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Nov 18, 2021, 10:13:21 AM11/18/21

to basil...@googlegroups.com

Dear All

I have a few questions about the notation used for various tensor components used in Basilisk.

Please help me with respect to the assumptions / notations.

1) From the definition of a tensor in the header file "common.h", I came to know that a tensor is represented with the help of three vectors x, y, and z.

My question is whether the individual vectors are assumed column-wise or row-wise in the matrix for the tensor?

2) The velocity gradient tensor

-------------------------------------

Whether \nabla U = [du/dx du/dy] OR \nabla U = [du/dx dv/dx]

[dv/dx dv/dy] [du/dy dv/dy]

3) The viscous stress tensor

----------------------------------

Let \tau_{ij} be the viscous stress tensor

then whether the subscript j denotes the plane normal OR the force component?

Accordingly, the subscript i will denote either the plane normal or the force component?

4) The eigenvector matrix R of conformation tensor for the viscoelastic flow solver

-----------------------------------

Whether the eigenvectors are stored column-wise or row-wise?

5) Lastly, how is the product between two matrices, say A and B, defined?

This will depend on how the components of tensors appear in the matrix form.

Please help me in this regard.

Thank you all.

Regards

--

Dr. Shyam Sunder Yadav

Assistant Professor

Mechanical Engineering

BITS Pilani

09902346342

http://www.bits-pilani.ac.in/pilani/ssyadav/Profile

Assistant Professor

Mechanical Engineering

BITS Pilani

09902346342

http://www.bits-pilani.ac.in/pilani/ssyadav/Profile

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Nov 19, 2021, 7:53:13 AM11/19/21

to basilisk-fr

Hi Shyam,

El jueves, 18 de noviembre de 2021 a la(s) 16:13:21 UTC+1, ss.y...@pilani.bits-pilani.ac.in escribió:

Dear AllI have a few questions about the notation used for various tensor components used in Basilisk.Please help me with respect to the assumptions / notations.1) From the definition of a tensor in the header file "common.h", I came to know that a tensor is represented with the help of three vectors x, y, and z.My question is whether the individual vectors are assumed column-wise or row-wise in the matrix for the tensor?

Not sure...

2) The velocity gradient tensor-------------------------------------Whether \nabla U = [du/dx du/dy] OR \nabla U = [du/dx dv/dx][dv/dx dv/dy] [du/dy dv/dy]

I struggle a little bit with it. Finally, I realized that in most of the viscoelatic literature it is used

\nabla U = [du/dx dv/dx]

[du/dy dv/dy]

3) The viscous stress tensor----------------------------------Let \tau_{ij} be the viscous stress tensorthen whether the subscript j denotes the plane normal OR the force component?Accordingly, the subscript i will denote either the plane normal or the force component?

Does not matter because it is symmetric.

4) The eigenvector matrix R of conformation tensor for the viscoelastic flow solver-----------------------------------Whether the eigenvectors are stored column-wise or row-wise?

by column

[v1 | v2]

each column is an eigenvector.

5) Lastly, how is the product between two matrices, say A and B, defined?This will depend on how the components of tensors appear in the matrix form.

c= a*b being

a =[axx axy]

[ayx ayy]
[

b =[bxx bxy]

[byx byy]

would be the usual "row by column"

cxx = axx*bxx + axy *byx

....

Message has been deleted

Nov 19, 2021, 11:30:02 AM11/19/21

to basilisk-fr

Thanks Prof. Jose!

Just like your previous posts/replies, the current one is also really helpful...

I have following observations, I will go by the order of my original questions...

1) Whether the individual vectors are assumed column-wise or row-wise in the matrix for the tensor?

Consider the following product in the log-conform.h: Ψ=R⋅log(Λ)⋅R^T

If we represent the eigenvectors column-wise then R = [R.x.x R.y.x] and log(Λ) = [ log(Λ1) 0]

[R.x.y R.y.y] [0 log(Λ1)]

Therefore Ψ=R⋅log(Λ)⋅R^T = [ log(Λ1)*R.x.x^2 + log(Λ2)*R.y.x^2 R.x.x*R.x.y* log(Λ1) + R.y.x*R.y.y* log(Λ2)]

[ R.x.x*R.x.y* log(Λ1) + R.y.x*R.y.y* log(Λ2) log(Λ1)*R.x.y^2 + log(Λ2)*R.y.y^2]

This raises the 5th point: how is the product between two matrices, say A and B, defined?

