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Soviet Union	according stanley fischer william easterly growth could faster.calculation per capita income twice higher considering amount investment education population.authors attribute poor performance low productivity capital.steven rosenfielde states standard living declined due stalins despotism.brief improvement death lapsed stagnation.in mikhail gorbachev attempted reform revitalize economy program perestroika.
Soviet Union	policies relaxed state control enterprises replace market incentives resulting sharp decline output.economy already suffering reduced petroleum export revenues started collapse.prices still fixed property still largely stateowned countrys dissolution.period world war ii collapse soviet gdp ppp secondlargest world third second half although percapita basis behind first world countries.compared countries similar percapita gdp soviet union experienced significant growth.in country human development index .
Soviet Union	placing high category human development.thirdhighest eastern bloc behind czechoslovakia east germany th world countries.energy need fuel declined soviet union per ruble gross social product per ruble industrial product.start decline grew rapidly gradually slowed .grew even slower .. david wilson historian believed gas industry would account soviet fuel production end century.
Soviet Union	theory come fruition ussrs collapse.ussr theory would continued economic growth rate –.soviet energy fields.however energy sector faced many difficulties among countrys high military expenditure hostile relations first world.in soviet union pipeline network kilometres mi crude oil another kilometres mi natural gas.petroleum petroleumbased products natural gas metals wood agricultural products variety manufactured goods primarily machinery arms military equipment exported.
Soviet Union	ussr heavily relied fossil fuel exports earn hard currency.peak largest producer secondlargest exporter crude oil surpassed saudi arabia.science technology soviet union placed great emphasis science technology within economy however remarkable soviet successes technology producing worlds first space satellite typically responsibility military.lenin believed ussr would never overtake developed world remained technologically backward upon founding.soviet authorities proved commitment lenins belief developing massive networks research development organizations.
Soviet Union	early soviets awarded chemistry phds women compared united states.soviet scientists among worlds besttrained specialists several areas energy physics selected areas medicine mathematics welding military technologies.due rigid state planning bureaucracy soviets remained far behind technologically chemistry biology computers compared first world.soviet government opposed persecuted geneticists favour lysenkoism pseudoscience rejected scientific community soviet union abroad supported stalins inner circles.implemented ussr china resulted reduced crop yields widely believed contributed great chinese famine.under reagan administration project socrates determined soviet union addressed acquisition science technology manner radically different us using.
Soviet Union	case us economic prioritization used indigenous research development means acquire science technology private public sectors.contrast ussr offensively defensively maneuvering acquisition use worldwide technology increase competitive advantage acquired technology preventing us acquiring competitive advantage.however technologybased planning executed centralized governmentcentric manner greatly hindered flexibility.exploited us undermine strength soviet union thus foster reform.transport transport vital component countrys economy.
Soviet Union	economic centralization late led development infrastructure massive scale notably establishment aeroflot aviation enterprise.country wide variety modes transport land water air.however due inadequate maintenance much road water soviet civil aviation transport outdated technologically backward compared first world.soviet rail transport largest intensively used world also better developed western counterparts.late early soviet economists calling construction roads alleviate burdens railways improve soviet government budget.street network automotive industry remained underdeveloped dirt roads common outside major cities.
Soviet Union	soviet maintenance projects proved unable take care even roads country had.earlytomids soviet authorities tried solve road problem ordering construction new ones.meanwhile automobile industry growing faster rate road construction.underdeveloped road network led growing demand public transport.despite improvements several aspects transport sector still riddled problems due outdated infrastructure lack investment corruption bad decisionmaking.soviet authorities unable meet growing demand transport infrastructure services.the soviet merchant navy one largest world.
Soviet Union	demographics excess deaths throughout world war russian civil war including postwar famine amounted combined total million million million –.postwar soviet population million smaller would prewar demographic growth continued.according catherine merridale ... reasonable estimate would place total number excess deaths whole period somewhere around million.the birth rate ussr decreased .per thousand .mainly due increasing urbanization rising average age marriages.
Soviet Union	mortality rate demonstrated gradual decrease well – .per thousand ..general birth rates southern republics transcaucasia central asia considerably higher northern parts soviet union cases even increased post–world war ii period phenomenon partly attributed slower rates urbanistion traditionally earlier marriages southern republics.soviet europe moved towards subreplacement fertility soviet central asia continued exhibit population growth well replacementlevel fertility.the late witnessed reversal declining trajectory rate mortality ussr especially notable among men working age also prevalent russia predominantly slavic areas country.
Soviet Union	analysis official data late showed worsening lates early adult mortality began improve again.infant mortality rate increased ...researchers regarded rise mostly real consequence worsening health conditions services.
Soviet Union	rises adult infant mortality explained defended soviet officials soviet government stopped publishing mortality statistics ten years.soviet demographers health specialists remained silent mortality increases lates publication mortality data resumed researchers could delve real causes.women fertility lenin state made explicit commitments promote equality men women.many early russian feminists ordinary russian working women actively participated revolution many affected events period new policies.beginning october lenins government liberalized divorce abortion laws decriminalized homosexuality recriminalized permitted cohabitation ushered host reforms.
Soviet Union	however without birth control new system produced many broken marriages well countless outofwedlock children.epidemic divorces extramarital affairs created social hardships soviet leaders wanted people concentrate efforts growing economy.giving women control fertility also led precipitous decline birth rate perceived threat countrys military power.stalin reversed liberal laws ushering pronatalist era lasted decades.by russia became first great power grant women right vote.heavy casualties world war ii women outnumbered men russia ratio.
Soviet Union	contributed larger role women played russian society compared great powers time.education anatoly lunacharsky became first peoples commissar education soviet russia.beginning soviet authorities placed great emphasis elimination illiteracy.lefthanded children forced write right hand soviet school system.literate people automatically hired teachers.
Soviet Union	short period quality sacrificed quantity.stalin could announce illiteracy eliminated.throughout social mobility rose sharply attributed reforms education.aftermath world war ii countrys educational system expanded dramatically tremendous effect.nearly children access education exception living remote areas.
Soviet Union	nikita khrushchev tried make education accessible making clear children education closely linked needs society.education also became important giving rise new man.citizens directly entering workforce constitutional right job free vocational training.the education system highly centralized universally accessible citizens affirmative action applicants nations associated cultural backwardness.however part general antisemitic policy unofficial jewish quota applied leading institutions higher education subjecting jewish applicants harsher entrance examinations.brezhnev era also introduced rule required university applicants present reference local komsomol party secretary.
