# A Pore Scale Model for Osmotic Flow: Homogenization and Lattice Boltzmann Simulations

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## Abstract

Osmosis is the phenomenon of spontaneous passage of solvents through a membrane that is permeable to the solvent but is completely or partially impermeable to solute particles. On a macroscopic scale, it is well understood how a difference in concentration of solute across a membrane gives rise to an osmotic pressure that may induce a flow through the membrane. On the pore scale inside the membrane, however, the ongoing processes are less well understood. In this paper, a model is presented for how osmotic effects on the pore scale are induced by forces acting on the solute from the membrane material. Furthermore, homogenization results rigorously derived elsewhere by one of the authors (Heintz and Piatnitski in Netw Heterog Media 11(3):585–610, 2016) are presented, and an implementation of the homogenized model using the lattice Boltzmann method is described. The homogenization results provide a means to compute macroscopic parameters determining the osmotic flow through a porous material, in particular the so called reflection coefficient. The numerical results show excellent agreement with theoretical results for straight cylindrical channels and also illustrate the applicability of the method to periodic porous media.

## Keywords

Osmosis Pore scale model Porous media Lattice Boltzmann method## 1 Introduction

The purpose of the present paper is to present in an accessible manner a model describing osmosis at various length scales for porous media that are partially permeable to solute particles, as well as to investigate numerically using the lattice Boltzmann approach the applicability of this model. The model describes the effect of forces acting between the material of a porous medium and electrically neutral solute particles on the transport of the solvent and the solute.

Osmosis is the phenomenon of spontaneous passage of water or other solvents through a membrane that is permeable to the solvent but is impermeable to solute particles. If such a semipermeable membrane separates pure solvent from a solution, the pure solvent will move through the membrane, making the solution at the opposite side of the membrane more dilute. The osmotic pressure across the membrane can be measured by measuring the external counter-pressure that needs to be applied to stop the flow of solvent.

The phenomenon of osmosis was first observed by the French experimental physicist Jean-Antoine Nollet in 1748 for membranes in nature, but was studied in detail by the German plant physiologist Wilhelm Pfeffer only in 1877. The term osmose or osmosis was introduced by the British chemist Thomas Graham in 1854.

*c*is the number density of solute particles,

*T*is temperature, and \(k_B\) is the Boltzmann constant. Van’t Hoff’s work, including in particular this formula, was rewarded with the first Nobel Prize in Chemistry in 1901.

Porous membranes are often not completely impermeable to solute particles, but depending on the size of pores, only partially obstruct the passage of particles. The effect of osmotic pressure is in this case not concentrated only at the outer surfaces of the membrane, but is distributed within its volume. Therefore, several phenomena combine to determine the transport of solute and solvent through the membrane in this case. The question about the nature of osmosis in such intermediate regimes is interesting for many applications in biology (Kedem and Katchalsky 1962; Elmoazzen et al. 2009) and in such modern technologies as desalination (Cath et al. 2006; Zhao et al. 2012) and sustainable power generation (Logan and Elimelech 2012).

*V*distributed along the surface of the porous material. The equations, which together with appropriate boundary and initial conditions describe the fluid velocity

*u*and pressure

*p*, and the solute number density

*c*under the influence of the potential, are

*D*is the diffusion constant, and \(\mu \) the mobility of solvent molecules. Note in particular the force term \(-\,c \nabla V\) in the Stokes equation, which arises from the friction between the solute particles and the fluid.

At the macroscopic level, a homogenized system of partial differential equations for limits of the pressure and velocity of the solvent and for the concentration of solute was derived rigorously by Heintz and Piatnitski (2016) using the two-scale limit approach (Allaire 1992), in the case when the porous solid microstructure is periodic with period \(\varepsilon \ll 1.\) The homogenization procedure is convenient to carry out for the concentration scaled by the Maxwellian distribution associated with the potential forces between the porous media and the solute particles. An effective Darcy’s type system of equations for the flow under osmotic pressure distributed within the porous medium and the formula (18) for the distribution of osmotic pressure inside the porous media in the limit of small \(\varepsilon \) gives a quantitative answer about the nature of the osmotic transport of neutral solutes at the microscopic level. The coefficients in the derived homogenized equations relate the values of the phenomenological coefficients in (2) with particular properties of the flow at the pore level.

