Effect of lubrication in the non-Reynolds regime due to the non-negligible gap on the fluid permeation through a membrane

Mass transfer through a permeable membrane is a widely observed phenomenon in biological environments, for example, oxygen transport by red blood cells (RBCs), gas exchange in the alveoli, and nutrient absorption. Under healthy biological conditions, RBCs flow almost continually, and the lubrication between the RBCs may alleviate the shear stress on the membrane. Alternatively, the pressure developed due to lubrication could become significant in a narrow gap region between an RBC and the vessel wall (or between RBCs). The magnitude of the force due to lubrication was roughly estimated for a rigid spherical particle (Cooley and O'Neill 1969, O'Neill and Majumdar 1970, Jefferey and Onishi 1984, Dance and Maxey 2003); the lubrication force on the particle develops in logarithmic proportion to the surface-to-surface distance for the tangential-mode motion, and inversely proportional to the surface-to-surface distance for the normal-mode motion. Therefore, considering that the permeate flux is proportional to the pressure discontinuity across the membrane (Katchalsky and Curran 1966), lubrication may be significant in the mass transfer in biological flows.

However, the lubrication-induced mass transfer is difficult to predict or analyse by numerical simulation. One of the difficulties is solving the pressure in the inter-particle gap under a limited spatial resolution, as described in the following. Let Δ denote the grid spacing, U the relative surface speed, and µ the viscous coefficient. In addition, ε = h/D_p is introduced, where D_p and h are the particle diameter and gap width in the inter-particle region, respectively. The lubrication pressure in a non-spherical narrow gap is described by the Reynolds lubrication equation (Reynolds 1886) when ε is much smaller than unity and when the Reynolds number (based on D_p and U) multiplied by ε² is also much smaller than unity. Suppose the resolvable inter-particle gap width h ( = εD_p ) is in the order of Δ (i.e. $10^0\sim{h}/{\Delta} = {\epsilon D_p}/{\Delta}$), the lateral length scale $\sqrt{\epsilon}D_p$, along which the gap pressure varies largely between the orders of the reference pressure $P_0\,(\stackrel{\mathrm{def}}{ = }\mu UD_p^{-1})$ and the lubrication pressure $\epsilon^{-2}P_0$ (Leal 2007), should be covered with the following number of grid points: ${\sqrt{\epsilon}D_p}/{\Delta} \sim \epsilon^{-1/2}$. This value is the lower bound of the necessary number of grid points, and the necessary resolution for reproducing the lubrication pressure increases upon decreasing ε, resulting in prohibitively large computations. However, for a computationally feasible mesh size, the insufficient resolution in the lubrication pressure and, therefore, the pressure discontinuity across the membrane result in poor prediction of the permeate flux.

Another setback of lubrication-induced mass transfer is that the applicability of the Reynolds lubrication equation is strongly restricted for the non-negligible value of ε (Gu et al 2018). For example, Takeishi and Imai (2017) studied the trapping event of micro and nano particles (as a candidate of efficient drug carrier) in the RBC–RBC regions and reported that the inter-cellular distance was in the range of $\epsilon\gtrsim0.1$ for the single-file motion of the RBCs in capillaries. Although a lubrication theory with compressibility of the porous media (i.e. the endothelial glycocalyx at 0.1–0.4 [µm] thickness) in the capillary (5–6 [µm] in diameter) was introduced for ε ∼ O[10⁻²] (Feng and Weinbaum 2000), this theory (as well as the ideal classical lubrication theory) may not be applicable for larger ε cases. Therefore, there may be a number of cases where the ideal condition for the Reynolds lubrication equation is violated. In this paper, we study the permeation caused by the lubrication with a non-negligible gap scale, i.e. non-Reynolds lubrication (Takeuchi and Gu 2019).

Tazaki et al (2020) indicate the possible effect of the non-Reynolds lubrication due to the non-negligible ε in a study of fluid permeation through a membrane moving tangentially in a parallel channel. Their analysis under the condition of infinitesimal permeability showed that while the Reynolds lubrication analysis indicated the order of magnitude of permeate flux as $J_y^\mathrm{(Re)}/U \propto \epsilon^{-1}\mathcal{L}$, the contribution due to the non-Reynolds regime on the permeate flux was ${J_y^{\textrm{(non-Re)}}}/U \propto \epsilon^1 \mathcal{L}$, where $\mathcal{L}$ is a non-dimensionalised permeability. Therefore, considering the cases with large ε in industrial and biological applications, the permeate flux resulting from non-Reynolds lubrication due to the non-negligible $\epsilon$ may be fundamentally important for mass transfer through a membrane.

