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A mass flow meter, also known as an inertial flow meter, is a device that measures mass flow rate of a fluid traveling through a tube. The mass flow rate is the mass of the fluid traveling past a fixed point per unit time. The mass flow meter does not measure the volume per unit time (e.g. cubic meters per second) passing through the device; it measures the mass per unit time (e.g. kilograms per second) flowing through the device. Volumetric flow rate is the mass flow rate divided by the fluid density. If the density is constant, then the relationship is simple. If the fluid has varying density, then the relationship is not simple. For example, the density of the fluid may change with temperature, pressure, or composition. The fluid may also be a combination of phases such as a fluid with entrained bubbles. Actual density can be determined due to dependency of sound velocity on the controlled liquid concentration. == Operating principle of a Coriolis flow meter == The Coriolis flow meter is based on the Coriolis force, which bends rotating objects depending on their velocity. There are two basic configurations of Coriolis flow meter: the curved tube flow meter and the straight tube flow meter. This article discusses the curved tube design. The animations on the right do not represent an actually existing Coriolis flow meter design. The purpose of the animations is to illustrate the operating principle, and to show the connection with rotation. Fluid is being pumped through the mass flow meter. When there is mass flow, the tube twists slightly. The arm through which fluid flows away from the axis of rotation must exert a force on the fluid, to increase its angular momentum, so it bends backwards. The arm through which fluid is
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pushed back to the axis of rotation must exert a force on the fluid to decrease the fluid's angular momentum again, hence that arm will bend forward. In other words, the inlet arm (containing an outwards directed flow), is lagging behind the overall rotation, the part which in rest is parallel to the axis is now skewed, and the outlet arm (containing an inwards directed flow) leads the overall rotation. The animation on the right represents how curved tube mass flow meters are designed. The fluid is led through two parallel tubes. An actuator (not shown) induces equal counter vibrations on the sections parallel to the axis, to make the measuring device less sensitive to outside vibrations. The actual frequency of the vibration depends on the size of the mass flow meter, and ranges from 80 to 1000 Hz. The amplitude of the vibration is too small to be seen, but it can be felt by touch. When no fluid is flowing, the motion of the two tubes is symmetrical, as shown in the left animation. The animation on the right illustrates what happens during mass flow: some twisting of the tubes. The arm carrying the flow away from the axis of rotation must exert a force on the fluid to accelerate the flowing mass to the vibrating speed of the tubes at the outside (increase of absolute angular momentum), so it is lagging behind the overall vibration. The arm through which fluid is pushed back towards the axis of movement must exert a force on the fluid to decrease the fluid's absolute angular speed (angular momentum) again, hence that arm leads the overall vibration. The inlet arm and the outlet arm vibrate with the same frequency as the overall vibration, but when there is mass flow the two vibrations
|
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"page_id": 1575643,
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are out of sync: the inlet arm is behind, the outlet arm is ahead. The two vibrations are shifted in phase with respect to each other, and the degree of phase-shift is a measure for the amount of mass that is flowing through the tubes and line. == Density and volume measurements == The mass flow of a U-shaped Coriolis flow meter is given as: Q m = K u โ I u ฯ 2 2 K d 2 ฯ {\displaystyle Q_{m}={\frac {K_{u}-I_{u}\omega ^{2}}{2Kd^{2}}}\tau } where Ku is the temperature dependent stiffness of the tube, K is a shape-dependent factor, d is the width, ฯ is the time lag, ฯ is the vibration frequency, and Iu is the inertia of the tube. As the inertia of the tube depend on its contents, knowledge of the fluid density is needed for the calculation of an accurate mass flow rate. If the density changes too often for manual calibration to be sufficient, the Coriolis flow meter can be adapted to measure the density as well. The natural vibration frequency of the flow tubes depends on the combined mass of the tube and the fluid contained in it. By setting the tube in motion and measuring the natural frequency, the mass of the fluid contained in the tube can be deduced. Dividing the mass on the known volume of the tube gives us the density of the fluid. An instantaneous density measurement allows the calculation of flow in volume per time by dividing mass flow with density. == Calibration == Both mass flow and density measurements depend on the vibration of the tube. Calibration is affected by changes in the rigidity of the flow tubes. Changes in temperature and pressure will cause the tube rigidity to change, but these can be compensated for
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{
"page_id": 1575643,
"source": null,
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through pressure and temperature zero and span compensation factors. Additional effects on tube rigidity will cause shifts in the calibration factor over time due to degradation of the flow tubes. These effects include pitting, cracking, coating, erosion or corrosion. It is not possible to compensate for these changes dynamically, but efforts to monitor the effects may be made through regular meter calibration or verification checks. If a change is deemed to have occurred, but is considered to be acceptable, the offset may be added to the existing calibration factor to ensure continued accurate measurement. == See also == Coriolis effect Flow measurement Gaspard-Gustave Coriolis Oscillating U-tube == References == == External links == Lecture slides on flow measurement, University of Minnesota
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{
"page_id": 1575643,
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"title": "Mass flow meter"
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A taxonomic treatment is a section in a scientific publication documenting the features of a related group of organisms or taxa. Treatments have been the building blocks of how data about taxa are provided, ever since the beginning of modern taxonomy by Linnaeus 1753 for plants and 1758 for animals. Each scientifically described taxon has at least one taxonomic treatment. In todayโs publishing, a taxonomic treatment tag is used to delimit such a section. It allows to make this section findable, accessible, interoperable and reusable FAIR data. This is implemented in the Biodiversity Literature Repository, where upon deposition of the treatment a persistent DataCite digital object identifier (DOI) is minted. This includes metadata about the treatment, the source publication and other cited resources, such as figures cited in the treatment. This DOI allows a link from a taxonomic name usage to the respective scientific evidence provided by the author(s), both for human and machine consumption. Treatments are considered data and thus copyright is not applicable and thus can be made available even from closed access publications. == Etymology == The term taxonomic treatment has been coined because the term description has two meanings in species or taxonomic descriptions. One is equivalent to treatment, the second as subsection in treatments describing the taxon, complementing diagnosis, materials examined, distribution, conservation and other subsections. == History == This term has been introduced during a national US NSF digital library project, and has been further developed into Taxpub, a taxonomy specific version of the Journal Article Tag Suite by Plazi, National Center for Biotechnology Information, and Pensoft Publishers. It was prototyped by the taxonomic journal ZooKeys, which adopted Taxpub from its volume 50 onwards, followed by PhytoKeys. Taxpub is now used by journals published by Pensoft Publishers, the European Journal of Taxonomy, the Consortium
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"title": "Taxonomic treatment"
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of European Taxonomic Facilities (CETAF), and the National Museum of Natural History, France. The TreatmentBank service provided by Plazi to convert taxonomic publications into FAIR data provides access to over 500,000 taxonomic treatments, including over 7,700 treatments for new described species in 2020. They will eventually become accessible in BLR after passing quality control to avoid artifacts due to the complex conversion of unstructured, mainly PDF based publications. == References ==
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{
"page_id": 67504866,
"source": null,
"title": "Taxonomic treatment"
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Astrobiology (also xenology or exobiology) is a scientific field within the life and environmental sciences that studies the origins, early evolution, distribution, and future of life in the universe by investigating its deterministic conditions and contingent events. As a discipline, astrobiology is founded on the premise that life may exist beyond Earth. Research in astrobiology comprises three main areas: the study of habitable environments in the Solar System and beyond, the search for planetary biosignatures of past or present extraterrestrial life, and the study of the origin and early evolution of life on Earth. The field of astrobiology has its origins in the 20th century with the advent of space exploration and the discovery of exoplanets. Early astrobiology research focused on the search for extraterrestrial life and the study of the potential for life to exist on other planets. In the 1960s and 1970s, NASA began its astrobiology pursuits within the Viking program, which was the first US mission to land on Mars and search for signs of life. This mission, along with other early space exploration missions, laid the foundation for the development of astrobiology as a discipline. Regarding habitable environments, astrobiology investigates potential locations beyond Earth that could support life, such as Mars, Europa, and exoplanets, through research into the extremophiles populating austere environments on Earth, like volcanic and deep sea environments. Research within this topic is conducted utilising the methodology of the geosciences, especially geobiology, for astrobiological applications. The search for biosignatures involves the identification of signs of past or present life in the form of organic compounds, isotopic ratios, or microbial fossils. Research within this topic is conducted utilising the methodology of planetary and environmental science, especially atmospheric science, for astrobiological applications, and is often conducted through remote sensing and in situ missions. Astrobiology also concerns
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the study of the origin and early evolution of life on Earth to try to understand the conditions that are necessary for life to form on other planets. This research seeks to understand how life emerged from non-living matter and how it evolved to become the diverse array of organisms we see today. Research within this topic is conducted utilising the methodology of paleosciences, especially paleobiology, for astrobiological applications. Astrobiology is a rapidly developing field with a strong interdisciplinary aspect that holds many challenges and opportunities for scientists. Astrobiology programs and research centres are present in many universities and research institutions around the world, and space agencies like NASA and ESA have dedicated departments and programs for astrobiology research. == Overview == The term astrobiology was first proposed by the Russian astronomer Gavriil Tikhov in 1953. It is etymologically derived from the Greek แผฯฯฯฮฟฮฝ, "star"; ฮฒฮฏฮฟฯ, "life"; and -ฮปฮฟฮณฮฏฮฑ, -logia, "study". A close synonym is exobiology from the Greek ฮฮพฯ, "external"; ฮฒฮฏฮฟฯ, "life"; and -ฮปฮฟฮณฮฏฮฑ, -logia, "study", coined by American molecular biologist Joshua Lederberg; exobiology is considered to have a narrow scope limited to search of life external to Earth. Another associated term is xenobiology, from the Greek ฮพฮญฮฝฮฟฯ, "foreign"; ฮฒฮฏฮฟฯ, "life"; and -ฮปฮฟฮณฮฏฮฑ, "study", coined by American science fiction writer Robert Heinlein in his 1954 work The Star Beast; xenobiology is now used in a more specialised sense, referring to 'biology based on foreign chemistry', whether of extraterrestrial or terrestrial (typically synthetic) origin. While the potential for extraterrestrial life, especially intelligent life, has been explored throughout human history within philosophy and narrative, the question is a verifiable hypothesis and thus a valid line of scientific inquiry; planetary scientist David Grinspoon calls it a field of natural philosophy, grounding speculation on the unknown in known scientific theory. The modern field
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of astrobiology can be traced back to the 1950s and 1960s with the advent of space exploration, when scientists began to seriously consider the possibility of life on other planets. In 1957, the Soviet Union launched Sputnik 1, the first artificial satellite, which marked the beginning of the Space Age. This event led to an increase in the study of the potential for life on other planets, as scientists began to consider the possibilities opened up by the new technology of space exploration. In 1959, NASA funded its first exobiology project, and in 1960, NASA founded the Exobiology Program, now one of four main elements of NASA's current Astrobiology Program. In 1971, NASA funded Project Cyclops, part of the search for extraterrestrial intelligence, to search radio frequencies of the electromagnetic spectrum for interstellar communications transmitted by extraterrestrial life outside the Solar System. In the 1960s-1970s, NASA established the Viking program, which was the first US mission to land on Mars and search for metabolic signs of present life; the results were inconclusive. In the 1980s and 1990s, the field began to expand and diversify as new discoveries and technologies emerged. The discovery of microbial life in extreme environments on Earth, such as deep-sea hydrothermal vents, helped to clarify the feasibility of potential life existing in harsh conditions. The development of new techniques for the detection of biosignatures, such as the use of stable isotopes, also played a significant role in the evolution of the field. The contemporary landscape of astrobiology emerged in the early 21st century, focused on utilising Earth and environmental science for applications within comparate space environments. Missions included the ESA's Beagle 2, which failed minutes after landing on Mars, NASA's Phoenix lander, which probed the environment for past and present planetary habitability of microbial life on Mars
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"page_id": 2787,
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and researched the history of water, and NASA's Curiosity rover, currently probing the environment for past and present planetary habitability of microbial life on Mars. == Theoretical foundations == === Planetary habitability === Astrobiological research makes a number of simplifying assumptions when studying the necessary components for planetary habitability. Carbon and Organic Compounds: Carbon is the fourth most abundant element in the universe and the energy required to make or break a bond is at just the appropriate level for building molecules which are not only stable, but also reactive. The fact that carbon atoms bond readily to other carbon atoms allows for the building of extremely long and complex molecules. As such, astrobiological research presumes that the vast majority of life forms in the Milky Way galaxy are based on carbon chemistries, as are all life forms on Earth. However, theoretical astrobiology entertains the potential for other organic molecular bases for life, thus astrobiological research often focuses on identifying environments that have the potential to support life based on the presence of organic compounds. Liquid water: Liquid water is a common molecule that provides an excellent environment for the formation of complicated carbon-based molecules, and is generally considered necessary for life as we know it to exist. Thus, astrobiological research presumes that extraterrestrial life similarly depends upon access to liquid water, and often focuses on identifying environments that have the potential to support liquid water. Some researchers posit environments of water-ammonia mixtures as possible solvents for hypothetical types of biochemistry. Environmental stability: Where organisms adaptively evolve to the conditions of the environments in which they reside, environmental stability is considered necessary for life to exist. This presupposes the necessity of a stable temperature, pressure, and radiation levels; resultantly, astrobiological research focuses on planets orbiting Sun-like red dwarf stars. This
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"page_id": 2787,
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is because very large stars have relatively short lifetimes, meaning that life might not have time to emerge on planets orbiting them; very small stars provide so little heat and warmth that only planets in very close orbits around them would not be frozen solid, and in such close orbits these planets would be tidally locked to the star; whereas the long lifetimes of red dwarfs could allow the development of habitable environments on planets with thick atmospheres. This is significant as red dwarfs are extremely common. (See also: Habitability of red dwarf systems). Energy source: It is assumed that any life elsewhere in the universe would also require an energy source. Previously, it was assumed that this would necessarily be from a Sun-like star, however with developments within extremophile research contemporary astrobiological research often focuses on identifying environments that have the potential to support life based on the availability of an energy source, such as the presence of volcanic activity on a planet or moon that could provide a source of heat and energy. It is important to note that these assumptions are based on our current understanding of life on Earth and the conditions under which it can exist. As our understanding of life and the potential for it to exist in different environments evolves, these assumptions may change. == Methods == === Studying terrestrial extremophiles === Astrobiological research concerning the study of habitable environments in our solar system and beyond utilises methods within the geosciences. Research within this branch primarily concerns the geobiology of organisms that can survive in extreme environments on Earth, such as in volcanic or deep sea environments, to understand the limits of life, and the conditions under which life might be able to survive on other planets. This includes, but is not limited
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to: Deep-sea extremophiles: Researchers are studying organisms that live in the extreme environments of deep-sea hydrothermal vents and cold seeps. These organisms survive in the absence of sunlight, and some are able to survive in high temperatures and pressures, and use chemical energy instead of sunlight to produce food. Desert extremophiles: Researchers are studying organisms that can survive in extreme dry, high temperature conditions, such as in deserts. Microbes in extreme environments: Researchers are investigating the diversity and activity of microorganisms in environments such as deep mines, subsurface soil, cold glaciers and polar ice, and high-altitude environments. === Researching Earth's present environment === Research also regards the long-term survival of life on Earth, and the possibilities and hazards of life on other planets, including: Biodiversity and ecosystem resilience: Scientists are studying how the diversity of life and the interactions between different species contribute to the resilience of ecosystems and their ability to recover from disturbances. Climate change and extinction: Researchers are investigating the impacts of climate change on different species and ecosystems, and how they may lead to extinction or adaptation. This includes the evolution of Earth's climate and geology, and their potential impact on the habitability of the planet in the future, especially for humans. Human impact on the biosphere: Scientists are studying the ways in which human activities, such as deforestation, pollution, and the introduction of invasive species, are affecting the biosphere and the long-term survival of life on Earth. Long-term preservation of life: Researchers are exploring ways to preserve samples of life on Earth for long periods of time, such as cryopreservation and genomic preservation, in the event of a catastrophic event that could wipe out most of life on Earth. === Finding biosignatures on other worlds === Emerging astrobiological research concerning the search for planetary biosignatures
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of past or present extraterrestrial life utilise methodologies within planetary sciences. These include: The study of microbial life in the subsurface of Mars: Scientists are using data from Mars rover missions to study the composition of the subsurface of Mars, searching for biosignatures of past or present microbial life. The study of liquid bodies on icy moons: Discoveries of surface and subsurface bodies of liquid on moons such as Europa, Titan and Enceladus showed possible habitability zones, making them viable targets for the search for extraterrestrial life. As of September 2024, missions like Europa Clipper and Dragonfly are planned to search for biosignatures within these environments. The study of the atmospheres of planets: Scientists are studying the potential for life to exist in the atmospheres of planets, with a focus on the study of the physical and chemical conditions necessary for such life to exist, namely the detection of organic molecules and biosignature gases; for example, the study of the possibility of life in the atmospheres of exoplanets that orbit red dwarfs and the study of the potential for microbial life in the upper atmosphere of Venus. Telescopes and remote sensing of exoplanets: The discovery of thousands of exoplanets has opened up new opportunities for the search for biosignatures. Scientists are using telescopes such as the James Webb Space Telescope and the Transiting Exoplanet Survey Satellite to search for biosignatures on exoplanets. They are also developing new techniques for the detection of biosignatures, such as the use of remote sensing to search for biosignatures in the atmosphere of exoplanets. === Talking to extraterrestrials === SETI and CETI Scientists search for signals from intelligent extraterrestrial civilizations using radio and optical telescopes within the discipline of extraterrestrial intelligence communications (CETI). CETI focuses on composing and deciphering messages that could theoretically be understood
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{
"page_id": 2787,
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by another technological civilization. Communication attempts by humans have included broadcasting mathematical languages, pictorial systems such as the Arecibo message, and computational approaches to detecting and deciphering 'natural' language communication. While some high-profile scientists, such as Carl Sagan, have advocated the transmission of messages, theoretical physicist Stephen Hawking warned against it, suggesting that aliens may raid Earth for its resources. === Investigating the early Earth === Emerging astrobiological research concerning the study of the origin and early evolution of life on Earth utilises methodologies within the palaeosciences. These include: The study of the early atmosphere: Researchers are investigating the role of the early atmosphere in providing the right conditions for the emergence of life, such as the presence of gases that could have helped to stabilise the climate and the formation of organic molecules. The study of the early magnetic field: Researchers are investigating the role of the early magnetic field in protecting the Earth from harmful radiation and helping to stabilise the climate. This research has immense astrobiological implications where the subjects of current astrobiological research like Mars lack such a field. The study of prebiotic chemistry: Scientists are studying the chemical reactions that could have occurred on the early Earth that led to the formation of the building blocks of life- amino acids, nucleotides, and lipids- and how these molecules could have formed spontaneously under early Earth conditions. The study of impact events: Scientists are investigating the potential role of impact events- especially meteorites- in the delivery of water and organic molecules to early Earth. The study of the primordial soup: Researchers are investigating the conditions and ingredients that were present on the early Earth that could have led to the formation of the first living organisms, such as the presence of water and organic molecules, and how
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"page_id": 2787,
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these ingredients could have led to the formation of the first living organisms. This includes the role of water in the formation of the first cells and in catalysing chemical reactions. The study of the role of minerals: Scientists are investigating the role of minerals like clay in catalysing the formation of organic molecules, thus playing a role in the emergence of life on Earth. The study of the role of energy and electricity: Scientists are investigating the potential sources of energy and electricity that could have been available on the early Earth, and their role in the formation of organic molecules, thus the emergence of life. The study of the early oceans: Scientists are investigating the composition and chemistry of the early oceans and how it may have played a role in the emergence of life, such as the presence of dissolved minerals that could have helped to catalyse the formation of organic molecules. The study of hydrothermal vents: Scientists are investigating the potential role of hydrothermal vents in the origin of life, as these environments may have provided the energy and chemical building blocks needed for its emergence. The study of plate tectonics: Scientists are investigating the role of plate tectonics in creating a diverse range of environments on the early Earth. The study of the early biosphere: Researchers are investigating the diversity and activity of microorganisms in the early Earth, and how these organisms may have played a role in the emergence of life. The study of microbial fossils: Scientists are investigating the presence of microbial fossils in ancient rocks, which can provide clues about the early evolution of life on Earth and the emergence of the first organisms. == Research == The systematic search for possible life outside Earth is a valid multidisciplinary scientific endeavor. However,
|
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"page_id": 2787,
