An immersed membrane method for mass transfer across flexible semipermeable membranes in flows

Flexible membranes are involved in various biological and engineering systems, and the mass transfer across semipermeable membranes often plays important roles in such systems. To simulate the mass transfer process across flexible semipermeable membranes, considerable technical difficulties exist due to the particular relationship between the trans-membrane mass flux and concentration jump across membrane. In this paper, we propose an immersed membrane method to address these challenges. Following the key conception and treatment in the classical immersed boundary method for flow-structure interaction, we replace the sharp membrane interface with an immersed layer of fluid along the membrane. The immersed membrane layer has no influence on the membrane deformation and fluid flow; however, it adds extra mass transfer resistance according to the original membrane permeability. With this replacement, the mass transfer process can be solved with a uniform numerical scheme over the same Eulerian mesh, and the complicated interface treatment for the interface condition across membrane can be avoided. The method is then validated by comparing numerical results to theoretical solutions for steady and unsteady systems with flat and curved membranes and excellent agreement is observed. Several demonstration simulations are also performed to illustrate that the membrane effect on mass transfer has been correctly represented in our method and that our method could be useful for simulating mass transfer with deformable membranes.