In this study, we focus on the contribution of non-Reynolds lubrication on the permeate flux through a corrugated membrane under the tangential motion. Permeate fluxes under both Reynolds and non-Reynolds lubrication regimes are modelled, and the nonlinear dependence of geometric parameters on the permeation flux in the non-Reynolds lubrication regime are discussed through a comparison with a fully validated numerical prediction.

The fluid is an incompressible Newtonian fluid of constant density, ρ, and constant viscosity, µ. The governing equations of the fluid are the equation of continuity and the Navier–Stokes equation:

$$\begin{align} &\nabla \cdot \boldsymbol{u} = 0, \end{align} \tag{ 1 }$$

$$\begin{align} &\frac{\partial\boldsymbol{u}}{\partial t} + \boldsymbol{u} \cdot \nabla \boldsymbol{u} = -\frac{\nabla p}{\rho} + \frac{\mu}{\rho} \nabla^2 \boldsymbol{u}, \end{align} \tag{ 2 }$$

where $\boldsymbol{u}$, t, and p are the velocity, time, and pressure, respectively.

The volumetric flux of a pure fluid, i.e. a fluid without any solute, across the membrane is modelled by the following equation (Katchalsky and Curran 1966):

$$\begin{equation} \boldsymbol{J}_n = L_\mathrm{p} \left[\!\!\left[\,{p}\,\right]\!\!\right] \boldsymbol{n}, \end{equation} \tag{ 3 }$$

where $\boldsymbol{n}$ is the unit normal vector pointing from the rear-side of the membrane Ω₁ to the front-side Ω₂, $L_\mathrm{p}$ the permeable coefficient, and $\left[\!\!\left[\,{p}\,\right]\!\!\right]$ is the hydrostatic-pressure jump (i.e. discontinuity) across the membrane calculated using the limiting pressure values at the interface, p₁ and p₂, on the respective sides of the membrane as $\left[\!\!\left[\,{p}\,\right]\!\!\right] = p_1 - p_2$.

Equations (1)–(3) are numerically solved on a rectangular mesh (i.e. non-conforming to the membrane geometry) using the same method as provided in Takeuchi et al (2019) and Tazaki et al (2020); the method guarantees the consistency between the incompressible velocity and pressure fields (Sato et al 2016, Takeuchi et al 2018). In addition, the method enables reproduction of the sharpness of the pressure discontinuity across the membrane (Takeuchi et al 2019), which was identified as an important subject for simulating fluid permeation by Miyauchi et al (2015, 2017).

We study the permeation across an infinitesimally thin, corrugated rigid membrane in a two-dimensional parallel channel moving at a constant velocity, U₀, as illustrated in figure 1. The geometry and prescribed motion of the corrugated membrane are given as follows:

$$\begin{equation} y = h_1(x,t) = h_0\big( 1+\delta\cos[k(x-U_0t)] \big) , \end{equation} \tag{ 4 }$$

where $k\,( = 2\pi/\ell)$ is the wavenumber, h₀ the half channel width and δ is a parameter between 0 and 1. The corrugation has an infinite extension in the x direction.

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In this study, for simplicity, the permeation is solved by transforming the frame attached to the moving membrane: the top- and bottom-flat walls travel at −U₀ along the centre axis of the stationary corrugated membrane. The variables observed on this membrane-fixed frame are distinguished with the superscript '*'.

For the ideal geometric conditions of corrugation, the amplitude of corrugation h₀ δ is sufficiently small with respect to the half channel width h₀ (i.e. $\delta \ll 1$) and h₀ is much smaller than the wavelength of the corrugation (i.e. $h_0 \ll 2\pi/k$). The latter condition is equivalent to the aspect ratio $\varepsilon\, ( = h_0/\ell = kh_0/2\pi)$ being sufficiently small, i.e. $\varepsilon\ll1$.

