Physica D 39 (1989) 38-42

North-Holland. Amsterdam

THREE-DIMENSIONAL VORTEX WITH A SPIRAL FILAMENT IN A CHEMICALLY ACTIVE MEDIUM

K.I. AGLADZE, V.I. KRINSKY and A.V. PANFILOV lnstrttrte of B~ologrcd Pl~ysics, USSR Academr of Sciences, Pushchino. MO.SCOM Regrow, 142 292. 1iSSR

H. LINDE and L. KUHNERT Zentrulrnstrtut JCr Ph~ysikahsche Chemie der Akudemie der WissmschuJten dcr DDR.

DDR-I 199. Rerlirl. Germun Democroric Repuhlrc

Received 2 November 1988

Accepted 17 February 1989

Communicated by A.V. Holden

A new technique, the self-completion technique, for obtaining 3D vortices with a predetermined configuration of the filament has been used in experiments with the B-Z reagent. The technique has made it possible to obtain a 3D autowave

vortex with a spiral filament, which has the form of a stationary spiral, emitting waves inwards and outwards. In the course of

time, the spiral shortens to give rise to a region inside it, in which the waves move to the center and vanish there.

1. Introduction

Three-dimensional autowaves were discovered

in the Belousov-Zhabotinskii (B-Z) reaction as

long ago as 1973 [l]. They were subsequently

studied both theoretically (by topological [2] and

numerical [3-51 methods) and experimentally (in

heart tissue [6] and in the B-Z reaction [7-91).

However, up to the present time no experimen-

tal routine has been developed to obtain predeter-

mined 3D structures and one had to be content

with those occurring by chance. The present work

uses a new method proposed in [3, lo] for obtain-

ing 3D autowave structures- the method of self-

completion. It has the advantage of being easily

utilized in experiments on chemically active media

and of making possible 3D vortices with filaments

oriented in space in a complicated manner.

2 lziEii3 ,L- ,I - I- ,

a b

Fig. 1. Self-completion of a flat wave to a simple scroll: (a)

initial position; (b) steady-state pattern

2. Theorem on self-completion of 3D vortices

(from (3, lo])

By self-completion of 3D vortices, the following

process is meant. Assume a 3D medium (1) in

which autowaves propagate (fig. l(a)). Let it make

contact with a 3D autowaves medium (2) whose

elements are at rest. The following statement can

be formulated concerning the wave pattern gener-

0167-2789/89/$03.50 0 Elsevier Science Publishers B.V.

(North-Holland Physics Publishing Division)

K.I. Agladze et al./3D vortex wrth a spiralfilament 39

ated by this “composite” medium ((I) + (2)): self-

completion results in a 3D vortex whose filament’s

position coincides with that of an autowave at the

boundary surface of the two media at the initial

instant of time. In brief, the wave becomes the

filament of a vortex. The vortex filaments that

were present in the medium will, naturally, re-

main.

Although at the moment it does not seem possi-

ble to prove rigorously the above statement, it is

obvious from the physical point of view. Let us

consider it by way of two examples. If a flat wave

is propagating in medium (1) the result of self-

completion will be a simple scroll (fig. 1). But if

the initial condition is given by a ring-shaped

wave, the result will be a vortex ring.

In a thin layer of the B-Z reaction two types of

waves spontaneously arise, spiral and concentric

waves. As follows from the aforesaid, self-comple-

tion of a concentric wave results in a vortex ring.

A system of concentric waves propagating from

one and the same center produces a series of

vortex rings fitted into one another and rotating in

the same direction. What happens on self-comple-

tion of the second (commonest) wave pattern- a

spiral wave? The question is an intriguing one

since the spiral wave is topologically unclosed.

This paper seeks to find an answer to it.

3. Experimental procedure

To obtain the original 2D autowave pattern, the

B-Z reaction was performed in a thin gel layer

with an immobilized catalyst [ll]. The layer was

covered with a solution containing all the reac-

tants except for the catalyst. The chemical reac-

tion waves could therefore propagate only within

the gel layer and not in the solution. To allow the

wave to enter the solution, the soluble catalyst was

added to it; the solution was mixed and then kept

undisturbed.

We used a SiO,-gel with immobilized ferroin

Fe(phen), (1.2 X 10e3 M in liquid) and a solution

of 0.3 M NaBrO,; 0.3 M H,SO,; 0.2 M

CH,(COOH),; 0.1 M KBr and 3 x 10M3 M

Fe(phen),, respectively. The thickness of the gel

layer was 1.2 mm and of the solution 2 mm.

