Divergence of a dot product of tensor and vector

Question

Hei, I am trying to derive energy equation from Navier-Stokes equation and I come across this: $$\nabla.(\sigma.v)=(\nabla.\sigma).v +\sigma:\nabla v$$ $\sigma $ is the stress tensor

V :is the velocity vector

Could anyone thankfully explain this and if that is correct?

Answer 1

Let us consider an orthonormal basis of the euclidean space. The divergence reads \begin{aligned} \nabla\cdot (\sigma\cdot v) &= (\sigma_{ij} v_j)_{,i} \\ &= \sigma_{ij,i} v_j + \sigma_{ij} v_{j,i} \\ &= \sigma_{ji,i} v_j + \sigma_{ji} v_{j,i} \\ &= (\nabla\cdot\sigma)\cdot v + \sigma : \nabla v \end{aligned} using Einstein notation and the symmetry of the stress tensor.