Ryuji Takaki
Tokyo University of Agriculture and Technology,
Koganei, Tokyo 1848588, Japan
Fax: +81423857204
takaki@cc.tuat.ac.jp
(Received: 26.12.1998}
Abstract: Slow mixing of a viscous fluid due to rotating cylinders dipped in the fluid is computed based on the basic law of viscous fluid dynamics. Several dye spots added to the fluid show chaotic but attractive patterns as the cylinders are rotated in turn. Both two and threedimensional cases are tried. In the twodimensional case the distribution of fluid displacement vectors has both elliptic and hyperbolic points, suggesting production of chaos. From the dye pattern in the threedimensional case a solid model was roduced with a help of a sculpture artist. This process is looked upon as a new method of creating sculptures. At the same time the artist made up his own shape by hand inspired by the computed dye shape. Differences between the dye shape (scientific shape) and the hand made shape (artificial shape) are discussed. Keywords: Chaotic Mixing, Viscous Fluid, Rheoart, Sculpture 
1. Introduction It is well known that repeated unsteady motions of walls of a container filled with a highly viscous fluid or unsteady motions of solid bodies immersed in the fluid can produce a chaotic mixing in the fluid, which is called "Lagrangian turbulence" (Ottino, 1989a,b). It can be visualized by adding some dyes with the same viscosity beforehand. The main interest in this phenomenon has been the mechanism of chaotic behavior as a result of accumulation of dye displacements. In fact, resultant dye patterns are similar to the Poincare maps characteristic of chaotic behavior of dynamical systems. On the other hand, this phenomenon has been attracting fluiddynamists because of the beauty of resulting patterns. This fact may suggest that this phenomenon can be applied as a new method of creating artworks. The term "rheo" in the title means a flow, and "rheoart" should mean an art produced by application of the fluid dynamics. Since there are a lot of flow patterns in the nature, there should be also many kinds of rheoart. However, in this paper we confine ourselves to the viscous mixing noted above, simply because it is relatively easy to simulate this flow.
The present author has recently made an attempt to apply 2D (twodimensional) chaotic
mixing to produce attractive patterns by numerical method (Takaki
& Tomioka, 1997). In thepresent paper this attempt is applied
to other 2D configurations of cylinders and also extended to a 3D configuration.
Furthermore, it is tried to make up solid models form the computed results
with a help of a sculpture artist, Mr. Y. Nakatsugawa. The purpose
of this paper is to present results of these studies, and to make a fundamental
discussion on similarities and differences between art and science.
2. Method of Computation
The 2D
chaotic mixing is produced by a method similar to that first introduced
by Aref (1984). In his computation two circular cylinders were set
vertically in a container filled with a viscous fluid (see Fig. 1a).
They are rotated around their axes very slowly by certain angles in turn,
so that only one cylinder is rotated while the oher is at rest. The
fluid around a rotating cylinder moves according to the theory of
slow flows, i.e. the velocity induced by the rotation is inversely
proportional to the distance from the cylinder axis.
In the present paper cases with two and four equal cylinders are tried (see Figs. 1a and 1b). Rotation angles of cylinders and the order of their motions can be chosen arbitrarily. But, a common rotation angle is chosen and the sense of rotation of each cylinder and the order of motions are left arbitrary. When a flow field is produced by rotation of one cylinder, the flow is modified by existence of other rest cylinders because the fluid should go round these objects. However, in this study their effects are neglected in order to simplify the computation of fluid motion. It is allowed when mutual distances between cylinders are much larger than their diameter.
In a typical case of two cylinders, which are denoted with surfices 1 and 2,
respectively (Fig. 1a), they are rotated by angles in an order,
This expression show that the following four processes constitute one step of operation: cylinder 1 is rotated clockwise by 180^{°}, cylinder 2 clockwise by 180^{°}, cylinder 1 anticlockwise by 180^{°}, then cylinder 2 anticlockwise by 180^{°}. Note that the clockwise rotation is denoted by a positive angle. This operation is repeated by arbitrary number of steps so long as computing time is available.
In the case with four cylinders a similar expression of processes is employed
(Fig. 1b). In this case one step of processes is expressed as
At the beginning several spots of dyes with many colors are placed on the fluid surface, where the viscosity of the dye is assumed equal to that of the fluid, so that the dyes move according to the same rule as for the fluid. Number, shapes, sizes and positions of initial dye spots vcan be given also arbitrarily. But, by tryanderror one may find that most interesting patterns will be obtained when several spots are placed within the central region, so that initial spots are surrounded by cylinders. In the first case with two cylinders seven dye spots are placed in hexagonal configuration (Fig. 3). In the case of four cylinders a lot of spots are placed in square grid configuration (Fig. 4).
Similar processes are made in the case of 3D viscous mixing. In this study
two equal cylinders are set perpendicularly in a deep container of a viscous
fluid with their axes apart, and one spherical dye is placed between the
two cylinders as an initial condition. Each cylinder is rotated one
after another by 180^{°} always in the same direction, i.e.
The fluid is assumed to
go round the rotating cylinder in a plane perpendicular to the cylinder
axis, where the effect of the other cylinder is neglected as mentioned
above.
