2.1. Governing equations

The ratio between the vertical and horizontal domain lengths is $\epsilon = {\hat {X}} /{\hat {Z}}$. Using the parameter values given in table 1, we find that $\epsilon \ll 1$. Moreover, the values in table 1 yield a small reduced Reynolds number $\epsilon ^2Re =\epsilon ^2\hat {\rho }\hat {W}_0 {\hat {Z}}/{\hat {\mu }}$. We therefore use the Stokes flow equations to describe the flow in $\hat {\varOmega }_1$ and $\hat {\varOmega }_2$:

(2.3a)\begin{gather} \hat{\mu} \hat{\nabla}^2 \hat{{\boldsymbol{u}}} - \hat{\boldsymbol\nabla} {\hat{p}} = 0, \end{gather}

(2.3b)\begin{gather}\hat{\boldsymbol\nabla} \boldsymbol{\cdot} \hat{{\boldsymbol{u}}} = 0, \end{gather}

where $\hat {{\boldsymbol {u}}}({\hat {x}},{\hat {z}})=({\hat {u}}({\hat {x}},{\hat {z}}),\hat {w}({\hat {x}},{\hat {z}}))$ is the velocity, ${\hat {p}}$ is the pressure and $\hat {\mu }$ is the constant fluid viscosity. We seek steady solutions in both subdomains denoting the variables in ${\hat {\varOmega }}_1$ by $({\hat {u}}_1,\hat {w}_1,{\hat {p}}_1)$ and those in ${\hat {\varOmega }}_{2}$ by $({\hat {u}}_{2},\hat {w}_{2},{\hat {p}}_{2})$.

As described above, the flow enters the domain along ${\hat {z}}=0$ into ${\hat {\varOmega }}_1$, passes through the membrane at ${\hat {x}} ={\hat {m}}({\hat {z}})$ and exits along ${\hat {z}}={\hat {Z}}$ from ${\hat {\varOmega }}_{2}$, driven by a pressure gradient. The boundary conditions at the inlet and outlet are

(2.4a)\begin{gather} {\hat{p}}_1 = {\hat{p}}_{{in}} \quad \mbox{at } {\hat{z}}=0,\quad {\hat{x}}\in[0,{\hat{m}}(0)], \end{gather}

(2.4b)\begin{gather}{\hat{p}}_{2} = {\hat{p}}_{{out}} \quad \mbox{at } {\hat{z}}={\hat{Z}},\quad {\hat{x}}\in[{\hat{m}}({\hat{Z}})+{\hat{h}}, {\hat{X}}]. \end{gather}

There are closed ends at ${\hat {z}}={\hat {Z}}$ in ${\hat {\varOmega }}_{1}$ and at ${\hat {z}}=0$ at ${\hat {\varOmega }}_{2}$ through which the flow cannot penetrate. The following boundary conditions enforce no penetration as well as no slip

(2.5a)\begin{gather} \hat{{\boldsymbol{u}}}_1 = \textbf{0} \quad \mbox{at } {\hat{z}}={\hat{Z}},\quad {\hat{x}}\in[0,{\hat{m}}({\hat{Z}})], \end{gather}

(2.5b)\begin{gather}\hat{{\boldsymbol{u}}}_{2} = \textbf{0} \quad \mbox{at } {\hat{z}}=0,\quad {\hat{x}}\in[{\hat{m}}(0)+{\hat{h}}, {\hat{X}}]. \end{gather}

Since the entire filter consists of a periodic array of modules, we impose symmetry conditions across ${\hat {x}}=0$ and ${\hat {x}}={\hat {X}}$, yielding

(2.6a)\begin{gather} {\hat{u}}_1 = 0, \quad \frac{{\partial} \hat{w}_1}{{\partial} {\hat{x}}} = 0 \quad \mbox{at } {\hat{x}}=0, \end{gather}

(2.6b)\begin{gather}{\hat{u}}_{2} = 0, \quad \frac{{\partial} \hat{w}_{2}}{{\partial} {\hat{x}}} = 0 \quad \mbox{at } {\hat{x}}={\hat{X}}. \end{gather}

The normal flow through the filter is modelled as a porous flow (i.e. governed by the pressure difference and permeance). Thus, employing Darcy flow across the membrane provides the boundary condition for the velocity in ${\hat {\varOmega }}_1$

(2.7)\begin{equation} \hat{{\boldsymbol{u}}}_1 \boldsymbol{\cdot} {\boldsymbol{n}} = {\hat{\kappa}} \left[{\hat{p}}_1({\hat{m}},{\hat{z}}) - {\hat{p}}_{2}({\hat{m}}+{\hat{h}},{\hat{z}})\right] \quad \mbox{at } {\hat{x}}={\hat{m}}({\hat{z}}), \end{equation}

where ${\boldsymbol {n}}=(\cos {\hat {\beta }}, \sin {\hat {\beta }})$ is the unit normal vector to the membrane and ${\hat {\kappa }} = {\hat {k}}/\hat {\mu } {\hat {h}}$ is the membrane permeance, where ${\hat {k}}$ is the permeability of the membrane. Note that the flow resistance due to the membrane is ${\hat {R}} = 1/{\hat {\kappa }}$.

