Writing a PDE spread over multiple lines with elements description in LaTeX [closed] - latex

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for my master thesis I have to include a long PDE which has to be distributed over multiple lines for space and understanding purposes. Each term has its own description.
The final goal is to reproduce the equation in the pictureturbulent energy equation.
My LaTeX code (overleaf) is the following:
\begin{multline}
%right-end side
%first term
\left.\rho c\left(\frac{\partial\overline{T}}{\partial t}+\overline{u}\frac{\partial\overline{T}}{\partial x}+\overline{v}\frac{\partial\overline{T}}{\partial y}+\overline{w}\frac{\partial\overline{T}}{\partial z}\right) \right\} \quad \text{advection}\\
%left-end side
%first term
=\left.\lambda\left(\frac{\partial^2\overline{T}}{\partial x^2}+\frac{\partial^2\overline{T}}{\partial x^2}+\frac{\partial^2\overline{T}}{\partial x^2}\right) \right\} \quad \text{diffusion}\\
%second term
\left.-\rho c\left(\frac{\partial\overline{u^\prime T^\prime}}{\partial x}+\frac{\partial\overline{v^\prime T^\prime}}{\partial y}+\frac{\partial\overline{w^\prime T^\prime}}{\partial z}\right) \right\} \quad \text{turbulent heat transport} \\
%third term
+\left.\mu\left(2\left(\frac{\partial\overline{u}}{\partial x}\right)^2+2\left(\frac{\partial\overline{v}}{\partial y}\right)^2+2\left(\frac{\partial\overline{w}}{\partial z}\right)^2 \\
+\left(\frac{\partial\overline{v}}{\partial z}+\frac{\partial\overline{w}}{\partial y}\right)^2\right) \\ \right\} \quad \text{direct dissipation}
%fourth term
+\left.\mu\left[2\bar{\left(\frac{\partial u\prime}{\partial x}\right)^2}+2\bar{\left(\frac{\partial v\prime}{\partial y}\right)^2}+2\bar{\left(\frac{\partial w}{\partial z}\right)^2} \\
+\bar{\left(\frac{\partial u\prime}{\partial y}+\frac{\partial v\prime}{\partial x}\right)^2}+\bar{\left(\frac{\partial u\prime}{\partial z}+\frac{\partial w\prime}{\partial x}\right)^2}\\
+\bar{\left(\frac{\partial v\prime}{\partial z}+\frac{\partial w\prime}{\partial y}\right)^2}\right] \right\} \quad \text{turbulent dissipation}
\end{multline}
Differently from the example in the figure, I also have to develop the last term.
The code that I've written still miss the alignment of the names of the different terms and if a single term of the PDE is distributed over multiple lines, then the curly brackets are not represented.
Can anyone please help me? Thanks.

Answer 1

Please, always provide the full code. It is particularly difficult to track issues in LaTeX code without having all the details.
I changed environment to align* and added a few inner aligned environments. The former splits lines into both equation and annotation parts.
All the right braces are a little tricky but you can always scale \left.\right. manually using an arbitrary vertical bar. If you change equations you might need to reapply the size. \left(...\right) macros are not recommended as they introduce many issues including inconsistent spacing and scaling. The \mleftright package corrects some of them but it is still better to use \Bigl[ or \Biggr\} etc.
The code
\documentclass{article}
\usepackage{mathtools}
\newlength\lefteqskip
\begin{document}
\lefteqskip3.5em
\begin{align*}
%right-end side
%first term
\mathrlap{\rho c
\left(
\frac{\partial\overline{T}}{\partial t}
+\overline{u}\frac{\partial\overline{T}}{\partial x}
+\overline{v}\frac{\partial\overline{T}}{\partial y}
+\overline{w}\frac{\partial\overline{T}}{\partial z}
\right)}
\phantom{={}}
%\hspace{0.5\lefteqskip} % indents LHS
&&&\biggr\} \quad \text{advection} \\
%left-end side
%first term
={}&{}\lambda
\left(
\frac{\partial^2\overline{T}}{\partial x^2}
+\frac{\partial^2\overline{T}}{\partial x^2}
+\frac{\partial^2\overline{T}}{\partial x^2}
\right)
&&\biggr\} \quad \text{diffusion}\\
%second term
&{}-\rho c
\left(
\frac{\partial\overline{u^\prime T^\prime}}{\partial x}
+\frac{\partial\overline{v^\prime T^\prime}}{\partial y}
+\frac{\partial\overline{w^\prime T^\prime}}{\partial z}
\right)
&&\biggr\} \quad \text{turbulent heat transport} \\
%third term
&{}
\begin{aligned}
&{}+\mu\left(
2\left(
\frac{\partial\overline{u}}{\partial x}
\right)^2
+ 2\left(
\frac{\partial\overline{v}}{\partial y}
\right)^2\right. \\
&\left.\hspace{\lefteqskip}
{}+ 2\left(
\frac{\partial\overline{w}}{\partial z}
\right)^2
+ \left(
\frac{\partial\overline{v}}{\partial z}
+\frac{\partial\overline{w}}{\partial y}
\right)^2
\right)
\end{aligned}
&&\hspace{-3pt}
\left.\rule[-7.5ex]{0pt}{15ex}\right\}
\quad
\text{direct dissipation} \\
% % %fourth term
&\begin{aligned}
&{}+\mu\left[
2\bar{\left(
\frac{\partial u\prime}{\partial x}
\right)^2}
+2\bar{\left(
\frac{\partial v\prime}{\partial y}
\right)^2}
\right. \\
&\hspace{\lefteqskip}
+ 2\bar{\left(
\frac{\partial w}{\partial z}
\right)^2}
+ \bar{\left(
\frac{\partial u\prime}{\partial y}
+\frac{\partial v\prime}{\partial x}
\right)^2} \\
&\left.\hspace{\lefteqskip}
+\bar{\left(
\frac{\partial u\prime}{\partial z}
+\frac{\partial w\prime}{\partial x}
\right)^2}
+\bar{\left(
\frac{\partial v\prime}{\partial z}
+\frac{\partial w\prime}{\partial y}
\right)^2}
\right]
\end{aligned}
&&\hspace{-3pt}
\left.\rule[-11ex]{0pt}{22ex}\right\}
\quad
\text{turbulent dissipation}
\end{align*}
\end{document}