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hypotheses and predictions as to its existence and origin vary widely, and at the present, the development of hypotheses firmly grounded on science may be considered astrobiology's most concrete practical application. It has been proposed that viruses are likely to be encountered on other life-bearing planets, and may be present even if there are no biological cells. === Research outcomes === As of 2024, no evidence of extraterrestrial life has been identified. Examination of the Allan Hills 84001 meteorite, which was recovered in Antarctica in 1984 and originated from Mars, is thought by David McKay, as well as few other scientists, to contain microfossils of extraterrestrial origin; this interpretation is controversial. Yamato 000593, the second largest meteorite from Mars, was found on Earth in 2000. At a microscopic level, spheres are found in the meteorite that are rich in carbon compared to surrounding areas that lack such spheres. The carbon-rich spheres may have been formed by biotic activity according to some NASA scientists. On 5 March 2011, Richard B. Hoover, a scientist with the Marshall Space Flight Center, speculated on the finding of alleged microfossils similar to cyanobacteria in CI1 carbonaceous meteorites in the fringe Journal of Cosmology, a story widely reported on by mainstream media. However, NASA formally distanced itself from Hoover's claim. According to American astrophysicist Neil deGrasse Tyson: "At the moment, life on Earth is the only known life in the universe, but there are compelling arguments to suggest we are not alone." === Elements of astrobiology === ==== Astronomy ==== Most astronomy-related astrobiology research falls into the category of extrasolar planet (exoplanet) detection, the hypothesis being that if life arose on Earth, then it could also arise on other planets with similar characteristics. To that end, a number of instruments designed to detect Earth-sized exoplanets have
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{
"page_id": 2787,
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been considered, most notably NASA's Terrestrial Planet Finder (TPF) and ESA's Darwin programs, both of which have been cancelled. NASA launched the Kepler mission in March 2009, and the French Space Agency launched the COROT space mission in 2006. There are also several less ambitious ground-based efforts underway. The goal of these missions is not only to detect Earth-sized planets but also to directly detect light from the planet so that it may be studied spectroscopically. By examining planetary spectra, it would be possible to determine the basic composition of an extrasolar planet's atmosphere and/or surface. Given this knowledge, it may be possible to assess the likelihood of life being found on that planet. A NASA research group, the Virtual Planet Laboratory, is using computer modeling to generate a wide variety of virtual planets to see what they would look like if viewed by TPF or Darwin. It is hoped that once these missions come online, their spectra can be cross-checked with these virtual planetary spectra for features that might indicate the presence of life. An estimate for the number of planets with intelligent communicative extraterrestrial life can be gleaned from the Drake equation, essentially an equation expressing the probability of intelligent life as the product of factors such as the fraction of planets that might be habitable and the fraction of planets on which life might arise: N = R โ ร f p ร n e ร f l ร f i ร f c ร L {\displaystyle N=R^{*}~\times ~f_{p}~\times ~n_{e}~\times ~f_{l}~\times ~f_{i}~\times ~f_{c}~\times ~L} where: N = The number of communicative civilizations R* = The rate of formation of suitable stars (stars such as the Sun) fp = The fraction of those stars with planets (current evidence indicates that planetary systems may be common for stars like
|
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the Sun) ne = The number of Earth-sized worlds per planetary system fl = The fraction of those Earth-sized planets where life actually develops fi = The fraction of life sites where intelligence develops fc = The fraction of communicative planets (those on which electromagnetic communications technology develops) L = The "lifetime" of communicating civilizations However, whilst the rationale behind the equation is sound, it is unlikely that the equation will be constrained to reasonable limits of error any time soon. The problem with the formula is that it is not used to generate or support hypotheses because it contains factors that can never be verified. The first term, R*, number of stars, is generally constrained within a few orders of magnitude. The second and third terms, fp, stars with planets and fe, planets with habitable conditions, are being evaluated for the star's neighborhood. Drake originally formulated the equation merely as an agenda for discussion at the Green Bank conference, but some applications of the formula had been taken literally and related to simplistic or pseudoscientific arguments. Another associated topic is the Fermi paradox, which suggests that if intelligent life is common in the universe, then there should be obvious signs of it. Another active research area in astrobiology is planetary system formation. It has been suggested that the peculiarities of the Solar System (for example, the presence of Jupiter as a protective shield) may have greatly increased the probability of intelligent life arising on Earth. ==== Biology ==== Biology cannot state that a process or phenomenon, by being mathematically possible, has to exist forcibly in an extraterrestrial body. Biologists specify what is speculative and what is not. The discovery of extremophiles, organisms able to survive in extreme environments, became a core research element for astrobiologists, as they are important
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to understand four areas in the limits of life in planetary context: the potential for panspermia, forward contamination due to human exploration ventures, planetary colonization by humans, and the exploration of extinct and extant extraterrestrial life. Until the 1970s, life was thought to be entirely dependent on energy from the Sun. Plants on Earth's surface capture energy from sunlight to photosynthesize sugars from carbon dioxide and water, releasing oxygen in the process that is then consumed by oxygen-respiring organisms, passing their energy up the food chain. Even life in the ocean depths, where sunlight cannot reach, was thought to obtain its nourishment either from consuming organic detritus rained down from the surface waters or from eating animals that did. The world's ability to support life was thought to depend on its access to sunlight. However, in 1977, during an exploratory dive to the Galapagos Rift in the deep-sea exploration submersible Alvin, scientists discovered colonies of giant tube worms, clams, crustaceans, mussels, and other assorted creatures clustered around undersea volcanic features known as black smokers. These creatures thrive despite having no access to sunlight, and it was soon discovered that they form an entirely independent ecosystem. Although most of these multicellular lifeforms need dissolved oxygen (produced by oxygenic photosynthesis) for their aerobic cellular respiration and thus are not completely independent from sunlight by themselves, the basis for their food chain is a form of bacterium that derives its energy from oxidization of reactive chemicals, such as hydrogen or hydrogen sulfide, that bubble up from the Earth's interior. Other lifeforms entirely decoupled from the energy from sunlight are green sulfur bacteria which are capturing geothermal light for anoxygenic photosynthesis or bacteria running chemolithoautotrophy based on the radioactive decay of uranium. This chemosynthesis revolutionized the study of biology and astrobiology by revealing that
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life need not be sunlight-dependent; it only requires water and an energy gradient in order to exist. Biologists have found extremophiles that thrive in ice, boiling water, acid, alkali, the water core of nuclear reactors, salt crystals, toxic waste and in a range of other extreme habitats that were previously thought to be inhospitable for life. This opened up a new avenue in astrobiology by massively expanding the number of possible extraterrestrial habitats. Characterization of these organisms, their environments and their evolutionary pathways, is considered a crucial component to understanding how life might evolve elsewhere in the universe. For example, some organisms able to withstand exposure to the vacuum and radiation of outer space include the lichen fungi Rhizocarpon geographicum and Rusavskia elegans, the bacterium Bacillus safensis, Deinococcus radiodurans, Bacillus subtilis, yeast Saccharomyces cerevisiae, seeds from Arabidopsis thaliana ('mouse-ear cress'), as well as the invertebrate animal Tardigrade. While tardigrades are not considered true extremophiles, they are considered extremotolerant microorganisms that have contributed to the field of astrobiology. Their extreme radiation tolerance and presence of DNA protection proteins may provide answers as to whether life can survive away from the protection of the Earth's atmosphere. Jupiter's moon, Europa, and Saturn's moon, Enceladus, are now considered the most likely locations for extant extraterrestrial life in the Solar System due to their subsurface water oceans where radiogenic and tidal heating enables liquid water to exist. The origin of life, known as abiogenesis, distinct from the evolution of life, is another ongoing field of research. Oparin and Haldane postulated that the conditions on the early Earth were conducive to the formation of organic compounds from inorganic elements and thus to the formation of many of the chemicals common to all forms of life we see today. The study of this process, known as prebiotic
|
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chemistry, has made some progress, but it is still unclear whether or not life could have formed in such a manner on Earth. The alternative hypothesis of panspermia is that the first elements of life may have formed on another planet with even more favorable conditions (or even in interstellar space, asteroids, etc.) and then have been carried over to Earth. The cosmic dust permeating the universe contains complex organic compounds ("amorphous organic solids with a mixed aromatic-aliphatic structure") that could be created naturally, and rapidly, by stars. Further, a scientist suggested that these compounds may have been related to the development of life on Earth and said that, "If this is the case, life on Earth may have had an easier time getting started as these organics can serve as basic ingredients for life." More than 20% of the carbon in the universe may be associated with polycyclic aromatic hydrocarbons (PAHs), possible starting materials for the formation of life. PAHs seem to have been formed shortly after the Big Bang, are widespread throughout the universe, and are associated with new stars and exoplanets. PAHs are subjected to interstellar medium conditions and are transformed through hydrogenation, oxygenation and hydroxylation, to more complex organicsโ"a step along the path toward amino acids and nucleotides, the raw materials of proteins and DNA, respectively". In October 2020, astronomers proposed the idea of detecting life on distant planets by studying the shadows of trees at certain times of the day to find patterns that could be detected through observation of exoplanets. == Philosophy == David Grinspoon called astrobiology a field of natural philosophy. Astrobiology intersects with philosophy by raising questions about the nature and existence of life beyond Earth. Philosophical implications include the definition of life itself, issues in the philosophy of mind and cognitive
|
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"page_id": 2787,
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science in case intelligent life is found, epistemological questions about the nature of proof, ethical considerations of space exploration, along with the broader impact of discovering extraterrestrial life on human thought and society. Dunรฉr has emphasized philosophy of astrobiology as an ongoing existential exercise in individual and collective self-understanding, whose major task is constructing and debating concepts such as the concept of life. Key issues, for Dunรฉr, are questions of resource money and monetary planning, epistemological questions regarding astrobiological knowledge, linguistics issues about interstellar communication, cognitive issues such as the definition of intelligence, along with the possibility of interplanetary contamination. Persson also emphasized key philosophical questions in astrobiology. They include ethical justification of resources, the question of life in general, the epistemological issues and knowledge about being alone in the universe, ethics towards extraterrestrial life, the question of politics and governing uninhabited worlds, along with questions of ecology. For von Hegner, the question of astrobiology and the possibility of astrophilosophy differs. For him, the discipline needs to bifurcate into astrobiology and astrophilosophy since discussions made possible by astrobiology, but which have been astrophilosophical in nature, have existed as long as there have been discussions about extraterrestrial life. Astrobiology is a self-corrective interaction among observation, hypothesis, experiment, and theory, pertaining to the exploration of all natural phenomena. Astrophilosophy consists of methods of dialectic analysis and logical argumentation, pertaining to the clarification of the nature of reality. ล ekrst argues that astrobiology requires the affirmation of astrophilosophy, but not as a separate cognate to astrobiology. The stance of conceptual speciesm, according to ล ekrst, permeates astrobiology since the very name astrobiology tries to talk about not just biology, but about life in a general way, which includes terrestrial life as a subset. This leads us to either redefine philosophy, or consider the need for
|
{
"page_id": 2787,
"source": null,
"title": "Astrobiology"
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astrophilosophy as a more general discipline, to which philosophy is just a subset that deals with questions such as the nature of the human mind and other anthropocentric questions. Most of the philosophy of astrobiology deals with two main questions: the question of life and the ethics of space exploration. Kolb specifically emphasizes the question of viruses, for which the question whether they are alive or not is based on the definitions of life that include self-replication. Schneider tried to defined exo-life, but concluded that we often start with our own prejudices and that defining extraterrestrial life seems futile using human concepts. For Dick, astrobiology relies on metaphysical assumption that there is extraterrestrial life, which reaffirms questions in the philosophy of cosmology, such as fine-tuning or the anthropic principle. == Rare Earth hypothesis == The Rare Earth hypothesis postulates that multicellular life forms found on Earth may actually be more of a rarity than scientists assume. According to this hypothesis, life on Earth (and more, multi-cellular life) is possible because of a conjunction of the right circumstances (galaxy and location within it, planetary system, star, orbit, planetary size, atmosphere, etc.); and the chance for all those circumstances to repeat elsewhere may be rare. It provides a possible answer to the Fermi paradox which wonders: if extraterrestrial aliens are common, why aren't they obvious? It is apparently in opposition to the principle of mediocrity, assumed by famed astronomers Frank Drake, Carl Sagan, and others. The principle of mediocrity suggests that life on Earth is not exceptional, and it is more than likely to be found on innumerable other worlds. == Missions == Research into the environmental limits of life and the workings of extreme ecosystems is ongoing, enabling researchers to better predict what planetary environments might be most likely to harbor
|
{
"page_id": 2787,
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life. Missions such as the Phoenix lander, Mars Science Laboratory, ExoMars, Mars 2020 rover to Mars, and the Cassini probe to Saturn's moons aim to further explore the possibilities of life on other planets in the Solar System. Viking program The two Viking landers each carried four types of biological experiments to the surface of Mars in the late 1970s. These were the only Mars landers to carry out experiments looking specifically for metabolism by current microbial life on Mars. The landers used a robotic arm to collect soil samples into sealed test containers on the craft. The two landers were identical, so the same tests were carried out at two places on Mars' surface; Viking 1 near the equator and Viking 2 further north. The result was inconclusive, and is still disputed by some scientists. Norman Horowitz was the chief of the Jet Propulsion Laboratory bioscience section for the Mariner and Viking missions from 1965 to 1976. Horowitz considered that the great versatility of the carbon atom makes it the element most likely to provide solutions, even exotic solutions, to the problems of survival of life on other planets. However, he also considered that the conditions found on Mars were incompatible with carbon based life. Beagle 2 Beagle 2 was an unsuccessful British Mars lander that formed part of the European Space Agency's 2003 Mars Express mission. Its primary purpose was to search for signs of life on Mars, past or present. Although it landed safely, it was unable to correctly deploy its solar panels and telecom antenna. EXPOSE EXPOSE is a multi-user facility mounted in 2008 outside the International Space Station dedicated to astrobiology. EXPOSE was developed by the European Space Agency (ESA) for long-term spaceflights that allow exposure of organic chemicals and biological samples to outer space
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"page_id": 2787,
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in low Earth orbit. Mars Science Laboratory The Mars Science Laboratory (MSL) mission landed the Curiosity rover that is currently in operation on Mars. It was launched 26 November 2011, and landed at Gale Crater on 6 August 2012. Mission objectives are to help assess Mars' habitability and in doing so, determine whether Mars is or has ever been able to support life, collect data for a future human mission, study Martian geology, its climate, and further assess the role that water, an essential ingredient for life as we know it, played in forming minerals on Mars. Tanpopo The Tanpopo mission is an orbital astrobiology experiment investigating the potential interplanetary transfer of life, organic compounds, and possible terrestrial particles in the low Earth orbit. The purpose is to assess the panspermia hypothesis and the possibility of natural interplanetary transport of microbial life as well as prebiotic organic compounds. Early mission results show evidence that some clumps of microorganism can survive for at least one year in space. This may support the idea that clumps greater than 0.5 millimeters of microorganisms could be one way for life to spread from planet to planet. ExoMars rover ExoMars is a robotic mission to Mars to search for possible biosignatures of Martian life, past or present. This astrobiological mission was under development by the European Space Agency (ESA) in partnership with the Russian Federal Space Agency (Roscosmos); it was planned for a 2022 launch; however, technical and funding issues and the Russian invasion of Ukraine have forced ESA to repeatedly delay the rover's delivery to 2028. Mars 2020 Mars 2020 successfully landed its rover Perseverance in Jezero Crater on 18 February 2021. It will investigate environments on Mars relevant to astrobiology, investigate its surface geological processes and history, including the assessment of its past
|
{
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habitability and potential for preservation of biosignatures and biomolecules within accessible geological materials. The Science Definition Team is proposing the rover collect and package at least 31 samples of rock cores and soil for a later mission to bring back for more definitive analysis in laboratories on Earth. The rover could make measurements and technology demonstrations to help designers of a human expedition understand any hazards posed by Martian dust and demonstrate how to collect carbon dioxide (CO2), which could be a resource for making molecular oxygen (O2) and rocket fuel. Europa Clipper Europa Clipper is a mission launched by NASA on 14 October 2024 that will conduct detailed reconnaissance of Jupiter's moon Europa beginning in 2030, and will investigate whether its internal ocean could harbor conditions suitable for life. It will also aid in the selection of future landing sites. Dragonfly Dragonfly is a NASA mission scheduled to land on Titan in 2036 to assess its microbial habitability and study its prebiotic chemistry. Dragonfly is a rotorcraft lander that will perform controlled flights between multiple locations on the surface, which allows sampling of diverse regions and geological contexts. === Proposed concepts === Icebreaker Life Icebreaker Life is a lander mission that was proposed for NASA's Discovery Program for the 2021 launch opportunity, but it was not selected for development. It would have had a stationary lander that would be a near copy of the successful 2008 Phoenix and it would have carried an upgraded astrobiology scientific payload, including a 1-meter-long core drill to sample ice-cemented ground in the northern plains to conduct a search for organic molecules and evidence of current or past life on Mars. One of the key goals of the Icebreaker Life mission is to test the hypothesis that the ice-rich ground in the polar regions
|
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has significant concentrations of organics due to protection by the ice from oxidants and radiation. Journey to Enceladus and Titan Journey to Enceladus and Titan (JET) is an astrobiology mission concept to assess the habitability potential of Saturn's moons Enceladus and Titan by means of an orbiter. Enceladus Life Finder Enceladus Life Finder (ELF) is a proposed astrobiology mission concept for a space probe intended to assess the habitability of the internal aquatic ocean of Enceladus, Saturn's sixth-largest moon. Life Investigation For Enceladus Life Investigation For Enceladus (LIFE) is a proposed astrobiology sample-return mission concept. The spacecraft would enter into Saturn orbit and enable multiple flybys through Enceladus' icy plumes to collect icy plume particles and volatiles and return them to Earth on a capsule. The spacecraft may sample Enceladus' plumes, the E ring of Saturn, and the upper atmosphere of Titan. Oceanus Oceanus is an orbiter proposed in 2017 for the New Frontiers mission No. 4. It would travel to the moon of Saturn, Titan, to assess its habitability. Oceanus' objectives are to reveal Titan's organic chemistry, geology, gravity, topography, collect 3D reconnaissance data, catalog the organics and determine where they may interact with liquid water. Explorer of Enceladus and Titan Explorer of Enceladus and Titan (E2T) is an orbiter mission concept that would investigate the evolution and habitability of the Saturnian satellites Enceladus and Titan. The mission concept was proposed in 2017 by the European Space Agency. == See also == == Citations == == General references == The International Journal of Astrobiology Archived 25 July 2008 at the Wayback Machine, published by Cambridge University Press, is the forum for practitioners in this interdisciplinary field. Astrobiology, published by Mary Ann Liebert, Inc., is a peer-reviewed journal that explores the origins of life, evolution, distribution, and destiny in the
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universe. Mix, Lucas; Cady, Sherry L. and McKay, Christopher; eds. (2024). The Astrobiology Primer 3.0 is a special issue of the Astrobiology journal compiled by 12 editors and 60 authors that provides an overview of the current state of research in astrobiology. Chapter 1: The Astrobiology Primer 3.0 Chapter 2: What is Life? Chapter 3: The Origins and Evolution of Planetary Systems Chapter 4: A Geological and Chemical Context for the Origins of Life on Early Earth Chapter 5: Major Biological Innovations in the History of Life on Earth Chapter 6: The Breadth and Limits of Life on Earth Chapter 7: Assessing Habitability Beyond Earth Chapter 8: Searching for Life Beyond Earth Chapter 9: Life as We Don't Know It Chapter 10: Planetary Protection - History, Science and the Future Chapter 11: Astrobiology Education, Engagement and Resources Catling, David C. (2013). Astrobiology: A Very Short Introduction. Oxford: Oxford University Press. ISBN 978-0-19-958645-5. Cockell, Charles S. (2015). Astrobiology: Understanding Life in the Universe. NJ: Wiley-Blackwell. ISBN 978-1-118-91332-1. Kolb, Vera M., ed. (2015). Astrobiology: An Evolutionary Approach. Boca Raton: CRC Press. ISBN 978-1-4665-8461-7. Kolb, Vera M., ed. (2019). Handbook of Astrobiology. Boca Raton: CRC Press. ISBN 978-1-138-06512-3. Loeb, Avi (2021). Extraterrestrial: The First Sign of Intelligent Life Beyond Earth. Houghton Mifflin Harcourt. ISBN 978-0358278146 Dick, Steven J.; James Strick (2005). The Living Universe: NASA and the Development of Astrobiology. Piscataway, NJ: Rutgers University Press. ISBN 978-0-8135-3733-7. Grinspoon, David (2004) [2003]. Lonely planets. The natural philosophy of alien life. New York: ECCO. ISBN 978-0-06-018540-4. Mautner, Michael N. (2000). Seeding the Universe with Life: Securing Our Cosmological Future (PDF). Washington D. C.: Legacy Books. ISBN 978-0-476-00330-9. Jakosky, Bruce M. (2006). Science, Society, and the Search for Life in the Universe. Tucson: University of Arizona Press. ISBN 978-0-8165-2613-0. Lunine, Jonathan I. (2005). Astrobiology. A Multidisciplinary
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Approach. San Francisco: Pearson Addison-Wesley. ISBN 978-0-8053-8042-2. Gilmour, Iain; Mark A. Sephton (2004). An introduction to astrobiology. Cambridge: Cambridge Univ. Press. ISBN 978-0-521-83736-1. Ward, Peter; Brownlee, Donald (2000). Rare Earth: Why Complex Life is Uncommon in the Universe. New York: Copernicus. ISBN 978-0-387-98701-9. Chyba, C. F.; Hand, K. P. (2005). "ASTROBIOLOGY: The Study of the Living Universe". Annual Review of Astronomy and Astrophysics. 43 (1): 31โ74. Bibcode:2005ARA&A..43...31C. doi:10.1146/annurev.astro.43.051804.102202. S2CID 2084246. == Further reading == Domagal-Goldman, Shawn D.; et al. (2016). Domagal-Dorman, Shawn (ed.). "The Astrobiology Primer v2.0". Astrobiology. 16 (8): 561โ653. Bibcode:2016AsBio..16..561D. doi:10.1089/ast.2015.1460. PMC 5008114. PMID 27532777. S2CID 4425585. D. Goldsmith, T. Owen, The Search For Life in the Universe, Addison-Wesley Publishing Company, 2001 (3rd edition). ISBN 978-1891389160 Andy Weir's 2021 novel, Project Hail Mary, centers on astrobiology. == External links == Astrobiology.nasa.gov UK Centre for Astrobiology Archived 20 October 2012 at the Wayback Machine Spanish Centro de Astrobiologรญa Astrobiology Research at The Library of Congress Astrobiology Survey โ An introductory course on astrobiology Summary - Search For Life Beyond Earth Archived 24 April 2023 at the Wayback Machine (NASA; 25 June 2021)
|
