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]
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]
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