In this section, we assume that the permeability of the membrane is infinitesimal, i.e. $L_\mathrm{p}\to 0$. Hereafter, the permeate flux under the infinitesimal permeability is called the asymptotic permeate flux.

Assuming a negligible amplitude of corrugation, i.e. $\delta\ll1$, the normal vector on the membrane surface is almost parallel to the y axis. Therefore, in the limit of zero amplitude (δ → 0), only the y component of the permeate flux $\boldsymbol{J}_n$ is predominant; the y component is represented as $J_y(x)\,( = \boldsymbol{J}_n\cdot\boldsymbol{e}_y)$, where $\boldsymbol{e}_y$ is the unit vector in the y direction (see figure 1). Solving the lubrication equation, J_y is modelled as a function of the membrane geometry.

When ε and the Reynolds number $\mathrm{Re} = \rho U_0 \ell/\mu$ satisfy $\varepsilon\ll1$ and $\varepsilon^2 \mathrm{Re} \ll 1$, the flow in the regions between the corrugated membrane and flat walls can be described by the Reynolds lubrication equation. Notably, the gap pressure is uniform in the wall normal direction, varying only in the x direction. Meanwhile, in the case where ε is not sufficiently small, the Reynolds lubrication equation ceases to be applicable, and the pressure in the gap also varies in the wall-normal direction; this situation is referred to as 'non-Reynolds lubrication due to the non-negligible ε'. However, if $\varepsilon\lesssim1, \varepsilon^2\ll1$ and $\mathrm{Re}\ll1$ are satisfied and both the gradient and curvature of the surface profile are small, a lubrication model with a higher-order effect, i.e. the extended lubrication model (Takeuchi and Gu 2019), is applicable. For the surface profile in equation (4), the gradient and curvature of the corrugation are evaluated as h₀ δk ( = 2πδε) and $h_0{\delta}k^2\ ( = 2\pi\delta\varepsilon^2/h_0)$, respectively, indicating that δε should be sufficiently small.

In this section, we highlight the effect of the non-Reynolds lubrication due to non-negligible ε on the pressure discontinuity and permeate flux, and we compare the results with those obtained by the Reynolds lubrication equation and numerical simulation.

4.1. Asymptotic flux for infinitesimal permeability by Reynolds equation

Following Tazaki et al (2020), the asymptotic permeate flux by the Reynolds equation is briefly reviewed in the following.

For a membrane of infinitesimal permeability (i.e. $L_\mathrm{p}\to 0$, and, therefore, J_y  → 0), the pressures on the lower and upper sides (Ω₁ and Ω₂, respectively) of the corrugation, $p_1^{(0)}(x)$ and $p_2^{(0)}(x)$, follow independent Reynolds lubrication equations. Here, the superscript (0) denotes the 0th order approximation, i.e. the case described by the Reynolds lubrication equation. Hereafter, the value observed from the moving frame of reference on the membrane is denoted with the superscript '$\ast$', e.g. $x^* = x-U_0t$. Then, the membrane geometry and pressure are transformed as follows:

$$\begin{equation*} h_1(x,t) = h_1^*(x^*) = h_0\big( 1+\delta\cos[kx^*] \big),\quad p_1^{(0)}(x,t) = {p_1^{(0)*}}(x^*), \end{equation*}$$

where $p_1^{(0)*}$ obeys the following Reynolds lubrication equation:

$$\begin{equation} \frac{\mathrm{d}}{\mathrm{d}x^*}\left( \frac{{h_1^*}^3}{12\mu}\frac{\mathrm{d}p_1^{(0)*}}{\mathrm{d}x^*} \right) = -\frac{0-(-U_0)}2\frac{\mathrm{d}h_1^*}{\mathrm{d}x^*} . \end{equation} \tag{ 5 }$$

