$$
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\x}{\boldsymbol{x}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\w}{\boldsymbol{w}}
\newcommand{\rpos}{\boldsymbol{r}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\F}{\boldsymbol{F}}
\newcommand{\stress}{\boldsymbol{\sigma}}
\newcommand{\I}{\boldsymbol{I}}
\newcommand{\U}{\boldsymbol{U}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\ii}{\boldsymbol{i}}
\newcommand{\jj}{\boldsymbol{j}}
\newcommand{\kk}{\boldsymbol{k}}
\newcommand{\ir}{\boldsymbol{i}_r}
\newcommand{\ith}{\boldsymbol{i}_{\theta}}
$$