Dispersion-enhanced solute transport in a cell-seeded hollow fibre membrane bioreactor

Abstract

We present a matched asymptotic analysis of the fluid flow and solute transport in a small aspect ratio hollow fibre membrane bioreactor. A two-dimensional domain is assumed for simplicity, enabling greater understanding of the typical behaviours of the system in a setup which is analytically tractable. The model permits analysis related to Taylor dispersion problems, and allows us to predict the dependence of the mean solute uptake and solute exposure time on key parameters such as the inlet fluid fluxes, porous membrane porosity and cell layer porosity and width, which could be controlled or measured experimentally.

Appendices

Appendix 1

In Sect. 2, we introduced a two-dimensional model of a HFMB which included the regions near the up- and downstream ECS ports. We then discussed the existence of inner regions near these ports in Sect. 3, and focussed on finding the solution valid in the outer region. Here, we analyse the up- and down-stream inner regions and determine the matching conditions needed to close the outer problem. We shall then outline the boundary layer analysis of the combined-order equation formulated in Sect. 3.8.

1.1 1.1 Analysis of the up- and down-stream inner regions

We return to the dimensionless Eqs. (19)–(22) and rescale in order to move into the left-hand inner region in which $x = \text{ O }(\varepsilon )$. We choose this scaling as it promotes the x-derivatives that were neglected in the outer region due to the lubrication scaling, and since we would expect both x- and y-derivatives to contribute to the leading-order behaviour in the inner region. Hence we set $x = \varepsilon X$ and $v_i = V_i / \varepsilon$ (for $i=\mathrm {l,m,w,f}$). For clarity, we also set , $p_i(x,y,t) = P_i(X,y,t)$, $c_i(x,y,t) = C_i(X,y,t)$, $u_i(x,y,t) = U_i(X,y,t)$ ($i=\mathrm {l,m,w,f}$) and $\mathbf {\nabla }_\mathrm {X} = \big (\partial /\partial {X},\;\partial /\partial {y}\big )$, $\nabla ^2_\mathrm {X} = \partial ^2/\partial {X}^2 + \partial ^2/\partial {y}^2$. We again use subscripts on dependent variables to indicate their domain of applicability from Fig. 2. The flow equations are now

$$\begin{aligned} \frac{\partial U_i}{\partial X} + \frac{\partial V_i}{\partial y} = 0,\quad -\frac{\partial P_i}{\partial X} + \varepsilon \nabla ^2_\mathrm {X} U_i = 0,\quad -\frac{\partial P_i}{\partial y} + \varepsilon \nabla ^2_\mathrm {X} V_i = 0,\quad i=\mathrm {l},\mathrm {f}, \end{aligned}$$

(125a-c)

$$\begin{aligned} U_\mathrm {m} = -\varepsilon \kappa _\mathrm {m}\frac{\partial P_\mathrm {m}}{\partial X},\quad V_\mathrm {m} = - \varepsilon \kappa _\mathrm {m}\frac{\partial P_\mathrm {m}}{\partial y},\quad \nabla ^2_\mathrm {X}P_\mathrm {m} = 0, \end{aligned}$$

(126a-c)

$$\begin{aligned} U_\mathrm {w} = -\varepsilon \kappa _\mathrm {w}\frac{\partial P_\mathrm {w}}{\partial X},\quad V_\mathrm {w} = - \varepsilon \kappa _\mathrm {w}\frac{\partial P_\mathrm {w}}{\partial y},\quad \nabla ^2_\mathrm {X}P_\mathrm {w} = 0, \end{aligned}$$

(127a-c)

while the boundary conditions become

$$\begin{aligned} \frac{\partial U_\mathrm {l}}{\partial y} = 0,\quad V_\mathrm {l} = 0\quad \mathrm {on}\quad y=0; \end{aligned}$$

(128)

$$\begin{aligned} \begin{array}{l} U_\mathrm {l} = 0, \quad V_\mathrm {l} = -\varepsilon ^2\kappa _\mathrm {m}\phi _\mathrm {m}\displaystyle \frac{\partial P_\mathrm {m}}{\partial y},\\ P_\mathrm {l} + \displaystyle \frac{2\varepsilon }{3}\left( \frac{\partial U_\mathrm {l}}{\partial X}-2\frac{\partial V_\mathrm {l}}{\partial y}\right) = P_\mathrm {m}\quad \mathrm {on}\quad y=1; \end{array} \end{aligned}$$

