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Reynold's Transport Theorem

  1. Apr 26, 2016 #1
    1. The problem statement, all variables and given/known data
    cap2.PNG

    2. Relevant equations
    one dimensional Reynold's transport theorem
    upload_2016-4-26_13-33-18.png


    3. The attempt at a solution

    upload_2016-4-26_13-33-49.png


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

    upload_2016-4-26_13-46-13.png

    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.
    upload_2016-4-26_13-42-33.png

    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
    helpppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
     
  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
    cap5.PNG

    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?

    cap5.PNG
     

    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.


    cp3.PNG

    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:

    upload_2016-4-28_2-44-59.png

    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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