Lin & Segel's standing gradient problem revisited: A lesson in mathematical modeling and asymptotics

We revisit a physiological standing gradient problem of Lin and Segel from their landmark text on mathematical modeling [C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences, SIAM, Philadelphia, 1988] with a view to giving it an up-to-date perspective. In particular, via an alternative nondimensionalization, we show that the problem can be analyzed using the tools of singular perturbation theory and matched asymptotic expansions. In the spirit of the aforementioned authors, the development is didactic in style. Solving the problem requires many of the necessary skills of continuous modern mathematical modeling: formulation from a physical description of the process, scaling and asymptotic simplification, and solution using advanced perturbation (boundary layer) techniques.