(129a-c)

$$\begin{aligned} \phi _\mathrm {m}\kappa _\mathrm {m}\displaystyle \frac{\partial P_\mathrm {m}}{\partial y} = \phi _\mathrm {w}\kappa _\mathrm {w}\displaystyle \frac{\partial P_\mathrm {w}}{\partial y},\quad P_\mathrm {m} = P_\mathrm {w}\quad \mathrm {on}\quad y=1+h_2; \end{aligned}$$

(130)

$$\begin{aligned} \begin{array}{l} U_\mathrm {f} = 0,\quad -\varepsilon ^2\phi _\mathrm {w}\kappa _\mathrm {w}\displaystyle \frac{\partial P_\mathrm {w}}{\partial y} = V_\mathrm {f},\\ P_\mathrm {w} = P_\mathrm {f} + \displaystyle \frac{2\varepsilon }{3}\left( \frac{\partial U_\mathrm {f}}{\partial X} - 2\frac{\partial V_\mathrm {f}}{\partial y}\right) \quad \mathrm {on}\quad y=H-h_4; \end{array} \end{aligned}$$

(131a-c)

$$\begin{aligned} U_\mathrm {f} = V_\mathrm {f} = 0\quad \mathrm {on}\quad y=H; \end{aligned}$$

(132)

and, as in Sect. 2.2, we assume that there is a prescribed two-dimensional flux $Q_\mathrm {l,in}$ at the lumen inlet, and $Q_\mathrm {f,in}$ at each of the upstream ECS ports. We note that the equations in (125) for the lumen and upper fluid layer are only valid at $\text{ O }(1)$ here, since the inertial terms would appear at $\text{ O }(\varepsilon )$ (see discussion in Sect. 2.1). However, this does not affect our analysis since only the leading-order inner velocities are required, as will be seen below.

Equations (125b,c) tell us that the leading-order pressure in the lumen and upper fluid layer is independent of both X and y. Hence, without needing to solve for the leading- and first-order velocities in the lumen and upper fluid layer, we can further see from Eqs. (126b), (127b) that $V_\mathrm {m},V_\mathrm {w} = \text{ O }(\varepsilon ^2)$, and so there is no flux across either the lumen/membrane or cell layer/upper fluid interfaces at $\text{ O }(1)$ or $\text{ O }(\varepsilon )$.

We determine the fluid flux conditions by considering how fluid transfers between layers. From inspection of boundary conditions (129b) and (131b), we can see that there is no flux of fluid across either the lumen/membrane or cell layer/upper fluid layer interfaces at $\text{ O }(1)$ or $\text{ O }(\varepsilon )$. Thus, the flux out of the inner region in the lumen and upper fluid layer must equal the flux into the outer region up to $\text{ O }(\varepsilon )$ in these sections, and hence, the correct inlet conditions to impose on the outer flow solution at leading order and first order in Sect. 3.5 are given by (37), (38), (47) and (51).

The inner problem for the solute concentration is given by

$$\begin{aligned}&\varepsilon \,\mathrm {Pe}\left( \varepsilon \frac{\partial C_\mathrm {l}}{\partial t} + \mathbf {\nabla }_\mathrm {X}\cdot (C_\mathrm {l}\mathbf {U}_\mathrm {l})\right) = \nabla ^2_\mathrm {X}C_\mathrm {l}, \end{aligned}$$

(133)

$$\begin{aligned}&\varepsilon ^2\,\mathrm {Pe}\left( \frac{\partial C_\mathrm {m}}{\partial t} + \mathbf {\nabla }_\mathrm {X}\cdot (C_\mathrm {m}\mathbf {U}_\mathrm {m})\right) = \nabla ^2_\mathrm {X}C_\mathrm {m},\end{aligned}$$

(134)

$$\begin{aligned}&\varepsilon ^2\,\mathrm {Pe}\left( \frac{\partial C_\mathrm {w}}{\partial t} + \mathbf {\nabla }_\mathrm {X}\cdot (C_\mathrm {w}\mathbf {U}_\mathrm {w})\right) = \nabla ^2_\mathrm {X}C_\mathrm {w} - \frac{\varepsilon \mathcal {R}}{\phi _\mathrm {w}},\end{aligned}$$

