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I was wondering if anyone could help me with the following problem.

If $H=\frac{P}{\rho}+\frac{v^{2}}{2}+gz$, Then $H$ is constant along streamlines

and if $\nabla \times v=0$, then $H$ is constant throughout the flow.

I don't really understand how to do this.

Gibberish
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  • What about the proof on the wikipedia? https://en.wikipedia.org/wiki/Bernoulli%27s_principle#Derivations_of_the_Bernoulli_equation – Botond May 23 '18 at 17:43
  • I dont see how i would do that – Gibberish May 23 '18 at 17:45
  • What are $H$, $P$, $\rho$, $v$, $g$, and $z$ and how do they relate to each other? – user7530 May 23 '18 at 17:50
  • $P(x,y,z,t)$ is the pressure, $\rho(x,y,z,t)$ is the density $v(x,y,z,t)$ is the velocity and $g$ is the acceleration due to gravity $z$ is the height, where the $z$ axis is directed up – Gibberish May 23 '18 at 17:56
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    It looks like the flow is steady, incompressible and inviscid? – Chee Han May 23 '18 at 19:24
  • @CheeHan That is correct sorry i should have added that. – Gibberish May 23 '18 at 19:25
  • This can be found in many classical fluid dynamics book such as Elementary Fluid Dynamics by David Acheson; see Section 1.3. – Chee Han May 23 '18 at 19:28
  • @CheeHan I have that book, i will have a look now. Thank you! – Gibberish May 23 '18 at 19:28
  • This is classical standard Bernoulli fluid pressure law. Please refer to several books in the library. It connects static and dynamic pressures with vertical column height if it is a liquid. – Narasimham May 24 '18 at 02:15

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