A flux balance through the membrane provides the corresponding boundary condition for the velocity in ${\hat {\varOmega }}_{2}$

(2.8)\begin{equation} \hat{{\boldsymbol{u}}}_{1}|_{{\hat{x}}={\hat{m}}} = \hat{{\boldsymbol{u}}}_{2}|_{{\hat{x}}={\hat{m}}+{\hat{h}}}. \end{equation}

Finally, we need conditions at the permeable membrane layer. While Beavers & Joseph (Reference Beavers and Joseph1967) provide an appropriate tangential slip-flow boundary condition, Griffiths, Howell & Shipley (Reference Griffiths, Howell and Shipley2013) show that including slip does not have a significant effect on the flow. Hence, for simplicity, we impose no slip on both sides of the membrane

(2.9a)\begin{gather} \hat{{\boldsymbol{u}}}_1 \boldsymbol{\cdot} {\boldsymbol{t}} = 0 \quad \mbox{at } {\hat{x}}={\hat{m}}({\hat{z}}), \end{gather}

(2.9b)\begin{gather}\hat{{\boldsymbol{u}}}_{2}\boldsymbol{\cdot} {\boldsymbol{t}} = 0 \quad \mbox{at } {\hat{x}}={\hat{m}}({\hat{z}})+{\hat{h}}, \end{gather}

where ${\boldsymbol {t}}=(-\sin {\hat {\beta }}, \cos {\hat {\beta }})$ is the unit tangent vector to the membrane.

2.2. Dimensionless model

We non-dimensionalise the system (2.3)–(2.9) by introducing the following scalings:

(2.10a–f)\begin{equation} {\hat{z}} = {\hat{Z}} z, \quad {\hat{x}} = \epsilon {\hat{Z}} x, \quad {\hat{\beta}} = \epsilon \beta, \quad {\hat{u}} = \epsilon \hat{W}_0 u, \quad \hat{w} = \hat{W}_0 w, \quad {\hat{p}} = \frac{{\hat{\mu}} \hat{W}_0}{\epsilon^2 {\hat{Z}}} p+{\hat{p}}_{{out}}; \end{equation}

recalling that $\epsilon = {\hat {X}}/{\hat {Z}}$. Note that the pressure difference is directly related to the velocity scale $W_0$ through (2.10a–f). Applying (2.10a–f) to (2.3) and taking the limit $\epsilon \to 0$ yields

(2.11a–c)\begin{equation} p_x = 0, \quad w_{xx} - p_z = 0, \quad u_x + w_z = 0. \end{equation}

Hence, in both subdomains the lubrication equations govern the flow.

The leading-order position of the membrane (2.1) becomes

(2.12)\begin{equation} m(z) = a - \tfrac{1}{2}h + \beta\left(\tfrac{1}{2}-z\right), \end{equation}

where $m={\hat {m}}/{\hat {X}}$, $a={\hat {a}}/{\hat {X}}$, $h={\hat {h}}/{\hat {X}}$ and the dimensionless membrane angle is $\beta \in [0,1]$. The corresponding dimensionless subdomains from (2.2) are given by

(2.13a)\begin{gather} \varOmega_1 = \{ x\in[0,m(z)],\ z\in[0,1]\}, \end{gather}

(2.13b)\begin{gather}\varOmega_{2} = \{ x \in [m(z)+h, 1],\ z\in[0,1]\}. \end{gather}

The dimensionless domain is identical to that shown in figure 1(b) with the hats removed and with the dimensionless vertical and horizontal lengths such that $X=Z=1$. Recall that, while the domain size specified by $Z$ and $X$ is fixed, the membrane length varies with $\beta$.

Applying the scaling (2.10a–f) to the boundary conditions prescribed above, the pressure conditions (2.4) become

(2.14a)\begin{gather} p_1 = 1 \quad \mbox{at } z=0,\quad x\in[0,m(0)], \end{gather}

(2.14b)\begin{gather}p_{2} = 0 \quad \mbox{at } z=1,\quad x\in[m(1)+h, 1], \end{gather}

and the boundary conditions at the closed walls (2.5) become

(2.15a)\begin{gather} w_1 = 0 \quad \mbox{at } z=1,\quad x\in[0,m(1)], \end{gather}

(2.15b)\begin{gather}w_{2} = 0 \quad \mbox{at } z=0,\quad x\in[m(0)+h, 1]. \end{gather}

We note that information regarding the tangential velocity on the closed walls in (2.15) is lost since we are working in the lubrication regime. The dimensionless symmetry conditions from (2.6) are

(2.16a)\begin{gather} u_1 = 0, \quad \frac{{\partial} w_1}{{\partial} x} = 0 \quad \mbox{at } x=0, \end{gather}

(2.16b)\begin{gather}u_{2} = 0, \quad \frac{{\partial} w_{2}}{{\partial} x} = 0 \quad \mbox{at } x = 1. \end{gather}

We define the dimensionless permeance by $\kappa = {\hat {\mu }} {\hat {\kappa }}/\epsilon ^3 {\hat {Z}}$, and the dimensionless permeability by $k = {\hat {k}}/\epsilon ^3 {\hat {Z}} {\hat {X}}$. With these definitions, the dimensionless permeance is $\kappa = k/h$. The Darcy flow condition through the membrane (2.7) becomes