(135)

$$\begin{aligned}&\varepsilon \,\mathrm {Pe}\left( \varepsilon \frac{\partial C_\mathrm {f}}{\partial t} + \mathbf {\nabla }_\mathrm {X}\cdot (C_\mathrm {f}\mathbf {U}_\mathrm {f})\right) = \nabla ^2_\mathrm {X}C_\mathrm {f}, \end{aligned}$$

(136)

for which the boundary conditions are

$$\begin{aligned}&\frac{\partial C_\mathrm {l}}{\partial y} = 0\quad \mathrm {on}\quad y=0, \end{aligned}$$

(137)

$$\begin{aligned}&C_\mathrm {l} = C_\mathrm {m},\quad \frac{\partial C_\mathrm {l}}{\partial y} = \phi _\mathrm {m}\frac{\partial C_\mathrm {m}}{\partial y}\quad \mathrm {on}\quad y=1;\end{aligned}$$

(138)

$$\begin{aligned}&C_\mathrm {m} = C_\mathrm {w},\quad \phi _\mathrm {m}\frac{\partial C_\mathrm {m}}{\partial y} = \phi _\mathrm {w}\frac{\partial C_\mathrm {w}}{\partial y}\quad \mathrm {on}\quad y=1+h_2;\end{aligned}$$

(139)

$$\begin{aligned}&C_\mathrm {w} = C_\mathrm {f},\quad \phi _\mathrm {w}\frac{\partial C_\mathrm {w}}{\partial y} = \frac{\partial C_\mathrm {f}}{\partial y}\quad \mathrm {on}\quad y=1+h_2+h_3; \end{aligned}$$

(140)

$$\begin{aligned}&\frac{\partial C_\mathrm {f}}{\partial y} = 0\quad \mathrm {on} \quad y=H; \end{aligned}$$

(141)

together with the inlet conditions

$$\begin{aligned} C_\mathrm {l} = c_\mathrm {l,in}\quad \mathrm {at}\ A,\quad C_\mathrm {f} = c_\mathrm {f,in}\quad \mathrm {at} \quad B. \end{aligned}$$

(142)

As in the outer region, we expand all variables in powers of $\varepsilon$, so that at leading order, we obtain (dropping the subscript 0)

$$\begin{aligned} \begin{aligned}&\varepsilon \,\mathrm {Pe}\,\mathbf {\nabla }_\mathrm {X}\cdot \left( C_{\mathrm {l}}\mathbf {U}_\mathrm {l}\right) = \nabla ^2_\mathrm {X}C_\mathrm {l},\quad \nabla ^2_\mathrm {X}C_\mathrm {m} = 0,\\&\nabla ^2_\mathrm {X}C_\mathrm {w}=0,\quad \varepsilon \,\mathrm {Pe}\,\mathbf {\nabla }_\mathrm {X}\cdot \left( C_\mathrm {f}\mathbf {U}_\mathrm {f}\right) = \nabla ^2_\mathrm {X}C_\mathrm {f}. \end{aligned} \end{aligned}$$

(143)

We note that for $c_\mathrm {l,in} \ne c_\mathrm {f,in}$ this full inner problem must be solved in order to determine $C_i$ ($i=\mathrm {l},\mathrm {m},\mathrm {w},\mathrm {f}$). However, as seen in Sect. 3.6, the leading-order inner concentration must be independent of y as we leave the inner region and therefore can be determined by appealing to global conservation of mass without knowing the solution of this leading-order inner problem. In particular, a far-field analysis of the system (143) subject to (142) consistent with global conservation of mass implies that, as $X \rightarrow \infty$,

$$\begin{aligned} C_{i_0} \sim c_\mathrm {in} = \frac{Q_\mathrm {l,in}c_\mathrm {l,in}+Q_\mathrm {f,in}c_\mathrm {f,in}}{Q_\mathrm {l,in}+Q_\mathrm {f,in}},\quad i=\mathrm {l},\mathrm {m},\mathrm {w},\mathrm {f}, \end{aligned}$$

(144)

and hence, this gives the correct inlet condition (59) for the $\text{ O }(1)$ solute in the outer region.

