Editing Bernoulli's principle (section)

=== 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.