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Geoffrey J. Gordon is a professor at the Machine Learning Department at Carnegie Mellon University in Pittsburgh and director of research at the Microsoft Montrรฉal lab. He is known for his research in statistical relational learning (a subdiscipline of artificial intelligence and machine learning) and on anytime dynamic variants of the A* search algorithm. His research interests include multi-agent planning, reinforcement learning, decision-theoretic planning, statistical models of difficult data (e.g. maps, video, text), computational learning theory, and game theory. Gordon received a B.A. in computer science from Cornell University in 1991, and a PhD at Carnegie Mellon in 1999. == References ==
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"page_id": 58067684,
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"title": "Geoffrey J. Gordon"
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The molecular formula C20H16O5 (molar mass: 336.33 g/mol, exact mass: 336.0998 u) may refer to: Alpinumisoflavone Psoralidin
|
{
"page_id": 26348259,
"source": null,
"title": "C20H16O5"
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The Piola transformation maps vectors between Eulerian and Lagrangian coordinates in continuum mechanics. It is named after Gabrio Piola. == Definition == Let F : R d โ R d {\displaystyle F:\mathbb {R} ^{d}\rightarrow \mathbb {R} ^{d}} with F ( x ^ ) = B x ^ + b , B โ R d , d , b โ R d {\displaystyle F({\hat {x}})=B{\hat {x}}+b,~B\in \mathbb {R} ^{d,d},~b\in \mathbb {R} ^{d}} an affine transformation. Let K = F ( K ^ ) {\displaystyle K=F({\hat {K}})} with K ^ {\displaystyle {\hat {K}}} a domain with Lipschitz boundary. The mapping p : L 2 ( K ^ ) d โ L 2 ( K ) d , q ^ โฆ p ( q ^ ) ( x ) := 1 | det ( B ) | โ
B q ^ ( x ^ ) {\displaystyle p:L^{2}({\hat {K}})^{d}\rightarrow L^{2}(K)^{d},\quad {\hat {q}}\mapsto p({\hat {q}})(x):={\frac {1}{|\det(B)|}}\cdot B{\hat {q}}({\hat {x}})} is called Piola transformation. The usual definition takes the absolute value of the determinant, although some authors make it just the determinant. Note: for a more general definition in the context of tensors and elasticity, as well as a proof of the property that the Piola transform conserves the flux of tensor fields across boundaries, see Ciarlet's book. == See also == PiolaโKirchhoff stress tensor RaviartโThomas basis functions RaviartโThomas Element == References ==
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"page_id": 37358308,
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"title": "Piola transformation"
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An extended periodic table theorizes about chemical elements beyond those currently known and proven. The element with the highest atomic number known is oganesson (Z = 118), which completes the seventh period (row) in the periodic table. All elements in the eighth period and beyond thus remain purely hypothetical. Elements beyond 118 will be placed in additional periods when discovered, laid out (as with the existing periods) to illustrate periodically recurring trends in the properties of the elements. Any additional periods are expected to contain more elements than the seventh period, as they are calculated to have an additional so-called g-block, containing at least 18 elements with partially filled g-orbitals in each period. An eight-period table containing this block was suggested by Glenn T. Seaborg in 1969. The first element of the g-block may have atomic number 121, and thus would have the systematic name unbiunium. Despite many searches, no elements in this region have been synthesized or discovered in nature. According to the orbital approximation in quantum mechanical descriptions of atomic structure, the g-block would correspond to elements with partially filled g-orbitals, but spinโorbit coupling effects reduce the validity of the orbital approximation substantially for elements of high atomic number. Seaborg's version of the extended period had the heavier elements following the pattern set by lighter elements, as it did not take into account relativistic effects. Models that take relativistic effects into account predict that the pattern will be broken. Pekka Pyykkรถ and Burkhard Fricke used computer modeling to calculate the positions of elements up to Z = 172, and found that several were displaced from the Madelung rule. As a result of uncertainty and variability in predictions of chemical and physical properties of elements beyond 120, there is currently no consensus on their placement in the extended periodic
|
{
"page_id": 68326,
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table. Elements in this region are likely to be highly unstable with respect to radioactive decay and undergo alpha decay or spontaneous fission with extremely short half-lives, though element 126 is hypothesized to be within an island of stability that is resistant to fission but not to alpha decay. Other islands of stability beyond the known elements may also be possible, including one theorised around element 164, though the extent of stabilizing effects from closed nuclear shells is uncertain. It is not clear how many elements beyond the expected island of stability are physically possible, whether period 8 is complete, or if there is a period 9. The International Union of Pure and Applied Chemistry (IUPAC) defines an element to exist if its lifetime is longer than 10โ14 seconds (0.01 picoseconds, or 10 femtoseconds), which is the time it takes for the nucleus to form an electron cloud. As early as 1940, it was noted that a simplistic interpretation of the relativistic Dirac equation runs into problems with electron orbitals at Z > 1/ฮฑ โ 137.036 (the reciprocal of the fine-structure constant), suggesting that neutral atoms cannot exist beyond element 137, and that a periodic table of elements based on electron orbitals therefore breaks down at this point. On the other hand, a more rigorous analysis calculates the analogous limit to be Z โ 168โ172 where the 1s subshell dives into the Dirac sea, and that it is instead not neutral atoms that cannot exist beyond this point, but bare nuclei, thus posing no obstacle to the further extension of the periodic system. Atoms beyond this critical atomic number are called supercritical atoms. == History == Elements beyond the actinides were first proposed to exist as early as 1895, when Danish chemist Hans Peter Jรธrgen Julius Thomsen predicted that thorium
|
{
"page_id": 68326,
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"title": "Extended periodic table"
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and uranium formed part of a 32-element period which would end at a chemically inactive element with atomic weight 292 (not far from the 294 for the only known isotope of oganesson). In 1913, Swedish physicist Johannes Rydberg similarly predicted that the next noble gas after radon would have atomic number 118, and purely formally derived even heavier congeners of radon at Z = 168, 218, 290, 362, and 460, exactly where the Aufbau principle would predict them to be. In 1922, Niels Bohr predicted the electronic structure of this next noble gas at Z = 118, and suggested that the reason why elements beyond uranium were not seen in nature was because they were too unstable. The German physicist and engineer Richard Swinne published a review paper in 1926 containing predictions on the transuranic elements (he may have coined the term) in which he anticipated modern predictions of an island of stability: he first hypothesised in 1914 that half-lives should not decrease strictly with atomic number, but suggested instead that there might be some longer-lived elements at Z = 98โ102 and Z = 108โ110, and speculated that such elements might exist in the Earth's core, in iron meteorites, or in the ice caps of Greenland where they had been locked up from their supposed cosmic origin. By 1955, these elements were called superheavy elements. The first predictions on properties of undiscovered superheavy elements were made in 1957, when the concept of nuclear shells was first explored and an island of stability was theorized to exist around element 126. In 1967, more rigorous calculations were performed, and the island of stability was theorized to be centered at the then-undiscovered flerovium (element 114); this and other subsequent studies motivated many researchers to search for superheavy elements in nature or attempt to
|
{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
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synthesize them at accelerators. Many searches for superheavy elements were conducted in the 1970s, all with negative results. As of April 2022, synthesis has been attempted for every element up to and including unbiseptium (Z = 127), except unbitrium (Z = 123), with the heaviest successfully synthesized element being oganesson in 2002 and the most recent discovery being that of tennessine in 2010. As some superheavy elements were predicted to lie beyond the seven-period periodic table, an additional eighth period containing these elements was first proposed by Glenn T. Seaborg in 1969. This model continued the pattern in established elements and introduced a new g-block and superactinide series beginning at element 121, raising the number of elements in period 8 compared to known periods. These early calculations failed to consider relativistic effects that break down periodic trends and render simple extrapolation impossible, however. In 1971, Fricke calculated the periodic table up to Z = 172, and discovered that some elements indeed had different properties that break the established pattern, and a 2010 calculation by Pekka Pyykkรถ also noted that several elements might behave differently than expected. It is unknown how far the periodic table might extend beyond the known 118 elements, as heavier elements are predicted to be increasingly unstable. Glenn T. Seaborg suggested that practically speaking, the end of the periodic table might come as early as around Z = 120 due to nuclear instability. == Predicted structures of an extended periodic table == There is currently no consensus on the placement of elements beyond atomic number 120 in the periodic table. All hypothetical elements are given an International Union of Pure and Applied Chemistry (IUPAC) systematic element name, for use until the element has been discovered, confirmed, and an official name is approved. These names are typically not
|
{
"page_id": 68326,
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used in the literature, and the elements are instead referred to by their atomic numbers; hence, element 164 is usually not called "unhexquadium" or "Uhq" (the systematic name and symbol), but rather "element 164" with symbol "164", "(164)", or "E164". === Aufbau principle === At element 118, the orbitals 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, 5s, 5p, 5d, 5f, 6s, 6p, 6d, 7s and 7p are assumed to be filled, with the remaining orbitals unfilled. A simple extrapolation from the Aufbau principle would predict the eighth row to fill orbitals in the order 8s, 5g, 6f, 7d, 8p; but after element 120, the proximity of the electron shells makes placement in a simple table problematic. === Fricke === Not all models show the higher elements following the pattern established by lighter elements. Burkhard Fricke et al., who carried out calculations up to element 184 in an article published in 1971, also found some elements to be displaced from the Madelung energy-ordering rule as a result of overlapping orbitals; this is caused by the increasing role of relativistic effects in heavy elements (They describe chemical properties up to element 184, but only draw a table to element 172.) Fricke et al.'s format is more focused on formal electron configurations than likely chemical behaviour. They place elements 156โ164 in groups 4โ12 because formally their configurations should be 7d2 through 7d10. However, they differ from the previous d-elements in that the 8s shell is not available for chemical bonding: instead, the 9s shell is. Thus element 164 with 7d109s0 is noted by Fricke et al. to be analogous to palladium with 4d105s0, and they consider elements 157โ172 to have chemical analogies to groups 3โ18 (though they are ambivalent on whether elements 165 and 166 are more like group 1
|
{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
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and 2 elements or more like group 11 and 12 elements, respectively). Thus, elements 157โ164 are placed in their table in a group that the authors do not think is chemically most analogous. === Nefedov === Nefedov, Trzhaskovskaya, and Yarzhemskii carried out calculations up to 164 (results published in 2006). They considered elements 158 through 164 to be homologues of groups 4 through 10, and not 6 through 12, noting similarities of electron configurations to the period 5 transition metals (e.g. element 159 7d49s1 vs Nb 4d45s1, element 160 7d59s1 vs Mo 4d55s1, element 162 7d79s1 vs Ru 4d75s1, element 163 7d89s1 vs Rh 4d85s1, element 164 7d109s0 vs Pd 4d105s0). They thus agree with Fricke et al. on the chemically most analogous groups, but differ from them in that Nefedov et al. actually place elements in the chemically most analogous groups. Rg and Cn are given an asterisk to reflect differing configurations from Au and Hg (in the original publication they are drawn as being displaced in the third dimension). In fact Cn probably has an analogous configuration to Hg, and the difference in configuration between Pt and Ds is not marked. === Pyykkรถ === Pekka Pyykkรถ used computer modeling to calculate the positions of elements up to Z = 172 and their possible chemical properties in an article published in 2011. He reproduced the orbital order of Fricke et al., and proposed a refinement of their table by formally assigning slots to elements 121โ164 based on ionic configurations. In order to bookkeep the electrons, Pyykkรถ places some elements out of order: thus 139 and 140 are placed in groups 13 and 14 to reflect that the 8p1/2 shell needs to fill, and he distinguishes separate 5g, 8p1/2, and 6f series. Fricke et al. and Nefedov et al. do
|
{
"page_id": 68326,
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"title": "Extended periodic table"
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not attempt to break up these series. === Kulsha === Computational chemist Andrey Kulsha has suggested two forms of the extended periodic table up to 172 that build on and refine Nefedov et al.'s versions up to 164 with reference to Pyykkรถ's calculations. Based on their likely chemical properties, elements 157โ172 are placed by both forms as eighth-period congeners of yttrium through xenon in the fifth period; this extends Nefedov et al.'s placement of 157โ164 under yttrium through palladium, and agrees with the chemical analogies given by Fricke et al. Kulsha suggested two ways to deal with elements 121โ156, that lack precise analogues among earlier elements. In his first form (2011, after Pyykkรถ's paper was published), elements 121โ138 and 139โ156 are placed as two separate rows (together called "ultransition elements"), related by the addition of a 5g18 subshell into the core, as according to Pyykkรถ's calculations of oxidation states, they should, respectively, mimic lanthanides and actinides. In his second suggestion (2016), elements 121โ142 form a g-block (as they have 5g activity), while elements 143โ156 form an f-block placed under actinium through nobelium. Thus, period 8 emerges with 54 elements, and the next noble element after 118 is 172. === Smits et al. === In 2023 Smits, Dรผllmann, Indelicato, Nazarewicz, and Schwerdtfeger made another attempt to place elements from 119 to 170 in the periodic table based on their electron configurations. The configurations of a few elements (121โ124 and 168) did not allow them to be placed unambiguously. Element 145 appears twice, some places have double occupancy, and others are empty. == Searches for undiscovered elements == === Synthesis attempts === Attempts have been made to synthesise the period 8 elements up to unbiseptium, except unbitrium. All such attempts have been unsuccessful. An attempt to synthesise ununennium, the first period 8
|
{
"page_id": 68326,
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}
|
element, is ongoing as of 2025. ==== Ununennium (E119) ==== The synthesis of element 119 (ununennium) was first attempted in 1985 by bombarding a target of einsteinium-254 with calcium-48 ions at the superHILAC accelerator at Berkeley, California: 25499Es + 4820Ca โ 302119* โ no atoms No atoms were identified, leading to a limiting cross section of 300 nb. Later calculations suggest that the cross section of the 3n reaction (which would result in 299119 and three neutrons as products) would actually be six hundred thousand times lower than this upper bound, at 0.5 pb. From April to September 2012, an attempt to synthesize the isotopes 295119 and 296119 was made by bombarding a target of berkelium-249 with titanium-50 at the GSI Helmholtz Centre for Heavy Ion Research in Darmstadt, Germany. Based on the theoretically predicted cross section, it was expected that an ununennium atom would be synthesized within five months of the beginning of the experiment. Moreover, as berkelium-249 decays to californium-249 (the next element) with a short half-life of 327 days, this allowed elements 119 and 120 to be searched for simultaneously. 24997Bk + 5022Ti โ 299119* โ no atoms The experiment was originally planned to continue to November 2012, but was stopped early to make use of the 249Bk target to confirm the synthesis of tennessine (thus changing the projectiles to 48Ca). This reaction of 249Bk + 50Ti was predicted to be the most favorable practical reaction for formation of element 119, as it is rather asymmetrical, though also somewhat cold. (254Es + 48Ca would be superior, but preparing milligram quantities of 254Es for a target is difficult.) Nevertheless, the necessary change from the "silver bullet" 48Ca to 50Ti divides the expected yield of element 119 by about twenty, as the yield is strongly dependent on the asymmetry
|
{
"page_id": 68326,
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"title": "Extended periodic table"
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of the fusion reaction. Due to the predicted short half-lives, the GSI team used new "fast" electronics capable of registering decay events within microseconds. No atoms of element 119 were identified, implying a limiting cross section of 70 fb. The predicted actual cross section is around 40 fb, which is at the limits of current technology. The team at RIKEN in Wakล, Japan began bombarding curium-248 targets with a vanadium-51 beam in January 2018 to search for element 119. Curium was chosen as a target, rather than heavier berkelium or californium, as these heavier targets are difficult to prepare. The 248Cm targets were provided by Oak Ridge National Laboratory. RIKEN developed a high-intensity vanadium beam. The experiment began at a cyclotron while RIKEN upgraded its linear accelerators; the upgrade was completed in 2020. Bombardment may be continued with both machines until the first event is observed; the experiment is currently running intermittently for at least 100 days a year. The RIKEN team's efforts are being financed by the Emperor of Japan. The team at the JINR plans to attempt synthesis of element 119 in the future, probably via the 243Am + 54Cr reaction, but a precise timeframe has not been publicly released. ==== Unbinilium (E120) ==== Following their success in obtaining oganesson by the reaction between 249Cf and 48Ca in 2006, the team at the Joint Institute for Nuclear Research (JINR) in Dubna started similar experiments in MarchโApril 2007, in hope of creating element 120 (unbinilium) from nuclei of 58Fe and 244Pu. Isotopes of unbinilium are predicted to have alpha decay half-lives of the order of microseconds. Initial analysis revealed that no atoms of element 120 were produced, providing a limit of 400 fb for the cross section at the energy studied. 24494Pu + 5826Fe โ 302120* โ no atoms
|
{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
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The Russian team planned to upgrade their facilities before attempting the reaction again. In April 2007, the team at the GSI Helmholtz Centre for Heavy Ion Research in Darmstadt, Germany, attempted to create element 120 using uranium-238 and nickel-64: 23892U + 6428Ni โ 302120* โ no atoms No atoms were detected, providing a limit of 1.6 pb for the cross section at the energy provided. The GSI repeated the experiment with higher sensitivity in three separate runs in AprilโMay 2007, JanuaryโMarch 2008, and SeptemberโOctober 2008, all with negative results, reaching a cross section limit of 90 fb. In JuneโJuly 2010, and again in 2011, after upgrading their equipment to allow the use of more radioactive targets, scientists at the GSI attempted the more asymmetrical fusion reaction: 24896Cm + 5424Cr โ 302120 โ no atoms It was expected that the change in reaction would quintuple the probability of synthesizing element 120, as the yield of such reactions is strongly dependent on their asymmetry. Three correlated signals were observed that matched the predicted alpha decay energies of 299120 and its daughter 295Og, as well as the experimentally known decay energy of its granddaughter 291Lv. However, the lifetimes of these possible decays were much longer than expected, and the results could not be confirmed. In AugustโOctober 2011, a different team at the GSI using the TASCA facility tried a new, even more asymmetrical reaction: 24998Cf + 5022Ti โ 299120* โ no atoms This was also tried unsuccessfully the next year during the aforementioned attempt to make element 119 in the 249Bk+50Ti reaction, as 249Bk decays to 249Cf. Because of its asymmetry, the reaction between 249Cf and 50Ti was predicted to be the most favorable practical reaction for synthesizing unbinilium, although it is also somewhat cold. No unbinilium atoms were identified, implying a limiting
|
{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
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|
cross-section of 200 fb. Jens Volker Kratz predicted the actual maximum cross-section for producing element 120 by any of these reactions to be around 0.1 fb; in comparison, the world record for the smallest cross section of a successful reaction was 30 fb for the reaction 209Bi(70Zn,n)278Nh, and Kratz predicted a maximum cross-section of 20 fb for producing the neighbouring element 119. If these predictions are accurate, then synthesizing element 119 would be at the limits of current technology, and synthesizing element 120 would require new methods. In May 2021, the JINR announced plans to investigate the 249Cf+50Ti reaction in their new facility. However, the 249Cf target would have had to be made by the Oak Ridge National Laboratory in the United States, and after the Russian invasion of Ukraine began in February 2022, collaboration between the JINR and other institutes completely ceased due to sanctions. Consequently, the JINR now plans to try the 248Cm+54Cr reaction instead. A preparatory experiment for the use of 54Cr projectiles was conducted in late 2023, successfully synthesising 288Lv in the 238U+54Cr reaction, and the hope is for experiments to synthesise element 120 to begin by 2025. Starting from 2022, plans have also been made to use 88-inch cyclotron in the Lawrence Berkeley National Laboratory (LBNL) in Berkeley, California, United States to attempt to make new elements using 50Ti projectiles. First, the 244Pu+50Ti reaction was tested, successfully creating two atoms of 290Lv in 2024. Since this was successful, an attempt to make element 120 in the 249Cf+50Ti reaction is planned to begin in 2025. The Lawrence Livermore National Laboratory (LLNL), which previously collaborated with the JINR, will collaborate with the LBNL on this project. ==== Unbiunium (E121) ==== The synthesis of element 121 (unbiunium) was first attempted in 1977 by bombarding a target of uranium-238
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{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
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with copper-65 ions at the Gesellschaft fรผr Schwerionenforschung in Darmstadt, Germany: 23892U + 6529Cu โ 303121* โ no atoms No atoms were identified. ==== Unbibium (E122) ==== The first attempts to synthesize element 122 (unbibium) were performed in 1972 by Flerov et al. at the Joint Institute for Nuclear Research (JINR), using the heavy-ion induced hot fusion reactions: 23892U + 66,6830Zn โ 304, 306122* โ no atoms These experiments were motivated by early predictions on the existence of an island of stability at N = 184 and Z > 120. No atoms were detected and a yield limit of 5 nb (5,000 pb) was measured. Current results (see flerovium) have shown that the sensitivity of these experiments were too low by at least 3 orders of magnitude. In 2000, the Gesellschaft fรผr Schwerionenforschung (GSI) Helmholtz Center for Heavy Ion Research performed a very similar experiment with much higher sensitivity: 23892U + 7030Zn โ 308122* โ no atoms These results indicate that the synthesis of such heavier elements remains a significant challenge and further improvements of beam intensity and experimental efficiency is required. The sensitivity should be increased to 1 fb in the future for better quality results. Another unsuccessful attempt to synthesize element 122 was carried out in 1978 at the GSI Helmholtz Center, where a natural erbium target was bombarded with xenon-136 ions: nat68Er + 13654Xe โ 298, 300, 302, 303, 304, 306122* โ no atoms In particular, the reaction between 170Er and 136Xe was expected to yield alpha-emitters with half-lives of microseconds that would decay down to isotopes of flerovium with half-lives perhaps increasing up to several hours, as flerovium is predicted to lie near the center of the island of stability. After twelve hours of irradiation, nothing was found in this reaction. Following a similar unsuccessful attempt
|
{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
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|
to synthesize element 121 from 238U and 65Cu, it was concluded that half-lives of superheavy nuclei must be less than one microsecond or the cross sections are very small. More recent research into synthesis of superheavy elements suggests that both conclusions are true. The two attempts in the 1970s to synthesize element 122 were both propelled by the research investigating whether superheavy elements could potentially be naturally occurring. Several experiments studying the fission characteristics of various superheavy compound nuclei such as 306122* were performed between 2000 and 2004 at the Flerov Laboratory of Nuclear Reactions. Two nuclear reactions were used, namely 248Cm + 58Fe and 242Pu + 64Ni. The results reveal how superheavy nuclei fission predominantly by expelling closed shell nuclei such as 132Sn (Z = 50, N = 82). It was also found that the yield for the fusion-fission pathway was similar between 48Ca and 58Fe projectiles, suggesting a possible future use of 58Fe projectiles in superheavy element formation. ==== Unbiquadium (E124) ==== Scientists at GANIL (Grand Accรฉlรฉrateur National d'Ions Lourds) attempted to measure the direct and delayed fission of compound nuclei of elements with Z = 114, 120, and 124 in order to probe shell effects in this region and to pinpoint the next spherical proton shell. This is because having complete nuclear shells (or, equivalently, having a magic number of protons or neutrons) would confer more stability on the nuclei of such superheavy elements, thus moving closer to the island of stability. In 2006, with full results published in 2008, the team provided results from a reaction involving the bombardment of a natural germanium target with uranium ions: 23892U + nat32Ge โ 308, 310, 311, 312, 314124* โ fission The team reported that they had been able to identify compound nuclei fissioning with half-lives > 10โ18 s.
