Fluid mechanics — Question about this vector diagram

[denniszhao]

Homework Statement

why is dF=1/2dpdA (at the bottom right of the pic)

Relevant Equations

F=pA
where F is force, p is pressure and A is area.

 

[haruspex]

You'll need to explain a few things first.
Describe what is going on the right hand diagram.
Is the left hand diagram for the same set up or a different situation?
How exactly is dF_s defined?

 

[denniszhao]

You'll need to explain a few things first.
Describe what is going on the right hand diagram.
Is the left hand diagram for the same set up or a different situation?
How exactly is dF_s defined?

Sorry for not providing more details.

 

[denniszhao]

You'll need to explain a few things first.
Describe what is going on the right hand diagram.
Is the left hand diagram for the same set up or a different situation?
How exactly is dF_s defined?

dFs is defined as the force due to the pressure on side walls. The left diagram shows the streamtube and differs from the right diagram which assumes the original pressure to be zero.

 

[haruspex]

dFs is defined as the force due to the pressure on side walls. The left diagram shows the streamtube and differs from the right diagram which assumes the original pressure to be zero.

Hmmm...
In the left hand diagram the forces on the ends are pA and (p+dp)(A+dA), giving a difference pdA+Adp.
In the right hand diagram the difference is only Adp, no?

 

[haruspex]

dFs is defined as the force due to the pressure on side walls. The left diagram shows the streamtube and differs from the right diagram which assumes the original pressure to be zero.

The author is looking along the streamtube from the narrow end. From that perspective, the side walls appear as an annulus of area dA. The author is taking the average pressure over that as dp/2, so ends up with dAdp/2 as the net force parallel to the streamline.
But I don't think that's right. If we consider that annulus as concentric thinner annuli the pressure near the thin end of the tube acts over a smaller area than at the far end. I would expect something like 2dAdp/3 as the net force. I'll try to check that.

 

[denniszhao]

The author is looking along the streamtube from the narrow end. From that perspective, the side walls appear as an annulus of area dA. The author is taking the average pressure over that as dp/2, so ends up with dAdp/2 as the net force parallel to the streamline.
But I don't think that's right. If we consider that annulus as concentric thinner annuli the pressure near the thin end of the tube acts over a smaller area than at the far end. I would expect something like 2dAdp/3 as the net force. I'll try to check that.

thanks it makes sense that the average pressure applied on the side is dp/2

 

[haruspex]

thanks it makes sense that the average pressure applied on the side is dp/2

Yes, I just checked it and 1/2 seems to be right if we assume the difference in radii is small compared with the smaller radius. But it is something that needs to be checked. If you set the small end radius to 0 then it is 2/3.

 

[denniszhao]

Yes, I just checked it and 1/2 seems to be right if we assume the difference in radii is small compared with the smaller radius. But it is something that needs to be checked. If you set the small end radius to 0 then it is 2/3.

Thank you so much! It is really helpful and I appreciate your patience and help!