(2.17)\begin{equation} u_1 =\kappa [\kern0.08em p_1(m,z) - p_{2}(m(z)+h,z)] \quad \mbox{at } x=m(z), \end{equation}

where the dimensionless membrane resistance is $R=1/\kappa$. The flux-balance boundary condition across the membrane (2.8) becomes

(2.18)\begin{equation} u_{1}|_{x=m} = u_{2}|_{x=m+h}. \end{equation}

Finally, the dimensionless no-slip condition (2.9), exploiting the small angle of the membrane, is

(2.19a)\begin{gather} w_1 = 0 \quad \mbox{at } x=m(z), \end{gather}

(2.19b)\begin{gather}w_{2} = 0 \quad \mbox{at } x=m(z)+h. \end{gather}

2.3. Model reduction

Since $p_1=p_1(z)$ and $p_2=p_2(z)$ from the momentum equation in the $x$-direction (2.11a), we can reduce the full problem (2.11)–(2.19) to modified Reynolds equations. Since we have two coupled domains, we derive two coupled second-order ordinary differential equations (ODEs) for the pressure; one ODE for each subdomain. The details are as follows.

In the region $\varOmega _1$ we solve the momentum equation in the $z$-direction (2.11b) and the continuity equation (2.11c) with the symmetry and no-slip boundary conditions (2.16a) and (2.19a) to find the following relationships between the velocities and the pressure:

(2.20a)\begin{gather} u_1 = \tfrac{1}{2}x\left[(\kern0.08em p_1' m^2)' - \tfrac{1}{3}x^2 p_{1}''\right], \end{gather}

(2.20b)\begin{gather}w_1 = \tfrac{1}{2}\kern0.08em p_1' (x^2 - m^2), \end{gather}

where $'$ denotes differentiation with respect to $z$ and recalling that $m=m(z)$ given in (2.12). We derive an equation for the pressure in $\varOmega _1$ by applying the Darcy flow boundary condition at the membrane (2.17)

(2.21)\begin{equation} \tfrac{1}{3}\left(\kern0.08em p_1'm^3 \right)' = \kappa\left(\kern0.08em p_1 - p_{2}\right). \end{equation}

The inlet pressure and closed end at $z=1$ from (2.14a) and (2.15a) provide the two necessary boundary conditions for the pressure in $\varOmega _1$

(2.22a)\begin{gather} p_1 = 1 \quad \mbox{at } z=0, \end{gather}

(2.22b)\begin{gather}p_1' = 0 \quad \mbox{at } z=1. \end{gather}

Similarly, in $\varOmega _{2}$, we solve the momentum (in the $z$-direction) and continuity equations (2.11b) and (2.11c) using the symmetry and no-slip boundary conditions (2.16b) and (2.19b) to derive

(2.23a)\begin{gather} u_{2} = \tfrac{1}{2}(x-1)\left[\left(\kern0.08em p_2' (m+h-1)^2 \right)'- \tfrac{1}{3}(x-1)^2 p_{2}''\right], \end{gather}

(2.23b)\begin{gather}w_{2} = \tfrac{1}{2} p_{2}' \left[ (x-1)^2 - (m+h-1)^2\right]. \end{gather}

The flux-balance boundary condition across the membrane (2.18) provides an equation for the pressure in $\varOmega _2$

(2.24)\begin{equation} \tfrac{1}{3}\left[p_{2}'(m+h-1)^3 \right]' = \kappa\left(\kern0.08em p_1 - p_{2}\right). \end{equation}

The governing equation (2.24) is coupled to the following boundary conditions, which arise from the outlet pressure (2.14b) and closed end at $z=0$ (2.15b):

(2.25a)\begin{gather} p_{2} = 0 \quad \mbox{at } z=1, \end{gather}

(2.25b)\begin{gather}p_{2}' = 0 \quad \mbox{at } z=0. \end{gather}

Thus, the problem is reduced to solving (2.21) and (2.22) for $p_1$ in $\varOmega _1$ and (2.24) and (2.25) for $p_{2}$ in $\varOmega _{2}$. The parameters in the reduced system are the membrane permeance $\kappa$, the membrane thickness $h$ and the position of the membrane described by $a$ and $\beta$ in (2.12).

Our goal is to understand how to maximise the flux through the filter for an applied pressure difference across the filter. As such, it is helpful to define the dimensionless flux through the system, calculated at the inlet $z=0$ as follows:

(2.26)\begin{equation} Q = \int_{0}^m w_1|_{z=0}\,{\textrm{d}x} ={-}\tfrac{1}{3} \left(\kern0.08em p_1'm^3\right) |_{z=0}. \end{equation}

The system parameters are $a$, $\beta$, $\kappa$ and $h$; our goal is therefore to find the system set-up that maximises $Q(a,\beta ,\kappa ,h)$. We note that the analysis of this system is much clearer in the limit of small $h$. Moreover, this limit is a regular perturbation and a finite $h$ will only slightly change the quantitative results. We consequently consider the case $h = 0$ in § 3. However, as is realistic for the physical problem, we examine the quantitative changes that arise from accounting for $h>0$ in § 4.