Soviet Union	according statistics number higher education students per population ussr compared us.nationalities ethnic groups soviet union ethnically diverse country distinct ethnic groups.total population country estimated million .according estimate majority population russians .followed ukrainians .
Soviet Union	uzbeks ..all citizens ussr ethnic affiliation.ethnicity person chosen age sixteen childs parents.parents agree child automatically assigned ethnicity father.partly due soviet policies smaller minority ethnic groups considered part larger ones mingrelians georgia classified linguistically related georgians.ethnic groups voluntarily assimilated others brought force.
Soviet Union	russians belarusians ukrainians east slavic orthodox shared close cultural ethnic religious ties groups not.multiple nationalities living territory ethnic antagonisms developed years.members various ethnicities participated legislative bodies.organs power like politburo secretariat central committee etc.formally ethnically neutral reality ethnic russians overrepresented although also nonrussian leaders soviet leadership joseph stalin grigory zinoviev nikolai podgorny andrei gromyko.soviet era significant number ethnic russians ukrainians migrated soviet republics many settled there.
Soviet Union	according last census russian diaspora soviet republics reached million.health revolution health conditions significantly behind developed countries.lenin later noted either lice defeat socialism socialism defeat lice.soviet principle health care conceived peoples commissariat health .health care controlled state would provided citizens free charge revolutionary concept time.
Soviet Union	article soviet constitution gave citizens right health protection free access health institutions ussr.leonid brezhnev became general secretary soviet healthcare system held high esteem many foreign specialists.changed however brezhnevs accession mikhail gorbachevs tenure leader health care system heavily criticized many basic faults quality service unevenness provision.minister health yevgeniy chazov th congress communist party soviet union highlighting successes doctors hospitals world recognized systems areas improvement felt billions soviet rubles squandered.after revolution life expectancy age groups went up.statistic seen socialist system superior capitalist system.
Soviet Union	improvements continued statistics indicated life expectancy briefly surpassed united states.life expectancy started decline possibly alcohol abuse.time infant mortality began rise.government stopped publishing statistics matter.trend partly explained number pregnancies rising drastically asian part country infant mortality highest declining markedly developed european part soviet union.
Soviet Union	dentistry soviet dental technology dental health considered notoriously bad.average yearold cavities fillings missing teeth.toothpaste often available toothbrushes conform standards modern dentistry.language lenin government gave small language groups writing systems.development writing systems highly successful even though flaws detected.
Soviet Union	later days ussr countries multilingual situation implemented similar policies.serious problem creating writing systems languages differed dialectally greatly other.language given writing system appeared notable publication would attain official language status.many minority languages never received writing system therefore speakers forced second language.examples government retreated policy notably stalin education discontinued languages widespread.
Soviet Union	languages assimilated another language mostly russian.world war ii minority languages banned speakers accused collaborating enemy.as widely spoken soviet unions many languages russian de facto functioned official language language interethnic communication russian язык межнационального общения assumed de jure status official national language .religion christianity islam highest number adherents among religious citizens.eastern christianity predominated among christians russias traditional russian orthodox church largest christian denomination.soviet unions muslims sunnis shias concentrated azerbaijan ssr.
Soviet Union	smaller groups included roman catholics jews buddhists variety protestant denominations especially baptists lutherans.religious influence strong russian empire.russian orthodox church enjoyed privileged status church monarchy took part carrying official state functions.immediate period following establishment soviet state included struggle orthodox church revolutionaries considered ally former ruling classes.in soviet law freedom hold religious services constitutionally guaranteed although ruling communist party regarded religion incompatible marxist spirit scientific materialism.practice soviet system subscribed narrow interpretation right fact used range official measures discourage religion curb activities religious groups.the council peoples commissars decree establishing russian sfsr secular state also decreed teaching religion places subjects general instruction taught forbidden.citizens may teach may taught religion privately.
Soviet Union	among restrictions adopted included express prohibitions range church activities including meetings organized bible study.christian nonchristian establishments shut thousands s. many churches synagogues mosques operating closed.the soviet union officially secular state governmentsponsored program forced conversion atheism conducted doctrine state atheism.government targeted religions based state interests organized religions never outlawed religious property confiscated believers harassed religion ridiculed atheism propagated schools.government founded league militant atheists intensify propaganda campaign.accordingly although personal expressions religious faith explicitly banned strong sense social stigma imposed formal structures mass media generally considered unacceptable members certain professions teachers state bureaucrats soldiers openly religious.
Soviet Union	persecution accelerated following stalins rise power revival orthodoxy fostered government world war ii soviet authorities sought control russian orthodox church rather liquidate it.first five years soviet power bolsheviks executed russian orthodox bishops russian orthodox priests.many others imprisoned exiled.believers harassed persecuted.seminaries closed publication religious material prohibited.
Soviet Union	churches remained open existence world war i.convinced religious antisovietism become thing past looming threat war stalin regime began shifting moderate religion policy late s. soviet religious establishments overwhelmingly rallied support war effort world war ii.amid accommodations religious faith german invasion churches reopened.radio moscow began broadcasting religious hour historic meeting stalin orthodox church leader patriarch sergius moscow held .stalin support majority religious people ussr even late s. general tendency period increase religious activity among believers faiths.under nikita khrushchev state leadership clashed churches – period atheism emphasized educational curriculum numerous state publications promoted atheistic views.period number churches fell number synagogues dropped .
Soviet Union	number working mosques also declined falling within decade.religious institutions remained monitored soviet government churches synagogues temples mosques given leeway brezhnev era.official relations orthodox church government warmed point brezhnev government twice honored orthodox patriarch alexy order red banner labour.poll conducted soviet authorities recorded soviet population active religious believers.legacy legacy ussr remains controversial topic.socioeconomic nature communist states ussr especially stalin also much debated varyingly labelled form bureaucratic collectivism state capitalism state socialism totally unique mode production.the ussr implemented broad range policies long period time large amount conflicting policies implemented different leaders.
Soviet Union	positive view whilst others critical towards country calling repressive oligarchy.opinions ussr complex changed time different generations different views matter well soviet policies corresponding separate time periods history.leftists largely varying views ussr.whilst leftists anarchists libertarian socialists agree give workers control means production centralized oligarchy others positive opinions bolshevik policies vladimir lenin.many antistalinist leftists anarchists extremely critical soviet authoritarianism repression.
