The effect of osmotic pressure on the flow of solutions through semi-permeable hollow fibers

We study the laminar flow of binary liquid mixture, whose components are a Newtonian fluid (solvent) and a solute, in a hollow fiber. The fiber walls are porous, but the pores size is small enough preventing the solute molecules to be transported across the membrane. This produces an osmotic pressure that offers resistance (in many cases non negligible) to the fluid cross flow.We model the phenomenon by means of a two scale procedure, introducing a small parameter ε≪1ε≪1 (the ratio between the radius and the length of the fiber) and expanding all relevant quantities in powers of εε. Thus, considering just the ε0ε0 order approximation, we obtain a set of two equations governing both the solute concentration and the pressure along the fibers.Performing some numerical simulations we point out the critical importance of the osmosis in order to make the device working properly.