The derivation of the above equation is detailed in Tipei (1962) and Leal (2007). The pressure on the original frame of reference, which is fixed in space, is given as follows (Takeuchi et al 2019, Tazaki et al 2020):

$$\begin{equation} p_1^{(0)}(x,t)=- \frac{6 P_0}{\varepsilon k^2}\frac{h_1(x,t)/h_0+1}{2+\delta^2}\frac{\partial h_1^{-1}(x,t)}{\partial x}, \end{equation} \tag{ 6 }$$

where $P_0 = \mu U_0/\ell$, and $\varepsilon^{-2}P_0$ is the typical magnitude of the lubrication pressure. From both the geometric symmetry between Ω₁ and Ω₂ (see figure 1) and the one-dimensionality of the gap pressure $p^{(0)}_i\ (i = 1,2)$, the pressure in Ω₂ is given as $p_2^{(0)}(x,t) = p_1^{(0)}(x+\pi/k,t)$. Then, the asymptotic permeate flux is described as follows:

$$\begin{equation} J_y^{(0)}(x,t) = L_\mathrm{p}\left[\!\!\left[{\,p^{(0)} }\right]\!\!\right] = L_\mathrm{p}\big( p_1^{(0)}(x,t)-p_2^{(0)}(x,t) \big) . \end{equation} \tag{ 7 }$$

Substituting equation (6) into the above equation and using $h_2(x,t) = h_1(x+\pi/k,t)$ $= 2h_0-h_1(x,t)$, the asymptotic permeate flux is given as:

$$\begin{align} J_y^{(0)}(x,t) & = -\frac{24\mathcal{L} U_0}{\pi\varepsilon} \frac{\delta}{2+\delta^2} \frac{\sin[k(x-U_0t)]}{(1-\delta^2\cos^2[k(x-U_0t)])^2} , \end{align} \tag{ 8 }$$

where $\mathcal{L}$ is the non-dimensional permeability defined as ${L_\mathrm{p} \mu} / H$ based on the full channel width H (see figure 1).

4.2. Adjusting component of the pressure through non-Reynolds lubrication

For a non-negligible value of ε, the above-mentioned theory is not applicable, and the lubrication pressure must be corrected by considering higher-order terms, i.e. the terms in O[ε²], in the governing equations.

In the extended lubrication model (Takeuchi and Gu 2019), the non-Reynolds lubrication pressure due to the non-negligible ε is decomposed into the base component and the adjusting component, $p_{\textrm{adj}}(x,y)$. Here, the base component, i.e. $p_{\textrm{Re}}$, is the solution of the Reynolds lubrication equation, equation (5); i.e. $p_{\textrm{Re}} = p_1^{(0)}$. Given $p_{\textrm{Re}}$, the adjusting component is calculated as follows (Takeuchi and Gu 2019):

$$\begin{eqnarray} p_{\textrm{adj}}(x,y)& = &-\mu \frac{\partial u_\mathrm{cp}}{\partial x}, \end{eqnarray} \tag{ 9 }$$

where $u_\mathrm{cp}$ is the velocity of the locally constructed Couette–Poiseuille flow:

$$\begin{eqnarray} u_\mathrm{cp}(x,y)& = &-\frac{y(h_1(x)-y)}{2\mu}\frac{\mathrm{d}p_{\textrm{Re}}(x)}{\mathrm{d}x}+\frac{y}{h_1(x)}U_{\textrm{r}} , \end{eqnarray} \tag{ 10 }$$

where $U_{\textrm{r}}$ is the relative velocity of the walls in the x direction. Equation (9) implies that the adjusting component is the stress caused by the spatial variation of $u_\mathrm{cp}$ driven by the local Reynolds-lubrication pressure, $p_{\textrm{Re}}$, and the wall movements. Using the base and adjusting components of the pressure, the lubrication pressure in the non-Reynolds regime is expressed as $p(x,y) = p_{\textrm{Re}}(x)+p_{\textrm{adj}}(x,y)$. The validity of the above model was established in Takeuchi and Gu (2019).

