example.tex

\documentclass{article}
\usepackage{amsmath}
\usepackage{dakling}

\title{Example use of macros}
\input{definLocalExample.tex}

\begin{document}
\section{test}
\begin{align}
    % \pd[3]{\umHigherLong{23}}{x} = 0
    % \diffGeneric[2]{y}{x}{d} = 0
\umHigherLong{ij}
\end{align} 


\section{RANS equations}
\begin{align}
    \label{eq:contirans}
    \pd{\um{i}}{x_i} &= 0,\\
    \label{eq:momrans}
    \ncm{i} = 
    \pd{\um{i}}{t}
    + \um{j} \pd{\um{i}}{x_j}
    + \pd{\pm}{x_i}
    - \nu \pd[2]{\um{i}}{x_j}
    + \pd{\uf{ij}}{x_j}
    &= 0.
\end{align} 
\section{\(R_{ij}\) transport equation}
\begin{align}
\pd{\uf{ij}}{t}
     \nonumber
    + \um{k} \pd{\uf{ij}}{x_k}
    + \uf{ik} \pd{\um{j}}{x_k}
    + \uf{jk} \pd{\um{i}}{x_k}
    - \mean{\pf \left( \pd{\uf{i}}{x_j} + \pd{\uf{j}}{x_i} \right)}
    + 2 \nu \mean{ \pd{\uf{i}}{x_k} \pd{\uf{j}}{x_k} }
    \\
    \label{eq:reystress}
    + \pd{}{x_k}
    \left( 
        \uf{ijk}
        + \mean{(\delta_{jk} \uf{i} + \delta_{ik} \uf{j}) \pf}
        - \nu \pd{\uf{ij}}{x_k}
    \right)
    =0.
\end{align}    

\section{\(k\)-Equation in \komegamodel}%
\label{sec:k_equation_in_komega_model}
\begin{align}
    \label{eq:kequationOrig}
    \pd{k}{t} + \um{j} \pd{k}{x_j} &= M_{ij} \pd{\um{i}}{x_j} - \beta \omega k + \pdl{\left(\nu + \nut \sigma_{\text{kin}} \right) \pd{k}{x_l}}{x_l},\\
\end{align}

\section{Euler symmetries}
\begin{align}
    \label{eq:symmGloUnavgTime}
    & \gloSymmT : \enskip
    &&\tra{t} t + \grp, \enskip
    &&\tra{x_i} x_i, \enskip
    &&\tra{U_i} U_i, \enskip
    &&\tra{P} P;
    \\ 
    \label{eq:symmGloUnavgSpace}
    & \gloSymmX :  \enskip
    &&\tra{t} t, \enskip
    &&\tra{x_i} x_i + \grp_i, \enskip
    &&\tra{U_i} U_i, \enskip
    &&\tra{P} P;
    \\
    \label{eq:symmGloUnavgRot}
    & \gloSymmRot : \enskip
    &&\tra{t} t, \enskip
    &&\tra{x_i} x_j Q_{ij}^{[\alpha]}, \enskip
    &&\tra{U_i} U_j Q_{ij}^{[\alpha]}, \enskip
    &&\tra{P} P;\\
    \label{eq:symmGloUnavgGal}
    & \gloSymmGal : \enskip
    &&\tra{t} t, \enskip
    &&\tra{x_i} x_i + \grp_i t, \enskip
    &&\tra{U_i} {U_i} + \grp_i, \enskip
    &&\tra{P} P;\\
    \label{eq:symmGloUnavgPress}
    & \gloSymmP : \enskip
    &&\tra{t} t, \enskip
    &&\tra{x_i} x_i, \enskip
    &&\tra{U_i} {U_i}, \enskip
    &&\tra{P} P + \grp f(t);\\
    \label{eq:symmGloUnavgScal1}
    & \gloSymmScI : \enskip
    &&\tra{t} t, \enskip
    &&\tra{x_i} x_i e^{\grp}, \enskip
    &&\tra{U_i} {U_i} e^{\grp}, \enskip
    &&\tra{P} P e^{2 \grp};\\
    \label{eq:symmGloUnavgScal2}
    & \gloSymmScII : \enskip
    &&\tra{t} e^{\grp}t, \enskip
    &&\tra{x_i} x_i, \enskip
    &&\tra{U_i} {U_i} e^{-\grp}, \enskip
    &&\tra{P} P e^{-2 \grp};
\end{align}
\end{document}