We now move on to the $\text{ O }(\varepsilon )$ problem for the concentration, and note that we do not state the equations for the fluid velocities at this order as we only require the incompressibility condition on $\mathbf {U}_{\mathrm {l}_1}$ and $\mathbf {U}_{\mathrm {f}_1}$ for our reductions. The $\text{ O }(\varepsilon )$ governing equations for the solute concentration are (reintroducing the subscript 0, 1 for clarity)

$$\begin{aligned}&\varepsilon \,\mathrm {Pe}\mathbf {\nabla }_\mathrm {X}\cdot \left( C_{\mathrm {l}_0}\mathbf {U}_{\mathrm {l}_1} + C_{\mathrm {l}_1}\mathbf {U}_{\mathrm {l}_0}\right) = \nabla ^2_\mathrm {X}C_{\mathrm {l}_1},\quad \nabla ^2_\mathrm {X}C_{\mathrm {m}_1} = 0,\nonumber \\&\nabla ^2_\mathrm {X}C_{\mathrm {w}_1} = \frac{\mathcal {R}}{\phi _\mathrm {w}},\quad \varepsilon \,\mathrm {Pe}\,\mathbf {\nabla }_\mathrm {X}\cdot \left( C_{\mathrm {f}_0}\mathbf {U}_{\mathrm {f}_1} + C_{\mathrm {f}_1}\mathbf {U}_{\mathrm {f}_0}\right) = \nabla ^2_\mathrm {X}C_{\mathrm {f}_1}. \end{aligned}$$

(145)

In general, (145) will need to be solved numerically in order to determine the correct matching condition for the outer solution, something which we do not pursue here. We note, however, that solution of this system would be further complicated by the appearance of $\mathbf {U}_{\mathrm {l}_1}$ and $\mathbf {U}_{\mathrm {f}_1}$, since inertia is not negligible at $\text{ O }(\varepsilon )$ in the inner region. However, progress can be made by again considering the far-field limit as $X \rightarrow \infty$.

We begin by making the assumption that the leading-order fluid flow in both the lumen and upper fluid layer is of Poiseuille form as it leaves the inner region. That is, as $X \rightarrow \infty$,

$$\begin{aligned} \mathbf {U}_{\mathrm {l}_0} \sim \left( \frac{3}{2}Q_\mathrm {l,in}(1-y^2),0\right) ,\quad \mathbf {U}_{\mathrm {f}_0} \sim \left( \frac{6}{h_4^3}Q_\mathrm {f,in}(y-H)(H-h_4-y),0\right) . \end{aligned}$$

(146)

Substituting these expressions into the far-field limits of the $\text{ O }(1)$ lumen and upper fluid layer equations for the solute concentration in (143), we find that the $\text{ O }(1)$ far-field concentration is constant in space, and so the terms involving $\mathbf {U}_{\mathrm {l}_1}$, $\mathbf {U}_{\mathrm {f}_1}$ disappear as they satisfy the continuity equation. Substituting in the far field forms for $\mathbf {U}_{\mathrm {l}_0}$ and $\mathbf {U}_{\mathrm {f}_0}$, analysis of (145) subject to the $\text{ O }(\varepsilon )$ boundary conditions from (137) to (141) then implies that

$$\begin{aligned}&C_{\mathrm {l}_1} \sim c_\infty (t) - \alpha X + f_\mathrm {l}(y), \end{aligned}$$

(147)

$$\begin{aligned}&C_{\mathrm {m}_1} \sim c_\infty (t) - \alpha X + f_\mathrm {m}(y), \end{aligned}$$

(148)

$$\begin{aligned}&C_{\mathrm {w}_1} \sim c_\infty (T) - \alpha X + f_\mathrm {w}(y), \end{aligned}$$

(149)

$$\begin{aligned}&C_{\mathrm {f}_1} \sim c_\infty (t) - \alpha X + f_\mathrm {f}(y), \end{aligned}$$

(150)

where

$$\begin{aligned}&f_\mathrm {l} = -\frac{\varepsilon \,\mathrm {Pe}Q_\mathrm {l,in}\alpha }{8}(6y^2-y^4) + \frac{\varepsilon \,\mathrm {Pe}}{2}C'_0(t)y^2 + c_{\infty }(t), \end{aligned}$$

