Effective permeability conditions for diffusive transport through impermeable membranes with gaps

The paper derives, in the long–thin channel limit, the steady effective permeability Peff = 1 / (L/ε + (2δ/π)(log(1/(8ε)) + 1)) via matched asymptotics and matching across inner, boundary-layer, and outer regions, see equations (40a)–(40b) in the paper . It further provides O(1) aspect-ratio results using a conformal map and elliptic integrals, yielding Peff = (1/δ) π log(1/(4 ε^2 π^2 λ d^2)) in Appendix A (equation (45)) and shows the large-a limit recovers the long–thin formula (equation (49)) . For the unsteady case, the paper derives two effective transmission laws with memory (equations (61)–(62)), including the O(ε) flux-jump driven by odd modes and an average-flux law with Peff [c] plus Volterra kernels involving time derivatives of (c± and flux combinations) weighted by δ/π(log(1/(8ε))+1) , with early-time Euler–Maclaurin approximations (equation (63)) . The candidate solution reproduces exactly these formulas (steady, unsteady, and O(1) aspect ratio), uses the same asymptotic region decomposition, and correctly identifies the additive resistance structure (channel plus two identical access contributions), consistent with the paper’s series-resistance interpretation in dimensional form (equations (66)–(72)) . Minor stylistic differences (dual-series versus conformal-mapping for the mixed boundary layer) do not change the substance. Hence both are correct and follow substantially the same proof strategy.