Fluids: Reducing Diameter: Speed???

[justadream]

Is it just me or do the principles outlined in the two flashcards (these are from wikipremed physics) contradict each other?

The first one basically says that flow speed is related to diameter by a factor of 4. The second one says they are related by a factor of 2. So which is it?

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[Cawolf]

These cards are both accurate - the top card is speaking about velocity while the bottom card is speaking about volume flow rate.

 

[justadream]

@Cawolf

For the bottom card, it says "flow speed would go up by the square" (whereas volume flux is proportional to the fourth power)

 

[Cawolf]

Yes, that seems right. Maybe I am missing where you are seeing the error?

 

[justadream]

@Cawolf
I'm a bit confused myself but here is what I am getting at:

If I change the radius of a pipe by a factor of 2 (e.g., reduce radius from 2m to 1m), what happens to the flow speed (as in, velocity)?

The first card says that flow speed will increase by 4 times.

The second card seems to suggest that the flow speed and radius are related by a SQUARE factor (not to the fourth). I might be reading that wrong but it says "If the volume flux is proportional to the fourth power of the pipe radius, flow speed goes up by the square"

When it says "flow speed goes up by the square", I think it is comparing it to the pipe radius.

 

[Cawolf]

If you decrease the radius by 50%, the velocity will increase 4x - this is true.

It seems like it is just talking about how R^4 in the VFR formula while R^2 in the continuity equation.

It may be poorly worded - but I still don't see where the confusion lies. It seems like you know the formulas - why get caught up on the details of the card?

If there is a specific situation you can't calculate, I am happy to help.

 

[justadream]

@Cawolf

I guess I'm thinking that if you change the radius, don't you also change the volume flow rate?*

"If you decrease the radius by 50%, the velocity will increase 4x - this is true." - Isn't this assuming constant flow rate?

*Poiseulle's Law has a radius term

 

[Cawolf]

Yes and Yes.

Have you tried working it out on paper though? You will see that if you solve Pousille's equation for a change in radius by 50%, and solve them for velocity - the velocity relationship holds true.

Everything you said is correct. I don't see what you are asking.

 

[justadream]

@Cawolf

lol okay so I followed your advice and I think the math worked out (as you can see below):

Let's say Original Volume Flow Rate = 32, Original Area = 2, so initial velocity is 16 (32 /2 = 16)

If I halve the radius, volume flow rate decreases by (1/2)^4 = 1/16
Thus, new volume flow rate = 32 * (1/16) = 2 = Q2

New Cross-Sectional Area = (1/2)^2 * 2 = 1/2 = A2

New Velocity = Q2 / A2 = 4/1 = 4

But I'm a bit confused about why it works out. Doesn't one of the formulas (A1V1=A2V2 or Pouis) not assume ideal fluid flow?

 

[Cawolf]

I think you are over thinking these fluid formulas again.

Pousille's allows for the calculation of Q in a fluid with some viscosity through a circular pipe.

The continuity equation relates the velocity and area of a pipe at different spots - it seems to work for any fluid assuming that the friction/viscosity/etc. is constant.

 

[justadream]

@Cawolf

"I think you are over thinking these fluid formulas again."
lol when I do the questions, I always think I am underthinking them ("hmm...I must be missing something")

Oh, so neither of the equations requires ideal fluid flow. Is it Bernoulli's that does?

 

[Cawolf]

Oh, so neither of the equations requires ideal fluid flow. Is it Bernoulli's that does?

Yup.

 

[justadream]

@Cawolf
Upon reviewing this topic again, I am confused (again).

I have posted the front and back of the first flashcard below.

Basically, we established that if you decrease radius by a factor of 2 (cut the radius in half), then flow speed will increase by a factor of 4.

In other words, radius and flow speed are inversely related. HOWEVER, this flashcard seems to show that as you increase radius, flow speed increases. Thus, they are not inversely related?

So from all of this, here is what I gather:

IF VOLUME FLOW RATE IS CONSTANT
Halving the radius will cause the speed to be 4 times as big.

IF VOLUME FLOW RATE IS NON-CONSTANT
Increasing the radius will cause flow speed to go up by the square.*

*I'm not exactly sure what "going up by the square" means since it appears in the graph in the picture that as the radius increases, the velocity increases by decreasing amounts.

 

[justadream]

Bump!

Please help master of fluids (aka @Cawolf)

 

[Cawolf]

Yes, those are two non-comparable situations.

As you said, if VFR is constant at two points - then decreasing the radius at one point will result in a velocity increase at the other (continuity).

If we are just calculating the VFR at one point (Poiseuille's equation) then the VFR is proportial to R^4.

The graph shows velocity increasing with radius in a larger pipe.

 

[justadream]

@Cawolf
"If we are just calculating the VFR at one point (Poiseuille's equation) then the VFR is proportial to R^4."

Do you mean proportional to the R^2? The flashcard explains why it is not to the 4th power (answer is D, not A)?

 

[Cawolf]

The VFR is proportional to R^4 while the velocity is proportional to R^2.

Let's use VFR = Q and velocity = v.

v = Q/A = Q/piR^2 = R^4/R^2 = R^2

(this is not exact of course, but it shows the relationship)

 

[justadream]

@Cawolf

So doubling the radius should result in a quadrupling of the velocity in that case.

Okay so I think I can finally write the 2 laws of fluid velocities:

IF VOLUME FLOW RATE IS CONSTANT
Halving the radius will cause the speed to be 4 times as big.

IF VOLUME FLOW RATE IS NON-CONSTANT
Doubling the radius will cause flow speed to be 2^2 = 4 times as big.

 

[Czarcasm]

@Cawolf

So doubling the radius should result in a quadrupling of the velocity in that case.

Okay so I think I can finally write the 2 laws of fluid velocities:

IF VOLUME FLOW RATE IS CONSTANT
Halving the radius will cause the speed to be 4 times as big.

IF VOLUME FLOW RATE IS NON-CONSTANT
Doubling the radius will cause flow speed to be 2^2 = 4 times as big.

Sounds right. I always think of a hose. If water is flowing through the hose (constant flow rate), we can alter the velocity water shoots out by pinching the hose. Intuitively, decreasing the diameter will force water to shoot out (to maintain constant flow rate - water has to exit somewhere, right?). So in other words, some constant (flow rate) = area x velocity. Because area is ~r^2, doubling diameter will double the radius, but quadruple the area. Again to maintain flow rate, velocity would have to decrease 4x as much. In the second scenario, you decide to buy an entirely new hose (the one firefighters use). This time, the diameter is significantly bigger. As a result, when you turn the faucet on, we'll have more fluid flowing per second (increeased flow rate) and indirectly, this will also result in more fluid passing within a given area per second.

 

[Cawolf]

@Cawolf

IF VOLUME FLOW RATE IS CONSTANT
Halving the radius will cause the speed to be 4 times as big.

IF VOLUME FLOW RATE IS NON-CONSTANT
Doubling the radius will cause flow speed to be 2^2 = 4 times as big.

Yes - just the constant terminology is very specific - just be able to apply prn.