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Homework Help: Reynold's Transport Theorem

  1. Apr 26, 2016 #1
    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.

    Can anyone help please?
  2. jcsd
  3. Apr 26, 2016 #2
    help please .. I just want to pass (and learn) :cry:
  4. Apr 26, 2016 #3
  5. Apr 27, 2016 #4
    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.
  6. Apr 27, 2016 #5

    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?


    Attached Files:

  7. Apr 27, 2016 #6
    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?
  8. Apr 27, 2016 #7
    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.
  9. Apr 27, 2016 #8
    $$(\rho v f)_{x_2}-(\rho vf)_{x_1}=\int_{x_1}^{x_2}{\frac{\partial (\rho vf)}{\partial x}dx}$$
  10. Apr 28, 2016 #9
    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...
  11. Apr 28, 2016 #10
  12. Apr 28, 2016 #11
  13. Apr 28, 2016 #12
    Oh... that makes sense. Thank you so much!!
  14. Apr 28, 2016 #13
    So you got it now?
  15. Apr 28, 2016 #14
    Yes I got it now. It simplified quite easily with your help. Thank you.
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