Near-exact resistance of an orifice in a 2D membrane blocked by a cylindrical obstruction

A near-exact analytical expression is presented for the Ohmic resistance of an orifice in a 2D membrane separating two infinitely large conductive reservoirs and obstructed by an infinitely long cylinder. The expression is obtained by constructing a curvilinear coordinate system adapted to the obstructed geometry, with constant-coordinate surfaces mapping the system boundaries, and by integrating the resistive contributions of infinitesimally thin constant-coordinate slices. Although these surfaces do not correspond exactly to equipotential surfaces, the resulting expression provides a rigorous lower bound on the resistance and deviates from the exact resistance by <0.17% over the full range of admissible obstruction sizes. This near-exact expression can further be used to model the access region of obstructed finite-length cylindrical channels, providing an analytical tool for single-molecule sensing applications.