maybe most of you know that there could be a material derivative for vector like
$(\vec{A}\cdot\nabla)\vec{B}$
I recently encounter an expression where I need to evaluate a similar but a somewhat different stuff, now the expression is
$(\vec{\Sigma}\cdot\nabla)\cdot\vec{v}$
Here $\Sigma$ is the stress tensor of fluid while $\vec{v}$ is the velocity of fluid. This expression occurs in one of the paper "Clement, Maurice J. "Hydrodynamical simulations of rotating stars. I-A model for subgrid-scale flow." The Astrophysical Journal 406 (1993): 651-660.", equation (10)
We do know that there are general expression for material derivative for vector field in any coordinate system, in particular the one I interested is cylindrical coordinates.
It seems that relevant topic and sources about the 2nd expression is quite hard to find, and in particular I would like to know is there any general expression for that? I tried a cartesian counterpart and tried to do coordinate transformation, and I do not think I get the right answer
Thank you everyone