In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous fluid substances. These balance equations arise from applying Isaac Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term and a pressure term—hence describing viscous flow; the main difference between them and the simpler Euler equations for inviscid flow is that Navier–Stokes equations factor in the Froude limit and are not conservation equations, but rather a dissipative system, in the sense that they cannot be put into the quasilinear homogeneous form: y t + A y x = 0. The Navier–Stokes equations are useful because they describe the physics of many phenomena of scientific and engineering interest, they may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing. The Navier–Stokes equations, in their full and simplified forms, help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, many other things.

Coupled with Maxwell's equations, they can be used to study magnetohydrodynamics. The Navier–Stokes equations are of great interest in a purely mathematical sense. Despite their wide range of practical uses, it has not yet been proven whether solutions always exist in three dimensions and, if they do exist, whether they are smooth – i.e. they are infinitely differentiable at all points in the domain. These are called the Navier–Stokes existence and smoothness problems; the Clay Mathematics Institute has called this one of the seven most important open problems in mathematics and has offered a US$1 million prize for a solution or a counterexample. The solution of the equations is a flow velocity, it is a vector field - to every point in a fluid, at any moment in a time interval, it gives a vector whose direction and magnitude are those of the velocity of the fluid at that point in space and at that moment in time. It is studied in three spatial dimensions and one time dimension, although the two dimensional case is useful as a model, higher-dimensional analogues are of both pure and applied mathematical interest.

Once the velocity field is calculated, other quantities of interest such as pressure or temperature may be found using dynamical equations and relations. This is different from what one sees in classical mechanics, where solutions are trajectories of position of a particle or deflection of a continuum. Studying velocity instead of position makes more sense for a fluid. In particular, the streamlines of a vector field, interpreted as flow velocity, are the paths along which a massless fluid particle would travel; these paths are the integral curves whose derivative at each point is equal to the vector field, they can represent visually the behavior of the vector field at a point in time. The Navier–Stokes momentum equation can be derived as a particular form of the Cauchy momentum equation, whose general convective form is D u D t = 1 ρ ∇ ⋅ σ + g By setting the Cauchy stress tensor σ to be the sum of a viscosity term τ and a pressure term − p I we arrive at where D D t is the material derivative, defined as ∂ ∂ t + u ⋅ ∇, ρ is the density, u is the flow velocity, ∇ ⋅ is the divergence, p is the pressure, t is time, τ is the deviatoric stress tensor, which has order two, g represents body accelerations acting on the continuum, for example gravity, inertial accelerations, electrostatic accelerations, so on,In this form, it is apparent that in the assumption of an inviscid fluid -no deviatoric stress- Cauchy equations reduce to the Euler equations.

Assuming conservation of mass we can use the mass continuity equation, ∂ ρ ∂ t + ∇ ⋅ = 0 to arrive to the conservation form of the equations of motion. This is written: where ⊗ is the outer product: u ⊗ v = u v T; the left side of the equation describes acceleration, may be composed of time-dependent and convective components. The right side of the equation is in effect a summation of hydrostatic effects, the divergence of deviatoric stress and body forces. All

Leila J. Rupp is a historian and professor of Feminist Studies at the University of California, Santa Barbara, she is an alumna of Bryn Mawr College, a member of the Seven Sisters women's colleges, where she received her bachelor's degree in 1972 as well as Ph. D. in 1976, both in history. Her areas of interest include women's movements, LGBT and women's history, she was the editor of the Journal of Women's History from 1996 to 2004. An out lesbian, her partner is Verta Taylor. Leila J. Rupp: Transnational Women's Movements, European History Online, Mainz: Institute of European History, 2011, retrieved: June 22, 2011. Leila J. Rupp, Sapphistries: A Global History of Love Between Women. Leila J. Rupp and Verta Taylor, Drag Queens at the 801 Cabaret. Xiii, 256 p.: ill.. ISBN 978-0-226-73158-2 Leila J. Rupp, A Desired Past: A Short History of Same-Sex Love in America. Vytou_ená minulost. Leila J. Rupp, Worlds of Women: The Making of an International Women's Movement. Leila J. Rupp and Verta Taylor, Survival in the Doldrums: The American Women's Rights Movement, 1945 to the 1960s.