Microscopic models of this type for flows with distributed osmotic pressure were considered in the one-dimensional case for flows in long channels by Anderson and Malone (1974) and Anderson et al. (1982) and were developed also by Wyman and Kostin (1973), Guell (1991), and Guell and Brenner (1996). They were applied to simple geometries by Zhang et al. (2006), Jensen et al. (2009), and Yant et al. (1986). However, studies of microscopic models for osmotic pressure in general porous geometries with electrically neutral solute particles have to the best of our knowledge not been previously performed.

Related mathematical and numerical problems for the Nernst–Planck–Poisson and the Nernst–Planck–Poisson–Stokes systems for nonstationary electrokinetic models were considered by Looker and Carnie (2006), Schmuck (2009), Schmuck (2011), and Allaire et al. (2010).

In this paper, the pore scale mathematical model and the homogenization results obtained by Heintz and Piatnitski (2016) for electrically neutral solute particles are described. In particular, a procedure to compute the reflection coefficient \(\sigma \) for a periodic porous material by solving cell problems in a periodic unit cell is presented. Furthermore, the implementation of the model using the lattice Boltzmann method is described, and some computational results validating the method by comparing to the results of Anderson and Malone (1974) are shown, together with an application to a more general periodic porous medium.

## 2 Mathematical Model

Throughout this paper, we suppose that the boundary of the porous structure is a periodic surface. The unit periodic cell is denoted by *Y*. Without loss of generality, we suppose that \(Y=\left[ 0,1\right] ^{N}\). We denote by \(Y_{F} \) an empty part of *Y* not filled by the solid material and assume that its periodic extension is connected. In what follows, we refer to \(Y_{F} \) as the fluid part of the porous medium. \(Y_{S}=Y\backslash Y_{F}\) denotes the solid part of the structure in *Y*. We also let \(Y^{\varepsilon } = \varepsilon Y\) denote periodicity cell scaled by the small parameter \(\varepsilon \).

The solute number density *c* satisfies the Nernst–Planck type advection–diffusion Eq. (4b), see also (Probstein 2003), with drift defined in terms of the potential \(V_{\varepsilon }\) acting close to the solid boundaries of the porous structure. We here consider the stationary case, i.e., with \(\partial c/\partial t = 0\). Let *V* be a periodic potential on the unit cell *Y*. We define the scaled potential by \(V_{\varepsilon }(x)=V\left( \frac{x}{\varepsilon }\right) \). We also apply zero normal flux boundary conditions for *c* on the solid boundary \(\varGamma _{\varepsilon }\) and fixed values for *c* on the inflow and outflow boundaries \(S_{1}^{\varepsilon }\) and \(S_{2}^{\varepsilon }\), defined as \(S_{i}^{\varepsilon }=S_{i}\cap \overline{\varOmega }_{\varepsilon }\), \(i=1,2 \).

The fluid velocity *u* and pressure *p* of the solvent are described by the stationary Stokes equations with the osmotic force \(-c\nabla V_{\varepsilon }\) arising from the friction between the solute particles and the fluid. The velocity *u* satisfies nonslip boundary conditions on the solid boundary \(\varGamma _{\varepsilon }\), and we impose boundary conditions for the pressure *p* on the inflow and outflow boundaries \(S_{1}^{\varepsilon }\) and \(S_{2}^{\varepsilon }\) as constant values \(\overline{P}_{1}\) and \(\overline{P}_{2}\), and for the tangential component of velocity as \(u_{\tau }=0\).

