Why the differential sequence switch at the first term of viscous shearing of N-S eq

TommyWind · November 13, 2020, 03:19

Hello Everyone, I am a student to look for reasonable explanation to answer the doubt in my mind. Thank you all!

In the attached picture of Curric's text book, he didn't explain why he just switch the differential sequence partial Xi and partial Xj to get the first term of viscous shear term eliminated again by incompressible in order to have the neat Laplacian term to solve the problem with better chance.

I know there is Fubuni's theorem in triple integral.
I know continuum (continuity) is the base assumption of fluid dynamics.

However, I am wondering if I can always do this kind of switch in fluid dynamics????????

Can anyone help? Thanks again!

TommyWind

Gerry Kan · November 13, 2020, 04:51

Dear TommyWind:

This is a subtle point in using tensor notation. You can switch tensor indices in any way deemed convenient if it still retains the summation that it is meant to represent.

As an example: While I was deriving some term in a turbulence model, I ended up getting a term which looked like this

$\left(\frac{\partial{u'_i}}{\partial{x_j}}\right)^2 + \left(\frac{\partial{u'_j}}{\partial{x_i}}\right)^2$

Each of these two terms expand to the same expression (I will let you do this if you need some convincing). So you can simply collect them by switching the

and

indices in the second term:

$2\left(\frac{\partial{u'_i}}{\partial{x_j}}\right)^2$

Hope that helps, Gerry.

FMDenaro · November 13, 2020, 05:51

Quote:

Originally Posted by TommyWind

Hello Everyone, I am a student to look for reasonable explanation to answer the doubt in my mind. Thank you all!

In the attached picture of Curric's text book, he didn't explain why he just switch the differential sequence partial Xi and partial Xj to get the first term of viscous shear term eliminated again by incompressible in order to have the neat Laplacian term to solve the problem with better chance.

I know there is Fubuni's theorem in triple integral.
I know continuum (continuity) is the base assumption of fluid dynamics.

However, I am wondering if I can always do this kind of switch in fluid dynamics????????

Can anyone help? Thanks again!

TommyWind

In the incompressible flow model, the continuity equation is simply dui/dxi=0. Use this divergence-free constraint in rewriting the term Div(2mu*S) where S is the symmetric part of the velocity gradient with zero trace.

sbaffini · November 13, 2020, 07:03

Not sure who this Curric is, he surely is a good guy but, I must admit that this is one of the most appalling presentations of this matter I have ever seen so far.

For your mental sanity, I suggest you to switch to some more common reference.

TommyWind · November 13, 2020, 07:09

Dear Gerry,

Thanks a lot for your explanation.

Will it explain your example when the matrix (of two tensors) is symmetry?

Quote

You can switch tensor indices in any way you deemed convenient if "it still retains the summation" that it is meant to represent.

End of Quote

The key point is you have evidence or proof that it still retains the summation.

That means we have to prove sock first and shoe second is equal to shoe first and sock second in this case.

In your example symmetry works, but not for my question. Otherwise, they are all eliminated by incompressible.

The changing of partial differential sequences are always some physics meaning behind in fluid.

But, I think I just found the reason when I go back to the definition of τ_ij.

τ_ij = μ ( ∇V_i + ∇V_i^t ) therefore symmetry

τ_12 = ∂/∂x(μ(∂u/∂y+∂v/∂x)) = τ_21 = ∂/∂y(μ(∂v/∂x+∂u/∂y))

Should I make any mislead or mistake, please let me know.

P.S. I will study my turbulence flow next semester.

By the way, what kind math tool do you use here?

Thank you Gerry!

All the best,

TommyWind

Gerry Kan · November 13, 2020, 07:56

Dear TommyWind:

To be honest I have not thought about symmetry, but this is an interesting point. In the Curric case, there is a switch in indices in the differential, that is:

$\frac{\partial{u_i}}{\partial{x_i}\partial{x_j}} = \frac{\partial{u_i}}{\partial{x_j}\partial{x_i}}$

I believe your question hinges on this. If this is not correct please let me know.

Going back to the above term, this is effectively saying (in 2D) for a fixed

, say $x_j \equiv y$ (to be in line with the Curric convention, where each equation is fixed at

instead of

):

$\frac{\partial{u}}{\partial{x}\partial{y}}+\frac{\partial{v}}{\partial{y}\partial{y}} = \frac{\partial{u}}{\partial{y}\partial{x}}+\frac{\partial{v}}{\partial{y}\partial{y}}$

The only thing that "symmetry" would be brought in question is whether

$\frac{\partial{u}}{\partial{x}\partial{y}} = \frac{\partial{u}}{\partial{y}\partial{x}}$

There is where I am not 100% certain. I believe there are rare special cases, from a mathematical point of view, where this might not be the case. However, for most naturally occurring phenomena the differential order can be regarded as interchangeable.

I am not sure if this answers your question, though. But please keep in mind that index notation is a shorthand, and is not an implicit indicator of tensor symmetry.

Gerry.

P.S. - You can enter Latex commands directly in the forum posts using the math formatting tags. For instance:

Code:

{math}\frac{\partial{u_i}}{\partial{x_i}\partial{x_j}}{/math}

NOTE - replace the braces around math and /math with square brackets

will give you

$\frac{\partial{u_i}}{\partial{x_i}\partial{x_j}}$

TommyWind · November 13, 2020, 09:41

Dear Gerry,

You are right!
That is exactly my question.
You did answer it.

And I am glad to learn from you that most naturally occurring phenomena the differential order can be regarded as interchangeable.

The problem of me is I need to know exactly the physics meaning behind the math. Otherwise, I can't really remember it.

This is a very nice forum to know you and other people.
And I finally know I need to learn Latex without second choice.

