Derivation of a macroscopic model for nutrient uptake by hairy-roots

In this article the process of nutrient uptake by a single root branch is studied. We consider diffusion and active transport of nutrients dissolved in water. The uptake happens on the surface of thin root hairs distributed periodically and orthogonal to the root surface. Water velocity is defined by the Stokes equations. We derive a macroscopic model for nutrient uptake by a hairy-root from microscopic descriptions using homogenization techniques. The macroscopic model consists of a reaction-diffusion equation in the domain with hairs and a diffusion-convection equation in the domain without hairs. The macroscopic water velocity is described by the Stokes system in the domain without hairs, with no-slip condition on the boundary between domain with hairs and free fluid.