There is an equation (from Chapman-Enskog theory) that uses index notation:
$$\sum_i w_ie_{i\alpha}e_{i\beta}e_{i\gamma}e_{i\mu}= c_s^4(\delta_{\alpha\beta}\delta_{\gamma\mu}+ \delta_{\alpha\gamma}\delta_{\beta\mu}+ \delta_{\alpha\mu}\delta_{\gamma\beta})$$
Based on this formulation, we want to find the value of:
\begin{align} \partial_\gamma\sum_i\frac{e_{i\alpha} u_\alpha}{c_s^2} e_{i\alpha}e_{i\beta}e_{i\gamma} &= c_s^2 \partial_\gamma\ u_\alpha(\delta_{\alpha\beta}\delta_{\alpha\gamma}+ \delta_{\alpha\gamma}\delta_{\alpha\beta}+ \delta_{\alpha\alpha}\delta_{\gamma\beta}) \\ &= c_s^2\partial_\gamma(u_\beta\delta_{\alpha\gamma}+ u_\gamma\delta_{\alpha\beta+} u_\alpha\delta_{\gamma\beta}) \\ &= c_s^2(\partial_\alpha u_\beta+\partial_\beta u_\alpha+\partial_\gamma u_\gamma\delta_{\alpha\beta}) \end{align}
If we write it in nabla operator notation, it is actually a strain tensor:
$$c_s^2(\nabla\mathbf{u}+\nabla\mathbf{u}^T+(\nabla\cdot\mathbf{u})\mathbf{I}).$$
Now, I have a question about this derivation. Can I convert all the above notation into nabla operator form? For example, can I consider $\delta_{\alpha\beta}\delta_{\gamma\mu}=\mathbf{I}\otimes\mathbf{I}$?
Then, the first relation equation becomes: $\sum_i w_i \mathbf{e}_i\mathbf{e}_i\mathbf{e}_i\mathbf{e}_i=3c_s^4\mathbf{I}\otimes\mathbf{I}$, and we will then have the final equation:
$$c_s^2\nabla\cdot(\mathbf{u}\cdot(3\mathbf{I}\otimes\mathbf{I}) )=3c_s^2\nabla\mathbf{u}???$$
I know it is wrong for sure, but could you help me identify where the mistake is? Should I have written $\delta_{\alpha\beta}\delta_{\gamma\mu}=1i_\alpha i_\beta i_\gamma i_\mu$?
It is related to high-order tensor calculus. The problem is from the relation equation.
$\sum_i w_i\mathbf{e}_i\mathbf{e}_i\mathbf{e}_i=c_s^4(\mathbf{I}+\bar{\mathbf{I}}+\mathbf{I}\otimes\mathbf{I})$,
where $\mathbf{I}:\mathbf{A}=\mathbf{A}$, $\mathbf{\bar{I}}:\mathbf{A}=\mathbf{A}^T$, and $\mathbf{I}\otimes\mathbf{I}:\mathbf{A}=tr(\mathbf{A})$ are three kind of 4th-order identity matrix. Here $\mathbf{A}$ is a random 4th-order matrix.
Then, the function becomes $c_s^2(\nabla\cdot\mathbf{I}\otimes\mathbf{u} +(\nabla\cdot \mathbf{I} \otimes\mathbf{u})^T +\nabla\cdot\mathbf{u}\otimes\mathbf{I})=c_s^2(\nabla\mathbf{u}+\nabla\mathbf{u}^T+(\nabla\cdot\mathbf{u})\mathbf{I})$