A dot product between a vector and a tensor

Question

I'd like to understand how to write $\mathbf{u}\cdot\nabla\mathbf{u}$ in open form, where $\mathbf{u}$ is the two dimensional velocity vector, and $\nabla$ is the gradient operator. I'd be glad if you could help me with this. Regards

Answer 1

It is intended as an inner product between $\mathbf{u}$ and the operator $\nabla$: $$ \mathbf{u}\cdot\nabla=\sum_{i=1}^2u_i\partial_i=u_1\frac{\partial}{\partial x_1}+u_2\frac{\partial}{\partial x_2} $$ applied to each component of the following vector field, in the present case again the vector $\mathbf{u}.$

So we have $$ \sum_{i=1}^2u_i\partial_iu_j=\sum_{i=1}^2u_i\frac{\partial u_j}{\partial x_i} $$ or more explicitly, with $\mathbf{u}=(u,v)$ $$ (uu_x+vu_y,uv_x+vv_y) $$