Soviet Union	much criticism receives centered around massacres soviet union centralized hierarchy present ussr mass political repression well violence towards government critics political dissidents leftists.critics also point towards failure implement substantial worker cooperatives implementing worker liberation well corruption soviet authoritarian nature.many russians former soviet citizens nostalgia ussr pointing towards infrastructure built soviet times increased job security increased literacy rate increased caloric intake supposed ethnic pluralism enacted soviet union well political stability.russian revolution also seen positive light well leadership lenin nikita khrushchev later ussr although many view joseph stalins rule positive country.armenia respondents said ussr collapse good said harm.kyrgyzstan respondents said collapse ussr good said harm.
Soviet Union	rating sociological group poll ukrainian respondents positive opinion soviet leader leonid brezhnev ruled soviet union .much admiration ussr comes failings modern postsoviet governments control oligarchs corruption outdated sovietera infrastructure well rise dominance organised crime collapse ussr directly leading nostalgia it.the – period world war ii still known russia great patriotic war.war became topic great importance cinema literature history lessons school mass media arts.result massive losses suffered military civilians conflict victory day celebrated may still one important emotional dates russia.former soviet republics post soviet republics negative view ussr although unanimity matter.
Soviet Union	large part due holodomor ethnic ukrainians negative view it.russianspeaking ukrainians ukraines southern eastern regions positive view ussr.countries internal conflict also nostalgia ussr especially refugees postsoviet conflicts forced flee homes displaced.nostalgia less admiration country policies longing return homes live poverty.many russian enclaves former ussr republics transnistria general positive remembrance it.
Soviet Union	political left lefts view ussr complex.leftists regard ussr example state capitalism oligarchical state leftists admire vladimir lenin russian revolution.council communists generally view ussr failing create class consciousness turning corrupt state elite controlled society.anarchists critical country labeling soviet system red fascism.soviets actively destroyed anarchist organizations anarchist communities labeling anarchists enemies people.factors contributing animosity towards ussr included soviet invasion anarchist free territory suppression anarchist kronstadt rebellion response norilsk uprising prisoners created radical system government based cooperatives direct democracy gulag.anarchist organizations unions also banned spanish civil war republican government orders soviet government.
Soviet Union	due anarchists generally hold large animosity towards ussr.culture culture soviet union passed several stages ussrs existence.first decade following revolution relative freedom artists experimented several different styles find distinctive soviet style art.lenin wanted art accessible russian people.hand hundreds intellectuals writers artists exiled executed work banned nikolay gumilyov shot alleged conspiring bolshevik regime yevgeny zamyatin.the government encouraged variety trends.
Soviet Union	art literature numerous schools traditional others radically experimental proliferated.communist writers maxim gorky vladimir mayakovsky active time.means influencing largely illiterate society films received encouragement state much director sergei eisensteins best work dates period.during stalins rule soviet culture characterized rise domination governmentimposed style socialist realism trends severely repressed rare exceptions mikhail bulgakovs works.many writers imprisoned killed.following khrushchev thaw censorship diminished.time distinctive period soviet culture developed characterized conformist public life intense focus personal life.
Soviet Union	greater experimentation art forms permissible resulting production sophisticated subtly critical work.regime loosened emphasis socialist realism thus instance many protagonists novels author yury trifonov concerned problems daily life rather building socialism.underground dissident literature known samizdat developed late period.architecture khrushchev era mostly focused functional design opposed highly decorated style stalins epoch.music response increasing popularity forms popular music like jazz west many jazz orchestras permitted throughout ussr notably melodiya ensemble named principle record label ussr.in second half gorbachevs policies perestroika glasnost significantly expanded freedom expression throughout country media press.
Soviet Union	sport founded july moscow sovetsky sport first sports newspaper soviet union.the soviet olympic committee formed april ioc recognized new body th session.year soviet representative konstantin andrianov became ioc member ussr officially joined olympic movement.summer olympics helsinki thus became first olympic games soviet athletes.soviet union biggest rival united states summer olympics winning six nine appearances games also topping medal tally winter olympics six times.soviet unions olympics success attributed large investment sports demonstrate superpower image political influence global stage.the soviet union national ice hockey team nearly every world championship olympic tournament never failed medal international ice hockey federation iihf tournament competed.the advent statesponsored fulltime amateur athlete eastern bloc countries eroded ideology pure amateur put selffinanced amateurs western countries disadvantage.
Soviet Union	soviet union entered teams athletes nominally students soldiers working profession – reality state paid many competitors train fulltime basis.nevertheless ioc held traditional rules regarding amateurism.a report committee australian senate claimed hardly medal winner moscow games certainly gold medal winner ... one sort drug another usually several kinds.moscow games might well called chemists games.a member ioc medical commission manfred donike privately ran additional tests new technique identifying abnormal levels testosterone measuring ratio epitestosterone urine.twenty percent specimens tested including sixteen gold medalists would resulted disciplinary proceedings tests official.results donikes unofficial tests later convinced ioc add new technique testing protocols.
Soviet Union	first documented case blood doping occurred summer olympics runner transfused two pints blood winning medals m.documentation obtained revealed soviet unions plans statewide doping system track field preparation summer olympics los angeles.dated decision boycott games document detailed existing steroids operations program along suggestions enhancements.dr. sergei portugalov institute physical culture prepared communication directed soviet unions head track field.portugalov later became one leading figures involved implementation russian doping summer olympics.environment official soviet environmental policy always attached great importance actions human beings actively improve nature.
Soviet Union	lenins quote communism soviet power electrification country many respects summarizes focus modernization industrial development.first fiveyear plan stalin proceeded industrialize country costs.values environmental nature protection completely ignored struggle create modern industrial society.stalins death focused environmental issues basic perception value environmental protection remained same.the soviet media always focused vast expanse land virtually indestructible natural resources.made feel contamination uncontrolled exploitation nature problem.
Soviet Union	soviet state also firmly believed scientific technological progress would solve problems.official ideology said socialism environmental problems could easily overcome unlike capitalist countries seemingly could solved.soviet authorities almost unwavering belief man could transcend nature.however authorities admit environmental problems ussr explained problems way socialism yet fully developed pollution socialist society temporary anomaly would resolved socialism developed.the chernobyl disaster first major accident civilian nuclear power plant.unparalleled world resulted large number radioactive isotopes released atmosphere.
