Periodic Nonlinear Diffusion: An Integral Relation and its Physical Consequences

Abstract

A general integral relation in steady periodic nonlinear diffusion is established through a Kirchhoff transformation. The time average e of the transformed concentration e satisfies Laplace's equation and may therefore be evaluated readily. Results for e not only serve as simple and exact checks on detailed numerical solutions of the nonlinear diffusion equation but also provide, immediately and exactly, the principal information often sought about these solutions. The results are especially simple in many steady periodic nonlinear diffusion problems where the time average of the flux density is everywhere zero and i!j is constant. The method is applied to examples in fluid mechanics and geophysics: spatially periodic laminar boundary layers, tidal influence and seasonal stream effects on groundwater levels, and soil temperature waves.