# Deriving Bernoulli's equation from Navier-Stokes equation [closed]

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.

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

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• 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 :) – Sanya Nov 1 '16 at 16:21
• 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: $-\nabla p+\varrho \boldsymbol{b}=\varrho\left(\frac{\partial v}{\partial t}+v\nabla v\right)$ – Anne4 Nov 1 '16 at 16:26
• 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 $\ \boldsymbol{b}$? – Luca Nov 1 '16 at 19:57
• 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. – Chet Miller Nov 2 '16 at 10:50
• 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. – Chet Miller Nov 2 '16 at 10:58