Soviet Union	radioactive doses scattered relatively far.new cases thyroid cancer reported incident led relatively low number deaths data .however longterm effects accident unknown.another major accident kyshtym disaster.after fall ussr discovered environmental problems greater soviet authorities admitted.kola peninsula one places clear problems.
Soviet Union	around industrial cities monchegorsk norilsk nickel example mined forests destroyed contamination northern parts russia affected emissions.people west also interested radioactive hazards nuclear facilities decommissioned nuclear submarines processing nuclear waste spent nuclear fuel.also known early ussr transported radioactive material barents sea kara sea later confirmed russian parliament.crash k kursk submarine west raised concerns.past accidents involving submarines k k k k k k komsomolets.
Soviet Union	see also notes references bibliography ambler john shaw denis j.b. symons leslie .soviet east european transport problems.taylor francis.isbn .comrie bernard .languages soviet union.
Soviet Union	cambridge university press cup archive.isbn .davies robert wheatcroft stephen .industrialisation soviet russia volume years hunger soviet agriculture –.palgrave macmillan.isbn .fischer louis .
Soviet Union	life lenin.london weidenfeld nicolson.fischer stanley easterly william .soviet economic decline historical republican data pdf.world bank.archived pdf original march .
Soviet Union	retrieved october .janz denis .world christianity marxism.new york oxford university press.isbn .lane david stuart .soviet society perestroika.
Soviet Union	routledge.isbn .leggett george .cheka lenins political police.oxford oxford university press.isbn .lewin moshe .
Soviet Union	lenins last struggle.translated sheridan smith a. m. london faber faber.rayfield donald .stalin hangmen authoritative portrait tyrant served him.viking press.isbn .service robert .
Soviet Union	lenin biography.london macmillan.isbn .simon gerard .church state opposition u.s.s.r. berkeley los angeles university california press.isbn .volkogonov dmitri .
Soviet Union	lenin life legacy.translated shukman harold.london harpercollins.isbn .white james d. .lenin practice theory revolution.
Soviet Union	european history perspective.basingstoke england palgrave.isbn .wilson david .demand energy soviet union.taylor francis.
Soviet Union	isbn .world bank oecd .study soviet economy.vol..international monetary fund.
Soviet Union	isbn .palat madhavan k. .social identities revolutionary russia.uk palgrave.isbn .retrieved may .warshofsky lapidus gail .
Soviet Union	women soviet society equality development social change.berkeley ca university california press.isbn .wheatcroft stephen .scale nature german soviet repression mass killings – pdf.europeasia studies.
Comic relief	–.doi.. jstor .reading external links wikimedia atlas soviet unionimpressions soviet russia john deweya country study soviet union formerdefinition comic relief usually means releasing emotional tension resulting comic episode interposed midst serious tragic elements drama.comic relief often takes form bumbling wisecracking sidekick hero villain work fiction.sidekick used comic relief usually comment absurdity heros situation make comments would inappropriate character taken seriously.characters may use comic relief means irritate others keep confident.use sometimes comic relief characters appear fiction comic.
Comic relief	generally occurs work enters dramatic moment character continues comical regardless.greek tragedy allow comic relief within drama tradition concluding series several tragic performances humorous satyr play.even elizabethan critic sidney following horace’s ars poetica pleaded exclusion comic elements tragic drama.renaissance england marlowe among university wits introduced comic relief presentation crude scenes doctor faustus following native tradition interlude usually introduced two tragic plays.fact classical tradition mingling tragic comic allowed.examples william shakespeare deviated classical tradition used comic relief hamlet macbeth othello merchant venice romeo juliet.
Navier–Stokes equations	porter scene macbeth gravedigger scene hamlet gulling roderigo provide immense comic relief.mockery fool king lear may also regarded comic relief.referencesdeveloped several decades progressively building theories navier – stokes.the navier–stokes equations mathematically express conservation momentum conservation mass newtonian fluids.sometimes accompanied equation state relating pressure temperature density.arise applying isaac newtons second law fluid motion together assumption stress fluid sum diffusing viscous term proportional gradient velocity pressure term—hence describing viscous flow.difference closely related euler equations navier–stokes equations take viscosity account euler equations model inviscid flow.result navier–stokes parabolic equation therefore better analytic properties expense less mathematical structure e.g.
Navier–Stokes equations	never completely integrable.the navier–stokes equations useful describe physics many phenomena scientific engineering interest.may used model weather ocean currents water flow pipe air flow around wing.navier–stokes equations full simplified forms help design aircraft cars study blood flow design power stations analysis pollution many things.coupled maxwells equations used model study magnetohydrodynamics.the navier–stokes equations also great interest purely mathematical sense.despite wide range practical uses yet proven whether smooth solutions always exist three dimensions—i.e.
Navier–Stokes equations	infinitely differentiable even bounded points domain.called navier–stokes existence smoothness problem.clay mathematics institute called one seven important open problems mathematics offered us million prize solution counterexample.flow velocity solution equations flow velocity.vector field—to every point fluid moment time interval gives vector whose direction magnitude velocity fluid point space moment time.
Navier–Stokes equations	usually studied three spatial dimensions one time dimension although two spatial dimensional steadystate cases often used models higherdimensional analogues studied pure applied mathematics.velocity field calculated quantities interest pressure temperature may found using dynamical equations relations.different one normally sees classical mechanics solutions typically trajectories position particle deflection continuum.studying velocity instead position makes sense fluid although visualization purposes one compute various trajectories.particular streamlines vector field interpreted flow velocity paths along massless fluid particle would travel.
Navier–Stokes equations	paths integral curves whose derivative point equal vector field represent visually behavior vector field point time.general continuum equations navier–stokes momentum equation derived particular form cauchy momentum equation whose general convective form isby setting cauchy stress tensor σ sum viscosity term τ deviatoric stress pressure term −pi volumetric stress arrive atwhereddt material derivative defined ∂∂t u ⋅ ∇ρ densityu flow velocity∇ ⋅ divergencep pressuret timeτ deviatoric stress tensor order g represents body accelerations acting continuum example gravity inertial accelerations electrostatic accelerations onin form apparent assumption inviscid fluid – deviatoric stress – cauchy equations reduce euler equations.assuming conservation mass use mass continuity equation simply continuity equationto arrive conservation form equations motion.often writtenwhere ⊗ outer productthe left side equation describes acceleration may composed timedependent convective components also effects noninertial coordinates present.right side equation effect summation hydrostatic effects divergence deviatoric stress body forces gravity.all nonrelativistic balance equations navier–stokes equations derived beginning cauchy equations specifying stress tensor constitutive relation.expressing deviatoric shear stress tensor terms viscosity fluid velocity gradient assuming constant viscosity cauchy equations lead navier–stokes equations below.