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{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
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This result suggests a strong stabilizing effect at Z = 124 and points to the next proton shell at Z > 120, not at Z = 114 as previously thought. A compound nucleus is a loose combination of nucleons that have not arranged themselves into nuclear shells yet. It has no internal structure and is held together only by the collision forces between the target and projectile nuclei. It is estimated that it requires around 10โ14 s for the nucleons to arrange themselves into nuclear shells, at which point the compound nucleus becomes a nuclide, and this number is used by IUPAC as the minimum half-life a claimed isotope must have to potentially be recognised as being discovered. Thus, the GANIL experiments do not count as a discovery of element 124. The fission of the compound nucleus 312124 was also studied in 2006 at the tandem ALPI heavy-ion accelerator at the Laboratori Nazionali di Legnaro (Legnaro National Laboratories) in Italy: 23290Th + 8034Se โ 312124* โ fission Similarly to previous experiments conducted at the JINR (Joint Institute for Nuclear Research), fission fragments clustered around doubly magic nuclei such as 132Sn (Z = 50, N = 82), revealing a tendency for superheavy nuclei to expel such doubly magic nuclei in fission. The average number of neutrons per fission from the 312124 compound nucleus (relative to lighter systems) was also found to increase, confirming that the trend of heavier nuclei emitting more neutrons during fission continues into the superheavy mass region. ==== Unbipentium (E125) ==== The first and only attempt to synthesize element 125 (unbipentium) was conducted in Dubna in 1970โ1971 using zinc ions and an americium-243 target: 24395Am + 66, 6830Zn โ 309, 311125* โ no atoms No atoms were detected, and a cross section limit of 5 nb was determined.
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{
"page_id": 68326,
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"title": "Extended periodic table"
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This experiment was motivated by the possibility of greater stability for nuclei around Z ~ 126 and N ~ 184, though more recent research suggests the island of stability may instead lie at a lower atomic number (such as copernicium, Z = 112), and the synthesis of heavier elements such as element 125 will require more sensitive experiments. ==== Unbihexium (E126) ==== The first and only attempt to synthesize element 126 (unbihexium), which was unsuccessful, was performed in 1971 at CERN (European Organization for Nuclear Research) by Renรฉ Bimbot and John M. Alexander using the hot fusion reaction: 23290Th + 8436Kr โ 316126* โ no atoms High-energy (13โ15 MeV) alpha particles were observed and taken as possible evidence for the synthesis of element 126. Subsequent unsuccessful experiments with higher sensitivity suggest that the 10 mb sensitivity of this experiment was too low; hence, the formation of element 126 nuclei in this reaction is highly unlikely. ==== Unbiseptium (E127) ==== The first and only attempt to synthesize element 127 (unbiseptium), which was unsuccessful, was performed in 1978 at the UNILAC accelerator at the GSI Helmholtz Center, where a natural tantalum target was bombarded with xenon-136 ions: nat73Ta + 13654Xe โ 316, 317127* โ no atoms === Searches in nature === A study in 1976 by a group of American researchers from several universities proposed that primordial superheavy elements, mainly livermorium, elements 124, 126, and 127, could be a cause of unexplained radiation damage (particularly radiohalos) in minerals. This prompted many researchers to search for them in nature from 1976 to 1983. A group led by Tom Cahill, a professor at the University of California at Davis, claimed in 1976 that they had detected alpha particles and X-rays with the right energies to cause the damage observed, supporting the presence of these
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{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
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elements. In particular, the presence of long-lived (on the order of 109 years) nuclei of elements 124 and 126, along with their decay products, at an abundance of 10โ11 relative to their possible congeners uranium and plutonium, was conjectured. Others claimed that none had been detected, and questioned the proposed characteristics of primordial superheavy nuclei. In particular, they cited that any such superheavy nuclei must have a closed neutron shell at N = 184 or N = 228, and this necessary condition for enhanced stability only exists in neutron deficient isotopes of livermorium or neutron rich isotopes of the other elements that would not be beta-stable unlike most naturally occurring isotopes. This activity was also proposed to be caused by nuclear transmutations in natural cerium, raising further ambiguity upon this claimed observation of superheavy elements. On April 24, 2008, a group led by Amnon Marinov at the Hebrew University of Jerusalem claimed to have found single atoms of 292122 in naturally occurring thorium deposits at an abundance of between 10โ11 and 10โ12 relative to thorium. The claim of Marinov et al. was criticized by a part of the scientific community. Marinov claimed that he had submitted the article to the journals Nature and Nature Physics but both turned it down without sending it for peer review. The 292122 atoms were claimed to be superdeformed or hyperdeformed isomers, with a half-life of at least 100 million years. A criticism of the technique, previously used in purportedly identifying lighter thorium isotopes by mass spectrometry, was published in Physical Review C in 2008. A rebuttal by the Marinov group was published in Physical Review C after the published comment. A repeat of the thorium experiment using the superior method of Accelerator Mass Spectrometry (AMS) failed to confirm the results, despite a 100-fold better
|
{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
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sensitivity. This result throws considerable doubt on the results of the Marinov collaboration with regard to their claims of long-lived isotopes of thorium, roentgenium and element 122. It is still possible that traces of unbibium might only exist in some thorium samples, although this is unlikely. The possible extent of primordial superheavy elements on Earth today is uncertain. Even if they are confirmed to have caused the radiation damage long ago, they might now have decayed to mere traces, or even be completely gone. It is also uncertain if such superheavy nuclei may be produced naturally at all, as spontaneous fission is expected to terminate the r-process responsible for heavy element formation between mass number 270 and 290, well before elements beyond 120 may be formed. A recent hypothesis tries to explain the spectrum of Przybylski's Star by naturally occurring flerovium and element 120. == Predicted properties of eighth-period elements == Element 118, oganesson, is the heaviest element that has been synthesized. The next two elements, elements 119 and 120, should form an 8s series and be an alkali and alkaline earth metal, respectively. Beyond element 120, the superactinide series is expected to begin, when the 8s electrons and the filling of the 8p1/2, 7d3/2, 6f, and 5g subshells determine the chemistry of these elements. Complete and accurate CCSD calculations are not available for elements beyond 122 because of the extreme complexity of the situation: the 5g, 6f, and 7d orbitals should have about the same energy level, and in the region of element 160, the 9s, 8p3/2, and 9p1/2 orbitals should also be about equal in energy. This will cause the electron shells to mix so that the block concept no longer applies very well, and will also result in novel chemical properties that will make positioning some of
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{
"page_id": 68326,
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these elements in a periodic table very difficult. === Chemical and physical properties === ==== Elements 119 and 120 ==== The first two elements of period 8 will be ununennium and unbinilium, elements 119 and 120. Their electron configurations should have the 8s orbital being filled. This orbital is relativistically stabilized and contracted; thus, elements 119 and 120 should be more like rubidium and strontium than their immediate neighbours above, francium and radium. Another effect of the relativistic contraction of the 8s orbital is that the atomic radii of these two elements should be about the same as those of francium and radium. They should behave like normal alkali and alkaline earth metals (albeit less reactive than their immediate vertical neighbours), normally forming +1 and +2 oxidation states, respectively, but the relativistic destabilization of the 7p3/2 subshell and the relatively low ionization energies of the 7p3/2 electrons should make higher oxidation states like +3 and +4 (respectively) possible as well. ==== Superactinides ==== The superactinides may range from elements 121 through 157, which can be classified as the 5g and 6f elements of the eighth period, together with the first 7d element. In the superactinide series, the 7d3/2, 8p1/2, 6f5/2 and 5g7/2 shells should all fill simultaneously. This creates very complicated situations, so much so that complete and accurate CCSD calculations have been done only for elements 121 and 122. The first superactinide, unbiunium or eka-actinium (element 121), should be similar to lanthanum and actinium: its main oxidation state should be +3, although the closeness of the valence subshells' energy levels may permit higher oxidation states, just as in elements 119 and 120. Relativistic stabilization of the 8p subshell should result in a ground-state 8s28p1 valence electron configuration for element 121, in contrast to the ds2 configurations of lanthanum and
|
{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
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actinium; nevertheless, this anomalous configuration does not appear to affect its calculated chemistry, which remains similar to that of actinium. Its first ionization energy is predicted to be 429.4 kJ/mol, which would be lower than those of all known elements except for the alkali metals potassium, rubidium, caesium, and francium: this value is even lower than that of the period 8 alkali metal ununennium (463.1 kJ/mol). Similarly, the next superactinide, unbibium or eka-thorium (element 122), may be similar to cerium and thorium, with a main oxidation state of +4, but would have a ground-state 7d18s28p1 or 8s28p2 valence electron configuration, unlike thorium's 6d27s2 configuration. Hence, its first ionization energy would be smaller than thorium's (Th: 6.3 eV; element 122: 5.6 eV) because of the greater ease of ionizing unbibium's 8p1/2 electron than thorium's 6d electron. The collapse of the 5g orbital itself is delayed until around element 125 (unbipentium or eka-neptunium); the electron configurations of the 119-electron isoelectronic series are expected to be [Og]8s1 for elements 119 through 122, [Og]6f1 for elements 123 and 124, and [Og]5g1 for element 125 onwards. In the first few superactinides, the binding energies of the added electrons are predicted to be small enough that they can lose all their valence electrons; for example, unbihexium (element 126) could easily form a +8 oxidation state, and even higher oxidation states for the next few elements may be possible. Element 126 is also predicted to display a variety of other oxidation states: recent calculations have suggested a stable monofluoride 126F may be possible, resulting from a bonding interaction between the 5g orbital on element 126 and the 2p orbital on fluorine. Other predicted oxidation states include +2, +4, and +6; +4 is expected to be the most usual oxidation state of unbihexium. The superactinides from unbipentium (element
|
{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
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|
125) to unbiennium (element 129) are predicted to exhibit a +6 oxidation state and form hexafluorides, though 125F6 and 126F6 are predicted to be relatively weakly bound. The bond dissociation energies are expected to greatly increase at element 127 and even more so at element 129. This suggests a shift from strong ionic character in fluorides of element 125 to more covalent character, involving the 8p orbital, in fluorides of element 129. The bonding in these superactinide hexafluorides is mostly between the highest 8p subshell of the superactinide and the 2p subshell of fluorine, unlike how uranium uses its 5f and 6d orbitals for bonding in uranium hexafluoride. Despite the ability of early superactinides to reach high oxidation states, it has been calculated that the 5g electrons will be most difficult to ionize; the 1256+ and 1267+ ions are expected to bear a 5g1 configuration, similar to the 5f1 configuration of the Np6+ ion. Similar behavior is observed in the low chemical activity of the 4f electrons in lanthanides; this is a consequence of the 5g orbitals being small and deeply buried in the electron cloud. The presence of electrons in g-orbitals, which do not exist in the ground state electron configuration of any currently known element, should allow presently unknown hybrid orbitals to form and influence the chemistry of the superactinides in new ways, although the absence of g electrons in known elements makes predicting superactinide chemistry more difficult. In the later superactinides, the oxidation states should become lower. By element 132, the predominant most stable oxidation state will be only +6; this is further reduced to +3 and +4 by element 144, and at the end of the superactinide series it will be only +2 (and possibly even 0) because the 6f shell, which is being filled at
|
{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
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that point, is deep inside the electron cloud and the 8s and 8p1/2 electrons are bound too strongly to be chemically active. The 5g shell should be filled at element 144 and the 6f shell at around element 154, and at this region of the superactinides the 8p1/2 electrons are bound so strongly that they are no longer active chemically, so that only a few electrons can participate in chemical reactions. Calculations by Fricke et al. predict that at element 154, the 6f shell is full and there are no d- or other electron wave functions outside the chemically inactive 8s and 8p1/2 shells. This may cause element 154 to be rather unreactive with noble gas-like properties. Calculations by Pyykkรถ nonetheless expect that at element 155, the 6f shell is still chemically ionizable: 1553+ should have a full 6f shell, and the fourth ionization potential should be between those of terbium and dysprosium, both of which are known in the +4 state. Similarly to the lanthanide and actinide contractions, there should be a superactinide contraction in the superactinide series where the ionic radii of the superactinides are smaller than expected. In the lanthanides, the contraction is about 4.4 pm per element; in the actinides, it is about 3 pm per element. The contraction is larger in the lanthanides than in the actinides due to the greater localization of the 4f wave function as compared to the 5f wave function. Comparisons with the wave functions of the outer electrons of the lanthanides, actinides, and superactinides lead to a prediction of a contraction of about 2 pm per element in the superactinides; although this is smaller than the contractions in the lanthanides and actinides, its total effect is larger due to the fact that 32 electrons are filled in the deeply buried
|
{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
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5g and 6f shells, instead of just 14 electrons being filled in the 4f and 5f shells in the lanthanides and actinides, respectively. Pekka Pyykkรถ divides these superactinides into three series: a 5g series (elements 121 to 138), an 8p1/2 series (elements 139 to 140), and a 6f series (elements 141 to 155), also noting that there would be a great deal of overlapping between energy levels and that the 6f, 7d, or 8p1/2 orbitals could well also be occupied in the early superactinide atoms or ions. He also expects that they would behave more like "superlanthanides", in the sense that the 5g electrons would mostly be chemically inactive, similarly to how only one or two 4f electrons in each lanthanide are ever ionized in chemical compounds. He also predicted that the possible oxidation states of the superactinides might rise very high in the 6f series, to values such as +12 in element 148. Andrey Kulsha has called the elements 121 to 156 "ultransition" elements and has proposed to split them into two series of eighteen each, one from elements 121 to 138 and another from elements 139 to 156. The first would be analogous to the lanthanides, with oxidation states mainly ranging from +4 to +6, as the filling of the 5g shell dominates and neighbouring elements are very similar to each other, creating an analogy to uranium, neptunium, and plutonium. The second would be analogous to the actinides: at the beginning (around elements in the 140s) very high oxidation states would be expected as the 6f shell rises above the 7d one, but after that the typical oxidation states would lower and in elements in the 150s onwards the 8p1/2 electrons would stop being chemically active. Because the two rows are separated by the addition of a complete
|
{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
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5g18 subshell, they could be considered analogues of each other as well. As an example from the late superactinides, element 156 is expected to exhibit mainly the +2 oxidation state, on account of its electron configuration with easily removed 7d2 electrons over a stable [Og]5g186f148s28p21/2 core. It can thus be considered a heavier congener of nobelium, which likewise has a pair of easily removed 7s2 electrons over a stable [Rn]5f14 core, and is usually in the +2 state (strong oxidisers are required to obtain nobelium in the +3 state). Its first ionization energy should be about 400 kJ/mol and its metallic radius approximately 170 picometers. With a relative atomic mass of around 445 u, it should be a very heavy metal with a density of around 26 g/cm3. ==== Elements 157 to 166 ==== The 7d transition metals in period 8 are expected to be elements 157 to 166. Although the 8s and 8p1/2 electrons are bound so strongly in these elements that they should not be able to take part in any chemical reactions, the 9s and 9p1/2 levels are expected to be readily available for hybridization. These 7d elements should be similar to the 4d elements yttrium through cadmium. In particular, element 164 with a 7d109s0 electron configuration shows clear analogies with palladium with its 4d105s0 electron configuration. The noble metals of this series of transition metals are not expected to be as noble as their lighter homologues, due to the absence of an outer s shell for shielding and also because the 7d shell is strongly split into two subshells due to relativistic effects. This causes the first ionization energies of the 7d transition metals to be smaller than those of their lighter congeners. Theoretical interest in the chemistry of unhexquadium is largely motivated by theoretical predictions
|
{
"page_id": 68326,
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that it, especially the isotopes 472164 and 482164 (with 164 protons and 308 or 318 neutrons), would be at the center of a hypothetical second island of stability (the first being centered on copernicium, particularly the isotopes 291Cn, 293Cn, and 296Cn which are expected to have half-lives of centuries or millennia). Calculations predict that the 7d electrons of element 164 (unhexquadium) should participate very readily in chemical reactions, so that it should be able to show stable +6 and +4 oxidation states in addition to the normal +2 state in aqueous solutions with strong ligands. Element 164 should thus be able to form compounds like 164(CO)4, 164(PF3)4 (both tetrahedral like the corresponding palladium compounds), and 164(CN)2โ2 (linear), which is very different behavior from that of lead, which element 164 would be a heavier homologue of if not for relativistic effects. Nevertheless, the divalent state would be the main one in aqueous solution (although the +4 and +6 states would be possible with stronger ligands), and unhexquadium(II) should behave more similarly to lead than unhexquadium(IV) and unhexquadium(VI). Element 164 is expected to be a soft Lewis acid and have Ahrlands softness parameter close to 4 eV. It should be at most moderately reactive, having a first ionization energy that should be around 685 kJ/mol, comparable to that of molybdenum. Due to the lanthanide, actinide, and superactinide contractions, element 164 should have a metallic radius of only 158 pm, very close to that of the much lighter magnesium, despite its expected atomic weight of around 474 u which is about 19.5 times the atomic weight of magnesium. This small radius and high weight cause it to be expected to have an extremely high density of around 46 gยทcmโ3, over twice that of osmium, currently the most dense element known, at 22.61 gยทcmโ3;
|
{
"page_id": 68326,
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"title": "Extended periodic table"
}
|
