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Turbulence

Turbulence

:For the film of the same name, see Turbulence

In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by semi-random, stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time. Flow that is not turbulent is called laminar flow. The (dimensionless) Reynolds number characterizes whether flow conditions lead to laminar or turbulent flow. The structure of turbulent flow was described by the Russian mathematician Andrey Kolmogorov. Consider the flow of water over a simple smooth object, such as a sphere. At very low speeds the flow is laminar; i.e., the flow is smooth (though it may involve vortices on a large scale). As the speed increases, at some point the transition is made to turbulent ("chaotic") flow. In turbulent flow, unsteady vortices appear on many scales and interact with each other. Drag due to boundary layer skin friction increases. The structure and location of boundary layer separation often changes, sometimes resulting in a reduction of overall drag. Because laminar-turbulent transition is governed by Reynolds number, the same transition occurs if the size of the object is gradually increased, or the viscosity of the fluid is decreased, or if the density of the fluid is increased. When designing piping systems, turbulent flow requires a higher input of energy from a pump (or fan) than laminar flow. However, for applications such as heat exchangers and reaction vessels, turbulent flow is essential for good heat transfer and mixing.

Examples of turbulence

- Flow from a faucet or tap goes through several stages as the faucet is opened:
- Off. No flow.
- Dripping flow. Water emerges in discrete drops.
- Laminar flow. Water flows in a jet with a steady surface. In this regime, the surface of the jet is symmetric and steady. Its surface is so smooth that the shaft of water looks like a transparent lens.
- Turbulent flow. Above a certain critical flow rate, the water jet becomes unsteady and asymmetric. The stream ceases to be transparent, due to the extreme convolution of the surface of the fluid.
- Smoke rising from a cigarette -- for the first few centimetres it remains laminar, and then becomes unstable and turbulent.
- Flow over a golf ball. This can be best understood by considering the golf ball to be stationary, with air flowing over it. If the golf ball was smooth, the boundary layer flow over the front of the sphere would be laminar at typical conditions. However, the boundary layer would separate early, as the pressure gradient switched from favorable(pressure decreasing in the flow direction) to unfavorable(pressure increasing in the flow direction), creating a large region of low pressure behind the ball that creates high form drag. To prevent this from happening, the surface is dimpled to perturb the boundary layer and promote transition to turbulence. This results in higher skin friction, but moves the point of boundary layer separation rearward, resulting in lower form drag and lower overall drag.
- The mixing of warm and cold air in the atmosphere by wind, which causes poor astronomical seeing (the blurring of images seen through the atmosphere) According to an apocryphal story, Werner Heisenberg was asked what he would ask God, given the opportunity. His reply was: "When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first." A similar witticism has been attributed to Horace Lamb (who had published a noted text book on Hydrodynamics) - his choice being quantum mechanics (instead of relativity) and turbulence.

External links

- [http://ctr.stanford.edu/ Center for Turbulence Research], Stanford University
- [http://www.iop.org/EJ/S/UNREG/journal/1468-5248/1 Journal of Turbulence] - [http://www.iop.org/ Institute of Physics]
- http://turb.seas.ucla.edu/~jkim/sciam/turbulence.html - Scientific American article Category:Aerodynamics Category:Chaos theory Category:Fluid dynamics Category:Transport phenomena ms:Gelora ja:乱流

Turbulence (film)

Turbulence is a 1997 film directed by Robert Butler which stars Ray Liotta and Lauren Holly. Ryan Weaver (Liotta) is a serial killer who is being transported with another prisoner, Stubbs (Gleeson), on a commercial Boeing 747. Stubbs kills the U.S. Marshals assigned to guard them on the flight. Both the pilot and co-pilot are also killed in the struggle. And Weaver kills Stubbs. The passengers are locked up in a pantry, leaving only flight attendant, Teri Halloran (Holly), alone with Weaver. Halloran must try to land the plane safely while trying to keep away from Weaver.

Cast

- Ray Liotta — Ryan Weaver
- Lauren Holly — Teri Halloran
- Brendan Gleeson — Stubbs
- Hector Elizondo — Lt. Aldo Hines
- Rachel Ticotin — Rachel Taper
- Jeffrey DeMunn — Brooks

External link

- Category:1997 films

Stochastic

Stochastic, from the Greek "stochos" or "goal", means of, relating to, or characterized by conjecture; conjectural; random. The antonym is astochastic. A stochastic process is one whose behavior is non-deterministic in that the next state of the environment is not fully determined by the previous state of the environment.

Mathematical theory

In mathematics, specifically in probability theory, the field of stochastic processes has for some decades been a major area of research. See that article for more. A stochastic matrix is a matrix that has non-negative real entries that sum to 1 in each column.

Artificial intelligence

In artificial intelligence stochastic programs work by using probabilistic methods to solve problems, as in simulated annealing, neural networks and genetic algorithms. A problem itself may be stochastic as well, as in planning under uncertainty. A deterministic environment is much simpler for an agent to deal with.

Natural science

An example of a stochastic process in the natural world is pressure in a gas. Even though each molecule is moving deterministically, a collection of them is unpredictable (this is an example of chaos arising from order). A large enough set of molecules will exhibit stochastic characteristics, such as filling the container, exerting equal pressure, diffusing along concentration gradients, etc. (this is an example of order arising from chaos). These are emergent properties of the system.

Music

In music stochastic elements are randomly generated elements created by strict mathematical processes. Stochastic processes can be used in music either to compose a fixed piece, or produced in performance. Stochastic music was pioneered by Iannis Xenakis, who used probability, game theory, group theory, set theory, and Boolean algebra, and frequently used computers to produce his scores. Earlier, John Cage and others had composed aleatoric or indeterminate music, which is created by chance processes but does not have the strict mathematical basis (Cage's Music of Changes, for example, uses a system of charts based on the I-Ching).

Visual arts

In the visual arts, Yoshiyuki Abe[http://www.pli.jp], has mastered the art of creation through stochastic process. His work uses geometric objects, mostly the surfaces of hyperbolic paraboloids, and the processing of stochastic elements. In his words: "No matter how you use a computer, or whichever computer you use, to create an art work is not easy. Nevertheless, I believe artists can find a new horizon in his/her creative activities by having the experience of using geometric object and/or stochastic process. For artists who want to create mathematical art through algorithm-driven parameter control, the essential element for success is artistic serendipity. This is the interesting fact of art in the perfect mathematical space."

Color reproduction

When color reproductions are made, the image is separated into its component colors by taking multiple photographs filtered for each color. One resultant film or plate represents each of the cyan, magenta, yellow, and black data. Color printing is a binary system, where ink is either present or not present, so all color separations to be printed must be translated into dots at some stage of the workflow. Traditional linescreens which are amplitude modulated had problems with moire but were used until stochastic screening became available. A stochastic (or frequency modulated) dot pattern creates a more photorealistic image.

Language and linguistics

In usage-based linguistic theories, where it is argued that competence, or langue, is based on performance, or parole, in the sense that linguistic knowledge is based on frequency of experience, grammar is often said to be probabilistic and variable rather than fixed and absolute. This is so, because one's competence changes in accordance with ones experience with linguistic units. This way, the frequency of usage-events determines one's knowledge of the language in question.

Convection

Convection is the transfer of heat by currents within a fluid. It may arise from temperature differences either within the fluid or between the fluid and its boundary, other sources of density variations (such as variable salinity), or from the application of an external motive force. It is one of the three primary mechanisms of heat transfer, the others being conduction and radiation. Convection occurs in atmospheres, oceans, and planetary mantles.

Free and forced convection

In heat transfer, a distinction is made between free and forced convection. Free convection is convection in which motion of the fluid arises solely due to the temperature differences existing within the fluid. Example: hot air rising off the surface of a radiator. Image:Convection cells.png The basic premise behind free convection is that heated matter becomes more buoyant and "rises"; while cooler material "sinks". Free convection occurs in any liquid or gas which expands or contracts in response to changing temperatures when it is exposed to multiple temperatures in an acceleration field such as gravity or a centrifuge. The local changes in density results in buoyancy forces that cause currents in the fluid. In zero gravity, because buoyancy no longer becomes a factor, free convection does not occur. Forced convection happens when motion of the fluid is imposed externally (such as by a pump or fan). Example: a fan-powered heater, where a fan blows cool air past a heating element, heating the air. When a person blows on their food to cool it, he/she is using forced convection. fan

Convection at a surface

In both of the previous examples, an engineer would often be interested in the rate of heat transfer from the hot 'source' surface to the fluid medium. The local convective heat flux of a fluid passing over a surface is expressed as :q' = h (T_s - T_∞); :q' local heat flux (dq/dA) :h local convection coefficient :T_s surface temperature :T_∞ quiescent or ambient temperature The total heat transfer over a surface is then calculated as the integral of q", :q = ∫A_sq' dA_s :A_s area of the surface :q total heat transfer rate (units of energy/time) This then leads to a definition of average convection coefficient, h-bar, defined from :q = h-bar A_s (T_s - T_∞) Studies of forced convection lead to a close inspection of the flow in the boundary layer of the fluid. :See also: Fluid dynamics, Nusselt number, Grashof number, and Heat transfer coefficient.

