I've been working on computational fluid dynamics and have come across the following term in index notation:
$$\frac{\partial u_\mathrm{i}}{\partial x_\mathrm{m}}\frac{\partial u_\mathrm{j}}{\partial x_\mathrm{m}}$$
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:
$$(\mathbf{\nabla}\otimes\mathbf{u})\odot(\mathbf{\nabla}\otimes\mathbf{u})$$
If one exists, or some other form. In the above, $\nabla \equiv \partial / \partial x_\mathrm{m}$, $\mathbf{u}$ is a Cartesian 3-vector, and $\otimes$ is the direct product.
Thanks!
Best Answer
The matrix $\nabla\otimes u$ is defined by $(\nabla\otimes u)_{im}:=\frac{\partial u_i}{\partial x_m}$. The given expression is $(\nabla u)_{im}(\nabla\otimes u)_{jm}=(\nabla\otimes u(\nabla\otimes u)^T)_{ij}$, so the matrix we need is just $(\nabla\otimes u)(\nabla\otimes u)^T$.