Instability

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A ball on the top of a hill is an unstable situation. Unstable3.svg
A ball on the top of a hill is an unstable situation.

In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds. [1] Not all systems that are not stable are unstable; systems can also be marginally stable or exhibit limit cycle behavior.

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In structural engineering, a structure can become unstable when excessive load is applied. Beyond a certain threshold, structural deflections magnify stresses, which in turn increases deflections. This can take the form of buckling or crippling. The general field of study is called structural stability.

Atmospheric instability is a major component of all weather systems on Earth.

Instability in control systems

In the theory of dynamical systems, a state variable in a system is said to be unstable if it evolves without bounds. A system itself is said to be unstable if at least one of its state variables is unstable.

In continuous time control theory, a system is unstable if any of the roots of its characteristic equation has real part greater than zero (or if zero is a repeated root). This is equivalent to any of the eigenvalues of the state matrix having either real part greater than zero, or, for the eigenvalues on the imaginary axis, the algebraic multiplicity being larger than the geometric multiplicity.[ clarification needed ] The equivalent condition in discrete time is that at least one of the eigenvalues is greater than 1 in absolute value, or that two or more eigenvalues are equal and of unit absolute value.

Instability in solid mechanics

Fluid instabilities

Hydrodynamics simulation of the Rayleigh-Taylor instability HD-Rayleigh-Taylor.gif
Hydrodynamics simulation of the Rayleigh–Taylor instability
Unstable flow structure generated from the collision of two impinging jets. Fluid Instability.jpg
Unstable flow structure generated from the collision of two impinging jets.

Fluid instabilities occur in liquids, gases and plasmas, and are often characterized by the shape that form; they are studied in fluid dynamics and magnetohydrodynamics. Fluid instabilities include:

Plasma instabilities

Plasma instabilities can be divided into two general groups (1) hydrodynamic instabilities (2) kinetic instabilities. Plasma instabilities are also categorised into different modes – see this paragraph in plasma stability.

Instabilities of stellar systems

Galaxies and star clusters can be unstable, if small perturbations in the gravitational potential cause changes in the density that reinforce the original perturbation. Such instabilities usually require that the motions of stars be highly correlated, so that the perturbation is not "smeared out" by random motions. After the instability has run its course, the system is typically "hotter" (the motions are more random) or rounder than before. Instabilities in stellar systems include:

Joint instabilities

The most common residual disability after any sprain in the body is instability. Mechanical instability includes insufficient stabilizing structures and mobility that exceed the physiological limits. Functional instability involves recurrent sprains or a feeling of giving way of the injured joint. [5] Injuries cause proprioceptive deficits and impaired postural control in the joint. Individuals with muscular weakness, occult instability, and decreased postural control are more susceptible to injury than those with better postural control. Instability leads to an increase in postural sway, the measurement of the time and distance a subject spends away from an ideal center of pressure. The measurement of a subject's postural sway can be calculated through testing center of pressure (CoP), which is defined as the vertical projection of center of mass on the ground. Investigators have theorized that if injuries to joints cause deafferentation, the interruption of sensory nerve fibers, and functional instability, then a subject's postural sway should be altered. [6] Joint stability can be enhanced by the use of an external support system, like a brace, to alter body mechanics. The mechanical support provided by a brace provides cutaneous afferent feedback in maintaining postural control and increasing stability.

Notes

  1. "Definition of INSTABILITY". www.merriam-webster.com. Retrieved 23 April 2018.
  2. "Definition of BAROCLINIC INSTABILITY". www.merriam-webster.com. Retrieved 23 April 2018.
  3. Shengtai Li, Hui Li. "Parallel AMR Code for Compressible MHD or HD Equations". Los Alamos National Laboratory. Archived from the original on 2016-03-03. Retrieved 2006-05-31.CS1 maint: uses authors parameter (link)
  4. Merritt, D.; Sellwood, J. (1994), "Bending Instabilities of Stellar Systems", The Astrophysical Journal, 425: 551–567, Bibcode:1994ApJ...425..551M, doi:10.1086/174005
  5. Guskiewicz, K. M.; Perrin, David H. (1996). "Effect of Orthotics on Postural Sway Following Inversion Ankle Sprain". Journal of Orthopedic and Sports Physical Therapy. 23 (5): 326–331. doi: 10.2519/jospt.1996.23.5.326 . PMID   8728531.
  6. Pintsaar, A.; Brynhildsen, J.; Tropp, H. (1996). "Postural Corrections after Standardised Perturbations of Single Limb Stance: Effect of Training and Orthotic Devices in Patients with Ankle Instability". British Journal of Sports Medicine . 30 (2): 151–155. doi:10.1136/bjsm.30.2.151. PMC   1332381 . PMID   8799602.

