What is the equation for viscous heating?

Question

I have been reading a paper which talks about viscosity in a plasma. They say that the viscous force is given by the divergence of the viscous stress tensor, $\boldsymbol{\nabla}\cdot\boldsymbol{\sigma}$. Therefore, the viscous heating rate must be given the dot product of the velocity with this force, $-\boldsymbol{v}\cdot\boldsymbol{\nabla}\cdot\boldsymbol{\sigma}$. They say that the heating rate is given by $\boldsymbol{\sigma}:\boldsymbol{\nabla v}.$ Therefore, the following must be true $$\boldsymbol{v}\cdot\boldsymbol{\nabla}\cdot\boldsymbol{\sigma}=-\boldsymbol{\sigma}:\boldsymbol{\nabla v},$$ however, they do not prove this result. Do you know how we can prove this result? Under what conditions does this equation hold?