Editing Bernoulli's principle (section)

=== 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