Related Research Articles

Stability may refer to:

Plasma stability measure of how likely a perturbation in a plasma is to be damped out

The stability of a plasma is an important consideration in the study of plasma physics. When a system containing a plasma is at equilibrium, it is possible for certain parts of the plasma to be disturbed by small perturbative forces acting on it. The stability of the system determines if the perturbations will grow, oscillate, or be damped out.

Baroclinity measure of misalignment between the gradients of pressure and density in a fluid

In fluid dynamics, the baroclinity of a stratified fluid is a measure of how misaligned the gradient of pressure is from the gradient of density in a fluid. In meteorology a baroclinic atmosphere is one for which the density depends on both the temperature and the pressure; contrast this with a barotropic atmosphere, for which the density depends only on the pressure. In atmospheric terms, the barotropic zones of the Earth are generally found in the central latitudes, or tropics, whereas the baroclinic areas are generally found in the mid-latitude/polar regions.

Kelvin–Helmholtz instability phenomenon of fluid mechanics

The Kelvin–Helmholtz instability can occur when there is velocity shear in a single continuous fluid, or where there is a velocity difference across the interface between two fluids. An example is wind blowing over water: The instability constant is shown to manifest itself through waves on a water surface. In addition to a water surface, clouds, the ocean, Saturn's bands, Jupiter's Red Spot, and the sun's corona also display this instability.

Rayleigh–Taylor instability Unstable behavior of two contacting fluids of different densities

The Rayleigh–Taylor instability, or RT instability, is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid. Examples include the behavior of water suspended above oil in the gravity of Earth, mushroom clouds like those from volcanic eruptions and atmospheric nuclear explosions, supernova explosions in which expanding core gas is accelerated into denser shell gas, instabilities in plasma fusion reactors and inertial confinement fusion.

The potential temperature of a parcel of fluid at pressure is the temperature that the parcel would attain if adiabatically brought to a standard reference pressure , usually 1,000 hPa (1,000 mb). The potential temperature is denoted and, for a gas well-approximated as ideal, is given by

In fluid dynamics, the Taylor number (Ta) is a dimensionless quantity that characterizes the importance of centrifugal "forces" or so-called inertial forces due to rotation of a fluid about an axis, relative to viscous forces.

Ferrofluids can be used to transfer heat, since heat and mass transport in such magnetic fluids can be controlled using an external magnetic field.

Standing human position in which the body is held in an upright position

Standing, also referred to as orthostasis, is a human position in which the body is held in an upright ("orthostatic") position and supported only by the feet. Although seemingly static, the body rocks slightly back and forth from the ankle in the sagittal plane. The sagittal plane bisects the body into right and left sides. The sway of quiet standing is often likened to the motion of an inverted pendulum.

Balance (ability) ability to maintain the line of gravity of a body within the base of support with minimal postural sway

In biomechanics, balance is an ability to maintain the line of gravity of a body within the base of support with minimal postural sway. Sway is the horizontal movement of the centre of gravity even when a person is standing still. A certain amount of sway is essential and inevitable due to small perturbations within the body or from external triggers. An increase in sway is not necessarily an indicator of dysfunctional balance so much as it is an indicator of decreased sensorimotor control.

Atmospheric instability

Atmospheric instability is a condition where the Earth's atmosphere is generally considered to be unstable and as a result the weather is subjected to a high degree of variability through distance and time. Atmospheric stability is a measure of the atmosphere's tendency to discourage or deter vertical motion, and vertical motion is directly correlated to different types of weather systems and their severity. In unstable conditions, a lifted thing, such as a parcel of air will be warmer than the surrounding air at altitude. Because it is warmer, it is less dense and is prone to further ascent.

In physics and engineering, kinetics is the branch of classical mechanics that is concerned with the relationship between motion and its causes, specifically, forces and torques. Since the mid-20th century, the term "dynamics" has largely superseded "kinetics" in physics textbooks, though the term is still used in engineering.