*u*, and drift term flux \(\mu c \nabla V_{\varepsilon }\). Here, \(\theta _{2}\ge 0\) is a constant, and the function \(\beta _{\varepsilon }(x)\) is defined as

*T*is absolute temperature and \(k_B\) is the Boltzmann constant, and thus van’t Hoff’s formula (1) for osmotic pressure can in our notation be rewritten as

*c*.

Only the difference \(\delta \overline{P}=\overline{P}_{1}-\overline{P}_{2}\) between pressure values at the inflow and outflow boundaries \(S_{1}\) and \(S_{2}\) has physical meaning and is controlled.

## 3 Homogenized Model

In this section, the results by Heintz and Piatnitski (2016) about the homogenized macroscopic limit problem are repeated for completeness. They were derived rigorously by Heintz and Piatnitski (2016) by passing to the two-scale homogenized limit (Allaire 1992; Nguetseng 1989) in the weak integral form of the microscopic system of Eqs. (5)–(6). In practical terms, it means that when the period \(\varepsilon \) tends to zero, the functions \(\theta \), *u*, *p* describing the complete picture of the transport tend to functions of two variables *x* and \(y=\frac{x}{\varepsilon } \), of which the first describes smooth dependence on the macroscopic scale and the second describes periodic oscillations at the microscopic scale.

We use in this applied paper the term convergence not specifying its rigorous meaning that might differ in particular cases and refer to the paper (Heintz and Piatnitski 2016) for mathematically strict formulations and proofs of these results. Considering these limits, we extend the unknowns \(\theta \), *u*, *p* keeping the same notations, into the solid part of the porous media: Velocity and scaled concentration are extended by zero, and pressure by its average value over the fluid part of the periodicity cell.

We consider here the case of small Péclet numbers on the microscale, more precisely \(Pe \rightarrow 0\) when \(\varepsilon \rightarrow 0\), where \(Pe = u L /D\) for a characteristic length *L*, where \(L \propto \epsilon \) on the microscale. In this case, the equation for the limit \(\varTheta \) of the scaled concentration \(\theta = c\beta _{\varepsilon }^{-1}\) is decoupled from the limit equation for the flow. Still, \(\theta \) plays a role in the Stokes part of the system, and its limit \(\varTheta \) enters a homogenized Darcy’s type equation for the flow.

In the case when the Péclet number is large, which is not considered here, the limit macroscopic equations become nonlinear and nonlocal and rigorous limit results are more complicated than here. The study of this case is ongoing.

For the case of small Péclet numbers, the main result for the scaled concentration \(\theta \) is as follows (see Heintz and Piatnitski 2016 for proof and more details).

### Theorem 1

*Y*:

For the fluid velocity *u* and pressure *p*, the following convergence result holds (again, see Heintz and Piatnitski 2016 for proof).

### Lemma 1

*x*in \(\varOmega \) and periodic with respect to

*y*in the unit cell

*Y*, such that \(\eta \varepsilon ^{-2}u\) and

*p*converge in the two-scale sense to these functions.

*y*is substituted by \(\left( \frac{x}{\varepsilon }\right) .\)

*Y*that are specified below.

We note that a formula similar to (18) was derived by Anderson and Malone (1974) in the one-dimensional case for an infinitely long cylindric channel.

In the following theorem, a Darcy’s type law with distributed osmotic forces is formulated for two-scale limits of the unknown variables.

As was proved by Heintz and Piatnitski (2016), one can in our situation (with small *Pe*) separate the macrosopic *x*-variable from the microscopic *y*-variable in the two-scale homogenized system (see Heintz and Piatnitski (2016)) for the two-scale limits \(u_{0}(x,y)\) and \(p_{0}(x,y)\) and reduce it to a microscopic periodic cell problem with respect to the *y*-variable on the periodic cell *Y* and to a homogenized macroscopic problem with respect to the *x*-variable only in the domain \(\varOmega \).