Hereafter, $p_{\textrm{adj}}$ is alternatively denoted by $p_1^{(2)}$, indicating that this pressure is the higher-order (i.e. ε² order) correction to $p_1^{(0)}\,( = p_{\textrm{Re}})$. Using equation (6), which is obtained under $L_\mathrm{p}\to 0$, the form of $p_{\textrm{adj}}$ is given as follows:

$$\begin{eqnarray} p_1^{(2)*} & =& p_{\textrm{adj}}^*(x^*,y)\nonumber\\[4pt] & =& -P_0 \frac{4\pi \varepsilon \delta}{2+\delta ^2}\frac{y}{h_0}\nonumber\\[4pt] && \times\frac{ \left(4 \delta ^2\!+\!\left(\delta ^2+2\right) \delta \cos (k x^*)-1\right) (2 \delta \cos (k x^*)+2 -3 y/h_0)}{ (1+\delta \cos (k x^*))^4} \sin (k x^*). \end{eqnarray} \tag{ 11 }$$

Subsequently, using equation (3), the adjusting asymptotic flux under the non-Reynolds lubrication regime is constructed as follows:

$$\begin{eqnarray} J_y^{(2)*}(x^*) \ \stackrel{\mathrm{def}}{ = }\ \boldsymbol{J}_\mathrm{adj}^*\cdot\boldsymbol{e}_y & = & L_\mathrm{p}\left[ p_1^{(2)*}(x^*,y)-p_1^{(2)*}(x^*+\pi/k,y) \right]_{y = h_1^*} \end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray} & = & 16\pi\varepsilon \mathcal{L} U_0 \frac{\delta }{2+\delta ^2 } \frac{4 \delta ^2+\left(2 \delta ^2-5\right) \delta ^2 \cos ^2(k x^*)-1} {\left[1-\delta ^2 \cos ^2(k x^*)\right]^2} \sin (k x^*) . \end{eqnarray} \tag{ 13 }$$

On comparing equation (8) with equation (13), $J_y^{(0)}$ and $J_y^{(2)}$ are different by two orders of magnitude in ε, which originates from the different magnitudes of $p_1^{(0)}$ and $p_1^{(2)}$. Eventually, the full component of the flux is given as

$$\begin{equation} J_y^*(x^*) = J_y^{(0)*} + J_y^{(2)*} . \end{equation} \tag{ 14 }$$

The dependency of the parameters on the asymptotic permeation is clearly demonstrated in the permeate flow rate calculated for the half wave length (i.e. $\ell/2 = \pi/k$):

$$\begin{equation*} Q_{\mathrm{h}\textrm{-}\mathrm{wl}}^{(\,j)} = \int_{\pi/k}^{\,2\pi/k} J_y^{(\,j)*}\, \mathrm{d}{x^*} \quad (\,j = 0, 2). \end{equation*}$$

For the 0th and 2nd order solutions:

$$\begin{eqnarray*} Q_{\mathrm{h}\textrm{-}\mathrm{wl}}^{(0)}& = & \frac{24 \mathcal{L} U_0 }{\pi k \varepsilon } \frac{\delta +(1-\delta^2) \tanh^{-1}\delta} {(1-\delta^2)(2+\delta^2)} , \\ Q_{\mathrm{h}\textrm{-}\mathrm{wl}}^{(2)}& = & \frac{32 \pi \mathcal{L} U_0 \varepsilon}{k} \frac{3 \delta - \left(2+\delta^2\right) \tanh^{-1}\delta }{2+\delta^2 } , \end{eqnarray*}$$

the total permeate flow rate is evaluated by the Taylor expansion with respect to δ as follows:

$$\begin{equation} Q_{\mathrm{h}\textrm{-}\mathrm{wl}}^{(0)}+ Q_{\mathrm{h}\textrm{-}\mathrm{wl}}^{(2)} = 8 \frac{\mathcal{L} U_0}{k} \left( \frac{3}{\pi\varepsilon} + 2\pi\varepsilon \right)\delta +\frac{\mathcal{L} U_0}{k}\, O\left[\delta^3\right] . \end{equation} \tag{ 15 }$$

The lowest-order term in equation (15) shows that the permeate flow rate is proportional to the corrugation amplitude (δ) and the wave length (k⁻¹). Although the dependence on the aspect ratio (ε) is more complex, the effects of the Reynolds and the non-Reynolds lubrications appear as the ε⁻¹ and ε¹ terms, respectively. This analysis is valid under the condition of $\delta,\varepsilon\ll1$ and $\mathcal{L}\to 0$; our preliminary analysis shows that δ in the range of $\delta\lt1$ exhibits a weakly non-linear effect on the permeation, while a finite value of $\mathcal{L}$ induces a strong non-linear response.

