Let $u = (u_1,u_2,u_3)^T$ be a velocity vector.
Then, I know $\nabla u$ is a matrix whose $(i,j)$ entry is $\partial_{x_j} u_i$,
In the momentum equation of fluid dynamics, we have $u \cdot \nabla u$, which I think should be $u^T\nabla u$ in analogy with the usual dot product of two vectors, but this does not give the correct terms. Why is there notational inconsistency?