Solutions¶

Using Macros

You should have ended up with the following macros and prose:

In [3]:

%%pdflatex
\newcommand{\viscosity}{\mu}
\newcommand{\viscosityMean}{\overline{\mu}}
\newcommand{\laplacian}{\nabla^2}
\newcommand{\bracs}[1]{\left( #1 \right)}
\newcommand{\density}{\rho} %or whatever you changed this to if you don't like rho!
\newcommand{\pdiff}[1]{\partial_{#1}} %notice that you need the curly-braces {} around #1, because you want the whole expression that is put in to be a subscript
\newcommand{\normDist}[2]{\mathcal{N}\left(#1, {#2}^2 \right)} %you could also use \bracs{#1, {#2}^2} if you were feeling lazy, too!
In this paper we are interested in solving the following system of equations;
\begin{align} \label{eq:NSEqn}
    \density\pdiff{t}v + \density \bracs{v\cdot\nabla}v &= -\nabla P + \gamma \density + \viscosity \laplacian v,
\end{align}
for randomly realised viscosity $\viscosity\sim\normDist{\viscosityMean}{\sigma}$ and constant density $\density$.
The vector field $\gamma$ and pressure gradient $P$ are assumed known, and along with $v$ are functions of the spatial variable $x$ and temporal variable $t$.
Demonstrating existence, uniqueness and regularity of the solution to \eqref{eq:NSEqn} in the event that $\viscosity=0$ and hence the $\laplacian v$ vanishing are trivial, and so we retain this term in order to formulate the main theorm of our paper.
We also introduce randomness in the viscosity parameter $\viscosity$, because we found that without this, it was also a simple task to determine the aforementioned existence, uniqueness and regularity properties.

Out[3]:

%%pdflatex
\newcommand{\viscosity}{\mu}
\newcommand{\viscosityMean}{\overline{\mu}}
\newcommand{\laplacian}{\nabla^2}
\newcommand{\bracs}[1]{\left( #1 \right)}
\newcommand{\density}{\rho} %or whatever you changed this to if you don't like rho!
\newcommand{\pdiff}[1]{\partial_{#1}} %notice that you need the curly-braces {} around #1, because you want the whole expression that is put in to be a subscript
\newcommand{\normDist}[2]{\mathcal{N}\left(#1, {#2}^2 \right)} %you could also use \bracs{#1, {#2}^2} if you were feeling lazy, too!
In this paper we are interested in solving the following system of equations;
\begin{align} \label{eq:NSEqn}
    \density\pdiff{t}v + \density \bracs{v\cdot\nabla}v &= -\nabla P + \gamma \density + \viscosity \laplacian v,
\end{align}
for randomly realised viscosity $\viscosity\sim\normDist{\viscosityMean}{\sigma}$ and constant density $\density$.
The vector field $\gamma$ and pressure gradient $P$ are assumed known, and along with $v$ are functions of the spatial variable $x$ and temporal variable $t$.
Demonstrating existence, uniqueness and regularity of the solution to \eqref{eq:NSEqn} in the event that $\viscosity=0$ and hence the $\laplacian v$ vanishing are trivial, and so we retain this term in order to formulate the main theorm of our paper.
We also introduce randomness in the viscosity parameter $\viscosity$, because we found that without this, it was also a simple task to determine the aforementioned existence, uniqueness and regularity properties.

%%pdflatex
\newcommand{\viscosity}{\mu}
\newcommand{\viscosityMean}{\overline{\mu}}
\newcommand{\laplacian}{\nabla^2}
\newcommand{\bracs}[1]{\left( #1 \right)}
\newcommand{\density}{\rho} %or whatever you changed this to if you don't like rho!
\newcommand{\pdiff}[1]{\partial_{#1}} %notice that you need the curly-braces {} around #1, because you want the whole expression that is put in to be a subscript
\newcommand{\normDist}[2]{\mathcal{N}\left(#1, {#2}^2 \right)} %you could also use \bracs{#1, {#2}^2} if you were feeling lazy, too!
In this paper we are interested in solving the following system of equations;
\begin{align} \label{eq:NSEqn}
    \density\pdiff{t}v + \density \bracs{v\cdot\nabla}v &= -\nabla P + \gamma \density + \viscosity \laplacian v,
\end{align}
for randomly realised viscosity $\viscosity\sim\normDist{\viscosityMean}{\sigma}$ and constant density $\density$.
The vector field $\gamma$ and pressure gradient $P$ are assumed known, and along with $v$ are functions of the spatial variable $x$ and temporal variable $t$.
Demonstrating existence, uniqueness and regularity of the solution to \eqref{eq:NSEqn} in the event that $\viscosity=0$ and hence the $\laplacian v$ vanishing are trivial, and so we retain this term in order to formulate the main theorm of our paper.
We also introduce randomness in the viscosity parameter $\viscosity$, because we found that without this, it was also a simple task to determine the aforementioned existence, uniqueness and regularity properties.