Atmospheric convection

In the case of Earth's atmosphere, solar radiation heats the Earth's surface, and this heat is then transferred to the air by convection. When a layer of air receives enough heat from the Earth's surface, it expands, becomes less dense and is pushed upward by buoyancy. Colder, heavier air sinks under it and is then warmed, expands, and rises. The warm rising air cools as it reaches the higher, cooler regions of the atmosphere and becomes denser. Since it cannot sink through the rising air beneath it, it moves laterally and then begins to sink. When it reaches the surface again it is heated, and is drawn back into the original rising column. These convection currents cause local breezes, winds, thermals, cyclones and thunderstorms, and at a larger scale, produce the global atmospheric circulation features. A single region of air with a rising and falling current is called a convection cell. Heat is lost from the rising air when it radiates into space. :See also: weather.

Oceanic convection

Solar radiation also affects the oceans. Warm water from the Equator tends to circulate toward the poles, while cold polar water heads towards the Equator. Oceanic convection is also frequently driven by density differences due to varying salinity, known as thermohaline convection, and is of crucial importance in the global thermohaline circulation. In this case it is quite possible for relatively warm, saline water to sink, and colder, fresher water to rise, reversing the normal transport of heat.

Mantle convection

Convection, within a mantle, can cause continental drift.

Pressure

:For the psychological or political context, see Peer pressure. Pressure (symbol: p) is the force per unit area acting on a surface in a direction perpendicular to that surface. Mathematically: :

p = F/A\,

where p is the pressure, F is the normal force, and A is the area. Pressure is transmitted to solid boundaries or across arbitrary sections of fluid normal to these boundaries or sections at every point. It is a fundamental parameter in thermodynamics and it is conjugate to volume. A closely related quantity is the stress tensor σ which relates the vector force F to the vector area A via :

\mathbf=\mathbf

This tensor may be divided up into a scalar part (pressure) and a traceless tensor part shear. The shear tensor gives the force in directions parallel to the surface, usually due to viscous or frictional forces. The stress tensor is sometimes called the pressure tensor, but in the following, the term "pressure" will refer only to the scalar pressure. shear

Example

As an example of varying pressures, a finger can be pressed against a wall without making any lasting impression; however, the same finger pushing a thumbtack can easily damage the wall. Although the force applied to the surface is the same, the thumbtack applies more pressure because the point concentrates that force into a smaller area. Pressure is transmitted to solid boundaries or across arbitrary sections of fluid normal to these boundaries or sections at every point. Unlike stress, pressure is defined as a scalar quantity. The gradient of pressure is force density. In the human body, baroreceptors monitor blood pressure.

Relative or gauge pressure

For gases, pressure is sometimes measured, not as an absolute pressure, but relative to atmospheric pressure; such measurements are sometimes called gauge pressure. An example of this is the air pressure in a car tire, which might be said to be "220 kPa," but is actually 220 kPa above atmospheric pressure. Since atmospheric pressure at sea level is about 100 kPa, the absolute pressure in the tire is therefore about 320 kPa. In technical work, this is written "a gauge pressure of 220 kPa." Where space is limited, such as on gauges, name plates, graph labels, and table headings, the use of a modifier in parentheses, such as "kPa (gauge)" or "kPa (absolute)," is permitted. In non-SI technical work, a gauge pressure is sometimes written as "32 psig," though the other methods explained above that avoid attaching characters to the unit of pressure are preferred [http://physics.nist.gov/Pubs/SP811/sec07.html#7.4 ¹].

Scalar nature of pressure

In static gas, the gas as a whole does not appear to move, the individual molecules of the gas, which we cannot see, are in constant random motion. Because we are dealing with an extremely large number of molecules and because the motion of the individual molecules is random in every direction, we do not detect any motion. If we enclose the gas within a container, we detect a pressure in the gas from the molecules colliding with the walls of our container. We can put the walls of our container anywhere inside the gas, and the force per unit area (the pressure) is the same. We can shrink the size of our "container" down to an infinitely small point, and the pressure has a single value at that point. Therefore, pressure is a scalar quantity, not a vector quantity. It has a magnitude but no direction associated with it. Pressure acts in all directions at a point inside a gas. At the surface of a gas, the pressure force acts perpendicular to the surface.

Hydrostatic pressure

Hydrostatic pressure is the pressure due to the weight of a fluid. :p = ρgh where ρ (rho) is density of the fluid, g is acceleration due to gravity, and h is height of the fluid above the point being measured. See also Pascal's law.

Stagnation pressure

Stagnation pressure is the pressure a fluid exerts when it is forced to stop moving. Consequently, although a fluid moving at higher speed will have a lower static pressure, it may have a higher stagnation pressure when forced to a standstill. Static pressure and stagnation pressure are related by the Mach number of the fluid. In addition, there can be differences in pressure due to differences in the elevation (height) of the fluid. See Bernoulli's equation. The pressure of a moving fluid can be measured using a Pitot probe, or one of its variations such as a Kiel probe or Cobra probe, connected to a manometer. Depending on where the inlet holes are located on the probe, it can measure static pressure or stagnation pressure.

Units

The SI unit for pressure is the pascal (Pa), equal to one newton per square metre (N·m^-2 or kg·m^-1·s^-2). This special name for the unit was added in 1971; before that, pressure in SI was expressed in units such as N/m². Non-SI measures (still in use in some parts of the world) include the pound-force per square inch (psi) and the bar. The cgs unit of pressure is the barye (ba). It is equal to 1 dyn·cm^-2. Pressure is still sometimes expressed in kgf/cm² or grams-force/cm² (sometimes as kg/cm² and g/cm² without properly identifying the force units). But using the names kilogram, gram, kilogram-force, or gram-force (or their symbols) as a unit of force is expressly forbidden in SI; the unit of force in SI is the newton (N). The technical atmosphere (symbol: at) is 1 kgf/cm². Some meteorologists prefer the hectopascal (hPa) for atmospheric air pressure, which is equivalent to the older unit millibar (mbar). Similar pressures are given in kilopascals (kPa) in practically all other fields, where the hecto prefix is hardly ever used. In Canadian weather reports, the normal unit is kPa. The obsolete unit inch of mercury (inHg) is still sometimes used in the United States. Blood pressure is still measured in millimetres of mercury in most of the world, and lung pressures in centimeters of water are still common. These obsolete manometric units of pressure are based on the pressure exerted by the weight of some "standard" fluid under some "standard" gravity. They are effectively attempts to define a unit for expressing the readings of a manometer. When millimetres or inches of mercury are used today, they have precise definitions that can be expressed in terms of SI units. The water-based units depend on the density of water, a measured, rather than defined, quantity. The standard atmosphere (atm) is an established constant. It is approximately equal to typical air pressure at earth mean sea level and is defined as follows. :standard atmosphere = 101325 Pa = 101.325 kPa = 1013.25 hPa. A rule of thumb commonly used by scuba divers is that one atmosphere is approximately equal to the pressure exerted by ten metres of water. Non-SI units presently or formerly in use include the following.
- atmosphere.
- manometric units:
- centimetre, inch, and millimetre of mercury (Torr).
- millimetre, centimetre, metre, inch, and foot of water.
- imperial units:
- kip, ton-force (short), ton-force (long), pound-force, ounce-force, and poundal per square inch.
- pound-force, ton-force (short), and ton-force (long) per square foot.
- non-SI metric units:
- bar, millibar.
- kilogram-force, or kilopond, per square centimetre (technical atmosphere).
- gram-force and tonne-force (metric ton-force) per square centimetre.
- barye (dyne per square centimetre).
- kilogram-force and tonne-force per square metre.
- sthene per square metre (pieze).