### Theorem 2

*i*-th coordinate axis.

*P*(

*x*) and \(\varTheta (x)\) becomes

*U*(

*x*) is expressed explicitly in terms of

*P*and \(\varTheta \) as in (20), the coefficient matrix \(A_{\mathrm {eff}}\) is computed from Eqs. (15)–(16), and \(B_P\) and \(B_{\varPi }\) are computed from Eqs. (21)–(23).

We note that the limit macroscopic system (24) consists of a decoupled effective diffusion equation and a Darcy type equation with an additional flux term \(B_{\varPi } \frac{D }{\mu }\nabla \varTheta (x) \) representing the effect of the distributed osmotic pressure.

## 4 Numerical Simulations

### 4.1 Lattice Boltzmann Method

The lattice Boltzmann method (Guo and Shu 2013) was used to solve both the full model (5)–(6) and the cell problems (22)–(23) and (16) in the homogenized model. All simulations were performed in three space dimensions.

*f*is

*q*is the number of discrete velocities in the model, \(c_i\) are the discrete velocity vectors,

*A*is a collision operator, and \(f^{\text {(eq)}}\) is the equilibrium distribution. To solve the Stokes equations, the standard Navier–Stokes equilibrium

*f*as

*V*.

The two-relaxation-time collision operator (Ginzburg et al. 2008) was used, with the parameter \(\varLambda =3/16\), which eliminates the viscosity dependence of the computed permeability. The relaxation parameter \(\lambda _e\) was given different values but in most cases set to \(-0.5\).

*f*is related to \(\nabla \theta \), see also (Ginzburg 2005). The time-dependent equation solved using the lattice Boltzmann method is therefore

When solving the cell problem (16), an external flux \(\beta (x) e_i\) in the *i*:th direction was added by setting \(U=\beta e_i\) in (32).

The Neumann boundary conditions were implemented using the ghost cell approach described in Gebäck and Heintz (2014), with \(\beta (x)\) extended continuously to the outside of the domain.

The two-relaxation-time collision operator was used also for the advection–diffusion equation, but here the parameter value \(\varLambda = 1/4\) was used, corresponding to the “optimal diffusion” setting in Ginzburg (2005). A space-dependent diffusion constant \(\beta (x) D\) was obtained by assigning different values of the parameter \(\lambda _o\) locally, with \(\beta (x) D = -c_s^2 (1/\lambda _o + 1/2)\).

*d*(

*x*) is the distance function, describing for each pore location the nearest distance to the solid surface. The shape of the potential for varying \(\delta \) is shown in Fig. 2. This form of the potential was chosen since it is fast decaying (if \(\delta \) is small), while remaining finite.

*V*(

*x*) obtains values between 0 and

*A*. The function \(\beta (x) = \exp (-V(x))\) was then computed (where

*kT*is included in the constant

*A*) and used in the simulations. For the pore scale equations, \(\nabla \beta \) was computed using central finite differences and used in the force term.

### 4.2 Results

*Reflection coefficients for a straight cylinder* In order to validate our model and the simulations, reflection coefficients were computed according to (26) for a straight circular cylinder for comparison with the theoretical results by Anderson and Malone (1974). To achieve a hard size-exclusion potential, a value of \(\delta = 0.001\) was used in (35), and the shift *a* was varied. The strength *A* was set to 4, a rather low value chosen since numerical instabilities may occur when \(\beta \) is too close to zero. For comparison, results with a soft, slowly decaying potential with \(\delta =3\) are also shown. The flow and diffusion were computed in the *x*-direction in a domain of \(100\times 50\times 50\) voxels containing a cylinder of radius \(R=22\).

The results are shown in Fig. 3 and show a very good agreement for the hard potential with the results from Anderson and Malone (1974). Some small differences can be seen for large *a*, when the cylinder is almost impermeable to the solute. This is due to the finite potential used, with \(A=4\), which results in a nonzero permeation even when the potential extends over the entire cylinder.

