Steady state diffusion in tubular structures: Assessment of one-dimensional models

• Published in: European journal of applied mathematics Vol. 34; no. 2; pp. 262 - 279
• Main Authors: MARTIN, P. A., SKVORTSOV, A. T.
• Format: Journal Article
• Language: English
• Published: Cambridge Cambridge University Press 01.04.2023
• Subjects: Applied mathematics; Approximation; Boundary conditions; Decay rate; Eigenvectors; Homogenization; One dimensional models; Ordinary differential equations; Partial differential equations; Potential flow; Steady state models; Trapping; Tubes
• ISSN: 0956-7925; 1469-4425
• DOI: 10.1017/S0956792522000110

Steady-state diffusion in long axisymmetric structures is considered. The goal is to assess one-dimensional approximations by comparing them with axisymmetric eigenfunction expansions. Two problems are considered in detail: a finite tube with one end that is partly absorbing and partly reflecting; and two finite coaxial tubes with different cross-sectional radii joined together abruptly. Both problems may be modelled using effective boundary conditions, containing a parameter known as the trapping rate. We show that trapping rates depend on the lengths of the finite tubes (and that they decay slowly as these lengths increase) and we show how trapping rates are related to blockage coefficients, which are well known in the context of potential flow along tubes of infinite length.