External links

- [http://calc.skyrocket.de/en/ Online unit converter] - conversion of many different units.
- [http://avc.comm.nsdlib.org/cgi-bin/wiki_grade_interface.pl?An_Exercise_In_Air_Pressure An exercise in air pressure]
- [http://www.grc.nasa.gov/WWW/K-12/airplane/pressure.html Pressure being a scalar quantity] Category:Diving Category:Meteorology Category:Physical quantity Category:Thermodynamics ko:압력 ms:Tekanan ja:圧力

Laminar flow

Laminar flow, sometimes known as streamline flow, is when a fluid flows in parallel layers, with no disruption between the layers. In fluid dynamics, laminar flow is a flow regime characterized by high momentum diffusion, low momentum convection, and pressure and velocity independence from time. It is the opposite of turbulent flow. The (dimensionless) Reynolds number characterizes whether flow conditions lead to laminar or turbulent flow. For example, consider the flow of air over an airplane wing. The boundary layer is a very thin sheet of air lying over the surface of the wing (and, for that matter, all other surfaces of the airplane). Because air has viscosity, this layer of air tends to adhere to the wing. As the wing moves forward through the air, the boundary layer at first flows smoothly over the streamlined shape of the airfoil. Here the flow is called the laminar layer. As the boundary layer approaches the centre of the wing, it begins to lose speed due to skin friction, and it becomes thicker and turbulent. Here it is called the turbulent layer. The process of a laminar boundary layer becoming turbulent is known as boundary layer transition.The point at which the boundary layer changes from laminar to turbulent is called the transition point. Where the boundary layer becomes turbulent, drag due to skin friction is relatively high. As speed increases, the transition point tends to move forward. As the angle of attack increases, the transition point also tends to move forward. One way to limit the size and effect of the turbulent region is to use swept-back "delta" wings. This is particularly important in supersonic aircraft.

Experiments

A famous experiment involving laminar flow uses two concentric glass cylinders with the gap filled with glycerin and a drop of ink placed in the fluid. When the outer cylinder is turned, the drop is drawn out into a thread that eventually becomes so thin that it disappears from view. At this point the ink molecules are said to be "enfolded" in the glycerin. If the cylinder is then turned in the opposite direction, the thread reforms and then becomes a drop. This experiment is typically used to show implicit order, but also nicely demonstrates the properties of laminar flow.

Reynolds number

The Reynolds number is the most important dimensionless number in fluid dynamics and provides a criterion for determining dynamic similitude. Where two similar objects in perhaps different fluids with possibly different flowrates have similar fluid flow around them, they are said to be dynamically similar. It is named after Osborne Reynolds (1842–1912), who proposed it in 1883. Typically it is given as follows: :

\mathit =

or :

\mathit =  \; .

With:
- v_s - mean fluid velocity,
- L - characteristic length (equal to diameter 2r if a cross-section is circular),
- μ - (absolute) dynamic fluid viscosity,
- ν - kinematic fluid viscosity: ν = μ / ρ,
- ρ - fluid density. The Reynolds number is the ratio of inertial forces (v_sρ) to viscous forces (μ/L) and is used for determining whether a flow will be laminar or turbulent. Laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion, while turbulent flow, on the other hand, occurs at high Reynolds numbers and is dominated by inertial forces, producing random eddies, vortices and other flow fluctuations. The transition between laminar and turbulent flow is often indicated by a critical Reynolds number (Re_crit), which depends on the exact flow configuration and must be determined experimentally. Within a certain range around this point there is a region of gradual transition where the flow is neither fully laminar nor fully turbulent, and predictions of fluid behaviour can be difficult. For example, within circular pipes the critical Reynolds number is generally accepted to be 2300, where the Reynolds number is based on the pipe diameter and the mean velocity v_s within the pipe, but engineers will avoid any pipe configuration that falls within the range of Reynolds numbers from about 2000 to 4000 to ensure that the flow is either laminar or turbulent.

The similarity of flows

In order for two flows to be similar they must have the same geometry and equal Reynolds numbers. When comparing fluid behaviour at homologous points in a model and a full-scale flow, the following holds: :

\mathit^ = \mathit \; , \quad\quad   =  \; ,

where quantities marked with
- concern the flow around the model and the others the real flow. This allows us to perform experiments with reduced models in water channels or wind tunnels, and correlate the data to the real flows. Note that true dynamic similarity may require matching other dimensionless numbers as well, such as the Mach number used in compressible flows, or the Froude number that governs free-surface flows. Some flows involve more dimensionless parameters than can be practically satisfied with the available apparatus and fluids (preferably air or water), so one is forced to decide which parameters are most important. This is why good experimental modelling requires a fair amount of experience and good judgement.

Reynolds number sets the smallest scales of turbulent motion

In a turbulent flow, there is a range of scales of the fluid motions, sometimes called eddies. A single packet of fluid moving with a bulk velocity is called an eddy. The size of the largest scales (eddies) are set by the overall geometry of the flow. For instance, in an industrial smoke-stack, the largest scales of fluid motion are as big as the diameter of the stack itself. The size of the smallest scales is set by the Reynolds number. As Reynolds number increases, smaller and smaller scales of the flow are visible. In the smoke-stack, the smoke may appear to have many very small bumps or eddies, in addition to large bulky eddies. In this sense, the Reynolds number is an indicator of the range of scales in the flow. The higher the Reynolds number, the greater the range of scales. What is the explanation for this phenomenon? A large Reynolds number indicates that viscous forces are not important to the flow. With a low level of viscosity, the smallest scales of fluid motion are undamped -- there is not enough viscosity to dissipate their motions. In contrast, a low Reynolds number indicates that viscosity is important to the flow dynamics. The smallest scales are damped out, and only the larger scales remain. Next time you look at a turbulent flow, try to pick out the smallest and biggest scales of fluid motion. Is the Reynolds number big or small?

Example on the importance of Reynolds number

If an aeroplane needs testing of its wing, one can make a scaled down small model of the wing and test the wing as a table top model in the lab with the same Reynolds number the actual air plane is subjected to. The results of the lab model will be similar to that of the actual plane wing results. Thus we need not bring a plane into the lab to test it actually. This is the example of "dynamic similarity." This is what Reynolds number is all about.

Typical values of Reynolds number

- Spermatazoa ~ 1×10⁻²
- Blood flow in brain ~ 1×10²
- Blood flow in aorta ~ 1×10³ Onset of turbulent flow ~ 2.3×10³
- Person swimming ~ 4×10⁶
- Aircraft ~ 1×10⁷
- Blue whale ~ 3×10⁸
- A large ship (RMS Queen Elizabeth 2) ~ 5×10⁹

References

- Rott, N., “Note on the history of the Reynolds number,” Annual Review of Fluid Mechanics, Vol. 22, 1990, pp. 1–11.
- Zagarola, M.V. and Smits, A.J., “Experiments in High Reynolds Number Turbulent Pipe Flow.” AIAApaper #96-0654, 34th AIAA Aerospace Sciences Meeting, Reno, Nevada, January 15 - 18, 1996.
- Jermy M., “Fluid Mechanics A Course Reader,” Mechanical Engineering Dept., University of Canterbury, 2005, pp. d5.10.

External links

- [http://www-rcf.usc.edu/~alexeenk/GDT/index.html Gas Dynamics Toolbox] Calculate Reynolds number for mixtures of gases using VHS model Category:Dimensionless numbers Category:Fluid dynamics ja:レイノルズ数

Andrey Kolmogorov

Andrey Nikolaevich Kolmogorov (Андре́й Никола́евич Колмого́ров) (kahl-mah-GAW-raff) (April 25, 1903 in Tambov - October 20, 1987 in Moscow) was a Soviet mathematician who made major advances in the fields of probability theory and topology. He worked early in his career on intuitionistic logic, and Fourier series. He also worked on turbulence, and classical mechanics; and was a founder of algorithmic complexity theory. Kolmogorov worked at Moscow State University. He studied under Nikolai Luzin, earning his Ph.D. in 1925, in 1931 he became the professor of the university. In 1939 he received the title of academician of the USSR Academy of Sciences. Quote: :"The theory of probability as mathematical discipline can and should be developed from axioms in exactly the same way as Geometry and Algebra."

