Editing Bernoulli's principle (section)

== Compressible flow equation ==
Bernoulli developed his principle from observations on liquids, and Bernoulli's equation is valid for ideal fluids: those that are inviscid, incompressible and subjected only to conservative forces. It is sometimes valid for the flow of gases as well, provided that there is no transfer of kinetic or potential energy from the gas flow to the compression or expansion of the gas. If both the gas pressure and volume change simultaneously, then work will be done on or by the gas. In this case, Bernoulli's equation in its incompressible flow form cannot be assumed to be valid. However, if the gas process is entirely [[isobaric process|isobaric]], or [[isochoric process|isochoric]], then no work is done on or by the gas (so the simple energy balance is not upset). According to the gas law, an isobaric or isochoric process is ordinarily the only way to ensure constant density in a gas. Also the gas density will be proportional to the ratio of pressure and absolute [[temperature]]; however, this ratio will vary upon compression or expansion, no matter what non-zero quantity of heat is added or removed. The only exception is if the net heat transfer is zero, as in a complete thermodynamic cycle or in an individual [[Isentropic process|isentropic]] (frictionless [[Adiabatic process|adiabatic]]) process, and even then this reversible process must be reversed, to restore the gas to the original pressure and specific volume, and thus density. Only then is the original, unmodified Bernoulli equation applicable. In this case the equation can be used if the flow speed of the gas is sufficiently below the [[speed of sound]], such that the variation in density of the gas (due to this effect) along each streamline can be ignored. Adiabatic flow at less than [[Mach number|Mach]] 0.3 is generally considered to be slow enough.<ref>{{cite book|last=White |first=Frank M. |title=Fluid Mechanics |edition=6th |publisher=McGraw-Hill International Edition |page=602}}</ref>

It is possible to use the fundamental principles of physics to develop similar equations applicable to compressible fluids. There are numerous equations, each tailored for a particular application, but all are analogous to Bernoulli's equation and all rely on nothing more than the fundamental principles of physics such as Newton's laws of motion or the [[first law of thermodynamics]].

=== Compressible flow in fluid dynamics ===
For a compressible fluid, with a [[Barotropic fluid|barotropic]] [[equation of state]], and under the action of conservative forces,<ref name="ClarkeCarswell2007">{{cite book |last1=Clarke |first1=Cathie |last2=Carswell |first2=Bob |author-link2=Bob Carswell |title=Principles of Astrophysical Fluid Dynamics |url=https://books.google.com/books?id=a2SW7XJ89H0C&pg=PA161 |year=2007 |publisher=Cambridge University Press |isbn=978-1-139-46223-5 |page=161}}</ref>
<math display="block">\frac {v^2}{2}+ \int_{p_1}^p \frac {\mathrm{d}\tilde{p}}{\rho\left(\tilde{p}\right)} + \Psi = \text{constant (along a streamline)}</math>
where:
*{{mvar|p}} is the pressure
*{{mvar|ρ}} is the density and {{math|''ρ''(''p'')}} indicates that it is a function of pressure
*{{mvar|v}} is the flow speed
*{{math|Ψ}} is the potential associated with the conservative force field, often the [[gravitational potential]]

In engineering situations, elevations are generally small compared to the size of the Earth, and the time scales of fluid flow are small enough to consider the equation of state as adiabatic. In this case, the above equation for an [[ideal gas]] becomes:<ref name="Clancy1975" />{{rp|at= § 3.11}}
<math display="block">\frac {v^2}{2}+ gz + \left(\frac {\gamma}{\gamma-1}\right) \frac {p}{\rho} = \text{constant (along a streamline)}</math>
where, in addition to the terms listed above:
*{{mvar|γ}} is the [[Heat capacity ratio|ratio of the specific heats]] of the fluid
*{{mvar|g}} is the acceleration due to gravity
*{{mvar|z}} is the elevation of the point above a reference plane

In many applications of compressible flow, changes in elevation are negligible compared to the other terms, so the term {{mvar|gz}} can be omitted. A very useful form of the equation is then:
<math display="block">\frac {v^2}{2}+\left( \frac {\gamma}{\gamma-1}\right)\frac {p}{\rho} = \left(\frac {\gamma}{\gamma-1}\right)\frac {p_0}{\rho_0}</math>

where:
*{{math|''p''<sub>0</sub>}} is the [[Stagnation pressure|total pressure]]
*{{math|''ρ''<sub>0</sub>}} is the total density

