# Fluid Mechanics (Time derivatives)

1. Dec 22, 2013

### Niller

Hi Guys.

I hope some of you are able to help me out. This is NOT a homework.

I have to explain the difference between the three different rate of change (time derivatives) of a
property (such as mass, momentum, internal energy, etc) in a continuum?

1 )The partial time derivative
2) The material time derivative
3) The total time derivative

Can some of you help me out? Im very lost at this field, and I hope some of you can help me a little bit og maybe guide me through.

2. Dec 22, 2013

### AlephZero

This is explained in any textbook on fluid mechanics.

If you have a specific question about something in a book (or web site) that you don't understand, somebody will probably try to explain it. But people here are not going to write a chapter of a textbook, just to answer your question.

3. Dec 23, 2013

### Niller

Hi again.

Yes I have read a a couple of books, and tryed to search the net?

I only get this out of it:

1) âˆ‚/âˆ‚t

2) D/Dt

3) d/dt

And I read that total derivative and material derivative is "the same thing" ?

I really hope some of you can help me a bit.

4. Dec 23, 2013

### Niller

Maybe someone can explain me the difference between material and total derivative ?

Isnt it the same?

5. Dec 23, 2013

### Staff: Mentor

Yes, it's the same. Some people use d/dt and some people use D/Dt. Can you articulate what these mean physically? Can you articulate what âˆ‚/âˆ‚t means physically?

Chet

6. Dec 23, 2013

Just make sure you aren't confusing a total derivative with an ordinary derivative. That's why it's common to use the D/Dt notation with material/total/convective derivatives.

7. Dec 23, 2013

### Niller

Hi Again Guys.

Maybe you can correct me in this:

We are interessted in rates of change of K (velocity of acc)

Material Derivative: D(K)/Dt. Tells us how K changes as time changes AND how K changes as it moves through space. (Similar to Langrian Description)

partial time derivative: âˆK‚/âˆ‚t. Tells us how K changes as time Changes at a specific point. we hold (x,y,x) as constants? (Similar to Eulerican Description)

Total Time derivative: Similar to Material Derivative? Right?

I have read a note saying following:

Rate of change of a property (3 time derivative)

1) Powered Boat in a lake ( = Total Derivative ) dK/dt (changed based on boat velocity)

2) Drifting Boat in a lake ( = Material Derivative) DK/Dt (Changes based on river velocity)

3) Tied Boat in a lake ( = Partial Derivative) âKˆ‚/âˆ‚t.

--

The changes they are talking about, I assume is particles in the water? Right?
I have a hard time to see and understand the difference of 1 and 2. Maybe some of you can explain me this? :S

8. Dec 25, 2013

### Staff: Mentor

Maybe it would help to think of it in terms of two coordinate systems:
1. The "laboratory" coordinate system, x,y,z
2. A moving coordinate system x',y',z' that moves (and deforms) with the flowing fluid particles

We are going to do a transformation between the two coordinate systems to see what we get.
We're going to let the laboratory coordinate x represent the position at time t of a fluid particle that was at position x=x', y=y', z=z' at time t = 0. Similarly for y and z. So were are going to express the laboratory coordinates of all the fluid particles at time t parametrically in terms of their laboratory coordinates at time t = 0. In that way, the coordinates of the particles at time zero act as labels (identifiers) for the material particles which they carry along with them for all times. So, the trajectory of each particle of fluid can be represented as:

x = x(t,x',y',z')
y= y(t, x',y',z')
z= z(t, x',y',z')

with
x' = x(0,x',y',z')
y'= y(0, x',y',z')
z'= z(0, x',y',z')

This is simply a transformation between a moving coordinate system and a fixed coordinate system. The kinematics of the fluid flow are totally specified once the functions x(t,x',y',z'), y(t,x',y',z'), and y(t,x',y',z') are specified.

Using this framework, let's start out by determining the velocities of all the fluid particles at time t. To do this, we first write,
$$dx=\frac{\partial x}{\partial t}dt+\frac{\partial x}{\partial x'}dx'+\frac{\partial x}{\partial y'}dy'+\frac{\partial x}{\partial z'}dz'$$
Now, if we are going to determine the velocity of a fluid particle, we are going to have to be holding its "material coordinates" x', y', and z' constant. Therefore, we must have that
$$v_x=\left(\frac{\partial x}{\partial t}\right)_{x',y',z'}$$
Similarly for the fluid velocities in the y and z directions.

Next, let's suppose that the temperature or the concentration of the flowing fluid is varying with time and position in the laboratory reference frame. In the case of concentration, for example, we can write C = C(t,x,y,z) and
$$dC=\frac{\partial C}{\partial t}dt+\frac{\partial C}{\partial x}dx+\frac{\partial C}{\partial y}dy+\frac{\partial C}{\partial z}dz$$
so that, at a given location in space (x, y,and are constant), the rate of change of concentration with respect to time is given by:
$$\frac{\partial C}{\partial t}=\left(\frac{\partial C}{\partial t}\right)_{x,y,z}$$
Next, let's determine what happens if we examine the rate of change of concentration with respect to time as measured by an observer traveling along with a material particle (i.e., holding x', y', and z' constant):
$$\left(\frac{\partial C}{\partial t}\right)_{x',y',z'}=\left(\frac{\partial C}{\partial t}\right)_{x,y,z}+\left(\frac{\partial C}{\partial x}\right)_{x,y,z}\left(\frac{\partial x}{\partial t}\right)_{x',y',z'}+\left(\frac{\partial C}{\partial y}\right)_{x,y,z}\left(\frac{\partial y}{\partial t}\right)_{x',y',z'}+\left(\frac{\partial C}{\partial z}\right)_{x,y,z}\left(\frac{\partial z}{\partial t}\right)_{x',y',z'}$$
But, from our equation for velocity, we then have:
$$\left(\frac{\partial C}{\partial t}\right)_{x',y',z'}=\left(\frac{\partial C}{\partial t}\right)_{x,y,z}+v_x\left(\frac{\partial C}{\partial x}\right)_{x',y',z'}+v_y\left(\frac{\partial C}{\partial y}\right)_{x',y',z'}+v_z\left(\frac{\partial C}{\partial z}\right)_{x',y',z'}$$

This is just the definition of the material derivative of C with respect to time.

$$\frac{DC}{Dt}=\left(\frac{\partial C}{\partial t}\right)_{x',y',z'}=\left(\frac{\partial C}{\partial t}\right)_{x,y,z}+v_x\left(\frac{\partial C}{\partial x}\right)_{x',y',z'}+v_y\left(\frac{\partial C}{\partial y}\right)_{x',y',z'}+v_z\left(\frac{\partial C}{\partial z}\right)_{x',y',z'}$$