Note that, from equations (8) and (13), both $J_y^{(0)\ast}/\mathcal{L}$ and $J_y^{(2)*}/\mathcal{L}$ become independent of $\mathcal{L}$. This is also the characteristics at the infinitesimal permeability; in this limit, $J_y/L_\mathrm{p}$ is equivalent to the $L_\mathrm{p}$-free pressure discontinuity $\left[\!\!\left[\,{p}\,\right]\!\!\right]$, as shown in equations (6) and (11). This $\mathcal{L}$-independent nature facilitates investigation of the converging (or asymptotic) trend of the numerical simulation result at small $\mathcal{L}$ values, as will be discussed in the next section.

The above analytical models are compared to the numerical simulation result. To adjust the condition to the numerical study by Tazaki et al (2020), Reynolds number is set at ρU₀ H/µ = 1.0 and ε is fixed at 0.1. The spatial resolution and the time increment are set at H/Δ = 40 (i.e. h₀/Δ = 20) and $\Delta\,t/(H/U_0) = 5\times10^{-5}$, respectively. The amplitude parameter and permeability are varied in the following ranges: δ = 0.1,  0.25,  0.45,  0.50 and $\mathcal{L} = 10^{-5},\ 10^{-4},\ 10^{-3},\ 10^{-2}$.

Figure 2 compares the non-dimensionalised permeate flux $J_y/U_0\mathcal{L}$ obtained by the numerical simulation (represented by symbols), the 0th order asymptotic model (equation (8); dashed line) and the full-component asymptotic model (equation (14); solid line). For all the δ cases, the numerical results show converging trends as $\mathcal{L}$ decreases. However, the distinct gaps exist between the numerical results at $\mathcal{L} = 10^{-5}$ and the 0th order model, as Tazaki et al (2020) reported. Meanwhile, the full-component fluxes agree well with the asymptotic numerical results, particularly for the cases of δ = 0.10 and 0.25, indicating that the effect of the non-Reynolds lubrication can be described with equations (9) and (12) for the small δ. For the larger δ cases (i.e. δ = 0.45 and 0.50), although the prediction is slightly overestimated, the 4% discrepancy at $kx^* = \pi/2$ (viz., $x^*/H = 5/4$) is still in the order of ε².

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The profiles of δ = 0.50 are characterised by dimples, i.e. weakened permeation between a pair of local peaks along the membrane, whereas the case of δ = 0.45 is selected as a value just below the lower bound of the δ range ($\delta\gt\delta_c = 0.466$) for having a dimpled profile. More details about the condition for the emergence of the dimples is explained in appendix A. Notably, for both δ cases, the profiles of $J_y^{(0)*}$ (dashed line) do not show dimples. However, the non-Reynolds component $J_y^{(2)*}$ provides adjustment that distinguishes the no-dimpled (δ = 0.45) and dimpled (δ = 0.50) cases in the permeation profiles.

To identify the applicable range of the permeability, an order analysis is conducted for the pressure and permeation flux. In the following, we introduce the reference velocities of the fluid in the longitudinal and transverse directions as U and V, respectively. The order of magnitude of the lubrication pressure under the predominant permeation would be lower than that under an infinitesimal permeability, i.e. $\varepsilon^{-2}P_0$. Then, for a finite permeability, the effect of $p_{\textrm{adj}}$ on the permeate flux is estimated. From the permeate flux $V\sim L_\mathrm{p} p_{\textrm{Re}}$, the pressure in the Reynolds lubrication regime is evaluated as

$$\begin{equation} p_{\textrm{Re}}\sim\frac{V}{L_\mathrm{p}} = \frac{\mu V/H}{\mu L_\mathrm{p}/H}\sim\frac{P_0}{\mathcal{L}}, \end{equation} \tag{ 16 }$$

where as the pressure due to the non-Reynolds effect is obtained as

$$\begin{eqnarray} p_{\textrm{adj}} &\sim& \frac{\mu}{\ell}\left[ \frac{H^2}{\mu} \frac 1{\ell} \frac{V}{L_\mathrm{p}}+\frac{H}{H}U \right] = \left( \frac{\varepsilon^2}{\mathcal{L}}+1 \right)P_0. \end{eqnarray} \tag{ 17 }$$