3.1. Slightly angled membrane: $\beta \ll 1$

We first consider the limit in which the membrane is slightly angled away from the horizontal. Since we are able to make analytical progress in this limit, this provides insight into the benefit of angling membranes. Within this limit, there are two important sub-limits, corresponding to the membrane being either far from or close to the symmetry line at $x=0$. Mathematically, these correspond to $a = O(1)$, for a centred membrane, and $a = O(\beta )$, for a membrane positioned near the corner of the domain.

3.1.1. Membranes positioned close to the centre: $a = O(1)$

We start by considering the limit $\beta \ll 1$ with $a = O(1)$, where the membranes are slightly angled but well separated. This limit is a regular perturbation of the problem considered in Herterich et al. (Reference Herterich, Xu, Field, Vella and Griffiths2017), but imposing a constraint on the pressure difference across the filter rather than the flux. We pose the following asymptotic expansions:

(3.1a)\begin{gather} p_1 = p_{10} + \beta p_{11} + O(\beta^2), \end{gather}

(3.1b)\begin{gather}p_{2} = p_{20} + \beta p_{21} + O(\beta^2). \end{gather}

Substituting (3.1) into the governing equations (2.21) and (2.22) for $p_1$ and (2.24) and (2.25) for $p_2$ enables derivation of explicit analytic expressions for $p_{10}$, $p_{20}$, $p_{11}$ and $p_{21}$

(3.2a)\begin{gather} p_{10} = a_{11} + a_{12} z + a_{13} \cosh(Mz) + a_{14} \cosh(M(z-1)), \end{gather}

(3.2b)\begin{gather}p_{20} = a_{21} + a_{22} z + a_{23} \cosh(Mz) + a_{24} \cosh(M(z-1)), \end{gather}

(3.2c)\begin{gather}p_{11} = b_{11}(z) + b_{12}(z) z + b_{13}(z) \cosh(Mz) + b_{14}(z) \cosh(M(z-1)), \end{gather}

(3.2d)\begin{gather}p_{21} = b_{21}(z) + b_{22}(z) z + b_{23}(z) \cosh(Mz) + b_{24}(z) \cosh(M(z-1)), \end{gather}

for constants $a_{ij}$ and functions $b_{ij}(z)$ where $i= 1,2$ and $j=1,\ldots ,4$. The details of the calculation and explicit expressions for $a_{ij}$ and $b_{ij}$ are given in Appendix A.1.

Using the asymptotic expansion (3.1) we can derive an expansion for the flux from (2.26)

(3.3)\begin{equation} Q = Q_0+\beta Q_1 + O(\beta^2). \end{equation}

From the solutions $p_{10}$, $p_{20}$, $p_{11}$ and $p_{21}$ in (3.2) we have an explicit analytic form for $Q_0$ and $Q_1$. Importantly, this procedure results in $Q_1 >0$. Therefore, tilting a horizontal membrane (so that $\beta >0$), will always improve the flux. This can be understood physically by noting that tilting elongates the membrane, increasing its surface area and, consequently, providing an easier transport route through the domain.

3.1.2. Membranes positioned close to the corner: $a = O(\beta )$

We now consider the limit where $\beta \ll 1$ and $a = O(\beta )$, where the membrane is still slightly angled but is now positioned close to the corner of the domain (i.e. pairwise close). This regime is not a sub-limit of that studied in § 3.1.1 above, but a distinct limit in of itself. This can be seen mathematically by noting that the asymptotic results in § 3.1.1 may switch asymptotic orders as $a\to 0$. Here, the presence of an apparent corner in the domain is important but the finite extent of the full domain cannot be realised. We derive the following analytical solutions to the governing equations (2.21), (2.22) and (2.24), (2.25) in this regime

(3.4a)\begin{gather} p_1 = \exp\left[ - \left(\frac{3\kappa}{\left(a+\frac{1}{2}\beta\right)^3}\right)^{1/2} z\right], \end{gather}

(3.4b)\begin{gather}p_2 = 0. \end{gather}

The details of the relevant boundary layer analysis are presented in Appendix A.2. The solutions (3.4) show that the pressure drop in this parameter regime only occurs in $\varOmega _1$ over a boundary layer of size $\beta ^{3/2}$. We examine the effect of this result on the resulting flux below in § 3.1.3.

3.1.3. Numerical solutions

In this section we present numerical results to (2.21) and (2.22) for $p_1$ in $\varOmega _1$ and (2.24) and (2.25) for $p_2$ in $\varOmega _2$, using $h=0$ and $\kappa =1$. We solve this system for $p_1$ and $p_2$ using MATLAB's bvp4c solver (figure 2a). Using the solution for the pressures, we can calculate the velocity field in each subdomain using (2.20) and (2.23). A helpful way to visualise the qualitative behaviour of the flow through the domain is through its streamlines (figure 2b). We also present the flow field predicted by our asymptotic results (3.1) and (3.2) for $\beta \ll 1$, $a=O(1)$, derived in § 3.1.1, which shows excellent agreement with the numerical predictions.

Figure 2.

Behaviour of the system governed by (2.21), (2.22) and (2.24), (2.25) for $\beta = 0.05$, $a = 0.5$, $\kappa =1$ and $h=0$. The pressure profiles $p_1$, $p_2$ are shown in (a). The membrane position (dotted green line) as given in (2.12) and streamlines are shown in (b). We show both numerical solutions (solid) and asymptotic solutions (3.1) and (3.2) (dashed) and see excellent agreement.

