The Pressure Distribution in a Narrowing Tube with Increasing Fluid Velocity

[Johny Prime]

Homework Statement

Question:
For a fluid flowing within a horizontal tube, if the diameter of the tube narrows, will the fluid's flow rate increase, resulting in increased pressure in the narrowing and explain your reasoning.

Homework Equations

Volume flow rate = volume/time
Volume = Area x Distance
Velocity = Distance/ time

The Attempt at a Solution

Sub in Volume = A x D
So, Flow rate = (Area x distance)/time
simplifies down to Flow rate= Area X Velocity

As the volume flow rate is constant, (A1 xV1 = A2 x V2)
if Area gets smaller (the tube narrowing) then Velocity must increase (water goes faster)

What I do not understand, is how does this affect the pressure? Is it to do with Kinetic energy?
Could someone please explain why it results in higher pressure?
Thanks

 

[haruspex]

What I do not understand, is how does this affect the pressure?

Think about a parcel of the fluid entering the narrow section. If it is to gain speed, how is that to happen, according to Newton?

 

[Johny Prime]

If it is to gain speed, how is that to happen, according to Newton?

I'm not quite sure what you mean, is it because the fluid is going faster and exerting more force on the inside walls of the tube?

 

[haruspex]

I'm not quite sure what you mean, is it because the fluid is going faster and exerting more force on the inside walls of the tube?

What, according to Newton, makes a mass move faster, or change velocity at all?

 

[Johny Prime]

What, according to Newton, makes a mass move faster, or change velocity at all?

A force, right? So that means that there's more force acting on the same area and that would increase the pressure since Pressure = Force/ Area?
Would that be correct? I think I'm seeing the connection now.

 

[haruspex]

A force, right? So that means that there's more force acting on the same area and that would increase the pressure since Pressure = Force/ Area?
Would that be correct? I think I'm seeing the connection now.

Getting there, but draw yourself a free body diagram for the parcel of water. It is in the middle of the flow, say, in the narrowing part of the tube, so it is accelerating. Which way is the net force acting?

 

[Johny Prime]

The net force would be facing the way that the fluid is flowing, right?

 

[Johny Prime]

Getting there, but draw yourself a free body diagram for the parcel of water. It is in the middle of the flow, say, in the narrowing part of the tube, so it is accelerating. Which way is the net force acting?

The net force would be facing the way that the fluid is flowing, right? The fluid is moving in that direction right?

 

[haruspex]

The net force would be facing the way that the fluid is flowing, right? The fluid is moving in that direction right?

For it to be getting faster, yes. What does this tell you about pressures?

 

[Johny Prime]

For it to be getting faster, yes. What does this tell you about pressures?

The pressure would have to increase, right? If it's going faster, then there's Acceleration so according to the formula: Force = Mass X Acceleration, Acceleration is increasing and that would mean that Force is increasing, which would result in Pressure going up via the formula Pressure = Force/ Area. Is that sound enough explanation to answer the question fully?

 

[haruspex]

The pressure would have to increase, right? If it's going faster, then there's Acceleration so according to the formula: Force = Mass X Acceleration, Acceleration is increasing and that would mean that Force is increasing, which would result in Pressure going up via the formula Pressure = Force/ Area. Is that sound enough explanation to answer the question fully?

No. You drew a free body diagram of the parcel of water, yes? You can draw it as a cube. Say the tube it is in is narrower to the right and it is accelerating to the right. Where on the cube is the greatest pressure?