Scaling Interpretation for 2-D Continuity PDE: What Does UH/L Represent?

  • Context: Graduate 
  • Thread starter Thread starter member 428835
  • Start date Start date
  • Tags Tags
    Interpretation Scaling
Click For Summary

Discussion Overview

The discussion revolves around the scaling interpretation of a 2-D continuity partial differential equation (PDE), specifically examining the physical implications of the relationship between velocities in a flowing channel. Participants explore the significance of the scaling factors related to channel dimensions and mainstream velocity.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant seeks to understand the physical meaning of the expression ##U H/L \sim v##, questioning whether it implies that vertical velocity is maximized at this value.
  • Another participant suggests that the expression indicates the typical y velocity is smaller than the x velocity by a factor on the order of H/L.
  • There is a request for clarification on the term "typical," with a suggestion that y velocities are on the order of H/L times smaller than x velocities, acknowledging the difficulty in precision.
  • A participant mentions the use of similar scaling techniques in momentum equations and provides an example related to force balance in a jumping scenario, indicating that this method can simplify calculations by eliminating less significant terms.
  • One participant references their learning experience and a specific paper related to dimensional analysis, indicating a divergence in approaches or understanding of the technique.

Areas of Agreement / Disagreement

Participants express differing interpretations of the scaling implications, particularly regarding the term "typical" and the significance of the scaling factors. The discussion remains unresolved with multiple competing views on the interpretation of the scaling in the context of the PDE.

Contextual Notes

There are limitations in precision regarding the definitions of velocity scaling and the assumptions underlying the use of dimensional analysis in this context. The discussion does not resolve the mathematical steps or the implications of the scaling factors.

member 428835
Hi PF!

I'm doing some scaling over a PDE and I understand the math side of things but I do not understand the physical side of what we are finding.

For example, suppose we have some PDE, say 2-D continuity for it's simplicity ##u_x + v_y = 0##. Let ##L## be the length of a side of a flowing channel and the height of the channel be ##H##. Now if ##x## scales as ##L## and ##y## scales as ##H## and if the mainstream velocity coming into the channel is ##U## then we may write ##U H/L \sim v##. What is actually being said here? That the vertical velocity is maximized as ##U H/L##? Please help!

Thanks a ton!
 
Physics news on Phys.org
It seems to be saying that the typical y velocity will be smaller than the x velocity by a factor on the order of H/L.

Chet
 
What do you mean by "typical"?

They also use this technique with a time derivative as well, like in the momentum equation. My professor has said that this technique can save a lot of math, like if you are are doing a force balance for someone jumping out of an airplane, and you are concerned with initial velocity, this technique can eliminate "less important" terms.
 
joshmccraney said:
What do you mean by "typical"?
Maybe typical was a poor choice of term. Maybe it would have been better to say that the y velocities will be on the order of H/L times smaller than the x velocities. It is difficult to be more precise with something like this.
They also use this technique with a time derivative as well, like in the momentum equation. My professor has said that this technique can save a lot of math, like if you are are doing a force balance for someone jumping out of an airplane, and you are concerned with initial velocity, this technique can eliminate "less important" terms.
I don't follow what you are saying here. The way I learned dimensional analysis was taught to me by S. W. Churchill at the University of Michigan in 1963. See the famous paper by Hellums and Churchill in AIChE Journal (1964)

Chet
 
  • Like
Likes   Reactions: member 428835
Thanks, I'll look into it! You're awesome Chet!
 

Similar threads

Graduate 't Hooft cellular automaton interpretation
  • · Replies 12 ·
Replies
12
Views
5K
Chemistry What is "gauge pressure" and how does a manometer work?
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Renormalizability conditions for a real scalar field in d dimensions
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Recover Hamilton equation from 2-form defined on phase space
  • · Replies 3 ·
Replies
3
Views
2K
What Is the Period of Oscillations in a U-Tube with Varying Radii?
  • · Replies 12 ·
Replies
12
Views
7K
Calibrating a rain gauge (problem interpreting question)
  • · Replies 8 ·
Replies
8
Views
2K
Challenge Intermediate Math Challenge - September 2018
  • · Replies 28 ·
Replies
28
Views
7K
Challenge Math Challenge - August 2021
  • · Replies 46 ·
2
Replies
46
Views
9K
Undergrad How specifically does an accelerated uniform rod Lorentz contract?
  • · Replies 55 ·
2
Replies
55
Views
5K
Challenge Math Challenge - November 2018
  • · Replies 175 ·
6
Replies
175
Views
28K