We compare the flux (defined in (2.26)) predicted by the numerical results to that predicted by our asymptotic analysis for the two cases of $a= O(1)$ (in § 3.1.1) and $a = O(\beta )$ (in § 3.1.2). The physically possible positions depend on the angle of the membrane: $a \in [\beta /2, 1-\beta /2]$. We note, however, that values of $a$ close to the boundaries of this interval are difficult to resolve numerically due to the presence of corners in the domain. We therefore solve for values of $a \in [ \beta /2+\delta _a, 1-\beta /2-\delta _a]$ for some small $\delta _a > 0$, and for all the results presented herein we take $\delta _a = 0.01$. To visualise the results we sample values of $\beta \ll 1$ and plot the resulting flux as a function of $a$.

There is excellent agreement between the numerical and analytic results for $a= O(1)$ derived in § 3.1.1 (figure 3a). Importantly, we see that the flux is symmetric about $a=a^*:= 0.5$ where the midpoint of the membrane coincides with the midpoint of the domain. Moreover, for small $\beta$, the optimal flux occurs exactly at $a=a^*$, and so it is optimal to centre the membrane in the domain.

Figure 3.

Flux $Q$ given by (2.26) calculated from (2.21), (2.22) and (2.24), (2.25). Numerical solutions (solid line) compared with asymptotic results for $\beta \ll 1$ and $a= O(1)$ given by (3.1) and (3.2) (dashed line) and for $\beta ,\ a \ll 1$ given by (3.4) (dotted). Results shown for $a=O(1)$ in (a) and for $a\ll 1$ in (b) with $\kappa =1$, $h=0$ and $\beta = 0.04,\ 0.1,\ 0.16,\ 0.22$.

As we increase $\beta$ (while still keeping $\beta$ small), the curve flattens around $a=a^*$ and the flux becomes fairly insensitive to variations in $a$ near $a^*$ (figure 3a). Eventually, $\partial ^2Q/\partial a^2(a^*,\beta ,1,0)>0$, indicating that the optimal location is no longer at $a=a^*$. However, the benefits of this optimal design over a centred design for $\beta \ll 1$ are marginal. In figure 3(b), we focus on the parameter regime where $a\ll 1$. As in the case when $a=O(1)$, we find that increasing the angle leads to higher fluxes. Moreover, our asymptotic results from § 3.1.2 show good agreement with the numerical results until $\beta$ gets too large.

The analysis of this section reveals two results: first, we can increase the flux through the device for the same applied pressure difference by tilting a horizontal membrane; second, for a small tilt, the flux is maximised when the membrane is located centrally, but as the membrane angle is increased the optimal location moves away from the centre. These observations motivate us to broaden our search to study the behaviour of the system for wider angled membranes in the following section.

3.2. Investigating the full angle domain

We now investigate the full angle domain, and examine the dependency of the optimal membrane position on the membrane angle. We continue to use $h=0$ and $\kappa =1$ in this section, but consider general values of these in § 4. We present numerical solutions to (2.21) and (2.22) for $p_1$ in $\varOmega _1$ and (2.24) and (2.25) for $p_2$ in $\varOmega _2$ for all values of $a$ and $\beta$ in figure 4. Note that $\beta \in [0,1]$ and $a\in [\beta /2, 1-\beta /2]$. As mentioned in the previous section, numerical results are difficult to resolve for this system set-up when corners are present in the domain. We therefore explore the solution landscape for $\beta \in [0,1-\delta _\beta ]$, for some small $\delta _\beta >0$ and extrapolate our results to the full domain $\beta \in [0,1]$. In this paper, for all results shown, we take $\delta _\beta = 0.04$. Our domain for $a$ remains the same as before: $a \in [\beta /2+\delta _a, 1-\beta /2-\delta _a]$, and we extrapolate our conclusions to $a \in [\beta /2, 1-\beta /2]$.

Figure 4.

Solutions to (2.21), (2.22) and (2.24), (2.25). Numerical results (black, solid) and asymptotic solutions from (3.1) and (3.2) (orange, dashed) for $\beta \in [0,1]$, $a \in [\beta /2, 1-\beta /2]$ with $\kappa =1$ and $h=0$. The dash-dotted lines indicate $\beta _b$ and the dotted lines indicate $\beta _c$. Panel (a) shows the flux $Q$ (2.26) for equally spaced $\beta \in [0, 0.87]$ inclusive, and (b) gives the contour plot of $Q$ in $(a,\beta )$-space. The optimal membrane position $a_{max}$ corresponding to the maximum flux achievable $Q_{max}$ (3.6) as a function of angle $\beta$ are shown in (c) and (d) respectively.

In § 3.1.3 we show that the flux increases with $\beta$ for small $\beta$, and this trend remains true for larger values of $\beta$ (figure 4a). Moreover, in the same limit, we also show that the membrane position that maximises the flux moves away from the centre for increasing $\beta$. This trend continues in the full angle domain presented here; the optimal membrane position moves significantly off-centre as $\beta$ increases, bifurcating into two distinct optima symmetric about the centre of the domain for larger values of $\beta$ (figures 4(a) and 4(b)).