element 164 should be the second most dense element in the first 172 elements in the periodic table, with only its neighbor unhextrium (element 163) being more dense (at 47 gยทcmโ3). Metallic element 164 should have a very large cohesive energy (enthalpy of crystallization) due to its covalent bonds, most probably resulting in a high melting point. In the metallic state, element 164 should be quite noble and analogous to palladium and platinum. Fricke et al. suggested some formal similarities to oganesson, as both elements have closed-shell configurations and similar ionisation energies, although they note that while oganesson would be a very bad noble gas, element 164 would be a good noble metal. Elements 165 (unhexpentium) and 166 (unhexhexium), the last two 7d metals, should behave similarly to alkali and alkaline earth metals when in the +1 and +2 oxidation states, respectively. The 9s electrons should have ionization energies comparable to those of the 3s electrons of sodium and magnesium, due to relativistic effects causing the 9s electrons to be much more strongly bound than non-relativistic calculations would predict. Elements 165 and 166 should normally exhibit the +1 and +2 oxidation states, respectively, although the ionization energies of the 7d electrons are low enough to allow higher oxidation states like +3 for element 165. The oxidation state +4 for element 166 is less likely, creating a situation similar to the lighter elements in groups 11 and 12 (particularly gold and mercury). As with mercury but not copernicium, ionization of element 166 to 1662+ is expected to result in a 7d10 configuration corresponding to the loss of the s-electrons but not the d-electrons, making it more analogous to the lighter "less relativistic" group 12 elements zinc, cadmium, and mercury. ==== Elements 167 to 172 ==== The next six elements on the
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{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
}
|
periodic table are expected to be the last main-group elements in their period, and are likely to be similar to the 5p elements indium through xenon. In elements 167 to 172, the 9p1/2 and 8p3/2 shells will be filled. Their energy eigenvalues are so close together that they behave as one combined p-subshell, similar to the non-relativistic 2p and 3p subshells. Thus, the inert-pair effect does not occur and the most common oxidation states of elements 167 to 170 are expected to be +3, +4, +5, and +6, respectively. Element 171 (unseptunium) is expected to show some similarities to the halogens, showing various oxidation states ranging from โ1 to +7, although its physical properties are expected to be closer to that of a metal. Its electron affinity is expected to be 3.0 eV, allowing it to form H171, analogous to a hydrogen halide. The 171โ ion is expected to be a soft base, comparable to iodide (Iโ). Element 172 (unseptbium) is expected to be a noble gas with chemical behaviour similar to that of xenon, as their ionization energies should be very similar (Xe, 1170.4 kJ/mol; element 172, 1090 kJ/mol). The only main difference between them is that element 172, unlike xenon, is expected to be a liquid or a solid at standard temperature and pressure due to its much higher atomic weight. Unseptbium is expected to be a strong Lewis acid, forming fluorides and oxides, similarly to its lighter congener xenon. Because of some analogy of elements 165โ172 to periods 2 and 3, Fricke et al. considered them to form a ninth period of the periodic table, while the eighth period was taken by them to end at the noble metal element 164. This ninth period would be similar to the second and third period in having no transition
|
{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
}
|
metals. That being said, the analogy is incomplete for elements 165 and 166; although they do start a new s-shell (9s), this is above a d-shell, making them chemically more similar to groups 11 and 12. ==== Beyond element 172 ==== Beyond element 172, there is the potential to fill the 6g, 7f, 8d, 10s, 10p1/2, and perhaps 6h11/2 shells. These electrons would be very loosely bound, potentially rendering extremely high oxidation states reachable, though the electrons would become more tightly bound as the ionic charge rises. Thus, there will probably be another very long transition series, like the superactinides. In element 173 (unsepttrium), the outermost electron might enter the 6g7/2, 9p3/2, or 10s subshells. Because spinโorbit interactions would create a very large energy gap between these and the 8p3/2 subshell, this outermost electron is expected to be very loosely bound and very easily lost to form a 173+ cation. As a result, element 173 is expected to behave chemically like an alkali metal, and one that might be far more reactive than even caesium (francium and element 119 being less reactive than caesium due to relativistic effects): the calculated ionisation energy for element 173 is 3.070 eV, compared to the experimentally known 3.894 eV for caesium. Element 174 (unseptquadium) may add an 8d electron and form a closed-shell 1742+ cation; its calculated ionisation energy is 3.614 eV. Element 184 (unoctquadium) was significantly targeted in early predictions, as it was originally speculated that 184 would be a proton magic number: it is predicted to have an electron configuration of [172] 6g5 7f4 8d3, with at least the 7f and 8d electrons chemically active. Its chemical behaviour is expected to be similar to uranium and neptunium, as further ionisation past the +6 state (corresponding to removal of the 6g electrons) is
|
{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
}
|
likely to be unprofitable; the +4 state should be most common in aqueous solution, with +5 and +6 reachable in solid compounds. === End of the periodic table === The number of physically possible elements is unknown. A low estimate is that the periodic table may end soon after the island of stability, which is expected to center on Z = 126, as the extension of the periodic and nuclide tables is restricted by the proton and the neutron drip lines and stability toward alpha decay and spontaneous fission. One calculation by Y. Gambhir et al., analyzing nuclear binding energy and stability in various decay channels, suggests a limit to the existence of bound nuclei at Z = 146. Other predictions of an end to the periodic table include Z = 128 (John Emsley) and Z = 155 (Albert Khazan). ==== Elements above the atomic number 137 ==== It is a "folk legend" among physicists that Richard Feynman suggested that neutral atoms could not exist for atomic numbers greater than Z = 137, on the grounds that the relativistic Dirac equation predicts that the ground-state energy of the innermost electron in such an atom would be an imaginary number. Here, the number 137 arises as the inverse of the fine-structure constant. By this argument, neutral atoms cannot exist beyond atomic number 137, and therefore a periodic table of elements based on electron orbitals breaks down at this point. However, this argument presumes that the atomic nucleus is pointlike. A more accurate calculation must take into account the small, but nonzero, size of the nucleus, which is predicted to push the limit further to Z โ 173. ===== Bohr model ===== The Bohr model exhibits difficulty for atoms with atomic number greater than 137, for the speed of an electron in
|
{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
}
|
a 1s electron orbital, v, is given by v = Z ฮฑ c โ Z c 137.04 {\displaystyle v=Z\alpha c\approx {\frac {Zc}{137.04}}} where Z is the atomic number, and ฮฑ is the fine-structure constant, a measure of the strength of electromagnetic interactions. Under this approximation, any element with an atomic number of greater than 137 would require 1s electrons to be traveling faster than c, the speed of light. Hence, the non-relativistic Bohr model is inaccurate when applied to such an element. ===== Relativistic Dirac equation ===== The relativistic Dirac equation gives the ground state energy as E = m c 2 1 + Z 2 ฮฑ 2 ( n โ ( j + 1 2 ) + ( j + 1 2 ) 2 โ Z 2 ฮฑ 2 ) 2 , {\displaystyle E={\frac {mc^{2}}{\sqrt {1+{\dfrac {Z^{2}\alpha ^{2}}{{\bigg (}{n-\left(j+{\frac {1}{2}}\right)+{\sqrt {\left(j+{\frac {1}{2}}\right)^{2}-Z^{2}\alpha ^{2}}}{\bigg )}}^{2}}}}}},} where m is the rest mass of the electron. For Z > 137, the wave function of the Dirac ground state is oscillatory, rather than bound, and there is no gap between the positive and negative energy spectra, as in the Klein paradox. More accurate calculations taking into account the effects of the finite size of the nucleus indicate that the binding energy first exceeds 2mc2 for Z > Zcr probably between 168 and 172. For Z > Zcr, if the innermost orbital (1s) is not filled, the electric field of the nucleus will pull an electron out of the vacuum, resulting in the spontaneous emission of a positron. This diving of the 1s subshell into the negative continuum has often been taken to constitute an "end" to the periodic table, but in fact it does not impose such a limit, as such resonances can be interpreted as Gamow states. Nonetheless, the accurate description of
|
{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
}
|
such states in a multi-electron system, needed to extend calculations and the periodic table past Zcr โ 172, are still open problems. Atoms with atomic numbers above Zcr โ 172 have been termed supercritical atoms. Supercritical atoms cannot be totally ionised because their 1s subshell would be filled by spontaneous pair creation in which an electron-positron pair is created from the negative continuum, with the electron being bound and the positron escaping. However, the strong field around the atomic nucleus is restricted to a very small region of space, so that the Pauli exclusion principle forbids further spontaneous pair creation once the subshells that have dived into the negative continuum are filled. Elements 173โ184 have been termed weakly supercritical atoms as for them only the 1s shell has dived into the negative continuum; the 2p1/2 shell is expected to join around element 185 and the 2s shell around element 245. Experiments have so far not succeeded in detecting spontaneous pair creation from assembling supercritical charges through the collision of heavy nuclei (e.g. colliding lead with uranium to momentarily give an effective Z of 174; uranium with uranium gives effective Z = 184 and uranium with californium gives effective Z = 190). Even though passing Zcr does not mean elements can no longer exist, the increasing concentration of the 1s density close to the nucleus would likely make these electrons more vulnerable to K electron capture as Zcr is approached. For such heavy elements, these 1s electrons would likely spend a significant fraction of time so close to the nucleus that they are actually inside it. This may pose another limit to the periodic table. Because of the factor of m, muonic atoms become supercritical at a much larger atomic number of around 2200, as muons are about 207 times as
|
{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
}
|
heavy as electrons. ===== Quark matter ===== It has also been posited that in the region beyond A > 300, an entire "continent of stability" consisting of a hypothetical phase of stable quark matter, comprising freely flowing up and down quarks rather than quarks bound into protons and neutrons, may exist. Such a form of matter is theorized to be a ground state of baryonic matter with a greater binding energy per baryon than nuclear matter, favoring the decay of nuclear matter beyond this mass threshold into quark matter. If this state of matter exists, it could possibly be synthesized in the same fusion reactions leading to normal superheavy nuclei, and would be stabilized against fission as a consequence of its stronger binding that is enough to overcome Coulomb repulsion. Calculations published in 2020 suggest stability of up-down quark matter (udQM) nuggets against conventional nuclei beyond A ~ 266, and also show that udQM nuggets become supercritical earlier (Zcr ~ 163, A ~ 609) than conventional nuclei (Zcr ~ 177, A ~ 480). === Nuclear properties === ==== Magic numbers and the island of stability ==== The stability of nuclei decreases greatly with the increase in atomic number after curium, element 96, so that all isotopes with an atomic number above 101 decay radioactively with a half-life under a day. No elements with atomic numbers above 82 (after lead) have stable isotopes. Nevertheless, because of reasons not very well understood yet, there is a slight increased nuclear stability around atomic numbers 110โ114, which leads to the appearance of what is known in nuclear physics as the "island of stability". This concept, proposed by University of California professor Glenn Seaborg, explains why superheavy elements last longer than predicted. Calculations according to the HartreeโFockโBogoliubov method using the non-relativistic Skyrme interaction have proposed
|
{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
}
|
Z = 126 as a closed proton shell. In this region of the periodic table, N = 184, N = 196, and N = 228 have been suggested as closed neutron shells. Therefore, the isotopes of most interest are 310126, 322126, and 354126, for these might be considerably longer-lived than other isotopes. Element 126, having a magic number of protons, is predicted to be more stable than other elements in this region, and may have nuclear isomers with very long half-lives. It is also possible that the island of stability is instead centered at 306122, which may be spherical and doubly magic. Probably, the island of stability occurs around Z = 114โ126 and N = 184, with lifetimes probably around hours to days. Beyond the shell closure at N = 184, spontaneous fission lifetimes should drastically drop below 10โ15 seconds โ too short for a nucleus to obtain an electron cloud and participate in any chemistry. That being said, such lifetimes are very model-dependent, and predictions range across many orders of magnitude. Taking nuclear deformation and relativistic effects into account, an analysis of single-particle levels predicts new magic numbers for superheavy nuclei at Z = 126, 138, 154, and 164 and N = 228, 308, and 318. Therefore, in addition to the island of stability centered at 291Cn, 293Cn, and 298Fl, further islands of stability may exist around the doubly magic 354126 as well as 472164 or 482164. These nuclei are predicted to be beta-stable and decay by alpha emission or spontaneous fission with relatively long half-lives, and confer additional stability on neighboring N = 228 isotones and elements 152โ168, respectively. On the other hand, the same analysis suggests that proton shell closures may be relatively weak or even nonexistent in some cases such as 354126, meaning that such nuclei
|
{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
}
|
might not be doubly magic and stability will instead be primarily determined by strong neutron shell closures. Additionally, due to the enormously greater forces of electromagnetic repulsion that must be overcome by the strong force at the second island (Z = 164), it is possible that nuclei around this region only exist as resonances and cannot stay together for a meaningful amount of time. It is also possible that some of the superactinides between these series may not actually exist because they are too far from both islands, in which case the periodic table might end around Z = 130. The area of elements 121โ156 where periodicity is in abeyance is quite similar to the gap between the two islands. Beyond element 164, the fissility line defining the limit of stability with respect to spontaneous fission may converge with the neutron drip line, posing a limit to the existence of heavier elements. Nevertheless, further magic numbers have been predicted at Z = 210, 274, and 354 and N = 308, 406, 524, 644, and 772, with two beta-stable doubly magic nuclei found at 616210 and 798274; the same calculation method reproduced the predictions for 298Fl and 472164. (The doubly magic nuclei predicted for Z = 354 are beta-unstable, with 998354 being neutron-deficient and 1126354 being neutron-rich.) Although additional stability toward alpha decay and fission are predicted for 616210 and 798274, with half-lives up to hundreds of microseconds for 616210, there will not exist islands of stability as significant as those predicted at Z = 114 and 164. As the existence of superheavy elements is very strongly dependent on stabilizing effects from closed shells, nuclear instability and fission will likely determine the end of the periodic table beyond these islands of stability. The International Union of Pure and Applied Chemistry (IUPAC)
|
{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
}
|
defines an element to exist if its lifetime is longer than 10โ14 seconds, which is the time it takes for the nucleus to form an electron cloud. However, a nuclide is generally considered to exist if its lifetime is longer than about 10โ22 seconds, which is the time it takes for nuclear structure to form. Consequently, it is possible that some Z values can only be realised in nuclides and that the corresponding elements do not exist. It is also possible that no further islands actually exist beyond 126, as the nuclear shell structure gets smeared out (as the electron shell structure already is expected to be around oganesson) and low-energy decay modes become readily available. In some regions of the table of nuclides, there are expected to be additional regions of stability due to non-spherical nuclei that have different magic numbers than spherical nuclei do; the egg-shaped 270Hs (Z = 108, N = 162) is one such deformed doubly magic nucleus. In the superheavy region, the strong Coulomb repulsion of protons may cause some nuclei, including isotopes of oganesson, to assume a bubble shape in the ground state with a reduced central density of protons, unlike the roughly uniform distribution inside most smaller nuclei. Such a shape would have a very low fission barrier, however. Even heavier nuclei in some regions, such as 342136 and 466156, may instead become toroidal or red blood cell-like in shape, with their own magic numbers and islands of stability, but they would also fragment easily. ==== Predicted decay properties of undiscovered elements ==== As the main island of stability is thought to lie around 291Cn and 293Cn, undiscovered elements beyond oganesson may be very unstable and undergo alpha decay or spontaneous fission in microseconds or less. The exact region in which half-lives exceed
|
{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
}
|
one microsecond is unknown, though various models suggest that isotopes of elements heavier than unbinilium that may be produced in fusion reactions with available targets and projectiles will have half-lives under one microsecond and therefore may not be detected. It is consistently predicted that there will exist regions of stability at N = 184 and N = 228, and possibly also at Z ~ 124 and N ~ 198. These nuclei may have half-lives of a few seconds and undergo predominantly alpha decay and spontaneous fission, though minor beta-plus decay (or electron capture) branches may also exist. Outside these regions of enhanced stability, fission barriers are expected to drop significantly due to loss of stabilization effects, resulting in fission half-lives below 10โ18 seconds, especially in evenโeven nuclei for which hindrance is even lower due to nucleon pairing. In general, alpha decay half-lives are expected to increase with neutron number, from nanoseconds in the most neutron-deficient isotopes to seconds closer to the beta-stability line. For nuclei with only a few neutrons more than a magic number, binding energy substantially drops, resulting in a break in the trend and shorter half-lives. The most neutron deficient isotopes of these elements may also be unbound and undergo proton emission. Cluster decay (heavy particle emission) has also been proposed as an alternative decay mode for some isotopes, posing yet another hurdle to identification of these elements. === Electron configurations === The following are expected electron configurations of elements 119โ174 and 184. The symbol [Og] indicates the probable electron configuration of oganesson (Z = 118), which is currently the last known element. The configurations of the elements in this table are written starting with [Og] because oganesson is expected to be the last prior element with a closed-shell (inert gas) configuration, 1s2 2s2 2p6 3s2 3p6
|
{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
}
|
3d10 4s2 4p6 4d10 4f14 5s2 5p6 5d10 5f14 6s2 6p6 6d10 7s2 7p6. Similarly, the [172] in the configurations for elements 173, 174, and 184 denotes the likely closed-shell configuration of element 172. Beyond element 123, no complete calculations are available and hence the data in this table must be taken as tentative. In the case of element 123, and perhaps also heavier elements, several possible electron configurations are predicted to have very similar energy levels, such that it is very difficult to predict the ground state. All configurations that have been proposed (since it was understood that the Madelung rule probably stops working here) are included. The predicted block assignments up to 172 are Kulsha's, following the expected available valence orbitals. There is, however, not a consensus in the literature as to how the blocks should work after element 138. == See also == Table of nuclides Hypernucleus Neutronium == References == == Further reading == Kaldor, U. (2005). "Superheavy ElementsโChemistry and Spectroscopy". Encyclopedia of Computational Chemistry. doi:10.1002/0470845015.cu0044. ISBN 978-0470845011. Seaborg, G. T. (1968). "Elements Beyond 100, Present Status and Future Prospects". Annual Review of Nuclear Science. 18: 53โ152. Bibcode:1968ARNPS..18...53S. doi:10.1146/annurev.ns.18.120168.000413. Scerri, Eric. (2011). A Very Short Introduction to the Periodic Table, Oxford University Press, Oxford. OUP Oxford. ISBN 978-0-19-958249-5. == External links == Holler, Jim. "Images of g-orbitals". University of Kentucky. Archived from the original on 2016-03-03. Retrieved 2016-03-03. Rihani, Jeries A. "The extended periodic table of the elements". Retrieved 2009-02-02. Scerri, Eric. "Eric Scerri's website for the elements and the periodic table". Retrieved 2013-03-26.
