Chaotic bubbles within

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

and

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, occur in cases when there are any dynamic processes that generate bubbles that are

nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...

. Many exhibit mathematically chaotic patterns consistent with

chaos theory Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have co ...

. In most systems, they arise out of a forcing

pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...

that encounters some kind of resistance or

shear Shear may refer to: Textile production *Animal shearing, the collection of wool from various species **Sheep shearing *The removal of nap during wool cloth production Science and technology Engineering *Shear strength (soil), the shear strength ...

factor, but the details vary depending on the particular context. The most widely known example is bubbles in various forms of

liquid A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, a ...

. 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 Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...

(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 Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...

of the dynamics of bubbles in liquids, with important contributions from Werner Lauterborn (1976). Lauterborn and Cramer (1981) also applied chaos theory to

acoustics Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...

, 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 fluid A non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity, i.e., constant viscosity independent of stress. In non-Newtonian fluids, viscosity can change when under force to either more liquid or more solid. Ketchup, for exa ...

s 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 In lab experiments that study chaos theory, approaches designed to control chaos are based on certain observed system behaviors. Any chaotic attractor contains an infinite number of unstable, periodic orbits. Chaotic dynamics, then, consists of ...

), converts them to periodic oscillations, and has an important application to gas–solids in

fluidized bed reactor A fluidized bed reactor (FBR) is a type of reactor device that can be used to carry out a variety of multiphase chemical reactions. In this type of reactor, a fluid (gas or liquid) is passed through a solid granular material (usually a catalyst ...

s, also applicable to the

ammoxidation In organic chemistry, ammoxidation is a process for the production of nitriles () using ammonia () and oxygen (). It is sometimes called the SOHIO process, acknowledging that ammoxidation was developed at Standard Oil of Ohio. The usual substrat ...

propylene Propylene, also known as propene, is an unsaturated organic compound with the chemical formula CH3CH=CH2. It has one double bond, and is the second simplest member of the alkene class of hydrocarbons. It is a colorless gas with a faint petrole ...

acrylonitrile Acrylonitrile is an organic compound with the formula and the structure . It is a colorless, volatile liquid although commercial samples can be yellow due to impurities. It has a pungent odor of garlic or onions. In terms of its molecular ...

(Kaart, Schouten, and van den Bleek, 1999). Sarnobat et al.) study the behavior of

electrostatic fields An electric field (sometimes E-field) is the field (physics), physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the ...

on chaotic bubbling in attempt to control the chaos into a lower order periodicity.

References

Further reading