# Reynolds Transport Theorem Derivation Sign Enquiry

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1. Jul 31, 2017

### williamcarter

Hi,

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?

Last edited: Jul 31, 2017
2. Jul 31, 2017

### Staff: Mentor

$\vec{dA}$ is defined as the scalar dA multiplied by an outwardly directed unit vector from the control volume.

3. Jul 31, 2017

### williamcarter

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?

Thanks

Last edited: Jul 31, 2017
4. Jul 31, 2017

### Staff: Mentor

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.

5. Jul 31, 2017

### williamcarter

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

6. Jul 31, 2017

### Staff: Mentor

I don't understand the figures too well. Just remember what I said, and you will be OK.

7. Jul 31, 2017

### williamcarter

Alright, but why? In both cases the outwardly directed unit vector with the velocity vector are parallel. What makes it positive or negative?

8. Jul 31, 2017

### Staff: Mentor

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.