We define the position corresponding to the maximum achievable flux for a given angle by $a_{max}(\beta )$. As $\beta$ increases, there is a bifurcation point $\beta = \beta _b$ at which the single, centred optimal design bifurcates into two off-centre optima, which drift further from the centre toward the corners of the domain as $\beta$ increases further (figure 4c). Thus, the bifurcation point $\beta =\beta _b$ marks a fundamental change in the optimal filter design. Due to the reversibility of Stokes flow, if one off-centre optimum exists then a second optimal configuration must exist in a set-up that is symmetric about $a=a^*$.

At a second critical value $\beta = \beta _c > \beta _b$, the two optimal positions reach the corners of the physical domain, where they remain as $\beta$ increases further (figure 4c). We refer to $\beta _b$ and $\beta _c$ as the bifurcation and critical points, respectively. For $\beta > \beta _b$, we define the two optimal (off-centre) positions by $a^-$ and $a^+$ (with $a^- < a^+$ and $a^- + a^+ = 1$). When $\beta > \beta _c$, and the optimal membrane position consists of one membrane end residing at a corner of the domain, and we have $a^-=\beta /2$ and $a^+ = 1-\beta /2$.

As $\beta _b$ is the lowest value of $\beta$ at which there is a local minimum in the flux at $a=a^*$, and recalling that the flux is symmetric about $a = a^*$, $\beta _b$ is implicitly defined by the condition

(3.5)\begin{equation} \frac{{\partial}^2 Q}{{\partial} a^2}(a^*,\beta_b,\kappa,h) = 0, \end{equation}

for a given $\kappa$ and $h$. We determine $\beta _c$ by calculating the lowest value of $\beta$ for which the two off-centre optima are such that $a^-=\beta /2$ and $a^+ = 1-\beta /2$. Finally, to understand how the flux varies as these key values of $\beta$ are crossed, it is helpful to define the maximum flux achievable over $a$-space

(3.6)\begin{equation} Q_{max}(\beta, \kappa,h) = \mathop{\max}\limits_{a} Q(a,\beta,\kappa,h),\quad \mbox{for}\ a\in[\beta/2,1-\beta/2], \end{equation}

which occurs at $a = a_{max}(\beta ,\kappa ,h)$.

For a membrane with permeance $\kappa =1$ and thickness $h=0$, we see that $Q_{max}$ increases as $\beta$ increases (figure 4d). There does not appear to be any change in the qualitative behaviour of $Q_{max}$ around $\beta _b$, and this is consistent with the definition of $\beta _b$ in (3.5). That is, as the first and second derivatives of $Q$ with respect to $a$ both vanish at this critical point, $Q$ is fairly insensitive to $a$ near $a = a^*$, $\beta = \beta _b$. However, as $\beta$ passes through $\beta _c$, there is a change in the qualitative behaviour of $Q_{max}$. In particular, the slope of the curve decreases after this point, although we note that it does remain positive for these parameter values ($\kappa =1$, $h=0$). We will show later in § 4.1 that this is not always the case. Thus there are diminishing returns for $Q_{max}$ in terms of increasing $\beta$ beyond $\beta _c$.

In figures 4(c) and 4(d) we also present the small-$\beta$ results derived in § 3.1.1 alongside the numerical results from the full system. The asymptotic results agree very well with the numerical results for $a_{max}$ and $Q_{max}$ up to $\beta \approx 0.2$, and do predict the general trend thereafter, including the presence and effect of $\beta _b$ and $\beta _c$.

The results presented in this section show that angling the membrane away from horizontal can significantly increase the flux through a direct-flow device for the same applied pressure difference. Specifically, for a membrane with $\kappa =1$ and $h=0$, angling the membrane away from horizontal can provide a 40 % increase in the maximum achievable flux (figure 4d). Moreover, we found that, while the optimal position for slightly tilted angles (with small $\beta$) is in the centre of the domain with $a=a^*$, the optimal position for increasing membrane angles bifurcates into two off-centre positions that move to the corner of the domain. Finally, the analysis of this section reveals the optimal set-up that maximises the flux through the membrane with $\kappa =1$ and $h=0$ to be a membrane centred and diagonal across the full domain (with $a=a^*$ and $\beta = 1$). We now turn to the wider aim of seeking the maximum achievable flux for a membrane of finite thickness and varying permeance by varying both the membrane position $a$ and the angle $\beta$.

4.1. Vanishingly thin membranes: $h= 0$

For a vanishingly thin membrane, we seek the optimal position and angle of a membrane of specified permeance $\kappa$. We solve (2.21), (2.22) and (2.24), (2.25) numerically for $h=0$, and examine the flux (2.26) as we vary $\kappa$. Recall that the dimensionless resistance is simply the reciprocal of the permeance: $R = 1/\kappa$. Therefore, increasing the permeance decreases the resistance, and we observe the expected increase in the maximum flux achievable for a given $\beta$ (figure 5).

Figure 5.

Maximum flux achievable $Q_{max}$ given by (3.6) calculated from (2.21), (2.22), (2.24), (2.25) and (2.26). Results shown as a function of $\beta$ with $h=0$ and $\kappa =1$, 2, 5, 10, 20. The dashed line indicates $\beta _c$.

