# Eulerian vs Lagrangian

Hi PF!

I am reading about Eulerian vs Lagrangian perspectives. To me, it seems that Eulerian considers a volume and follows that volume (which may deform) through space. A Lagrangian frame of reference doesn't track volume, but instead specific particle matters.

Am I correct? If so, what are the advantages of each? Perhaps you have a toy problem or thought experiment where one frame of reference is superior to the other?

jedishrfu
Mentor
I found these youtube videos that describe the differences in approach:

and more formally here:

and a classic video from 50+ years ago:

jedishrfu
Mentor
It seems like you'd choose lagrangian if your sensors were traveling with the flow and Eulerian if your sensors were stationary with the flow passing through them.

The last video mentions use of weather balloons floating freely following the currents of the air and hence following a lagrangian frame of reference.

Also I found this tutorial that has some interesting stuff in it:

http://www.mne.psu.edu/cimbala/Learning/Fluid/Introductory/descriptions_of_fluid_flows.htm

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Chestermiller
Mentor
Eulerian means using a stationary control volume with material flowing in and out. Lagrangian means what you described as Eulerian in your first post: Lagrangian considers a volume and follows that volume (which may deform) through space. This is also sometimes called a material coordinate system, since it labels each particle within the volume by means of its coordinates at time zero.

Let me guess, Josh. You're studying deformational kinematics.

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jedishrfu
jedishrfu
Mentor
So I guess you could say Euler was a stick in the mud and Lagrange just went with the flow.

Two different philosophies of life.

Chestermiller
olivermsun
It seems like you'd choose lagrangian if your sensors were traveling with the flow and Eulerian if your sensors were stationary with the flow passing through them.
In those cases, Lagrangian and Eulerian (measurement) frames are chosen by the sensor. However, you might prefer to work with one or the other depending on whether the properties you are examining are traveling with the flow or stationary with the flow. (For example, away from boundaries, lots of stuff really happens "relative to the medium.")

jedishrfu
jedishrfu
Mentor
Also Eulerian is often chosen in laboratory setting where your sensors are fixed and not moving with the flow.

olivermsun
olivermsun
To clarify what I said in my previous post: even if the sensor is Eulerian (as it usually is), it is often advantageous to transform the measurements to a Lagrangian frame for analysis.

One classic example is measuring density profiles in a stratified fluid. If you use, e.g., a vertical array of sensors, and the fluid is moving, then you can get very "discontinuous" time series due to fine features that are advected past the sensor. When you shift to Lagrangian (or "semi-Lagrangian") coordinates, then the underlying structure often becomes much more clear.

Thank you all for the advice! That first video was actually hilarious! Stick in the mud And yep Chet, I'm studying continuum mechanics, so deformational kinematics is here too!

Chestermiller
Mentor
Thank you all for the advice! That first video was actually hilarious! Stick in the mud And yep Chet, I'm studying continuum mechanics, so deformational kinematics is here too!
Use of an embedded material coordinate system (Lagrangian) that moves with the material is essential to analyzing large deformation mechanics (kinematics, rheology, stress). The focus is on prediction the stress tensor so that stress-equilibrium equation can be applied.

olivermsun
Thanks for pointing that out!