(151)

$$\begin{aligned}&f_\mathrm {m} = -\varepsilon \,\mathrm {Pe}Q_\mathrm {l,in}\alpha \left( \frac{y-1}{\phi _\mathrm {m}}+\frac{5}{8}\right) + \varepsilon \,\mathrm {Pe}\,C'_0(t)\left[ \frac{y^2}{2} - \left( 1-\frac{1}{\phi _\mathrm {m}}\right) (y-1)\right] + c_{\infty }(t), \end{aligned}$$

(152)

$$\begin{aligned} f_\mathrm {w}&= -\varepsilon \,\mathrm {Pe}\,Q_\mathrm {l,in}\alpha \left( \frac{y-(1+h_2)}{\phi _\mathrm {w}} + \frac{h_2}{\phi _\mathrm {m}} + \frac{5}{8}\right) + \frac{\mathcal {R}}{2\phi _\mathrm {w}}(1+h_2-y)^2 \nonumber \\&\quad + \varepsilon \,\mathrm {Pe}\,C'_0(t)\left\{ \frac{y^2}{2} - (1+h_2)y + (1+h_2)^2 + \frac{\phi _\mathrm {m}}{\phi _\mathrm {w}}\left( h_2+\frac{1}{\phi _\mathrm {m}}\right) \left[ y-(1+h_2)\right] - h_2\left( 1-\frac{1}{\phi _\mathrm {m}}\right) \right\} + c_{\infty }(t), \end{aligned}$$

(153)

$$\begin{aligned} f_\mathrm {f}&= -\frac{\varepsilon \,\mathrm {Pe}\,Q_\mathrm {f,in}\alpha }{h_4^3}B_\mathrm {f}(y) - \varepsilon \,\mathrm {Pe}Q_\mathrm {l,in}\alpha \left( \frac{h_2}{\phi _\mathrm {m}} + \frac{h_3}{\phi _\mathrm {w}} + \frac{5}{8}\right) \nonumber \\&\quad + \varepsilon \,\mathrm {Pe}\,C'_0(t)\left[ \frac{y^2}{2} - Hy + (1+h_2)^2 + (H-h_4)(h_3+h_4) + \frac{h_3\phi _\mathrm {m}}{\phi _\mathrm {w}}\left( h_2+\frac{1}{\phi _\mathrm {m}}\right) - h_2\left( 1-\frac{1}{\phi _\mathrm {m}}\right) \right] \nonumber \\&\quad + \frac{h_3^2\mathcal {R}}{2\phi _\mathrm {w}} + c_{\infty }(t), \end{aligned}$$

(154)

$$\begin{aligned}&B_\mathrm {f}(y) := -\frac{y^4}{2} + (2H-h_4)y^3 - 3H(H-h_4)y^2-H^2(3h_4-2H)y - \frac{H^4}{2} + h_4H^3 - \frac{h_4^4}{2}, \end{aligned}$$

(155)

$$\begin{aligned}&\alpha (t) = \frac{h_3\mathcal {R} + \varepsilon \,\mathrm {Pe}\,\bar{h}C'_0(t)}{\varepsilon \,\mathrm {Pe}(Q_\mathrm {l,in}+Q_\mathrm {f,in})}, \end{aligned}$$

(156)

and $c_\infty (t)$ is a degree of freedom which, through matching with the outer solution, will determine the correct form for $\bar{c}_\mathrm {u}$. We note that the leading-order term in each of $C_{i_1}$ ($i=\mathrm {l},\mathrm {m},\mathrm {w},\mathrm {f}$) as $X \rightarrow \infty$ corresponds to a linear decay in concentration as a result of uniform uptake across all four sections. This reflects the fact that in the far field of the inner region as $X \rightarrow \infty$, transverse diffusion is sufficiently strong that the dominant contribution to the solute concentration is independent of y, as in the outer solution. In order to determine $c_\infty (t)$ (and hence, $\bar{c}_\mathrm {u}(t)$), it would be necessary to solve the full inner problem numerically at $\text{ O }(1)$ and $\text{ O }(\varepsilon )$. The exact form of $\bar{c}_\mathrm {u}$ will not affect the shape of the first-order solute concentrations in the outer region, but will determine their magnitude, and hence, whether or not these correction terms make a significant contribution to the leading-order concentration in the outer region.