There is a quantitative change in the profiles of $Q_{max}$ as a function of $\beta$ for increased $\kappa$ (figure 5). For small $\kappa$, the flux increases monotonically with $\beta$ and the optimal flux is achieved for $\beta = 1$ where the membrane is diagonal across the domain and centralised ($a=a^*$). For large $\kappa$, however, the highest flux is achieved when $\beta < 1$.

To examine at what position $a$ this optimal flux is achieved, in this section we consider the critical angle $\beta_c$. We present the analysis of how the corresponding bifurcation angle $\beta_b$ varies with $\kappa$ in Appendix B. Recall that, for $\beta \geq \beta _c$, the maximum flux is achieved when either end of the membrane is positioned in a corner of the domain. We find that the optimal flux is achieved for angles slightly larger than $\beta _c$, and thus, for large $\kappa$, the optimal membrane position involves one end being in a corner of the domain. The associated optimal angle is predicted by the model equations (2.21), (2.22), (2.24), (2.25) and (2.26) through maximising $Q_{max}$ (3.6) as shown in figure 5. This result highlights the underlying physics in the problem. When optimising the flux, there is an inherent trade-off between maximising the length of the membrane and maximising the available space in one subdomain. Maximising the length of the membrane maximises the available surface area for ease of transport through the domain. Maximising the available space in either subdomain increases the transmembrane pressure drop. We find that for small $\kappa$, where the flow resistance is increased, it is more important to maximise the surface area of the membrane. Thus the optimal set-up is for the membrane to be diagonal across the full domain, thereby maximising the membrane length (figure 6a). For large $\kappa$, however, the membrane resistance is lower. In this case, it is more important to maximise the transmembrane pressure drop by shifting the membranes into the corners with increased available space in either subdomain (figure 6b). Thus, for large $\kappa$, the optimal angle is slightly larger than $\beta _c$ which is due to the trade-off between elongating the membrane and maximising the available space in either subdomain (figure 5).

Figure 6.

Schematics of the two dimensionless set-ups associated with optimal flux. The first, achieved for low permeances, is a centred membrane diagonal across full domain with $\beta =1-h$ and $a=a^*$ as shown in (a). The second is an angled membrane in the corner with $\beta <1$ and $a=(\beta +h)/2$ as shown in (b). Note that the optimal configuration shown in (b) has an equivalent optimal configuration for the same optimal angle $\beta <1$ with $a = 1-(\beta +h)/2$. The full concertinaed device comprises repeated modules, and we illustrate this with faded schematics for the modules neighbouring the domain of consideration.

4.2. Membranes of finite thickness

We have so far restricted our attention to infinitely thin membranes, but in practice membranes have a finite albeit small thickness. We now relax this assumption to consider the effect of membrane thickness on the device behaviour. Hence, the new angle domain is $\beta \in [0,1-h]$. We solve (2.21), (2.22) and (2.24), (2.25) numerically. As before, to avoid numerical complications arising from the presence of corners, we solve for $\beta \in [0, 1-h-\delta _\beta ]$ and $a\in [(\beta +h)/2-\delta _a, 1- (\beta +h)/2-\delta _a)]$, for some small $\delta _\beta , \delta _a >0$, and extend our conclusions to the full domain with $\delta _\beta ,\delta _a \to 0$.

In § 4.2.1, we first specify the permeance and focus on the effect of varying $h$. In § 4.2.2, we then examine the combined effect of a finite thickness for varying permeance.

4.2.1. Investigating the effect of membrane thickness

We first examine the effect of a finite membrane thickness by varying $h$ while holding $\kappa$ fixed. This corresponds to filters with different thicknesses but equal permeance, or equivalently equal net resistance. A helpful metric for comparison is $Q_{max}(\beta ,\kappa ,h)$, defined in (3.6), for varying $h$. For fixed permeance (and resistance), with $\kappa =1$, the maximum flux increases monotonically with $\beta$ and decreases as $h$ increases (figure 7a). Fixing the resistance isolates the effect of the geometry; as the membrane thickness increases, the membrane physically takes up more space and consequently inhibits the flow. We then consider the case where we vary $h$ and set $\kappa = 0.01/h$. Experimentally, this corresponds to comparing membranes made of the same material, so the permeability is the same, but the permeance or resistance changes accordingly with thickness. Increasing the thickness and decreasing the permeance (i.e. increasing the resistance) while holding the permeability constant decreases the maximum flux (figure 7b).

Figure 7.

Maximum flux achievable $Q_{max}$ given by (3.6) calculated from (2.21), (2.22), (2.24), (2.25) and (2.26). Results shown for $h=0$, $0.025$, $0.05$, $0.075$, $0.1$ with fixed permeance $\kappa =1$ in (a), and for $h=0.01$, $0.02$, $0.03$, $0.04$, $0.05$ with $\kappa =0.01/h$ (i.e. fixed permeability) in (b).

