月曜解析セミナー：正宗 淳 氏（京都大学）Emergence of an unsaturated Darcy’s law and a Richards-type equation via homogenization of the Stokes-Chan-Hilliard

We derive macroscopic unsaturated flow laws from a microscopic Stokes-Cahn-Hilliard system posed in a periodically perforated porous medium. The model describes incompressible two-phase flow with diffuse interfaces and incorporates capillary effects through a chemical potential associated with a logarithmic free energy.　

Our main result shows that, in the homogenization limit, the averaged velocity satisfies a Darcy–Korteweg law. The effective permeability tensor is determined solely by the pore geometry, as in the saturated Stokes case, while the driving force contains both the pressure gradient and the capillary force. Consequently, the resulting macroscopic law is not, in general, reducible to the classical Darcy–Buckingham form driven by a scalar hydraulic potential. Combining this Darcy–Korteweg law with the homogenized Cahn–Hilliard equation, we obtain a generalized Richards-type system.

This is a joint effort with Rong LEI (Tohoku University).