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\begin{center}
\large{Quantifying Permeability and Slip Using Crossflow to Define Operating  Equations for Poly(vinyl alcohol)-Poly(lactide-co-glycolide) Microfiltration Membrane-Scaffold Bioreactors} \\

\large{Supplementary Material} \\

\normalsize{$\mbox{R.J. Shipley}^1$ , \quad $\mbox{S.L Waters}^{1}$ , \quad $\mbox{M.J. Ellis}^{2\star}$}
\end{center}

\noindent \footnotesize{$1$: Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, $24$--$29$ St. Giles', Oxford OX$1$ $3$LB. \\
$2$: Department of Chemical Engineering, University of Bath, Bath BA$2$ $7$AY. \\
Tel: $01225$ $384484$ \\
Fax: $01225$ $385713$ \\
Email: M.J.Ellis@bath.ac.uk \\
$\star$: Corresponding author}

\pagebreak

\appendix

\section{Reduction of the Full System of Equations}\label{sec:reduction}

First of all, we repeat the full system of equations that describe fluid flow throughout a single module comprised of the lumen, lumen wall and ECS. In the lumen,
\bq
\nab \cdot \rb{u}_l = 0 , \quad \rho \brac{\pd{\rb{u}_l}{t} + \brac{\rb{u}_l \cdot \nab } \rb{u}_l } = - \nab p_l + \mu \nabla^2 \rb{u}_l, \label{eq:NSlumen}
\eq
in the lumen wall,
\bq
\nab \cdot \rb{u}_w = 0 , \quad \rb{u}_w = - \frac{k}{\mu} \nab p_w , \label{eq:darcylumen}
\eq
and in the ECS
\bq
\nab \cdot \rb{u}_e = 0, \quad \rho \brac{\pd{\rb{u}_e}{t} + \brac{\rb{u}_e \cdot \nab } \rb{u}_e } = - \nab p_e + \mu \nabla^2 \rb{u}_e, \label{eq:NSecs}
\eq
where $\rho$ and $\mu$ are the fluid density and viscosity respectively. On the lumen/wall and wall/ECS boundaries we prescribe conservation of fluid flow and continuity of pressure, so that
\bq
\rb{u}_l \cdot \n_l = \rb{u}_w \cdot \n_l \ \mbox{and} \ p_l= p_w \ \mbox{on the lumen/wall boundary}, \quad \rb{u}_w \cdot \n_w = \rb{u}_e \cdot \n_w \ \mbox{and} \ p_w = p_e \ \mbox{on the wall/ECS boundary}, \label{eq:consmassbw} 
\eq
where $\n_l$ and $\n_w$ are the unit outward pointing normals to the lumen/wall and wall/ECS boundaries respectively. The slip boundary conditions on the lumen/wall and wall/ECS boundaries are
\begin{align}
\left[ \brac{\n_l \cdot \nab} \rb{u}_l \right] \cdot \boldsymbol{\tau}_l &= - \frac{\alpha}{\sqrt{k}}\brac{ \rb{u}_l - \rb{u}_w} \cdot \boldsymbol{\tau}_l  \quad \mbox{on the lumen/wall boundary}, \label{eq:beaverslumen} \\
\left[ \brac{\n_w \cdot \nab} \rb{u}_e \right] \cdot \boldsymbol{\tau}_w &= - \frac{\alpha}{\sqrt{k}} \brac{\rb{u}_e - \rb{u}_w} \cdot \boldsymbol{\tau}_w \quad \mbox{on the wall/ECS boundary}. \label{eq:beaversECS}
\end{align}
Here $\boldsymbol{\tau}_l$ and $\boldsymbol{\tau}_w$ are the unit tangential vectors to their respective surfaces, and $\alpha$ is a dimensionless slip constant that depends on the surface properties. 

No flux of fluid out of the ECS at $z=0$ and $z=L$ (the ECS is glued at the ends) yields
\bq
\rb{u}_e \cdot \rb{e}_z = 0 \quad \mbox{on} \ z=0,L, \label{eq:nofluxw}
\eq
where $\rb{e}_z$ is the unit vector in the $z$--direction. The boundary conditions on the outer ECS boundary are 
\bq
p_e = P_0 \ \mbox{and} \ \rb{u}_e \cdot \boldsymbol{\tau}_e = 0, \label{eq:ecsoutlet1}
\eq
where $P_0$ is atmospheric pressure. 

