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

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