*a*since the potential then has a tail that extends into the domain over a distance of approximately \(\delta \) and causes some hindrance to the flow.

*Full osmotic problem through a cylinder* In order to investigate the possibility to solve the full osmotic problem (5)–(6), and to investigate the resulting velocity profiles, a simulation was set up as shown in Fig. 4a. A straight cylinder connects two domains, where constant solute number densities were set as boundary values on the left and right boundaries, with a higher value on the left. For the Stokes equations, outflow boundary conditions were set on the right boundary, i.e., \(u_y = u_z = 0\) and \(\partial u_x / \partial x = 0\), while a fixed pressure \(p=0\) and zero tangential velocity \(u_y = u_z = 0\) were set on the left boundary. On the other boundaries of the simulation box, periodic boundary conditions were used. In Fig. 4b, the function \(\beta (x)\) is shown.

*Reflection coefficients for a periodic porous medium* To show the potential of the present homogenized model and the lattice Boltzmann implementation, simulations were also performed solving the cell problems for a more general porous medium. A geometry was created using diffusion-limited cluster aggregation (DLCA) of spherical particles (Lach-hab et al. 1996), which were allowed to aggregate and form a solid structure. This may, for example, model a material made of aggregated silica particles, with pores on the nanometer scale and up. The particular geometry used here was on a lattice of \(200^3\) voxels and had a porosity of 70% and an average pore diameter of 7.2 voxels.

*a*, and with \(\delta =1\) and 3, and \(A=2\) and 3. The results are shown in Fig. 5, together with a streamline plot showing the velocity field for the solution of the cell problem (23). Just as for the cylinder, the values for \(\sigma \) do not quite reach 1, since the potential is finite with rather low values. Also, \(\sigma \) does not become 0 when \(a=0\) because of the tail of the potential. Although the general shape of the curve is similar to the case of the cylinder, the details are quite different, reflecting the more complex structure. It is interesting to see that although the largest pores in the geometry have a radius close to 10, the maximum value of \(\sigma \) is reached at much smaller values for

*a*, since the potential blocks most paths through the structure even at smaller

*a*. The sharper potential with \(\delta =1\) yields a much sharper rise in \(\sigma \).

*a*and \(\delta \) for the potential. The tests were performed with \(A=3\) and \(\delta =3\) (at grid size 200). The results are shown in Fig. 6 for various values of

*a*and show a convergence for \(\sigma \) as the resolution increases. However, as there is a significant change in \(\sigma \) with increasing resolution for low values of

*a*, the results indicate that for improved accuracy, it is important to resolve the boundary layer where the potential is large.

## 5 Conclusions

We have presented the results of homogenization in a porous medium for a model describing the effects of osmotic pressure differences on the pore scale. The homogenized equations may be used to compute macroscopic solvent velocities induced by difference in concentration of solutes, but in particular also to compute the reflection coefficient \(\sigma \) for a periodic porous material.

The models have been implemented using the lattice Boltzmann method, and the simulation results show good agreement with theoretical results available for straight circular cylinders. The method is also flexible enough to be applicable to general porous materials, as has been illustrated for a particular porous geometry here. The method could also be applied to 3D structures of real porous materials when available, as has, for example, been done for pure diffusion in our previous work (Gebäck et al. 2015). An interesting future study would be to perform such simulations, and to validate with measurement data for various materials. Further work is also needed to derive correct expressions for the reflection coefficient for heterogeneous materials, including effects of boundary layers.

Future work also includes deriving homogenized equations for the case of large Péclet numbers, when there will be no decoupling of the equations for flow and advection–diffusion. This introduces nonlinearities and makes both the homogenization and the solving of both the cell problems and the macroscopic problem more challenging.

