Editing Bernoulli's principle (section)

{{Short description|Principle relating to fluid dynamics}}
{{about| Bernoulli's principle and Bernoulli's equation in fluid dynamics|Bernoulli's theorem in probability|law of large numbers|an unrelated topic in [[ordinary differential equation]]s|Bernoulli differential equation}}
[[File:VenturiFlow.png|right|thumb|A flow of air through a [[Venturi meter]]. The kinetic energy increases at the expense of the [[fluid pressure]], as shown by the difference in height of the two columns of water.]]
[[File:Venturi Tube en.webm|thumb|Video of a [[Venturi effect|Venturi meter]] used in a lab experiment]]
{{Continuum mechanics|fluid}}

'''Bernoulli's principle''' is a key concept in [[fluid dynamics]] that relates pressure, speed and height. For example, for a fluid flowing horizontally Bernoulli's principle states that an increase in the speed  occurs simultaneously with a decrease in [[static pressure|pressure]]<ref name="Clancy1975">{{cite book |last=Clancy |first=L.J. |author-link=Laurence Joseph Clancy |title=Aerodynamics |url=https://books.google.com/books?id=zaNTAAAAMAAJ |year=1975 |publisher=Wiley |isbn=978-0-470-15837-1}}</ref>{{rp|at=Ch.3}}<ref name="Batchelor2000">{{cite book |last=Batchelor |first=G.K. |author-link=George Batchelor |title=An Introduction to Fluid Dynamics |url=https://books.google.com/books?id=Rla7OihRvUgC&pg=PA156 |year=2000 |publisher=[[Cambridge University Press]] |location=Cambridge |isbn=978-0-521-66396-0}}</ref>{{rp|at= § 3.5|pp=156–164}} The principle is named after the Swiss mathematician and physicist [[Daniel Bernoulli]], who published it in his book ''[[Hydrodynamica]]'' in 1738.<ref>{{cite web | url =https://www.britannica.com/EBchecked/topic/658890/Hydrodynamica#tab=active~checked%2Citems~checked&title=Hydrodynamica%20–%20Britannica%20Online%20Encyclopedia | title=Hydrodynamica | access-date=2008-10-30 |publisher= Britannica Online Encyclopedia }}</ref> Although Bernoulli deduced that pressure decreases when the flow speed increases, it was [[Leonhard Euler]] in 1752 who derived '''Bernoulli's equation''' in its usual form.<ref name="Anderson2016">{{citation | title=Handbook of fluid dynamics | edition=2nd | editor-last=Johnson | editor-first=R.W. | chapter=Some reflections on the history of fluid dynamics | last=Anderson | first=J.D. | author-link=John D. Anderson | year=2016 | publisher=CRC Press | isbn=9781439849576|chapter-url={{google books|id=TQfYCwAAQBAJ|page=2-1| plainurl=yes | keywords=Anderson}} }}</ref><ref>{{citation | first1=O. | last1=Darrigol | first2=U. | last2=Frisch |title=From Newton's mechanics to Euler's equations | journal=Physica D: Nonlinear Phenomena | volume=237 | issue=14–17 | year=2008 | pages=1855–1869 | doi=10.1016/j.physd.2007.08.003 | bibcode=2008PhyD..237.1855D }}</ref>

Bernoulli's principle can be derived from the principle of [[conservation of energy]]. This states that, in a steady flow, the sum of all forms of energy in a fluid is the same at all points that are free of viscous forces. This requires that the sum of [[kinetic energy]], potential energy and [[internal energy]] remains constant.<ref name="Batchelor2000" />{{rp|at= § 3.5}} Thus an increase in the speed of the fluid—implying an increase in its kinetic energy—occurs with a simultaneous decrease in (the sum of) its potential energy (including the static pressure) and internal energy. If the fluid is flowing out of a reservoir, the sum of all forms of energy is the same because in a reservoir the energy per unit volume (the sum of pressure and [[gravitational potential]] {{math|''ρ''&thinsp;''g''&thinsp;''h''}}) is the same everywhere.<ref name="Streeter1966">{{cite book|last=Streeter|first=Victor Lyle |title=Fluid mechanics|url=https://books.google.com/books?id=G6RRAAAAMAAJ|year=1966|publisher=McGraw-Hill|location=New York}}</ref>{{rp|at=Example 3.5 and p.116}}