In the downstream inner region near $x=1$, the same scalings apply and we set $x = 1-\varepsilon X$. This gives the same system as the upstream inner region, with sign changes in front of single X-derivatives and the downstream boundary conditions

$$\begin{aligned} P_\mathrm {l} = P_\mathrm {d}\quad \mathrm {at} \quad C,\quad P_\mathrm {f} = 0\quad \mathrm {at} \quad D. \end{aligned}$$

(157)

We once again find that the leading-order pressure in the lumen and upper fluid layer is a function of time only, and by the above boundary conditions we can deduce that, at leading order, $P_\mathrm {l} \equiv P_\mathrm {d}$ and $P_\mathrm {f} \equiv 0$. By matching with the outer solution as $X \rightarrow \infty$, this gives the correct downstream boundary conditions to apply on the leading-order outer system in Sect. 3.5 as

$$\begin{aligned} p_\mathrm {l} = P_\mathrm {d},\quad p_\mathrm {f} = 0\quad \mathrm {at}\ x=1^-. \end{aligned}$$

(158)

This analysis is sufficient for our purposes; however, we note that the inner problem here would also need to be solved numerically if the full solution was required: for instance, if it was necessary to know the fluid flux and solute concentration leaving the lumen outlet and downstream ECS port in order to match with experimental measurements.

Appendix 2: Boundary layer analysis of the combined-order equation

In this Appendix, we revisit the combined-order solute concentration equation from Sect. 3.8. We analyse more closely our choice for the additional boundary condition (111),

$$\begin{aligned} \frac{\partial \bar{c}}{\partial x} = 0\quad \mathrm {at}\quad x=1^-, \end{aligned}$$

(159)

which is required to close the averaged system (107) and (110):

$$\begin{aligned} \begin{aligned}&\frac{\partial \bar{c}}{\partial t} + \frac{Q_\mathrm {l,in}+Q_\mathrm {f,in}}{\bar{h}}\frac{\partial \bar{c}}{\partial x} = D_\mathrm {eff}\frac{\partial ^2 \bar{c}}{\partial x^2} - \frac{h_3\mathcal {R}}{\varepsilon \,\mathrm {Pe}},\\&\bar{c} = \bar{h}(c_\mathrm {in} + \varepsilon c_\mathrm {u})\quad \mathrm {at}\quad x=0^+. \end{aligned} \end{aligned}$$

(160)

To investigate the error that arises from our choice (159), here we solve the combined-order equation in both the outer region and the boundary layer near $x=1$ up to and including $\text{ O }(\varepsilon )$ and compare the resulting composite solution with the globally averaged asymptotic solutions at $\text{ O }(1)$ and $\text{ O }(\varepsilon )$, namely $\bar{c}_0$ and $\bar{c}_1$. We assume throughout our analysis that, at the orders considered, the concentration is non-zero throughout the domain (i.e. large enough t and appropriate operating conditions). Otherwise, if the concentration is zero for some $0^+ < \xi < x < 1^-$ we note that the boundary condition (159) is trivially satisfied.

We first solve the asymptotic equations and boundary conditions at $\text{ O }(1)$ and $\text{ O }(\varepsilon )$ from (100) and write in terms of averaged concentrations. This yields

$$\begin{aligned} \bar{c}_0&= -\frac{\bar{h}h_3\mathcal {R}}{\varepsilon \,\mathrm {Pe}(Q_\mathrm {l,in}+Q_\mathrm {f,in})}x + \bar{h}c_\mathrm {in}\left( t - \frac{\bar{h}}{Q_\mathrm {l,in}+Q_\mathrm {f,in}}x\right) , \end{aligned}$$

(161)

$$\begin{aligned} \bar{c}_1&= \frac{\bar{h}D_\mathrm {eff}}{\varepsilon }\left( \frac{\bar{h}}{Q_\mathrm {l,in}+Q_\mathrm {f,in}}\right) ^3xc''_\mathrm {in}\left( t - \frac{\bar{h}}{Q_\mathrm {l,in}+Q_\mathrm {f,in}}x\right) + \bar{h}\bar{c}_\mathrm {u}\left( t - \frac{\bar{h}}{Q_\mathrm {l,in}+Q_\mathrm {f,in}}x\right) , \end{aligned}$$

(162)

where $\bar{c}_0 = \bar{h}c_0$ and $'$ denotes differentiation with respect to the argument of a function. We define the combined, averaged asymptotic solution up to and including $\text{ O }(\varepsilon )$ by $\bar{c}_\mathrm {as} := \bar{c}_0 + \varepsilon \bar{c}_1$, which will be compared to the composite solution to the combined-order equation later.