=== Compressible flow in thermodynamics ===
The most general form of the equation, suitable for use in thermodynamics in case of (quasi) steady flow, is:<ref name="Batchelor2000" />{{rp|at= § 3.5}}<ref name="LandauLifshitz1987">{{cite book |last1=Landau |first1=L.D. |author1-link=Lev Landau |last2=Lifshitz |first2=E.M. |author2-link=Evgeny Lifshitz |title=Fluid Mechanics |edition=2nd |series=[[Course of Theoretical Physics]] |publisher=Pergamon Press |year=1987 |isbn=978-0-7506-2767-2 }}</ref>{{rp|at=§ 5}}<ref name="VanWylenSonntag1965">{{cite book |last1=Van Wylen |first1=Gordon J. |author-link1=Gordon Van Wylen |last2=Sonntag |first2=Richard E. |title=Fundamentals of Classical Thermodynamics |url=https://books.google.com/books?id=Ahx-JsR3_OgC |year=1965 |publisher=John Wiley and Sons |location=New York}}</ref>{{rp|at=§ 5.9}}

<math display="block">\frac{v^2}{2} + \Psi + w = \text{constant}.</math>

Here {{mvar|w}} is the [[enthalpy]] per unit mass (also known as specific enthalpy), which is also often written as {{mvar|h}} (not to be confused with "head" or "height").

Note that
<math display="block">w =e + \frac{p}{\rho} ~~~\left(= \frac{\gamma}{\gamma-1} \frac{p}{\rho}\right)</math>
where {{mvar|e}} is the [[thermodynamics|thermodynamic]] energy per unit mass, also known as the [[specific energy|specific]] [[internal energy]]. So, for constant internal energy <math>e</math> the equation reduces to the incompressible-flow form.

The constant on the right-hand side is often called the Bernoulli constant and denoted {{mvar|b}}. For steady inviscid adiabatic flow with no additional sources or sinks of energy, {{mvar|b}} is constant along any given streamline. More generally, when {{mvar|b}} may vary along streamlines, it still proves a useful parameter, related to the "head" of the fluid (see below).

When the change in {{math|Ψ}} can be ignored, a very useful form of this equation is:
<math display="block">\frac{v^2}{2} + w = w_0</math>
where {{math|''w''<sub>0</sub>}} is total enthalpy. For a calorically perfect gas such as an ideal gas, the enthalpy is directly proportional to the temperature, and this leads to the concept of the total (or stagnation) temperature.

When [[shock wave]]s are present, in a [[frame of reference|reference frame]] in which the shock is stationary and the flow is steady, many of the parameters in the Bernoulli equation suffer abrupt changes in passing through the shock. The Bernoulli parameter remains unaffected. An exception to this rule is radiative shocks, which violate the assumptions leading to the Bernoulli equation, namely the lack of additional sinks or sources of energy.

=== Unsteady potential flow ===
For a compressible fluid, with a barotropic equation of state, the unsteady momentum conservation equation
<math display="block">\frac{\partial \vec{v}}{\partial t} + \left(\vec{v}\cdot \nabla\right)\vec{v} = -\vec{g} - \frac{\nabla p}{\rho}</math>

With the irrotational assumption, namely, the flow velocity can be described as the gradient {{math|∇''φ''}} of a velocity potential {{math|''φ''}}. The unsteady momentum conservation equation becomes
<math display="block">\frac{\partial \nabla \phi}{\partial t} + \nabla \left(\frac{\nabla \phi \cdot \nabla \phi}{2}\right) = -\nabla \Psi - \nabla \int_{p_1}^{p}\frac{d \tilde{p}}{\rho(\tilde{p})}</math>
which leads to
<math display="block">\frac{\partial \phi}{\partial t} + \frac{\nabla \phi \cdot \nabla \phi}{2} + \Psi + \int_{p_1}^{p}\frac{d \tilde{p}}{\rho(\tilde{p})} = \text{constant}</math>

In this case, the above equation for isentropic flow becomes:
<math display="block">\frac{\partial \phi}{\partial t} + \frac{\nabla \phi \cdot \nabla \phi}{2} + \Psi + \frac{\gamma}{\gamma-1}\frac{p}{\rho} = \text{constant}</math>