Experimentally we prescribe the volumetric flowrate of fluid into the lumen
\bq
\int_{\mbox{lumen cross-section}} \left. \rb{u}_l \cdot \n \right|_{z=0} \, \ds = Q_{l,in} ,
\eq
and the pressure at the lumen outlet,
\bq
p_l = P_1  \quad \mbox{on} \ z=L .
\eq

The volumetric flowrates of fluid leaving the lumen and ECS are given by
\bq
\int_{\mbox{lumen cross-section}} \left. \rb{u}_l \cdot \n \right|_{z=L} \, \ds = Q_{l,out} \quad \mbox{and} \quad \int_{\mbox{outer ECS boundary}}  \rb{u}_e \cdot \n \, \ds = Q_{e,out}. \label{eq:lumenECSoutlets}
\eq


Next, we mathematically reduce the full system of equations given by equations (\ref{eq:NSlumen})--(\ref{eq:lumenECSoutlets}) to determine analytical expressions for the flow velocities and pressure in each of the lumen, lumen wall and ECS.

Firstly, we assume that the fibre is positioned symmetrically in the extra-lumen space and ignore the asymmetrical position of the outlet. Therefore we move to a radially symmetric setup described in  cylindrical polar coordinates (defined by $x=r \cos \theta$, $y=r \sin \theta$ and $z=z$), neglecting $\theta$--dependence so that $\rb{u} = \brac{u_{r} \brac{r,z}, 0 , u_{z} \brac{r,z}}$. 

We non-dimensionalize the system to reduce the number of parameters and to enable us to estimate the relative importance of the various terms. We set
\bq
r = d r^{\prime}, \ z=L z^{\prime}, \ u_{z} = U u_{z}^{\prime}, \ u_{r} = \epsilon U u_{r}^{\prime}, \ p=P p^{\prime} + P_0 ,
\label{eq:scalings}
\eq
where $d$ is the lumen radius, $L$ is the fibre length, $U$ is a typical lumen flow velocity, $\epsilon=d/L$ is the aspect ratio of a fibre, $P$ is the pressure scale and $P_0 = 14.50$ psia is atmospheric pressure. The pressure scale $P$ must be chosen appropriately in each component of the module to maintain the correct balance of physical effects at leading order. In the lumen, we pick $P$ so that the pressure and viscous terms balance (as would be anticipated for flow in a thin channel),
\bq
P = \frac{\mu U }{\epsilon^2 L} .
\eq
Given that $\epsilon \ll 1$ and $\epsilon^2 \Rey \ll 1$, it is appropriate to use lubrication theory to simplify the system \cite{viscousflow}. The leading-order lubrication equations in the lumen are now given by
\bq
\frac{1}{r} \pd{}{r} \brac{r u_{l,r}} + \pd{u_{l,z}}{z} = 0, \quad \frac{1}{r} \pd{}{r} \brac{ r \pd{u_{l,z}}{r}} = \od{p_l}{z}, \label{eq:lumenlub}
\eq
where the lumen pressure $p_l$ only varies in the axial direction $z$ (\textit{i.e.} it is constant radially).

In the lumen wall, non-dimensionlization using the scalings (\ref{eq:scalings}) yields
\bq
\frac{1}{r} \pd{}{r} \brac{r u_{w,r}} + \pd{u_{w,z}}{z} = 0 , \quad u_{w,r} = - \frac{k P}{\mu \brac{d} \epsilon U} \pd{p_w}{r} , \quad u_{w,z} = -\frac{k P}{\mu L U} \pd{p_w}{z} . \label{eq:darcylub} 
\eq
Given that $k$ is unknown, the size of the coefficients in equations (\ref{eq:darcylub}) are undetermined. In the experimental setup, we adjust the lumen outlet pressure (using a clamp to provide back pressure) to provide a sizeable radial flow through the porous wall; therefore we choose the coefficient of $\partial p_w / \partial r$ to be $\order{1}$ through the pressure scaling
\bq
P = \frac{\mu d \epsilon U}{k} .
\label{eq:kappadefinition}
\eq
Assuming that $\epsilon^2 \ll 1$, the leading-order equations are
\bq
\frac{1}{r} \pd{}{r} \brac{r \pd{p_w}{r}} = 0 , \quad u_{w,r} = - \pd{p_w}{r} , \label{eq:walllub}
\eq
together with $u_{w,z} = \order{\epsilon^2}$. 