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{
"page_id": 68326,
"source": null,
"title": "Extended periodic table"
}
|
A DNA clamp, also known as a sliding clamp, is a protein complex that serves as a processivity-promoting factor in DNA replication. As a critical component of the DNA polymerase III holoenzyme, the clamp protein binds DNA polymerase and prevents this enzyme from dissociating from the template DNA strand. The clamp-polymerase proteinโprotein interactions are stronger and more specific than the direct interactions between the polymerase and the template DNA strand; because one of the rate-limiting steps in the DNA synthesis reaction is the association of the polymerase with the DNA template, the presence of the sliding clamp dramatically increases the number of nucleotides that the polymerase can add to the growing strand per association event. The presence of the DNA clamp can increase the rate of DNA synthesis up to 1,000-fold compared with a nonprocessive polymerase. == Structure == The DNA clamp is an ฮฑ+ฮฒ protein that assembles into a multimeric, six-domain ring structure that completely encircles the DNA double helix as the polymerase adds nucleotides to the growing strand. Each domain is in turn made of two ฮฒ-ฮฑ-ฮฒ-ฮฒ-ฮฒ structural repeats. The DNA clamp assembles on the DNA at the replication fork and "slides" along the DNA with the advancing polymerase, aided by a layer of water molecules in the central pore of the clamp between the DNA and the protein surface. Because of the toroidal shape of the assembled multimer, the clamp cannot dissociate from the template strand without also dissociating into monomers. The DNA clamp fold is found in bacteria, archaea, eukaryotes and some viruses. In bacteria, the sliding clamp is a homodimer composed of two identical beta subunits of DNA polymerase III and hence is referred to as the beta clamp. In archaea and eukaryotes, it is a trimer composed of three molecules of PCNA. The T4
|
{
"page_id": 6425322,
"source": null,
"title": "DNA clamp"
}
|
bacteriophage also uses a sliding clamp, called gp45 that is a trimer similar in structure to PCNA but lacks sequence homology to either PCNA or the bacterial beta clamp. === Bacterial === The beta clamp is a specific DNA clamp and a subunit of the DNA polymerase III holoenzyme found in bacteria. Two beta subunits are assembled around the DNA by the gamma subunit and ATP hydrolysis; this assembly is called the pre-initiation complex. After assembly around the DNA, the beta subunits' affinity for the gamma subunit is replaced by an affinity for the alpha and epsilon subunits, which together create the complete holoenzyme. DNA polymerase III is the primary enzyme complex involved in prokaryotic DNA replication. The gamma complex of DNA polymerase III, composed of ฮณฮดฮด'ฯฯ subunits, catalyzes ATP to chaperone two beta subunits to bind to DNA. Once bound to DNA, the beta subunits can freely slide along double stranded DNA. The beta subunits in turn bind the ฮฑฮต polymerase complex. The ฮฑ subunit possesses DNA polymerase activity and the ฮต subunit is a 3โ-5โ exonuclease. The beta chain of bacterial DNA polymerase III is composed of three topologically equivalent domains (N-terminal, central, and C-terminal). Two beta chain molecules are tightly associated to form a closed ring encircling duplex DNA. ==== As a drug target ==== Certain NSAIDs (carprofen, bromfenac, and vedaprofen) exhibit some suppression of bacterial DNA replication by inhibiting bacterial DNA clamp. === Eukaryotic and archaeal === The sliding clamp in eukaryotes is assembled from a specific subunit of DNA polymerase delta called the proliferating cell nuclear antigen (PCNA). The N-terminal and C-terminal domains of PCNA are topologically identical. Three PCNA molecules are tightly associated to form a closed ring encircling duplex DNA. The sequence of PCNA is well conserved between plants, animals and fungi, indicating
|
{
"page_id": 6425322,
"source": null,
"title": "DNA clamp"
}
|
a strong selective pressure for structure conservation, and suggesting that this type of DNA replication mechanism is conserved throughout eukaryotes. In eukaryotes, a homologous, heterotrimeric "9-1-1 clamp" made up of RAD9-RAD1-HUS1 (911) is responsible for DNA damage checkpoint control. This 9-1-1 clamp mounts onto DNA in the opposite direction. Archaea, probable evolutionary precursor of eukaryotes, also universally have at least one PCNA gene. This PCNA ring works with PolD, the single eukaryotic-like DNA polymerase in archaea responsible for multiple functions from replication to repair. Some unusual species have two or even three PCNA genes, forming heterotrimers or distinct specialized homotrimers. Archaeons also share with eukaryotes the PIP (PCNA-interacting protein) motif, but a wider variety of such proteins performing different functions are found. PCNA is also appropriated by some viruses. The giant virus genus Chlorovirus, with PBCV-1 as a representative, carries in its genome two PCNA genes (Q84513, O41056) and a eukaryotic-type DNA polymerase. Members of Baculoviridae also encode a PCNA homolog (P11038). === Caudoviral === The viral gp45 sliding clamp subunit protein contains two domains. Each domain consists of two alpha helices and two beta sheets โ the fold is duplicated and has internal pseudo two-fold symmetry. Three gp45 molecules are tightly associated to form a closed ring encircling duplex DNA. === Herpesviral === Some members of Herpesviridae encode a protein that has a DNA clamp fold but does not associate into a ring clamp. The two-domain protein does, however, associate with the viral DNA polymerase and also acts to increase processivity. As it does not form a ring, it does not need a clamp loader to be attached to DNA. == Assembly == Sliding clamps are loaded onto their associated DNA template strands by specialized proteins known as "sliding clamp loaders", which also disassemble the clamps after replication has
|
{
"page_id": 6425322,
"source": null,
"title": "DNA clamp"
}
|
completed. The binding sites for these initiator proteins overlap with the binding sites for the DNA polymerase, so the clamp cannot simultaneously associate with a clamp loader and with a polymerase. Thus the clamp will not be actively disassembled while the polymerase remains bound. DNA clamps also associate with other factors involved in DNA and genome homeostasis, such as nucleosome assembly factors, Okazaki fragment ligases, and DNA repair proteins. All of these proteins also share a binding site on the DNA clamp that overlaps with the clamp loader site, ensuring that the clamp will not be removed while any enzyme is still working on the DNA. The activity of the clamp loader requires ATP hydrolysis to "close" the clamp around the DNA. == References == == Further reading == Clamping down on pathogenic bacteriaโ how to shut down a key DNA polymerase complex. Quips at PDBe Archived 2013-08-02 at archive.today Watson JD, Baker TA, Bell SP, Gann A, Levine M, Losick R (2004). Molecular Biology of the Gene. San Francisco: Pearson/Benjamin Cummings. ISBN 978-0-8053-4635-0. == External links == SCOP DNA clamp fold CATH box architecture clamp+protein+DnaN,+E+coli at the U.S. National Library of Medicine Medical Subject Headings (MeSH)
|
{
"page_id": 6425322,
"source": null,
"title": "DNA clamp"
}
|
Chladni's law, named after Ernst Chladni, relates the frequency of modes of vibration for flat circular surfaces with fixed center as a function of the numbers m of diametric (linear) nodes and n of radial (circular) nodes. It is stated as the equation f = C ( m + 2 n ) p {\displaystyle f=C(m+2n)^{p}} where C and p are coefficients which depend on the properties of the plate. For flat circular plates, p is roughly 2, but Chladni's law can also be used to describe the vibrations of cymbals, handbells, and church bells in which case p can vary from 1.4 to 2.4. In fact, p can even vary for a single object, depending on which family of modes is being examined. == References == == External links == A Study of Vibrating Plates by Derek Kverno and Jim Nolen (Archived 27 July 2011)
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{
"page_id": 1903345,
"source": null,
"title": "Chladni's law"
}
|
In mycology, a sanctioned name is a name that was adopted (but not necessarily coined) in certain works of Christiaan Hendrik Persoon or Elias Magnus Fries, which are considered major points in fungal taxonomy. == Definition and effects == Sanctioned names are those, regardless of their authorship, that were used by Persoon in his Synopsis Methodica Fungorum (1801) for rusts, smuts and gasteromycetes, and in Fries's Systema Mycologicum (three volumes, published 1821โ1832) and Elenchus fungorum for all other fungi. A sanctioned name, as defined under article 15 of the International Code of Nomenclature for algae, fungi, and plants (previously, the International Code of Botanical Nomenclature) is automatically treated as if conserved against all earlier synonyms or homonyms. It can still, however, be conserved or rejected normally. == History == Because of the imprecision associated with assigning starting dates for fungi sanctioned in Fries' three Systema volumes, the Stockholm 1950 International Botanical Congress defined arbitrary or actual publication dates for the starting points to improve the stability of nomenclature. These dates were 1 May 1753 for Species Plantarum (vascular plants), 31 December 1801 for Synopsis Methodica Fungorum, 31 December 1820 for Flora der Vorweldt (fossil plants), and 1 January 1821 for the first volume of Systema. Because fungi defined in the second and third volumes lacked a starting-point book for reference, the Congress declared that these species, in addition to species defined in Fries' 1828 Elenchus Fungorum (a two-volume supplement to his System), had "privileged status". According to Korf, the term "sanctioned" was first used to indicate these privileged names by the Dutch mycologist Marinus Anton Donk in 1961. In 1982, changes in the International Code for Botanical Nomenclature (the Sydney Code) restored Linnaeus' 1753 Species Plantarum as the starting point for fungal nomenclature; however, protected status was given to all
|
{
"page_id": 27921142,
"source": null,
"title": "Sanctioned name"
}
|
names adopted by Persoon in his 1801 Synopsis, and by Fries in both the Systema and the Elenchus. Soon after, in 1983, Richard P. Korf proposed the now widely accepted "colon-author indication", whereby sanctioned names are indicated by including ": Pers." or ": Fr." when fully citing the species author. Formal approval of this convention was abolished in the 2018 revision to the Fungi chapter of the code. == References == Hawksworth, David L. (2001). "The Naming of Fungi". In David J. McLaughlin; Esther G. McLaughlin (eds.). The Mycota. Vol. 7, Part B: Systematics and Evolution. Berlin: Springer Verlag. pp. 171โ192. doi:10.1007/978-3-662-10189-6_6. ISBN 978-3-540-66493-2.
|
{
"page_id": 27921142,
"source": null,
"title": "Sanctioned name"
}
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In fluid dynamics, the Euler equations are a set of partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular, they correspond to the NavierโStokes equations with zero viscosity and zero thermal conductivity. The Euler equations can be applied to incompressible and compressible flows. The incompressible Euler equations consist of Cauchy equations for conservation of mass and balance of momentum, together with the incompressibility condition that the flow velocity is divergence-free. The compressible Euler equations consist of equations for conservation of mass, balance of momentum, and balance of energy, together with a suitable constitutive equation for the specific energy density of the fluid. Historically, only the equations of conservation of mass and balance of momentum were derived by Euler. However, fluid dynamics literature often refers to the full set of the compressible Euler equations โ including the energy equation โ as "the compressible Euler equations". The mathematical characters of the incompressible and compressible Euler equations are rather different. For constant fluid density, the incompressible equations can be written as a quasilinear advection equation for the fluid velocity together with an elliptic Poisson's equation for the pressure. On the other hand, the compressible Euler equations form a quasilinear hyperbolic system of conservation equations. The Euler equations can be formulated in a "convective form" (also called the "Lagrangian form") or a "conservation form" (also called the "Eulerian form"). The convective form emphasizes changes to the state in a frame of reference moving with the fluid. The conservation form emphasizes the mathematical interpretation of the equations as conservation equations for a control volume fixed in space (which is useful from a numerical point of view). == History == The Euler equations first appeared in published form in Euler's article "Principes gรฉnรฉraux du mouvement des fluides", published in
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"page_id": 396022,
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Mรฉmoires de l'Acadรฉmie des Sciences de Berlin in 1757 (although Euler had previously presented his work to the Berlin Academy in 1752). Prior work included contributions from the Bernoulli family as well as from Jean le Rond d'Alembert. The Euler equations were among the first partial differential equations to be written down, after the wave equation. In Euler's original work, the system of equations consisted of the momentum and continuity equations, and thus was underdetermined except in the case of an incompressible flow. An additional equation, which was called the adiabatic condition, was supplied by Pierre-Simon Laplace in 1816. During the second half of the 19th century, it was found that the equation related to the balance of energy must at all times be kept for compressible flows, and the adiabatic condition is a consequence of the fundamental laws in the case of smooth solutions. With the discovery of the special theory of relativity, the concepts of energy density, momentum density, and stress were unified into the concept of the stressโenergy tensor, and energy and momentum were likewise unified into a single concept, the energyโmomentum vector. == Incompressible Euler equations with constant and uniform density == In convective form (i.e., the form with the convective operator made explicit in the momentum equation), the incompressible Euler equations in case of density constant in time and uniform in space are: where: u {\displaystyle \mathbf {u} } is the flow velocity vector, with components in an N-dimensional space u 1 , u 2 , โฆ , u N {\displaystyle u_{1},u_{2},\dots ,u_{N}} , D ฮฆ D t = โ ฮฆ โ t + v โ
โ ฮฆ {\displaystyle {\frac {D{\boldsymbol {\Phi }}}{Dt}}={\frac {\partial {\boldsymbol {\Phi }}}{\partial t}}+\mathbf {v} \cdot \nabla {\boldsymbol {\Phi }}} , for a generic function (or field) ฮฆ {\displaystyle {\boldsymbol {\Phi
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}}} denotes its material derivative in time with respect to the advective field v {\displaystyle \mathbf {v} } and โ w {\displaystyle \nabla w} is the gradient of the specific (with the sense of per unit mass) thermodynamic work, the internal source term, and โ โ
u {\displaystyle \nabla \cdot \mathbf {u} } is the flow velocity divergence. g {\displaystyle \mathbf {g} } represents body accelerations (per unit mass) acting on the continuum, for example gravity, inertial accelerations, electric field acceleration, and so on. The first equation is the Euler momentum equation with uniform density (for this equation it could also not be constant in time). By expanding the material derivative, the equations become: โ u โ t + ( u โ
โ ) u = โ โ w + g , โ โ
u = 0. {\displaystyle {\begin{aligned}{\partial \mathbf {u} \over \partial t}+(\mathbf {u} \cdot \nabla )\mathbf {u} &=-\nabla w+\mathbf {g} ,\\\nabla \cdot \mathbf {u} &=0.\end{aligned}}} In fact for a flow with uniform density ฯ 0 {\displaystyle \rho _{0}} the following identity holds: โ w โก โ ( p ฯ 0 ) = 1 ฯ 0 โ p , {\displaystyle \nabla w\equiv \nabla \left({\frac {p}{\rho _{0}}}\right)={\frac {1}{\rho _{0}}}\nabla p,} where p {\displaystyle p} is the mechanic pressure. The second equation is the incompressible constraint, stating the flow velocity is a solenoidal field (the order of the equations is not causal, but underlines the fact that the incompressible constraint is not a degenerate form of the continuity equation, but rather of the energy equation, as it will become clear in the following). Notably, the continuity equation would be required also in this incompressible case as an additional third equation in case of density varying in time or varying in space. For example, with density nonuniform in space but constant in
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"page_id": 396022,
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time, the continuity equation to be added to the above set would correspond to: โ ฯ โ t = 0. {\displaystyle {\frac {\partial \rho }{\partial t}}=0.} So the case of constant and uniform density is the only one not requiring the continuity equation as additional equation regardless of the presence or absence of the incompressible constraint. In fact, the case of incompressible Euler equations with constant and uniform density discussed here is a toy model featuring only two simplified equations, so it is ideal for didactical purposes even if with limited physical relevance. The equations above thus represent respectively conservation of mass (1 scalar equation) and momentum (1 vector equation containing N {\displaystyle N} scalar components, where N {\displaystyle N} is the physical dimension of the space of interest). Flow velocity and pressure are the so-called physical variables. In a coordinate system given by ( x 1 , โฆ , x N ) {\displaystyle \left(x_{1},\dots ,x_{N}\right)} the velocity and external force vectors u {\displaystyle \mathbf {u} } and g {\displaystyle \mathbf {g} } have components ( u 1 , โฆ , u N ) {\displaystyle (u_{1},\dots ,u_{N})} and ( g 1 , โฆ , g N ) {\displaystyle \left(g_{1},\dots ,g_{N}\right)} , respectively. Then the equations may be expressed in subscript notation as: โ u i โ t + โ j = 1 N โ ( u i u j + w ฮด i j ) โ x j = g i , โ i = 1 N โ u i โ x i = 0. {\displaystyle {\begin{aligned}{\partial u_{i} \over \partial t}+\sum _{j=1}^{N}{\partial \left(u_{i}u_{j}+w\delta _{ij}\right) \over \partial x_{j}}&=g_{i},\\\sum _{i=1}^{N}{\partial u_{i} \over \partial x_{i}}&=0.\end{aligned}}} where the i {\displaystyle i} and j {\displaystyle j} subscripts label the N-dimensional space components, and ฮด i j {\displaystyle \delta _{ij}} is the Kroenecker delta. The use
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"page_id": 396022,
