# Speed/Velocity and Volume Flow Rate of Viscous Fluids

• Ophi-Siren-Kit
In summary, the conversion from centimeters to meters led to a speed/velocity of 0.031415927 for section 1, and 2.005352283 for section 2. The continuity equation was used to solve for pressure difference in question 4, and the results were 3001.764705882353 Pa.
Ophi-Siren-Kit
Homework Statement
A viscous fluid with viscosity η = 1.7 × 10^-3 Pa s is flowing horizontally in the pipe shown below. Section 1, 2, and 3 have radii of r1 = 10.0 cm, r2 = 8.0 cm, and r3 = 6.0 cm, respectively. All sections have the same length L = 8.1 m. The volume flow rate of the fluid in Section 1 is F1 = 0.063 m^3/s. The pressure differences across Section 2 and Section 3 are both 41 PA because these two split sections are parallel, just as in the DC circuit.

{I am given a diagram with a single pipe splitting into two pipes, flow arrows all pointed to the right, with the L sections being the flat horizontal areas, L sections not including the distance.}

(a) What is the speed of the fluid in Section 1?
(b) What is the volume flow rate in Section 2?
(c) What is the speed of the fluid in Section 2?
(d) What is the speed of the fluid in Section 3?
Relevant Equations
Volume Flow Rate(VFR) = Cross-sectional Area multiplied by Velocity (A*v)
[VFR=Av]

and/or

Volume Flow Rate = viscosity (η) times velocity (v) times cross-sectional area (A) all divided by length (L)
[VFR=ηvA/L]

Continuity Equation: A1V1=A2V2
where one area times velocity equaled another area times velocity
In my first attempt, I started off converting the radii of all three sections from centimeters (10, 8, 6) to meters (0.10 , 0.08 , 0.06), then used the VFR=Av formula to find the speed/velocity of section one.

VFR== 0.063 m^3/s
A== pi*r^2=pi*(10cm)^2=pi*(0.10)^2=pi*0.01 == 0.031415927
VFR/A=v ==2.005352283...

Where my attempts began to change was for the second (b) question. I originally went for just plugging the velocity found in question one with a different area, then moving on to question b -- but then trying to reverse-engineer to get question c, I just got the velocity I borrowed earlier.

VFR=(pi*(0.08)^2)v
v=2.005352283...
VFR=0.04032
{then reversed for the same thing due to using the same velocity as question a}

This pinged as a mental red flag for me, and so I went to the internet to search. While searching for possible other ways to solve for this, I was reminded of the continuity equation. So, using that to answer the second question by saying the volume flow rate for section 1 was equal to the VFR for section 2, I then moved to question 3 by saying the following:

VFR == 0.063 m^3/s
A==(pi*0.08^2)=(pi*0.0064)=0.020106193...
VFR/A=v == 3.133362942121689...

I tried, after getting both attempts at a third question (c) answer, to figure out if I needed continuity or something else for question 4, but I'm honestly stuck.

I tried to get a tutor to help me confirm if I did this right, and they gave me the second VFR equation to redo the first question with. The velocity number for the redo was 3001.764705882353, and with that being such a large number, I felt like it couldn't be correct.

I'm not sure if I was ever on the right track, or if all of these attempts are wrong. Please help.

Can you post an image of the diagram? That would help.
If VFR = Av, you can't then say VFR = ηvA/L. That would require viscosity to have the same dimensions as length, which it doesn't. (It also implies that flow rate is greater the higher the viscosity, which is counter-intuitive. Look up the Poiseuille equation.)
You can't assume that either the velocity or the VFR are the same between sections 1 and 2. What is true is that the total VFR for sections 2 and 3 is the same as that for section 1.

mjc123 said:
Can you post an image of the diagram? That would help.
If VFR = Av, you can't then say VFR = ηvA/L. That would require viscosity to have the same dimensions as length, which it doesn't. (It also implies that flow rate is greater the higher the viscosity, which is counter-intuitive. Look up the Poiseuille equation.)
You can't assume that either the velocity or the VFR are the same between sections 1 and 2. What is true is that the total VFR for sections 2 and 3 is the same as that for section 1.

Here's the picture!

Did you look up the Hagen-Poiseuille equation yet?

Chestermiller said:
Did you look up the Hagen-Poiseuille equation yet?
I know of the Poiseuille equation, but I don't know if I'm supposed to use it here. I had a fifth question included in the problem where I used the equation to find pressure difference. I didn't include it because I figured out how to do it on my own.

Ophi-Siren-Kit said:
I know of the Poiseuille equation, but I don't know if I'm supposed to use it here. I had a fifth question included in the problem where I used the equation to find pressure difference. I didn't include it because I figured out how to do it on my own.
Oh really? I'd be interested in seeing your solution.

## 1. What is the difference between speed and velocity of a viscous fluid?

The speed of a viscous fluid refers to the rate at which the fluid is moving, while velocity takes into account the direction of the fluid's movement. In other words, speed is a scalar quantity while velocity is a vector quantity.

## 2. How is the speed/velocity of a viscous fluid measured?

The speed/velocity of a viscous fluid can be measured using various techniques such as flow meters, pitot tubes, and laser Doppler anemometry. These methods involve measuring the flow rate of the fluid at a specific point or along a specific path.

## 3. What factors affect the speed/velocity of a viscous fluid?

The speed/velocity of a viscous fluid is affected by several factors including the viscosity of the fluid, the pressure gradient, the geometry of the flow, and the surface roughness of the boundaries. These factors can either increase or decrease the speed/velocity of the fluid.

## 4. How does the volume flow rate of a viscous fluid relate to its speed/velocity?

The volume flow rate of a viscous fluid is directly proportional to its speed/velocity. This means that as the speed/velocity of the fluid increases, the volume flow rate also increases. This relationship is described by the continuity equation, which states that the volume flow rate is equal to the product of the cross-sectional area and the speed/velocity of the fluid.

## 5. Can the speed/velocity and volume flow rate of a viscous fluid be controlled?

Yes, the speed/velocity and volume flow rate of a viscous fluid can be controlled through various methods such as changing the viscosity of the fluid, adjusting the pressure gradient, and altering the geometry of the flow. These techniques are often used in industrial processes to optimize the flow of viscous fluids.

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