Hydrodynamic stability

In fluid dynamics, hydrodynamic stability is the field which analyses the stability and the onset of instability of fluid flows. The study of hydrodynamic stability aims to find out if a given flow is stable or unstable, and if so, how these instabilities will cause the development of turbulence. The foundations of hydrodynamic stability, both theoretical and experimental, were laid most notably by Helmholtz, Kelvin, Rayleigh and Reynolds during the nineteenth century. These foundations have given many useful tools to study hydrodynamic stability. These include Reynolds number, the Euler equations, and the Navier–Stokes equations. When studying flow stability it is useful to understand more simplistic systems, e.g. incompressible and inviscid fluids which can then be developed further onto more complex flows. Since the 1980s, more computational methods are being used to model and analyse the more complex flows.

Energy dissipation and entropy production extremal principles are ideas developed within non-equilibrium thermodynamics that attempt to predict the likely steady states and dynamical structures that a physical system might show. The search for extremum principles for non-equilibrium thermodynamics follows their successful use in other branches of physics. According to Kondepudi (2008), and to Grandy (2008), there is no general rule that provides an extremum principle that governs the evolution of a far-from-equilibrium system to a steady state. According to Glansdorff and Prigogine, irreversible processes usually are not governed by global extremal principles because description of their evolution requires differential equations which are not self-adjoint, but local extremal principles can be used for local solutions. Lebon Jou and Casas-Vásquez (2008) state that "In non-equilibrium ... it is generally not possible to construct thermodynamic potentials depending on the whole set of variables". Šilhavý (1997) offers the opinion that "... the extremum principles of thermodynamics ... do not have any counterpart for [non-equilibrium] steady states ." It follows that any general extremal principle for a non-equilibrium problem will need to refer in some detail to the constraints that are specific for the structure of the system considered in the problem.

The Taylor–Goldstein equation is an ordinary differential equation used in the fields of geophysical fluid dynamics, and more generally in fluid dynamics, in presence of quasi-2D flows. It describes the dynamics of the Kelvin–Helmholtz instability, subject to buoyancy forces, for stably stratified fluids in the dissipation-less limit. Or, more generally, the dynamics of internal waves in the presence of a (continuous) density stratification and shear flow. The Taylor–Goldstein equation is derived from the 2D Euler equations, using the Boussinesq approximation.

Rayleigh–Bénard convection natural convection, occurring in a plane horizontal layer of fluid heated from below, in which the fluid develops a regular pattern of convection cells known as Bénard cells

Rayleigh-Bénard convection is a type of natural convection, occurring in a plane horizontal layer of fluid heated from below, in which the fluid develops a regular pattern of convection cells known as Bénard cells. Bénard–Rayleigh convection is one of the most commonly studied convection phenomena because of its analytical and experimental accessibility. The convection patterns are the most carefully examined example of self-organizing nonlinear systems.

Rayleighs equation (fluid dynamics)

In fluid dynamics, Rayleigh's equation or Rayleigh stability equation is a linear ordinary differential equation to study the hydrodynamic stability of a parallel, incompressible and inviscid shear flow. The equation is:

The interchange instability is a type of plasma instability seen in magnetic fusion energy that is driven by the gradients in the magnetic pressure in areas where the confining magnetic field is curved. The name of the instability refers to the action of the plasma changing position with the magnetic field lines without significant disturbance to the geometry of the external field. The instability causes flute-like structures to appear on the surface of the plasma, and thus the instability is also known as the flute instability. The interchange instability is a key issue in the field of fusion energy, where magnetic fields are used to confine a plasma in a volume surrounded by the field.

Combustion instability

Combustion instabilities are physical phenomena occurring in a reacting flow in which some perturbations, even very small ones, grow and then become large enough to alter the features of the flow in some particular way.

Conditional symmetric instability

Conditional symmetric instability, or CSI, is a form of convective instability in a fluid subject to temperature differences in a uniform rotation frame of reference while it is thermally stable in the vertical and dynamically in the horizontal. The instability in this case develop only in an inclined plane with respect to the two axes mentioned and that is why it can give rise to a so-called "slantwise convection" if the air parcel is almost saturated and moved laterally and vertically in a CSI area. This concept is mainly used in meteorology to explain the mesoscale formation of intense precipitation bands in an otherwise stable region, such as in front of a warm front. The same phenomenon is also applicable to oceanography.