# Reynold's Transport Theorem

• kev931210

## Homework Equations

one dimensional Reynold's transport theorem

## The Attempt at a Solution

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I started with this equation, and tried to expand it using the equation given in #2.

This is the farthest I have gotten so far. I got stuck from here. I do not know how to get from the shaded equation to the equation below.

Can anyone help please?

help please .. I just want to pass (and learn)

helpppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

Substitute ##v_1=\frac{dx_1}{dt}## and ##v_2=\frac{dx_2}{dt}## into your second equation. What does that give you? Incidentally, the ##f\rho##'s in this equation should be evaluated at 1 and 2.

kev931210

Sorry, there was some typo in my question. So, I ended up with the above equation using your advice, but it doesn't seem to get me any further. Both f and p are functions of (x,t), so it's hard for me to simplify easily. Is there any relation that can let me convert the above equation to the below equation?

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Maybe it would help if I wrote your expression using LateX, which you should learn from the PF tutorial:

$$\int_{x_1}^{x_2}{\left(\rho\frac{\partial f}{\partial t}+f\frac{\partial \rho}{\partial t}\right)dx}+(\rho v f)_{x_2}-(\rho vf)_{x_1}$$

Does this give you any ideas?

kev931210
No .. sorry I'm not an advanced student in Physics. Is there any physics theory or mathematical trick I can use from that point? I have no clue what to do from that point. I was stuck there for long time.

No .. sorry I'm not an advanced student in Physics. Is there any physics theory or mathematical trick I can use from that point? I have no clue what to do from that point. I was stuck there for long time.
$$(\rho v f)_{x_2}-(\rho vf)_{x_1}=\int_{x_1}^{x_2}{\frac{\partial (\rho vf)}{\partial x}dx}$$

kev931210
Sorry, I could not find the tutorial, and ended up not using latex to write the equations.

This is what I have based on your advise.

In order to derive the equation given by the question, the following equation needs to be true:

If f is independent of x, I can easily extract f, and prove the above equation by using the mass balance equation. However, f is a function of x and t, so I am not so sure how I can derive the above equation...

kev931210
Sorry, I could not find the tutorial, and ended up not using latex to write the equations.

View attachment 99882
The f' they are referring to is not ##=\partial f/\partial t##. It is $$f'=\frac{\partial f}{\partial t}+v\frac{\partial f}{\partial x}$$

kev931210
Oh... that makes sense. Thank you so much!

Oh... that makes sense. Thank you so much!
So you got it now?

Yes I got it now. It simplified quite easily with your help. Thank you.