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Bernoulli's principle
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== Incompressible flow equation == <!-- [[total pressure]] and [[energy head]] redirect to here --> In most flows of liquids, and of gases at low [[Mach number]], the [[density]] of a fluid parcel can be considered to be constant, regardless of pressure variations in the flow. Therefore, the fluid can be considered to be incompressible, and these flows are called [[incompressible flow]]s. Bernoulli performed his experiments on liquids, so his equation in its original form is valid only for incompressible flow. A common form of Bernoulli's equation is: {{NumBlk||<math display="block">\frac{v^2}{2} + gz + \frac{p}{\rho} = \text{constant}</math>|{{EquationRef|A}}}} where: *<math>v</math> is the fluid flow [[speed]] at a point, *<math>g</math> is the [[Gravitational constant|acceleration due to gravity]], *<math>z</math> is the [[elevation]] of the point above a reference plane, with the positive <math>z</math>-direction pointing upward—so in the direction opposite to the gravitational acceleration, *<math>p</math> is the [[static pressure]] at the chosen point, and *<math>\rho</math> is the [[density]] of the fluid at all points in the fluid. Bernoulli's equation and the Bernoulli constant are applicable throughout any region of flow where the energy per unit mass is uniform. Because the energy per unit mass of liquid in a well-mixed reservoir is uniform throughout, Bernoulli's equation can be used to analyze the fluid flow everywhere in that reservoir (including pipes or flow fields that the reservoir feeds) ''except'' where [[laminar flow|viscous forces]] [[Reynolds number|dominate]] and erode the energy per unit mass.<ref name="Streeter1966"/>{{rp|at=Example 3.5 and p.116}} The following assumptions must be met for this Bernoulli equation to apply:<ref name="Batchelor2000" />{{rp|page=265}} * the flow must be [[Fluid dynamics|steady]], that is, the flow parameters (velocity, density, etc.) at any point cannot change with time, * the flow must be incompressible—even though pressure varies, the density must remain constant along a streamline; * friction by [[Viscosity|viscous]] forces must be negligible. For [[conservative force]] fields (not limited to the [[gravitational field]]), Bernoulli's equation can be generalized as:<ref name="Batchelor2000" />{{rp|page=265}} <math display="block">\frac{v^2}{2} + \Psi + \frac{p}{\rho} = \text{constant}</math> where {{math|Ψ}} is the force potential at the point considered. For example, for the Earth's gravity {{math|1=Ψ = ''gz''}}. By multiplying with the fluid density {{mvar|ρ}}, equation ({{EquationNote|A}}) can be rewritten as: <math display="block">\tfrac{1}{2} \rho v^2 + \rho g z + p = \text{constant}</math> or: <math display="block">q + \rho g h = p_0 + \rho g z = \text{constant}</math> where *{{math|1=''q'' = {{sfrac|1|2}}''ρv''<sup>2</sup>}} is [[dynamic pressure]], *{{math|1=''h'' = ''z'' + {{sfrac|''p''|''ρg''}}}} is the [[piezometric head]] or [[hydraulic head]] (the sum of the elevation {{mvar|z}} and the [[pressure head]])<ref name="Mulley2004">{{cite book|last=Mulley|first=Raymond |title=Flow of Industrial Fluids: Theory and Equations|url=https://books.google.com/books?id=LuUZ0OG6EZMC|year=2004|publisher=CRC Press| isbn=978-0-8493-2767-4| pages=43–44}}</ref><ref name="Chanson2004">{{cite book|last=Chanson|first=Hubert |author-link=Hubert Chanson|title=Hydraulics of Open Channel Flow|url={{google books|id=VCNmKQI6GiEC|plainurl=yes|page=22|keywords=hydraulic head}}|year=2004|publisher=Elsevier|isbn=978-0-08-047297-3|page=22}}</ref> and *{{math|1=''p''<sub>0</sub> = ''p'' + ''q''}} is the [[stagnation pressure]] (the sum of the [[static pressure]] {{mvar|p}} and [[dynamic pressure]] {{mvar|q}}).<ref name="Oerteletal2004">{{Cite book | title=Prandtl's Essentials of Fluid Mechanics | first1=Herbert | last1=Oertel | first2=Ludwig | last2= Prandtl | first3=M. | last3=Böhle | first4=Katherine | last4=Mayes | publisher=Springer | year=2004 | isbn=978-0-387-40437-0 | pages=70–71|url=https://books.google.com/books?id=J2NBBJDO79MC&pg=PA70 }}</ref> The constant in the Bernoulli equation can be normalized. A common approach is in terms of '''total head''' or '''energy head''' {{mvar|H}}: <math display="block">H = z + \frac{p}{\rho g} + \frac{v^2}{2g} = h + \frac{v^2}{2g},</math> The above equations suggest there is a flow speed at which pressure is zero, and at even higher speeds the pressure is negative. Most often, gases and liquids are not capable of negative absolute pressure, or even zero pressure, so clearly Bernoulli's equation ceases to be valid before zero pressure is reached. In liquids—when the pressure becomes too low—[[cavitation]] occurs. The above equations use a linear relationship between flow speed squared and pressure. At higher flow speeds in gases, or for [[sound]] waves in liquid, the changes in mass density become significant so that the assumption of constant density is invalid. === Simplified form === In many applications of Bernoulli's equation, the change in the {{mvar|ρgz}} term is so small compared with the other terms that it can be ignored. For example, in the case of aircraft in flight, the change in height {{mvar|z}} is so small the {{mvar|ρgz}} term can be omitted. This allows the above equation to be presented in the following simplified form: <math display="block">p + q = p_0</math> where {{math|''p''<sub>0</sub>}} is called '''''total pressure''''', and {{mvar|q}} is '' [[dynamic pressure]]''.