Mathematical Modelling of the Phloem: The Importance of Diffusion on Sugar Transport at Osmotic Equilibrium

Abstract

Plants rely on the conducting vessels of the phloem to transport the products of photosynthesis from the leaves to the roots, or to any other organs, for growth, metabolism, and storage. Transport within the phloem is due to an osmotically-generated pressure gradient and is hence inherently nonlinear. Since convection dominates over diffusion in the main bulk flow, the effects of diffusive transport have generally been neglected by previous authors. However, diffusion is important due to boundary layers that form at the ends of the phloem, and at the leaf-stem and stem-root boundaries. We present a mathematical model of transport which includes the effects of diffusion. We solve the system analytically in the limit of high Münch number which corresponds to osmotic equilibrium and numerically for all parameter values. We find that the bulk solution is dependent on the diffusion-dominated boundary layers. Hence, even for large Péclet number, it is not always correct to neglect diffusion. We consider the cases of passive and active sugar loading and unloading. We show that for active unloading, the solutions diverge with increasing Péclet. For passive unloading, the convergence of the solutions is dependent on the magnitude of loading. Diffusion also permits the modelling of an axial efflux of sugar in the root zone which may be important for the growing root tip and for promoting symbiotic biological interactions in the soil. Therefore, diffusion is an essential mechanism for transport in the phloem and must be included to accurately predict flow.

Appendices

Appendix A: Scaled Equations and Boundary Conditions

The scaled fluid transport and concentration transport equations are

$$\begin{aligned}&\epsilon ^{1+\delta -\beta }\frac{\hbox {d}^2\bar{w}^i}{\hbox {d}z^2}=\epsilon ^{\delta -\beta }\bar{w}^i-\bar{H}\frac{\hbox {d}c^i}{\hbox {d}z},\end{aligned}$$

(44)

$$\begin{aligned}&\epsilon ^{\delta -\chi }\frac{\hbox {d}c^i}{\hbox {d}z}\bar{w}^i+\epsilon ^{\delta -\chi }c^i\frac{\hbox {d}\bar{w}^i}{\hbox {d}z}+\frac{1}{\bar{\hbox {Pe}}}\frac{\hbox {d}^2c^i}{\hbox {d}z^2}=-\epsilon ^{\gamma -\chi }\bar{F}^i, \end{aligned}$$

(45)

in the bulk regions. In the leaf, stem, and root zones, the left-hand-side boundaries are $z=0$, $z=z_1$, and $z=z_2$, respectively, and we rewrite $z$ in terms of a boundary layer variable $x^i$ as; $z=\epsilon ^{\alpha }x^L$ in the leaf, $z=z_1+\epsilon ^{\alpha }x^S$ in the stem, and $z=z_2+\epsilon ^{\alpha }x^R$ in the root. Therefore, the equations in the LHS boundary region are

$$\begin{aligned}&\epsilon ^{1-\alpha +\delta -\beta }\frac{\hbox {d}^2\bar{w}^i}{\hbox {d}x^{i\,2}}=\epsilon ^{\delta +\alpha -\beta }\bar{w}^i-\bar{H}\frac{\hbox {d}c^i}{\hbox {d}x^i},\end{aligned}$$

(46)

$$\begin{aligned}&\epsilon ^{\alpha +\delta -\chi }\frac{\hbox {d}c^i}{\hbox {d}x^i}\bar{w}^i+\epsilon ^{\alpha +\delta -\chi }c^i\frac{\hbox {d}\bar{w}^i}{\hbox {d}x^i}+\frac{1}{\bar{\hbox {Pe}}}\frac{\hbox {d}^2c^i}{\hbox {d}x^{i\,2}}=-\epsilon ^{\gamma +2\alpha -\chi }\bar{F}^i. \end{aligned}$$

(47)

In the leaf, stem, and root zones, the right-hand-side boundaries, respectively, are $z=z_1$, $z=z_2$, and $z=1$, and we rewrite $z$ in terms of a boundary layer variable $y^i$ as; $z=z_1-\epsilon ^{\alpha }y^L$ in the leaf, $z=z_2-\epsilon ^{\alpha }y^S$ in the stem, and $z=1-\epsilon ^{\alpha }y^R$ in the root. Therefore, the RHS boundary region equations are

$$\begin{aligned}&\epsilon ^{1-\alpha +\delta -\beta }\frac{\hbox {d}^2\bar{w}^i}{\hbox {d}y^{i\,2}}=\epsilon ^{\delta +\alpha -\beta }\bar{w}^i+\bar{H}\frac{\hbox {d}c^i}{\hbox {d}y^i},\end{aligned}$$

(48)

$$\begin{aligned}&-\epsilon ^{\alpha +\delta -\chi }\frac{dc^i}{dy^i}\bar{w}^i-\epsilon ^{\alpha +\delta -\chi }c^i\frac{d\bar{w}^i}{dy^i}+\frac{1}{\bar{Pe}}\frac{d^2c^i}{dy^{i\,2}}=-\epsilon ^{\gamma +2\alpha -\chi }\bar{F}^i. \end{aligned}$$

(49)

The boundary conditions at $z=0$ and $z=1$ become

$$\begin{aligned} \bar{w}^L&= 0,\qquad c^L=\phi ,\qquad \text {at}\qquad x^L=0,\end{aligned}$$

(50)

$$\begin{aligned} \bar{w}^R&= 0,\qquad \frac{\hbox {d}c^R}{\hbox {d}y^R}=\epsilon ^{\gamma +\alpha -\chi }\bar{\hbox {Pe}}\bar{f} \qquad \text {at}\qquad y^R=0. \end{aligned}$$