4.2.2. Quantifying the optimal design over experimental parameter space

Finally, we explore the full parameter space to understand what angle and position maximise the flux through the filter for a membrane of given thickness and permeance. We address this through consideration of the maximum achievable flux defined by

(4.1a,b) \begin{equation} \left.\begin{aligned} Q_{max}^*(\kappa,h) = \max_{a,\beta} Q(a,\beta,\kappa,h) \quad & \mbox{for} \quad \beta \in[0, 1-h], \nonumber\\ & \mbox{and} \ \ \, a\in[(\beta+h)/2,1-(\beta+h)/2], \end{aligned}\right\} \end{equation}

which we say occurs at $\beta = \beta _{max}^*$ and $a=a_{max}^*$. Note that our earlier definition of $Q_{max}$ in (3.6) was introduced to find the optimal position associated with the maximum achievable flux for a membrane of specified angle, thickness and permeance. The definition of $Q_{max}^*$ in (4.1), however, seeks the optimal angle and position for a membrane of specified thickness and permeance.

The optimal angle and position are shown in figure 8. We immediately see that the profiles are discontinuous and this corresponds to the jump between the optimal positions depicted in figure 6. For smaller $\kappa$, the optimal angle is $\beta _{max}^* = 1-h$ with the corresponding optimal position at $a_{max}^* = a^*$ (figure 8). (In fact, figure 8 shows that $\beta _{max}^* = 1-h-\delta _\beta$ which represents our numerically tested domain $\beta \in [0, 1-h-\delta _\beta ]$. We extrapolate our conclusions to the domain $\beta \in [0,1-h]$.) The set-up with $\beta_{\textit{max}}^* = 1-h$ is associated with the maximum achievable flux occurring for a centred membrane diagonal across the domain (as shown in figure 6a). As $\kappa$ increases, there is a critical value of $\kappa$ over which the optimal position jumps from the centre to the corner of the domain with the optimal angle occurring at $\beta _{max}^*< 1-h$, which is determined by our model, and the associated optimal position at $a_{max}^* = (\beta _{max}^*+h)/2$ (as shown in figure 6b). Sweeping across the full $\beta$-space (shown in figure 7) enables us to draw the conclusion that there exists a single optimal angle for all values of $\kappa$ except that at the transition point depicted by the discontinuity in figure 8(a).

Figure 8.

Angle $\beta _{max}^*$ (a) and position $a_{max}^*$ (b) associated with the maximum achievable flux $Q_{max}^*$ defined by (4.1) calculated from (2.21), (2.22), (2.24), (2.25) and (2.26). Results shown as a function of $\kappa$ for different thickness $h=0, 0.025, 0.05, 0.075, 0.1$.

Note that we have, without loss of generality, presented results for the branch of solutions corresponding to $a_{max}^* = (\beta _{max}^*+h)/2$ only, but each off-centre optimal position is associated with a secondary optimal position symmetric about $a=a^*$, at $a_{max}^* = 1-(\beta _{max}^*+h)/2$ for the same $\beta _{max}^*$. Thus, for $\kappa$ above the critical value, the maximum flux is achieved for an angled membrane in the corner of the domain and the angle is determined by our model. Increasing $h$ results in the critical jump occurring for smaller $\kappa$ (figure 8).

The maximum achievable flux $Q_{max}^*$ over a wide range of physical membrane properties is shown in figure 9. We observe that the global maximum $Q_{max}^*$ is achieved for $h \to 0$, $\kappa \to \infty$, with a horizontal membrane positioned in the corner (figures 8 and 9). Physically, this corresponds to maximally spaced infinitely thin membranes with zero resistance. In this case, the flux is maximised due to the pressure drop occurring almost entirely across the membrane. In practice, however, membrane design is limited in permeance and thickness. While figure 9 confirms that thinner, more permeable filters allow for greater flux, it provides a quantitative measure of how to design these filters for given membrane characteristics.

Figure 9.

Maximum achievable flux $Q_{max}^*$ in $(\kappa ,h)$-space defined by (4.1) calculated from (2.21), (2.22), (2.24), (2.25) and (2.26). The dashed line corresponds to the critical value of $\kappa$ at which there is a discontinuous jump from centred membranes to membranes in the corners. To the left of the dashed line, the maximum flux is achieved at centred membranes diagonal across the full domain; and to the right, the maximum flux is achieved at angled membranes in the corners of the domain.

Importantly, for any thickness and permeance, the maximum flux is achieved either for a centred membrane diagonal across the domain ($(\kappa ,h)$-values to the left of the dashed line in figure 9) or for an angled membrane in the corner of the domain ($(\kappa ,h)$-values to the right of the dashed line in figure 9); i.e. the dashed line in figure 9 corresponds to the critical value of $\kappa$ over which there is a discontinuous jump in optimal design. Thus, a practitioner may use the model developed herein for a membrane of specified thickness and permeance to find the maximum achievable flux and the required angle and position that the membrane should take to achieve this flux. Specifically, if the membrane properties are to the left of the dashed line in figure 9, the optimal configuration is centred and diagonal across the domain, with angle $\beta ^*_{max} = 1-h$ and position $a^*_{max} = a^*$ (as depicted in figure 6a). If the membrane properties are to the right of the dashed line in figure 9, the optimal configuration is in either corner, and the angle $\beta ^*_{max}$ associated with maximum flux is provided by figure 8(a). The resulting optimal position is then $a^*_{max}=(\beta ^*_{max}+h)/2$ (or equivalently $a^*_{max}=1-(\beta ^*_{max}+h)/2$) as shown in figure 6(b).