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

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.