Correct way to write pi buckingham theorem

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Homework Help Overview

The discussion revolves around the application of the Buckingham Pi theorem in fluid dynamics, specifically regarding the formulation of dimensionless groups. The original poster questions the assignment of specific dimensionless groups, particularly the reciprocal of the Reynolds number and its relationship to other variables in the context of the theorem.

Discussion Character

  • Conceptual clarification, Assumption checking, Mixed

Approaches and Questions Raised

  • Participants explore the implications of defining different dimensionless groups, questioning whether the reciprocal of the Reynolds number can be treated equivalently to the Reynolds number itself. There is discussion about the function relationships between these groups and the variables involved.

Discussion Status

The conversation is ongoing, with participants raising questions about the definitions and relationships of the functions involved. There is no clear consensus, as various interpretations of the problem are being explored, and some participants are seeking clarification on the definitions of the functions.

Contextual Notes

Participants note that the rearrangement of variables may be for convenience, and the necessity of knowing the explicit form of the function relating the dimensionless groups is under discussion. There is an acknowledgment of the complexity involved in defining these relationships accurately.

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Homework Statement


in this problem , the author make π1 = D(dp/ dx) / ρ( V^2) , and make π3 as μ/ ρVD , how if i want to make μ/ ρVD (reciprocal of reynold number ) as π1 and make D(dp/ dx) / ρ( V^2) as π3 ?

Homework Equations

The Attempt at a Solution


since we know that π1 is function of ( π2 , π3 )
is it necessary to change μ/ ρVD (reciprocal of reynold number ) to reynold number (ρVD / μ ) ?
which is correct ? Re = f ( D(dp/ dx) / ρ( V^2) , Ks/ D ) or μ/ ρVD = f ( D(dp/ dx) / ρ( V^2) , Ks/ D ) ? which is correct ?
 

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wow , thi is considered as advanced physics question ?
 
I don't see any difference between taking the reciprocal of Reynold's number and taking actually Reynold's number as ##\pi_1##.
 
MathematicalPhysicist said:
I don't see any difference between taking the reciprocal of Reynold's number and taking actually Reynold's number as ##\pi_1##.
Why?
 
If you take ##\pi_1 =f (\pi_2 , \pi_3)## for some function ##f## of ##\pi_2 , \pi_3## in which case you can find a function ##g## such that ##\pi_1^{-1} = g(\pi_2 , \pi_3)##; so it doesn't matter which one you choose your ##\pi_1## to be, your function will of course be different for different cases, but you don't seem to know what is your function ##f##, right?

You didn't state what is your precise problem here?
 
MathematicalPhysicist said:
If you take ##\pi_1 =f (\pi_2 , \pi_3)## for some function ##f## of ##\pi_2 , \pi_3## in which case you can find a function ##g## such that ##\pi_1^{-1} = g(\pi_2 , \pi_3)##; so it doesn't matter which one you choose your ##\pi_1## to be, your
function will of course be different for different cases, but you don't seem to know what is your function ##f##, right?
It's pi Buckingham theorem, can I still do so?
You didn't state what is your precise problem here?
 
Re = f ( D(dp/ dx) / ρ( V^2) , Ks/ D ) or μ/ ρVD which is 1 / Re = f ( D(dp/ dx) / ρ( V^2) , Ks/ D ) ? which is correct ?
 
Do you know what is f here?
 
MathematicalPhysicist said:
Do you know what is f here?
Ff means function, where pi1 is the function of pi2 and pi3...
 
  • #10
I mean is f given explicitly?
 
  • #11
MathematicalPhysicist said:
I mean is f given explicitly?
? What do you mean?
 
  • #12
I mean do you know how is f given? I mean do you know what is f(\pi_2, \pi_3) what is this function of \pi_2 and \pi_3?
 
  • #13
MathematicalPhysicist said:
I mean do you know how is f given? I mean do you know what is f(\pi_2, \pi_3) what is this function of \pi_2 and \pi_3?
Dun know
 
  • #14
As it mentioned in one of the pics you rearrange only for convenience, i.e. it doesn't matter if you take Reynold's number or the reciprocal of Reynold's number as a function of the other dimensionless variables since you can always take the reciprocal of the function if you have ##Re = f(\pi_2 , \pi_3)## then you can take ##1/Re = 1/f = g(\pi_2,\pi_3)##.

If on the other hand ##f## were given then you'd know how to rearrange the equation.
 
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