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of Einstein notation (where the sum is implied by repeated indices instead of sigma notation) is also frequent. === Properties === Although Euler first presented these equations in 1755, many fundamental questions or concepts about them remain unanswered. In three space dimensions, in certain simplified scenarios, the Euler equations produce singularities. Smooth solutions of the free (in the sense of without source term: g=0) equations satisfy the conservation of specific kinetic energy: โ โ t ( 1 2 u 2 ) + โ โ
( u 2 u + w u ) = 0. {\displaystyle {\partial \over \partial t}\left({\frac {1}{2}}u^{2}\right)+\nabla \cdot \left(u^{2}\mathbf {u} +w\mathbf {u} \right)=0.} In the one-dimensional case without the source term (both pressure gradient and external force), the momentum equation becomes the inviscid Burgers' equation: โ u โ t + u โ u โ x = 0. {\displaystyle {\partial u \over \partial t}+u{\partial u \over \partial x}=0.} This model equation gives many insights into Euler equations. === Nondimensionalisation === In order to make the equations dimensionless, a characteristic length r 0 {\displaystyle r_{0}} , and a characteristic velocity u 0 {\displaystyle u_{0}} , need to be defined. These should be chosen such that the dimensionless variables are all of order one. The following dimensionless variables are thus obtained: u โ โก u u 0 , r โ โก r r 0 , t โ โก u 0 r 0 t , p โ โก w u 0 2 , โ โ โก r 0 โ . {\displaystyle {\begin{aligned}u^{*}&\equiv {\frac {u}{u_{0}}},&r^{*}&\equiv {\frac {r}{r_{0}}},\\[5pt]t^{*}&\equiv {\frac {u_{0}}{r_{0}}}t,&p^{*}&\equiv {\frac {w}{u_{0}^{2}}},\\[5pt]\nabla ^{*}&\equiv r_{0}\nabla .\end{aligned}}} and of the field unit vector: g ^ โก g g . {\displaystyle {\hat {\mathbf {g} }}\equiv {\frac {\mathbf {g} }{g}}.} Substitution of these inversed relations in Euler equations, defining the Froude number, yields (omitting the * at
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apix): Euler equations in the Froude limit (no external field) are named free equations and are conservative. The limit of high Froude numbers (low external field) is thus notable and can be studied with perturbation theory. === Conservation form === The conservation form emphasizes the mathematical properties of Euler equations, and especially the contracted form is often the most convenient one for computational fluid dynamics simulations. Computationally, there are some advantages in using the conserved variables. This gives rise to a large class of numerical methods called conservative methods. The free Euler equations are conservative, in the sense they are equivalent to a conservation equation: โ y โ t + โ โ
F = 0 , {\displaystyle {\frac {\partial \mathbf {y} }{\partial t}}+\nabla \cdot \mathbf {F} ={\mathbf {0} },} or simply in Einstein notation: โ y j โ t + โ f i j โ r i = 0 i , {\displaystyle {\frac {\partial y_{j}}{\partial t}}+{\frac {\partial f_{ij}}{\partial r_{i}}}=0_{i},} where the conservation quantity y {\displaystyle \mathbf {y} } in this case is a vector, and F {\displaystyle \mathbf {F} } is a flux matrix. This can be simply proved. At last Euler equations can be recast into the particular equation: === Spatial dimensions === For certain problems, especially when used to analyze compressible flow in a duct or in case the flow is cylindrically or spherically symmetric, the one-dimensional Euler equations are a useful first approximation. Generally, the Euler equations are solved by Riemann's method of characteristics. This involves finding curves in plane of independent variables (i.e., x {\displaystyle x} and t {\displaystyle t} ) along which partial differential equations (PDEs) degenerate into ordinary differential equations (ODEs). Numerical solutions of the Euler equations rely heavily on the method of characteristics. == Incompressible Euler equations == In convective form the incompressible
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Euler equations in case of density variable in space are: where the additional variables are: ฯ {\displaystyle \rho } is the fluid mass density, p {\displaystyle p} is the pressure, p = ฯ w {\displaystyle p=\rho w} . The first equation, which is the new one, is the incompressible continuity equation. In fact the general continuity equation would be: โ ฯ โ t + u โ
โ ฯ + ฯ โ โ
u = 0 , {\displaystyle {\partial \rho \over \partial t}+\mathbf {u} \cdot \nabla \rho +\rho \nabla \cdot \mathbf {u} =0,} but here the last term is identically zero for the incompressibility constraint. === Conservation form === The incompressible Euler equations in the Froude limit are equivalent to a single conservation equation with conserved quantity and associated flux respectively: y = ( ฯ ฯ u 0 ) ; F = ( ฯ u ฯ u โ u + p I u ) . {\displaystyle \mathbf {y} ={\begin{pmatrix}\rho \\\rho \mathbf {u} \\0\end{pmatrix}};\qquad {\mathbf {F} }={\begin{pmatrix}\rho \mathbf {u} \\\rho \mathbf {u} \otimes \mathbf {u} +p\mathbf {I} \\\mathbf {u} \end{pmatrix}}.} Here y {\displaystyle \mathbf {y} } has length N + 2 {\displaystyle N+2} and F {\displaystyle \mathbf {F} } has size ( N + 2 ) N {\displaystyle (N+2)N} . In general (not only in the Froude limit) Euler equations are expressible as: โ โ t ( ฯ ฯ u 0 ) + โ โ
( ฯ u ฯ u โ u + p I u ) = ( 0 ฯ g 0 ) . {\displaystyle {\frac {\partial }{\partial t}}{\begin{pmatrix}\rho \\\rho \mathbf {u} \\0\end{pmatrix}}+\nabla \cdot {\begin{pmatrix}\rho \mathbf {u} \\\rho \mathbf {u} \otimes \mathbf {u} +p\mathbf {I} \\\mathbf {u} \end{pmatrix}}={\begin{pmatrix}0\\\rho \mathbf {g} \\0\end{pmatrix}}.} === Conservation variables === The variables for the equations in conservation form are not yet optimised. In fact we could
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define: y = ( ฯ j 0 ) ; F = ( j j โ 1 ฯ j + p I j ฯ ) , {\displaystyle {\mathbf {y} }={\begin{pmatrix}\rho \\\mathbf {j} \\0\end{pmatrix}};\qquad {\mathbf {F} }={\begin{pmatrix}\mathbf {j} \\\mathbf {j} \otimes {\frac {1}{\rho }}\,\mathbf {j} +p\mathbf {I} \\{\frac {\mathbf {j} }{\rho }}\end{pmatrix}},} where j = ฯ u {\displaystyle \mathbf {j} =\rho \mathbf {u} } is the momentum density, a conservation variable. where f = ฯ g {\displaystyle \mathbf {f} =\rho \mathbf {g} } is the force density, a conservation variable. == Euler equations == In differential convective form, the compressible (and most general) Euler equations can be written shortly with the material derivative notation: where the additional variables here is: e {\displaystyle e} is the specific internal energy (internal energy per unit mass). The equations above thus represent conservation of mass, momentum, and energy: the energy equation expressed in the variable internal energy allows to understand the link with the incompressible case, but it is not in the simplest form. Mass density, flow velocity and pressure are the so-called convective variables (or physical variables, or lagrangian variables), while mass density, momentum density and total energy density are the so-called conserved variables (also called eulerian, or mathematical variables). If one expands the material derivative, the equations above become: โ ฯ โ t + u โ
โ ฯ + ฯ โ โ
u = 0 , โ u โ t + u โ
โ u + โ p ฯ = g , โ e โ t + u โ
โ e + p ฯ โ โ
u = 0. {\displaystyle {\begin{aligned}{\partial \rho \over \partial t}+\mathbf {u} \cdot \nabla \rho +\rho \nabla \cdot \mathbf {u} &=0,\\[1.2ex]{\frac {\partial \mathbf {u} }{\partial t}}+\mathbf {u} \cdot \nabla \mathbf {u} +{\frac {\nabla p}{\rho }}&=\mathbf {g} ,\\[1.2ex]{\partial e \over \partial
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t}+\mathbf {u} \cdot \nabla e+{\frac {p}{\rho }}\nabla \cdot \mathbf {u} &=0.\end{aligned}}} === Incompressible constraint (revisited) === Coming back to the incompressible case, it now becomes apparent that the incompressible constraint typical of the former cases actually is a particular form valid for incompressible flows of the energy equation, and not of the mass equation. In particular, the incompressible constraint corresponds to the following very simple energy equation: D e D t = 0. {\displaystyle {\frac {De}{Dt}}=0.} Thus for an incompressible inviscid fluid the specific internal energy is constant along the flow lines, also in a time-dependent flow. The pressure in an incompressible flow acts like a Lagrange multiplier, being the multiplier of the incompressible constraint in the energy equation, and consequently in incompressible flows it has no thermodynamic meaning. In fact, thermodynamics is typical of compressible flows and degenerates in incompressible flows. Basing on the mass conservation equation, one can put this equation in the conservation form: โ ฯ e โ t + โ โ
( ฯ e u ) = 0 , {\displaystyle {\partial \rho e \over \partial t}+\nabla \cdot (\rho e\mathbf {u} )=0,} meaning that for an incompressible inviscid nonconductive flow a continuity equation holds for the internal energy. === Enthalpy conservation === Since by definition the specific enthalpy is: h = e + p ฯ . {\displaystyle h=e+{\frac {p}{\rho }}.} The material derivative of the specific internal energy can be expressed as: D e D t = D h D t โ 1 ฯ ( D p D t โ p ฯ D ฯ D t ) . {\displaystyle {De \over Dt}={Dh \over Dt}-{\frac {1}{\rho }}\left({Dp \over Dt}-{\frac {p}{\rho }}{D\rho \over Dt}\right).} Then by substituting the momentum equation in this expression, one obtains: D e D t = D h D t โ 1 ฯ ( p
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โ โ
u + D p D t ) . {\displaystyle {De \over Dt}={Dh \over Dt}-{\frac {1}{\rho }}\left(p\nabla \cdot \mathbf {u} +{Dp \over Dt}\right).} And by substituting the latter in the energy equation, one obtains that the enthalpy expression for the Euler energy equation: D h D t = 1 ฯ D p D t . {\displaystyle {Dh \over Dt}={\frac {1}{\rho }}{Dp \over Dt}.} In a reference frame moving with an inviscid and nonconductive flow, the variation of enthalpy directly corresponds to a variation of pressure. === Thermodynamics of ideal fluids === In thermodynamics the independent variables are the specific volume, and the specific entropy, while the specific energy is a function of state of these two variables. For a thermodynamic fluid, the compressible Euler equations are consequently best written as: where: v {\displaystyle v} is the specific volume u {\displaystyle \mathbf {u} } is the flow velocity vector s {\displaystyle s} is the specific entropy In the general case and not only in the incompressible case, the energy equation means that for an inviscid thermodynamic fluid the specific entropy is constant along the flow lines, also in a time-dependent flow. Basing on the mass conservation equation, one can put this equation in the conservation form: โ ฯ s โ t + โ โ
( ฯ s u ) = 0 , {\displaystyle {\partial \rho s \over \partial t}+\nabla \cdot (\rho s\mathbf {u} )=0,} meaning that for an inviscid nonconductive flow a continuity equation holds for the entropy. On the other hand, the two second-order partial derivatives of the specific internal energy in the momentum equation require the specification of the fundamental equation of state of the material considered, i.e. of the specific internal energy as function of the two variables specific volume and specific entropy: e = e (
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v , s ) . {\displaystyle e=e(v,s).} The fundamental equation of state contains all the thermodynamic information about the system (Callen, 1985), exactly like the couple of a thermal equation of state together with a caloric equation of state. === Conservation form === The Euler equations in the Froude limit are equivalent to a single conservation equation with conserved quantity and associated flux respectively: y = ( ฯ j E t ) ; F = ( j 1 ฯ j โ j + p I ( E t + p ) 1 ฯ j ) , {\displaystyle \mathbf {y} ={\begin{pmatrix}\rho \\\mathbf {j} \\E^{t}\end{pmatrix}};\qquad {\mathbf {F} }={\begin{pmatrix}\mathbf {j} \\{\frac {1}{\rho }}\mathbf {j} \otimes \mathbf {j} +p\mathbf {I} \\\left(E^{t}+p\right){\frac {1}{\rho }}\mathbf {j} \end{pmatrix}},} where: j = ฯ u {\displaystyle \mathbf {j} =\rho \mathbf {u} } is the momentum density, a conservation variable. E t = ฯ e + 1 2 ฯ u 2 {\textstyle E^{t}=\rho e+{\frac {1}{2}}\rho u^{2}} is the total energy density (total energy per unit volume). Here y {\displaystyle \mathbf {y} } has length N + 2 and F {\displaystyle \mathbf {F} } has size N(N + 2). In general (not only in the Froude limit) Euler equations are expressible as: where f = ฯ g {\displaystyle \mathbf {f} =\rho \mathbf {g} } is the force density, a conservation variable. We remark that also the Euler equation even when conservative (no external field, Froude limit) have no Riemann invariants in general. Some further assumptions are required However, we already mentioned that for a thermodynamic fluid the equation for the total energy density is equivalent to the conservation equation: โ โ t ( ฯ s ) + โ โ
( ฯ s u ) = 0. {\displaystyle {\partial \over \partial t}(\rho s)+\nabla \cdot (\rho s\mathbf {u} )=0.} Then the conservation equations
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"page_id": 396022,
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in the case of a thermodynamic fluid are more simply expressed as: where S = ฯ s {\displaystyle S=\rho s} is the entropy density, a thermodynamic conservation variable. Another possible form for the energy equation, being particularly useful for isobarics, is: โ H t โ t + โ โ
( H t u ) = u โ
f โ โ p โ t , {\displaystyle {\frac {\partial H^{t}}{\partial t}}+\nabla \cdot \left(H^{t}\mathbf {u} \right)=\mathbf {u} \cdot \mathbf {f} -{\frac {\partial p}{\partial t}},} where H t = E t + p = ฯ e + p + 1 2 ฯ u 2 {\textstyle H^{t}=E^{t}+p=\rho e+p+{\frac {1}{2}}\rho u^{2}} is the total enthalpy density. == Quasilinear form and characteristic equations == Expanding the fluxes can be an important part of constructing numerical solvers, for example by exploiting (approximate) solutions to the Riemann problem. In regions where the state vector y varies smoothly, the equations in conservative form can be put in quasilinear form: โ y โ t + A i โ y โ r i = 0 . {\displaystyle {\frac {\partial \mathbf {y} }{\partial t}}+\mathbf {A} _{i}{\frac {\partial \mathbf {y} }{\partial r_{i}}}={\mathbf {0} }.} where A i {\displaystyle \mathbf {A} _{i}} are called the flux Jacobians defined as the matrices: A i ( y ) = โ f i ( y ) โ y . {\displaystyle \mathbf {A} _{i}(\mathbf {y} )={\frac {\partial \mathbf {f} _{i}(\mathbf {y} )}{\partial \mathbf {y} }}.} This Jacobian does not exist where the state variables are discontinuous, as at contact discontinuities or shocks. === Characteristic equations === The compressible Euler equations can be decoupled into a set of N+2 wave equations that describes sound in Eulerian continuum if they are expressed in characteristic variables instead of conserved variables. In fact the tensor A is always diagonalizable. If the eigenvalues (the
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"page_id": 396022,
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case of Euler equations) are all real the system is defined hyperbolic, and physically eigenvalues represent the speeds of propagation of information. If they are all distinguished, the system is defined strictly hyperbolic (it will be proved to be the case of one-dimensional Euler equations). Furthermore, diagonalisation of compressible Euler equation is easier when the energy equation is expressed in the variable entropy (i.e. with equations for thermodynamic fluids) than in other energy variables. This will become clear by considering the 1D case. If p i {\displaystyle \mathbf {p} _{i}} is the right eigenvector of the matrix A {\displaystyle \mathbf {A} } corresponding to the eigenvalue ฮป i {\displaystyle \lambda _{i}} , by building the projection matrix: P = [ p 1 , p 2 , . . . , p n ] . {\displaystyle \mathbf {P} =\left[\mathbf {p} _{1},\mathbf {p} _{2},...,\mathbf {p} _{n}\right].} One can finally find the characteristic variables as: w = P โ 1 y . {\displaystyle \mathbf {w} =\mathbf {P} ^{-1}\mathbf {y} .} Since A is constant, multiplying the original 1-D equation in flux-Jacobian form with Pโ1 yields the characteristic equations: โ w i โ t + ฮป j โ w i โ r j = 0 i . {\displaystyle {\frac {\partial w_{i}}{\partial t}}+\lambda _{j}{\frac {\partial w_{i}}{\partial r_{j}}}=0_{i}.} The original equations have been decoupled into N+2 characteristic equations each describing a simple wave, with the eigenvalues being the wave speeds. The variables wi are called the characteristic variables and are a subset of the conservative variables. The solution of the initial value problem in terms of characteristic variables is finally very simple. In one spatial dimension it is: w i ( x , t ) = w i ( x โ ฮป i t , 0 ) . {\displaystyle w_{i}(x,t)=w_{i}\left(x-\lambda _{i}t,0\right).} Then the solution in terms
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"page_id": 396022,
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of the original conservative variables is obtained by transforming back: y = P w , {\displaystyle \mathbf {y} =\mathbf {P} \mathbf {w} ,} this computation can be explicited as the linear combination of the eigenvectors: y ( x , t ) = โ i = 1 m w i ( x โ ฮป i t , 0 ) p i . {\displaystyle \mathbf {y} (x,t)=\sum _{i=1}^{m}w_{i}\left(x-\lambda _{i}t,0\right)\mathbf {p} _{i}.} Now it becomes apparent that the characteristic variables act as weights in the linear combination of the jacobian eigenvectors. The solution can be seen as superposition of waves, each of which is advected independently without change in shape. Each i-th wave has shape wipi and speed of propagation ฮปi. In the following we show a very simple example of this solution procedure. === Waves in 1D inviscid, nonconductive thermodynamic fluid === If one considers Euler equations for a thermodynamic fluid with the two further assumptions of one spatial dimension and free (no external field: g = 0): โ v โ t + u โ v โ x โ v โ u โ x = 0 , โ u โ t + u โ u โ x โ e v v v โ v โ x โ e v s v โ s โ x = 0 , โ s โ t + u โ s โ x = 0. {\displaystyle {\begin{aligned}{\partial v \over \partial t}+u{\partial v \over \partial x}-v{\partial u \over \partial x}&=0,\\[1.2ex]{\partial u \over \partial t}+u{\partial u \over \partial x}-e_{vv}v{\partial v \over \partial x}-e_{vs}v{\partial s \over \partial x}&=0,\\[1.2ex]{\partial s \over \partial t}+u{\partial s \over \partial x}&=0.\end{aligned}}} If one defines the vector of variables: y = ( v u s ) , {\displaystyle \mathbf {y} ={\begin{pmatrix}v\\u\\s\end{pmatrix}},} recalling that v {\displaystyle v} is the specific volume, u {\displaystyle u} the flow speed,
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"page_id": 396022,