Fig. 3 shows an initial dye pattern and its deformations after several steps in the case of two cylinders due to the fourprocess operation (Eq. (1)). Readers might agree that delicate and beautiful patterns are obtained. The resulting patterns have two complicated swirling centers, each of which by close observation is seen to be a mixture of rotations of opposite senses. The whole pattern does not have a symmetry inspite of the symmetrical configuration of cylinders. It is owing without doubt to the asymmetry of operation, i.e. the right cylinder is rotated first. This coexistence of large scale order and asymmetry seems to give an attractiveness to these patterns.
The displacement
vector field corresponding to one step of fourprocess operation is shown
in Fig. 5 (left). In this figure the angle of rotation is increased
to 360^{°}
so that fluid dispalenments are seen more clearly.
Some trajecroties are also drawn on the map. The complicated swirling
centers are located near both to the hyperbolic point (saddle point) and
the elliptic point (vortex) in the map. It is in accordance with
the common fact established in the chaos dynamics. It is interesting
to point out that the swirling centers are nearer to the hyperbolic points
(not the elliptic points) inspite of their swirling nature!
Resulting patterns in the fourcylinder case are shown in Fig. 4. They have
4fold symmetries corresponding to the square configuration of cylinders,
but the symmetry is broken owing to the asymmetry of operation, as in the
twocylinder case. The fluid displacement map, shown in Fig. 5 (right),
is much complicated, but one can recognize six hyperbolic points and eight
elliptic points. The two hyperbolic centers within the cylinder square
seem to contribute much in formation of highly chaotic mixture at the center.
The hyperbilic points located out of the cylinder square are located near
to the outer swirling centers as in the twocylinder case.
4. Results of 3D cases
Some of dye patterns up to the 9th step in the 3D case are shown in Fig. 6.
The 9th pattern is displayed once more in Fig. 6(below) by the use of a
software of raytracing. As is seen from these patterns, the
dye continues to be wound to the two cylinders and aquires a long thin
tail at one end. Appearance of a thin tail is observed also in 2D
cases and not so surprising, although it is more complicated in 3D cases.
As is mentioned in Sec.1, this 3D simulation can be applied to art creation, i.e. a sculpture. A shape constructed from the 3D simulation is called a "scientific model", because it is also a scientific result. A scientific model was constructed to a solid model fairly faithfully with a help of a sculpture artist, Y. Nakatsugawa. A more precise discussion will be given on the meaning of "scientific" in the next section in terms of "repeatability".
Along with this scientific model the present author asked the artist to create
a solid model with a new shape according to his inspiration from this model,
which is called an "artificial model" (see Fig. 7). The term "artificial"
means that the model reflects the artist's desire and concept but the repeatability
is not expected, as is usual in hand works.
5. Discussions Results shown above would suggest that the simulation of chaotic mixing of a viscous fluid can provide a new method of art production. Especially, the 2D simulation is made by the use of a conventional softwares and is open to anyone who can use a personal computer. On the other hand, production of scuptures from computer simulation is still difficult at present for general people, because they must be engaged in computer programming and must have partners to help them in making solid models from computer data. As for the solid model production from computed data there is a possibility to utilize a facility for photosolidifying reaction, so that computed surface shapes are automatically reproduced to plastic models. The present author is considering to try this method in near future. This author has been using the term "art" too easily without discussing its meaning. Here, some discussions are given on the common features and differences of science and art on the basis of the author's experiences during activity of this work. This problem can be treated from variety of aspects and discusser's standpoints, but the present author would like to put a stress on the following points. Science and art are similar to each other in:
On the other hand, science and art are different in the processes of creations as listed below:
These two differences look very much critical, hence science and art have been supposed to belong to different categories of human activities. However, in spite of these differences they have a close relation as ointed out above. Now, as for the rheoart proposed here, it is considered to belong to the category of science, because the process of production is that of science. In fact, initial conditions are given and algorithms of dye deformations are determined first, then the later processes proceed automatically without any artificial manipulation. There is still a room for the human sensitivity to play a role, by choosing better results from scientific results and adjusting initial conditions or parameters. This way of creation is called here "rheoart". However, we need an appropriate facility in order to display results of rheoart. In the present work we have no problem with the 2D cases, because convetional color printers are enough for that purpose. On the other hand, displays of 3D works are not straight forward. In fact the present author needed a cooperation with a scuplture artist to make solid objects. Is the lightsolodification technique becomes more popular, this problem will be solved. The present author is considering to try this way in future. It is hoped that the rheoart will become possible by much easier method so that many people can use it. A true art should be based on simple technologies; Picasso used the same tools which we are using now. Acknowledgements The present author would like to express his cordial thanks to Mr. S. Tomioka for his long term helps in computer simulations and to Mr. F. Nakatsugawa for his kind efforts to make up solid objects from computed data. The present work would not have been completed without their helps. Aref, H., 1984, Stirring by chaotic advection, J. Fluid Mech. 143, 121. Ottino, J.M., 1989a, The Kinematics of Mixing, Cambridge Univ. Press. Ottino, J.M., 1989b, The mixing of fluids, Scientific American, January, 4049. Takaki, R., 1994, Proposal of a new kind of art "Rheoart", FORMA, 9, 203208.
