2.1 Poroelasticity

The net stress tensor in a poroelastic medium containing moving fluid is the sum of an elastic stress tensor in the solid matrix and the stress tensor in the fluid. We assume here that the medium is isotropic and homogeneous. The elastic contribution to the stress tensor in Cartesian coordinates $(x_1,\,x_2,\, x_3)$ is

(2.1) $$ \begin{align} \tau^{e}_{ik}=\lambda_{\mathrm{eff}}\,({{\boldsymbol{\nabla}}}\cdot\mathbf{u})\;\delta_{ik}+2\,\mu_{\mathrm{eff}}\,E_{ik}, \end{align} $$

where $\mathbf {u}$ is the displacement vector, ${{\boldsymbol {\nabla }}}$ is the del vector operator and $E_{ik}$ is the strain tensor

$$ \begin{align*} E_{ik}={1\over2}\;\bigg({\partial u_i\over\partial x_k}+{\partial u_k\over\partial x_i}\bigg). \end{align*} $$

The effective Lamé constants, $\lambda _{\mathrm {eff}}$ and $\mu _{\mathrm {eff}}$ are different from the Lamé constants of the material of the matrix. It is assumed that the stress in the fluid averages, on a length scale of many pore sizes, to an isotropic pore fluid pressure P with stress tensor

$$ \begin{align*} \tau^{f}_{ik}=-P\,\delta_{ik}. \end{align*} $$

The net stress tensor of the porous elastic medium is therefore

(2.2) $$ \begin{align} \tau_{ik}=\tau^{e}_{ik}+\tau^{f}_{ik}=(-P+\lambda_{\mathrm{eff}}\,{{\boldsymbol{\nabla}}}\,{\boldsymbol{\cdot}}\,\mathbf{u})\,\delta_{ik}+ \mu_{\mathrm{eff}}\,\bigg({\partial u_i\over\partial x_k}+{\partial u_k\over\partial x_i}\bigg); \end{align} $$

the inertia and body force terms due to gravity are neglected. Under steady conditions, the net stress tensor satisfies the equations of static equilibrium

(2.3) $$ \begin{align} {\partial\;\;\over\partial x_k}\;\tau_{ki}=0,\quad i=1,2,3. \end{align} $$

Substituting (2.2) into (2.3) leads to the Navier displacement equation for static equilibrium

(2.4) $$ \begin{align} \mu_{\mathrm{eff}} \nabla^2 \mathbf{u}+(\lambda_{\mathrm{eff}}+\mu_{\mathrm{eff}})\,{{\boldsymbol{\nabla}}}({{\boldsymbol{\nabla}}}\,{\boldsymbol{\cdot}}\,\mathbf{u})={{\boldsymbol{\nabla}}} P. \end{align} $$

The fluid flux $\mathbf {q}$ is the volume flow rate per unit surface area through the porous medium. Under the steady-state conditions of interest here, Darcy’s law applies so that

(2.5) $$ \begin{align} \mathbf{q}=-{K\over\eta}\;{{\boldsymbol{\nabla}}}\,P, \end{align} $$

where $\eta $ is the viscosity of the fluid and K is the permeability of the medium. Additionally, we need to impose mass conservation for the fluid which, under steady-state conditions (and neglecting fluid compressibility) gives

(2.6) $$ \begin{align} {{\boldsymbol{\nabla}}}\,{\boldsymbol{\cdot}}\,\mathbf{q}=0. \end{align} $$

Poroelastic theory [Reference Howell, Kozyreff and Ockendon4] takes into account the effect of solid matrix compressibility on the flow through the matrix, and assumes that the permeability depends on the displacement gradient [Reference Köry, Krupp, Please and Griffiths5, Reference Parker, Mehta and Caro7], so that, assuming a linear constitutive equation and a one-dimensional displacement $u(x)$ , we write

(2.7) $$ \begin{align} K=k+\alpha^*_n\;{{d} u\over{d} x\;}, \end{align} $$

where k is the permeability of the material in its undeformed state. The poroelastic parameter $\alpha ^*$ is assumed to be positive (or zero) for the mask situation of interest here, so that the permeability reduces as the material compresses. The elastic constants $\lambda _{\mathrm {eff}}, \mu _{\mathrm {eff}}$ and $\eta $ are assumed to remain constant. Note that the elastic and fluid flow equations are coupled in the poroelastic model.

The effect of compression is to change the pore size in the matrix which is closely related to the porosity, $\phi $ , of the medium. An example of the relationship used in practice is the Kozeney–Carman equation

$$ \begin{align*} K=\frac{K_0\phi^3}{(1-\phi)^2},\quad {{d} K\over{d}\phi}={K_0(3-\phi)\phi^2\over(1-\phi)^3},\quad 0<\phi<1, \end{align*} $$

according to which the permeability is an increasing function of $\phi $ . Whilst porosity is a major factor for determining particle capture, we will not be addressing this issue in this paper, so we will not need to impose a specific constitutive equation between the permeability and the porosity; it suffices to work directly with permeabilities.

2.2 The model equations

We return to the double-mask problem (see Figure 1). Since the model is assumed to be one-dimensional, all quantities depend only on x. The quantities of Mask 1 are denoted by suffix 1 and Mask 2 by suffix 2. Thus,

$$ \begin{align*} \mathbf{u}_n=(u_n(x), 0,0),\quad P_n=P_n(x),\quad n=1,2. \end{align*} $$

We take the fluid flux to be positive in the negative x-direction so that the fluid flux from the mouth and nose into the mask is positive. Hence,

(2.8) $$ \begin{align} \mathbf{q}_n=(-q_n(x),0,0),\quad n=1,2, \end{align} $$

where $q_n(x)\geq 0$ . To simplify the notation, $\lambda _{\mathrm {eff}}$ and $\mu _{\mathrm {eff}}$ will be denoted by $\lambda $ and $\mu $ , respectively. Since the same fluid flows through both masks, the fluid viscosity is denoted simply by $\eta $ .