Excerpt reprinted in Perspectives on the American Past. Leila J. Rupp, Mobilizing Women for War: German and American Propaganda, 1939-1945

The Herald is a seven-day morning daily newspaper published in Sharon, covering Mercer County and the greater Shenango Valley area of Pennsylvania. It is owned by Community Newspaper Holdings Inc. Montgomery, Ala; the Sharon Herald, a weekly newspaper, began publication April 14, 1864, founded by R. C. and James Frey. It was converted to a daily newspaper nearly 45 years April 12, 1909; the newspaper's office at the foot of Pitt Street in Sharon was washed into the Shenango River during a flood in March 1913. The newspaper missed only four issues and resumed publication with temporary production first in Farrell and for about a month at the printing plant of the New Castle News before a new office and pressroom were set up on Chestnut Street in Sharon in the Willsonia Building; the Herald merged with its main competitor, the Sharon News-Telegraph, May 13, 1935. The News-Telegraph incorporated the old Farrell Sharon Telegraph; the new newspaper kept The Sharon Herald as its name but production moved to the News-Telegraph building two blocks away on South Dock Street.

The newspaper diversified during the mid-20th century and owning Sharon radio station WPIC from 1938 to 1959. Hubbard Press ceased publication in 2012. On Feb. 23, 1970, The Herald dropped "Sharon" from its nameplate to reflect its countywide audience. The next year, it was bought by Inc. a division of Dow Jones & Company. Outside ownership brought the first Sunday edition, on September 9, 1990, the launch of sharonherald.com in May 1996, making it one of the first Ottaway papers to be online. The Herald's Digital Edition, a subscription-only facsimile of the print edition available as Adobe Acrobat PDF pages, debuted in November 2001 as one of the first of its kind in the country. Both the website and digital edition were developed by in-house talent. In 1981, printing was moved from an aging Scott letterpress at the main office at 52 South Dock Street to the new Dow Jones & Company offset printing plant six miles away in Shenango Township, built to print regional editions of the parent company's The Wall Street Journal.

Page negatives photographed of pasted-up pages – and paginated pages sent directly to an imagesetter that produced page negatives – were driven to the plant for platemaking. CNHI, the current owners, bought The Herald effective March 29, 2002. In the fall of 2002, The Herald switched to morning publication, it was not the first time. The Sharon Herald had been published both as a morning daily and at the New Castle News' printing plant as a professional courtesy for about a month after the flood of 1913 until The Herald re-established a pressroom in downtown Sharon and an afternoon publication cycle; the Herald is printed at the CNHI-owned West Penn Printing plant near New Castle in neighboring Lawrence County, Pennsylvania. The Herald publishes a twice-weekly newspaper, Allied News, covering outlying areas of eastern and southeastern Mercer County. A 14-unit, single-wide DGM press configured with two full-color towers at West Penn Printing, built in five months on a brownfield on Sampson Street outside New Castle and placed in operation in early 2003, replaced an aging Goss Metro offset press, installed in 1968 at the News' downtown headquarters.

Printing of The Herald, Allied News and Hubbard Press had been transferred to the downtown New Castle press in August 2002 from the TKS press line at the Wall Street Journal printing plant in West Middlesex, owned by The Herald's former parent company, Dow Jones & Company Inc. It had been printed there since the spring of 1981. In April 2016 the media watchdog group, iMediaEthics.org, published How Not to Report on Suicide: Sharon Herald Story Crossed the Line. The article begins, "Readers were shocked after daily Pennsylvania newspaper The Sharon Herald published an insensitive and gratuitous story that horrifically detailed a local man’s death by suicide." Numerous community members registered their outrage at The Herald"s lack of ethics on the paper's Facebook page. Criticisms of The Herald's coverage include disregard accepted journalistic standards for covering suicide; the Herald Website CNHI Website http://www.imediaethics.org/case-study-report-suicide-sharon-herald/

The Hebrew word onah, which does not occur in the Bible means "time period" or "season." In the context of the Jewish laws of niddah, it refers to a day or a night. Each 24-hour day thus consists of two onot; the daytime onah begins at "netz hachamah" and ends at "shekiat hachamah". The night-time onah lasts from sunset until sunrise. Marital relations are forbidden on an onah; this is called an "onat perishah". The term onah can refer to the length of the menstrual cycle. Halachically, one assumes; the term "mitzvat onah" refers to a husband's conjugal obligations toward his wife and is used as a halachic euphemism for marital relations. Some say: The mitzvah of onah – marital intimate relations – is defined in the Mishnah as fixed, regular times that a married couple may not be intimate together; the times for conjugal duty prescribed in the Torah are: for men of every day. These are the rulings of R. Eliezer. R. Joseph learnt: Her flesh implies close bodily contact, that he must not treat her in the manner of the Persians who perform their conjugal duties in their clothes.

