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Reynolds Transport Theorem Derivation Sign Enquiry

  1. Jul 31, 2017 #1

    Our lecturer explained us the Reynold Transport theorem, its derivation , but I don't get where the - sign in control surface 1 comes from? He said that the Area goes in opposite direction compared with this system.
    I can't visualise this on our picture.

    Can you please help me understand why we have the negative sign on the control surface 1 and at the end of the theorem we have +ve everywhere?
    The pictures are attached below
    Fig1-Illustrates the - sign enquiry with regards to control surface 1. Why is it - here? and not +

    Fig2-Illustrates the final form of the Reynolds Transport theorem where all signs are +, why?

    How to know when the Area is in same direction as the system velocity ? and how to know when the Area goes opposite direction with regards to the system velocity?

    Thank you in advance.
    Last edited: Jul 31, 2017
  2. jcsd
  3. Jul 31, 2017 #2
    ##\vec{dA}## is defined as the scalar dA multiplied by an outwardly directed unit vector from the control volume.
  4. Jul 31, 2017 #3
    Alright, thank you but why in control surface 1 is -ve and in control surface 3 is + ve? How can we sense that?

    How to know when is in same direction as the system velocity and when is opposite?

    Last edited: Jul 31, 2017
  5. Jul 31, 2017 #4
    The dot product of the outwardly directed unit vector with the velocity vector is either positive or negative. If flow is entering, then it comes out negative; if flow is leaving, then it comes out positive.
  6. Jul 31, 2017 #5
    Thank you, how come it ends up as positive(control surface1) in figure/picture 2? Because initially control surface 1 was negative but in picture 2 in final form of Reynold Transport Theorem ends up as +ve
  7. Jul 31, 2017 #6
    I don't understand the figures too well. Just remember what I said, and you will be OK.
  8. Jul 31, 2017 #7
    Alright, but why? In both cases the outwardly directed unit vector with the velocity vector are parallel. What makes it positive or negative?
  9. Jul 31, 2017 #8
    in a region where fluid flow is entering the control volume, the unit outwardly directed normal dotted with the (inwardly directed) velocity vector is negative.
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