Only the x-components of the equations of poroelasticity are not identically zero. The x-component of the Navier displacement equation (2.4) in each layer is

(2.9) $$ \begin{align} (\lambda_n+2\mu_n)\;{{d}^2u_n\over {d} x^2\;}={{d} P_n\over{d} x\;},\quad n=1,2. \end{align} $$

The x-component of Darcy’s law, (2.5), is

(2.10) $$ \begin{align} q_n(x)={K_n\over\eta}\;{{d} P_n\over{d} x\;},\quad n=1,2. \end{align} $$

The permeability of the mask materials is assumed to depend linearly on the displacement gradient ${\textstyle {d} u/\textstyle {d} x}$ . So we have from (2.7)

$$ \begin{align*} K_n=k_n+\alpha^*_n\;{{d} u_n\over{d} x\;}, \quad n=1,2, \end{align*} $$

where $k_n$ is the permeability of the medium in its undeformed state and is a positive constant, and we have assumed that the constant $\alpha ^*_n \ge 0$ . We write

(2.11) $$ \begin{align} K_n=k_n\bigg(1+\alpha_n\;{{d} u_n\over{d} x\;}\bigg), \end{align} $$

where $\alpha _n=\alpha ^*_n/k_n$ , which we will refer to as the “poroelastic ratio”. This ratio is dimensionless and is a suitable small perturbation parameter to determine compressibility effects since for the rigid mask, $\alpha _n=0$ . Equation (2.10) becomes

(2.12) $$ \begin{align} q_n(x)={k_n\over\eta}\bigg(1+\alpha_n\;{{d} u_n\over{d} x\;}\bigg)\;{{d} P_n\over{d} x\;}. \end{align} $$

The conservation of mass equation (2.6) reduces to

(2.13) $$ \begin{align} {{d} q_n\over{d} x\;}=0, \quad n=1,2. \end{align} $$

Next, we consider the boundary conditions. Mask 1 is attached to a porous grid at $x=0$ . The displacement at $x=0$ must therefore be zero:

(2.14) $$ \begin{align} u_1(0)=0. \end{align} $$

At $x=L$ , the end is not compressed and therefore is free of applied stress. Thus, from (2.1),

(2.15) $$ \begin{align} \tau^{e}_{xx}(L)=(\lambda_2+2\mu_2)\;{{d} u_2\over{d} x\;}\;(L)=0. \end{align} $$

The pressure at $x=0$ and $x=L$ is $P_{\!\text {out}}$ and $P_{\!\text {in}}$ which are constants. Hence,

(2.16) $$ \begin{align} P_1(0)=P_{\!\text{out}},\quad P_2(L)=P_{\!\text{in}}. \end{align} $$

The pressure $P_{\!\text {in}}$ is prescribed, but we will see that the fluid flux and the pressure $P_{\!\text {out}}$ cannot both be prescribed.

Finally, there are matching conditions at the interface between the masks at $x=L_1$ . We assume that the two masks are in perfect contact, and therefore at $x=L_1$ , the displacement is continuous, so

(2.17) $$ \begin{align} u_1(L_1)=u_2\;(L_1), \end{align} $$

and the fluid flux is continuous, so

(2.18) $$ \begin{align} q_1(L_1)=q_2\;(L_1). \end{align} $$

The normal elastic stress is also continuous at $x=L_1$ ,

(2.19) $$ \begin{align} \tau^{e}_{xx}\;(L_1-)=\tau^{e}_{xx}(L_1+), \end{align} $$

which, from (2.1), gives the matching condition

(2.20) $$ \begin{align} (\lambda_1+2\mu_1)\;{{d} u_1\over{d} x\;}\;(L_1)=(\lambda_2+2\mu_2)\;{{d} u_2\over{d} x\;}\;(L_1). \end{align} $$

Since the stress in the fluid is continuous at $x=L_1$ , the pore fluid pressure is continuous. Hence,

(2.21) $$ \begin{align} P_1(L_1)=P_2(L_1). \end{align} $$

We will be interested in the flow for which $P_{\!\text {in}}>P_{\!\text {out}}$ . The mask is then under compression since the porous grid is fixed at $x=0$ . The fluid flows in the negative x-direction and the x-component of the fluid flux in (2.8), $q_n(x)$ , is positive. The model also applies for $P_{\!\text {out}}>P_{\!\text {in}}$ .

The problem is to solve the six equations (2.9), (2.12) and (2.13) for the six quantities $u_n(x)$ , $P_n(x)$ and $q_n(x)$ , where $n=1$ and 2, subject to the boundary conditions (2.14), (2.15) and (2.16), and the matching conditions (2.17), (2.18), (2.20) and (2.21).

We do not make the equations dimensionless, because we find that the solution can be expressed in terms of the ratios of $\alpha $ , k and $\lambda +2\mu $ in the two layers.

3.1 Solution extraction

From (2.13),

$$ \begin{align*} q_n(x)=q_{n0}, \quad n=1,2, \end{align*} $$

where $q_{n0}$ is a constant. However, from the matching condition (2.18),

$$ \begin{align*} q_{10}=q_{20}=q_0, \end{align*} $$

where $q_0$ is a constant.

Equation (2.12) becomes

(3.1) $$ \begin{align} {{d} P_n\over{d} x\;\;}={q_0\eta\over k_n}\;\bigg[1+\alpha_n\;{{d} u_n\over{d} x\;}\bigg]^{-1}, \end{align} $$

and by inserting (3.1) into (2.9), we obtain for $u_n$ the second-order differential equation

(3.2) $$ \begin{align} {{d}^2u_n\over{d} x^2\;} + \alpha_n\;{{d} u_n\over {d} x\;}\;{{d}^2u_n\over{d} x^2\;}={q_0\,\eta\over k_n(\lambda_n+2\mu_n)}. \end{align} $$

Equation (3.2) can be rewritten as

$$ \begin{align*} {{d}^2u_n\over{d} x^2\;}+{\alpha_n\over 2}\;{{d}\;\over{d} x}\bigg(\bigg({{d} u_n\over {d} x\;}\bigg)^2\bigg)={q_0\,\eta\over k_n(\lambda_n+2\mu_n)}, \end{align*} $$

and by integrating with respect to x, we obtain

(3.3) $$ \begin{align} {\alpha_n\over 2}\;\bigg({{d} u_n\over{d} x\;}\bigg)^2+{{d} u_n\over{d} x\;}-\bigg(A_n+{q_0\,\eta\over k_n(\lambda_n+2\mu_n)}\;x\bigg)=0, \end{align} $$

where $A_n$ is a constant. Equation (3.3) is a quadratic equation for ${\textstyle {d} u_n/\textstyle {d} x}.$ Hence,

(3.4) $$ \begin{align} {{d} u_n\over{d} x\;}=-{1\over\alpha_n}\, \pm\, {1\over\alpha_n}\;\bigg[1+2\alpha_n\;\bigg(A_n+{q_0\,\eta\over k_n(\lambda_n+2\mu_n)}\;x\bigg)\bigg]^{{1/2}}. \end{align} $$

To decide which sign to take in (3.4), we expand (3.4) for small values of $\alpha _n$ and compare with the corresponding result for a rigid two-layer mask. Now,

(3.5) $$ \begin{align} {{d} u_n\over{d} x\;}=-{1\over\alpha_n}\,\pm\,{1\over\alpha_n}\,\pm\;\bigg(A_n+{q_0\,\eta\over k_n(\lambda_n+2\mu_n)}\;x\bigg)+\mbox{O}\,(\alpha)\quad \text{as } \alpha\to 0. \end{align} $$