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s {\displaystyle s} the specific entropy, the corresponding jacobian matrix is: A = ( u โ v 0 โ e v v v u โ e v s v 0 0 u ) . {\displaystyle {\mathbf {A} }={\begin{pmatrix}u&-v&0\\-e_{vv}v&u&-e_{vs}v\\0&0&u\end{pmatrix}}.} At first one must find the eigenvalues of this matrix by solving the characteristic equation: det ( A ( y ) โ ฮป ( y ) I ) = 0 , {\displaystyle \det(\mathbf {A} (\mathbf {y} )-\lambda (\mathbf {y} )\mathbf {I} )=0,} that is explicitly: det [ u โ ฮป โ v 0 โ e v v v u โ ฮป โ e v s v 0 0 u โ ฮป ] = 0. {\displaystyle \det {\begin{bmatrix}u-\lambda &-v&0\\-e_{vv}v&u-\lambda &-e_{vs}v\\0&0&u-\lambda \end{bmatrix}}=0.} This determinant is very simple: the fastest computation starts on the last row, since it has the highest number of zero elements. ( u โ ฮป ) det [ u โ ฮป โ v โ e v v v u โ ฮป ] = 0. {\displaystyle (u-\lambda )\det {\begin{bmatrix}u-\lambda &-v\\-e_{vv}v&u-\lambda \end{bmatrix}}=0.} Now by computing the determinant 2ร2: ( u โ ฮป ) ( ( u โ ฮป ) 2 โ e v v v 2 ) = 0 , {\displaystyle (u-\lambda )\left((u-\lambda )^{2}-e_{vv}v^{2}\right)=0,} by defining the parameter: a ( v , s ) โก v e v v , {\displaystyle a(v,s)\equiv v{\sqrt {e_{vv}}},} or equivalently in mechanical variables, as: a ( ฯ , p ) โก โ p โ ฯ . {\displaystyle a(\rho ,p)\equiv {\sqrt {\partial p \over \partial \rho }}.} This parameter is always real according to the second law of thermodynamics. In fact the second law of thermodynamics can be expressed by several postulates. The most elementary of them in mathematical terms is the statement of convexity of the fundamental equation of state, i.e. the hessian matrix of the specific
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"page_id": 396022,
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}
|
energy expressed as function of specific volume and specific entropy: ( e v v e v s e v s e s s ) , {\displaystyle {\begin{pmatrix}e_{vv}&e_{vs}\\e_{vs}&e_{ss}\end{pmatrix}},} is defined positive. This statement corresponds to the two conditions: { e v v > 0 e v v e s s โ e v s 2 > 0 {\displaystyle \left\{{\begin{aligned}e_{vv}&>0\\[1.2ex]e_{vv}e_{ss}-e_{vs}^{2}&>0\end{aligned}}\right.} The first condition is the one ensuring the parameter a is defined real. The characteristic equation finally results: ( u โ ฮป ) ( ( u โ ฮป ) 2 โ a 2 ) = 0 {\displaystyle (u-\lambda )\left((u-\lambda )^{2}-a^{2}\right)=0} That has three real solutions: ฮป 1 ( v , u , s ) = u โ a ( v , s ) , ฮป 2 ( u ) = u , ฮป 3 ( v , u , s ) = u + a ( v , s ) . {\displaystyle \lambda _{1}(v,u,s)=u-a(v,s),\qquad \lambda _{2}(u)=u,\qquad \lambda _{3}(v,u,s)=u+a(v,s).} Then the matrix has three real eigenvalues all distinguished: the 1D Euler equations are a strictly hyperbolic system. At this point one should determine the three eigenvectors: each one is obtained by substituting one eigenvalue in the eigenvalue equation and then solving it. By substituting the first eigenvalue ฮป1 one obtains: ( a โ v 0 โ e v v v a โ e v s v 0 0 a ) ( v 1 u 1 s 1 ) = 0. {\displaystyle {\begin{pmatrix}a&-v&0\\-e_{vv}v&a&-e_{vs}v\\0&0&a\end{pmatrix}}{\begin{pmatrix}v_{1}\\u_{1}\\s_{1}\end{pmatrix}}=0.} Basing on the third equation that simply has solution s1=0, the system reduces to: ( a โ v โ a 2 / v a ) ( v 1 u 1 ) = 0 {\displaystyle {\begin{pmatrix}a&-v\\-a^{2}/v&a\end{pmatrix}}{\begin{pmatrix}v_{1}\\u_{1}\end{pmatrix}}=0} The two equations are redundant as usual, then the eigenvector is defined with a multiplying constant. We choose as right eigenvector: p 1 = ( v
|
{
"page_id": 396022,
"source": null,
"title": "Euler equations (fluid dynamics)"
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|
a 0 ) . {\displaystyle \mathbf {p} _{1}={\begin{pmatrix}v\\a\\0\end{pmatrix}}.} The other two eigenvectors can be found with analogous procedure as: p 2 = ( e v s 0 โ ( a v ) 2 ) , p 3 = ( v โ a 0 ) . {\displaystyle \mathbf {p} _{2}={\begin{pmatrix}e_{vs}\\0\\-\left({\frac {a}{v}}\right)^{2}\end{pmatrix}},\qquad \mathbf {p} _{3}={\begin{pmatrix}v\\-a\\0\end{pmatrix}}.} Then the projection matrix can be built: P ( v , u , s ) = ( p 1 , p 2 , p 3 ) = ( v e v s v a 0 โ a 0 โ ( a v ) 2 0 ) . {\displaystyle \mathbf {P} (v,u,s)=(\mathbf {p} _{1},\mathbf {p} _{2},\mathbf {p} _{3})={\begin{pmatrix}v&e_{vs}&v\\a&0&-a\\0&-\left({\frac {a}{v}}\right)^{2}&0\end{pmatrix}}.} Finally it becomes apparent that the real parameter a previously defined is the speed of propagation of the information characteristic of the hyperbolic system made of Euler equations, i.e. it is the wave speed. It remains to be shown that the sound speed corresponds to the particular case of an isentropic transformation: a s โก ( โ p โ ฯ ) s . {\displaystyle a_{s}\equiv {\sqrt {\left({\partial p \over \partial \rho }\right)_{s}}}.} === Compressibility and sound speed === Sound speed is defined as the wavespeed of an isentropic transformation: a s ( ฯ , p ) โก ( โ p โ ฯ ) s , {\displaystyle a_{s}(\rho ,p)\equiv {\sqrt {\left({\partial p \over \partial \rho }\right)_{s}}},} by the definition of the isoentropic compressibility: K s ( ฯ , p ) โก ฯ ( โ p โ ฯ ) s , {\displaystyle K_{s}(\rho ,p)\equiv \rho \left({\partial p \over \partial \rho }\right)_{s},} the soundspeed results always the square root of ratio between the isentropic compressibility and the density: a s โก K s ฯ . {\displaystyle a_{s}\equiv {\sqrt {\frac {K_{s}}{\rho }}}.} ==== Ideal gas ==== The sound speed in an ideal gas
|
{
"page_id": 396022,
"source": null,
"title": "Euler equations (fluid dynamics)"
}
|
depends only on its temperature: a s ( T ) = ฮณ T m . {\displaystyle a_{s}(T)={\sqrt {\gamma {\frac {T}{m}}}}.} Since the specific enthalpy in an ideal gas is proportional to its temperature: h = c p T = ฮณ ฮณ โ 1 T m , {\displaystyle h=c_{p}T={\frac {\gamma }{\gamma -1}}{\frac {T}{m}},} the sound speed in an ideal gas can also be made dependent only on its specific enthalpy: a s ( h ) = ( ฮณ โ 1 ) h . {\displaystyle a_{s}(h)={\sqrt {(\gamma -1)h}}.} == Bernoulli's theorem for steady inviscid flow == Bernoulli's theorem is a direct consequence of the Euler equations. === Incompressible case and Lamb's form === The vector calculus identity of the cross product of a curl holds: v ร ( โ ร F ) = โ F ( v โ
F ) โ v โ
โ F , {\displaystyle \mathbf {v\ \times } \left(\mathbf {\nabla \times F} \right)=\nabla _{F}\left(\mathbf {v\cdot F} \right)-\mathbf {v\cdot \nabla } \mathbf {F} \ ,} where the Feynman subscript notation โ F {\displaystyle \nabla _{F}} is used, which means the subscripted gradient operates only on the factor F {\displaystyle \mathbf {F} } . Lamb in his famous classical book Hydrodynamics (1895), still in print, used this identity to change the convective term of the flow velocity in rotational form: u โ
โ u = 1 2 โ ( u 2 ) + ( โ ร u ) ร u , {\displaystyle \mathbf {u} \cdot \nabla \mathbf {u} ={\frac {1}{2}}\nabla \left(u^{2}\right)+(\nabla \times \mathbf {u} )\times \mathbf {u} ,} the Euler momentum equation in Lamb's form becomes: โ u โ t + 1 2 โ ( u 2 ) + ( โ ร u ) ร u + โ p ฯ = g = โ u โ t + 1 2 โ
|
{
"page_id": 396022,
"source": null,
"title": "Euler equations (fluid dynamics)"
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|
( u 2 ) โ u ร ( โ ร u ) + โ p ฯ . {\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+{\frac {1}{2}}\nabla \left(u^{2}\right)+(\nabla \times \mathbf {u} )\times \mathbf {u} +{\frac {\nabla p}{\rho }}=\mathbf {g} ={\frac {\partial \mathbf {u} }{\partial t}}+{\frac {1}{2}}\nabla \left(u^{2}\right)-\mathbf {u} \times (\nabla \times \mathbf {u} )+{\frac {\nabla p}{\rho }}.} Now, basing on the other identity: โ ( p ฯ ) = โ p ฯ โ p ฯ 2 โ ฯ , {\displaystyle \nabla \left({\frac {p}{\rho }}\right)={\frac {\nabla p}{\rho }}-{\frac {p}{\rho ^{2}}}\nabla \rho ,} the Euler momentum equation assumes a form that is optimal to demonstrate Bernoulli's theorem for steady flows: โ ( 1 2 u 2 + p ฯ ) โ g = โ p ฯ 2 โ ฯ + u ร ( โ ร u ) โ โ u โ t . {\displaystyle \nabla \left({\frac {1}{2}}u^{2}+{\frac {p}{\rho }}\right)-\mathbf {g} =-{\frac {p}{\rho ^{2}}}\nabla \rho +\mathbf {u} \times (\nabla \times \mathbf {u} )-{\frac {\partial \mathbf {u} }{\partial t}}.} In fact, in case of an external conservative field, by defining its potential ฯ: โ ( 1 2 u 2 + ฯ + p ฯ ) = โ p ฯ 2 โ ฯ + u ร ( โ ร u ) โ โ u โ t . {\displaystyle \nabla \left({\frac {1}{2}}u^{2}+\phi +{\frac {p}{\rho }}\right)=-{\frac {p}{\rho ^{2}}}\nabla \rho +\mathbf {u} \times (\nabla \times \mathbf {u} )-{\frac {\partial \mathbf {u} }{\partial t}}.} In case of a steady flow the time derivative of the flow velocity disappears, so the momentum equation becomes: โ ( 1 2 u 2 + ฯ + p ฯ ) = โ p ฯ 2 โ ฯ + u ร ( โ ร u ) . {\displaystyle \nabla \left({\frac {1}{2}}u^{2}+\phi +{\frac {p}{\rho }}\right)=-{\frac {p}{\rho ^{2}}}\nabla \rho +\mathbf {u} \times (\nabla \times \mathbf {u} ).} And by
|
{
"page_id": 396022,
"source": null,
"title": "Euler equations (fluid dynamics)"
}
|
projecting the momentum equation on the flow direction, i.e. along a streamline, the cross product disappears because its result is always perpendicular to the velocity: u โ
โ ( 1 2 u 2 + ฯ + p ฯ ) = โ p ฯ 2 u โ
โ ฯ . {\displaystyle \mathbf {u} \cdot \nabla \left({\frac {1}{2}}u^{2}+\phi +{\frac {p}{\rho }}\right)=-{\frac {p}{\rho ^{2}}}\mathbf {u} \cdot \nabla \rho .} In the steady incompressible case the mass equation is simply: u โ
โ ฯ = 0 , {\displaystyle \mathbf {u} \cdot \nabla \rho =0,} that is the mass conservation for a steady incompressible flow states that the density along a streamline is constant. Then the Euler momentum equation in the steady incompressible case becomes: u โ
โ ( 1 2 u 2 + ฯ + p ฯ ) = 0. {\displaystyle \mathbf {u} \cdot \nabla \left({\frac {1}{2}}u^{2}+\phi +{\frac {p}{\rho }}\right)=0.} The convenience of defining the total head for an inviscid liquid flow is now apparent: b l โก 1 2 u 2 + ฯ + p ฯ , {\displaystyle b_{l}\equiv {\frac {1}{2}}u^{2}+\phi +{\frac {p}{\rho }},} which may be simply written as: u โ
โ b l = 0. {\displaystyle \mathbf {u} \cdot \nabla b_{l}=0.} That is, the momentum balance for a steady inviscid and incompressible flow in an external conservative field states that the total head along a streamline is constant. === Compressible case === In the most general steady (compressible) case the mass equation in conservation form is: โ โ
j = ฯ โ โ
u + u โ
โ ฯ = 0. {\displaystyle \nabla \cdot \mathbf {j} =\rho \nabla \cdot \mathbf {u} +\mathbf {u} \cdot \nabla \rho =0.} Therefore, the previous expression is rather u โ
โ ( 1 2 u 2 + ฯ + p ฯ ) = p ฯ โ
|
{
"page_id": 396022,
"source": null,
"title": "Euler equations (fluid dynamics)"
}
|
โ
u . {\displaystyle \mathbf {u} \cdot \nabla \left({\frac {1}{2}}u^{2}+\phi +{\frac {p}{\rho }}\right)={\frac {p}{\rho }}\nabla \cdot \mathbf {u} .} The right-hand side appears on the energy equation in convective form, which on the steady state reads: u โ
โ e = โ p ฯ โ โ
u . {\displaystyle \mathbf {u} \cdot \nabla e=-{\frac {p}{\rho }}\nabla \cdot \mathbf {u} .} The energy equation therefore becomes: u โ
โ ( e + p ฯ + 1 2 u 2 + ฯ ) = 0 , {\displaystyle \mathbf {u} \cdot \nabla \left(e+{\frac {p}{\rho }}+{\frac {1}{2}}u^{2}+\phi \right)=0,} so that the internal specific energy now features in the head. Since the external field potential is usually small compared to the other terms, it is convenient to group the latter ones in the total enthalpy: h t โก e + p ฯ + 1 2 u 2 , {\displaystyle h^{t}\equiv e+{\frac {p}{\rho }}+{\frac {1}{2}}u^{2},} and the Bernoulli invariant for an inviscid gas flow is: b g โก h t + ฯ = b l + e , {\displaystyle b_{g}\equiv h^{t}+\phi =b_{l}+e,} which can be written as: u โ
โ b g = 0. {\displaystyle \mathbf {u} \cdot \nabla b_{g}=0.} That is, the energy balance for a steady inviscid flow in an external conservative field states that the sum of the total enthalpy and the external potential is constant along a streamline. In the usual case of small potential field, simply: u โ
โ h t โผ 0. {\displaystyle \mathbf {u} \cdot \nabla h^{t}\sim 0.} === Friedmann form and Crocco form === By substituting the pressure gradient with the entropy and enthalpy gradient, according to the first law of thermodynamics in the enthalpy form: v โ p = โ T โ s + โ h , {\displaystyle v\nabla p=-T\nabla s+\nabla h,} in the convective form of
|
{
"page_id": 396022,
"source": null,
"title": "Euler equations (fluid dynamics)"
}
|
Euler momentum equation, one arrives to: D u D t = T โ s โ โ h . {\displaystyle {\frac {D\mathbf {u} }{Dt}}=T\nabla \,s-\nabla \,h.} Friedmann deduced this equation for the particular case of a perfect gas and published it in 1922. However, this equation is general for an inviscid nonconductive fluid and no equation of state is implicit in it. On the other hand, by substituting the enthalpy form of the first law of thermodynamics in the rotational form of Euler momentum equation, one obtains: โ u โ t + 1 2 โ ( u 2 ) + ( โ ร u ) ร u + โ p ฯ = g , {\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+{\frac {1}{2}}\nabla \left(u^{2}\right)+(\nabla \times \mathbf {u} )\times \mathbf {u} +{\frac {\nabla p}{\rho }}=\mathbf {g} ,} and by defining the specific total enthalpy: h t = h + 1 2 u 2 , {\displaystyle h^{t}=h+{\frac {1}{2}}u^{2},} one arrives to the CroccoโVazsonyi form (Crocco, 1937) of the Euler momentum equation: โ u โ t + ( โ ร u ) ร u โ T โ s + โ h t = g . {\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+(\nabla \times \mathbf {u} )\times \mathbf {u} -T\nabla s+\nabla h^{t}=\mathbf {g} .} In the steady case the two variables entropy and total enthalpy are particularly useful since Euler equations can be recast into the Crocco's form: u ร โ ร u + T โ s โ โ h t = g , u โ
โ s = 0 , u โ
โ h t = 0. {\displaystyle {\begin{aligned}\mathbf {u} \times \nabla \times \mathbf {u} +T\nabla s-\nabla h^{t}&=\mathbf {g} ,\\\mathbf {u} \cdot \nabla s&=0,\\\mathbf {u} \cdot \nabla h^{t}&=0.\end{aligned}}} Finally if the flow is also isothermal: T โ s = โ ( T s ) ,
|
{
"page_id": 396022,
"source": null,
"title": "Euler equations (fluid dynamics)"
}
|
{\displaystyle T\nabla s=\nabla (Ts),} by defining the specific total Gibbs free energy: g t โก h t + T s , {\displaystyle g^{t}\equiv h^{t}+Ts,} the Crocco's form can be reduced to: u ร โ ร u โ โ g t = g , u โ
โ g t = 0. {\displaystyle {\begin{aligned}\mathbf {u} \times \nabla \times \mathbf {u} -\nabla g^{t}&=\mathbf {g} ,\\\mathbf {u} \cdot \nabla g^{t}&=0.\end{aligned}}} From these relationships one deduces that the specific total free energy is uniform in a steady, irrotational, isothermal, isoentropic, inviscid flow. == Discontinuities == The Euler equations are quasilinear hyperbolic equations and their general solutions are waves. Under certain assumptions they can be simplified leading to Burgers equation. Much like the familiar oceanic waves, waves described by the Euler Equations 'break' and so-called shock waves are formed; this is a nonlinear effect and represents the solution becoming multi-valued. Physically this represents a breakdown of the assumptions that led to the formulation of the differential equations, and to extract further information from the equations we must go back to the more fundamental integral form. Then, weak solutions are formulated by working in 'jumps' (discontinuities) into the flow quantities โ density, velocity, pressure, entropy โ using the RankineโHugoniot equations. Physical quantities are rarely discontinuous; in real flows, these discontinuities are smoothed out by viscosity and by heat transfer. (See NavierโStokes equations) Shock propagation is studied โ among many other fields โ in aerodynamics and rocket propulsion, where sufficiently fast flows occur. To properly compute the continuum quantities in discontinuous zones (for example shock waves or boundary layers) from the local forms (all the above forms are local forms, since the variables being described are typical of one point in the space considered, i.e. they are local variables) of Euler equations through finite difference methods generally too
|
{
"page_id": 396022,
"source": null,
"title": "Euler equations (fluid dynamics)"
}
|
many space points and time steps would be necessary for the memory of computers now and in the near future. In these cases it is mandatory to avoid the local forms of the conservation equations, passing some weak forms, like the finite volume one. === RankineโHugoniot equations === Starting from the simplest case, one consider a steady free conservation equation in conservation form in the space domain: โ โ
F = 0 , {\displaystyle \nabla \cdot \mathbf {F} =\mathbf {0} ,} where in general F is the flux matrix. By integrating this local equation over a fixed volume Vm, it becomes: โซ V m โ โ
F d V = 0 . {\displaystyle \int _{V_{m}}\nabla \cdot \mathbf {F} \,dV=\mathbf {0} .} Then, basing on the divergence theorem, we can transform this integral in a boundary integral of the flux: โฎ โ V m F d s = 0 . {\displaystyle \oint _{\partial V_{m}}\mathbf {F} \,ds=\mathbf {0} .} This global form simply states that there is no net flux of a conserved quantity passing through a region in the case steady and without source. In 1D the volume reduces to an interval, its boundary being its extrema, then the divergence theorem reduces to the fundamental theorem of calculus: โซ x m x m + 1 F ( x โฒ ) d x โฒ = 0 , {\displaystyle \int _{x_{m}}^{x_{m+1}}\mathbf {F} (x')\,dx'=\mathbf {0} ,} that is the simple finite difference equation, known as the jump relation: ฮ F = 0 . {\displaystyle \Delta \mathbf {F} =\mathbf {0} .} That can be made explicit as: F m + 1 โ F m = 0 , {\displaystyle \mathbf {F} _{m+1}-\mathbf {F} _{m}=\mathbf {0} ,} where the notation employed is: F m = F ( x m ) . {\displaystyle \mathbf {F} _{m}=\mathbf {F} (x_{m}).}
|
{
"page_id": 396022,
"source": null,
"title": "Euler equations (fluid dynamics)"
}
|
Or, if one performs an indefinite integral: F โ F 0 = 0 . {\displaystyle \mathbf {F} -\mathbf {F} _{0}=\mathbf {0} .} On the other hand, a transient conservation equation: โ y โ t + โ โ
F = 0 , {\displaystyle {\partial y \over \partial t}+\nabla \cdot \mathbf {F} =\mathbf {0} ,} brings to a jump relation: d x d t ฮ u = ฮ F . {\displaystyle {\frac {dx}{dt}}\,\Delta u=\Delta \mathbf {F} .} For one-dimensional Euler equations the conservation variables and the flux are the vectors: y = ( 1 v j E t ) , {\displaystyle \mathbf {y} ={\begin{pmatrix}{\frac {1}{v}}\\j\\E^{t}\end{pmatrix}},} F = ( j v j 2 + p v j ( E t + p ) ) , {\displaystyle \mathbf {F} ={\begin{pmatrix}j\\vj^{2}+p\\vj\left(E^{t}+p\right)\end{pmatrix}},} where: v {\displaystyle v} is the specific volume, j {\displaystyle j} is the mass flux. In the one dimensional case the correspondent jump relations, called the RankineโHugoniot equations, are:< d x d t ฮ ( 1 v ) = ฮ j , d x d t ฮ j = ฮ ( v j 2 + p ) , d x d t ฮ E t = ฮ ( j v ( E t + p ) ) . {\displaystyle {\begin{aligned}{\frac {dx}{dt}}\Delta \left({\frac {1}{v}}\right)&=\Delta j,\\[1.2ex]{\frac {dx}{dt}}\Delta j&=\Delta (vj^{2}+p),\\[1.2ex]{\frac {dx}{dt}}\Delta E^{t}&=\Delta (jv(E^{t}+p)).\end{aligned}}} In the steady one dimensional case the become simply: ฮ j = 0 , ฮ ( v j 2 + p ) = 0 , ฮ ( j ( E t ฯ + p ฯ ) ) = 0. {\displaystyle {\begin{aligned}\Delta j&=0,\\[1.2ex]\Delta \left(vj^{2}+p\right)&=0,\\[1.2ex]\Delta \left(j\left({\frac {E^{t}}{\rho }}+{\frac {p}{\rho }}\right)\right)&=0.\end{aligned}}} Thanks to the mass difference equation, the energy difference equation can be simplified without any restriction: ฮ j = 0 , ฮ ( v j 2 + p ) = 0 , ฮ h t =
|
{
"page_id": 396022,
"source": null,
"title": "Euler equations (fluid dynamics)"
}
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