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UPDATE


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

$$F=\{{F_x},{F_y},{F_z}\}$$

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!

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1 Answer 1

up vote 1 down vote accepted
+100

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!

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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
1  
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

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