4. Results

Fig. 2 presents a series of frames illustrating the

conversion of a spiral wave into a 3D vortex

structure. It is seen that the spiral changes its

rotational velocity and becomes part of a vortex

which emits waves inwards and outwards from the

spiral (frames 3,6); the waves collide and annul

each other (frames 4,7). This results in a conspicu-

ous periodic alteration of the spiral thickness

(period T - 1 min, as in the original rotating 2D

spiral). It is also seen that the spiral contracts,

beginning from the center. The frames of fig. 3

show the behaviour of a 3D spiral structure at the

late stages of the development. It is seen that the

spiral continues to contract and inside it a pecu-

liar drop-shaped region is formed, which is ex-

panding with time. The waves do not originate in

this region, they enter it from outside, move to its

center and disappear there to form an inverted, or

absorbing, center characteristic of the central part

of a vortex.

The contraction velocity of the vortex filament

is 0.7 X 10e3 cm/s. This value can be measured to

no better than lo-20% since during the contrac-

tion of a spiral its shape undergoes alteration. This

alteration is most pronounced in the center of the

spiral. Because of this, the very concept of the

local velocity of filament contraction cannot be

accurately defined. The filament contraction veloc-

ity was found to be constant within the same

accuracy (20%) in different experiments and for

different spirals in one and the same experiment.

5. Discussion

Fig. 4 presents a scheme that makes clear the

mechanism of the occurrence of an empty expand-

ing region in the center of a vortex with a spiral

filament. The solid line (A) shows the filament of a

spiral wave rotating in a thin gel layer. In fig. 4(a)

(top view), it appears as a point and the spiral

K.I. Agludze et al./ 30 vortex with a spmljknent

Fig. 3. Cinegram of the late stage of the conversion (18 min after the onset of experiment)

wave rotating around it is shown by a thin line.

Fig. 4(b) shows the configuration of the filament

after the wave was allowed to escape from the gel

to the solution on top of it: the spiral wave on the

boundary surface of the two media has become

the filament of a vortex (solid line). Fig. 4(c)

demonstrates the shortening of the filament with

time (“elasticity” effect), which is proportional to

its curvature [12, 31 as in the case of a vortex in

liquid helium. This process results in the emer-

gence of an expanding region free of the filament

(fig. 4(c): 1,2,3).

It is evident that

filament determined

the shortening of the vortex

by its curvature provides an

41

Fig. 4. Scheme of evolution of a 30 vortex with a spiral

filament. A, B, C: dish with reagent, side view; a, b, c: top view. Solid line: vortex filament; thin line: wave. Numbers 1, 2 and 3

in C indicate successive positions of the filament.

velocity of shortening of its filament is found from

the expression [12, 13, 51

V= -2qD/r, (1)

where D is the diffusion coefficient. As seen from

fig. 4(c), the filament constitutes a quarter of the

vortex ring and its velocity of shortening is equal

to

V= - f~D/r. (2)

The radius r, as seen from fig. 4(c), is close to the

gel layer thickness. The diffusion coefficient D for

the SiO,-gel is unknown, but it must be close to

that for water; in any case, not greater than it.

Using the values of D I 2 x 10e5 cm2/s and r =

1.2 X 10 _ ’ cm we find the value for the velocity of

shortening of the filament, u I 2.6 x lo- 4 cm/s.

Comparing it with the experimental value, u =

7 x lop4 cm/s, gives an almost 3-fold difference,

which cannot be attributed to an inaccuracy of

measurement. The causes of the difference may be

the following: (i) Formula (1) is inapplicable here.

Note that it is valid in the case of weakly nonlin-

ear systems (X-w system) [13] and for systems

with any kinetics where the vortex ring radius

r z+ A. X being the wavelength [5]. However, in

the experiment, X = 1.4 mm and the layer thick-

ness h = 1.2 mm. (ii) Other mechanisms of fila-

ment shortening may be involved along with that

described above; for instance, mechanisms related

to inhomogeneity of the medium.

The experiments described above aim at devel-

oping a theory of evolution of strongly curved

fragments of vortex filaments with a curvature

radius r - A. Advances in handling three-dimen-

sional autowave media will make such measure-

ments possible.

The method of self-completion is successfully

used, for example, in the case of two combined

agar pieces soaked with the B-Z reagent [14]. In

this case, it is possible to obtain simple and par-

tially twisted vortex rings as well as r-shaped

vortex filaments and to trace their evolution with

time.

In conclusion, it may be noted that experiments

with the B-Z reagent in 3D are, as a rule, heavily

hampered by the gas bubbles being evolved during

the course of reaction. To avoid such interference,

tightly sealed test-tubes were used in [8] in order

that the increasing pressure inside the tube should

prevent the gas bubbles from growing. When the

reaction is run in the SiO,-gel, no gas bubbles are

formed, which makes the system insensitive to

mechanical disturbances and, therefore, suitable

for free manipulation. Exchange of substances by diffusion between

the gel and the stirred solution (renewable, if

necessary) enables an observation of wave propa-

gation for an arbitrarily long time, in contrast to

the classical methods [15].

Acknowledgement

We thank Dr. H. Brandstadter for great help in

preparation of the immobilized catalyst in SiO,-gel.

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