(51)

The continuity conditions at the leaf-stem boundary are

$$\begin{aligned}&c^L=c^S,\qquad \epsilon ^{\delta }\bar{w}^L=\epsilon ^{\delta }\bar{w}^S,\end{aligned}$$

(52)

$$\begin{aligned}&\epsilon ^{\delta }c^L\bar{w}^L+\frac{\epsilon ^{\chi }}{\bar{\hbox {Pe}}}\frac{\hbox {d}c^L}{\hbox {d}z}=\epsilon ^{\delta }c^S\bar{w}^S+\frac{\epsilon ^{\chi }}{\bar{\hbox {Pe}}}\frac{\hbox {d}c^S}{\hbox {d}z},\qquad \epsilon ^{\delta }\frac{\hbox {d}\bar{w}^L}{\hbox {d}z}=\epsilon ^{\delta }\frac{\hbox {d}\bar{w}^S}{\hbox {d}z}. \end{aligned}$$

(53)

The continuity conditions at the stem-root boundary are

$$\begin{aligned}&c^S=c^R,\qquad \epsilon ^{\delta }\bar{w}^S=\epsilon ^{\delta }\bar{w}^R,\end{aligned}$$

(54)

$$\begin{aligned}&\epsilon ^{\delta }c^S\bar{w}^S+\frac{\epsilon ^{\chi }}{\bar{\hbox {Pe}}}\frac{\hbox {d}c^S}{\hbox {d}z}=\epsilon ^{\delta }c^R\bar{w}^R+\frac{\epsilon ^{\chi }}{\bar{\hbox {Pe}}}\frac{\hbox {d}c^R}{\hbox {d}z},\qquad \epsilon ^{\delta }\frac{\hbox {d}\bar{w}^S}{\hbox {d}z}=\epsilon ^{\delta }\frac{\hbox {d}\bar{w}^R}{\hbox {d}z}. \end{aligned}$$

(55)

Appendix B: Analytical Solution Procedure

Here, we give the procedure to determine $c^i$ and $\bar{w}^i$ at $O(\epsilon ^0)$ and $O(\epsilon ^{\frac{1}{2}})$. The overall method involves solving the fluid transport and concentration transport equations in each region, applying the boundary conditions at $z=0$ and $z=1$, matching the solutions across the regions to obtain composite solutions in each zone, and finally applying continuity conditions at $z=z_1$ and $z=z_2$ to fully determine the solutions. We start at Region 1 which is in the leaf zone.

1.1 Region 1

This region represents the LHS boundary layer in the leaf zone. The solution in Region 1 has to match to the solution in Region 2 and also satisfy the boundary conditions

$$\begin{aligned}&c_0^L+\epsilon ^{\frac{1}{2}}c_1^L+\epsilon c_2^L+O(\epsilon ^{\frac{3}{2}})=\phi ,\qquad \text { at } x^L=0 \end{aligned}$$

(56)

$$\begin{aligned}&\bar{w}_0^L+\epsilon ^{\frac{1}{2}}\bar{w}_1^L+\epsilon \bar{w}_2^L+O(\epsilon ^{\frac{3}{2}})=0,\qquad \text { at } x^L=0. \end{aligned}$$

(57)

The $O(\epsilon ^0)$ terms of Eqs. (21) and (22) are

$$\begin{aligned} -\bar{H}\frac{\hbox {d}c_0^L}{\hbox {d}x^L}=0,\qquad \frac{1}{\bar{\hbox {Pe}}}\frac{\hbox {d}^2c_0^L}{\hbox {d}x^{L\,2}}=0, \end{aligned}$$

(58)

and applying the boundary condition (56), the solution for $c_0^L$ in Region 1 is $c_0^L=\phi$. The $O(\epsilon ^\frac{1}{2})$ terms of Eqs. (21) and (22) are

$$\begin{aligned} -\bar{H}\frac{\hbox {d}c_1^L}{\hbox {d}x^L}=0,\qquad \frac{1}{\bar{\hbox {Pe}}}\frac{\hbox {d}^2c_1^L}{\hbox {d}x^{L\,2}}=0, \end{aligned}$$

(59)

and applying the boundary condition (56), the solution for $c_1^L$ in Region 1 is $c_1^L=0$. The $O(\epsilon )$ terms of Eqs. (21) and (22) are

$$\begin{aligned}&\frac{\hbox {d}^2\bar{w}_0^L}{\hbox {d}x^{L\,2}}=\bar{w}_0^L-\bar{H}\frac{\hbox {d}c_2^L}{\hbox {d}x^L},\end{aligned}$$

(60)

$$\begin{aligned}&\frac{\hbox {d}c_0^L}{\hbox {d}x^L}\bar{w}_0^L+c_0^L\frac{\hbox {d}\bar{w}_0^L}{\hbox {d}x^L}+\frac{1}{\bar{\hbox {Pe}}}\frac{\hbox {d}^2c_2^L}{\hbox {d}x^{L\,2}}=0, \end{aligned}$$

(61)

and substituting $c_0^L=\phi$ into (61) and integrating, we obtain $\tfrac{\mathrm{{d}}c_2^L}{\mathrm{{d}}x^L}=-\bar{\hbox {Pe}}\phi \bar{w}_0^L+k_1$ where $k_1$ is an unknown constant. Substituting back into (60) yields

$$\begin{aligned} \frac{\hbox {d}^2\bar{w}_0^L}{\hbox {d}x^{L\,2}}=\bar{w}_0^L\left( 1+\bar{H}\bar{\hbox {Pe}}\phi \right) -\bar{H}k_1. \end{aligned}$$

(62)

Solving (62), applying the boundary condition (57), and neglecting any exponentially growing terms, we obtain

$$\begin{aligned} \bar{w}_0^L=\frac{k_1\bar{H}}{1+\bar{H}\bar{\hbox {Pe}}\phi }\left( 1-\hbox {e}^{-\sqrt{1+\bar{H}\bar{\hbox {Pe}}\phi }x^L}\right) . \end{aligned}$$

(63)

The constant $k_1$ can only be determined by applying continuity conditions between the plant zones but this can only occur once the composite solution is determined for the entirety of the leaf zone. We therefore proceed to Region 2 where we repeat this solution procedure.