However, for a rigid mask, $\alpha _n=0$ and solving in the same way as for a compressible mask, we find that, instead of (3.5),

$$ \begin{align*} {{d} u_n\over{d} x\;}=A_{0n} +{q_0\,\eta\over k_n(\lambda_n+2\mu_n)}\;x, \end{align*} $$

where $A_{0n}$ is a constant. By letting $\alpha _n\to 0$ in (3.5), we see that the “+” sign must be taken in (3.4). The constant $A_n$ could depend on $\alpha _n$ , but in such a way that it tends to the finite constant $A_{0n}$ as $\alpha _n\to 0$ . Equation (3.4) becomes

(3.6) $$ \begin{align} {{d} u_n\over{d} x}=-{1\over\alpha_n}+{1\over\alpha_n}\bigg[1+2\alpha_n\;\bigg(A_n+{q_0\,\eta\over k_n(\lambda_n+2\mu_n)}\;x\bigg)\bigg]^{{1/2}}, \end{align} $$

and by integrating again, we obtain for the displacement

(3.7) $$ \begin{align} u_n(x)=-{x\over\alpha_n}+{k_n(\lambda_n+2\mu_n)\over 3\,\alpha^2_n\,q_0\,\eta}\bigg[1+2\alpha_n\;\bigg(A_n+{q_0\,\eta\over k_n(\lambda_n+2\mu_n)}\;x\bigg)\bigg]^{{3/2}}+B_n, \end{align} $$

where $B_n$ is a constant.

The pressure in the fluid, $P_n(x)$ , is obtained by substituting (3.6) into (3.1). This gives

$$ \begin{align*} {{d} P_n\over{d} x\;\,}={q_0\,\eta\over k_n}\bigg[1+2\alpha_n\bigg(A_n+{q_0\,\eta\over k_n(\lambda_n+2\mu_n)}\;x\bigg)\bigg]^{-{1/2}}, \end{align*} $$

and by integrating with respect to x, we obtain

(3.8) $$ \begin{align} P_n(x)={(\lambda_n+2\mu_n)\over\alpha_n}\bigg[1+2\alpha_n\;\bigg(A_n+{q_0\eta\over k_n(\lambda_n+2\mu_n)}\;x\bigg)\bigg]^{{1/2}}+C_n , \end{align} $$

where $C_n$ is a constant.

We now apply the boundary conditions. Imposing the boundary conditions (2.14) on (3.7), (2.15) on (3.6) and the first condition in (2.16) on (3.8) gives

(3.9) $$ \begin{align} {k_1(\lambda_1+2\mu_1)\over 3\alpha^2_1\,q_0\,\eta}\;(1+2\alpha_1\,A_1)^{{1/2}}+B_1=0, \end{align} $$

(3.10) $$ \begin{align} A_2=-{q_0\,\eta\,L\over k_2(\lambda_2+2\mu_2)}, \end{align} $$

(3.11) $$ \begin{align} {(\lambda_1+2\mu_1)\over\alpha_1}\;(1+2\alpha_1\;A_1)^{{1/2}}+C_1=P_{\!\text{out}}. \end{align} $$

Imposing the second pressure boundary condition in (2.16) on (3.8) and using (3.10) gives

(3.12) $$ \begin{align} C_2=P_{\!\text{in}}-{(\lambda_2+2\mu_2)\over\alpha_2}. \end{align} $$

It remains to impose the matching conditions at the interface $x=L_1$ . By using (3.7) for $u_n(x)$ and replacing $A_2$ in terms of $q_0$ by (3.10), the matching condition (2.17) becomes

(3.13) $$ \begin{align} &-{L_1\over\alpha_1}+{k_1(\lambda_1+2\mu_1)\over 3\alpha_1^2\,q_0\,\eta}\;\bigg[1+2\alpha_1\,\bigg(A_1+{q_0\,\eta\,L_1\over k_1(\lambda_1+2\mu_1)}\bigg)\bigg]^{{3/2}}+B_1\nonumber\\ &\quad=-{L_1\over\alpha_2}+{k_2(\lambda_2+2\mu_2)\over 3\alpha_2^2\,q_0\,\eta}\;\bigg[1-{2\alpha_2\,q_0\,\eta(L-L_1)\over k_2(\lambda_2+2\mu_2)}\bigg]^{{3/2}}+B_2. \end{align} $$

By using (3.6) and eliminating $A_2$ with (3.10), the matching condition (2.20) becomes

(3.14) $$ \begin{align} &(\lambda_1+2\mu_1)\bigg[-{1\over\alpha_1}+{1\over\alpha_1}\bigg[1+2\alpha_1\bigg(A_1+{q_0\,\eta\, L_1\over k_1(\lambda_1+2\mu_1)}\bigg)\bigg]^{{1/2}}\bigg]\nonumber\\ &\quad=(\lambda_2+2\mu_2)\bigg[-{1\over\alpha_2}+{1\over\alpha_2}\bigg[1-{2\alpha_2\,q_0\,\eta(L-L_1)\over k_2(\lambda_2+2\mu_2)}\bigg]^{{1/2}}\bigg]. \end{align} $$

The remaining matching condition (2.21) becomes, using (3.8) for $P_n(x)$ , (3.10) for $A_2$ and (3.12) for $C_2$ ,

(3.15) $$ \begin{align} &{( \lambda_1+2\mu_1)\over\alpha_1} \bigg[1+2\alpha_1\bigg( A_1 +{q_0\,\eta_1\,L_1\over k_1(\lambda_1+2\mu_1)}\bigg)\bigg]^{{1/2}}+C_1 \nonumber \\[6pt] &\quad={(\lambda_2+2\mu_2)\over\alpha_2}\;\bigg[1-{2\alpha_2\,q_0\,\eta(L-L_1)\over k_2(\lambda_2+2\mu_2)}\bigg]^{{1/2}} -{(\lambda_2+2\mu_2)\over\alpha_2}+P_{\!\text{in}}. \end{align} $$

There are eight quantities, $A_1$ , $B_1$ , $C_1$ , $A_2$ , $B_2$ , $C_2$ , $q_0$ and $P_{\!\text {out}}$ , and there are seven equations, (3.9)–(3.15). Since $P_{\!\text {in}}$ is prescribed, the constant $C_2$ is given by (3.12). The constant $A_2$ is determined in terms of $q_0$ from (3.10), while $A_1$ is determined in terms of $q_0$ from the matching condition (3.14). With $A_1$ determined in terms of $q_0$ , $B_1$ and $C_1$ are obtained in terms of $q_0$ from (3.9) and (3.15), while $B_2$ is now obtained in terms of $q_0$ from (3.13). Since $A_1$ and $C_1$ can be expressed in terms of $q_0$ , the remaining boundary condition (3.11) is a relation between $q_0$ and $P_{\!\text {out}}$ . The fluid flux $q_0$ and the exit pressure $P_{\!\text {out}}$ cannot both be prescribed. Either $q_0$ or $P_{\!\text {out}}$ is prescribed. The quantity not prescribed is then determined from (3.11).