This provides support for R. Huna who laid down that a husband who said, ‘I will not unless she wears her clothes and I mine’, must divorce her and give her her ketubah. R. Yaakov Emden: One should ease his wife's mind and make her happy, prepare her and nurture her with words that make her happy so that she feels passionate towards him. According to the Rabad enumerated four permitted "kavvanot" for sexual relations with rewards in the world to come: for procreation, for welfare of the fetus, for a wife’s desire, that a man has desire to act promiscuously and relieves that through intercourse with his wife, yet the last one is a lesser reward. If He does not show any strength, has sex anytime he wants, this would not be rewarded. Maimonides and other rationalists saw sexuality and desire as an animal drive, not something for the rational man, he says: “The sense of touch, a disgrace to us leads to indulge in eating and sensuality”, etc. There was a deep reaction to this about a generation after the Rambam.

The growing movement of Kabbalah and other schools of the region had a negative reaction to this passage of Maimonides quoting Aristotle. The Nahmanides in "Iggeret Ha-kodesh" says: "But we who have the Torah and believe that God created all in his wisdom created anything inherently ugly or unseemly. If we were to say that intercourse is repulsive we blaspheme God who made the genitals"

Dorset, is a county in South West England on the English Channel coast. The ceremonial county comprises the area covered by the non-metropolitan county, governed by Dorset County Council, together with the unitary authorities of Poole and Bournemouth. Dorset is an average sized county with an area of 2,653 square kilometres. Around half of Dorset's population lives in the South East Dorset conurbation; the rest of the county is rural with a low population density. Dorset has a long history of a rich culture; the county contains 1,500 scheduled ancient monuments, including the iron age hillfort, Maiden Castle. It is famed in literature as the birthplace of Thomas Hardy and has been an inspiration to several authors including Enid Blyton who used the local landscape in many of her books; the local people have their own regional dialect, still spoken in parts. The county hosts many annual events and fairs including the Great Dorset Steam Fair near Blandford, purported to be the largest outdoor event in Europe.

As a rural county, Dorset has fewer major cultural institutions than larger or more densely populated areas. Major venues for concerts and theatre include Poole Borough Council's Lighthouse arts centre, Bournemouth's BIC and Pavilion Theatre, Verwood's Hub, Wimborne's Tivoli Theatre, the Pavilion theatre in Weymouth. Dorset's most famous cultural institution is the Bournemouth Symphony Orchestra, founded in 1893 and now one of the country's most celebrated orchestras. Since 1974, Dorset Opera have been running a opera summer school. An annual production at Bryanston School near Blandford has a chorus of 50-80 pupils performing alongside soloists of international repute, a full professional orchestra with an acclaimed conductor and director. In 2011 this turned into Dorset Opera Festival; the Yetties are a folk music group who take their name from the Dorset village of Yetminster, their childhood home. The Yetties now live in Sherborne and they draw on their personal experiences of country life for much of their material.

Some of the songs they sing recall what life was like when they were kids helping the farmers at harvest time, scrumping when the farmers weren't looking, raiding the countryside for food and eating rabbit for every meal. Dorset is the birthplace of Greg Lake and Robert Fripp. Dorset's dramatic coastline, manor houses and gardens have featured in a number films and television productions including unsurprisingly, adaptations of The Mayor of Casterbridge, Tess of the D'Urbervilles and Far from the Madding Crowd. Lyme Regis was the home of author John Fowles and the 1980 film The French Lieutenant's Woman starring Meryl Streep and Jeremy Irons, based on his novel, was filmed there. Mapperton House and its gardens have been the setting for many period dramas such as the 1996 Emma, starring Gwyneth Paltrow, from Jane Austen's novel. Dorset's history is celebrated in more than 30 specialist museums throughout the county. There are museums dedicated to fashion, plastic design, mineral extraction, water supply and electricity.

