Coupled Model of a Biological Fluid Filtration Through a Flat Layer with Due Account for Barodiffusion

Abstract

This paper describes the model of liquid flow in porous layer taking into account the finite compressibility and concentration expansion associated with both diffusion and the additional mechanism of the pressure change in the pore space volume. The equations for component fluxes are determined within the framework of thermodynamics of irreversible processes and include the phenomenon of component transfer under the action of a pressure gradient. This new transfer mechanism differs from the transfer by convective fluid flow. Thus, the model is fully coupled. As an example illustrating the role of barodiffusion and interaction of diffusion and filtration, the problem of fluid transport through a flat porous layer is considered. In limiting particular cases, the problem is reduced to known models, but in general case it has peculiarities which required detailed investigation. Firstly, dimensionless complexes including characteristic physical scales of different phenomena are identified. Secondly, an iterative algorithm for numerical solution of the formulated nonlinear problem was developed. Further the influence of taken into account effects on flow characteristics and concentration distribution is investigated numerically. It was found that in convective and diffusive flow modes, the influence of dimensionless complexes responsible for the interrelation of different phenomena is different. New qualitative regularities in the distribution of component concentrations and flow velocity are distinguished. It is found that barodiffusion has the most significant influence on the flow in the diffusion regime. Its role also increases with increase of compressibility of liquid. The detected effects can have an application to biological systems, for example, to the transfer of substances through membranes.

Appendix

We describe below the solution algorithm for the following equation system

$$\begin{aligned} & \delta \frac{{{\text{d}}^{2} C}}{{{\text{d}}\xi^{2} }} - Y\frac{{{\text{d}}C}}{{{\text{d}}\xi }} + gC = 0, \\ & \frac{{{\text{d}}^{2} \overline{p}}}{{{\text{d}}\xi^{2} }} - Z\frac{{{\text{d}}\overline{p}}}{{{\text{d}}\xi }} = 0, \\ \end{aligned}$$

where the notation is entered

$$\begin{aligned} & Y = \omega \left( {1 + \alpha_{\rm C} C} \right)\left( {\frac{{{\text{d}}\overline{p}}}{{{\text{d}}\xi }}} \right)^{2} + \alpha_{\rm C} \delta \frac{{{\text{d}}C}}{{{\text{d}}\xi }} - \frac{{{\text{d}}\overline{p}}}{{{\text{d}}\xi }}\left( {\overline{\beta }_{\rm T} \delta + 1} \right) = Y\left( {C,u} \right), \\ & g = \omega \overline{\beta }_{\rm T} \left( {\frac{{{\text{d}}\overline{p}}}{{{\text{d}}\xi }}} \right)^{3} = g\left( u \right), \\ & Z = \alpha_{\rm C} \frac{{{\text{d}}C}}{{{\text{d}}\xi }} - \overline{\beta }_{\rm T} \frac{{{\text{d}}\overline{p}}}{{{\text{d}}\xi }} = Z\left( {\frac{{{\text{d}}C}}{{{\text{d}}\xi }},u} \right). \\ \end{aligned}$$

For an arbitrary point i of the difference grid, we have Taylor expansions in small integration step to the right and to the left of this point:

$$\begin{aligned} & C_{i + 1} = C_{i} + \left( {\frac{{{\text{d}}C_{i} }}{{{\text{d}}\xi }}} \right)h + \left( {\frac{{{\text{d}}^{2} C_{i} }}{{{\text{d}}\xi^{2} }}} \right)\frac{{h^{2} }}{2} + \cdots , \\ & C_{i - 1} = C_{i} - \left( {\frac{{{\text{d}}C_{i} }}{{{\text{d}}\xi }}} \right)h + \left( {\frac{{{\text{d}}^{2} C_{i} }}{{{\text{d}}\xi^{2} }}} \right)\frac{{h^{2} }}{2} + \cdots . \\ \end{aligned}$$

