By solving Darcy's law, over an incompressible fluid, we were able to simulate the pressure and velocity profiles which would be generated using our simplified geometry of the human eye. In Darcy's law for porous media, given by Equation \ref{eq:PI_Darcys_law}, is used to model fluid flow within the vitreous, $K$ and $\eta$ were taken from Ferreira et al. 2017 \cite{Ferreira2017}, with values of $8.4\times 10^{-14}$ m$^{2}$ and $10^{-3}$ kg m$^{-1}$ s$^{-1}$, respectively.
\begin{align} \label{eq:PI_Darcys_law}
\vec{v} = - \frac{K}{\eta} \nabla p
\end{align}
By assuming the vitreal fluid is incompressible we are able to generate a steady state pressure profile, using Equation \ref{eq:PI_incomp}, with suitable boundary conditions. In Ferreira et al. 2017 \cite{Ferreira2017} it was found that the pressure differential over the vitreous body was roughly 10 Pa, with an anterior and posterior pressure of 2000 Pa and 1990 Pa, respectively, these values were used as boundary conditions on $\delta\Omega_2$ and $\delta\Omega_1$.
\begin{align} \label{eq:PI_incomp}
\nabla \cdot \vec{v} = 0
%%%
\quad\Rightarrow\quad
%%%
\nabla^2 p = 0
\end{align}
The resultant pressure and velocity magnitude profiles can be seen in Figures \ref{fig:PI_pressure_profile} and \ref{fig:PI_velocity_profile}. By interpolating the velocity profile mesh points over a uniform spatial grid we were able to take a spatial median, which gave us a median vitreal velocity of $5.6\times10^{-7}$ cm/sec, a retinal surface velocity of $1.1\times 10^{-7}$ cm/sec, as well as a global minimum of $3.0\times10^{-9}$ cm/sec. By taking the length scale of the human eye as 1 cm and diffusion as $10^{-6}$ cm$^2$/sec, this generates Peclet numbers $\ll1$, indicating fluid transport is primarily due to diffusion.

% pressure profile
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{PI_pressure_profile.png}
\caption{Simulated vitreal pressure profile, derived using Equation \ref{eq:PI_incomp}, with units Pa}
\label{fig:PI_pressure_profile}
\end{figure}

% velocity profile
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{PI_velocity_profile.png}
\caption{Simulated vitreal velocity magnitude profile, derived using Equation \ref{eq:PI_Darcys_law}, with units cm/sec}
\label{fig:PI_velocity_profile}
\end{figure}\textbf{}