His portraits by Dmitry Gordeev

Bibliography

- Selected works of A.N. Kolmogorov / edited by V.M. Tikhomirov; translated from the Russian by V.M. Volosov. 3 volumes. Dordrecht; Boston : Kluwer Academic Publishers, c1991-c1993 ISBN 9027727953 .

External links

- [http://www.kolmogorov.com/ The Legacy of Andrei Nikolaevich Kolmogorov] Curriculum Vitae and Biography. Kolmogorov School. Ph.D. students and descendants of A.N. Kolmogorov. A.N. Kolmogorov works, books, papers, articles. Photographs and Portraits of A.N. Kolmogorov.
- [http://www.probabilityandfinance.com/articles/04.pdf The origins and legacy of Kolmogorov's Grundbegriffe]
-
- [http://www.geometry.net/scientists/kolmogorov_andrey.php Collection of links to Kolmogorov resources]
- [http://www.kolmogorov.pms.ru/ Andrei Nikolaevich Kolmogorov] (in Russian)
- [http://www.pms.ru/ Kolmogorov School] at Moscow University
- Kolmogorov, Andrey Kolmogorov, Andrey Kolmogorov, Andrey Kolmogorov, Andrey Nikolaevich Kolmogorov, Andrey Nikolaevich ja:アンドレイ・コルモゴロフ th:อันเดรย์ คอลโมโกรอฟ

Chaos theory

:For the album by Jumpsteady, see Chaos Theory (album). : For the video game, see Splinter Cell: Chaos Theory. In mathematics and physics, chaos theory deals with the behavior of certain nonlinear dynamical systems that (under certain conditions) exhibit the phenomenon known as chaos, most famously characterised by sensitivity to initial conditions (see butterfly effect). As a result of this sensitivity, the observed behavior of physical systems that exhibit chaos appears to be random, even though the model of the system is 'deterministic' in the sense that it is well defined and contains no random parameters. Examples of such systems include the atmosphere, the solar system, plate tectonics, turbulent fluids, economies, and population growth. turbulent Systems that exhibit mathematical chaos are deterministic and thus orderly in some sense; this technical use of the word chaos is at odds with common parlance, which suggests complete disorder. See the article on chaos for a discussion of the origin of the word in mythology, and other uses. When it is said that chaos theory studies deterministic systems, it is necessary to mention a related field of physics called quantum chaos theory which studies non-deterministic systems following the laws of quantum mechanics.

Description of the theory

A non-linear dynamical system can, in general, exhibit one or more of the following types of behavior:
- forever at rest
- forever expanding (only for unbounded systems)
- periodic motion
- quasi-periodic motion
- chaotic motion The type of behavior a system may exhibit depends on the initial state of the system and the values of its parameters, if any. The most difficult type of behavior to characterize and predict is chaotic motion, a non-periodic complex motion which has given name to the theory.

Chaotic motion

In order to classify the behavior of a system as chaotic, the system must exhibit the following properties:
- it must be sensitive to initial conditions
- it must be topologically transitive
- its periodic orbits must be dense Sensitivity to initial conditions means that two points in such a system may move in vastly different trajectories in their phase space even if the difference in their initial configurations is very small. The systems behave identically only if their initial configurations were exactly the same. An example of such sensitivity is the so-called "butterfly effect", whereby the flapping of a butterfly's wings is imagined to create tiny changes in the atmosphere which over the course of time cause it to diverge from what it would have been and potentially cause something as dramatic as a tornado to occur. The butterfly flapping its wings represents a small change in the initial condition of the system which causes a chain of events leading to large-scale phenomena like tornadoes. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different. Other commonly-known examples of chaotic motion are the mixing of colored dyes and airflow turbulence. Sensitivity to initial conditions is related to the Lyapunov exponent. Transitivity means that application of the transformation on any given Interval

I_1

stretches it until it overlaps with any other given Interval

I_2

. Transitivity, dense periodic points, and sensitivity to initial conditions can all be extended to an arbitrary metric space. J. Banks and colleagues showed in 1992 that in the setting of a general metric space, transitivity and dense periodic points together imply sensitivity to initial conditions. This elementary but unexpected fact prompted Bau-Sen Du, of the Institute of Mathematics, Academia Sinica, Taiwan to define a stronger version of sensitive dependence - extreme sensitive dependence - which is not a consequence of transitivity and dense periodic points. Extreme sensitive dependence means, roughly, that points close together separate and converge infinitely often, as is often the case in examples of chaotic dynamical systems.

Attractors

One way of visualizing chaotic motion, or indeed any type of motion, is to make a phase diagram of the motion. In such a diagram time is implicit and each axis represents one dimension of the state. For instance, one might plot the position of a pendulum against its velocity. A pendulum at rest will be plotted as a point and a one in periodic motion will be plotted as a simple closed curve. When such a plot forms a closed curve, the curve is called an orbit. Our pendulum has an infinite number of such orbits, forming a pencil of nested ellipses about the origin. Often phase diagrams reveal that most state trajectories wind up approaching some common limit. The system ends up doing the same motion for all initial states in a region around the motion, almost as though the system is attracted to that motion. Such attractive motion is fittingly called an attractor for the system and is very common for forced dissipative systems. For instance, if we attach a damper to our pendulum, no matter what its initial position and velocity it will wind up being at rest - or more correctly: it will reach rest at the limit. The trajectories on the phase diagram will all spiral in towards the middle, rather than forming sets of ovals. This point in the middle - the state when the pendulum is at rest - is called an "attractor". Attractors are often associated with dissipative systems like this, where some element (the damper) dissipates energy. Such an attractor may be called a "point attractor". Not all attractors are points. Some are simple loops, or more complex doubled loops (for which you need more than two degrees of freedom). And some are actually fractals: the so called "strange attractors". Systems with loop attractors exhibit periodic motion. Those with more complex split loops tend to exhibit quasiperiodic motion. And systems with strange attractors tend to exhibit chaotic behavior. At any point on the phase diagram, the system will tend to evolve to another neighbouring state in some sort of deterministic way. If our pendulum is at a particular position and travelling with a particular velocity, we can calculate what its (infinitesimally) "next" position and velocity will be. That is, we can treat our phase diagram as being a vector field, and use vector calculus to understand it. Attractors in our phase diagram are simply those regions with a negative divergence.

Strange attractors

While most of the motion types mentioned above give rise to very simple attractors, such as points and circle-like curves called limit cycles, chaotic motion gives rise to what are known as strange attractors, attractors that can have great detail and complexity. For instance, a simple three-dimensional model of the Lorenz weather system gives rise to the famous Lorenz attractor. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because not only was it one of the first, but it is one of the most complex and as such gives rise to a very interesting pattern which looks like the wings of a butterfly. Another such attractor is the Rössler Map, which experiences period-two doubling route to chaos, like the logistic map. Strange attractors occur in both continuous dynamical systems (such as the Lorenz system) and in some discrete systems (such as the Hénon map). Other discrete dynamical systems have a repelling structure called a Julia set which forms at the boundary between basins of attraction of fixed points - Julia sets can be thought of as strange repellers. Both strange attractors and Julia sets typically have a fractal structure. The Poincaré-Bendixson theorem shows that a strange attractor can only arise in a continuous dynamical system if it has three or more dimensions. However, no such restriction applies to discrete systems, which can exhibit strange attractors in two or even one dimensional systems.

History

The roots of chaos theory date back to about 1900, in the studies of Henri Poincaré on the problem of the motion of three objects in mutual gravitational attraction, the so-called three-body problem. Poincaré found that there can be orbits which are nonperiodic, and yet not forever increasing nor approaching a fixed point. Later studies, also on the topic of nonlinear differential equations, were carried out by G.D. Birkhoff, A.N. Kolmogorov, M.L. Cartwright, J.E. Littlewood, and Stephen Smale. Except for Smale, who was perhaps the first pure mathematician to study nonlinear dynamics, these studies were all directly inspired by physics: the three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case of Kolmogorov, and radio engineering in the case of Cartwright and Littlewood. Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio circuits without the benefit of a theory to explain what they were seeing. Chaos theory progressed more rapidly after mid-century, when it first became evident for some scientists that linear theory, the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments like that of the logistic map. The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand. Electronic computers made these repeated calculations practical. One of the earliest electronic digital computers, ENIAC, was used to run simple weather forecasting models. An early pioneer of the theory was Edward Lorenz whose interest in chaos came about accidentally through his work on weather prediction in 1961. Lorenz was using a basic computer, a Royal McBee LGP-30, to run his weather simulation. He wanted to see a sequence of data again and to save time he started the simulation in the middle of its course. He was able to do this by entering a printout of the data corresponding to conditions in the middle of his simulation which he had calculated last time. To his surprise the weather that the machine began to predict was completely different to the weather calculated before. Lorenz tracked this down to the computer printout. The printout rounded variables off to a 3-digit number, but the computer worked with 6-digit numbers. This difference is tiny and the consensus at the time would have been that it should have had practically no effect. However Lorenz had discovered that small changes in initial conditions produced large changes in the long-term outcome. The term chaos as used in mathematics was coined by the applied mathematician James A. Yorke. The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos theory continues to be a very active area of research.

