The Material/Lagrangian Derivative

yhsa · 11-04-2009: 11th April 2009 15:03

Hello,

I'm having difficulty with some fluid dynamics, specifically the material derivative, i.e. $\frac{DX}{Dt}$ where X can be a number of variables.

If I go through the theory and point out where I'm getting confused, perhaps a maths genius can help me out?

Ok...

Consider a parcel, of say air, moving along a vector field. Rather than considering the rate of change of air at each point in a fixed coordinate system, we follow the parcel and describe the rates of change of various variables of that parcel as it moves along.

Let the parcel of air have temperature T at a position x(t), y(t), z(t) such that T = T(x, y, z, t).

So,

$dT = \frac{\partial T}{\partial x}dx + \frac{\partial T}{\partial y}dy + \frac{\partial T}{\partial z} dz + \frac{\partial T}{\partial t}dt$

<--- Confused from here--->
Right. My notes then say "the material derivative is $\displaystyle\lim_{\partial t \to 0} (\frac{\partial T}{\partial t})$
Ok...I don't really understand what happens there, and the next step is...

$\frac{DT}{Dt} = \frac{\partial T}{\partial x} \frac{Dx}{Dt} + \frac{\partial T}{\partial y} \frac{Dy}{Dt} + \frac{\partial T}{\partial z} \frac{Dz}{Dt} + \frac{\partial T}{\partial t}$

<---until here--->

So what's happening there? The next step makes sense...

$\frac{DT}{Dt} = u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z} + \frac{\partial T}{\partial t}$

$= \frac{\partial T}{\partial t} + \textbf{u}.\nabla T$

So it's the bit in the middle, with the limit, that really confuses me at the moment.

Any help would be greatly appreciated!

ashy (from his revision account while his main account is blocked by request

)

yusufu · 11-04-2009: 11th April 2009 15:42

Originally Posted by yhsa

<--- Confused from here--->
Right. My notes then say "the material derivative is $\displaystyle\lim_{\partial t \to 0} (\frac{\partial T}{\partial t})$

Does this say $\frac{\delta T}{\delta t}$ or $\frac{\partial T}{\partial t}$ in your notes?

yhsa · 11-04-2009: 11th April 2009 16:16

Originally Posted by yusufu

Does this say $\frac{\delta T}{\delta x}$ or $\frac{\partial T}{\partial x}$ in your notes?

$\frac{\delta t}{\delta x}$ , you're right, sorry. Similarly it's $\displaystyle\lim_{\delta t \to 0}$

It should also be:

$dT = \frac{\partial T}{\partial x}\delta x + \frac{\partial T}{\partial y}\delta y + \frac{\partial T}{\partial z} \delta z + \frac{\partial T}{\partial t}\delta t$

I don't really understand the difference, though

yusufu · 11-04-2009: 11th April 2009 17:31

Originally Posted by yhsa

$\frac{\delta t}{\delta x}$ , you're right, sorry. Similarly it's $\displaystyle\lim_{\delta t \to 0}$

It should also be:

$dT = \frac{\partial T}{\partial x}\delta x + \frac{\partial T}{\partial y}\delta y + \frac{\partial T}{\partial z} \delta z + \frac{\partial T}{\partial t}\delta t$

I don't really understand the difference, though

$\frac{dT}{dt}$ is the limit of $\frac{\delta T}{\delta t}$ as ${\delta t} \to 0$ .

Just means you're actually taking the derivative.

yhsa · 11-04-2009: 11th April 2009 18:35

Originally Posted by yusufu

$\frac{dT}{dt}$ is the limit of $\frac{\delta T}{\delta t}$ as ${\delta t} \to 0$ .

Just means you're actually taking the derivative.

I am made of fail.

Thanks.

Send a PM to "ashy" and I'll rep you when I can access the account again (June 4th)

yusufu · 11-04-2009: 11th April 2009 18:41

Originally Posted by yhsa

I am made of fail.

Thanks.

Send a PM to "ashy" and I'll rep you when I can access the account again (June 4th)

lol. OK.

yusufu · 11-04-2009: 11th April 2009 18:43

Originally Posted by yhsa

I am made of fail.

Thanks.

Send a PM to "ashy" and I'll rep you when I can access the account again (June 4th)

ashy has chosen not to receive private messages or may not be allowed to receive private messages. Therefore you may not send your message to him/her.

Never mind.

yhsa · 11-04-2009: 11th April 2009 18:43

Originally Posted by yusufu

lol. OK.

When I do check it I'll be like "oo a PM" then laugh at myself for forgetting, despite writing this now.

What?

I don't get why the lecturer has to use capital D, now that I look at it. Seems a little...misleading. Never mind.

yhsa · 11-04-2009: 11th April 2009 18:45

Originally Posted by yusufu

ashy has chosen not to receive private messages or may not be allowed to receive private messages. Therefore you may not send your message to him/her.

Never mind.

Bah. Hmm. Should be a separate "ban by request" PM feature, really.

I shall find a way to remember do give you many hundreds of points

yusufu · 11-04-2009: 11th April 2009 19:20

Originally Posted by yhsa

Bah. Hmm. Should be a separate "ban by request" PM feature, really.

I shall find a way to remember do give you many hundreds of points

I've put this thread into my favourites and labelled it "Not to be opened until June 4th 2009".

yhsa · 11-04-2009: 11th April 2009 19:37

Originally Posted by yusufu

I've put this thread into my favourites and labelled it "Not to be opened until June 4th 2009".

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The Material/Lagrangian Derivative