We now consider the equation for the combined averaged formulation

$$\begin{aligned} \frac{\partial \bar{c}}{\partial t} + \frac{Q_\mathrm {l,in}+Q_\mathrm {f,in}}{\bar{h}}\frac{\partial \bar{c}}{\partial x} = D_\mathrm {eff}\frac{\partial ^2 \bar{c}}{\partial x^2} - \frac{h_3\mathcal {R}}{\varepsilon \,\mathrm {Pe}}, \end{aligned}$$

(163)

and the corresponding conditions

$$\begin{aligned} \bar{c} = \bar{h}(c_\mathrm {in} + \bar{c}_\mathrm {u})\quad \mathrm {at}\ x=0^+,\quad \frac{\partial \bar{c}}{\partial x}=0\quad \mathrm {at} \quad x=1^-. \end{aligned}$$

(164)

Firstly, we note that if we expand the combined-order concentration $\bar{c}$ in powers of $\varepsilon$ and solve (163) at $\text{ O }(1)$ and $\text{ O }(\varepsilon )$ subject to the inlet boundary condition at $x=0^+$ only, this yields ‘outer’ solutions which are identical to $\bar{c}_0$ and $\bar{c}_1$ in (161) and (162). For clarity of notation, when we form the composite solution later, we will denote this combined-order outer solution by $\bar{c}_\mathrm {out}$.

We now wish to find the inner solution in the boundary layer near $x=1$ at $\text{ O }(1)$ and $\text{ O }(\varepsilon )$. To do so, we rescale

$$\begin{aligned} x = 1-\varepsilon X\ \ (X<0),\quad \bar{c}(x,t) = \bar{C}(X,t), \end{aligned}$$

(165)

and (163) then becomes

$$\begin{aligned} \varepsilon \frac{\partial \bar{C}}{\partial t} - \frac{Q_\mathrm {l,in}+Q_\mathrm {f,in}}{\bar{h}}\frac{\partial \bar{C}}{\partial X} = \frac{D_\mathrm {eff}}{\varepsilon }\frac{\partial ^2 \bar{C}}{\partial X^2} - \frac{h_3\mathcal {R}}{\mathrm {Pe}}, \end{aligned}$$

(166)

subject to

$$\begin{aligned} \frac{\partial \bar{C}}{\partial X} = 0\quad \mathrm {at}\quad X=0, \end{aligned}$$

(167)

and matching with $\bar{c}$ as $X \rightarrow \infty$. Recalling that $D_\mathrm {eff} = \text{ O }(\varepsilon )$ and $\mathrm {Pe}= \text{ O }(1/\varepsilon )$, and expanding $\bar{C}$ in powers of $\varepsilon$ with the usual subscript notation, we find at $\text{ O }(1)$

$$\begin{aligned}&-\frac{Q_\mathrm {l,in}+Q_\mathrm {f,in}}{\bar{h}}\frac{\partial \bar{C}_0}{\partial X} = \frac{D_\mathrm {eff}}{\varepsilon }\frac{\partial ^2 \bar{C}_0}{\partial X^2},\nonumber \\&\frac{\partial \bar{C}_0}{\partial X} = 0\quad \mathrm {at}\quad X=0. \end{aligned}$$

(168)

Solving (168) implies that

$$\begin{aligned} \bar{C}_0 = \alpha (t), \end{aligned}$$

(169)

for some function $\alpha (t)$, which can be determined through matching with $\bar{c}_0$ as $x\rightarrow 1$ and $X \rightarrow \infty$. Using Van Dyke’s matching rule that the n-term-outer expansion of the m-term-inner expansion is equal to the m-term-inner expansion of the n-term-outer expansion gives for $m = n = 1$ the condition.

$$\begin{aligned} \bar{C}_0 = \alpha (t) = -\frac{\bar{h}h_3\mathcal {R}}{\varepsilon \,\mathrm {Pe}(Q_\mathrm {l,in}+Q_\mathrm {f,in})} + \bar{h}c_\mathrm {in}\left( t - \frac{\bar{h}}{Q_\mathrm {l,in}+Q_\mathrm {f,in}}\right) . \end{aligned}$$