Mathematical theory

Mathematicians have devised many additional ways to make quantitative statements about chaotic systems. These include:
- fractal dimension of the attractor
- Lyapunov exponents
- recurrence plots
- Poincaré maps
- bifurcation diagrams
- Transfer operator

Minimum complexity of a chaotic system

Many simple systems can also produce chaos without relying on differential equations, such as the logistic map, which is a difference equation (recurrence relation) that describes population growth over time. Even discrete systems, such as cellular automata, can heavily depend on initial conditions. Stephen Wolfram has investigated a cellular automaton with this property, termed by him rule 30.

Other examples of chaotic systems

- Double pendulum
- Logistic map
- Hénon map
- Lorenz model
- Smale horseshoe
- Dynamical billiards

References

Textbooks and technical works

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-
-
-
-
-
-
-
- "Wave Propagation in Ray-Chaotic Enclosures: Paradigms, Oddities and Examples", Vincenzo Galdi, et. al., IEEE Antennas and Propagation Magazine, February 2005, p. 62

Semitechnical and popular works

- The Beauty of Fractals, by H.-O. Peitgen and P.H. Richter
- Chance and Chaos, by David Ruelle
- Computers, Pattern, Chaos, and Beauty, by Clifford A. Pickover
- Fractals, by Hans Lauwerier
- Fractals Everywhere, by Michael Barnsley
- Order Out of Chaos, by Ilya Prigogine and Isabelle Stengers
- Chaos and Life, by Richard J Bird
- Does God Play Dice?, by Ian Stewart
- The Science of Fractal Images, by Heinz-Otto Peitgen and Dietmar Saupe, Eds.
- Explaining Chaos, by Peter Smith
- Chaos, by James Gleick
- Complexity, by M. Mitchell Waldrop
- Chaos, Fractals and Self-organisation, by Arvind Kumar
- Chaotic Evolution and Strange Attractors, by David Ruelle
- Sync: The emerging science of spontaneous order, by Steven Strogatz
- The Essence of Chaos, by Edward Lorenz
- Deep Simplicity, by John Gribbin

Popular culture

- Ian Malcolm, a character from the movie and book Jurassic Park, was a chaos theory mathematician.

External links

- http://www.nbi.dk/ChaosBook/
- [http://www.libraryreference.org/chaos.html Chaos Theory and Education]
- [http://www.imho.com/grae/chaos/chaos.html Chaos Theory: A Brief Introduction]
- [http://www.ae.uiuc.edu/lndvl Linear and Nonlinear Dynamics and Vibrations Laboratory at the University of Illinois]
- [http://hypertextbook.com/chaos/ The Chaos Hypertextbook]. An introductory primer on chaos and fractals.
- Chaos Theory in the Social Sciences, edited by L Douglas Kiel, Euel W Elliott.
- [http://www.cut-the-knot.org/blue/chaos.shtml Emergence of Chaos] at cut-the-knot
- Category:Non-linear systems ko:혼돈 이론 ja:カオス理論 simple:Chaos theory th:ทฤษฎีความอลวน

Drag (physics)

:This page is about forces which tend to slow a moving object. For other uses, see Drag (disambiguation). For a solid object moving through a fluid or gas, drag is the sum of all the aerodynamic or hydrodynamic forces in the direction of the external fluid flow. It therefore acts to oppose the motion of the object, and in a powered vehicle it is overcome by thrust. Types of drag are generally divided into three categories: parasitic drag, lift-induced drag and wave drag. Parasitic drag includes form drag, skin friction and interference drag. Lift-induced drag is only relevant when wings or a lifting body are present, and is therefore usually discussed only in the aviation perspective of drag. Beyond these two kinds of drag there is a third kind of drag, called wave drag, that occurs when the solid object is moving through the fluid at or near the speed of sound in that fluid. The overall drag of an object is characterized by a dimensionless number called the drag coefficient, and is calculated using the drag equation. Assuming a constant drag coefficient, drag will vary as the square of velocity. Thus, the resultant power needed to overcome this drag will vary as the cube of velocity. The standard equation for drag is one half the coefficient of drag multiplied by the fluid density, the cross sectional area of your specified item, and the square of the velocity Wind resistance is a layman's term used to describe drag. Its use is often vague, and is usually used in a relative sense (e.g. A badminton shuttlecock has more wind resistance than a squash ball).

Boundary layer

In physics and fluid mechanics, the boundary layer is that layer of fluid in the immediate vicinity of a bounding surface. In the atmosphere the boundary layer is the air layer near the ground affected by diurnal heat, moisture or momentum transfer to or from the surface. On an aircraft wing the boundary layer is the part of the flow close to the wing. The Boundary layer effect occurs at the field region in which all changes occur in the flow pattern. The boundary layer distorts surrounding nonviscous flow. It is a phenomenon of viscous forces. This effect is related to the Leidenfrost effect and the Reynolds number.

Aerodynamics

The aerodynamic boundary layer was discovered by Ludwig Prandtl at the beginning of the twentieth century and represents one of the greatest discoveries in the history of aerodynamics. It is particularly important in aerodynamics because it is directly responsible for the drag experienced by a body immersed in a fluid. In high-performance designs, such as sailplanes and commercial transport aircraft, much attention is paid to controlling the behavior of the boundary layer to minimize drag. Two effects must to be considered. First, the boundary layer adds to the effective thickness of the body, through the displacement thickness, hence increasing the pressure drag. Secondly, the shear forces at the surface of the wing create skin friction drag. At high Reynolds numbers, typical of full-sized aircraft, it is desirable to have a laminar boundary layer. This results in a lower skin friction due to the characteristic velocity profile of laminar flow. However, the boundary layer inevitably thickens and becomes less stable as the flow develops along the body, and eventually becomes turbulent, the process known as boundary layer transition. One way of dealing with this problem is to suck the boundary layer away through a porous surface. This can result in a reduction in drag, but is usually impractical due to the mechanical complexity involved. At lower Reynolds numbers, such as those seen with model aircraft, it is relatively easy to maintain laminar flow. This gives low skin-friction, which is desirable. However, the same velocity profile which gives the laminar boundary layer its low skin friction also causes it to be badly affected by adverse pressure gradients. As the pressure begins to recover over the rear part of the wing chord, a laminar boundary layer tends to separate from the surface. Such separation causes a large increase in the pressure drag, since it greatly increases the effective size of the wing section. In these cases, it can be advantageous to deliberately trip the boundary layer into turbulence at a point prior to the location of laminar separation, using a turbulator. The fuller velocity profile of the turbulent boundary layer allows it to sustain the adverse pressure gradient without separating. Thus, although the skin friction is increased, overall the drag is decreased. Special wing sections have also been designed which tailor the pressure recovery so that laminar separation is reduced or even eliminated. This represents an optimum compromise between the pressure drag from flow separation and skin friction the induced turbulence

Boundary layer equations

The deduction of the boundary layer equations was perhaps one of the most important advances in aerodynamics. Using an order of magnitude analysis, the well-known governing Navier-Stokes equations of viscous fluid flow can be greatly simplified within the boundary layer. Notably, the characteristic of the partial differential equations (PDE) becomes parabolic, rather than the elliptical form of the full Navier-Stokes equations. This greatly simplifies the solution of the equations. By making the boundary layer approximation, the flow is divided into an inviscid portion (which is easy to solve by a number of methods) and the boundary layer, which is governed by an easier to solve PDE. The Navier-Stokes equations for a two-dimensional steady incompressible flow in cartesian coordinates are given by :

+=0

u+v=- +(+)

u+v=- +(+)

where

\nu

is the kinematic viscosity of the fluid at a point; The approximation states that, for a sufficiently high Reynolds number the flow over a surface can be divided into an outer region of inviscid flow unaffected by viscosity (the majority of the flow), and a region close to the surface where viscosity is important (the boundary layer). Let

u

and

v

be streamwise and transverse(wall normal) velocities respectively inside the boundary layer. Using asymtotic analysis, it can be shown that the above equations of motion reduce within the boundary layer to become :

