Effect of lubrication in the non-Reynolds regime due to the non-negligible gap on the fluid permeation through a membrane

Abstract

To understand lubrication-induced membrane permeation, the effects of permeability and membrane geometry on lubrication pressure and permeate flux are studied in a range of a wall-membrane gap width wherein the effect of lubrication cannot be resolved using the Reynolds lubrication equation. The unresolvable lubrication effect (referred to as the non-Reynolds lubrication effect) is modelled by including a higher-order effect as the wall-tangential variation of the local Couette–Poiseuille velocity. Analytical prediction of the permeate flux is then validated with the fully validated numerical simulation. The result shows that, while the traditional Reynolds lubrication model underestimates the permeate flux, the permeation with the effect of the non-Reynolds lubrication is effectively improved in a small permeability range. Furthermore, the non-Reynolds lubrication model also enables reproduction of the characteristic variation in the permeate flux along the membrane. The effective range of the permeability for the non-Reynolds permeate model is discussed through the order analysis of the pressure terms in the Reynolds and non-Reynolds lubrication regimes.

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

“…3c,d,5c,d), stronger y-dependent trend is observed in the pressure field; in particular, high-and low-pressure regions near the flat plates are highlighted at t∕(H 0 ∕U 0 ) = 15.0 . The result indicates the presence of a regime that is not described by Reynolds lubrication equation, which is referred to as the non-Reynolds lubrication regime (Takeuchi et al 2021;Takeuchi and Gu 2019), hereafter.…”

“…the distance between the lowest/highest point of the circular membrane and the bottom/top wall) having a value of (H 0 − D p )∕2 = 0.25D p , 0.50D p , 1.0D p , respectively. Although the case of the smallest gap is already out of the range of the ideal lubrication condition ≪ 1 , our previous studies (Takeuchi and Gu 2019;Takeuchi et al 2021) showed that the lubrication phenomenon can still be described by including a higher-order correction for a non-negligible gap. Note that the effective longitudinal length scales of the lubrication region can be estimated to (Takeuchi et al 2021), and the effective scales for the above three cases are 0.5D p , 0.71D p , and 1.0D p , respectively, which are sufficiently smaller than the domain length l, indicating that the permeation induced by lubrication pressure is influenced little by the mirror images of the circular membranes due to the periodic boundary condition.…”

“…≲ 1 ), a higher-order lubrication model (Takeuchi and Gu 2019) is applied to describe the wall-normal distribution of pressure. Takeuchi et al (2021) showed the 2nd-order pressure for the corrugated membrane as: where the superscript " * " represents the value observed on the frame fixed at the membrane;…”

“…In the ideal lubrication state in a negligibly small gap region (between the interfaces), the pressure increases locally. However, there may be a number of cases in biological environments where the ideal condition for the lubrication theory is violated (Takeuchi et al 2021). For example, in relatively large gaps, the theory could deviate from the conventional Reynolds lubrication theory (Takeuchi and Gu 2019), resulting in an underestimation of solute and solvent permeations driven by the pressure difference.…”

“…For example, in relatively large gaps, the theory could deviate from the conventional Reynolds lubrication theory (Takeuchi and Gu 2019), resulting in an underestimation of solute and solvent permeations driven by the pressure difference. One of the difficulties associated with the numerical analysis of lubrication is that by introducing as the ratio of the gap width and reference length, the minimum number of grid points that are required to capture the pressure increase generated by lubrication is −1∕2 (Takeuchi et al 2021), and, the resolution, therefore, becomes insufficient on a coarse grid system; meanwhile, the fine grid system becomes computationally demanding. In addition, the lubrication pressure decreases owing to permeation, which makes the analysis or prediction difficult (Takeuchi et al 2021).…”