In the ECS, we similarly pick the pressure scale $P$ to maintain an order one pressure gradient in the radial direction,
\bq
P = \frac{\mu U}{L}.
\eq
Assuming that $\epsilon \ll 1$ and $\epsilon^2 \Rey \ll 1$, the leading-order lubrication equations in the ECS are then given by
\bq
\frac{1}{r} \pd{}{r} \brac{r u_{e,r}} + \pd{u_{e,z}}{z} = 0 , \quad \pdd{u_{e,r}}{r} + \frac{1}{r} \pd{u_{e,r}}{r} - \frac{u_{e,r}}{r^2} = \pd{p_e}{r} , \quad \frac{1}{r} \pd{}{r} \brac{ r \pd{u_{e,z}}{r}} =0 . \label{eq:ECSlub}
\eq

Next we consider the boundary conditions. In this radially-symmetrical setup, the lumen/wall, wall/ECS and ECS outer boundaries are given in dimensionless terms by $r=1$, $r=1+a$ and $r=1+a+b$ where $a=s/d$ and $b=l/d$. The unit outward pointing normal vectors to the lumen/wall and wall/ECS boundaries are $\n_l = \rb{e}_r$ and $\n_w= -\rb{e}_r$, whereas the unit tangential vectors are $\pmb{\tau}_l = \pmb{\tau}_w = \rb{e}_z$. Upon non-dimensionalization, the conservation of fluid flow boundary conditions in (\ref{eq:consmassbw}) yield
\bq
u_{l,r} = - \pd{p_w}{r} \quad \mbox{on} \ r=1 , \quad u_{e,r} = - \pd{p_w}{r} \quad \mbox{on} \ r=1+a. \label{eq:consmasslub}
\eq
The continuity of pressure boundary conditions in (\ref{eq:consmassbw}) become
\bq
p_l = \frac{\epsilon^2 \brac{d}^2}{k} p_w  \quad \mbox{on} \ r=1 , \quad p_w = \frac{k}{L d \epsilon} p_e  \quad \mbox{on} \ r=1+a .
\eq
We define $\kappa = \epsilon^2 \brac{d}^2 /  k = \order{1}$ so that 
\bq
p_l = \kappa p_w \ \mbox{on} \ r=1 , \quad \kappa p_w = \epsilon^2 p_e  \ \mbox{on} \ r=1+a.
\eq
The leading order pressure boundary conditions therefore reduce to
\bq
p_l = \kappa p_w  \ \mbox{on} \ r=1 , \quad p_w = 0  \ \mbox{on} \ r=1+a . \label{eq:bc4} 
\eq
Physically, this means that the bulk of the pressure drop from the lumen inlet to the ECS outlet occurs across the lumen wall. Consequently, the pressure in the ECS will be significantly smaller than that in the lumen and wall.