1.2 Region 2

This region represents the central bulk of the leaf zone, and the solution in this region has to match to both the solution in Region 1 and Region 3. The $O(\epsilon ^0)$ terms of Eqs.  (19) and (20) are

$$\begin{aligned} -\bar{H}\frac{\hbox {d}c_0^L}{\hbox {d}z}=0,\qquad \frac{1}{\bar{\hbox {Pe}}}\frac{\hbox {d}^2c_0^L}{\hbox {d}z^2}=0, \end{aligned}$$

(64)

such that $c_0^L$ in Region 2 is a constant, which upon matching to Region 1 gives $c_0^L=\phi$. The $O(\epsilon ^{\frac{1}{2}})$ terms of Eqs. (19) and (20) are

$$\begin{aligned} 0=\bar{w}_0^L-\bar{H}\frac{\hbox {d}c_1^L}{\hbox {d}z},\qquad \frac{\hbox {d}c_0^L}{\hbox {d}z}\bar{w}_0^L+c_0^L\frac{\hbox {d}\bar{w}_0^L}{\hbox {d}z}+\frac{1}{\bar{\hbox {Pe}}}\frac{\hbox {d}^2c_1^L}{\hbox {d}z^2}=-\bar{F}^L_0, \end{aligned}$$

(65)

where $\bar{F}^L_0=\bar{\eta }^Lc^L_0+\bar{\sigma }^L$. Substituting $c_0^L=\phi$, solving Eq. (65), and matching to the solution in Region 1, we find

$$\begin{aligned}&c_1^L=\frac{\bar{\hbox {Pe}}}{\bar{H}\bar{\hbox {Pe}}\phi +1}\left( -\bar{F}^L_0\frac{z^2}{2}+\frac{k_1}{\bar{\hbox {Pe}}} z\right) ,\end{aligned}$$

(66)

$$\begin{aligned}&\bar{w}_0^L=\frac{\bar{H}\bar{\hbox {Pe}}}{\bar{H}\bar{\hbox {Pe}}\phi +1}\left( -\bar{F}^L_0z+\frac{k_1}{\bar{\hbox {Pe}}}\right) . \end{aligned}$$

(67)

1.3 Region 3

This region represents the RHS boundary layer in the leaf zone. The solution in Region 3 has to match to the solution in Region 2 and satisfy continuity conditions at $z=z_1$. The $O(\epsilon ^0)$ terms of Eqs. (23) and (24) are

$$\begin{aligned} \bar{H}\frac{\hbox {d}c_0^L}{\hbox {d}y^L}=0,\qquad \frac{1}{\bar{\hbox {Pe}}}\frac{\hbox {d}^2c_0^L}{\hbox {d}y^{L\,2}}=0, \end{aligned}$$

(68)

such that $c_0^L$ in Region 3 is a constant, which upon matching to Region 2 gives $c_0^L=\phi$. The $O(\epsilon ^{\frac{1}{2}})$ terms of Eqs. (23) and (24) are

$$\begin{aligned} \bar{H}\frac{\hbox {d}c_1^L}{\hbox {d}y^L}=0,\qquad \frac{1}{\bar{\hbox {Pe}}}\frac{\hbox {d}^2c_1^L}{\hbox {d}y^{L\,2}}=0, \end{aligned}$$

(69)

which indicates that $c_1^L$ in Region 3 is a constant and we let $c_1^L=B^L_1$, where $B^L_1$ is an unknown constant. The $O(\epsilon )$ terms of Eqs. (23) and (24) are

$$\begin{aligned}&\frac{\hbox {d}^2\bar{w}_0^L}{\hbox {d}y^{L\,2}}=\bar{w}_0^L+\bar{H}\frac{\hbox {d}c_2^L}{\hbox {d}y^L},\end{aligned}$$

(70)

$$\begin{aligned} -&\frac{\hbox {d}c_0^L}{\hbox {d}y^L}\bar{w}_0^L-c_0^L\frac{\hbox {d}\bar{w}_0^L}{\hbox {d}y^L}+\frac{1}{\bar{\hbox {Pe}}}\frac{\hbox {d}^2c_2^L}{\hbox {d}y^{L\,2}}=0, \end{aligned}$$

(71)

and substituting $c_0^L=\phi$ into (71) and integrating, we obtain $\tfrac{\mathrm{d}c_2^L}{\mathrm{d}y^L}=-\bar{\hbox {Pe}}\phi \bar{w}_0^L+B^L_2$ where $B^L_2$ is an unknown constant. Substituting back into (70) yields

$$\begin{aligned} \frac{\hbox {d}^2\bar{w}_0^L}{\hbox {d}y^{L\,2}}=\bar{w}_0^L\left( 1+\bar{H}\bar{\hbox {Pe}}\phi \right) +\bar{H}B^L_2. \end{aligned}$$

(72)

