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I am trying to learn about the material (or particle) derivative in fluid dynamics. I was looking through this explanation, and they mention things at the beginning and the end that confuse me.

Lagrangian description: Picture a fluid flow where each fluid particle caries its own properties such as density, momentum, etc. As the particle advances its properties may change in time.

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Example of incompressible flow where $\frac{D\rho}{Dt}=0$: Assume a flow where the density of each fluid particle is constant in time. Be careful not to confuse this with $\frac{\partial\rho}{\partial t}=0$, which means that the density at a particular point in the flow is constant and would allow particles to change density as they flow from point to point. Also, do not confuse this with $\rho=const$, which for example does not allow a flow of two incompressible fluids.

I can understand the picture of each particle having its own position, velocity, etc. that involves spatial or time coordinates, but I am confused as to the discussion of particles carrying density, and how particles could change density as they flow. (Note that I understand the explanation of incompressible fluid flow given if the density of a particle can change, I am just confused as to what this really means).

So my question then is the following: What do we mean by particles in the Lagrangian description of fluid flow, and how could the density of these particles be changing? (I am new to looking at fluid dynamics, so if this is a really simple or too broad of a question I will gladly close the question. I hate bad or ignorant questions on this site).

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When we use the term "fluid particle" in this context, what we really mean is "fluid parcel," consisting of many many atoms/molecules, present in the limit of a very small spatial volume containing these atoms/molecules. If, at a given location is space, the density is not changing with time within a fixed tiny control volume (parcel), this means that the number of molecules/atoms within the control volume is not changing, and $\partial \rho/\partial t=0$.

If we are following the motion of a fluid parcel in a Lagrangian sense, what we mean is that the number of atoms/molecules within the moving parcel is constant, but the volume of the parcel can be increasing or decreasing, and the fluid density can be changing as a result. If the volume of the Lagrangian parcel does not change in a Lagrangian sense, we say that the density is constant with time following the motion of the parcel, and we represent this as $D\rho/Dt=0$

So, a Lagrangian parcel contains a fixed number of material "particles" within a tiny moving (and potentially changing parcel volume), while an Eulerian parcel consists of a fixed tiny parcel with a fixed and stationary spatial volume (and potentially changing number of "particles" within the parcel).

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  • $\begingroup$ "If the volume of the Lagrangian parcel does not change in a Lagrangian sense, we say that the density is constant with time following the motion of the parcel" Do you mean to say if the density of the Lagrangian parcel does not change? $\endgroup$ – Aaron Stevens Sep 28 '18 at 19:22
  • $\begingroup$ Yes, if it’s volume doesn’t change. $\endgroup$ – Chet Miller Sep 28 '18 at 19:39
  • $\begingroup$ Ah sorry, I was misreading things $\endgroup$ – Aaron Stevens Sep 28 '18 at 19:48
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In a nutshell:

  1. The Eulerian picture is when you have a reference frame fixed relatively to, say, the ground. The fluid flow is a velocity vector field ${\bf u}({\bf x},t)$ in this reference frame.

  2. The Lagrangian picture is closer to point mechanics: In point mechanics each point particle with position ${\bf r}_i(t)$ has a label $i$ whereas in the Lagrangian fluid picture each fluid parcel with position ${\bf r}({\bf a},t)$ has a label ${\bf a}$, except it's a continuous label, and it is assumed that a neighboring fluid parcel stays a neighbor, with neighboring label, say ${\bf a}+\delta{\bf a}$. The continuous label ${\bf a}$ constitutes the coordinate of the Lagrangian picture.

    The neighbor distance [and the density $\rho({\bf a},t)$ of fluid parcels] is in general allowed to vary with time $t$. However, for an incompressible fluid the density $\rho({\bf a},t)$ does not depend on $t$.

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