Expressing $\left( {\frac{{{\text{d}}C_{i} }}{{{\text{d}}\xi }}} \right)$ to the right and to the left of point i

$$\left( {\frac{{{\text{d}}C_{i} }}{{{\text{d}}\xi }}} \right)_{{{\text{right}}}} = \frac{{C_{i + 1} - C_{i} }}{h} - \left( {\frac{{{\text{d}}^{2} C_{i} }}{{{\text{d}}\xi^{2} }}} \right)\frac{h}{2}\quad {\text{and}}\quad \left( {\frac{{{\text{d}}C_{i} }}{{{\text{d}}\xi }}} \right)_{{{\text{left}}}} = \frac{{C_{i} - C_{i - 1} }}{h} + \left( {\frac{{{\text{d}}^{2} C_{i} }}{{{\text{d}}\xi^{2} }}} \right)\frac{h}{2}$$

and then substituting it into (45) we obtain

$$\begin{aligned} & \delta \frac{{{\text{d}}^{2} C}}{{{\text{d}}\xi^{2} }} - \frac{{Y_{i} + \left| {Y_{i} } \right|}}{2}\left( {\frac{{{\text{d}}C_{i} }}{{{\text{d}}\xi }}} \right)_{left} - \frac{{Y_{i} - \left| {Y_{i} } \right|}}{2}\left( {\frac{{{\text{d}}C_{i} }}{{{\text{d}}\xi }}} \right)_{right} + gC_{i} = 0\quad {\text{or}} \\ & \delta \frac{{{\text{d}}^{2} C}}{{{\text{d}}\xi^{2} }} - \frac{{Y_{i} + \left| {Y_{i} } \right|}}{2}\left( {\frac{{C_{i} - C_{i - 1} }}{h} + \left( {\frac{{{\text{d}}^{2} C_{i} }}{{{\text{d}}\xi^{2} }}} \right)\frac{h}{2}} \right) - \frac{{Y_{i} - \left| {Y_{i} } \right|}}{2}\left( {\frac{{C_{i + 1} - C_{i} }}{h} - \left( {\frac{{{\text{d}}^{2} C_{i} }}{{{\text{d}}\xi^{2} }}} \right)\frac{h}{2}} \right) + gC_{i} = 0. \\ \end{aligned}$$

Hence, Eq. (45) appears in the form:

$$\delta \frac{{{\text{d}}^{2} C}}{{{\text{d}}\xi^{2} }}\left[ {1 - \frac{h}{2\delta }\left| {Y_{i} } \right|} \right] - \frac{{Y_{i} + \left| {Y_{i} } \right|}}{2}\frac{{C_{i} - C_{i - 1} }}{h} - \frac{{Y_{i} - \left| {Y_{i} } \right|}}{2}\frac{{C_{i + 1} - C_{i} }}{h} + g_{i} C_{i} = 0$$

Since h ≪ 1, $\left( {1 - \frac{h}{2\delta }\left| {Y_{i} } \right|} \right) \approx \exp \left[ { - \frac{h}{2\delta }\left| {Y_{i} } \right|} \right]$, and the last equation can be rewritten as follow:

$$\begin{aligned} & C_{i + 1} \left[ {\exp \left[ { - \frac{h}{2\delta }\left| {Y_{i} } \right|} \right]\frac{\delta }{{h^{2} }} - \frac{{Y_{i} - \left| {Y_{i} } \right|}}{2h}} \right] - C_{i} \left[ {2\exp \left[ { - \frac{h}{2\delta }\left| {Y_{i} } \right|} \right]\frac{\delta }{{h^{2} }} + \frac{{\left| {Y_{i} } \right|}}{h}} \right] \\ & \quad + C_{i - 1} \left[ {\exp \left[ { - \frac{h}{2\delta }\left| {Y_{i} } \right|} \right]\frac{\delta }{{h^{2} }} + \frac{{Y_{i} + \left| {Y_{i} } \right|}}{2h}} \right] + g_{i} C_{i} = 0. \\ \end{aligned}$$

As a result, we get

$$a_{i} C_{i + 1} - d_{i} C_{i} + b_{i} C_{i - 1} + f_{i} = 0,$$

where

$$\begin{aligned} & a_{i} = \exp \left[ { - \frac{h}{2\delta }\left| {Y_{i} } \right|} \right]\frac{\delta }{{h^{2} }} - \frac{{Y_{i} - \left| {Y_{i} } \right|}}{2h}, \\ & b_{i} = \exp \left[ { - \frac{h}{2\delta }\left| {Y_{i} } \right|} \right]\frac{\delta }{{h^{2} }} + \frac{{Y_{i} + \left| {Y_{i} } \right|}}{2h}, \\ & d_{i} = 2\exp \left[ { - \frac{h}{2\delta }\left| {Y_{i} } \right|} \right]\frac{\delta }{{h^{2} }} + \frac{{\left| {Y_{i} } \right|}}{h},\quad f_{i} = C_{i} g_{i} , \\ \end{aligned}$$

which can be solved by the double-sweep method. A double-sweep method for particular cases of such systems (n = 1, 2) has been widely discussed in the literature (Hazewinkel 2013; Samarskij and Nikolaev 1989) All the values included into the coefficients follow from the previous iteration; the coefficients satisfy the following condition

$$\left| {a_{i} + b_{i} } \right| \le d_{i} ,$$

i.e. for every iteration, the sweep will be stable.