Solving for $\bar{w}_0^L$ and neglecting any exponentially growing terms, we obtain

$$\begin{aligned} \bar{w}_0^L=A_1\hbox {e}^{-\sqrt{1+\bar{H}\bar{\hbox {Pe}}\phi }y^L}-\frac{\bar{H}B^L_2}{\bar{H}\bar{\hbox {Pe}}\phi +1}. \end{aligned}$$

(73)

Matching to Region 2 gives

$$\begin{aligned} B^L_1=\frac{\bar{\hbox {Pe}}}{\bar{H}\bar{\hbox {Pe}}\phi +1}\left( -\frac{\bar{F}^L_0z_1^2}{2}+\frac{k_1}{\bar{\hbox {Pe}}}z_1\right) ,\qquad B^L_2=\bar{\hbox {Pe}}\bar{F}^L_0z_1-k_1. \end{aligned}$$

(74)

Combining the solutions for $c$ and $\bar{w}$ for all three regions within the leaf zone, we obtain the following composite solution for the leaf zone

$$\begin{aligned} c_0^L&=\phi ,\end{aligned}$$

(75)

$$\begin{aligned} c_1^L&=\frac{\bar{\hbox {Pe}}}{\kappa }\left( -\bar{F}^L_0\frac{z^2}{2}+\frac{k_1}{\bar{\hbox {Pe}}}z\right) ,\end{aligned}$$

(76)

$$\begin{aligned} \bar{w}_0^L&=\frac{\bar{H}\bar{\hbox {Pe}}}{\kappa }\left( -\bar{F}^L_0z+\frac{k_1}{\bar{\hbox {Pe}}}\left( 1-\hbox {e}^{-\sqrt{\kappa }\epsilon ^{-\frac{1}{2}}z}\right) \right) +A_1\hbox {e}^{-\sqrt{\kappa }\epsilon ^{-\frac{1}{2}}(z_1-z)}, \end{aligned}$$

(77)

where $\kappa =\bar{H}\bar{\hbox {Pe}}\phi +1$ and where $k_1$ and $A_1$ are unknown constants to be determined by applying continuity conditions at the zone boundaries. Before continuity can be applied, we first calculate the stem and root composite solutions. For brevity, we do not show the details here, but we do use the same procedure outlined in sections Region 1, Region 2, and Region 3 for the leaf zone.

1.4 Applying Continuity Conditions

Solving at $O(\epsilon ^0)$ in the remaining six regions, we determine that $c_0$ is constant everywhere. Assuming that $c_0$ is continuous across the zone boundaries, we find that $c_0^i=\phi$ for all nine regions. Solving at the next order, $O(\epsilon ^\frac{1}{2})$, we find the following composite solutions in the stem and root zones

$$\begin{aligned} c_1^S&=\frac{\bar{\hbox {Pe}}}{\kappa }\left( -\bar{F}^S_0\frac{z^2}{2}+k_2z+k_3\right) ,\end{aligned}$$

(78)

$$\begin{aligned} \bar{w}_0^S&=\frac{\bar{H}\bar{\hbox {Pe}}}{\kappa }\left( -\bar{F}^S_0z+k_2\right) +A_2\hbox {e}^{-\sqrt{\kappa }\epsilon ^{-\frac{1}{2}}(z-z_1)}+A_3\hbox {e}^{-\sqrt{\kappa }\epsilon ^{-\frac{1}{2}}(z_2-z),}\end{aligned}$$

(79)

$$\begin{aligned} c_1^R&=\frac{\bar{\hbox {Pe}}}{\kappa }\left( -\bar{F}^R_0\frac{z^2}{2}+(\bar{F}^R-\bar{f})z+k_4\right) ,\end{aligned}$$

(80)

$$\begin{aligned} \bar{w}_0^R&=\frac{\bar{H}\bar{\hbox {Pe}}}{\kappa }\left( -\bar{F}^R_0z+(\bar{F}^R_0-\bar{f})\right) +\frac{\bar{H}\bar{\hbox {Pe}}\bar{f}}{\kappa }\hbox {e}^{-\sqrt{\kappa }\epsilon ^{-\frac{1}{2}}(1-z)}+A_4\hbox {e}^{-\sqrt{\kappa }\epsilon ^{-\frac{1}{2}}(z-z_2)}, \end{aligned}$$

(81)

where $k_1$, $k_2$, $k_3$, $k_4$, $A_1$, $A_2$, $A_3$, $A_4$ are unknown constants to be determined from applying continuity boundary conditions.

As can be seen, the solutions for $\bar{w}_0^i$ contain exponential terms due to the boundary layer terms. This implies that terms of a higher order (for example $c_2^i$ and $\bar{w_1}^i$) can contribute to the flux continuity conditions due to the presence of $\epsilon ^{-\frac{1}{2}}$ within the exponential. For example, if we write

$$\begin{aligned} \bar{w}^i=\bar{w}^{i,\mathrm{bulk}}+\bar{w}^{i,\mathrm{exp}}\hbox {e}^{\epsilon ^{-\frac{1}{2}}f(z)},\qquad c^i=c^{i,\mathrm{bulk}}+c^{i,\mathrm{exp}}\hbox {e}^{\epsilon ^{-\frac{1}{2}}f(z)}, \end{aligned}$$