(170)

At $\text{ O }(\varepsilon )$ the system to solve is

$$\begin{aligned}&\frac{\partial \bar{C}_0}{\partial t} - \frac{Q_\mathrm {l,in}+Q_\mathrm {f,in}}{\bar{h}}\frac{\partial \bar{C}_1}{\partial X} = \frac{D_\mathrm {eff}}{\varepsilon }\frac{\partial ^2 \bar{C}_1}{\partial X^2} - \frac{h_3\mathcal {R}}{\varepsilon \,\mathrm {Pe}},\nonumber \\&\frac{\partial \bar{C}_1}{\partial X}=0\quad \mathrm {at}\quad X=0, \end{aligned}$$

(171)

which has solution

$$\begin{aligned} \bar{C}_1&= \frac{\bar{h}}{Q_\mathrm {l,in}+Q_\mathrm {f,in}}\left[ \frac{D_\mathrm {eff}\bar{h}}{\varepsilon (Q_\mathrm {l,in}+Q_\mathrm {f,in})}\mathrm {exp}\left( -\frac{\varepsilon (Q_\mathrm {l,in}+Q_\mathrm {f,in})}{\bar{h}D_\mathrm {eff}}X\right) \right. \nonumber \\&\quad \,\, \left. +\,X - \frac{D_\mathrm {eff}\bar{h}}{\varepsilon (Q_\mathrm {l,in}+Q_\mathrm {f,in})}\right] \left[ \bar{h}c_\mathrm {in}'\left( t-\frac{\bar{h}}{Q_\mathrm {l,in}+Q_\mathrm {f,in}}\right) + \frac{h_3\mathcal {R}}{\varepsilon \,\mathrm {Pe}}\right] + \beta (t), \end{aligned}$$

(172)

for some function $\beta (t)$. Van Dyke’s matching rule (as stated above) is automatically satisfied for $m=2,n=1$ and $m=1,n=2$, and for $m=n=2$ we find

$$\begin{aligned} \beta (t)&= \frac{D_\mathrm {eff}}{\varepsilon }\left( \frac{\bar{h}}{Q_\mathrm {l,in}+Q_\mathrm {f,in}}\right) ^2\left[ \bar{h}c_\mathrm {in}'\left( t - \frac{\bar{h}}{Q_\mathrm {l,in}+Q_\mathrm {f,in}}\right) + \frac{h_3\mathcal {R}}{\varepsilon \,\mathrm {Pe}} \right. \nonumber \\&\quad \,\, \left. + \, \frac{\bar{h}^2}{Q_\mathrm {l,in}+Q_\mathrm {f,in}}c_\mathrm {in}''\left( t - \frac{\bar{h}}{Q_\mathrm {l,in}+Q_\mathrm {f,in}}\right) \right] + \bar{h}\bar{c}_\mathrm {u}\left( t-\frac{\bar{h}}{Q_\mathrm {l,in}+Q_\mathrm {f,in}}\right) . \end{aligned}$$

(173)

From (170) and (172), we can thus obtain the combined-order inner solution, $\bar{C}_\mathrm {in} := \bar{C}_0 + \varepsilon \bar{C}_1$ and then also construct the composite solution to the combined-order equation,

$$\begin{aligned} \bar{c}_\mathrm {comp} := \bar{c}_\mathrm {out} + \bar{C}_\mathrm {in} - \bar{c}_\mathrm {over}, \end{aligned}$$

(174)

where $\bar{c}_\mathrm {over}$ is the ‘overlap’ between the outer and inner solutions, found from the (2 t. i.)(2 t. o.) terms. We can now compare the composite solution $\bar{c}_\mathrm {comp}$ to the combined-order Eq. (163) to the combined, averaged asymptotic solution, $\bar{c}_\mathrm {as}$. Given the analysis presented here, we expect the two solutions to match up to $\text{ O }(\varepsilon ^2)$, except in the boundary layer of width $\text{ O }(\varepsilon )$ near $x=1$ within which we expect an error of $\text{ O }(\varepsilon )$. Plots comparing the two solutions confirm that this is the case (results not shown), and hence, (159) is an appropriate choice of boundary condition for the combined-order system in Sect. 3.8.