Speed/Velocity and Volume Flow Rate of Viscous Fluids

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SUMMARY

The discussion centers on calculating the speed and volume flow rate (VFR) of viscous fluids using the formula VFR = Av, where A is the cross-sectional area. The user initially calculated the VFR for three sections with radii of 10 cm, 8 cm, and 6 cm, converting them to meters for accurate computation. Key equations referenced include the continuity equation and the Hagen-Poiseuille equation, which are essential for understanding fluid dynamics in varying cross-sections. The user expressed confusion regarding the application of viscosity in flow rate calculations and sought clarification on their methodology.

PREREQUISITES
  • Understanding of fluid dynamics principles, specifically the continuity equation.
  • Familiarity with the Hagen-Poiseuille equation for viscous flow.
  • Basic knowledge of calculus for area and volume calculations.
  • Ability to convert units, particularly from centimeters to meters.
NEXT STEPS
  • Study the Hagen-Poiseuille equation to understand its application in viscous fluid flow.
  • Learn about the continuity equation and its implications in fluid dynamics.
  • Explore the relationship between viscosity and flow rate in different fluid scenarios.
  • Practice calculating volume flow rates and velocities for various cross-sectional areas.
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Students and professionals in engineering, physics, and fluid mechanics who are working with viscous fluids and need to understand flow rate calculations and the impact of viscosity on fluid dynamics.

Ophi-Siren-Kit
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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.
 
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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.
1635892817772.png

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.
 

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