Eulerian Field vs LagrangianConceptual

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In summary, the conversation discusses the concept of the Eulerian frame of reference and its relationship to the Lagrangian frame of reference. The Eulerian frame of reference is described as having fixed coordinates in space, while the Lagrangian frame of reference follows the moving position of individual particles. The conversation also talks about deriving the acceleration field in the Eulerian frame of reference and clarifies the use of the term "fixed particle" to refer to a specific particle in the fluid. There is confusion about the use of the term "following" in the context of the Eulerian frame, as it seems to contradict the idea of fixed coordinates. Overall, the conversation aims to understand the difference between the Eulerian and Lagrangian frames of reference in
  • #1
Saladsamurai
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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:

...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:

[tex]\mathbf{a} = \frac{d\,\mathbf{V}}{d\,t} = \frac{\partial{V}}{\partial{t}} + (\mathbf{V}\cdot\nabla)\mathbf{V}[/tex]

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

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
 
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  • #2
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.
 
  • #3
foxjwill said:
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?
 
  • #4
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.
 
  • #5
In the lagrangian frame of reference, the origin is always at the specific particle, while in the eulerian frame of reference, it is not.
 
  • #6
foxjwill said:
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

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.
 
Last edited:

1. What is an Eulerian field?

An Eulerian field refers to a type of mathematical representation of a physical field, where the values of the field are fixed at specific points in space. This means that the field is observed at a fixed point and its properties are described at that point.

2. What is a Lagrangian conceptual?

A Lagrangian conceptual is a different mathematical approach to representing a physical field, where the values of the field are attached to individual particles that move through space and time. This means that the field is observed from the perspective of a specific particle and its properties change as it moves through space.

3. What is the difference between Eulerian and Lagrangian representations?

The main difference between Eulerian and Lagrangian representations is the way in which the field is observed and described. Eulerian fields are fixed in space, while Lagrangian conceptuals are attached to moving particles. This leads to different ways of understanding and analyzing physical phenomena.

4. When is it more appropriate to use an Eulerian field?

Eulerian fields are typically used when the focus is on understanding the overall behavior of a physical system, such as weather patterns or ocean currents. This approach is useful for studying large-scale phenomena that do not depend on the specific motion of individual particles.

5. When is it more appropriate to use a Lagrangian conceptual?

Lagrangian conceptuals are often used when the focus is on understanding the behavior of individual particles, such as in the case of fluid dynamics or celestial mechanics. This approach is useful for studying small-scale phenomena where the specific motion of particles is important.

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