+=0

u+v=- +

and the remarkable result that :

=0

The asymptotic analysis also shows that

v

, the wall normal velocity, is small compared with

u

the streamwise velocity, and that variations in properties in the streamwise direction are generally much lower than those in the wall normal direction. Since the static pressure

p

is independent of

y

, then pressure at the edge of the boundary layer is the pressure througout the boundary layer at a given streamwise position. The external pressure may be obtained through an application of Bernoulli's Equation. Let

u_0

be the fluid velocity outside the boundary layer, where

u

and

u_0

are both parallel. This gives upon substituting for

p

the following result :

u+v=u_0+

with the boundary condition :

+=0

For a flow in which the static pressure

p

also does not change in the direction of the flow then :

=0

u_0

remains constant. Therefore, the equation of motion simplifies to become :

u+v=

These approximations are used in a variety of practical flow problems of scientific and engineering interest. The above analysis is for any instantaneous laminar or turbulent boundary layer, but are used mainly in laminar flow studies since the mean flow is also the instantaneous flow because there are no velocity fluctuations present.

Turbulent boundary layers

The treatment of turbulent boundary layers is far more difficult due to the time-dependent variation of the flow properties. One of the most widely used techniques in which turbulent flows are tackled is to apply Reynolds decomposition. Here the instantaneous flow properties are decomposed into a mean and fluctuating component. Applying this technique to the boundary layer equations give the full turbulent boundary layer equations not often given in literature, viz. :

+=0

\overline+\overline=- +(+)-\frac(\overline)-\frac(\overline)

\overline+\overline=- +(+)-\frac(\overline)-\frac(\overline)

Using the same order-of-magnitude analysis as for the instantaneous equations, these turbulent boundary layer equations generally reduce to become in their classical form: :

+=0

\overline+\overline=- +-\frac(\overline)

=0

The additional term

\overline

in the turbulent boundary layer equations is known as the Reynolds shear stress and is unknown a priori. The solution of the turbulent boundary layer equations therefore necessitate the use of a turbulence model, which aims to express the Reynolds shear stress in terms of known flow variables or derivatives. The lack of accuracy and generality of such models is the single major obstacle which inhibits the successful prediction of turbulent flow properties in modern aerodynamics.

Boundary layer turbine

This effect was exploited in the Tesla turbine, patented by Nikola Tesla in 1913. It is referred to as a bladeless turbine because it uses the boundary layer effect and not a fluid impinging upon the blades as in a conventional turbine. Boundary layer turbines are also known as cohesion-type turbines, bladeless turbine, and Prandtl layer turbine (after Ludwig Prandtl).

External links

- [http://www.nsdl.arm.gov/Library/glosInsert non-formatted text heresary.shtml#boundary_layer National Science Digital Library - Boundary Layer]
- Moore, Franklin K., "[http://naca.larc.nasa.gov/reports/1953/naca-report-1124/ Displacement effect of a three-dimensional boundary layer]". NACA Report 1124, 1953.
- Benson, Tom, "[http://www.grc.nasa.gov/WWW/K-12/airplane/boundlay.html Boundary layer]". NASA Glenn Learning Technologies.
- [http://www.maths.man.ac.uk/~ruban/blsep.html Boundary layer separation]
- [http://eqworld.ipmnet.ru/en/solutions/npde/npde-toc5.pdf Boundary layer equations: Exact Solutions] - from EqWorld

Bibliography

- A.D. Polyanin and V.F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton - London, 2004. ISBN 1-58488-355-3
- A.D. Polyanin, A.M. Kutepov, A.V. Vyazmin, and D.A. Kazenin, Hydrodynamics, Mass and Heat Transfer in Chemical Engineering, Taylor & Francis, London, 2002. ISBN 0-415-27237-8
- Herrmann Schlichting, Klaus Gersten, E. Krause, H. Jr. Oertel, C. Mayes "Boundary-Layer Theory" 8th edition Springer 2004 ISBN: 3540662707 Category:Fluid dynamics ja:境界層

Viscosity

Viscosity is a measure of the resistance of a fluid to deformation under shear stress. It is commonly perceived as "thickness", or resistance to pouring. Viscosity describes a fluid's internal resistance to flow and may be thought of as a measure of fluid friction. Thus, water is "thin", having a low viscosity, while vegetable oil is "thick" having a high viscosity.

Newton's theory

When a shear stress is applied to a solid body, the body deforms until the deformation results in an opposing force to balance that applied, an equilibrium. However, when a shear stress is applied to a fluid, such as a wind blowing over the surface of the ocean, the fluid flows, and continues to flow while the stress is applied. When the stress is removed, in general, the flow decays due to internal dissipation of energy. The "thicker" the fluid, the greater its resistance to shear stress and the more rapid the decay of its flow. In general, in any flow, layers move at different velocities and the fluid's "thickness" arises from the shear stress between the layers that ultimately opposes any applied force. velocities velocities
Isaac Newton postulated that, for straight, parallel and uniform flow, the shear stress, τ, between layers is proportional to the velocity gradient, ∂u/∂y, in the direction perpendicular to the layers, in other words, the relative motion of the layers. :

\tau=\mu \frac

. Here, the constant μ is known as the coefficient of viscosity, viscosity, or dynamic viscosity. Many fluids, such as water and most gases, satisfy Newton's criterion and are known as Newtonian fluids. Non-Newtonian fluids exhibit a more complicated relationship between shear stress and velocity gradient than simple linearity. In many situations, we are concerned with the ratio of the viscous force to the inertial force, the latter characterised by the fluid density ρ. This ratio is characterised by the kinematic viscosity, defined as follows: :

\nu = \frac

. James Clerk Maxwell called viscosity fugitive elasticity because of the analogy that elastic deformation opposes shear stress in solids, while in viscous fluids, shear stress is opposed by rate of deformation. Viscosity is the principal means by which energy is dissipated in fluid motion, typically as heat.

Measurement of viscosity

Viscosity is measured with various types of viscometer, typically at 25°C (standard state).For the most of the fluids is a constant in a wide range of shear rates. The fluids without a constant viscosity are called Non-Newtonian fluids.

Units

Viscosity (dynamic viscosity)

The SI physical unit of dynamic viscosity (greek symbol: μ) is the pascal-second (Pa·s), which is identical to 1 N·s/m² or 1 kg/(m·s). In France there have been some attempts to establish the poiseuille (Pl) as a name for the Pa·s but without international success. Care must be taken in not confusing the poiseuille with the poise named after the same person! The cgs physical unit for dynamic viscosity is the poise (P) named after Jean Louis Marie Poiseuille. It is more commonly expressed, particularly in ASTM standards, as centipoise (cP). The centipoise is commonly used because water has a viscosity of 1.0020 cP (at 20 °C; the closeness to one is a convenient coincidence). 1 poise = 100 centipoise = 1 g/(cm·s) = 0.1 Pa·s. 1 centipoise = 1 mPa·s.

Kinematic viscosity

Kinematic viscosity (greek symbol: ν) has SI units (m²/s). The cgs physical unit for kinematic viscosity is the stokes (abbreviated S or St), named after George Gabriel Stokes . It is sometimes expressed in terms of centistokes (cS or cSt). In U.S. usage, stoke is sometimes used as the singular form. 1 stokes = 100 centistokes = 1 cm²/s = 0.0001 m²/s.

Molecular origins

The viscosity of a system is determined by how molecules constituting the system interact. There are no simple but correct expressions for the viscosity of a fluid. The simplest exact expressions are the Green-Kubo relations for the linear shear viscosity or the Transient Time Correlation Function expressions derived by Evans and Morriss in 1985. Although these expressions are each exact in order to calculate the viscosity of a dense fluid, using these relations requires the use of molecular dynamics computer [http://rsc.anu.edu.au/~evans/evansmorrissbook.htm simulation].

Gases

Viscosity in gases arises principally from the molecular diffusion that transports momentum between layers of flow. The kinetic theory of gases allows accurate prediction of the behaviour of gaseous viscosity, in particular that, within the regime where the theory is applicable:
- Viscosity is independent of pressure; and
- Viscosity increases as temperature increases.

