Different versions of Falkner-Skan equation

[fahraynk]

So I have seen 3 different version of the Falkner-Skan equations and I am wondering what is the difference between them?

The first version :
$$ F'''+FF''+\beta (1-F^{'2})=0\\\\
\beta=\frac{2m}{m+1}\\\\
U_e=aX^m $$

Second version:
$$F'''+\frac{m+1}{2}FF''+m(1-F^{'2})=0\\\\
U_e=aX^m $$

Third version :
$$F'''+(m+1)FF''+m(1-F^{'2})=0\\\\
U_e=aX^m $$

If I plus the formula for beta into the first equation I get $\frac{m+1}{2}(F'''+FF'')+m(1-F^{'2})=0$
Clearly its not the same.

 

[RheologyIsFun]

Do you have the sources where you saw the different versions?

If by plus $\beta$ you mean multiple by $\frac{1}{\beta}$ and $\beta$ is now defined as $\beta=\frac{2}{m+1}$, that looks okay. As for the difference between the F-S equations you're seeing, I'd first look at what assumptions were made during the derivation.

It'd be my guess that you'd find the answer there, or I could be totally wrong.

 

[fahraynk]

Thanks for your reply!

So the second version came from this book from some MIT open course
https://learning-modules.mit.edu/service/materials/groups/166456/files/3079b99e-d885-4950-a04a-21486cb994ec/link?errorRedirect=/materials/index.html
Chapter 3 Page 41of the above link.

The first version came from Schlichting's Boundary Layer Theory 9ed pg 169
But also I found it in the pages I posted below. (here there is a moving ramp in opposite direction of freestream and also suction at wall)

The third version came from a recitation of that class whose book was above. The only difference is it is not divided by a 2.

 

[boneh3ad]

The difference is the choice of similarity variable. The behavior of all of the equations is effectively the same but the variables have been normalized slightly differently. It's similar to how you can find two versions of the Blasius equation.