Homework Help: Reynold's Transport Theorem

1. Apr 26, 2016

kev931210

1. The problem statement, all variables and given/known data

2. Relevant equations
one dimensional Reynold's transport theorem

3. The attempt at a solution

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.

2. Apr 26, 2016

kev931210

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

3. Apr 26, 2016

kev931210

helpppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

4. Apr 27, 2016

Staff: Mentor

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.

5. Apr 27, 2016

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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6. Apr 27, 2016

Staff: Mentor

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?

7. Apr 27, 2016

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.

8. Apr 27, 2016

Staff: Mentor

$$(\rho v f)_{x_2}-(\rho vf)_{x_1}=\int_{x_1}^{x_2}{\frac{\partial (\rho vf)}{\partial x}dx}$$

9. Apr 28, 2016

kev931210

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

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...

10. Apr 28, 2016

Staff: Mentor

11. Apr 28, 2016

Staff: Mentor

12. Apr 28, 2016

kev931210

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

13. Apr 28, 2016

Staff: Mentor

So you got it now?

14. Apr 28, 2016

kev931210

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