Liquids

In liquids, the additional forces between molecules become important. This leads to an additional contribution to the shear stress though the exact mechanics of this are still controversial. Thus, in liquids:
- Viscosity is independent of pressure (except at very high pressure); and
- Viscosity tends to fall as temperature increases (for example, water viscosity goes from 1.79 cP to to 0.28 cP in the temperature range from 0°C to 100°C); see temperature dependence of liquid viscosity for more details. The dynamic viscosities of liquids are typically several orders of magnitude higher than dynamic viscosities of gases.

Viscosity of some common materials

Some dynamic viscosities of Newtonian fluids are listed below: Gases (at 0 °C): Liquids (at 20 °C): Fluids with variable compositions, such as honey, can have a wide range of viscosities.

Can solids have a viscosity?

It is commonly asserted that amorphous solids, such as glass, have viscosity, arguing on the basis that all solids flow, to some possibly minuscule extent, in response to shear stress. Advocates of such a view hold that the distinction between solids and liquids is unclear and that solids are simply liquids with a very high viscosity, typically greater than 10¹² Pa·s. This position is often adopted by supporters of the widely held idea that glass flow can be observed in old buildings. However, others argue that solids are, in general, elastic for small stresses while fluids are not. Even if solids flow at higher stresses, they are characterized by their low-stress behavior. Viscosity may be an appropriate characteristic for solids in a plastic regime. The situation becomes somewhat confused as the term viscosity is sometimes used for solid materials, for example Maxwell materials, to describe the relationship between stress and the rate of change of strain, rather than rate of shear. These distinctions may be largely resolved by considering the constitutive equations of the material in question, which take into account both its viscous and elastic behaviors. Materials for which both their viscosity and their elasticity are important in a particular range of deformation and deformation rate are called viscoelastic. In geology, earth materials that exhibit viscous deformation at least three times greater than their elastic deformation are sometimes called rheids. One example of solids flowing which has been observed since 1930 is the Pitch drop experiment.

Bulk viscosity

The trace of the stress tensor is often identified with the negative of one third of the thermodynamic pressure, which only depends upon the equilibrium state potentials like temperature and density. However, in general, the trace of the stress tensor is the sum of thermodynamic pressure contribution plus another contribution which is proportional to the divergence of the velocity field. This constant of proportionality is called the bulk viscosity.

Eddy viscosity

In the study of turbulence in fluids, a common practical strategy for calculation is to ignore the small-scale vortices (or eddies) in the motion and to calculate a large-scale motion with an eddy viscosity that characterizes the transport and dissipation of energy in the smaller-scale flow. Typical values of eddy viscosity used in modeling ocean circulation are in excess of 10⁷ Pa·s.

Fluidity

The reciprocal of viscosity is fluidity, usually symbolised by φ (=1/μ) or F (=1/η), depending on the convention used, measured in reciprocal poise (cm·s/g), sometimes called the rhe. Fluidity is seldom used in engineering practice. The concept of fluidity can be used to determine the viscosity of an ideal solution. For two components (a and b), the fluidity of a solution of a and b is: F ≈ [χ(a)F(a)] + [χ(b)F(b)] which is only slightly simpler than the equivalent equation in terms of viscosity: η ≈ 1/[χ(a)/η(a) +χ(b)/η(b)] Where χ = mole fration of a or b and η = the viscosity of pure a or b

Etymology

The word "viscosity" derives from the Latin word "viscum" for mistletoe. A viscous glue was made from mistletoe berries and used for lime-twigs to catch birds.

External links

- [http://www.unitconversion.org/unit_converter/viscosity-dynamic.html Online Dynamic Viscosity Converter] - convert between various units of dynamic viscosity, such as pascal second, kilogram-force second/square meter, pound-force second/square inch, poise, and so on
- [http://www.unitconversion.org/unit_converter/viscosity-dynamic-v.html Interactive Dynamic Viscosity Conversion Table] - convert selected unit to all other units of dynamic viscosity
- [http://www.unitconversion.org/unit_converter/viscosity-kinematic.html Online Kinematic Viscosity Converter] - convert between various units of kinematic viscosity, such as square meter/second, square foot/second ,stokes, and so on
- [http://www.unitconversion.org/unit_converter/viscosity-kinematic-v.html Interactive Kinematic Viscosity Conversion Table] - convert selected unit to all other units of kinematic viscosity
- [http://www-rcf.usc.edu/~alexeenk/GDT/index.html Gas Dynamics Toolbox] Calculate coefficient of viscosity for mixtures of gases using VHS model

Bibliography

- Massey, B S (1983) Mechanics of Fluids, fifth edition, ISBN 0442305524 Download free Viscosity- und Rheology E-book in English and German (PDF files):
- [http://www.haake.de/info/Rheology_GSchr_E.pdf Introduction to Rheology by Gebhard Schramm in English language]
- [http://www.haake.de/info/Rheology_GSchr_D.pdf Einführung in die Rheologie von Gebhard Schramm in deutscher Sprache] Category:Continuum mechanics Category:Fluid dynamics Category:Chemical properties Category:Physical quantity Category:Fluid mechanics ms:Kelikatan ja:粘度

Faucet

A tap is a valve for controlling the release of a liquid or gas. In British English the word is used for any everyday type of valve, particularly the fittings on bathtubs and sinks that would be called faucets elsewhere. In American English the usage is sometimes more specialised, with the term tap restricted to uses such as beer taps and the word "faucet" used for cold water outlets; although some Americans use "tap" in the broader sense as well.

Water taps

American English Water for baths, sinks and basins can be provided by separate hot and cold taps; this arrangement is common in the UK, particularly in bathrooms. In kitchens, and in the US and many other places, mixer taps are used instead. This is a single, more complex, valve whose handle moves up and down to control the amount of water flow and from side to side to control the temperature of the water (achieved by mixing the hot and cold water together). Latest designs do this using a built in thermostat. If separate taps are fitted, it may not be immediately clear which tap is hot and which is cold. The hot tap generally has a red indicator and/or be labeled H or Hot. The cold tap tends to have a blue or green indicator and/or be labeled C or Cold. (Note that the French for 'hot' is chaud, which starts with a 'C'.) Mixer taps may have a red-blue stripe or arrows indicating which side will give hot and which cold. In some countries there is a 'standard' arrangement of hot/cold taps: for example in the United States the hot tap is generally on the left. This convention applies in the UK too, but many installations exist where it has been ignored.

Beer taps

While in other contexts, depending on location, a "tap" may be a "faucet", "valve" or "spigot", the use of "tap" for beer is almost universal. This may be because the word was originally coined for the wooden valve in traditional barrels. A "beer tap" now may be one of several items: ; Pressure-dispense bar tap : Almost universally in modern times, bulk beer is supplied in kegs that are served with the aid of external pressure. In a normal bar dispense system, this pressure comes from a cylinder of carbon dioxide (or occasionally nitrogen) which forces the beer out of the keg and up a narrow tube to the bar. At the end of this tube is a valve built into a fixture (usually somewhat decorative) on the bar. This is the beer tap, and opening it with a small lever causes beer, pushed by the gas from the cylinder, to flow into the glass. ; Portable keg tap : Sometimes, beer kegs designed to be connected to the above system are instead used on their own, perhaps at a party or outdoor event. In this case, a self-contained portable tap is required that allows beer to be served straight from the keg. Because the keg system uses pressure to force the beer up and out of the keg, these taps must have a means of supplying it. The typical "picnic tap" uses a hand pump to push air into the keg; this will cause the beer to spoil faster but is perfectly acceptable when it will be consumed in a short time. Portable taps with small CO₂ cylinders are also available. nitrogen ; Cask beer tap : Beers brewed and served in the traditional way (typically real ale) do not use artificial gas. Taps for cask beer are simple on-off valves that are hammered into the end of the cask (see keystone for details). When beer is served directly from the cask ("by gravity"), as at beer festivals and some pubs, it simply flows out of the tap and into the glass. When the cask is stored in the cellar and served from the bar, as in most pubs, the beer line is screwed onto the tap and the beer is sucked through it by a hand-operated low-pressure pump on the bar. The taps used are the same, and in beer-line setups the first pint is often poured from the cask as for "gravity", for tasting, before the line is connected. Cask beer taps can be brass (now discouraged for fear of lead contamination), stainless steel (good, but expensive), plastic (acceptable, and cheaper), and wood (to be avoided if possible).