Similarly to the previous, we have

$$\begin{aligned} & \overline{p}_{i + 1} = \overline{p}_{i} + \left( {\frac{{{\text{d}}\overline{p}_{i} }}{{{\text{d}}\xi }}} \right)h + \left( {\frac{{{\text{d}}^{2} \overline{p}_{i} }}{{{\text{d}}\xi^{2} }}} \right)\frac{{h^{2} }}{2} + \cdots , \\ & \overline{p}_{i - 1} = \overline{p}_{i} - \left( {\frac{{{\text{d}}\overline{p}_{i} }}{{{\text{d}}\xi }}} \right)h + \left( {\frac{{{\text{d}}^{2} \overline{p}_{i} }}{{{\text{d}}\xi^{2} }}} \right)\frac{{h^{2} }}{2} + \cdots . \\ \end{aligned}$$

From here, we express $\left( {\frac{{{\text{d}}\overline{p}_{i} }}{{{\text{d}}\xi }}} \right)$ to the right and to the left of point i:

$$\left( {\frac{{{\text{d}}\overline{p}_{i} }}{{{\text{d}}\xi }}} \right)_{{{\text{right}}}} = \frac{{\overline{p}_{i + 1} - \overline{p}_{i} }}{h} - \left( {\frac{{{\text{d}}^{2} \overline{p}_{i} }}{{{\text{d}}\xi^{2} }}} \right)\frac{h}{2}\quad {\text{and}}\quad \left( {\frac{{{\text{d}}\overline{p}_{i} }}{{{\text{d}}\xi }}} \right)_{{{\text{left}}}} = \frac{{\overline{p}_{i} - \overline{p}_{i - 1} }}{h} + \left( {\frac{{{\text{d}}^{2} \overline{p}_{i} }}{{{\text{d}}\xi^{2} }}} \right)\frac{h}{2}.$$

Therefore,

$$\frac{{{\text{d}}^{2} \overline{p}}}{{{\text{d}}\xi^{2} }} - \frac{{Z_{i} + \left| {Z_{i} } \right|}}{2}\left( {\frac{{\overline{p}_{i} - \overline{p}_{i - 1} }}{h} + \left( {\frac{{{\text{d}}^{2} \overline{p}_{i} }}{{{\text{d}}\xi^{2} }}} \right)\frac{h}{2}} \right) - \frac{{Z_{i} - \left| {Z_{i} } \right|}}{2}\left( {\frac{{\overline{p}_{i + 1} - \overline{p}_{i} }}{h} - \left( {\frac{{{\text{d}}^{2} \overline{p}_{i} }}{{{\text{d}}\xi^{2} }}} \right)\frac{h}{2}} \right) = 0$$

or

$$\frac{{{\text{d}}^{2} \overline{p}}}{{{\text{d}}\xi^{2} }}\left[ {1 - \frac{{Z_{i} + \left| {Z_{i} } \right|}}{2}\frac{h}{2} + \frac{{Z_{i} - \left| {Z_{i} } \right|}}{2}\frac{h}{2}} \right] - \frac{{Z_{i} + \left| {Z_{i} } \right|}}{2}\left( {\frac{{\overline{p}_{i} - \overline{p}_{i - 1} }}{h}} \right) - \frac{{Z_{i} - \left| {Z_{i} } \right|}}{2}\left( {\frac{{\overline{p}_{i + 1} - \overline{p}_{i} }}{h}} \right) = 0$$

or

$$\frac{{{\text{d}}^{2} \overline{p}}}{{{\text{d}}\xi^{2} }}\left[ {1 - \frac{{\left| {Z_{i} } \right|}}{2}h} \right] - \frac{{Z_{i} + \left| {Z_{i} } \right|}}{2}\left( {\frac{{\overline{p}_{i} - \overline{p}_{i - 1} }}{h}} \right) - \frac{{Z_{i} - \left| {Z_{i} } \right|}}{2}\left( {\frac{{\overline{p}_{i + 1} - \overline{p}_{i} }}{h}} \right) = 0.$$