(82)

where $f(z)$ refers to a function of $z$, then the derivatives, $\frac{\mathrm{{d}}\bar{w}^i}{\mathrm{{d}}z}$ and $\frac{\mathrm{{d}}c^i}{\mathrm{{d}}z}$, which appear in the flux continuity conditions become

$$\begin{aligned} \frac{\hbox {d}\bar{w}^i}{\hbox {d}z}&=\epsilon ^{-\frac{1}{2}}\left( \frac{\hbox {d}f_0}{\hbox {d}z}\bar{w}_0^{i,\mathrm{exp}}\hbox {e}^{\epsilon ^{-\frac{1}{2}}f_0}\right) \nonumber \\&\quad +\,\epsilon ^0\left( \frac{\hbox {d}\bar{w}_0^{i,\mathrm{bulk}}}{\hbox {d}z}+\frac{\hbox {d}\bar{w}_0^{i,\mathrm{exp}}}{\hbox {d}z}\hbox {e}^{\epsilon ^{-\frac{1}{2}}f_0}+\frac{\hbox {d}f_1}{\hbox {d}z}\bar{w}_1^{i,\mathrm{exp}}\hbox {e}^{\epsilon ^{-\frac{1}{2}}f_1}\right) +O(\epsilon ^{\frac{1}{2}}),\end{aligned}$$

(83)

$$\begin{aligned} \frac{\hbox {d}c^i}{\hbox {d}z}&=\epsilon ^{-\frac{1}{2}}\left( \frac{\hbox {d}f_0}{\hbox {d}z}c_0^{i,\mathrm{exp}}\hbox {e}^{\epsilon ^{-\frac{1}{2}}f_0}\right) \nonumber \\&\quad +\,\epsilon ^0\left( \frac{\hbox {d}c_0^{i,\mathrm{bulk}}}{\hbox {d}z}+\frac{\hbox {d}c_0^{i,\mathrm{exp}}}{\hbox {d}z}\hbox {e}^{\epsilon ^{-\frac{1}{2}}f_0}+\frac{\hbox {d}f_1}{\hbox {d}z}c_1^{i,\mathrm{exp}}\hbox {e}^{\epsilon ^{-\frac{1}{2}}f_1}\right) +O(\epsilon ^{\frac{1}{2}}). \end{aligned}$$

(84)

Since $c_0^{i,\mathrm{exp}}=c_1^{i,\mathrm{exp}}=0$ and $\tfrac{\mathrm{{d}}c_0^{i,\mathrm{bulk}}}{\mathrm{{d}}z}=0$, the first contribution to $\tfrac{\mathrm{d}c^i}{\mathrm{d}z}$ is at $O(\epsilon ^{\frac{1}{2}})$. The consequence of this is that the exponential terms of $c_2^i$ first have to be calculated before continuity conditions can be applied and $c_1^i$ and $\bar{w}_0^i$ can be fully determined.

Solving for $c_2^i$ in all the Regions using the same procedure as in Region 1, Region 2 and Region 3, and then applying the continuity conditions, we find

$$\begin{aligned} k_1&= \bar{\hbox {Pe}}\left( z_1\left( \bar{F}^L_0-\bar{F}^S_0\right) +z_2\left( \bar{F}^S_0-\bar{F}^R_0\right) +\bar{F}^R_0-\bar{f}\right) ,\end{aligned}$$

(85)

$$\begin{aligned} k_2&= z_2(\bar{F}^S_0-\bar{F}^R_0)+\bar{F}^R_0-\bar{f},\end{aligned}$$

(86)

$$\begin{aligned} k_3&= \frac{z_1^2}{2}\left( \bar{F}^L_0-\bar{F}^S_0\right) ,\end{aligned}$$

(87)

$$\begin{aligned} k_4&= \frac{z_2^2}{2}\left( \bar{F}^S_0-\bar{F}^R_0\right) +\frac{z_1^2}{2}\left( \bar{F}^L_0-\bar{F}^S_0\right) ,\end{aligned}$$

(88)

$$\begin{aligned} A_1&= A_2=A_3=A_4=0, \end{aligned}$$

(89)

indicating that, at this order, there are no boundary layer terms near $z_1$ and $z_2$. Finally, we calculate the exponential terms of $c_3^i$ in all Regions to fully determine $c_2^i$ and $\bar{w}_1^i$, the solutions of which are given in Appendix . At this order, we find that boundary layer terms are non-zero at $z_1$ and $z_2$ and hence we terminate the calculation here.

Appendix C: Analytical Solution

Here, we provide the solutions for $c_0^i$, $c_1^i$, $c_2^i$, $c_3^{i}\,\bar{w}_0^i$ and $\bar{w}_1^i$

$$\begin{aligned} c_0^L&=c_0^S=c_0^R=\phi , \end{aligned}$$

(90)

$$\begin{aligned} c_1^L&=\frac{\bar{\hbox {Pe}}}{\kappa }\left( -\bar{F}^L_0\frac{z^2}{2}+\frac{k_1}{\bar{\hbox {Pe}}}z\right) , \end{aligned}$$

(91)

$$\begin{aligned} c_1^S&=\frac{\bar{\hbox {Pe}}}{\kappa }\left( -\bar{F}^S_0\frac{z^2}{2}+k_2z+k_3\right) ,\end{aligned}$$

(92)

$$\begin{aligned} c_1^R&=\frac{\bar{\hbox {Pe}}}{\kappa }\left( -\bar{F}^R_0\frac{z^2}{2}+(\bar{F}^R_0-\bar{f})z+k_4\right) , \end{aligned}$$