Bernoulli's principle can also be derived directly from [[Isaac Newton]]'s second [[Newton's laws of motion|law of motion]]. When a fluid is flowing horizontally from a region of high pressure to a region of low pressure, there is more pressure from behind than in front. This gives a net force on the volume, accelerating it along the streamline. {{efn|If the particle is in a region of varying pressure (a non-vanishing pressure gradient in the {{mvar|x}}-direction) and if the particle has a finite·size {{mvar|l}}, then the front of the particle will be 'seeing' a different pressure from the rear. More precisely, if the pressure drops in the {{mvar|x}}-direction ({{math|{{sfrac|d''p''|d''x''}} < 0}}) the pressure at the rear is higher than at the front and the particle experiences a (positive) net force. According to Newton's second law, this force causes an acceleration and the particle's velocity increases as it moves along the streamline... Bernoulli's equation describes this mathematically (see the complete derivation in the appendix).<ref name="Babinsky2003">{{citation | journal=Physics Education | first=Holger | last=Babinsky | date=November 2003 | title=How do wings work? | doi=10.1088/0031-9120/38/6/001 | bibcode = 2003PhyEd..38..497B | volume=38 | issue=6 | pages=497–503 | s2cid=1657792 }}</ref> }}{{efn|Acceleration of air is caused by pressure gradients. Air is accelerated in the direction of the velocity if the pressure goes down. Thus the decrease of pressure is the cause of a higher velocity.<ref name="Weltner">"{{Citation|last1=Weltner |first1=Klaus |last2=Ingelman-Sundberg |first2=Martin | title=Misinterpretations of Bernoulli's Law |url=http://user.uni-frankfurt.de/~weltner/Mis6/mis6.html |url-status=dead |archive-url=https://web.archive.org/web/20090429040229/http://user.uni-frankfurt.de/~weltner/Mis6/mis6.html |archive-date=April 29, 2009 }}</ref>}}{{efn|The idea is that as the parcel moves along, following a streamline, as it moves into an area of higher pressure there will be higher pressure ahead (higher than the pressure behind) and this will exert a force on the parcel, slowing it down. Conversely, if the parcel is moving into a region of lower pressure, there will be a higher pressure behind it (higher than the pressure ahead), speeding it up. As always, any unbalanced force will cause a change in momentum (and velocity), as required by Newton's laws of motion.<ref name="Denker2005">{{Cite web | title = 3 Airfoils and Airflow | last = Denker | first = John S. | work = See How It Flies | date = 2005 | access-date = 2018-07-27 | url = http://www.av8n.com/how/htm/airfoils.html }}</ref>
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Fluid particles are subject only to pressure and their own weight. If a fluid is flowing horizontally and along a section of a streamline, where the speed increases it can only be because the fluid on that section has moved from a region of higher pressure to a region of lower pressure; and if its speed decreases, it can only be because it has moved from a region of lower pressure to a region of higher pressure. Consequently, within a fluid flowing horizontally, the highest speed occurs where the pressure is lowest, and the lowest speed occurs where the pressure is highest.<ref>{{cite book|last1=Resnick |first1=R. |last2=Halliday |first2=D. |date=1960 |at=section 18–4 |title=Physics |publisher=John Wiley & Sons}}</ref>

Bernoulli's principle is only applicable for [[Isentropic process|isentropic flows]]: when the effects of [[irreversible process]]es (like [[turbulence]]) and non-[[adiabatic process]]es (e.g. [[thermal radiation]]) are small and can be neglected. However, the principle can be applied to various types of flow within these bounds, resulting in various forms of Bernoulli's equation. The simple form of Bernoulli's equation is valid for [[incompressible flow]]s (e.g. most [[liquid]] flows and [[gas]]es moving at low [[Mach number]]). More advanced forms may be applied to [[compressible flow]]s at higher Mach numbers.<!-- This was previously deleted and had to be restored. Please state the criteria for the use of Bernoulli's principle. If there are none, don't just delete it, state it or preferably explain it. -->