How do I expand Reynold's transport theorem using the given equation?

[kev931210]

Homework Statement

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?

 

[kev931210]

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

 

[kev931210]

helpppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

 

[Chestermiller]

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?

 

[Chestermiller]

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.

 

[Chestermiller]

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...

 

[Chestermiller]

$$\frac{\partial (\rho v f)}{\partial x}=f\frac{\partial (\rho v)}{\partial x}+\rho v\frac{\partial f}{\partial x}$$

https://www.physicsforums.com/help/latexhelp/

 

[Chestermiller]

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!

 

[Chestermiller]

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

So you got it now?

 

[kev931210]

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