## Notes

### Acknowledgements

The financial support from Vinnova through the VINN Excellence Center SuMo Biomaterials is gratefully acknowledged. The authors are also grateful to Andrey Piatnitski for fruitful discussions.

## References

- Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal.
**23**(6), 1482–1518 (1992). https://doi.org/10.1137/0523084 CrossRefGoogle Scholar - Allaire, G., Mikelić, A., Piatnitski, A.: Homogenization of the linearized ionic transport equations in rigid periodic porous media. J. Math. Phys.
**51**(12), 123,103, 18 (2010). https://doi.org/10.1063/1.3521555 CrossRefGoogle Scholar - Anderson, J.L., Lowell, M.E., Prieve, D.C.: Motion of a particle generated by chemical gradients Part 1. Non-electrolytes. J. Fluid Mech. (1982). https://doi.org/10.1017/S0022112082001542
- Anderson, J.L., Malone, D.M.: Mechanism of osmotic flow in porous membranes. Biophys. J.
**14**(12), 957–982 (1974). https://doi.org/10.1016/S0006-3495(74)85962-X CrossRefGoogle Scholar - Cath, T.Y., Childress, A.E., Elimelech, M.: Forward osmosis: principles, applications, and recent developments. J. Membr. Sci.
**281**, 70–87 (2006)CrossRefGoogle Scholar - Einstein, A.: Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik
**322**, 549–560 (1905). https://doi.org/10.1002/andp.19053220806 CrossRefGoogle Scholar - Elmoazzen, H.Y., Elliott, J.A., McGann, L.E.: Osmotic transport across cell membranes in nondilute solutions: a new nondilute solute transport equation. Biophys. J.
**96**(7), 2559–2571 (2009)CrossRefGoogle Scholar - Gebäck, T., Heintz, A.: A lattice Boltzmann method for the advection–diffusion equation with Neumann boundary conditions. Commun. Comput. Phys.
**15**(2), 487–505 (2014). https://doi.org/10.4208/cicp.161112.230713a CrossRefGoogle Scholar - Gebäck, T., Marucci, M., Boissier, C., Arnehed, J., Heintz, A.: Investigation of the effect of the tortuous pore structure on water diffusion through a polymer film using lattice Boltzmann simulations. J. Phys. Chem. B
**119**(16), 5220–5227 (2015). https://doi.org/10.1021/acs.jpcb.5b01953 CrossRefGoogle Scholar - Ginzburg, I.: Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation. Adv. Water Resour.
**28**(11), 1171–1195 (2005). https://doi.org/10.1016/j.advwatres.2005.03.004 CrossRefGoogle Scholar - Ginzburg, I., Verhaeghe, F., d’Humieres, D.: Two-relaxation-time lattice Boltzmann scheme: about parametrization, velocity, pressure and mixed boundary conditions. Commun. Comp. Phys.
**3**(2), 427–478 (2008)Google Scholar - Guell, D.: The Physical Mechanism of Osmosis and Osmotic Pressure-a Hydrodynamic Theory for Calculating the Osmotic Reflection Coefficient. Massachusetts Institute of Technology, Department of Chemical Engineering (1991)Google Scholar
- Guell, D., Brenner, H.: Physical mechanism of membrane osmotic phenomena. Ind. Eng. Chem. Res.
**35**(9), 3004–3014 (1996)CrossRefGoogle Scholar - Guo, Z., Shu, C.: Lattice Boltzmann Method and its Applications in Engineering, Advances in Computational Fluid Dynamics, vol. 3. World Scientific Publishing Co. Pte. Ltd., Hackensack (2013). https://doi.org/10.1142/8806 CrossRefGoogle Scholar
- Guo, Z., Zheng, C., Shi, B.: Discrete lattice effects on the forcing term in the lattice Boltzmann method. Phys. Rev. E