In addition there is the internationally important Tank Museum at Bovington, The Keep Military Museum, one of the top provincial military museums in the country. The Dorset County Museum in Dorchester was founded in 1846 and contains an extensive collection of exhibits covering, amongst other things: the Jurassic Coast, local writers, wildlife and local history. Dorset contains 190 conservation areas, over 30 listed parks and gardens, more than 1,500 Scheduled Ancient Monuments and 12,850 listed buildings. Of the 229 that are Grade I listed, 174 are churches or places of worship, from the longest church in England, Christchurch Priory, to one of the smallest, St Edwold's. Nine castles are listed: some were constructed as defensive fortresses such as Corfe and Christchurch Castle, others are mock castles such as Highcliffe and Lulworth; the county has a long history of human settlement and some notable archaeology, such as the hill forts of Maiden Castle, Badbury Rings and Hod Hill. A large defensive ditch, Bokerley Dyke, delayed the Saxon conquest of Dorset for up to 150 years.

There are numerous ancient burial sites and standing stones. Some of the more impressive include the stone circles at Kingston Russell, Hampton Hill and Nine Stones near Winterbourne Abbas. Just north of Portesham is a huge Neolithic dolmen known as the Hell Stone consisting of nine upright stones and a capstone. Dorset hosts a number of annual festivals and events including the Great Dorset Steam Fair near Blandford, purported to be the largest outdoor event in Europe, the Bournemouth Air Festival, a free show that attracts 1.3 million visitors. The Spirit of the Sea Maritime Festival aims to combine sporting activities, cultural events and entertainments in a celebration of Weymouth and Portland's close ties with the sea; the Dorset County Show, which in 2011 will be in its 170th year, is a celebration of Dorset's relationship with agriculture. The two-day event showcases local produce and livestock and attract

Christopher Michael Dawson is an Australian former professional rugby league footballer who played in the 1970s. Following the 1982 disappearance of his wife and two separate coronial inquests, the NSW Coroner determined that Lynette Dawson was dead and that her most murderer was her husband, Chris. After many years of stalled and failed investigations, on 5 December 2018 Chris Dawson was arrested by detectives from Queensland Police and extradited to New South Wales the following day. Dawson is on bail, pending a trial for the murder of Lynette Dawson. In June 2019 he formally pleaded not guilty to her murder. In February 2020, Dawson was committed to stand trial for the murder of Lynette. Chris Dawson was the main subject of an investigative podcast series, The Teacher's Pet, by journalist Hedley Thomas. Thomas investigated allegations into the disappearance of Lynette Dawson and brought Chris Dawson into renewed public and media focus through the podcast's broadcast in 2018; as of December 2018, the Gold Walkley-award winning podcast was downloaded 28 million times.

Dawson was born in New South Wales. He is the second born of twins. Chris Dawson attended Sydney Boys High School where he was a prefect and graduated in 1966. Both he and Paul Dawson played rugby union for Eastern Suburbs; the Dawson brothers switched codes to play rugby league for the Newtown Jets in 1972. Chris Dawson played in the second row along with Paul for the Newtown Jets club for five seasons; the Dawson brothers were part of the 1973 New South Wales Rugby League Club Championship winning team. Chris Dawson met Lynette Simms in 1965. After their rugby careers finished, both Chris and Paul Dawson became physical education teachers at public high schools on the north shore and northern beaches of Sydney. Dawson was teaching at Cromer High School in 1981 when he commenced a sexual relationship with a 16-year-old pupil, Joanne Curtis, while he was married to Lynette Dawson. Lynette Dawson went missing in January 1982, with Chris Dawson claiming she had left and joined a commune. Shortly after the disappearance of Lynette Dawson, Curtis moved in with Chris Dawson and they married in 1984.

Dawson and Curtis divorced in 1993. The Dawson brothers and families moved to Queensland in 1985 where Chris Dawson worked at Keebra Park State High School first later moved to Coombabah State High School to work with Paul Dawson. Lynette Dawson's body has never been found but two coronial inquests were conducted in 2001 and 2003 with both ruling that Lynette Dawson must be dead and was most murdered by a known person. In 2003 Dawson was working at St Ursula's College in Queensland. On 5 December 2018 Dawson was arrested for the murder of Lynette Dawson, he was extradited to Sydney on 6 December 2018 to face trial. He was refused bail and has been remanded in custody. On 17 December 2018 Dawson was granted his family paid A$1.5 million bail to be released. Appearing in the NSW Local Court in June 2019, Dawson pleaded not guilty to the murder of Lynette Dawson. In February 2020, Dawson was committed to stand trial for the murder of Lynette