For small steps of the difference grid we have

$$\frac{{{\text{d}}^{2} \overline{p}}}{{{\text{d}}\xi^{2} }}\exp \left( { - \frac{{\left| {Z_{i} } \right|}}{2}h} \right) - \frac{{Z_{i} + \left| {Z_{i} } \right|}}{2}\left( {\frac{{\overline{p}_{i} - \overline{p}_{i - 1} }}{h}} \right) - \frac{{Z_{i} - \left| {Z_{i} } \right|}}{2}\left( {\frac{{\overline{p}_{i + 1} - \overline{p}_{i} }}{h}} \right) = 0$$

or

$$\frac{{\overline{p}_{i + 1} - 2\overline{p}_{i} + \overline{p}_{i - 1} }}{{h^{2} }}\exp \left( { - \frac{{\left| {Z_{i} } \right|}}{2}h} \right) - \frac{{Z_{i} + \left| {Z_{i} } \right|}}{2}\left( {\frac{{\overline{p}_{i} - \overline{p}_{i - 1} }}{h}} \right) - \frac{{Z_{i} - \left| {Z_{i} } \right|}}{2}\left( {\frac{{\overline{p}_{i + 1} - \overline{p}_{i} }}{h}} \right) = 0$$

or

$$\overline{p}_{i + 1} \left[ {\frac{1}{{h^{2} }}\exp \left( { - \frac{{\left| {Z_{i} } \right|}}{2}h} \right) - \frac{{Z_{i} - \left| {Z_{i} } \right|}}{2h}} \right] - \overline{p}_{i} \left[ {\frac{2}{{h^{2} }}\exp \left( { - \frac{{\left| {Z_{i} } \right|}}{2}h} \right) + \frac{{\left| {Z_{i} } \right|}}{h}} \right] + \overline{p}_{i - 1} \left[ {\frac{1}{{h^{2} }}\exp \left( { - \frac{{\left| {Z_{i} } \right|}}{2}h} \right) + \frac{{Z_{i} + \left| {Z_{i} } \right|}}{2h}} \right] = 0.$$

As a result, we get

$$a_{i} p_{i + 1} - d_{i} p_{i} + b_{i} p_{i - 1} + f_{i} = 0,$$

where

$$\begin{aligned} & a_{i} = \frac{1}{{h^{2} }}\exp \left( { - \frac{{\left| {Z_{i} } \right|}}{2}h} \right) - \frac{{Z_{i} - \left| {Z_{i} } \right|}}{2h}, \\ & b_{i} = \frac{1}{{h^{2} }}\exp \left( { - \frac{{\left| {Z_{i} } \right|}}{2}h} \right) + \frac{{Z_{i} + \left| {Z_{i} } \right|}}{2h}, \\ & d_{i} = \frac{2}{{h^{2} }}\exp \left( { - \frac{{\left| {Z_{i} } \right|}}{2}h} \right) + \frac{{\left| {Z_{i} } \right|}}{h},\quad f_{i} = 0. \\ \end{aligned}$$

Table 2 presents final velocities and concentrations in chosen points after increasing the amount of divisions of the calculation domain (when decreasing the spatial step), obtained after solving set (41)–(43) by the method of simple iteration. Here, we see a good convergence even for the number of points n = 200. The calculations presented in Table below were performed at the following fixed parameters: $\alpha_{\rm C} = 0.57$; $\overline{p}_{2} = 0.7$; $\overline{\beta }_{\rm T} = 0.01$; $\delta = 0.05$; $\omega = 0.3$. The values of the parameters $\alpha_{\rm C}$ and $\overline{\beta }_{\rm T}$ were estimated for biological fluids such as saline and blood. For other biological fluids, the parameter $\overline{\beta }_{\rm T}$ will differ slightly, and the value of parameter $\alpha_{\rm C}$ depends on the size of the atoms in the liquid phase.

Table

Table 3 Example of convergence of iterations, n = 200

3 shows subsequent approximations for velocity and concentration for the partitions number n = 200 illustrating the convergence of the solution. Table data means that the result after 5–6 iterations can be regarded satisfactory.

The similar convergence check was made in different variation ranges of the model parameters and with different specified accuracy.