A relation between membrane permeability to a pure solvent (i.e., inverse of resistance to a flow through membrane) and flow rate is derived in a circular pipe driven by a constant pressure difference across the pipe length. Membrane is assumed to be non-deformable and zero thickness, and the permeate flux of pure solvent due to pressure discontinuity across the flat membrane is coupled with the governing equations for incompressible Newtonian fluid in the Stokes regime. The permeation flow rate (normalised by that of the Hagen-Poiseuille flow for the no-membrane case or with a membrane of infinite permeability) is represented as a function of a non-dimensional permeability including the aspect ratio of the pipe geometry. The relation is established through comparison with a fully-validated numerical simulation result: the numerical discretisation is based on our original discrete-forcing immersed boundary method, which guarantees (i) conservations of mass and momentum even in the immediate vicinity of the membrane surface and (ii) consistency between incompressible velocity and pressure fields. Inverse analysis of the above formula yields the permeability as function of the flow rate through the pipe, comprising three equations covering the entire permeability ranges (from low to high permeabilities). The established permeability formulae are expected to be useful for identifying effective permeabilities of membrane to single-component fluid or pure solvent in practical applications.

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.

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.

A concise and accurate prediction method is required for membrane permeability in chemical engineering and biological fields. As a preliminary study on this topic, we propose the concentration polarization model (CPM) of the permeate flux and flow rate under dominant effects of viscosity and solute diffusion. In this model, concentration polarization is incorporated for the solution flow through a semi-permeable membrane (i.e., permeable for solvent but not for solute) in a circular pipe. The effect of the concentration polarization on the flow field in a circular pipe under a viscous-dominant condition (i.e., at a low Reynolds number) is discussed by comparing the CPM with the numerical simulation results and infinitesimal Péclet number model (IPM) for the membrane permeability, strength of the osmotic pressure, and Péclet number. The CPM and IPM are confirmed to be a reasonable extension of the model for a pure fluid, which was proposed previously. The application range of the IPM is narrow because the advection of the solute concentration is not considered, whereas the CPM demonstrates superior applicability in a wide range of parameters, including the permeability coefficient, strength of the osmotic pressure, and Péclet number. This suggests the necessity for considering concentration polarization. Although the mathematical expression of the CPM is more complex than that of the IPM, the CPM exhibits a potential to accurately predict the permeability parameters for a condition in which a large permeate flux and osmotic pressure occur.