# Chaotic bubble

Many dynamic processes that generate bubbles are nonlinear, many exhibiting mathematically chaotic patterns consistent with chaos theory. In such cases, **chaotic bubbles** can be said to occur. In most systems they arise out of a forcing pressure that encounters some kind of resistance or shear factor, but the details vary depending on the particular context.

The most widely known example is bubbles in various forms of liquid. Although there may have been an earlier use of the term, it was used in 1987 specifically in connection with a model of the motion of a single bubble in a fluid subject to periodically driven pressure oscillations (Smereka, Birnir, and Banerjee, 1987). For an overview of models of single-bubble dynamics see Feng and Leal (1997). There is extensive literature on nonlinear analysis of the dynamics of bubbles in liquids, with important contributions from Werner Lauterborn (1976). Lauterborn and Cramer (1981) also applied chaos theory to acoustics, in which bubble dynamics play a crucial part. This includes analysis of chaotic dynamics in an acoustic cavitation bubble field in a liquid (Lauterborn, Holzfuss, and Bilio, 1994). The study of the role of shear stresses in non-Newtonian fluids has been done by Li, Mouline, Choplin, and Midoux (1997).

A somewhat related field, the study of controlling such chaotic bubble dynamics (control of chaos), converts them to periodic oscillations, and has an important application to gas–solids in fluidized bed reactors, also applicable to the ammoxidation of propylene to acrylonitrile (Kaart, Schouten, and van den Bleek, 1999). Sarnobat et al. (2000 & 2004) study the behavior of electrostatic fields on chaotic bubbling in attempt to control the chaos into a lower order periodicity.

An early attempted application that led to failure was in Alan H. Guth’s (1981) chaotic inflation theory of the early period of the universe. While he did not precisely use the term “chaotic bubbles,” his model involved “bubbles” in the original cosmic foam that collided chaotically. The model has since been modified due to the inability to find in the real universe some of the phenomena predicted by it, with improvements involving quantum fluctuations provided by Andrei Linde (1986).

In economics, bubbles are due to speculation in asset markets, causing an economic bubble. The first to apply the term in this context was J. Barkley Rosser, Jr. in 1991. While they did not use the term, Richard H. Day and Weihong Huang (1990) showed that the interaction of fundamentalist and trend-chasing traders could lead to chaotic dynamics in the price path of a speculative bubble. De Grauwe, Dewachter, and Embrechts (1983) applied such a model to foreign exchange rate dynamics.

## References

- P. Smereka, B. Birnir, and S. Banerjee. “Regular and Chaotic Bubble Oscillations in Periodically Driven Pressure Fields.”
*Physics of Fluids*November 1987, 30(11), pp. 3342–3350. - Z.C. Feng and L.G. Leal. “Nonlinear Bubble Dynamics.”
*Annual Review of Fluid Mechanics*January 1997, 29(1), pp. 201–243. - Werner Lauterborn. “Numerical Investigation of Nonlinear Oscillations of Gas Bubbles in Liquids.”
*Journal of the Acoustical Society of America*1976, 59(2), pp. 283–293. - Sarnobat, S.U., “Modification, Identification & Control of Chaotic Bubbling via Electrostatic Potential”, Masters Thesis, University of Tennessee, Knoxville, August 2000. online archive
- Sarnobat S.U. et al., "The impact of external electrostatic fields on gas–liquid bubbling dynamics", Chemical Engineering Science, Volume 59, Issue 1, January 2004, Pages 247–258 link
- W. Lauterborn and E. Cramer. “Subharmonic Route to Chaos Observed in Acoustics.”
*Physical Review Letters*1981, 47(20), pp. 1445–1448. - W. Lauterborn, J. Holzfuss, and A. Bilio. “Chaotic Behavior in Acoustic Cavitation.”
*IEEE Ultrasonics Symposium Proceedings*November 1994, 2(1–4), pp. 801–810. - H.Z. Li, Y. Mouline, L. Choplin, and N. Midoux. “Chaotic Bubble Coalescence in Non-Newtonian Fluids.”
*International Journal of Multiphase Flow*August 1997, 23(4), pp. 713–723. - Sander Kaart, Jaap C. Schouten, and Cor M. van den Bleek. “Improving Conversion and Selectivity of Catalytic Reactions in Bubbling Gas-Solid Fluidized Bed Reactors by Control of the Nonlinear Bubble Dynamics.”
*Catalysis Today*January 27, 1999, 48(1–4), pp. 185–194. - Alan H. Guth. “Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems.”
*Physical Review*D January 1981, 23(2), pp. 347–356. - Andrei D. Linde. “Eternally Existing Self-Reproducing Chaotic Inflationary Universe.”
*Physical Letters*B 1986, 175, pp. 395–400. - J. Barkley Rosser, Jr.
*From Catastrophe to Chaos: A General Theory of Economic Discontinuities*. Boston/Dordrecht: Kluwer Academic Publishers, 1991. - Richard H. Day and Weihong Huang. “Bulls, Bears, and Market Sheep.”
*Journal of Economic Behavior and Organization*December 1990, 14(3), pp. 299–329. - Paul De Grauwe, Hans Dewachter, and Mark Embrechts.
*Exchange Rate Theory: Chaotic Models of Foreign Exchange Rate Markets*. Oxford: Blackwell, 1993.