Application of Fluid Dynamics to Art Creation

Ryuji Takaki

Tokyo University of Agriculture and Technology,
Koganei, Tokyo 184-8588, Japan

Fax: +81-42-3857204


(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 three-dimensional cases are tried. In the two-dimensional case the distribution of fluid displacement vectors has both elliptic and hyperbolic points, suggesting production of chaos. From the dye pattern in the three-dimensional 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, Rheo-art, 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 fluid-dynamists 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 "rheo-art" 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 rheo-art. 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 (two-dimensional) 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.


Fig. 1. Configurations of cylinders and container walls for 2D viscous mixing: (a) two rotating cylinders, (b) four rotating cylynders.

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,

(W1, W2, W1, W2) = (120, 120, -120, -120).                 (1)

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 anti-clockwise by 180, then cylinder 2 anti-clockwise 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

(W1, W2, W3, W4, W1, W2, W3, W4,) = (180, 180, 180, 180, -180, -180, -180, -180 ),         (2)

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 try-and-error 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.

(W1, W2) = (180, 180).         (3)

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.


All mixing patterns in this study are obtained by the use of a personal computer (Power Macintosh 7500/100). In the 2D cases an image processing software (Photoshop, 2.5J) is used, especially its function of rotatory deformation was applied. This function does not seem to produce exact viscous deformations. But, it is not asked severely here, because the purpose of this paper is not to derive scientific results but to propose a method of art creation open to many people. In the 3D case the fluid motion due to the rotating cylinder is computed by a program written in C-language, where the dye surface is expressed by a grid network and motions of grid points are traced as the dye is deformed.

Fig. 2. Configuration of two cylinders for 3D viscous mixing. They are perpendicular to each other and their axes are apart by a distance of 5/3 times their diameter.

It is worth noting that it must be also interesting to undertake real experiments to produce mixing patterns, although there may arise some difficulties, for example in suppressing dye diffusion and initial fluctuation of dye inputs. In addition many other cases with various cylinder arrangements (or with other solid objects) and initial dye patterns could be tried to see whether they lead to attractive patterns. Readers are suggested to try themselves. At least the 2D mixing is easy and anyone can begin immediately, if he(she) has a personal computer.

3. Results of 2D cases

Fig. 3 shows an initial dye pattern and its deformations after several steps in the case of two cylinders due to the four-process 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 four-process 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!


Fig. 3. Initial dye pattern (upper left) and their deformations after several steps, in the two-cylinder case. The white spots indicate the cylinders.

Resulting patterns in the four-cylinder case are shown in Fig. 4. They have 4-fold symmetries corresponding to the square configuration of cylinders, but the symmetry is broken owing to the asymmetry of operation, as in the two-cylinder 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 two-cylinder case.


Fig. 4. Initial dye pattern (upper left) and their deformations after several steps in the four-cylinder case.


Fig. 5. Distributions of fluid displacements after one step of processes. Modes of rotations are given as in (a) Eq.(1) and (b) Eq.(2), respectively, but the angles are increased to 360 in both cases.

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 ray-tracing. 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.



Fig. 6. Results of 3D viscous mixing. The two cylinders by 180 in turn, each in the same direction: shapes at some steps (upper), ray-tracing image of the 9th step (lower).

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.


There are some clear differences between these two models. First, the artificial model lacks the thin tail. Second, the surface of the artificial model has a spherically concave part, which lacks in the scientific model. Third, the thickness distributions are different (not apparent from this picture), so that it tends to change monotonically in the scientific model, while the artificial model has wavy distribution. In other words the scientific model has a thickness distribution like that of a knife, while the shape of the artificial model is more biological. These characteristics of the artificial model may reflect the artist's ideas. When the present author showed these models to participants of a conference (Takaki & Tomioka, 1997), most of them preferred the artificial model to the scientific model, except one who is an engineer.

Fig. 7 Solid models produced by Y. Nakatsugawa. left: faithful copy of the 9th step (scientific model), right: a shape created by Y. Nakatsugawa (artificial model).

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 photo-solidifying 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:

  1. Their objects are to create new knowledges (new forms), and originalities are important.

  2. Their results should be presented to public and new human communications are triggered therewith.

  3. Utilities of results are expected by society, but scientists (artists) prefer to be free from the utilies.

On the other hand, science and art are different in the processes of creations as listed below:

  1. In science the repeatability is important, i.e. if the same conditions are given, the same results must be obtained. Therefore, especially in experimental works, artificial control of phenomena during an experimental run is avoided. In art, on the other hand, the human sensitivity to the artwork is important and repeatability is neglected, so that artificial control to improve the resullts is continuously made during production and no equivalent pieces of works are producued even by the same artist.

  2. The mental processes occurring in scientists and artists are just opposite to each other. The science is an activity to extract natural laws from real phenomena, which is to convert form to concept. On the other hand, the art is to produce real forms out of artist's imaginations, which is to convert concept to form.

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 rheo-art 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 "rheo-art".

However, we need an appropriate facility in order to display results of rheo-art. 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 light-solodification 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 rheo-art 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.


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, 1-21.

Ottino, J.M., 1989a, The Kinematics of Mixing, Cambridge Univ. Press.

Ottino, J.M., 1989b, The mixing of fluids, Scientific American, January, 40-49.

Takaki, R., 1994, Proposal of a new kind of art "Rheo-art", FORMA, 9, 203-208.

Takaki, R. and Tomioka, S., 1997, Mixing of viscous fluid and its application to art, Abstracts of 2nd Int. Conf. on Flow Interaction, Berlin, 1997, 3-5.