The slip conditions (\ref{eq:beaverslumen})--(\ref{eq:beaversECS}) become
\bq
\pd{u_{l,z}}{r} = - \hat{\alpha} \brac{\epsilon} \brac{u_{l,z}  + \epsilon^2 \pd{p_w}{z}}  \ \mbox{on} \ r=1, \quad \pd{u_{e,z}}{r} = \hat{\alpha} \brac{\epsilon} \brac{u_{e,z}  + \epsilon^2 \pd{p_w}{z}} \ \mbox{on} \ r=1+a,
\eq
where $\hat{\alpha} \brac{\epsilon} = d \alpha / \sqrt{k}$ is a dimensionless parameter that represents the importance of slip versus wall permeability on the tangential component of velocity in the lumen/ECS. The parameter $\alpha$ is unknown, and so we must decide which scaling of $\alpha$ with $\epsilon$ is appropriate; we pick $\hat{\alpha} = \order{1}$ as the distinguishing limit that retains the most physical features at leading order (and note that taking the limit $\hat{\alpha} \rightarrow \infty$ corresponds to the typical no-slip condition). Given that $\epsilon^2 \ll 1$, the leading-order slip boundary conditions are
\bq
\pd{u_{l,z}}{r} = - \hat{\alpha} u_{l,z} , \quad \pd{u_{e,z}}{r} = \hat{\alpha} u_{e,z} . \label{eq:sliplub}
\eq
Upon non-dimensionalization, the pressure and velocity conditions on the outer ECS boundary become
\bq
p_e = 0 \ \mbox{and} \ u_{e,z} = 0 \ \mbox{on} \ r=1+a+b. \label{eq:ecsoutlet}
\eq
The no-flux boundary conditions on the ECS walls (\ref{eq:nofluxw}) yield
\bq
u_{e,z} =0 \ \mbox{on} \ z=0, 1. 
\eq
The experimentally imposed boundary conditions become
\bq
\int_{r=0}^1 \left. r u_{l,z} \right|_{z=0} \, \ro{d}r = \frac{Q_{l,in}}{2 \pi d^2 U} = \hat{Q}_{l,in} , \quad p_l = \hat{P}_1 = \frac{\epsilon^2 L}{\mu U} \brac{P_1 - P_0} \quad \mbox{on} \ z=1, \label{eq:imposedlub}
\eq
together with $p_e = 0$ on the ECS outlet. The experimentally measured flowrates are now given by
\bq
\int_{r=0}^1 \left. r u_{l,z} \right|_{z=1} \, \ro{d}r = \frac{Q_{l,out}}{2 \pi d^2 U }= \hat{Q}_{l,out} , \quad \int_{z=0}^1 \left. r u_{e,r} \right|_{r=1+a+b} \, \ro{d}z = \frac{Q_{e,out}}{2 \pi L d U \epsilon} = \hat{Q}_{e,out}.
\eq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Model Solution}
Next we solve the leading-order equations that were derived in the previous Section. Given that the lumen pressure $p_l$ is independent of $r$, we can integrate the second equation of (\ref{eq:lumenlub}) to give
\bq
u_{l,z} = \frac{r^2}{4} \od{p_l}{z} + A \brac{z} \ln r + B \brac{z} .
\eq
The integration functions $A \brac{z}$ and $B \brac{z}$ are determined using the fact that $u_{l,z}$ must be defined at $r=0$, together with the slip boundary conditions for $u_{l,z}$ from (\ref{eq:sliplub}), respectively. This yields
\bq
u_{l,z} = \frac{1}{4 \hat{\alpha}} \od{p_l}{z} \brac{\hat{\alpha} r^2 - \hat{\alpha} - 2} .
\eq
The radial component of the velocity in the lumen, $u_{l,r}$ can now be determined from the first equation of (\ref{eq:lumenlub}) to give
\bq
u_{l,r} = - \frac{r}{16 \hat{\alpha}} \odd{p_l}{z} \brac{ \hat{\alpha} r^2 - 4 - 2 \hat{\alpha}} + \frac{C \brac{z}}{r}.
\eq
However, $u_{l,r}$ must be finite at $r=0$ therefore $C = 0$ and so
\bq
u_{l,r} = - \frac{r}{16 \hat{\alpha}} \odd{p_l}{z} \brac{ \hat{\alpha} r^2 - 4 - 2 \hat{\alpha}} .
\eq

Next we consider the lumen wall; the pressure equation of (\ref{eq:walllub}) may be integrated with respect to $r$ to give
\bq
p_w = D \brac{z} \ln r + E \brac{z}.
\eq
The integration functions $D \brac{z}$ and $E \brac{z}$ are determined using the conservation of mass boundary condition on $r=1$ from (\ref{eq:consmasslub}), together with the continuity of pressure boundary condition on $r=1$ from (\ref{eq:bc4}). This yields
\bq
p_w = - \frac{1}{16 \hat{\alpha}} \odd{p_l}{z} \brac{\hat{\alpha} + 4} \ln r + \frac{p_l}{\kappa} .
\eq
This also determines the radial component of the velocity in the wall as
\bq
u_{w,r} = \frac{1}{16 \hat{\alpha}} \odd{p_l}{z} \frac{\brac{\hat{\alpha} + 4}}{r}.
\eq
Finally, the wall pressure $p_w$ must satisfy boundary condition (\ref{eq:bc4}) on the wall $r=1+a$, and this provides a mechanism for determining the pressure in the lumen through
\bq
\odd{p_l}{z}  - \frac{16 \hat{\alpha}}{\brac{\hat{\alpha} +4} \ln \brac{1+a} \kappa} p_l = 0.
\label{eq:odepl}
\eq
Equation (\ref{eq:odepl}) is a second-order differential equation that may be solved for $p_l$, subject to two boundary conditions. These conditions are the lumen inlet flowrate and pressure outlet conditions (\ref{eq:imposedlub}) which simplify to
\begin{align}
\left. \od{p_l}{z} \right|_{z=0} = - \frac{16 \hat{Q}_{l,in} \hat{\alpha}}{4 + \hat{\alpha}}, \quad \left. p_l \right|_{z=1} = \hat{P}_1 .
\label{eq:plbcs}
\end{align}
Defining the coefficient
\bq
\lambda^2 = \frac{16 \hat{\alpha}}{\brac{\hat{\alpha} + 4} \ln \brac{1+a} \kappa},
\eq
then the solution of (\ref{eq:odepl})--(\ref{eq:plbcs}) is
\bq
p_l = \frac{\hat{P}_1}{\cosh \lambda} \cosh \brac{\lambda z} + \frac{16 \hat{Q}_{l,in} \hat{\alpha}}{\lambda \cosh \lambda \brac{ \hat{\alpha} + 4}} \sinh \brac{\lambda \brac{1-z}}.
\eq

