Index notation with Navier-Stokes equations

The divergence is a vector operator. This simply means that it is a differential operator that acts only on vectors. In this particular case, $$ {\rm div}\equiv\frac{\partial}{\partial x}\hat{x}+\frac{\partial}{\partial y}\hat{y}+\frac{\partial}{\partial z}\hat{z} $$ Which, assuming an implicit summation, $$ {\rm div}\equiv\frac{\partial}{\partial x_i}\hat{x}_i\tag{1} $$ Since the velocity field is a vector, $$ \mathbf{u}:=(u,\,v,\,w)=u\hat{x}+v\hat{y}+w\hat{z}\tag{2} $$ then the divergence of this is the dot product of the velocity and the vector operator: \begin{align} {\rm div}\mathbf u&=\left(\frac{\partial}{\partial x}\hat{x}+\frac{\partial}{\partial y}\hat{y}+\frac{\partial}{\partial z}\hat{z}\right)\cdot\left(u\hat{x}+v\hat{y}+w\hat{z}\right)\\ &=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}\\ &\equiv\frac{\partial u_i}{\partial x_i} \end{align} where $i$ is an index (1, 2, 3) that is implicitly summed over.

In the case of the Navier-Stokes equations, we have $$ \left(\frac{\partial}{\partial t}+\mathbf u\cdot{\rm div}\right)\mathbf u=\frac1\rho\nabla p+\nu\nabla^2\mathbf u + \frac1\rho\mathbf f $$ where the second term on the left is the one you are concerned with. First, you must take the dot-product of the velocity with the divergence operator (e.g., (2) dotted with (1)): $$ \mathbf u\cdot{\rm div}=u_j\frac{\partial}{\partial x_j} $$ which is a new operator. Then you can apply the vector $\mathbf u=u_i$ to this operator to get $$ \left(\mathbf u\cdot{\rm div}\right)\mathbf u=u_j\frac{\partial u_i}{\partial x_j} $$