What is the flow rate through a tapered tube?

[zeralda21]

Homework Statement

Water flows through a tapered tube. At one end of the tube, where the diameter is 20 mm, water flows in at a rate of 1.0 liter / s. At the other end, where the water flows out, the diameter is 15 mm. How large is the flow at that end?

Homework Equations

No equations needed.

The Attempt at a Solution

What I am trying to understand in this question is if the flow rate is dependent on the cross-sectional area or not. With these kind of questions I always find it interesting to study extreme values.

If I imagine a tube where one side has a very large cross-sectional area at one end while at the other end, the cross-sectional area is infinitesimal. Will the flow rate be the same? It sounds likely tough the pressure(F/A) will increase due to a lower cross-sectional area.

 

[rcgldr]

Mass flow is constant everywhere in the tube (otherwise mass would be accumulating), and you can assume that water is incompressable, which would mean volume flow is also constant.

 

[zeralda21]

Sounds OK to me. In the imagination above, is it correct that the pressure will vary if the cross-sectional area throughout the tube varies?

 

[rcgldr]

Sounds OK to me. In the imagination above, is it correct that the pressure will vary if the cross-sectional area throughout the tube varies?

Yes, but the problem only asks for the velocity of the flow. If the volume of flow is constant, then what happens to the velocity of the flow if the cross-sectional area changes?

 

[zeralda21]

Yes, but the problem only asks for the velocity of the flow. If the volume of flow is constant, then what happens to the velocity of the flow if the cross-sectional area changes?

I believe that if the cross-sectional area increases then the velocity of the flow will reduce. And if the cross-sectional area decreases then the velocity of the flow will increase, due to a higher pressure. That is what my intuition says. Is that correct?

 

[e^(i Pi)+1=0]

A narrower cross section will result in a higher velocity and a lower pressure. This is because the molecules of the fluid have lass time to impart force against the pipe as they flow past.

 

[rcgldr]

I believe that if the cross-sectional area increases then the velocity of the flow will reduce. And if the cross-sectional area decreases then the velocity of the flow will increase, due to a higher pressure.

Fluids accelerate from higher pressure zones to lower pressure zones. Assuming no external forces are involved (including friction from the pipe), then acceleration of the water is due to pressure differentials within the water. If the water accelerates from a lower velocity to a higher velocity, then happens with the pressure during the transition (during acceleration)?