Fluid Permeation Through A Membrane With Infinitesimal Permeability Under Reynolds Lubrication

Abstract

To understand the lubrication-dominated permeation through a membrane, numerical simulations of permeation through a moving corrugated permeable membrane is carried out with a fully validated numerical method. Through comparisons between the numerical results and the results of an asymptotic analysis of permeate flux (under an infinitesimal permeability condition) using Reynolds lubrication equation, the effect of permeation on lubrication and its inverse effect (i.e., the dependence of permeation on lubrication) are discussed. The linear and non-linear dependences of the relaxation of the lubrication pressure due to membrane permeation are identified. The effect of the tangential component of the permeate flux is evaluated by a linear analysis, and the limitation of Reynolds-type lubrication is discussed.

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

“…Using their "consistent direct discretisation" for the DF-IB approach, the no-slip and impermeable conditions on the interface were strictly imposed in a discrete sense, while satisfying the mass and momentum conservations, which enables capturing the sharp distribution of the velocity and pressure at the interface. The DF-IB method was extended to enable permeation of the solvent through the membrane by and Tazaki et al (2020); they showed discretisations that consider the permeable condition for the solvent at the interface. In the present study, we further deal…”

“…We further assume an infinitesimal limit for L p to isolate the effect of permeability to develop an asymptotic analytical model. Then, the pressure in Ω 1 is expressed as follows (Takeuchi and Gu 2019;Tazaki et al 2020):…”

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].…”

“…When the surface-to-surface distance is sufficiently small, the Reynolds lubrication equation [3] may be able to describe the lubrication pressure. Tazaki et al [4] mentioned that the order of magnitude of the permeate flux of the solvent driven by the Reynolds lubrication pressure is evaluated as J (Re) /U ∝ −1 L when the permeability is infinitesimal. Here, J (Re) is the permeate flux, U is the reference velocity, is the ratio of the gap width to the longitudinal length scale (see Fig.…”