I am reading papers about Liquid crystals.
Changyou Wang. “Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data” The equation $$\begin{cases} u_t + u \cdot \nabla u - \Delta u + \nabla P = -\nabla \cdot (\nabla d \otimes \nabla d) \quad &\text{in } \mathbb{R}^3 \times (0, +\infty), \\ \nabla \cdot u = 0 \quad &\text{in } \mathbb{R}^3 \times (0, +\infty), \\ d_t + u \cdot \nabla d = \Delta d + |\nabla d|^2 d &\quad \text{in } \mathbb{R}^3 \times (0, +\infty), \end{cases}$$
Fanghua Lin and Chun Liu. “Nonparabolic dissipative systems modeling the flow of liquid crystals”
$$\begin{cases} v_t + (v\cdot \nabla )v - \nu \Delta + \nabla p = -\lambda \nabla \cdot (\nabla d \odot \nabla d) \\ \nabla v = 0 \\ d_t + (v\cdot \nabla) d = \gamma (\Delta d - f(d)) \end{cases}$$
$d$ in both cases is $d:\mathbb{R}^3\to \mathbb{R}^3$. So $(\nabla d)_{ij}=\partial_i d_j$ is a rank-two tensor. In both papers they define it as $(\nabla d \otimes \nabla d)_{ij} = \partial_i d\cdot \partial_j d $ clearly thats a symmetric tensor so why did Wang used the $\otimes$ which not always symmetric?
Even if you do not have access to the papers, I would like to know when in physics do you use the symmetric tensor and when you use the regular tensor?
Also, the tensor of two one-rank tensors is a two-rank tensor $\begin{bmatrix}a_1\\a_2\\a_3\end{bmatrix}\otimes \begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix}=\begin{bmatrix}a_1b_1 & a_1b_2 & a_1b_3 \\ a_2b_1&a_2b_2&a_2b_3\\a_3b_1&a_3b_2&a_3b_3\end{bmatrix}$ , why for rank-two tensors is not it is again a rank-two tensor?