Where possible, the constants are expressed in terms of the ratio of the parameters in the two masks. The following results are obtained for the constants:

(3.16) $$ \begin{align} \!\!\!\!A_1=&-{q_o\,\eta\,L_1\over k_1(\lambda_1+2\mu_1)}-{1\over 2\alpha_1}\ \nonumber \\ &\quad +{1\over2\alpha_1}\bigg[1-{\alpha_1\over\alpha_2}\bigg({\lambda_2+2\mu_2\over\lambda_1+2\mu_1}\bigg)\bigg\{1-\bigg(1-{2\alpha_2\,q_0\,\eta\,L_2\over k_2(\lambda_2+2\mu_2)}\bigg)^{{1/2}}\bigg\}\bigg]^2,\qquad\qquad\quad\ \end{align} $$

(3.17) $$ \begin{align} \!\!C_1=P_{\!\text{in}}-{(\lambda_1+2\mu_1)\over\alpha_1},\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \end{align} $$

(3.18) $$ \begin{align} \kern-18pt B_1=-{k_1(\lambda_1+2\mu_1)\over 3\alpha_1^2\,q_0\,\eta}\;(1+2\alpha_1\,A_1)^{{3/2}},\quad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\end{align} $$

(3.19) $$ \begin{align} B_2&=L_1\bigg({1\over\alpha_2}-{1\over\alpha_1}\bigg)+ {k_1(\lambda_1+2\mu_1)\over3\alpha^2_1q_0\eta}\bigg[\bigg(1+2\alpha_1A_1+{2\alpha_1q_0\eta L_1\over k_1(\lambda_1+2\mu_1)}\bigg)^{{3/2}}-(1+2\alpha_1A_1)^{{3/2}}\bigg]\nonumber \\[.1cm] &\quad -{k_2(\lambda_2+2\mu_2)\over3\alpha^2_2\,q_0\,\eta}\;\bigg[1-{2\alpha_2\,q_0\,\eta\,(L-L_1)\over k_2(\lambda_2+2\mu_2)}\bigg]^{{3/2}}.\ \ \end{align} $$

We will be mainly interested in the permeabilities, $K_1$ and $K_2$ , which depend on $A_1$ and $A_2$ , and in the pressure difference, $P_{\!\text {in}}$ - $P_{\!\text {out}}$ , which depends on $A_1$ and $C_1$ . The displacements, $u_1$ and $u_2$ , depend on $B_1$ and $B_2$ , which depend on $A_1$ given by (3.16). Equation (3.11), which is the relation between $q_0$ and $P_{\!\text {out}}$ , becomes

(3.20) $$ \begin{align} P_{\!\text{in}}-P_{\!\text{out}}={(\lambda_1+2\mu_1)\over\alpha_1}\bigg[1-(1+2\alpha_1\,A_1)^{{1/2}}\bigg], \end{align} $$

where

(3.21) $$ \begin{align} 1+2\alpha_1\,A_1&=-{2\alpha_1\,q_0\,\eta\,L_1\over k_1(\lambda_1+2\mu_1)}\nonumber \\[4pt] &\quad +\bigg[1-{\alpha_1\over\alpha_2}\;{(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}+{\alpha_1\over\alpha_2}\;{(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}\;\bigg(1-{2\alpha_2\,q_0\,\eta\,L_2\over k_2(\lambda_2+2\mu_2)}\bigg)^{{1/2}}\bigg]^2. \end{align} $$

3.2 Permeability and fluid pressure

The permeability of each mask is obtained from (2.11) and (3.6) as

$$ \begin{align*} K_n(x)=k_n\bigg[1+2\alpha_n\;\bigg(A_n+{q_0\,\eta\over k_n(\lambda_n+2\mu_n)}\;x\bigg)\bigg]^{{1/2}},\quad n=1,2. \end{align*} $$

By using (3.21) for $A_1$ and (3.10) for $A_2$ , we find that

(3.22) $$ \begin{align} K_1(x)&=k_1\bigg[-{2\alpha_1\,q_0\,\eta\over k_1(\lambda_1+2\mu_1)}\;(L_1-x) \nonumber\\[4pt] &\quad + \bigg\{1-{\alpha_1\over\alpha_2}\;{(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}+{\alpha_1\over\alpha_2}\;{(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}\;\bigg(1-{2\alpha_2\,q_0\,\eta\,L_2\over k_2(\lambda_2+2\mu_2)}\bigg)^{{1/2}}\bigg\}^{2}\bigg]^{{1/2}} \end{align} $$

for $0\leq x\leq L_1$ , and

(3.23) $$ \begin{align} K_2(x)=k_2\bigg[1-{2\alpha_2\,q_0\,\eta\over k_2(\lambda_2+2\mu_2)}\;(L-x)\bigg]^{{1/2}} \end{align} $$

for $L_1\leq x\leq L$ .

The fluid pressure is given by (3.8) for $n=1$ and 2. By using (3.21), (3.10), (3.17) and (3.12) for $A_1$ , $A_2$ , $C_1$ and $C_2$ , it can be verified that

(3.24) $$ \begin{align} \displaystyle P_1(x)&=P_{\!\text{in}}-{(\lambda_1+2\mu_1)\over\alpha_1}\bigg[1-\bigg\{-{2\alpha_1\,q_0\,\eta\over k_1(\lambda_1+2\mu_1)}\;(L_1-x) \nonumber\\[4pt] &\quad + \bigg(1-{\alpha_1\over\alpha_2}\;{(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}+{\alpha_1\over\alpha_2}\;{(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}\;\bigg(1-{2\alpha_2\,q_0\,\eta\,L_2\over k_2(\lambda_2+2\mu_2)}\bigg)^{{1/2}}\bigg)^2 \bigg\}^{{1/2}}\bigg] \end{align} $$

for $0\leq x\leq L_1$ and

(3.25) $$ \begin{align} P_2(x)=P_{\!\text{in}}-{(\lambda_2+2\mu_2)\over\alpha_2}\bigg[1-\bigg(1-{2\alpha_2\,q_0\,\eta\over k_2(\lambda_2+2\mu_2)}\;(L-x)\bigg)^{{1/2}}\bigg] \end{align} $$

for $L_1\leq x\leq L.$

The difference between the fluid pressure at the entry at Mask 2, $x=L$ , and at the exit at Mask 1, $x=0$ , is given by (3.20). By using again (3.21), the following relation between $q_0$ and the pressure drop $\Delta P$ is obtained:

(3.26) $$ \begin{align} P_{\!\text{in}}-P_{\!\text{out}} &= \Delta P \nonumber\\ &={(\lambda_1+2\mu_1)\over\alpha_1}\bigg[1-\bigg\{-{2\alpha_1\,q_0\,\eta\,L_1\over k_1(\lambda_1+2\mu_1)} \nonumber\\ & \quad + \bigg(1-{\alpha_1\over\alpha_2}\;{(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}+{\alpha_1\over\alpha_2}\;{(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}\;\bigg(1-{2\alpha_2\,q_0\,\eta\,L_2\over k_2(\lambda_2+2\mu_2)}\bigg)^{{1/2}}\bigg)^2 \bigg\}^{1/2}\bigg]. \end{align} $$

It is readily verified that (3.26) agrees with (3.24) evaluated at $x=0$ .