If we do the Ψ=R⋅log(Λ)⋅R^T product in the following way: ( R^T * log(Λ) )*R then the answer is as mentioned in the log-conform.h file.

2) About velocity gradient in Basilisk?

\nabla U = [du/dx dv/dx] Thanks for the clarification Prof.

[du/dy dv/dy]

3) Notation for the viscous stress tensor?

In log-conform.h, we have the x direction force due to divergence of elastic stresses defined as: d \taup.x.x/dx + d \taup.x.y/dy

From the above it appears that in \taup_{ij}, j indicates the plane normal and i indicates the force direction...

=> d \taup.x.y/dy = the x direction stress component on a plane normal to y axis should vary with respect to y so as to create a force along x.

This is the case with divergence of Maxwell stress tensor as well in the EHD solver...

4) Whether the eigenvectors are stored column-wise or row-wise?

The notation is important for point 1 above.

5) How is the product between two matrices, say A and B, defined?

Again, important for point 1 above.

Again, Thank you Prof. Jose for the reply.

Regards

Just like your previous posts/replies, the current one is also really helpful...

I have following observations, I will go by the order of my original questions...

1) Whether the individual vectors are assumed column-wise or row-wise in the matrix for the tensor?

Consider the following product in the log-conform.h: Ψ=R⋅log(Λ)⋅R^T

If we represent the eigenvectors column-wise then R = [R.x.x R.y.x] and log(Λ) = [ log(Λ1) 0]

[R.x.y R.y.y] [0 log(Λ1)]

Therefore Ψ=R⋅log(Λ)⋅R^T = [ log(Λ1)*R.x.x^2 + log(Λ2)*R.y.x^2 R.x.x*R.x.y* log(Λ1) + R.y.x*R.y.y* log(Λ2)]

[ R.x.x*R.x.y* log(Λ1) + R.y.x*R.y.y* log(Λ2) log(Λ1)*R.x.y^2 + log(Λ2)*R.y.y^2]

But we have the following in log-conform.h

Ψ=R⋅log(Λ)⋅R^T = [ log(Λ1)*R.x.x^2 + log(Λ2)*R.x.y^2 R.x.x*R.y.x* log(Λ1) + R.x.y*R.y.y* log(Λ2)]

[ R.x.x*R.y.x* log(Λ1) + R.x.y*R.y.y* log(Λ2) log(Λ1)*R.y.x^2 + log(Λ2)*R.y.y^2]

[ R.x.x*R.y.x* log(Λ1) + R.x.y*R.y.y* log(Λ2) log(Λ1)*R.y.x^2 + log(Λ2)*R.y.y^2]

This raises the 5th point: how is the product between two matrices, say A and B, defined?

If we do the Ψ=R⋅log(Λ)⋅R^T product in the following way: ( R^T * log(Λ) )*R then the answer is as mentioned in the log-conform.h file.

2) About velocity gradient in Basilisk?

\nabla U = [du/dx dv/dx] Thanks for the clarification Prof.

[du/dy dv/dy]

3) Notation for the viscous stress tensor?

In log-conform.h, we have the x direction force due to divergence of elastic stresses defined as: d \taup.x.x/dx + d \taup.x.y/dy

From the above it appears that in \taup_{ij}, j indicates the plane normal and i indicates the force direction...

=> d \taup.x.y/dy = the x direction stress component on a plane normal to y axis should vary with respect to y so as to create a force along x.

This is the case with divergence of Maxwell stress tensor as well in the EHD solver...

4) Whether the eigenvectors are stored column-wise or row-wise?

The notation is important for point 1 above.

5) How is the product between two matrices, say A and B, defined?

Again, important for point 1 above.

Again, Thank you Prof. Jose for the reply.

Regards

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