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Homogenization approach to water transport in plant tissues with periodic microstructures

Water flow in plant tissues takes place in two different physical domains separated by semipermeable membranes: cell insides and cell walls. The assembly of all cell insides and cell walls are termed symplast and apoplast, respectively. Water transport is pressure driven in both, where osmosis plays an essential role in membrane crossing. In this paper, a microscopic model of water flow and transport of an osmotically active solute in a plant tissue is considered. The model is posed on the scale of a single cell and the tissue is assumed to be composed of periodically distributed cells. The flow in the symplast can be regarded as a viscous Stokes flow, while Darcy's law applies in the porous apoplast. Transmission conditions at the interface (semipermeable membrane) are obtained by balancing the mass fluxes through the interface and by describing the protein mediated transport as a surface reaction. Applying homogenization techniques, macroscopic equations for water and solute transport in a plant tissue are derived. The macroscopic problem is given by a Darcy law with a force term proportional to the difference in concentrations of the osmotically active solute in the symplast and apoplast; i.e. the flow is also driven by the local concentration difference and its direction can be different than the one prescribed by the pressure gradient.