The displacement components, $u_1(x)$ and $u_2(x)$ , are given by (3.7), where $B_1$ and $B_2$ are given by (3.18) and (3.19). The constants $B_1$ and $B_2$ are expressed in terms of $1+2\alpha _1\,A_1$ and can be expanded using again (3.21). We will not analyse $u_1(x)$ and $u_2(x)$ since it is sufficient to investigate the properties of the permeability and pressure difference across the double mask to understand the working of the double mask.

3.3 Scaling and nondimensionalization

We first introduce scaled physical quantities.

Note that (from (3.23)) the permeability $K_2(x)$ is real for values of x in the range $L_1\leq x\leq L$ , provided

(3.27) $$ \begin{align} q_0\leq q_s, \quad \text{where } q_s={k_2(\lambda_2+2\mu_2)\over 2\alpha_2\,\eta\,L_2}. \end{align} $$

We choose the following scales for the physical variables in both masks:

$$ \begin{align*} \begin{array}{ll} \text{fluid flux scale} = q_s, &\text{fluid pressure scale } P_s=\displaystyle {\lambda_1+2\mu_1\over\alpha_1},\\[.1cm] \text{permeability scale} =k_2, &\text{length scale} =L. \end{array} \end{align*} $$

We emphasize that the scales are not characteristic quantities. They are suitable scales that produce dimensionless variables and give a useful way to present the results. To obtain the actual pressure, for example, we would need to multiply the scaled result by  $P_s$ .

The scale $q_s$ is the maximum value for the fluid flux in Mask 2, and therefore in the double mask, for given values of $\alpha _2$ , $\lambda _2$ , $\mu _2$ , $k_2$ and $L_2$ . It depends only on the parameters and width of Mask 2. The pressure scale $P_s$ is a factor in the pressure difference (3.26) across the double mask. The scale $P_s$ for given values of $\alpha _1$ , $\lambda _1$ and $\mu _1$ depends only on the parameters in Mask 1. It is used to scale the pressure in both Mask 1 and Mask 2. The permeability scale, $k_2$ , is the permeability of Mask 2 in its undeformed state. The length L is based on the width of the double mask because we are interested, for example, in the pressure difference across the double mask.

We define the following scaled quantities:

$$ \begin{align*} &q^*_0={q_0\over q_s},\quad 0\leq q^*_0\leq 1, \\ &x^*={x\over L} \quad (0\leq x^*\leq 1),\quad L^*_1={L_1\over L},\quad L^*_2={L_2\over L}, \quad L^*_1+L^*_2=1,\\[.1cm] &K_1^*(x^*) ={K_1(x)\over k_2},\quad K^*_2(x^*)={K_2(x)\over k_2}, \\[.1cm] &P^*_1(x^*) ={P_1(x)\over P_s},\quad P^*_2(x^*)={P_2(x)\over P_s},\quad \Delta P^*={\Delta P \over P_s}. \end{align*} $$

From (3.22) and (3.23), the scaled permeabilities in Mask 1 and Mask 2 are

(3.28) $$ \begin{align} K^*_1(x^*)&={k_1\over k_2}\;\bigg[\bigg\{1-{\alpha_1\over\alpha_2}\;{(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}+{\alpha_1\over\alpha_2}\;{(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}\;(1-q^*_0)^{{1/2}}\bigg\}^2\nonumber\\ &\quad\phantom{\bigg |}-{\alpha_1\over\alpha_2}\;{(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}\;{k_2\over k_1}\;q_0^*\;{(L_1^*-x^*)\over(1-L^*_1)}\bigg]^{{1/2}}\phantom{are interested} \end{align} $$

for $0 \leq x^*\leq L^*_1$ , and

(3.29) $$ \begin{align} K^*_2(x^*) = \bigg[1-q^*_0\;{(1-x*)\over(1-L^*_1)}\bigg]^{{1/2}} \end{align} $$

for $L_1^*\leq x^*\leq 1.$ From (3.26), the scaled pressure difference across the double mask is

(3.30) $$ \begin{align} \Delta P ^{\star} &=1-\bigg[\bigg\{1-{\alpha_1\over\alpha_2}\;{(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}+{\alpha_1\over\alpha_2}\;{(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}\;(1-q^*_0)^{{1/2}}\bigg\}^2\nonumber\\ & \quad\phantom{\bigg |}-{\alpha_1\over\alpha_2}\;{(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}\;{k_2\over k_1}\;q_0^*\;{L_1^*\over(1-L^*_1)}\bigg]^{{1/2}}.\phantom{are interested} \end{align} $$

Equation (3.30) is a relation between $q^*_0$ and $\Delta P^*$ : only one of $q^*_0$ and $\Delta P^*$ can be specified. The other is obtained from (3.30).

The fluid pressure is continuous across the interface $x=L_1$ , and it is readily verified from (3.24) and (3.25) that $P_1(x)$ for $0\leq x\leq L_1$ and $P_2(x)$ for $L_1\leq x\leq L$ are increasing functions of x. The quantity of interest is the pressure difference (3.26) across the double mask. We will therefore not investigate $P_1(x)$ and $P_2(x)$ in each mask separately, and therefore we do not write $P_1(x)$ and $P_2(x)$ in scaled form.

The results will be fully analysed in Section 4. We give here a few elementary properties of the solutions. From (3.28) and (3.29),

(3.31) $$ \begin{align} \begin{aligned} {{d} K_1^*\over{d} x^*}&={1\over 2}\;{\alpha_1\over\alpha_2}\;{(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}\;{k_1\over k_2}\;{q_0^*\over K^*_1(x^*)(1-L^*_1)}>0, \quad 0\leq x^*\leq L_1^*, \\[.1cm] {{d} K_2^*\over{d} x^*}&={q^*_0\over 2K^*_2(x^*)(1-L^*_1)}>0, \quad L^*_1\leq x^*\leq 1. \end{aligned} \end{align} $$

Hence, $K^*_1(x^*)$ and $K^*_2(x^*)$ are increasing functions of $x^*$ . In general, the permeabilities $K_1^*(x^*)$ and $K_2^*(x^*)$ will not be equal at the interface $x^*=L^*_1$ . It can be verified that

(3.32) $$ \begin{align} K^*_1(L^*_1)=K^*_2(L^*_2) \end{align} $$

provided

(3.33) $$ \begin{align} q^*_0={\displaystyle\bigg(1-{k_1\over k_2}\bigg)\;\bigg(1+{k_1\over k_2}-{2k_1\over k_2}\;{\alpha_1\over\alpha_2}\;{(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}\bigg)\over \displaystyle\bigg(1-{k_1\over k_2}\;{\alpha_1\over\alpha_2}\;{(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}\bigg)^2} \end{align} $$

and

$$ \begin{align*} {k_1\over k_2}\;{\alpha_1\over\alpha_2}\; {(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}\not=1. \end{align*} $$

When

$$ \begin{align*} {\alpha_1\over\alpha_2}\;{(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}={k_2\over k_1}, \ \end{align*} $$

(3.32) is satisfied for all $0\leq q^*_0\leq 1$ provided $k_1=k_2$ .