Navier–Stokes equations	convective acceleration significant feature cauchy equation consequently continuum equations including euler navier–stokes presence convective acceleration effect acceleration flow respect space.individual fluid particles indeed experience timedependent acceleration convective acceleration flow field spatial effect one example fluid speeding nozzle.compressible flow remark cauchy stress tensor denoted σ instead τ general continuum equations incompressible flow section.the compressible momentum navier–stokes equation results following assumptions cauchy stress tensorthe stress galilean invariant depend directly flow velocity spatial derivatives flow velocity.stress variable tensor gradient ∇u.the stress linear variable σ∇u c ∇u c fourthorder tensor representing constant proportionality called viscosity elasticity tensor doubledot product.the fluid assumed isotropic gases simple liquids consequently v isotropic tensor furthermore since stress tensor symmetric helmholtz decomposition expressed terms two scalar lamé parameters bulk viscosity λ dynamic viscosity μ usual linear elasticitywhere identity tensor ε∇u ≡ ∇u ∇ut rateofstrain tensor ∇ ⋅ u rate expansion flow.decomposition explicitly defined assince trace rateofstrain tensor three dimensions isthe trace stress tensor three dimensions becomesso alternatively decomposing stress tensor isotropic deviatoric parts usual fluid dynamicsintroducing second viscosity ζwe arrive linear constitutive equation form usually employed thermal hydraulicsboth second viscosity ζ dynamic viscosity μ need constant – general depend two thermodynamics variables fluid contains single chemical species say example pressure temperature.
Navier–Stokes equations	equation makes explicit one transport coefficient conservation variables called equation state.the general navier–stokes equations becomeapart dependence pressure temperature second viscosity coefficient also depends process say second viscosity coefficient material property.instance case sound wave definitive frequency alternatively compresses expands fluid element seocnd viscosity coefficient depends frequency wave.dependence called dispersion.cases second viscosity ζ assumed constant case effect volume viscosity ζ mechanical pressure equivalent thermodynamic pressure demonstrated below.sinceone introduce modified pressure remove term corresponding second viscosity.
Navier–Stokes equations	however difference usually neglected time whenever dealing processes sound absorption attenuation shock waves second viscosity coefficeint becomes important explicitly assuming ζ .assumption setting ζ called stokes hypothesis.validity stokes hypothesis demonstrated monoatomic gas experimentally kinetic theory gases liquids stokes hypothesis general correct.stokes hypothesis navier–stokes equations becomeif dynamic viscosity μ also assumed constant equations simplified further.computing divergence stress tensor since divergence tensor ∇u ∇u divergence tensor ∇ut ∇∇ ⋅ u one finally arrives compressible general navier–stokes momentum equationwhere ddt material derivative.
Navier–Stokes equations	lefthand side changes conservation form navier–stokes momentum equationbulk viscosity assumed constant otherwise taken last derivative.convective acceleration term also written aswhere vector ∇ × u × u known lamb vector.for special case incompressible flow pressure constrains flow volume fluid elements constant isochoric flow resulting solenoidal velocity field ∇ ⋅ u .incompressible flow incompressible momentum navier–stokes equation results following assumptions cauchy stress tensorthe stress galilean invariant depend directly flow velocity spatial derivatives flow velocity.stress variable tensor gradient ∇u.the fluid assumed isotropic gases simple liquids consequently τ isotropic tensor furthermore since deviatoric stress tensor expressed terms dynamic viscosity μwhereis rateofstrain tensor.decomposition made explicit asdynamic viscosity μ need constant – incompressible flows depend density pressure.
Navier–Stokes equations	equation makes explicit one transport coefficient conservative variables called equation state.the divergence deviatoric stress given bybecause ∇ ⋅ u incompressible fluid.incompressibility rules density pressure waves like sound shock waves simplification useful phenomena interest.incompressible flow assumption typically holds well fluids low mach numbers say mach .modelling air winds normal temperatures.incompressible navier–stokes equations best visualised dividing densityif density constant throughout fluid domain words fluid elements density ρ ρ displaystyle rho rho havewhere 𝜈 μρ called kinematic viscosity.it well worth observing meaning term compare cauchy momentum equationthe higherorder term namely shear stress divergence ∇ ⋅ τ simply reduced vector laplacian term μ∇u.laplacian term interpreted difference velocity point mean velocity small surrounding volume.
Navier–Stokes equations	implies – newtonian fluid – viscosity operates diffusion momentum much way heat conduction.fact neglecting convection term incompressible navier–stokes equations lead vector diffusion equation namely stokes equations general convection term present incompressible navier–stokes equations belong class convection–diffusion equations.in usual case external field conservative fieldby defining hydraulic headone finally condense whole source one term arriving incompressible navier–stokes equation conservative external fieldthe incompressible navier–stokes equations conservative external field fundamental equation hydraulics.domain equations commonly less euclidean space orthogonal coordinate reference frame usually set explicit system scalar partial differential equations solved.dimensional orthogonal coordinate systems cartesian cylindrical spherical.expressing navier–stokes vector equation cartesian coordinates quite straightforward much influenced number dimensions euclidean space employed case also firstorder terms like variation convection ones also noncartesian orthogonal coordinate systems.
Navier–Stokes equations	higher order terms two coming divergence deviatoric stress distinguish navier–stokes equations euler equations tensor calculus required deducing expression noncartesian orthogonal coordinate systems.the incompressible navier–stokes equation composite sum two orthogonal equationswhere πs πi solenoidal irrotational projection operators satisfying πs πi fs fi nonconservative conservative parts body force.result follows helmholtz theorem also known fundamental theorem vector calculus.first equation pressureless governing equation velocity second equation pressure functional velocity related pressure poisson equation.the explicit functional form projection operator found helmholtz theoremwith similar structure d. thus governing equation integrodifferential equation similar coulomb biot–savart law convenient numerical computation.an equivalent weak variational form equation proved produce velocity solution navier–stokes equation given byfor divergencefree test functions w satisfying appropriate boundary conditions.projections accomplished orthogonality solenoidal irrotational function spaces.discrete form eminently suited finite element computation divergencefree flow shall see next section.