(93)

$$\begin{aligned} c_2^L&=\frac{\bar{\hbox {Pe}}}{\kappa }\left( \frac{k_5}{\bar{\hbox {Pe}}}z+\frac{\bar{H}\phi k_1}{\kappa ^{\frac{1}{2}}}\right) -\bar{H}\left( \frac{\bar{\hbox {Pe}}}{\kappa }\right) ^3\left( \frac{\bar{F}^{L\,2}_0z^4}{8}-\frac{\bar{F}^L_0k_1z^3}{2\bar{\hbox {Pe}}}+\frac{k_1^2z^2}{2\bar{\hbox {Pe}}^2}\right) \nonumber \\&\quad -\,\bar{\eta }^L\left( \frac{\bar{\hbox {Pe}}}{\kappa }\right) ^2\left( -\frac{\bar{F}^{L}_0z^4}{24}+\frac{k_1z^3}{\bar{\hbox {Pe}}6}\right) -\frac{k_1\bar{H}\bar{\hbox {Pe}}\phi }{\kappa ^{\frac{3}{2}}}\hbox {e}^{-\left( \frac{\kappa }{\epsilon }\right) ^{\frac{1}{2}}z},\end{aligned}$$

(94)

$$\begin{aligned} c_2^S&=\frac{\bar{\hbox {Pe}}}{\kappa }\left( k_6z+k_7\right) -\bar{\eta }^S\left( \frac{\bar{\hbox {Pe}}}{\kappa }\right) ^2\left( -\frac{\bar{F}^{S}_0z^4}{24}+\frac{k_2z^3}{6}+\frac{k_3z^2}{2}\right) \nonumber \\&\quad -\,\bar{H}\left( \frac{\bar{\hbox {Pe}}}{\kappa }\right) ^3\left( \frac{\bar{F}^{S\,2}_0z^4}{8}-\frac{\bar{F}^S_0k_2z^3}{2}+\frac{\left( k_2^2-\bar{F}^S_0k_3\right) z^2}{2}+k_2k_3z\right) ,\end{aligned}$$

(95)

$$\begin{aligned} c_2^R&=-\bar{H}\left( \frac{\bar{\hbox {Pe}}}{\kappa }\right) ^3\Bigg (\frac{\bar{F}^{R\,2}_0z^4}{8}-\frac{\left( \bar{F}^R_0-\bar{f}\right) \bar{F}^R_0z^3}{2} \nonumber \\&\quad +\,\frac{\left( \left( \bar{F}^R_0-\bar{f}\right) ^2-\bar{F}^R_0k_4\right) z^2}{2}+k_4\left( \bar{F}^R_0-\bar{f}\right) z\Bigg ) \nonumber \\&\quad -\,\bar{\eta }^R\left( \frac{\bar{\hbox {Pe}}}{\kappa }\right) ^2\left( -\frac{\bar{F}^{R}_0z^4}{24}+\frac{\left( \bar{F}^R_0-\bar{f}\right) z^3}{6}+\frac{k_4z^2}{2}-\left( \frac{\bar{F}^R_0}{3}-\frac{\bar{f}}{2}+k_4\right) z\right) \nonumber \\&\quad +\,k_8-\frac{\bar{H}\bar{\hbox {Pe}}^2\phi \bar{f}}{\kappa ^{\frac{3}{2}}}\hbox {e}^{-\left( \frac{\kappa }{\epsilon }\right) ^{\frac{1}{2}}\left( 1-z\right) }, \end{aligned}$$

(96)

The exponential terms of $c_3$ are

$$\begin{aligned} c_3^{L,\mathrm{exp}}&=-\frac{k_5\bar{H}\bar{\hbox {Pe}}\phi }{\kappa ^{\frac{3}{2}}}\hbox {e}^{-(\frac{\kappa }{\epsilon })^{\frac{1}{2}} z}-\frac{A_5\bar{\hbox {Pe}}\phi }{\kappa ^{\frac{1}{2}}} \hbox {e}^{-\left( \frac{\kappa }{\epsilon }\right) ^{\frac{1}{2}} (z_1-z)},\end{aligned}$$

(97)

$$\begin{aligned} c_3^{S,\mathrm{exp}}&=\frac{A_6\bar{\hbox {Pe}}\phi }{\kappa ^{\frac{1}{2}}}\hbox {e}^{-(\frac{\kappa }{\epsilon })^{\frac{1}{2}} (z-z_1)}-\frac{A_7\bar{\hbox {Pe}}\phi }{\kappa ^{\frac{1}{2}}} \hbox {e}^{-\left( \frac{\kappa }{\epsilon }\right) ^{\frac{1}{2}} (z_2-z)},\end{aligned}$$

(98)

$$\begin{aligned} c_3^{R,\mathrm{exp}}&=\frac{\bar{\hbox {Pe}}^3\bar{f}\bar{H}\left( \frac{\bar{F}^R_0}{2}-\bar{f}+k_4\right) }{\kappa ^{\frac{7}{2}}}\left( -1+\frac{\bar{H}\bar{\hbox {Pe}}}{2}(1-z)\phi \left( \frac{\kappa }{\epsilon }\right) ^{\frac{1}{2}}\right) \hbox {e}^{-\left( \frac{\kappa }{\epsilon }\right) ^{\frac{1}{2}} (1-z)} \nonumber \\&\quad +\frac{A_8\bar{\hbox {Pe}}\phi }{\kappa ^{\frac{1}{2}}} \hbox {e}^{-\left( \frac{\kappa }{\epsilon }\right) ^{\frac{1}{2}} (z-z_2)} ,\end{aligned}$$

(99)

$$\begin{aligned} \bar{w}_0^L&=\frac{\bar{H}\bar{\hbox {Pe}}}{\kappa }\left( -\bar{F}^L_0z\right) +\frac{k_1 \bar{H}}{\kappa }\left( 1-\hbox {e}^{-\left( \frac{\kappa }{\epsilon }\right) ^{\frac{1}{2}}z}\right) ,\end{aligned}$$

(100)

$$\begin{aligned} \bar{w}_0^S&=\frac{\bar{H}\bar{\hbox {Pe}}}{\kappa }\left( -\bar{F}^S_0z+k_2\right) ,\end{aligned}$$