Comparing equations (16) and (17), the adjusting pressure could become relatively large for a large permeability; particularly, in the range of $\mathcal{L}\gt1$, the adjusting pressure could become even larger than $p_{\textrm{Re}}$. To remain in the assumed range of the pressures, $p_{\textrm{adj}}/p_{\textrm{Re}}\sim\varepsilon^2$, the permeability has to be limited in the range of $\mathcal{L}\ll\varepsilon^2$. Considering that the analytical curves at $\mathcal{L} = 10^{-4}$ and 10⁻⁵ (corresponding to $\mathcal{L}/\varepsilon^2 = 10^{-2}$ and 10⁻³ for the present geometry of ε = 0.1) in figure 2 reasonably reproduce the numerical result, the above order analysis effectively includes the critical terms for the lubrication-induced permeation. For an even higher permeability range (e.g. $\mathcal{L}\geqslant10^{-3}$), different higher-order models for pressure and permeation may be necessary, which is the subject of ongoing study of the present authors.

Lubrication-induced permeate flux through a corrugated membrane was studied in a gap-width range wherein the Reynolds lubrication equation is not applicable (i.e. non-Reynolds lubrication regime). The effect of lubrication in the non-Reynolds regime was modelled using an extended lubrication model that considers the wall-normal variation of the pressure distribution.

While the traditional Reynolds lubrication model underestimated the permeate flux, particularly in a low-permeability range, the non-Reynolds lubrication model improved the prediction (towards the fully validated numerical result) for a small amplitude of corrugation. For a large amplitude of corrugation, the characteristic variation in permeate flux along the membrane (i.e. dimpled profile) was reproduced by including the non-Reynolds effect.

The effective range of the permeability for the non-Reynolds lubrication was discussed through the order analysis of the lubrication pressures. The results indicate the validity of the non-Reynolds lubrication model in relatively small permeabilities for problems with non-negligible gap width.

This work is partly supported by Grants-in-Aid for Scientific Research (B) Nos. 17H03174 and 19H02066, Grant-in-Aid for Early-Career Scientists No. 19K20659, and Grant-in-Aid for Challenging Research (Exploratory) No. 20K20972 of the Japan Society for the Promotion of Science (JSPS). One of the authors, S M, gratefully acknowledges the Leading Young Researcher Overseas Visit Program in Tohoku University.

The condition of δ for having a dimpled profile of asymptotic permeate flux (see figure 2(d)) is investigated. The full flux considering the pressure adjustment using the extended lubrication model is calculated as follows:

$$\begin{eqnarray*} J_y& =& J_y^{(0)}+J_y^{(2)} \\ & = & U_0 \frac{4 \mu L_p}{h_0} \frac{\delta}{2+\delta ^2}\\ && \times \frac{ (2\pi\varepsilon)^2 \left(4 \delta ^2-1\right) + (2\pi\varepsilon)^2 \delta ^2 \left(2\delta^2-5\right) \cos ^2(k x^*)-6} {2\pi\varepsilon \left(1-\delta ^2 \cos ^2(k x^*)\right)^2} \sin (k x^*) , \end{eqnarray*}$$

where $x^* = x-U_0t$. The profile of the flux could have a pair of dimples, i.e. weakened permeation between a pair of local peaks, at $kx^* = \pi/2$ and 3π/2 as observed in figure 2(d). The condition for no-dimpled profile is as follows:

$$\begin{eqnarray*} 0\ \lt\ \delta&\leqslant& \delta_c\ \stackrel{\mathrm{def}}{ = }\ \sqrt{ \frac{3 \left(3 \pi ^2 \varepsilon ^2+1\right) - \sqrt{61 \pi ^4 \varepsilon ^4+24 \pi ^2 \varepsilon ^2+9} }{20 \pi ^2 \varepsilon ^2} } . \end{eqnarray*}$$

In the present case, $\varepsilon = h_0/\ell = 0.1$, the critical value becomes $\delta_\mathrm{c} = 0.466$.

In addition, it is easily confirmed that the profile of $J_y^{(0)}$ does not have dimples in $0\lt\delta\leqslant1/2$ (${}^\forall\varepsilon$).