Why is ro = 0.2255 in fluid dynamics vortices?

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Homework Help Overview

The discussion revolves around a fluid dynamics problem involving a closed cylindrical tank containing water and its behavior when rotated at a constant angular velocity. The specific focus is on understanding the value of ro = 0.2255 as referenced in a professor's handwritten notes and its relation to the calculations performed by the original poster.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants are attempting to reconcile their calculations with a provided answer, questioning the accuracy of their own results compared to the professor's. There is a focus on understanding the variables involved, particularly the interpretation of ζ and its relation to angular velocity.

Discussion Status

The discussion is ongoing, with participants seeking clarification on the calculations and the definitions of variables. Some guidance has been offered regarding the interpretation of the formula, but no consensus has been reached on the correct approach or understanding of the problem.

Contextual Notes

Participants are working within the constraints of a homework assignment, which may limit the information available for discussion. There is an emphasis on understanding the definitions and implications of the variables used in the problem.

yecko
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Homework Statement
A closed cylindrical tank of 4 m in height and 2 m in diameter contains water to a depth of 3 m. When the cylinder is rotated at a constant angular velocity of 40 rad/s, show that nearly 0.71 m2 of the bottom surface of the tank is uncovered.
Relevant Equations
pressure surface, Bernoulli's equation
1607059199222.png

why is ro = 0.2255?
1607059272299.png


Thank you
 
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yecko said:
Homework Statement:: A closed cylindrical tank of 4 m in height and 2 m in diameter contains water to a depth of 3 m. When the cylinder is rotated at a constant angular velocity of 40 rad/s, show that nearly 0.71 m2 of the bottom surface of the tank is uncovered.
Relevant Equations:: pressure surface, Bernoulli's equation

View attachment 273668
why is ro = 0.2255?
View attachment 273669

Thank you
I would have assumed the handwritten calculation was yours, but your question implies it is someone else's. Please clarify.
 
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Sorry for being unclear.
The handwritten one is the answer written by prof. And the answer is same as final model answer in textbook.
However, when I substitute the number in the red bracket, I can only obtain 0.1519 from calculator instead of prof‘s answer 0.2255.
Have I done anything wrong? Thank you.
 
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yecko said:
Sorry for being unclear.
The handwritten one is the answer written by prof. And the answer is same as final model answer in textbook.
However, when I substitute the number in the red bracket, I can only obtain 0.1519 from calculator instead of prof‘s answer 0.2255.
Have I done anything wrong? Thank you.
OK, but you'll have to decipher the scrawl for me. It looks like ##\frac{64g}{\zeta^2}+4r_0^2=1##. If so, what is ##\zeta##?
 
1607081680932.png

From Kundu, by comparing the formula "Surfaces of constant pressure are given by ...", ζ written by Prof should be ω here, which implicates angular velocity. However, when I substite 40 rad/s, the answer is wrong. Am I having any misunderstanding with ζ?
 
for reference, this question is utilising the approach of "constant pressure surface"
1607082158657.png
 

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