Finally, we consider the ECS. Integrating the final equation of (\ref{eq:ECSlub}) for the axial velocity component $u_{e,z}$ yields
\bq
u_{e,z} = F \brac{z} \ln r + G \brac{z} .
\eq
Applying the slip boundary condition (\ref{eq:sliplub}) for $u_{e,z}$ together with the ECS velocity outlet condition (\ref{eq:ecsoutlet}) yields $F\brac{z} = 0 = G \brac{z}$ and consequently
\bq
u_{e,z} = 0,
\eq
so there is no axial component of velocity in the ECS. Integrating the first equation of (\ref{eq:ECSlub}) for $u_{e,r}$ yields
\bq
u_{e,r} = \frac{H \brac{z}}{r} .
\eq
Similarly, the pressure in the ECS is determined by integrating the second equation of (\ref{eq:ECSlub}) to give
\bq
p_e = K \brac{z} .
\eq
However, the pressure is zero at the outlet from (\ref{eq:ecsoutlet}) and therefore $K\brac{z} = 0$ so that
\bq
p_e = 0 ,
\eq
and the pressure is constant through the ECS. Finally, the conservation of mass boundary condition (\ref{eq:consmasslub}) for $u_{e,r}$ determines the final integration function $H\brac{z}$; this yields
\bq
u_{e,r} = \frac{\brac{\hat{\alpha} + 4}}{16 \hat{\alpha}} \odd{p_l}{z} \frac{1}{r}. 
\eq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pagebreak

\section{Raw Data}\label{rawdata}

\begin{table}[htb!]\centering
\begin{tabular}{|c|c|c|c|}
\hline
Time (mins) & Mass of Retentate & Mass of Permeate & Lumen Outlet Pressure (psig) \\ 
 &  Collected (g)  &  Collected (g)  & \\ \hline
$1$ & $2.91$ & $0.21$  & $2.3$   \\ \hline
$2$ & $2.93$ & $0.22$ &  $2.3$ \\ \hline
$3$ & $2.90$ & $0.23$ &   $2.3$ \\ \hline
$4$ & $2.89$ & $0.18$  &   $2.3$\\ \hline
$5$ & $2.93$ & $0.22$ &   $2.3$\\ \hline
$6$ & $2.94$ & $0.22$ &   $2.3$\\ \hline
\end{tabular}
\caption{Raw data for an inlet flowrate of $3.13$ ml/min}
\label{tab:raw3.25}
\end{table}

\begin{table}[h!]\centering
\begin{tabular}{|c|c|c|c|}
\hline
Time (mins) & Mass of Retentate & Mass of Permeate & Lumen Outlet Pressure (psig) \\ 
 &  Collected (g)  &  Collected (g)  & \\ \hline
$1$ & $4.36$ & $0.29$  & $3.0$   \\ \hline
$2$ & $4.39$ & $0.25$ &  $3.0$ \\ \hline
$3$ & $4.38$ & $0.30$ &   $3.2$ \\ \hline
$4$ & $4.33$ & $0.30$  &   $3.2$\\ \hline
$5$ & $4.31$ & $0.31$ &   $3.2$\\ \hline
$6$ & $4.38$ & $0.32$ &   $3.2$\\ \hline
$7$ & $4.38$ & $0.34$ &   $3.2$\\ \hline
\end{tabular}
\caption{Raw data for an inlet flowrate of $4.66$ ml/min}
\label{tab:raw4.8}
\end{table}

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