Transport of solute and solvent driven by lubrication pressure through non-deformable permeable membranes

Abstract

A discrete-forcing immersed boundary method with permeable membranes is developed to investigate the effect of lubrication on the permeations of solute and solvent through membrane. The permeation models are incorporated into the discretisation at the fluid cells including the membrane, and discretised equations for the pressure Poisson equation and convection–diffusion equation for the solute are represented with the discontinuities at the membrane. The validity of the proposed method is established by the convergence of the numerical results of the permeate fluxes (solute and solvent) to higher-order analytical models in a lubrication-dominated flow field. As a model of the mass exchange between inside and outside of a biological cell flowing in a capillary, a circular membrane is placed between parallel flat plates, and the effect of lubrication is investigated by varying the distance between the membrane and the walls. The pressure discontinuity near the wall is larger than that at the stagnation point, which is a highlighted effect of lubrication. In the case of a small gap, the solute transport is dominated by convection inside the circular membrane and by diffusion outside. Through the time variation of the concentration in the circular membrane, lubrication is shown to enhance mass transport from/to inside and outside the membrane.

Lubrication pressure model in a non-negligible gap for fluid permeation through a membrane with finite permeability

“…Equations ( 2)-( 4) are numerically solved on a rectangular mesh (i.e., nonconforming to the membrane geometry) by the discrete-forcing immersed-boundary (DF-IB) method provided in Refs. [4,16,17]. To simulate fluid permeation on a rectangular mesh, reproducing the sharpness of the pressure discontinuity across the membrane is critical [18,19].…”

“…[20,21] was developed to reproduce the sharpness in the pressure distribution at a solid surface by handling the mass and momentum conservations even in the immediate vicinity of the solid surface. The present method [4,16,17] solves the problem of permeation by incorporating the pressure discontinuity term into the pressure Poisson equation of the DF-IB method. A brief summary of the numerical method is provided in the Supplemental Material [22].…”

“…Note that according to Eqs. ( 7) and (17), J (0) with an even value for N ( 2) is constructed in the same function space as that in the (N − 1) case, resulting in no essential difference between the (N − 1) and N truncations in Eq. (17).…”

“…(7)] obtained from Eq. (17). For the case of the finite permeability L = 10 −2 , the Fourier decomposition method is expected to give a better approximation than Eq.…”

Flow Rate Prediction for a Semi-permeable Membrane at Low Reynolds Number in a Circular Pipe

“…The details of the numerical method and its validity for mass transport problems, including the semipermeable membrane, are described in the Supplementary material. A msore generalized numerical method considering the coupling with solute permeation is provided in Yamada et al (2021). All the governing equations in the simulation are non-dimensionalized by the maximum velocity V max in the no membrane case, pipe radius R, reference pressure V 2 max (with fluid density ), reference time R∕V max , and boundary concentration difference Δc .…”