Transient diffusion from high-capacity solute beacons

A useful protocol for maintaining long-lasting solute gradients involves a large post (“beacon”) which tends to associate with the solute in the surrounding solution. By replacing that solution with a clean one, solute diffuses out from the beacon. This transient process is significantly slowed down when the partition coefficient is large. The associated slow equilibration is naturally modeled using a quasi-steady description, with the diffusion equation being approximated by Laplace's equation. This simplification, however, results in an ill-posed problem in the two-dimensional geometries representative of realistic experiments. This failure has to do with the breakdown of the quasi-steady description at large distances away from the beacon. The singular limit of large partition coefficient is handled here using matched asymptotic expansions.