by a Grant-in-Aid for Science
ative value of 5: mean the up-flow (or inflow) and the
Research from the Ministry of Education, Culture and
down-flow (or outflow), respectively. The switching
Science, the Shimadzu Science Foundation, and the
between stationary flow (Modes I or II) in Fig. 4(al)
Nestl6 Science Promotion Committee.
and the oscillatory flow (Mode II1) in Fig. 4(a2) was reproduced in the simulation, as shown in Figs. 6(a) and (b). The transition point of the direction of fluid flow, lit,, changes depending on the parameter v ( = ~0 - / ~ , , ( n = I or 2)), which is related to the viscosity of fluid through the orifice, as shown in Fig. 6(c). The dependence of ht, on the viscosity of salt-water phase in the experiment (Fig. 4(b)) may correspond to that on parameter v theoretically (Fig. 6(c)); i.e., the hysteresis region depends on the degree of the autocatalytic process of oscillation. Oscillatory flow was maintained even if A H was slightly changed from zero, but was bifurcated over A H = 0.675 (t = 45) in Fig. 6(bl). Thus, the stationary flow (Mode I or II) and oscillatory flow (Mode Ill) were also quantitatively reproduced by the theoretical simulation.
4. Conclusion In two coupled salt-water oscillators, the mode of synchronization changes depending on the surface area of outer vessel. Various modes of synchronization can be categorized using the return map of the phase difference between the two oscillators. The manner in which synchronization depended on the surface area is reproduced by a theoretical simulation based on nonlinear differential equations. In experiments with two orifices in a single cup, either circular flow or oscillatory flow was observed as a function of the difference in height between the two orifices, and the region of circular flow changed depending on the viscosity of the salt-water phase. These phenomena for the two orifices in a single cup are also reproduced by the theoretical simulation.
References [ I [ A.T. Winfree, The Geometry of Biological Time, Springer, New York, 1980. [2] R.J. Field, E. K6r6s, R.M. Noyes, J. Am. Chem. Soc. 94 (1972) 8649. [3] R.J. Field, R.M. Noyes. J. Chem. Phys. 60 (1974) 1877. [4] M. Marek, 1. Stuchl, Biophys. Chem. 3 (1975) 241. [5] K. Nakajima, Y. Sawada, J. Chem. Phys. 72 (1980) 2231. [6] M.E Crowley, R. Epstein, J. Phys. Chem. 93 (1989) 2496. [7] J. Laplante, T. Erneux, J. Phys. Chem. 96 (1992) 4931. [8] N. Nishiyama. K. Eto. J. Chem. Phys. 100 (1994) 6977. [9] M. Yoshimoto. K. Yoshikawa. Y. Mori, Phys. Rev. E 47 (1993) 864. [10] J.J. Tyson, S. Kauffman, J. Math. Bio. 1 (1975) 289. 111] Y. Kuramoto, Prog. Theor. Phys. Suppl. 79 (1984) 223. 112[ Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer, New York, 1984. [13] H. Daido. Rev. Phys. Lett. 61 (1988) 231. [141 S. Omata. Y. Yamaguchi. H. Shimizu, Physica D 31 (1988) 397. 115] S. Martin, Geophys. Fluid Dyn. I (1970) 143. [16] K. Yoshikawa, S. Nakata, M. Yamanaka,T. Waki. J. Chem. Edu. 66 (1989) 205. 7] K. Yoshikawa, N. Oyama, M. Shoji, S. Nakata, Am. J. Phys. 59 (1991) 137. 8] K. Yoshikawa, K. Fukunaga, H. Kawakami. Chem. Phys. Len. 174 11990) 203. 1191 K. Yoshikawa. T. Yoshinaga, H. Kawakami. S. Nakata, Physica A 188 (1992) 243. 120] K. Yoshikawa, H. Kawakami, Appl. Math. (Japanese) 4 (1994) 238. [21] A.K. Das, R.C. Srivastava. J. Chem. Faraday Trans. 89 (1993) 905. [221 L. Adamciko'vfi. React. Kinet. Catal. Lett. 48 (1992) 649. [231 L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Course of Theoretical Physics, vol. 6, 2nd ed., Pergamon Press, London, 1987.