**65**(4, Part 2B), 046308 (2002). https://doi.org/10.1103/PhysRevE.65.046308 CrossRefGoogle Scholar - Heintz, A., Piatnitski, A.: Osmosis for non-electrolyte solvents in permeable periodic porous media. Netw. Heterog. Media
**11**(3), 585–610 (2016)CrossRefGoogle Scholar - Hunter, R.J.: Foundations of Colloid Science. Oxford University Press, Oxford (2004)Google Scholar
- Jensen, K.H., Rio, E., Rasmus Hansen, C.C., Bohr, T.: Osmotically driven pipe flows and their relation to sugar transport in plants. J. Fluid Mech.
**636**, 371–396 (2009)CrossRefGoogle Scholar - Kedem, O., Katchalsky, A.: Thermodynamic analysis of the permeability of biological membranes to non-electrolytes. Biochim. Biophys. Acta
**27**, 229–246 (1958)CrossRefGoogle Scholar - Kedem, O., Katchalsky, A.: Thermodynamics of flow processes in biological systems. Biophys. J.
**2**(2), 53–78 (1962)CrossRefGoogle Scholar - Lach-hab, M., Gonzalez, A., Blaisten-Barojas, E.: Concentration dependence of structural and dynamical quantities in colloidal aggregation: computer simulations. Phys. Rev. E
**54**(5), 5456–5462 (1996)CrossRefGoogle Scholar - Logan, B.E., Elimelech, M.: Membrane-based processes for sustainable power generation using water. Nature
**488**(7411), 313–319 (2012)CrossRefGoogle Scholar - Looker, J.R., Carnie, S.L.: Homogenization of the ionic transport equations in periodic porous media. Transp. Porous Media
**65**(1), 107–131 (2006). https://doi.org/10.1007/s11242-005-6080-9 CrossRefGoogle Scholar - Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal.
**20**(3), 608–623 (1989). https://doi.org/10.1137/0520043 CrossRefGoogle Scholar - Probstein, R.F.: Physicochemical Hydrodynamics: An Introduction. Wiley, London (2003)Google Scholar
- Schmuck, M.: Analysis of the Navier–Stokes–Nernst–Planck–Poisson system. Math. Models Methods Appl. Sci.
**19**(6), 993–1015 (2009). https://doi.org/10.1142/S0218202509003693 CrossRefGoogle Scholar - Schmuck, M.: Modeling and deriving porous media Stokes–Poisson–Nernst–Planck equations by a multi-scale approach. Commun. Math. Sci.
**9**(3), 685–710 (2011). https://doi.org/10.4310/CMS.2011.v9.n3.a3 CrossRefGoogle Scholar - van’t Hoff, J.: The role of osmotic pressure in the analogy between solutions and gases. Zeitschrift fur physikalische Chemie
**1**, 481–508 (1887)Google Scholar - von Smoluchowski, M.: Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen. Annalen der Physik
**326**, 756–780 (1906). https://doi.org/10.1002/andp.19063261405 CrossRefGoogle Scholar - Wyman, C.E., Kostin, M.D.: Anomalous osmosis: solutions to the Nernst–Planck and Navier–Stokes equations. J. Chem. Phys.
**59**(6), 3411–3413 (1973). https://doi.org/10.1063/1.1680484 CrossRefGoogle Scholar - Yant, Z.Y., Weinbaum, S., Pfeffer, R.: On the fine structure of osmosis including threedimensional pore entrance and exit behaviour. J. Fluid Mech.
**162**, 415–438 (1986)CrossRefGoogle Scholar - Zhang, X., Curry, F.R., Weinbaum, S.: Mechanism of osmotic flow in a periodic fiber array. Am. J. Physiol. Heart Circ. Physiol.
**290**(2), H844–52 (2006)CrossRefGoogle Scholar - Zhao, S., Zou, L., Tang, C.Y., Mulcahy, D.: Recent developments in forward osmosis: opportunities and challenges. J. Membr. Sci.
**396**, 1–21 (2012)CrossRefGoogle Scholar

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