Don't understand material derivative and and advective derivative

[cks]

Langrangian description: one essentially follows the history of individual fluid particles
(I understand this, it's like tracking a particle and find its position x=x(c,t) where c is the initial position at time t=0. ) So, an equation that gives the position of a particle in terms of its initial position and time t can be said the be described in Langrangian coordinate)

Eulerian description: one concentrates on what happens at a spatial point x.
(I also understand this, my teacher said it's like setting a fixed volume, and describe what happens inside this volume) (For example, maybe, the temperature at a point x, the pressure at a point x)

I can understand what are the meanings of the two explanatinos in the parentheses. but, I don't really understand how the Eulerian description in the following equation!

Here is the material derivative of following a fluid element

DF/Dt = delF/delt + v . grad (F)

where F is the physical property of the system.

The book says that DF/Dt is the material derivative, and delF/delt is the advective derivative(which I think is Eulerian description) , I don't understand why the books say so? It's not so intuitive to me.

 

[nizi]

I believe it may help you understand the material time derivative and spatial time derivative.

The material time derivative of the spatial field is in the following by use of the chain rule.

$$\frac{Df\left(\textbf{x},t\right)}{Dt}$$
$$=\frac{Df\left(\textbf{x}\left(\textbf{X},t\right),t\right)}{Dt}$$
$$=\left.\frac{\partial f\left(\textbf{x}\left(\textbf{X},t\right),t\right)}{\partial t}}\right|_{\textbf{x}} +\left.\frac{\partial f\left(\textbf{x}\left(\textbf{X},t\right),t\right)}{\partial \textbf{x}}}\right|_{t} \cdot\left.\frac{\partial \textbf{x}\left(\textbf{X},t\right)}{\partial t}}\right|_{\textbf{X}=\textbf{x}^{-1}\left(\textbf{x},t\right)}$$

where

$$\textbf{X}$$ :initial water particle position

$$\textbf{x}$$ :water particle position at time t

$$\textbf{x}=\textbf{x}\left(\textbf{X},t\right)$$ :water particle motion i.e.$$\textbf{X}=\textbf{x}\left(\textbf{X},0\right)$$
And the variables followed by the upright bar mean they're fixed when the partial differentiations are conducted.

 

[HallsofIvy]

I think this would do better in Physics rather than mathematics.

 

[arildno]

Langrangian description: one essentially follows the history of individual fluid particles
(I understand this, it's like tracking a particle and find its position x=x(c,t) where c is the initial position at time t=0. ) So, an equation that gives the position of a particle in terms of its initial position and time t can be said the be described in Langrangian coordinate)

Eulerian description: one concentrates on what happens at a spatial point x.
(I also understand this, my teacher said it's like setting a fixed volume, and describe what happens inside this volume) (For example, maybe, the temperature at a point x, the pressure at a point x)

I can understand what are the meanings of the two explanatinos in the parentheses. but, I don't really understand how the Eulerian description in the following equation!

Here is the material derivative of following a fluid element

DF/Dt = delF/delt + v . grad (F)

where F is the physical property of the system.

The book says that DF/Dt is the material derivative, and delF/delt is the advective derivative(which I think is Eulerian description) , I don't understand why the books say so? It's not so intuitive to me.

The "material derivative" is called "material", since it measures the acceleration of a MATERIAL (i.e, mass) particle.
Remember that Newton's second law of motion concerns the relation of forces and acceleration of mass particles,; an abstract concept like a "velocity field" is NOT a material particle, and hence, the relations between a velocity field and forces acting upon particles within that field is rather subtle.

As for "advective derivative", I confess I haven't heard that termn, but it is probably correct, since the other term is called the "convective derivative".

The "advective derivative", or "local derivative" as I like to call it, measures the rate of change of the velocity field AT A FIXED SPATIAL POINT.
Thus, it does NOT, over time measure the acceleration of a single particle, rather, it measures the relative velocity difference between different mass particles that happen to stray into that point at different times (relative, that is, to the time interval between their habitation there).

 

[cristo]

I think this would do better in Physics rather than mathematics.

Agreed. Moved to general physics.

 

[Andy Resnick]

Langrangian description: one essentially follows the history of individual fluid particles

<snip>
Eulerian description: one concentrates on what happens at a spatial point x.
<snip>
I can understand what are the meanings of the two explanatinos in the parentheses. but, I don't really understand how the Eulerian description in the following equation!

Here is the material derivative of following a fluid element

DF/Dt = delF/delt + v . grad (F)

where F is the physical property of the system.

The book says that DF/Dt is the material derivative, and delF/delt is the advective derivative(which I think is Eulerian description) , I don't understand why the books say so? It's not so intuitive to me.

There's a few concepts here. First, the difference between the Eulerian and Lagrangian points of view. Second, the meaning of the material derivative, advected derivative, etc.

The second point: The total/material derivative D/Dt = $\frac{\partial}{\partial t} + v\bullet\nabla$. Physically, this means the amount of change in a variable comes from two possible sources- a temporal change and a spatial change. For example, D(the weather)/Dt = $\frac{\partial (the weather)}{\partial t} + v\bullet\nabla (the weather)$. This means the weather can change in two ways: either I can sit still and the weather changes ($\frac{\partial (the weather)}{\partial t}$), or I can get on an airplane and fly someplace with different weather ($v\bullet\nabla (the weather)$).

There's more complex ways to define a total derivative as well- upper and lower convected derivatives. Those appear in Maxwell constitutive realtionships (Oldroyd models) and involve rotations as well.

The first point: There are two equivalent ways of picturing continuum mechanics, one in which a coordinate system is identified with and moves/deforms with a chosen region, or one where the coordinate system is static and volumes/points move through the coordinates. The two are related via the total derivative, because the Eulerian coordinates are allowed to both move and deform (stretch/shear).

Does that help?