Topic

# Instability

About: Instability is a research topic. Over the lifetime, 39564 publications have been published within this topic receiving 871497 citations.

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TL;DR: A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented in this article, with emphasis on comparisons between theory and quantitative experiments, and a classification of patterns in terms of the characteristic wave vector q 0 and frequency ω 0 of the instability.

Abstract: A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.

6,145 citations

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TL;DR: In this article, a linear analysis is presented of the instability, which is local and extremely powerful; the maximum growth rate which is of the order of the angular rotation velocity, is independent of the strength of the magnetic field.

Abstract: A broad class of astronomical accretion disks is presently shown to be dynamically unstable to axisymmetric disturbances in the presence of a weak magnetic field, an insight with consequently broad applicability to gaseous, differentially-rotating systems. In the first part of this work, a linear analysis is presented of the instability, which is local and extremely powerful; the maximum growth rate, which is of the order of the angular rotation velocity, is independent of the strength of the magnetic field. Fluid motions associated with the instability directly generate both poloidal and toroidal field components. In the second part of this investigation, the scaling relation between the instability's wavenumber and the Alfven velocity is demonstrated, and the independence of the maximum growth rate from magnetic field strength is confirmed.

4,265 citations

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TL;DR: In this article, the authors demonstrate the mechanism for a universal instability, the Arnold diffusion, which occurs in the oscillating systems having more than two degrees of freedom, which results in an irregular, or stochastic, motion of the system as if the latter were influenced by a random perturbation even though, in fact, the motion is governed by purely dynamical equations.

3,527 citations

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TL;DR: In this article, it was shown that a conical interface between two fluids can exist in equilibrium in an electric field, but only when the cone has a semi-vertical angle 49.3$^\circ$.

Abstract: The disintegration of drops in strong electric fields is believed to play an important part in the formation of thunderstorms, at least in those parts of them where no ice crystals are present. Zeleny showed experimentally that disintegration begins as a hydrodynamical instability, but his ideas about the mechanics of the situation rest on the implicit assumption that instability occurs when the internal pressure is the same as that outside the drop. It is shown that this assumption is false and that instability of an elongated drop would not occur unless a pressure difference existed. When this error is corrected it is found that a drop, elongated by an electric field, becomes unstable when its length is 1.9 times its equatorial diameter, and the calculated critical electric field agrees with laboratory experiments to within 1%. When the drop becomes unstable the ends develop obtuse-angled conical points from which axial jets are projected but the stability calculations give no indication of the mechanics of this process. It is shown theoretically that a conical interface between two fluids can exist in equilibrium in an electric field, but only when the cone has a semi-vertical angle 49.3$^\circ$. Apparatus was constructed for producing the necessary field, and photographs show that conical oil/water interfaces and soap films can be produced at the caloulated voltage and that their semi-vertical angles are very close to 49.3$^\circ$. The photographs give an indication of how the axial jets are produced but no complete analytical description of the process is attempted.

2,994 citations

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TL;DR: In this article, it was shown that when two superposed fluids of different densities are accelerated in a direction perpendicular to their interface, this surface is stable or unstable according to whether the acceleration is directed from the heavier to the lighter fluid or vice versa.

Abstract: It is shown that, when two superposed fluids of different densities are accelerated in a direction perpendicular to their interface, this surface is stable or unstable according to whether the acceleration is directed from the heavier to the lighter fluid or vice versa. The relationship between the rate of development of the instability and the length of wave-like disturbances, the acceleration and the densities is found, and similar calculations are made for the case when a sheet of liquid of uniform depth is accelerated.

2,839 citations