Reynolds Transport Theorem Derivation Sign Enquiry

In summary: Conversely, if fluid flow is leaving the control volume, the unit outwardly directed normal dotted with the (inwardly directed) velocity vector is positive.
  • #1
williamcarter
153
4
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
Capture.PNG

Fig1-Illustrates the - sign enquiry with regards to control surface 1. Why is it - here? and not +

Capture2.PNG

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.
 
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  • #2
##\vec{dA}## is defined as the scalar dA multiplied by an outwardly directed unit vector from the control volume.
 
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  • #3
Chestermiller said:
##\vec{dA}## is defined as the scalar dA multiplied by an outwardly directed unit vector from the control volume.
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
 
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  • #4
williamcarter said:
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
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.
 
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  • #5
Chestermiller said:
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.
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
williamcarter said:
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
I don't understand the figures too well. Just remember what I said, and you will be OK.
 
  • #7
Chestermiller said:
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.
Alright, but why? In both cases the outwardly directed unit vector with the velocity vector are parallel. What makes it positive or negative?
 
  • #8
williamcarter said:
Alright, but why? In both cases the outwardly directed unit vector with the velocity vector are parallel. What makes it positive or negative?
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.
 

What is the Reynolds Transport Theorem?

The Reynolds Transport Theorem is a fundamental concept in fluid mechanics that describes the change in a physical quantity within a control volume over time. It is commonly used to analyze the behavior of fluid flow systems.

How is the Reynolds Transport Theorem derived?

The Reynolds Transport Theorem is derived from the basic principles of conservation of mass and momentum. By applying these principles to a control volume, the theorem allows us to describe the change in a physical quantity within that volume over time.

What is the significance of the sign in the Reynolds Transport Theorem?

The sign in the Reynolds Transport Theorem represents the direction of the physical quantity being studied. It can be positive or negative, depending on whether the quantity is entering or leaving the control volume.

How is the Reynolds Transport Theorem used in practical applications?

The Reynolds Transport Theorem is used in a wide range of practical applications, such as in the design of pumps and turbines, the analysis of air and water flow in pipes and channels, and the study of ocean currents. It is also used in the development of mathematical models for predicting and controlling fluid flow systems.

What are some limitations of the Reynolds Transport Theorem?

One limitation of the Reynolds Transport Theorem is that it assumes the fluid being studied is continuous and homogeneous, and that the control volume is fixed. In reality, many fluids are not continuous and the control volume may be changing over time. Additionally, the theorem does not take into account turbulent flow, which can significantly affect the behavior of fluid systems.

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