Effective solvent-solute transport across micro-structured thin membranes

A multi-scale homogenization based model to simulate the transport of a passive solute across a membrane and the hydrodynamic of the surrounding fluid solvent is presented.

It provides a description of the micro- and macroscopic solvent behavior and solute convection and consists of some constraints to be satisfied by the solvent velocity $\mathbf{u}$ and solute concentration $c$, imposed within the fluid domain, over a virtual smooth surface passing through the center of each membrane pore

$$\mathbf{u}=-{\bf M}:{\bf{\Sigma}}^--{\bf N}:{\bf{\Sigma}}^+, \quad {c}=-{\bf X}\cdot{\bf{F}}^--{\bf Y}\cdot {\bf{F}}^+,$$ 

where $\mathbf{\Sigma}^{\pm}$ and $\mathbf{F}^\pm$ denotes the upward and downward solvent stresses and solute fluxes and ${\bf M}$, $\bf N$, $\bf X$ and $\bf Y$ represent the microscopic feedback of the solid skeleton on the macroscopic fields and can be computed once and for all at the pore-scale. 

The model shows that the membrane produces a jump in solvent stresses  and solute fluxes whose intensity and direction,

evaluated solving problems at the microscale, depend on the external transport phenomenon and on the pore geometry. To assert the validity of the macroscopic model developed, its solution is compared with the solution of the full-scale problem.