Gas taps

keystone Although a gas tap may be a valve that releases any gas, the word is most commonly used to refer to taps that control the flow of natural gas in the home (for gas fires) or in school science laboratories (for Bunsen burners).

Physics of taps

Most water and gas taps have adjustable flow. Turning the knob or working the lever sets the flow rate by adjusting the size of an opening in the valve assembly, giving rise to choked flow through the narrow opening in the valve. The choked flow rate is independent of the viscosity or temperature of the fluid or gas in the pipe, and depends only weakly on the supply pressure, so that flow rate is stable at a given setting. At intermediate flow settings the pressure at the valve restriction drops nearly to zero from the venturi effect; in water taps, this causes the water to boil momentarily at room temperature as it passes through the restriction. Bubbles of cool water vapor form and collapse at the restriction, causing the familiar hissing sound. At very low flow settings, the viscosity of the water becomes important and the pressure drop (and hissing noise) vanish; at full flow settings, parasitic drag in the pipes becomes important and the water again becomes quiet. Most taps use a soft washer which is screwed down onto a seat in order to stop the flow. This is called a "globe valve" in engineering and, while it gives a leak-proof seal and good fine adjustment of flow, the tortuous S-shaped path the water is forced to follow offers a significant obstruction to the flow. For high pressure domestic water systems this does not matter, but for low pressure systems where flowrate is important, such as a shower fed by a storage tank, a "stop tap" or, in engineering terms, a "gate valve" is preferred. This uses a metal disc the same diameter as the pipe which is screwed into place perpendicularly to the flow, cutting it off. There is no resistance to flow when the tap is fully open, but this type of tap rarely gives a perfect seal when closed. In the UK the latter type of tap normally has a wheel-shaped handle rather than a crutch or capstan handle. One reason that most beer taps are not designed for adjustable flow is that the beer itself is damaged by the pressure drop in a choked-flow valve: holding a beer tap partially open causes the beer to foam vigorously, ruining the pour. Category:Valves ja:蛇口

Cigarette

A cigarette is a small paper-wrapped cylinder (generally less than 120mm in length and 10mm in diameter) of cured and shredded or cut tobacco leaves. The cigarette is ignited at one end and allowed to smoulder for the purpose of inhalation of its smoke from the filtered end, inserted in the mouth. The term, as commonly used, typically refers to a tobacco cigarette, but can apply to similar devices containing other herbs, such as cannabis. A cigarette is distinguished from a cigar by its smaller size, use of processed leaf, and paper wrapping; cigars are typically composed entirely of whole leaf tobacco. Cigarettes were largely unknown in the English-speaking world before the Crimean War, when British soldiers began emulating their Ottoman Turkish comrades, who resorted to rolling their tobacco with newsprint.

Manufacture and ingredients

In practice, commercial cigarettes and cigarette tobaccos rarely contain pure tobacco. Manufacturers often use a tremendous variety of additives for a number of purposes, including maintaining blend consistency, improving perceived blend quality, as preservatives and even completely changing the organoleptic qualities of the tobacco smoke. While this is true for many brands of cigarettes, in Canada, the major cigarette brands all contain 100% natural virginia leaf - No Additives. Some cigarettes (known as kreteks, clove cigarettes, or simply cloves) have cloves blended with the tobacco. This is done to enhance the smoker's pleasure by numbing the mouth and lungs and providing a mild euphoric effect. Lower-quality clove cigarettes simply have a clove essence added to the tobacco. In addition to additives, cigarette tobaccos, especially lower-quality blends, are often highly physically processed. During the original processing of leaf for cigarettes, the leaves are deveined, and the lamina is shredded or cut. Since the leaf is relatively dry at this point, these processes result in a significant amount of tobacco dust. Manufacturing operations have developed procedures for collecting this dust and remaking it into usable material (known as reconstituted sheet tobacco). The removed leaf midveins, which are unsuitable for use in cigarettes in their natural state, were historically discarded or spread on fields, because of their high nitrogen content. Procedures have been developed, however, to "expand" the stems, and process them for inclusion in the cigarette blends. All these procedures allow cigarette manufacturers to produce as many cigarettes as possible using the least amount of raw materials as possible. The most common usage of the cigarette is tobacco smoke delivery. The second most common usage of the cigarette is for marijuana smoke delivery. The hand rolled cigarette is the most common form of marijuana cigarette. Marijuana users will usually twist the ends of the cigarette to prevent fine cut marijuana buds from falling out. Tobacco users who roll their own cigarettes, however, will usually not twist the cigarette at the ends; hand rolling tobacco is made in strands so it doesn't have a tendency to fall out. Some cigarette smokers roll their own cigarettes by wrapping loose cured tobacco in paper; most, however, purchase machine-made commercially available brands, generally sold in small cardboard packages of 10 or 20 cigarettes in the United States and UK or 25 in Canada. Commercial cigarettes usually contain a cellulose acetate or cotton filter through which the smoker inhales the cigarette's smoke; the filter serves to cool and somewhat clean the smoke. Recently, cigarette rolling machines are also becoming popular. One can purchase tobacco in pouches or cans, usually at half the price of what one would pay for the same amount pre-rolled. One can get a rolling machine that makes filterless, or "straight" cigarettes, or one can purchase a machine that packs the tobacco into a pre-rolled form with a filter. These filtered papers usually come in boxes of 200, while unfiltered papers will come in packs ranging from 12 to 64, and some contain even more.

Sale

rolling papers Before the Second World War many manufacturers gave away collectible cards, one in each packet of cigarettes. This practice was discontinued to save paper during the war, and was never generally reintroduced. On April 1, 1970 President Richard Nixon signed the Public Health Cigarette Smoking Act into law, banning cigarette advertisements on television in the United States starting on January 2, 1971. However, some tobacco companies attempted to circumvent the ban by marketing new brands of cigarettes as "little cigars;" examples included Tijuana Smalls, which came out almost immediately after the ban took effect, and Backwoods Smokes, which hit the market in the winter of 1973-1974 and whose ads used the slogan, "How can anything that looks so wild taste so mild?" The sale of cigarettes and other tobacco products to minors under 18 is now prohibited by law in all fifty states of the United States. In Alabama, Alaska and Utah the statutory age is 19, and legislation was pending as of 2004 in some other states, including California and New Jersey, to raise the age to 19, or even 21 in some cases. In Massachusetts, parents and guardians are allowed to give cigarettes to minors, but sales to minors are prohibited. Legislation was successfully passed on Long Island (New York) to raise the legal age in Suffolk county to 19, effective January 1st, 2005. Similar laws exist in many other countries as well. In Canada, most of the provinces require smokers to be 19 years of age to purchase cigarettes (except for Quebec, Saskatchewan, Manitoba and Alberta, where the age is 18). However, the minimum age only concerns the purchase of tobacco, not use. Alberta, however, does have a law which prohibits the possession or use of tobacco products by all persons under 18, punishable by a $100 fine. Australia has a nation-wide ban on the selling of all tobacco products to people under 18. In the UK, cigarettes can legally be sold only to people aged 16 and over. However it is not illegal for people under this age to buy (or attempt to buy) cigarettes, so only the retailer is breaking the law by selling to under 16s. However, while bans stand in most countries for sales to minors, it is still common for merchants to disregard such laws as they are tough to enforce. Often the profits from selling cigarettes to minors illegally are much greater than the fines paid out in very infrequent times when they are caught. Some police departments in the United States occasionally send a clearly underage child into a store where cigarettes are sold, and have the child attempt to purchase cigarettes. If the vendor sells them to the minor, the store is issued a fine. This is by far the most common way in which cigarette vendors are caught when they sell cigarettes to minors.

Online cigarette stores

Online stores have recently appeared that offer foreign cigarettes to internet buyers. As many jurisdictions place high taxes on tobacco sales, these could be seen as an effort to avoid paying duty or taxes. Some online cigarette stores exist to sell tax-free cigarettes inside one's own country of residence as well. The legality of these stores is being questioned currently in the United States. Federal lawmakers contend that these stores are clear tax evasions. Recently in

Turbulence

Examples of turbulence

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Turbulence (film)

Cast

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Stochastic

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Pressure

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Scalar nature of pressure

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Laminar flow

Experiments

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Reynolds number

The similarity of flows

Reynolds number sets the smallest scales of turbulent motion

Example on the importance of Reynolds number

Typical values of Reynolds number

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References

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Andrey Kolmogorov

His portraits by Dmitry Gordeev

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