Non-dimensionalizing the N-S equations

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In non-dimensionalizing the N-S equations, are the choices of the characteristic variable important? I am currently reading the Stokes approximation to the N-S equation and find two versions of Stokes equations. They have used different scaling in their derivation and I find no clue in understanding the importance of scaling and am wondering if someone here can help me out.

Please kindly read the figure attached.

While the version 1 has characterise the time by U and L, version2 uses the kinemtic viscosity to scale the time. Are there any difference between these two approaches? I don't know what kind of difference will be produced and what effect would different choice of scales bring out. Could anyone explain clearly?

And are the two versions of Stokes equation the same? If not, what are the differences?

I find no clue in non-dimensionalization....
 

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In non-dimensionalizing the N-S equations, are the choices of the characteristic variable important? I am currently reading the Stokes approximation to the N-S equation and find two versions of Stokes equations. They have used different scaling in their derivation and I find no clue in understanding the importance of scaling and am wondering if someone here can help me out.

Please kindly read the figure attached.

While the version 1 has characterise the time by U and L, version2 uses the kinemtic viscosity to scale the time. Are there any difference between these two approaches? I don't know what kind of difference will be produced and what effect would different choice of scales bring out. Could anyone explain clearly?

And are the two versions of Stokes equation the same? If not, what are the differences?

I find no clue in non-dimensionalization....
The only real difference between these two approaches is in the definition of the dimensionless time. The second approach looks more appropriate for considering the transient case.
 

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