3
$\begingroup$

I can prove the Bernoulli equation in an elementary way (that is, using the conservation of energy law). But I was asked to derive the Bernoulli equation from the Navier-Stokes equations, and I have no idea how to proceed. I tried searching on the Internet, but I found no good explanations.

I was told that Bernoulli equation is $${v^{2} \over 2}+gh+{p \over \varrho }={\mathrm {const}}$$ and in class we had this form of Navier Stokes: $$-\nabla p+\varrho \boldsymbol{b}=\varrho\left(\frac{\partial v}{\partial t}+v\nabla v\right).$$

If someone could show me how to derive Bernoulli equation from Navier-Stokes, it would be great.

$\endgroup$

closed as off-topic by user36790, Jon Custer, Gert, John Rennie, Qmechanic Nov 2 '16 at 12:02

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – Community, Jon Custer, Gert, John Rennie, Qmechanic
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Welcome on Physics SE :) It would be very helpful I think if you could add the explicit forms of the Navier Stokes and Bernoulli equations you are using to make sure that the answers are really helpful to you :) $\endgroup$ – Sanya Nov 1 '16 at 16:21
  • $\begingroup$ I was told that Bernoulli equation is ${v^{2} \over 2}+gh+{p \over \varrho }={\mathrm {const}}$, and in class we had this form of Navier Stokes: $-\nabla p+\varrho \boldsymbol{b}=\varrho\left(\frac{\partial v}{\partial t}+v\nabla v\right)$ $\endgroup$ – Anne4 Nov 1 '16 at 16:26
  • 1
    $\begingroup$ In the second equation you mean $-\nabla p+\varrho \boldsymbol{g}=\varrho\left(\frac{\partial v}{\partial t}+v\nabla v\right) $? These are the Euler equations, and the derivation you seek is in Landau, Lifshitz vol 6 "Fluid Mechanics", pages 9-10. It's quite straigthforward. Otherwise, what's $\ \boldsymbol{b}$? $\endgroup$ – Luca Nov 1 '16 at 19:57
  • $\begingroup$ In the version of the Euler equation that the OP wrote down, b is the general body force per unit mass. It doesn't necessarily have to be g. $\endgroup$ – Chet Miller Nov 2 '16 at 10:50
  • $\begingroup$ To get the Bernoulli equation from the Euler equation, the standard method is to dot the Euler equation with the velocity v and to then integrate with respect to t. This allows you to integrate along a streamline. Incidentally, those v's in the Euler equation should be vectors. See Bird, Stewart, and Lightfoot, Transport Phenomena. $\endgroup$ – Chet Miller Nov 2 '16 at 10:58