From (3.31) and (3.33),

(3.34) $$ \begin{align} \begin{aligned} K^*_1(0)& = \frac{k_1}{k_2}\;[F^2(q^*_0)-G(q^*_0)]^{{1/2}}, \\ \Delta P^* & = 1 - [F^2(q^*_0) - G(q^*_0)]^{{1/2}}, \end{aligned} \end{align} $$

where

(3.35) $$ \begin{align} F(q^*_0)&=1-{\alpha_1\over\alpha_2}\;{(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}+{\alpha_1\over\alpha_2}\; {(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}\;(1-q^*_0)^{{1/2}}, \end{align} $$

(3.36) $$ \begin{align} \hspace{-27pt} G(q^*_0)&={\alpha_1\over\alpha_2}\; {(\lambda_2+2\mu_2)\over(\lambda_1+2\mu_1)}\;{k_2\over k_1}\;q^*_0\; {L^*_1\over(1-L^*_1)},\qquad\qquad \end{align} $$

and

(3.37) $$ \begin{align} F(0)=1,\quad G(0)=0. \end{align} $$

From (3.37) and (3.34),

(3.38) $$ \begin{align} \Delta P^*=1-{k_2\over k_1}\;K^*_1\,(0). \end{align} $$

The pressure difference $\Delta P^*$ and the permeability $K^*_1(0)$ which are related through (3.38) are important scaled quantities in the understanding of the working of the double mask and are fully analysed in terms of the functions $F^2(q^*_0)$ and $G(q^*_0)$ in Section 4.

3.4 Parameter reduction

The dimensionless parameters involved are the zero flow permeability ratio ${k_1}/{k_2}$ , the “compressibility” ratio $\alpha _1/\alpha _2$ , the elastic modulus ratio $(\lambda _1+2 \mu _1)/)\lambda _2+2 \mu _2)$ and the filter thickness ratio $L_1^*/(1-L_1^*)$ . However, an examination of the results in Section 3.3 indicates that the compressibility and elastic parameters only occur in combination leaving just three dimensionless groups:

$$ \begin{align*} \Lambda_{21} = \frac{(\lambda_2+2 \mu_2)/\alpha_2}{(\lambda_1+2 \mu_1)/\alpha_1}, \quad k_{21}= \frac{k_2}{k_1}, \quad L^*_{12}= \frac{L_1^*}{1-L_1^*}. \end{align*} $$

The implication of this observation is that a changed flow behaviour can be achieved by either adjusting the permeability parameters $\alpha _i$ or by adjusting the elastic parameters $\lambda _i+ 2 \mu _i$ . In terms of this new set of parameters, the expressions for the scaled permeabilities are given by

$$ \begin{align*} & K^*_1 (x^{*})\equiv \dfrac{K_1(x^{*})}{k_2} = \dfrac{1}{k_{21}} \sqrt{F^2(q^*_0)-G^*(q^*_0,x^*)},\nonumber\\[.1cm] & K^*_2(x^*) \equiv \dfrac{K_2(x^*)}{k_2} = \sqrt{1- q^*_0 \bigg(\dfrac{1-x^{*}}{1-L^*_1}\bigg) }\phantom{queroF} \end{align*} $$

with

$$ \begin{align*} G(q^*_0, x^*)=k_{21} \Lambda_{21}\, q_0^*L^*_{12}(x^*),\quad L^*_{12}(x^*)=\bigg({L^*_1-x^*\over 1-L^*_1}\bigg). \end{align*} $$

The scaled pressure difference is given by

$$ \begin{align*} \Delta P^*=1-[F^2(q^*_0)-G(q^*_0,0)]^{{1/2}} \end{align*} $$

with

$$ \begin{align*} G(q^*_0,0)=G(q^*_0)=k_{21}\,\Lambda_{21}\,q^*_0\,L^*_{12},\quad F(q^*_0)=1-\Lambda_{21}\;(1-(1-q^*_0)^{{1/2}}). \end{align*} $$

It is useful to think of the above results as determining the (scaled) pressure drop $\Delta P^*>0$ required to drive a prescribed flow $q_0^*$ through the two masks under steady-state conditions.

4.1 Preliminary simulations

4.1.1 The $k_{21}=1, \Lambda _{21}=1$ case

Note that this case includes the case in which both filters are identical, which we will refer to as the single-filter case, but also includes cases with different values of $\alpha $ but compensating values of $\lambda + 2 \mu $ arranged so that the poroelastic parameter ratio $\Lambda _{21}$ remains unchanged; this feature may be significant in terms of mask design.

Plots of permeability through the mask (two filters) are displayed in Figure 2. We note that for zero flux conditions, compressibility effects disappear and the scaled permeability $K^*(x)=1$ , as required by the scaling. As influx $q_0^*$ levels increase from zero, the permeability decreases uniformly according to our poroelastic model under the mask compression circumstances

$$ \begin{align*} \Delta P^*=P_{\!\text{in}}^*-P_{\!\text{out}}^*>0 \end{align*} $$

of interest here.

This particular parameter set is special in that the (local) permeability is continuous through the mask; for all other parameter combinations, there is a discontinuity across the interface $x^*=L_1^*$ between the two filters. The reader will recall that the external face of the mask ( $x=0$ ) is rigidly constrained by a porous grid, so that the compression is greatest here and thus the permeability $K^*(x)$ reaches a minimum at $x^*=0$ and increases with distance from this outer mask face, as seen in Figure 2.

Of course no steady-state solution is possible if the permeability is zero anywhere within the mask, and the first location to realise such a zero flux state is the external face of the mask $x^*=0$ . Once this happens the flow “shuts down”, so that there is an abrupt change in through-flux from a maximum value (of $q^*_0=0.5$ for our present set of parameters, as seen in Figure 2) to zero. Higher steady-state flux levels than this maximum value are not possible basically because the pores in filter 1 have closed. Of course a higher pressure drop than $\Delta P^*=1$ can be applied to the mask, but the excess pressure will simply be taken up elastically by the filter fibres.

The mask consisting of the two filters can be thought of as an equivalent single mask with effective global (or bulk) permeability defined by the pressure drop across the two filters verses through-flux relation, which is determined by $K^*(0)$ , see (3.38). This effective global permeability will be dependent on the through-flux and will vary from $K^*(0)=1$ when $q_0^*=0$ to $K^*(0)=0$ when $q_0^*=0.5$ , see Figure 3(a). The associated pressure drop verses flux relation is shown in Figure 3(b); the maximal pressure drop ( $\Delta P^*=1$ ) is realized with a through-flux of $q_0^*=0.5$ . It should be noted that a mask consisting of a single filter (just filter 2) would allow the maximal through-flux $q_0^*=1$ , the thickness of the single-layer mask being half that of the double mask.

