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Eulerian Field vs Lagrangian::Conceptual

Okay. This is a very straight forward question. I believe that my text has an error or I am misunderstanding something.

It describes the Eulerian Field as:

Quote:
 ...our coordinates are fixed in space and we observe a particle of fluid as it passes by-- as if we had scribed a set of coordinate lines on a glass window in a wind tunnel. This is the eulerian frame of reference as opposed to the lagrangian which follows the moving position of individual particles.
Then we go on to derive the acceleration field in this eulerian field by taking the Total Derivative of the Velocity Field vector, which yields:

$$\mathbf{a} = \frac{d\,\mathbf{V}}{d\,t} = \frac{\partial{V}}{\partial{t}} + (\mathbf{V}\cdot\nabla)\mathbf{V}$$

Okay great..I get all of that. Here is where I croak. It then summarizes what we just did by saying:

Quote:
 We emphasize that this is the total time derivative that follows a particle of fixed identity, making it convenient for expressing laws of particle mechanics in the eulerian fluid field description. The operator d/dt is sometimes assigned a special symbol D/Dt to remind us that it contains four terms and follows a fixed particle.
This last quote keeps referring to "following a fixed particle" or "following a particle of fixed identity."

Isn't that by definition the Lagrangian frame? Or am I misinterpreting how they are using the word "following"?

Can someone clear up my confusion here?

Thank you,
Casey

 foxjwill Oct25-09 01:11 AM

Re: Eulerian Field vs Lagrangian::Conceptual

I think what the text means is "specific" particle, i.e. one particular particle in the fluid. I agree, though, that fixed was a bad choice of word.

Re: Eulerian Field vs Lagrangian::Conceptual

Quote:
 Quote by foxjwill (Post 2409350) I think what the text means is "specific" particle, i.e. one particular particle in the fluid. I agree, though, that fixed was a bad choice of word.
Yes. I was assuming that by "fixed particle" they mean a "specific particle."

My problem is that they are referring to a "fixed particle" but they are also saying that this is the eulerian approach. But I thought that the fixed particle approach was lagrangian?

Re: Eulerian Field vs Lagrangian::Conceptual

Any ideas on this one? I feel like I could move on, but I really want to understand what I am doing from here forward.

 foxjwill Oct25-09 03:45 PM

Re: Eulerian Field vs Lagrangian::Conceptual

In the lagrangian frame of reference, the origin is always at the specific particle, while in the eulerian frame of reference, it is not.

Re: Eulerian Field vs Lagrangian::Conceptual

Quote:
 Quote by foxjwill (Post 2410148) In the lagrangian frame of reference, the origin is always at the specific particle, while in the eulerian frame of reference, it is not.
Yes. I am quite aware of that. But that is not my question. Please look at what I am asking.

The whole point of my question is that I KNOW that the eulerian frame stays fixed and watches different fluid particle entering and leaving. So why do they say

Quote:
 We emphasize that this is the total time derivative that follows a particle of fixed identity, making it convenient for expressing laws of particle mechanics in the eulerian fluid field description.
The words in bold seem to contradict each other.

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