Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.


First solution:

I think the full solution is found here (6.1.1)-(6.1.4) and then add a definition of Divergence theorem, and then the pages between (2-7) in this link, if it's true, answer me and I will accept your answer

Second solution:

another possible solution is found here, please answer me on which pages it is found (and I will accept your answer and let you the bounty). I think it is found on pages 1-3 (till 2.19)

my question:

There is a theorem of Bernoulli integral (conservation of energy).

I want to lecture this topic and let the students an evidence for stationarity flow.

I considered a general potential (conservative) force


with the potential $\phi$ such that:

$$F = -\nabla \phi = \{ -\partial\phi/\partial x, -\partial\phi/\partial y, -\partial\phi/\partial z\}$$

I have to find a proof that along any streamline in a stationary flow,

$$ \frac{v^2}{2} + \frac{p}{\rho}+\phi={\rm constant}$$

I found this file, but it contains a proof for the Earth's gravity ($\phi$ = gz)

any help appreciated!

share|improve this question
add comment

1 Answer

up vote 1 down vote accepted

It is simple to prove.

You want to show that:

$v\cdot \nabla(\frac{v^2}{2} + \frac{p}{\rho}+\phi) = 0 $. This is equivalent to $\frac{v^2}{2} + \frac{p}{\rho}+\phi = {\rm constant} $ along stream lines.

Bernoulli theorem holds for invisid flows(i.e. Euler equations). So you know that $(v\cdot\nabla)v+\frac{1}{\rho}\nabla p+\nabla\phi=0$ (assuming stationary flow).

Let's write down everything in coordinates(using Einstein summation) and take dot product of $v$ and Euler equations.

$0=v\cdot0=v\cdot((v\cdot\nabla)v+\frac{1}{\rho}\nabla p+\nabla\phi)=v_i(v_j\partial_jv_i+\frac{1}{\rho}\partial_ip+\partial_i\phi)= \frac{1}{2}v_j\partial_j(v_iv_i)+\frac{1}{\rho}v_i\partial_ip+v_i\partial_i\phi) = v\cdot\nabla(\frac{v^2}{2} + \frac{p}{\rho}+\phi)$

I hope it is clear!

share|improve this answer
Plus if you really want reference. There is proof in your first link. Equations (2.13)-(2.16). But you have to show that you can rewrite Euler equations to form (2.13) that basically means to prove that vector identity given there. So I think that my proof is easier. $\\$ And do you really think that it is good idea to give lecture about this if you can't show such a simple theorem as Bernoulli's? –  tom Feb 14 '13 at 18:04
thank you very much Tom! I think I will show them the both solutions, so I have one proof (of you). about the second proof: do you have a link with a solution of rewriting Euler equations to form (2.13)? and then the second solution will be the rewriting Euler equations to form (2.13) and after that I will write (2.14)-(2.19). BTW, what's about (2.1)-(2.12)? I don't need this? about your question, The lecturer wants me to lecture about this topic.. so I will have to learn the both solutions (I will learn it by myself when I get them).. thank you very very much! –  Alon Shmiel Feb 14 '13 at 19:52
I updated my topic :) –  Alon Shmiel Feb 15 '13 at 9:50
Well equations (2.1)-(2.12) are just derivation of Euler equations I don't know if they are known to you or not. And about (2.13) I have to ask did you even try to prove that? Because in that pdf there is hint how to do that. You just use definition of $D/Dt$ and that vector identity. –  tom Feb 15 '13 at 10:22
thank you Tom! I didn't try to prove that.. what's about what I wrote in the topic, under the title: First solution ? –  Alon Shmiel Feb 15 '13 at 10:48
show 3 more comments

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.