Navier–Stokes equations	one able address question one specify pressuredriven poiseuille problems pressureless governing equation.the absence pressure forces governing velocity equation demonstrates equation dynamic one rather kinematic equation divergencefree condition serves role conservation equation.would seem refute frequent statements incompressible pressure enforces divergencefree condition.variational form incompressible navier–stokes equations strong form consider incompressible navier–stokes equations newtonian fluid constant density ρ domainwith boundarybeing γd γn portions boundary respectively dirichlet neumann boundary condition applied γd ∩ γn ∅u fluid velocity p fluid pressure f given forcing term n̂ outward directed unit normal vector γn σu p viscous stress tensor defined aslet μ dynamic viscosity fluid secondorder identity tensor εu strainrate tensor defined asthe functions g h given dirichlet neumann boundary data u initial condition.first equation momentum balance equation second represents mass conservation namely continuity equation.assuming constant dynamic viscosity using vectorial identityand exploiting mass conservation divergence total stress tensor momentum equation also expressed asmoreover note neumann boundary conditions rearranged weak form order find variational form navier–stokes equations firstly consider momentum equationmultiply test function v defined suitable space v integrate members respect domain ωcounterintegrating parts diffusive pressure terms using gauss theoremusing relations one getsin fashion continuity equation multiplied test function q belonging space q integrated domain ωthe space functions chosen follows considering test function v vanishes dirichlet boundary considering neumann condition integral boundary rearranged ashaving mind weak formulation navier–stokes equations expressed discrete velocity partitioning problem domain defining basis functions partitioned domain discrete form governing equation isit desirable choose basis functions reflect essential feature incompressible flow – elements must divergencefree.
Navier–Stokes equations	velocity variable interest existence stream function vector potential necessary helmholtz theorem.determine fluid flow absence pressure gradient one specify difference stream function values across channel line integral tangential component vector potential around channel flow given stokes theorem.discussion restricted following.we restrict discussion continuous hermite finite elements least firstderivative degreesoffreedom.one draw large number candidate triangular rectangular elements platebending literature.elements derivatives components gradient.
Navier–Stokes equations	gradient curl scalar clearly orthogonal given expressionsadopting continuous platebending elements interchanging derivative degreesoffreedom changing sign appropriate one gives many families stream function elements.taking curl scalar stream function elements gives divergencefree velocity elements.requirement stream function elements continuous assures normal component velocity continuous across element interfaces necessary vanishing divergence interfaces.boundary conditions simple apply.stream function constant noflow surfaces noslip velocity conditions surfaces.stream function differences across open channels determine flow.boundary conditions necessary open boundaries though consistent values may used problems.dirichlet conditions.the algebraic equations solved simple set course nonlinear requiring iteration linearized equations.similar considerations apply threedimensions extension immediate vector nature potential exists simple relation gradient curl case d. pressure recovery recovering pressure velocity field easy.
Navier–Stokes equations	discrete weak equation pressure gradient iswhere testweight functions irrotational.conforming scalar finite element may used.however pressure gradient field may also interest.case one use scalar hermite elements pressure.testweight functions gi one would choose irrotational vector elements obtained gradient pressure element.
Navier–Stokes equations	noninertial frame reference rotating frame reference introduces interesting pseudoforces equations material derivative term.consider stationary inertial frame reference k noninertial frame reference k′ translating velocity ut rotating angular velocity ωt respect stationary frame.navier–stokes equation observed noninertial frame becomeshere x u measured noninertial frame.first term parenthesis represents coriolis acceleration second term due centrifugal acceleration third due linear acceleration k′ respect k fourth term due angular acceleration k′ respect k. equations navier–stokes equations strictly statement balance momentum.fully describe fluid flow information needed much depending assumptions made.
Navier–Stokes equations	additional information may include boundary data noslip capillary surface etc.conservation mass balance energy andor equation state.continuity equation incompressible fluid regardless flow assumptions statement conservation mass generally necessary.achieved mass continuity equation given general form asor using substantive derivativefor incompressible fluid density along line flow remains constant time therefore divergence velocity always zero stream function incompressible fluid taking curl incompressible navier–stokes equation results elimination pressure.especially easy see cartesian flow assumed like degenerate case uz dependence anything z equations reduce todifferentiating first respect second respect x subtracting resulting equations eliminate pressure conservative force.
Navier–Stokes equations	incompressible flow defining stream function ψ throughresults mass continuity unconditionally satisfied given stream function continuous incompressible newtonian momentum mass conservation condense one equationwhere ∇ biharmonic operator ν kinematic viscosity ν μρ.also express compactly using jacobian determinantthis single equation together appropriate boundary conditions describes fluid flow taking kinematic viscosity parameter.note equation creeping flow results left side assumed zero.in axisymmetric flow another stream function formulation called stokes stream function used describe velocity components incompressible flow one scalar function.the incompressible navier–stokes equation differential algebraic equation inconvenient feature explicit mechanism advancing pressure time.consequently much effort expended eliminate pressure part computational process.stream function formulation eliminates pressure two dimensions expense introducing higher derivatives elimination velocity primary variable interest.
Navier–Stokes equations	properties nonlinearity navier–stokes equations nonlinear partial differential equations general case remain almost every real situation.cases onedimensional flow stokes flow creeping flow equations simplified linear equations.nonlinearity makes problems difficult impossible solve main contributor turbulence equations model.the nonlinearity due convective acceleration acceleration associated change velocity position.hence convective flow whether turbulent involve nonlinearity.example convective laminar nonturbulent flow would passage viscous fluid example oil small converging nozzle.
Navier–Stokes equations	flows whether exactly solvable often thoroughly studied understood.turbulence turbulence timedependent chaotic behavior seen many fluid flows.generally believed due inertia fluid whole culmination timedependent convective acceleration hence flows inertial effects small tend laminar reynolds number quantifies much flow affected inertia.believed though known certainty navier–stokes equations describe turbulence properly.the numerical solution navier–stokes equations turbulent flow extremely difficult due significantly different mixinglength scales involved turbulent flow stable solution requires fine mesh resolution computational time becomes significantly infeasible calculation direct numerical simulation.attempts solve turbulent flow using laminar solver typically result timeunsteady solution fails converge appropriately.
