# Correct way to write pi buckingham theorem

## 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 ?

## 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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## Answers and Replies

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wow , thi is considered as advanced physics question ?

MathematicalPhysicist
Gold Member
I don't see any difference between taking the reciprocal of Reynold's number and taking actually Reynold's number as $\pi_1$.

I don't see any difference between taking the reciprocal of Reynold's number and taking actually Reynold's number as $\pi_1$.
Why?

MathematicalPhysicist
Gold Member
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?

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 ?

MathematicalPhysicist
Gold Member
Do you know what is f here?

Do you know what is f here?
Ff means function, where pi1 is the function of pi2 and pi3...

MathematicalPhysicist
Gold Member
I mean is f given explicitly?

I mean is f given explicitly?
? What do you mean?

MathematicalPhysicist
Gold Member
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?

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

MathematicalPhysicist
Gold Member
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.