Correct way to write pi buckingham theorem

[foo9008]

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 ?

 

[foo9008]

wow , thi is considered as advanced physics question ?

 

[MathematicalPhysicist]

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

 

[foo9008]

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]

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?

 

[foo9008]

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?

 

[foo9008]

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]

Do you know what is f here?

 

[foo9008]

Do you know what is f here?

Ff means function, where pi1 is the function of pi2 and pi3...

 

[MathematicalPhysicist]

I mean is f given explicitly?

 

[foo9008]

I mean is f given explicitly?

? What do you mean?

 

[MathematicalPhysicist]

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?

 

[foo9008]

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]

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.