Navier–Stokes equations	counter timeaveraged equations reynoldsaveraged navier–stokes equations rans supplemented turbulence models used practical computational fluid dynamics cfd applications modeling turbulent flows.models include spalart–allmaras k–ω k–ε sst models add variety additional equations bring closure rans equations.large eddy simulation les also used solve equations numerically.approach computationally expensive—in time computer memory—than rans produces better results explicitly resolves larger turbulent scales.applicability together supplemental equations example conservation mass wellformulated boundary conditions navier–stokes equations seem model fluid motion accurately even turbulent flows seem average agree real world observations.the navier–stokes equations assume fluid studied continuum infinitely divisible composed particles atoms molecules moving relativistic velocities.
Navier–Stokes equations	small scales extreme conditions real fluids made discrete molecules produce results different continuous fluids modeled navier–stokes equations.example capillarity internal layers fluids appears flow high gradients.large knudsen number problem boltzmann equation may suitable replacement.failing one may resort molecular dynamics various hybrid methods.another limitation simply complicated nature equations.timetested formulations exist common fluid families application navier–stokes equations less common families tends result complicated formulations often open research problems.
Navier–Stokes equations	reason equations usually written newtonian fluids viscosity model linear truly general models flow kinds fluids blood exist.application specific problems navier–stokes equations even written explicitly specific fluids rather generic nature proper application specific problems diverse.partly enormous variety problems may modeled ranging simple distribution static pressure complicated multiphase flow driven surface tension.generally application specific problems begins flow assumptions initialboundary condition formulation may followed scale analysis simplify problem.parallel flow assume steady parallel onedimensional nonconvective pressuredriven flow parallel plates resulting scaled dimensionless boundary value problem isthe boundary condition slip condition.problem easily solved flow fieldfrom point onward quantities interest easily obtained viscous drag force net flow rate.
Navier–Stokes equations	radial flow difficulties may arise problem becomes slightly complicated.seemingly modest twist parallel flow would radial flow parallel plates involves convection thus nonlinearity.velocity field may represented function fz must satisfythis ordinary differential equation obtained navier–stokes equations written flow assumptions applied additionally pressure gradient solved for.nonlinear term makes difficult problem solve analytically lengthy implicit solution may found involves elliptic integrals roots cubic polynomials.issues actual existence solutions arise r .
Navier–Stokes equations	approximately √ parameter r reynolds number appropriately chosen scales.example flow assumptions losing applicability example difficulty high reynolds number flows.convection type natural convection described navier–stokes equation rayleigh–bénard convection.one commonly studied convection phenomena analytical experimental accessibility.exact solutions navier–stokes equations exact solutions navier–stokes equations exist.
Navier–Stokes equations	examples degenerate cases—with nonlinear terms navier–stokes equations equal zero—are poiseuille flow couette flow oscillatory stokes boundary layer.also interesting examples solutions full nonlinear equations exist jeffery–hamel flow von kármán swirling flow stagnation point flow landau–squire jet taylor–green vortex.note existence exact solutions imply stable turbulence may develop higher reynolds numbers.under additional assumptions component parts separated.threedimensional steadystate vortex solution steadystate example singularities comes considering flow along lines hopf fibration.let r constant radius inner coil.one set solutions given byfor arbitrary constants b. solution nonviscous gas compressible fluid whose density velocities pressure goes zero far origin.
Navier–Stokes equations	note solution clay millennium problem refers incompressible fluids ρ constant neither deal uniqueness navier–stokes equations respect turbulence properties.also worth pointing components velocity vector exactly pythagorean quadruple parametrization.choices density pressure possible velocity field viscous threedimensional periodic solutions two examples periodic fullythreedimensional viscous solutions described in.these solutions defined threedimensional torus l displaystyle mathbb l characterized positive negative helicity respectively.the solution positive helicity given bywhere k π l displaystyle kpi l wave number velocity components normalized average kinetic energy per unit mass u displaystyle u displaystyle .the pressure field obtained velocity field p p − ρ ‖ u ‖ displaystyle pprho boldsymbol u p displaystyle p ρ displaystyle rho reference values pressure density fields respectively.since solutions belong class beltrami flow vorticity field parallel velocity case positive helicity given ω k u displaystyle omega sqrt kboldsymbol u .solutions regarded generalization three dimensions classic twodimensional taylorgreen taylor–green vortex.wyld diagrams wyld diagrams bookkeeping graphs correspond navier–stokes equations via perturbation expansion fundamental continuum mechanics.
Navier–Stokes equations	similar feynman diagrams quantum field theory diagrams extension keldyshs technique nonequilibrium processes fluid dynamics.words diagrams assign graphs often turbulent phenomena turbulent fluids allowing correlated interacting fluid particles obey stochastic processes associated pseudorandom functions probability distributions.representations note formulas section make use singleline notation partial derivatives e.g.∂ x u displaystyle partial xu means partial derivative u respect x ∂ f θ displaystyle partial yftheta means secondorder partial derivative fθ respect y. cartesian coordinates general form navier–stokes velocity vector expanded u ux uy uz sometimes respectively named u v w may write vector equation explicitlynote gravity accounted body force values gx gy gz depend orientation gravity respect chosen set coordinates.the continuity equation readswhen flow incompressible ρ change fluid particle material derivative vanishes dρdt .continuity equation reduced tothus incompressible version navier–stokes equation second part viscous terms fall away see incompressible flow.this system four equations comprises commonly used studied form.
Navier–Stokes equations	though comparatively compact representations still nonlinear system partial differential equations solutions difficult obtain.cylindrical coordinates change variables cartesian equations yield following momentum equations r φ zthe gravity components generally constants however applications either coordinates chosen gravity components constant else assumed gravity counteracted pressure field example flow horizontal pipe treated normally without gravity without vertical pressure gradient.continuity equation isthis cylindrical representation incompressible navier–stokes equations second commonly seen first cartesian above.cylindrical coordinates chosen take advantage symmetry velocity component disappear.common case axisymmetric flow assumption tangential velocity uφ remaining quantities independent φ spherical coordinates spherical coordinates r φ θ momentum equations note convention used θ polar angle colatitude ≤ θ ≤ πmass continuity readthese equations could slightly compacted example factoring r viscous terms.
