Check an equality relating the divergence of a matrix and gradient of a vector field

Question

Going through a fluid mechanics book, I encountered this expression: $$\nabla \cdot (uu^T) = u \cdot \nabla u, $$ whereby $u$ is supposed to be the velocity vector field. I can not make sense of this. Considering $u$ as a vector, $uu^T$ will be a 3 by 3 matrix. For the left-hand side of the equation above I took the divergence by taking the divergence of the three column vectors of the matrix. For the right-hand side I do not know what the gradient of a vector field is defined and what the dot product there means. Can somebody explain all the terms and operations in the equation and prove it holds ? Thanks.

Answer 1

In terms of Cartesian coordinates and using the Einstein summation convention (sum over repeated indices):

$$\nabla \cdot(\mathbb{uu}^T) = \partial_i(u_iu_j) = (\partial_iu_i)u_j + u_i \partial_i u_j= (\nabla \cdot \mathbb{u})\mathbb{u} + \mathbb{u} \cdot \nabla \mathbb{u}$$

If the flow is incompressible, we have $\nabla \cdot \mathbb{u} = 0$ and we get

$$\nabla \cdot(\mathbb{uu}^T) =\mathbb{u} \cdot \nabla \mathbb{u}$$