Homogenisation of two fluid flow dependent on exudate concentration. from The effect of root exudates on rhizosphere water dynamics

The Cahn-Hilliard-Stokes equations are combined with the advection-diffusion equation by concentration dependent variables to model the pore scale movement of water, air and root exudates in soil. The supplementary material presents the application of multiscale homogenisation theory to derive a coupled set of equations to describe the processes on the macroscale, with parameters that are related to the underlying geometry.