I will keep studying what you and FMDenaro told me about tensor notation.
I slightly touched the 4th order tensor when I studied the viscous shear stresses in a finite volume.

Thank you Gerry!

All the best,

TommyWind

FMDenaro · November 13, 2020, 13:16

What I see, is a topic of the differential calculus, the mix derivative do commute under the

https://en.wikipedia.org/wiki/Symmet...nd_derivatives

This is a classical topic and has nothing to do with the physics of the NSE.

Just consider the regularity hypotheses.

November 13, 2020, 04:51		#2
Gerry Kan Senior Member Gerry Kan Join Date: May 2016 Posts: 373 Rep Power: 11	Dear TommyWind: This is a subtle point in using tensor notation. You can switch tensor indices in any way deemed convenient if it still retains the summation that it is meant to represent. As an example: While I was deriving some term in a turbulence model, I ended up getting a term which looked like this $\left(\frac{\partial{u'_i}}{\partial{x_j}}\right)^2 + \left(\frac{\partial{u'_j}}{\partial{x_i}}\right)^2$ Each of these two terms expand to the same expression (I will let you do this if you need some convincing). So you can simply collect them by switching the and indices in the second term: $2\left(\frac{\partial{u'_i}}{\partial{x_j}}\right)^2$ Hope that helps, Gerry. aero_head and TommyWind like this. Last edited by Gerry Kan; November 13, 2020 at 06:03.

November 13, 2020, 07:03		#4
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,192 Blog Entries: 29 Rep Power: 39	Not sure who this Curric is, he surely is a good guy but, I must admit that this is one of the most appalling presentations of this matter I have ever seen so far. For your mental sanity, I suggest you to switch to some more common reference. aero_head likes this.

November 13, 2020, 07:56		#6
Gerry Kan Senior Member Gerry Kan Join Date: May 2016 Posts: 373 Rep Power: 11	Dear TommyWind: To be honest I have not thought about symmetry, but this is an interesting point. In the Curric case, there is a switch in indices in the differential, that is: $\frac{\partial{u_i}}{\partial{x_i}\partial{x_j}} = \frac{\partial{u_i}}{\partial{x_j}\partial{x_i}}$ I believe your question hinges on this. If this is not correct please let me know. Going back to the above term, this is effectively saying (in 2D) for a fixed , say $x_j \equiv y$ (to be in line with the Curric convention, where each equation is fixed at instead of ): $\frac{\partial{u}}{\partial{x}\partial{y}}+\frac{\partial{v}}{\partial{y}\partial{y}} = \frac{\partial{u}}{\partial{y}\partial{x}}+\frac{\partial{v}}{\partial{y}\partial{y}}$ The only thing that "symmetry" would be brought in question is whether $\frac{\partial{u}}{\partial{x}\partial{y}} = \frac{\partial{u}}{\partial{y}\partial{x}}$ There is where I am not 100% certain. I believe there are rare special cases, from a mathematical point of view, where this might not be the case. However, for most naturally occurring phenomena the differential order can be regarded as interchangeable. I am not sure if this answers your question, though. But please keep in mind that index notation is a shorthand, and is not an implicit indicator of tensor symmetry. Gerry. P.S. - You can enter Latex commands directly in the forum posts using the math formatting tags. For instance: Code: {math}\frac{\partial{u_i}}{\partial{x_i}\partial{x_j}}{/math} NOTE - replace the braces around math and /math with square brackets will give you $\frac{\partial{u_i}}{\partial{x_i}\partial{x_j}}$ aero_head and TommyWind like this.

November 13, 2020, 09:41		#7
TommyWind New Member Lee Join Date: Nov 2020 Location: Taoyuan Posts: 3 Rep Power: 6	Dear Gerry, You are right! That is exactly my question. You did answer it. And I am glad to learn from you that most naturally occurring phenomena the differential order can be regarded as interchangeable. The problem of me is I need to know exactly the physics meaning behind the math. Otherwise, I can't really remember it. This is a very nice forum to know you and other people. And I finally know I need to learn Latex without second choice. I will keep studying what you and FMDenaro told me about tensor notation. I slightly touched the 4th order tensor when I studied the viscous shear stresses in a finite volume. Thank you Gerry! All the best, TommyWind Gerry Kan and aero_head like this.

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November 13, 2020, 07:09		#5
TommyWind New Member Lee Join Date: Nov 2020 Location: Taoyuan Posts: 3 Rep Power: 6	Dear Gerry, Thanks a lot for your explanation. Will it explain your example when the matrix (of two tensors) is symmetry? Quote You can switch tensor indices in any way you deemed convenient if "it still retains the summation" that it is meant to represent. End of Quote The key point is you have evidence or proof that it still retains the summation. That means we have to prove sock first and shoe second is equal to shoe first and sock second in this case. In your example symmetry works, but not for my question. Otherwise, they are all eliminated by incompressible. The changing of partial differential sequences are always some physics meaning behind in fluid. But, I think I just found the reason when I go back to the definition of τ_ij. τ_ij = μ ( ∇V_i + ∇V_i^t ) therefore symmetry τ_12 = ∂/∂x(μ(∂u/∂y+∂v/∂x)) = τ_21 = ∂/∂y(μ(∂v/∂x+∂u/∂y)) Should I make any mislead or mistake, please let me know. P.S. I will study my turbulence flow next semester. By the way, what kind math tool do you use here? Thank you Gerry! All the best, TommyWind

November 13, 2020, 13:16		#8
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,882 Rep Power: 73	What I see, is a topic of the differential calculus, the mix derivative do commute under the https://en.wikipedia.org/wiki/Symmet...nd_derivatives This is a classical topic and has nothing to do with the physics of the NSE. Just consider the regularity hypotheses.