Figure 2 The $k_{21}=1,\, \Lambda _{21}=1$ single-mask case. Local permeability variations $K^*(x)$ through the mask for increasing through-flux levels: $q_0^*=0$ (top, black), then 0.2 (red), 0.4 (green), 0.5 (lowest, blue). The maximum possible (scaled) flux through the mask is $q_0^*=0.5$ corresponding to the blue curve.

Figure 3 The $k_{21}=1,\, \Lambda _{21}=1$ single-mask case: (a) Global permeability versus flux results. Note that zero global permeability occurs with a maximum through-flux of $q_0^*=0.5$ . Note that the permeability is continuous across the interface ( $x=0.5$ ) in this special case. (b) Pressure drop versus flux results. Note that the maximum pressure drop $\Delta P^*=1$ occurs when the through-flux is maximal at $q_0^*=0.5$ .

4.1.2 The $k_{21}=1, \lambda _{21} \ne 1$ case

Note the abrupt drop in the (local) permeability $K^*(x)$ across the interface (here at $x=0.5$ ) between the two layers in the case when $\lambda _{21}=2$ ; see Figure 4(a). Flux shut down now occurs at a lower maximum flux level compared with the $\Lambda _{21}=1$ case of $q_0^*=0.26 < 0.5$ due to the increase in the poroelastic ratio. The associated pressure drop versus flux relationship is shown in Figure 4(b). Again, shut down is abrupt; flux levels reduce to zero if the pressure drop exceeds the maximal value $\Delta P^*=1$ .

Figure 4 The $k_{21}=1,\, \Lambda _{21}=2$ case. (a) Local permeability variations through the mask for increasing through-flux levels: $q_0^*=0$ top (black) curve, then 0.2 (red), 0.25 (green), 0.26 (blue, lowest). A (scaled) maximum flux level of $q_0^*=0.26$ is possible (blue curve). (b) The associated pressure drop versus flux relationship.

Figure 5 The $k_{21}=1, \Lambda_{21}$ variable case. (a) Global permeability verses flux results for $\Lambda_{21}= 0.1$ (top, red), then 0.2 (green), 0.38 (blue) $\cdots$ 2.0 (bottom, black). (b)The pressure drop verses flux relationship for $\Lambda_{21} = 0.1$ (bottom, red), then 0.2 (green), 0.38 (blue), $\cdots$ 2 (top, black). The blue curve with $\Lambda_{21} = 0.38$ separates out the two possible scenarios.

The effect of varying the poroelastic parameter ratio $\Lambda _{21}$ (with $k_{21}=1$ ) on the permeability and pressure drop is shown in Figure 5. There are two distinctly different cases.

1. (1) Either curves hit the maximal flux value of the $q^*=1$ barrier. This occurs for smaller poroelastic ratio situations ( $\Lambda _{21}=0.1$ (red), 0.2 (green)). Note that ${q^*=1}$ corresponds to an unscaled flux value of

   $$ \begin{align*} q_s={k_2(\lambda_2+2\mu_2)\over 2\alpha_2\,\eta\,L_2}; \end{align*} $$

   it is mask 2 (the inner mask) that controls the limiting behaviour.

2. (2) Or curves hit the maximal pressure drop barrier $\Delta P^*=1$ . This occurs for poroelastic values larger than $\Lambda _{21}=0.38$ . Note that this corresponds to a real (unscaled) pressure drop of $(\lambda _1+2\mu _1)/\alpha _1$ ; it is mask 1 (the outer mask) that controls the limiting behaviour.

The blue curve, corresponding to $\Lambda _{21}=0.38$ , separates the two cases.

4.1.3 The $k_{21}=2$ case

Qualitatively, the results in the $k_{21}=2$ case are similar to those obtained with $k_{21}=1$ . First, we present the $\Lambda _{21}=1$ case (see Figure 6). The maximum possible flux is $q_0^*=0.335<1$ which results if a maximal pressure drop of $\Delta P^*=1$ is applied.

Figure 6 The $k_{21}=2$ , $\Lambda_{21}=1$ case. (a) (Local) permeability variations through the mask for increasing through-flux levels varying from zero to cut-off ( $q_0^*=0$ (top, black), then 0.2 (red), 0.3 (green), 0.335 (bottom, blue)). Cut off occurs at a flux level of $q_0^*= 0.334$ (the blue curve). (b) The pressure drop verses flux relationship.

The results for variable poroelastic ratios ( $\Lambda _{21}$ ) are displayed in Figure 7.

Figure 7 The $k_{21}=2, \Lambda_{21}$ variable case. (a) Global permeability verses flux results for $\Lambda_{21}= 0.1$ (top, red), then 0.2 (green), 0.27 (blue), 0.5 (magenta), 1 (brown), 2 (bottom, black). (b) The pressure drop verses flux relationship for the same $\Lambda_{21}$ range; $\Lambda_{21}= 0.1$ (bottom, red) etc.. The (blue) $\Lambda_{21}=0.27$ case separates out the two possible scenarios.

4.1.4 Maximal through-flux levels

We have seen that through-flux levels are reduced by increases of either the zero flux permeability factor $k_{21}$ or the poroelastic parameter $\Lambda _{21}$ . This may be useful for design purposes, so the dependence of the maximum flux possible through the two masks as a function of the two parameters is of interest. We can obtain this by the equation $K^*(0)=0$ and solving for $q^*_0$ . Exact (but complicated) results are obtained and are plotted in Figure 8. Note that reduced maximum flux levels occur for larger values of $\Lambda _{21}$ . Evidently, the same maximum flux result can be obtained for a prescribed $k_{21}$ by adjusting $\Lambda _{21}$ , as seen in Figure 8.

Figure 8 Maximal flux levels for two masks as a function of $\Lambda_{21}$ for fixed values of $k_{21}=1$ (top, red), then $k_{21}=1.5$ (green) and $k_{21}=2$ (bottom, blue).

4.2 The general solution structure

As noted earlier, real steady-state solutions for mask flow exist in the range given by $q^{*}_0\le 1$ and $\Delta P^* \le 1$ , otherwise the permeability goes to zero or becomes complex, indicating that no steady-state flow is possible. In terms of the solution components $(F^2, G)$ defined by (3.35) and (3.36), real values for the pressure drop only exist if $F^2 \ge G$ . When $F^2=G$ , the scaled pressure reaches its maximal value of $P^*=1$ , with a global permeability of $K^*_1(0)=0$ . With this in mind, we plot in Figure 9 the functions $F^2(q_0^*)$ and $G(q_0^*)$ for a range of values of the poroelastic parameters $k_{21}$ and $\Lambda _{21}$ . Note that $G(q_0^*)$ is a linear function of $q^*_0$ , whereas $F^2(q_0^*)$ can either curve upwards or downwards depending on the values of the zero flux permeability ratio $k_{21}$ and the poroelastic ratio $\Lambda _{21}$ . This means that, over the allowable flux range ${0 \le q_0^* \le 1}$ , there will be for $F^2(q_0^*)-G(q_0^*)=0$ either no solutions, one solution or two solutions for $q_0^*$ , depending on the poroelastic parameter values.