(101)

$$\begin{aligned} \bar{w}_0^R&=\frac{\bar{H}\bar{\hbox {Pe}}}{\kappa }\left( -\bar{F}^R_0z+(\bar{F}^R_0-\bar{f})\right) +\frac{\bar{H}\bar{\hbox {Pe}}\bar{f}}{\kappa }\hbox {e}^{-\left( \frac{\kappa }{\epsilon }\right) ^{\frac{1}{2}}(1-z)},\end{aligned}$$

(102)

$$\begin{aligned} \bar{w}_1^L&=\frac{\bar{H}k_5}{\kappa }-\bar{H}^2\left( \frac{\bar{\hbox {Pe}}}{\kappa }\right) ^3\left( \frac{\bar{F}^{L\,2}_0z^3}{2}-\frac{3\bar{F}^L_0k_1z^2}{2\bar{\hbox {Pe}}}+\frac{k_1^2z}{\bar{\hbox {Pe}}^2}\right) \nonumber \\&\quad -\,\bar{\eta }^L\bar{H}\left( \frac{\bar{\hbox {Pe}}}{\kappa }\right) ^2\left( -\frac{\bar{F}^{L}_0z^3}{6}+\frac{k_1z^2}{\bar{\hbox {Pe}}2}\right) -\frac{\bar{H}k_5}{\kappa }\hbox {e}^{-\left( \frac{\kappa }{\epsilon }\right) ^{\frac{1}{2}}z}+A_5\hbox {e}^{-\left( \frac{\kappa }{\epsilon }\right) ^{\frac{1}{2}}\left( z_1-z\right) },\end{aligned}$$

(103)

$$\begin{aligned} \bar{w}_1^S&=\frac{\bar{H}\bar{\hbox {Pe}}}{\kappa }k_6-\bar{H}^2\left( \frac{\bar{\hbox {Pe}}}{\kappa }\right) ^3\left( \frac{\bar{F}^{S\,2}_0z^3}{2}-\frac{3\bar{F}^S_0k_2z^2}{2}+{\left( k_2^2-\bar{F}^S_0k_3\right) z}+k_2k_3\right) \nonumber \\&\quad -\,\bar{\eta }^S\bar{H}\left( \frac{\bar{\hbox {Pe}}}{\kappa }\right) ^2\left( -\frac{\bar{F}^{S}_0z^3}{6}+\frac{k_2z^2}{2}+{k_3z}\right) \nonumber \\&\quad +A_6\hbox {e}^{-\left( \frac{\kappa }{\epsilon }\right) ^{\frac{1}{2}}\left( z-z_1\right) }+A_7\hbox {e}^{-\left( \frac{\kappa }{\epsilon }\right) ^{\frac{1}{2}}\left( z_2-z\right) },\end{aligned}$$

(104)

$$\begin{aligned} \bar{w}_1^R&=-\bar{H}^2\left( \frac{\bar{\hbox {Pe}}}{\kappa }\right) ^3\Bigg (\frac{\bar{F}^{R\,2}_0z^3}{2}-\frac{3\left( \bar{F}^R_0-\bar{f}\right) \bar{F}^R_0z^2}{2} \nonumber \\&\quad +\,\left( \left( \bar{F}^R_0-\bar{f}\right) ^2-\bar{F}^R_0k_4\right) z+k_4\left( \bar{F}^R_0-\bar{f}\right) \Bigg )+A_8\hbox {e}^{-\left( \frac{\kappa }{\epsilon }\right) ^{\frac{1}{2}}\left( z-z_2\right) } \nonumber \\&\quad -\,\bar{\eta }^R\bar{H}\left( \frac{\bar{\hbox {Pe}}}{\kappa }\right) ^2\left( -\frac{\bar{F}^{R}_0z^3}{6}+\frac{\left( \bar{F}^R_0-\bar{f}\right) z^2}{2}+{k_4z}-\left( \frac{\bar{F}^R_0}{3}-\frac{\bar{f}}{2}+k_4\right) \right) \nonumber \\&\quad -\,\bar{H}^2\bar{f}\left( \frac{\bar{\hbox {Pe}}}{\kappa }\right) ^3\left( \frac{\bar{F}^R_0}{2}-\bar{f}+k_4\right) \left( 1+\frac{\left( \frac{\kappa }{\epsilon }\right) ^{\frac{1}{2}}\left( 1-z\right) }{2}\right) \hbox {e}^{-\left( \frac{\kappa }{\epsilon }\right) ^{\frac{1}{2}}\left( 1-z\right) }, \end{aligned}$$

(105)

The constants $k_1$ to $k_4$ and $A_1$ to $A_4$ are given in Eq. (85) to Eq. (89), and $k_5$, $k_6$, $k_7$, $k_8$, $A_5$, $A_6$, $A_7$, $A_8$ are given by

$$\begin{aligned} A_5&=\frac{\bar{H}\bar{\hbox {Pe}}}{2\kappa ^{\frac{3}{2}}}\left( \bar{F}^L_0-\bar{F}^S_0\right) ,\qquad A_6=A_5,\end{aligned}$$

(106)

$$\begin{aligned} A_7&=\frac{\bar{H}\bar{\hbox {Pe}}}{2\kappa ^{\frac{3}{2}}}\left( \bar{F}^S_0-\bar{F}^R_0\right) ,\qquad A_7=A_8,\end{aligned}$$

