From what I know, in fluid dynamics, the Eulerian approach studies points that are fixed in space and time, while the Lagrangian approach studies individual particles and follows them. Is it correct to say that the Lagrangian approach is a microscopic point of view, and the Eulerian one is a macroscopic one?
If not, can you provide examples of Eulerian systems with a microscopic point of view, or Lagrangian ones with a macroscopic view?
Answer
The Lagrangian or Eulerian formulations make no specific distinction in size—they merely relate to whether you are tracking the properties at a point or volume of space (Eulerian), or the properties of a point or volume of “something” which moves along with whatever flow field that causes said “something” to move (Lagrangian).
One could, for example, construct a Eulerian control volume analysis associated with a microscopic spherical volume—nothing about the mathematics or physics says such an analysis fails at specific scales. Likewise, one could construct a Lagrangian analysis around a block of solid material—which is in fact the way solid mechanicists develop mathematical models of elasticity.
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