Navier–Stokes equations	however would undesirably alter structure laplacian quantities.navier–stokes equations use games navier–stokes equations used extensively video games order model wide variety natural phenomena.simulations smallscale gaseous fluids fire smoke often based seminal paper realtime fluid dynamics games jos stam elaborates one methods proposed stams earlier famous paper stable fluids .stam proposes stable fluid simulation using navier–stokes solution method coupled unconditionally stable semilagrangian advection scheme first proposed .more recent implementations based upon work run game systems graphics processing unit gpu opposed central processing unit cpu achieve much higher degree performance.many improvements proposed stams original work suffers inherently high numerical dissipation velocity mass.an introduction interactive fluid simulation found acm siggraph course fluid simulation computer animation.see also citations general references acheson d. j. elementary fluid dynamics oxford applied mathematics computing science series oxford university press isbn batchelor g. k. introduction fluid dynamics cambridge university press isbn currie i. g. fundamental mechanics fluids mcgrawhill isbn v. girault p. a. raviart.
Navier–Stokes equations	finite element methods navier–stokes equations theory algorithms.springer series computational mathematics.springerverlag .landau l. d. lifshitz e. m. fluid mechanics vol.course theoretical physics volume nd revised ed.pergamon press isbn oclc polyanin a. d. kutepov a. m. vyazmin a. v. kazenin d. a. hydrodynamics mass heat transfer chemical engineering taylor francis london isbn rhyming inge l. dynamique des fluides presses polytechniques et universitaires romandessmits alexander j. physical introduction fluid mechanics wiley isbn temam roger navier–stokes equations theory numerical analysis acm chelsea publishing isbn white frank m. viscous fluid flow mcgrawhill isbn external links simplified derivation navier–stokes equationsthreedimensional unsteady form navier–stokes equations glenn research center nasa
Pakistan	worlds fifthmost populous country population almost million worlds secondlargest muslim population.pakistan rdlargest country area spanning square kilometres square miles.kilometre mile coastline along arabian sea gulf oman south bordered india east afghanistan west iran southwest china northeast.separated narrowly tajikistan afghanistans wakhan corridor north also shares maritime border oman.pakistan site several ancient cultures including yearold neolithic site mehrgarh balochistan indus valley civilisation bronze age extensive civilisations old world.region comprises modern state pakistan realm multiple empires dynasties including achaemenid briefly alexander great seleucid maurya kushan gupta umayyad caliphate southern regions hindu shahi ghaznavids delhi sultanate mughals durranis sikh empire british east india company rule recently british indian empire .spurred pakistan movement sought homeland muslims british india election victories allindia muslim league pakistan gained independence partition british indian empire awarded separate statehood muslimmajority regions accompanied unparalleled mass migration loss life.
Pakistan	initially dominion british commonwealth pakistan officially drafted constitution emerged declared islamic republic.exclave east pakistan seceded new country bangladesh ninemonthlong civil war.following four decades pakistan ruled governments whose descriptions although complex commonly alternated civilian military democratic authoritarian relatively secular islamist.pakistan elected civilian government adopted parliamentary system periodic elections.pakistan regional middle power nation worlds sixthlargest standing armed forces.declared nuclearweapons state ranked amongst emerging growthleading economies large rapidlygrowing middle class.
Pakistan	pakistans political history since independence characterized periods significant economic military growth well political economic instability.ethnically linguistically diverse country similarly diverse geography wildlife.however country continues face challenges including poverty illiteracy corruption terrorism.pakistan member united nations shanghai cooperation organisation organisation islamic cooperation commonwealth nations south asian association regional cooperation islamic military counterterrorism coalition designated major nonnato ally united states.etymology name pakistan means literally land abounding pure land pure abound urdu persian.
Pakistan	references word پاک pāk meaning pure persian pashto.suffix ـستان transliterated english stan persian means place abounding place anything abounds.the name country coined choudhry rahmat ali pakistan movement activist published pamphlet never using acronym thirty million muslim brethren live pakistan referring names five northern regions british raj punjab afghania kashmir sindh baluchistan.history early medieval age earliest ancient human civilisations south asia originated areas encompassing presentday pakistan.earliest known inhabitants region soanian lower paleolithic stone tools found soan valley punjab.indus region covers present day pakistan site several successive ancient cultures including neolithic mehrgarh bronze age indus valley civilisation – bce harappa mohenjodaro.the vedic period – bce characterised indoaryan culture period vedas oldest scriptures associated hinduism composed culture later became well established region.
Pakistan	multan important hindu pilgrimage centre.vedic civilisation flourished ancient gandhāran city takṣaśilā taxila punjab founded around bce.successive ancient empires kingdoms ruled region persian achaemenid empire around bce alexander greats empire bce maurya empire founded chandragupta maurya extended ashoka great bce.indogreek kingdom founded demetrius bactria – bce included gandhara punjab reached greatest extent menander – bce prospering grecobuddhist culture region.taxila one earliest universities centres higher education world established late vedic period th century bce.
Pakistan	school consisted several monasteries without large dormitories lecture halls religious instruction provided individualistic basis.ancient university documented invading forces alexander great also recorded chinese pilgrims th th century ce.at zenith rai dynasty – ce sindh ruled region surrounding territories.pala dynasty last buddhist empire dharmapala devapala stretched across south asia bangladesh northern india pakistan.islamic conquest arab conqueror muhammad bin qasim conquered sindh ce.pakistan governments official chronology claims time foundation pakistan laid concept pakistan arrived th century.
Pakistan	early medieval period – ce witnessed spread islam region.period sufi missionaries played pivotal role converting majority regional buddhist hindu population islam.upon defeat turk hindu shahi dynasties governed kabul valley gandhara presentday khyber pakhtunkwa western punjab th th centuries ce several successive muslim empires ruled region including ghaznavid empire – ce ghorid kingdom delhi sultanate – ce.lodi dynasty last delhi sultanate replaced mughal empire – ce.the mughals introduced persian literature high culture establishing roots indopersian culture region.region modernday pakistan key cities mughal period lahore thatta chosen site impressive mughal buildings.
Pakistan	early th century region remained mughal empire.in th century slow disintegration mughal empire hastened emergence rival powers maratha confederacy later sikh empire well invasions nader shah iran durrani empire afghanistan .growing political power british bengal yet reached territories modern pakistan.colonial period none territory modern pakistan ruled british european powers karachi small fishing village mud fort guarding harbour taken held enclave port military base first afghan war soon followed.rest sindh taken following decades first east india company postsepoy mutiny – direct rule queen victoria british empire took country partly wars also treaties.main wars baloch talpur dynasty ended battle miani sindh anglosikh wars – angloafghan wars –.

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