"A fluids acceleration $a(\bar{x},t)$ is given by the material derivative of the fluid velocity $u(\bar{x},t)$,
$$ a(\bar{x},t) = \frac{D\bar{u}}{Dt} =\frac{\partial \bar{u}}{\partial t} + \bar{u} \cdot \nabla\bar{u}" $$
Assume $\bar{u} = u_i \bar{x}_i $ in an orthonormal basis $ \{\bar{x}_i\} $, then $ \nabla \bar{u} $ is defined as $ (\nabla \bar{u})_{ij} = \frac{\partial u_i}{\partial x_j}$.
What is then $\bar{u} \cdot \nabla\bar{u}$ defined as?
$(\bar{u} \cdot \nabla\bar{u})_i = u_j \frac{\partial u_i}{\partial x_j}$, $u_j \frac{\partial u_j}{\partial x_i}$ or neither?
For a vector-valued function, there are two different but closely related derivatives $$\eqalign{ J &= \frac{\partial y}{\partial x} \qquad\implies\quad J_{ik} &= \frac{\partial y_i}{\partial x_k} \\ J^T &= \nabla y \\ }$$ Depending on which one you use, the differential is written in two different but equivalent ways $$dy = J\cdot dx \;=\; dx\cdot\nabla y$$ or in index notation $$dy_i = J_{ik}\; dx_k \;=\; dx_k\,\big(\nabla y\big)_{ki} \\$$