<ref>{{cite web|title = Bernoulli's Equation| publisher = NASA Glenn Research Center| url =http://www.grc.nasa.gov/WWW/K-12/airplane/bern.htm| archive-url =https://archive.today/20120731182454/http://www.grc.nasa.gov/WWW/K-12/airplane/bern.htm| url-status =dead| archive-date =2012-07-31|access-date = 2009-03-04 }}</ref> Many authors refer to the pressure {{mvar|p}} as static pressure to distinguish it from total pressure {{math|''p''<sub>0</sub>}} and dynamic pressure {{mvar|q}}. In ''Aerodynamics'', L.J. Clancy writes: "To distinguish it from the total and dynamic pressures, the actual pressure of the fluid, which is associated not with its motion but with its state, is often referred to as the static pressure, but where the term pressure alone is used it refers to this static pressure."<ref name="Clancy1975" />{{rp|at= § 3.5}} The simplified form of Bernoulli's equation can be summarized in the following memorable word equation:<ref name="Clancy1975" />{{rp|at= § 3.5}} {{block indent | em = 1.5 | text = Static pressure + Dynamic pressure = Total pressure.}} Every point in a steadily flowing fluid, regardless of the fluid speed at that point, has its own unique static pressure {{mvar|p}} and dynamic pressure {{mvar|q}}. Their sum {{math|''p'' + ''q''}} is defined to be the total pressure {{math|''p''<sub>0</sub>}}. The significance of Bernoulli's principle can now be summarized as "total pressure is constant in any region free of viscous forces". If the fluid flow is brought to rest at some point, this point is called a stagnation point, and at this point the static pressure is equal to the [[stagnation pressure]]. If the fluid flow is [[irrotational flow|irrotational]], the total pressure is uniform and Bernoulli's principle can be summarized as "total pressure is constant everywhere in the fluid flow".<ref name="Clancy1975" />{{rp|at=Equation 3.12}} It is reasonable to assume that irrotational flow exists in any situation where a large body of fluid is flowing past a solid body. Examples are aircraft in flight and ships moving in open bodies of water. However, Bernoulli's principle importantly does not apply in the [[boundary layer]] such as in flow through long [[Pipe flow|pipes]]. === Unsteady potential flow === The Bernoulli equation for unsteady potential flow is used in the theory of [[Wind wave|ocean surface waves]] and [[acoustics]]. For an irrotational flow, the [[flow velocity]] can be described as the [[gradient]] {{math|∇''φ''}} of a [[velocity potential]] {{mvar|φ}}. In that case, and for a constant density {{mvar|ρ}}, the [[momentum]] equations of the [[Euler equations (fluid dynamics)|Euler equations]] can be integrated to:<ref name="Batchelor2000" />{{rp|page=383}}<math display="block">\frac{\partial \varphi}{\partial t} + \tfrac12 v^2 + \frac{p}{\rho} + gz = f(t),</math> which is a Bernoulli equation valid also for unsteady—or time dependent—flows. Here {{math|{{sfrac|∂''φ''|∂''t''}}}} denotes the [[partial derivative]] of the velocity potential {{mvar|φ}} with respect to time {{mvar|t}}, and {{math|1=''v'' = {{abs|∇''φ''}}}} is the flow speed. The function {{math|''f''(''t'')}} depends only on time and not on position in the fluid. As a result, the Bernoulli equation at some moment {{mvar|t}} applies in the whole fluid domain. This is also true for the special case of a steady irrotational flow, in which case {{mvar|f}} and {{math|{{sfrac|∂''φ''|∂''t''}}}} are constants so equation ({{EquationNote|A}}) can be applied in every point of the fluid domain.<ref name="Batchelor2000" />{{rp|page=383}} Further {{math|''f''(''t'')}} can be made equal to zero by incorporating it into the velocity potential using the transformation:<math display="block">\Phi = \varphi - \int_{t_0}^t f(\tau)\, \mathrm{d}\tau,</math> resulting in: <math display="block">\frac{\partial \Phi}{\partial t} + \tfrac12 v^2 + \frac{p}{\rho} + gz = 0.</math> Note that the relation of the potential to the flow velocity is unaffected by this transformation: {{math|1=∇Φ = ∇''φ''}}. The Bernoulli equation for unsteady potential flow also appears to play a central role in [[Luke's variational principle]], a variational description of free-surface flows using the [[Lagrangian mechanics]].
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