Figure 9 Plots for $F^2(q_0^*, k_{21}, \Lambda _{21})$ (solid curves) and $G(q_0^*, k_{21}, \Lambda _{21})$ (dashed curves) for $k_{21}=1$ , and $\Lambda _{21}=0.3$ (black), 1 (red) and 3 (green). The curves for $\Lambda _{21}=0.3$ (black) do not intersect. Those for $\Lambda _{21}=1$ (red) intersect at one point. Those for $\Lambda _{21}=3$ (green) intersect at two points.

Figure 10 The critical $k_{21}$ curve ( $k_{21}^{crit}(\Lambda _{21})$ ) given by (4.1) and solution branches in the $L_1^*=1/2$ case. (a) The critical curve splits the parameter space into three regions (left, above and right). (b) Solution curves corresponding to $k_{21}=1$ . The red curve (with $\Lambda _{21}=0.3$ ) is in the small lambda range, the blue curve ( $\Lambda _{21}=1$ ) is in the medium lambda range, with the green curve ( $\Lambda _{21}=0.38$ ) splitting the two solution zones. The (two) magenta curves correspond to ( $\Lambda _{21}=3$ ) are in the large lambda range.

In Figure 9, we plot G and $F^2$ for a conductivity ratio $k_{21}=1$ , and a range of values of the poroelastic ratio $\Lambda _{21}$ . For increasing values of $\Lambda _{21}$ , the $F^2$ and G curves first cross at $q_0^*=1$ so that a transition between the different solution structures can be determined by equating $F^2(1)$ to $G(1)$ . This gives the result that, if the thickness adjusted permeability ratio $k_{21}(L_1^*/(1-L_1^*))$ is greater than the critical value

(4.1) $$ \begin{align} k_{21}^{\text{crit}}(\Lambda_{21})= \frac{(1-\Lambda_{21})^2}{\Lambda_{21}}, \end{align} $$

then there is just one solution for $q_0^*$ (given by $q_0^*=0.126$ in the $k_{21}=1$ case). The critical $k_{21}$ curve is plotted in Figure 10. Above this curve (the shaded region), there is just one solution for $q_0^*$ , to the left of this critical curve, there are no solutions, while to the right, there are two solutions. In the case in which $k_{21}=1$ , the regions are defined by the small lambda range $(0,0.38)$ , the medium range $(0.38,2.6)$ and the high lambda range $>2.6$ , see Figure 10(a). The associated $\Delta P^*(q^*_0)$ solution curves are displayed in Figure 10(b). In the small lambda range, there is a single solution (the red curve) with flux levels increasing in response to the pressure drop $\Delta P^*$ until the maximum through-flux of $q_0^*=1$ is reached (asymptotically) with $\Delta P^*<1$ ; the maximum possible pressure drop is not reached. In the medium lambda range, there is a single solution (shaded region blue curve) with flux levels increasing with the applied pressure drop until this reaches its maximal value $\Delta P^*=1$ , with a value of $q_0^*$ less than its maximal value of unity. As described earlier, shut down occurs for higher pressure drops. This medium lambda range includes the $k_{21}=1, \Lambda _{21}=1$ single-filter case. In the large lambda range ( $\Lambda _{21}=3$ (magenta)), there are two solution branches.

Examples of solutions in the small and medium lambda range have been described before. All these solutions continuously evolve from an initial no flow ${(\Delta P^*, q_0^*)=(0,0)}$ state. The small $q_0^*$ branch in the large lambda case also evolves from an initial no flow state, but the second (large $q_0^*$ ) branch does not connect onto this zero flux state. In Figure 11(a), we have plotted pressure drop versus flux results in the $k_{21}=1$ case with $\Lambda _{21}=2.6$ , 3 and 4 (the large lambda range). Note that $\Lambda _{21}=2.6$ lies on the border of the large/medium parameter range and there is a single solution branch (the red curve) which matches the solutions in the neighbouring medium lambda parameter range. For larger values of $\Lambda _{21}$ , a second (large $q_0^*$ ) branch opens up (the blue curve), indicating that under steady-state circumstances, the same pressure drop can result in either a small or a large through-flux. Evidently, the flow behaviour through the mask will be different in the two cases.

Figure 11 Two possible solutions in the large $\Lambda_{21}$ case with $k_{21}=1$ . (a) $\Delta P^*(q_0^*)$ . The red curve corresponds to $\Lambda_{21}=2.6$ which lies on the critical curve so there is just one branch. The blue curve ( $\Lambda_{21}=3$ ) and magenta ( $\Lambda_{21}=4$ ) curves correspond to $\Lambda_{21}>2.6$ ; there are two branches. (b) Local permeability variations through the mask for the two possible solution branches with $\Lambda_{21}=4$ . The black curves (left bottom and right top with $q_0^*=0.126$ ) correspond to the normal (small $q_0^*$ ) branch, the red curves (left top and right bottom with $q_0^*=0.929$ ) to the large flux branch.

Figure 11(a) displays the pressure drop versu flux relation for $\Lambda _{21}$ values close to the transitional value of $\Lambda _{21}=2.6$ (red curve). One can see the two branches (blue, magenta) opening up for $\Lambda _{21}= 3$ and 4; both branches move to the left as $\Lambda _{21}$ increases.

To examine this second branch situation further, we determine permeability variations through the mask for the two solutions corresponding to a pressure drop of $\Delta P^*=0.951$ close to maximal pressure drop of unity. The corresponding flux levels are $q_0^*=0.131$ and $q^*_0=0.928$ . The results for the (local) permeability variations through the masks are displayed in Figure 11(b). In the (normal) small lambda branch case (the black curve), the variations in permeability within filter 2 are moderate with larger variations through the external filter 1, whereas in the large $q_0^*$ branch, there are large variations in permeability through both filters. As indicated earlier, the small flux branch results from a (gradual) increase in pressure drop from zero, so this is indeed the situation one would normally expect. If, however, the pressure drop across the mask is at its maximal value of unity (and so is “at” shut down), then the through-flux may either be at a maximal value or zero, and a small change in the applied pressure may cause the solution to switch branches.

In explanation, the external pressure drop on the mask can be either taken up in filter 1 fibres or filter 2 fibres, and what happens depends on history. A slow build up in pressure forcing will likely result in the low flux solution whereas an abrupt pressure change will compress filter 2 before filter 1 responds, giving rise to the high-flux result.