(107)

$$\begin{aligned} k_5&=\frac{\bar{\eta }^L\bar{\hbox {Pe}}^2z_1^2}{2\kappa }\left( z_1\left( \frac{2}{3}\bar{F}^L_0-\bar{F}^S_0\right) +z_2\left( \bar{F}^S_0-\bar{F}^R_0\right) +\bar{F}^R_0-\bar{f}\right) \nonumber \\&\quad +\,\frac{(z_1-z_2)\bar{\eta }^S\bar{\hbox {Pe}}^2}{2\kappa }\Big (-z_1^2\bar{F}^L_0+\frac{2}{3}\bar{F}^S_0(z_1-z_2)(2z_1+z_2) \nonumber \\&\quad +\,\bar{F}^R_0(z_1+z_2)(z_2-1)+\bar{f}(z_1+z_2)\Big ) \nonumber \\&\quad +\,\frac{\bar{\eta }^R\bar{\hbox {Pe}}^2(z_2-1)}{2\kappa }\Big (-z_1^2\bar{F}^L_0+\bar{F}^S_0(z_1+z_2)(z_1-z_2) \nonumber \\&\quad +\,\frac{2}{3}\bar{F}^R_0(2z_2+1)(z_2-1)+\bar{f}(z_2+1)\Big ),\end{aligned}$$

(108)

$$\begin{aligned} k_6&=\frac{\bar{\eta }^S\bar{\hbox {Pe}}z_2}{2\kappa }\left( z_1^2\bar{F}^L_0-\frac{1}{3}\bar{F}^S_0(3z_1^2-2z_2^2)-\bar{F}^R_0z_2(z_2-1)-z_2\bar{f}\right) \nonumber \\&\quad +\,\frac{\bar{\eta }^R\bar{\hbox {Pe}}(z_2-1)}{2\kappa }\Big (-z_1^2\bar{F}^L_0+\bar{F}^S_0(z_1+z_2)(z_1-z_2) \nonumber \\&\quad +\,\frac{2}{3}\bar{F}^R_0(2z_2+1)(z_2-1)+\bar{f}(z_2+1)\Big ),\end{aligned}$$

(109)

$$\begin{aligned} k_7&=-\frac{1}{8\kappa ^2}\bar{\hbox {Pe}}^2\bar{H}z_1^4(\bar{F}^L_0-\bar{F}^S_0)^2 \nonumber \\&\quad +\,\frac{\bar{H}\bar{\hbox {Pe}}\phi }{\kappa ^{\frac{1}{2}}}\left( z_1(\bar{F}^L_0-\bar{F}^S_0)+z_2(\bar{F}^S_0-\bar{F}^R_0)+\bar{F}^R_0-\bar{f}\right) \nonumber \\&\quad +\,\frac{\bar{\eta }^L\bar{\hbox {Pe}}z_1^3}{24\kappa }\left( z_1(5\bar{F}^L_0-8\bar{F}^S_0)+8z_2(\bar{F}^S_0-\bar{F}^R_0)+8\bar{F}^R_0-8\bar{f}\right) \nonumber \\&\quad -\,\frac{\bar{\eta }^S\bar{\hbox {Pe}}z_1^3}{24\kappa }\left( z_1(6\bar{F}^L_0-9\bar{F}^S_0)+8z_2(\bar{F}^S_0-\bar{F}^R_0)+8\bar{F}^R_0-8\bar{f}\right) , \end{aligned}$$

(110)

$$\begin{aligned} k_8&=-\frac{\bar{\hbox {Pe}}^3\bar{H}}{8\kappa ^3}\left( z_1^2(\bar{F}^L_0-\bar{F}^S_0)+z_2^2(\bar{F}^S_0-\bar{F}^R_0)\right) ^2 \nonumber \\&\quad +\,\frac{\bar{H}\bar{\hbox {Pe}}^2\phi }{\kappa ^{\frac{3}{2}}}\left( z_1(\bar{F}^L_0-\bar{F}^S_0)+z_2(\bar{F}^S_0-\bar{F}^R_0)+\bar{F}^R_0-\bar{f}\right) \nonumber \\&\quad -\,\frac{\bar{\eta }^R\bar{\hbox {Pe}}^2z_2^2}{24\kappa ^2}\left( 6z_1^2(\bar{F}^L_0-\bar{F}^S_0)+z_2^2(6\bar{F}^S_0-9\bar{F}^R_0)+8z_2(\bar{F}^R_0-\bar{f})\right) \nonumber \\&\quad +\,\frac{\bar{\eta }^L\bar{\hbox {Pe}}^2z_1^3}{24\kappa ^2}\left( z_1(5\bar{F}^L_0-8\bar{F}^S_0)+8z_2(\bar{F}^S_0-\bar{F}^R_0)+8\bar{F}^R_0-8\bar{f}\right) \nonumber \\&\quad +\,\frac{\bar{\eta }^S\bar{\hbox {Pe}}^2(z_1-z_2)}{24\kappa ^2}\Big (\bar{F}^S_0(9z_1^2+10z_1z_2+5z_2^2)(z_1-z_2) \nonumber \\&\quad -\,6z_1^2\bar{F}^L_0(z_1+z_2) \nonumber \\&\quad +\,8\bar{F}^R_0(z_2-1)(z_1^2+z_1z_2+z_2^2)+8\bar{f}(z_1^2+z_1z_2+z_2^2)\Big ), \end{aligned}$$

(111)

where $\bar{F}^L_0=\bar{\eta }^Lc^L_0+\bar{\sigma }^L,\qquad \bar{F}^S_0=\bar{\eta }^Sc^S_0+\bar{\sigma }^S,\qquad \bar{F}^R_0=\bar{\eta }^Rc^R_0+\bar{\sigma }^R$.