hazbijav6

2022-10-09

I've been working on computational fluid dynamics and have come across the following term in index notation:

$\frac{\mathrm{\partial}{u}_{\mathrm{i}}}{\mathrm{\partial}{x}_{\mathrm{m}}}\frac{\mathrm{\partial}{u}_{\mathrm{j}}}{\mathrm{\partial}{x}_{\mathrm{m}}}$

However, I'm having a hard time finding a vector notation equivalent to this operation. This is definitely not the inner product or outer product, but kind of like a "right" inner product. Has anyone come across any term like this and its vector notation equivalent?

To be more precise, I would like to know the operation $\odot $ in:

$(\mathrm{\nabla}\otimes \mathbf{u})\odot (\mathrm{\nabla}\otimes \mathbf{u})$

If one exists, or some other form. In the above, $\mathrm{\nabla}\equiv \mathrm{\partial}/\mathrm{\partial}{x}_{\mathrm{m}}$, u is a Cartesian 3-vector, and $\otimes $ is the direct product.

$\frac{\mathrm{\partial}{u}_{\mathrm{i}}}{\mathrm{\partial}{x}_{\mathrm{m}}}\frac{\mathrm{\partial}{u}_{\mathrm{j}}}{\mathrm{\partial}{x}_{\mathrm{m}}}$

However, I'm having a hard time finding a vector notation equivalent to this operation. This is definitely not the inner product or outer product, but kind of like a "right" inner product. Has anyone come across any term like this and its vector notation equivalent?

To be more precise, I would like to know the operation $\odot $ in:

$(\mathrm{\nabla}\otimes \mathbf{u})\odot (\mathrm{\nabla}\otimes \mathbf{u})$

If one exists, or some other form. In the above, $\mathrm{\nabla}\equiv \mathrm{\partial}/\mathrm{\partial}{x}_{\mathrm{m}}$, u is a Cartesian 3-vector, and $\otimes $ is the direct product.

graulhavav9

Beginner2022-10-10Added 14 answers

The matrix $\mathrm{\nabla}\otimes u$ is defined by $(\mathrm{\nabla}\otimes u{)}_{im}:=\frac{\mathrm{\partial}{u}_{i}}{\mathrm{\partial}{x}_{m}}$. The given expression is $(\mathrm{\nabla}u{)}_{im}(\mathrm{\nabla}\otimes u{)}_{jm}=(\mathrm{\nabla}\otimes u(\mathrm{\nabla}\otimes u{)}^{T}{)}_{ij}$, so the matrix we need is just $(\mathrm{\nabla}\otimes u)(\mathrm{\nabla}\otimes u{)}^{T}$

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