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Thermal Energy Equation Term - Chain Rule

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  1. Dec 18, 2017 #1
    1. The problem statement, all variables and given/known data
    I am going through a derivation of the thermal energy equation for a fluid and am stumped on one of the steps. Specifically, the text I am using converts the term:

    P/ρ*(Dρ/Dt)

    to:

    ρ*D/Dt(P/ρ) - DP/Dt

    where:
    ρ = density
    P = pressure
    D/Dt = material derivative

    The text says this is done using the chain rule of differentiation but I can't derive it myself. I'm far removed from calculus so maybe i'm missing something simple but any help would be appreciated.

    2. Relevant equations


    3. The attempt at a solution
     
  2. jcsd
  3. Dec 18, 2017 #2

    Charles Link

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    Homework Helper
    Gold Member

    One of the two expressions you have needs an extra minus sign. Momentarily, I will show the calculus of the second expression with the chain rule... @kevman90 Do you know how to take the derivative of ## \frac{d(uv)}{dt}##? It is ## u (\frac{dv}{dt}) +v(\frac{du}{dt}) ##. In this case, ## u=P ## and ## v=1/\rho ##. With the chain rule, ## \frac{dv}{dt}=(\frac{dv}{d \rho}) (\frac{d \rho}{dt}) ##. Do you know how to compute ## \frac{d v}{d \rho} ## ? With that, you should be able to process the second expression that you have, but I think you will find that it equals the minus of your first expression.
     
    Last edited: Dec 18, 2017
  4. Dec 18, 2017 #3

    berkeman

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    Staff: Mentor

    Sorry @Charles Link -- I was in the process of deleting the OP and warning for not showing enough work. But if you want to give a couple hints, that's probably okay.
     
    Last edited: Dec 18, 2017
  5. Dec 18, 2017 #4
    This makes sense - didn't think about using the product rule. I will work through it later but I think I've got it. Also my mistake with the minus sign I forgot to include it out in front of the first term. Thanks!
     
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