Analysis of osmotically driven pipe flows with application to conifer needles

The long-distance sugar transport in vascular plants takes place in the phloem vascular tissue, roughly a collection of pipes (sieve elements) with semipermeable walls, where the flow is driven by osmotic uptake of water from the neighboring xylem vascular tissue. In analytical models of sugar transport through the sieve elements, interaction with the xylem and axial sugar diffusion are typically not taken into account. However, in the transfusion tissue surrounding the vasculature of conifer needles, the typical length scales are so short and the geometry so complex that interaction with the neighboring xylem cells and axial diffusion become important and can even be dominant. We discuss the limitations in the sugar transport induced by the opposing pressure gradient in the xylem, and we introduce appropriate dimensionless numbers in the form of Peclet and Nusselt numbers (from diffusion and sugar flux respectively) characterizing the flow. For values of the parameters appropriate to the transfusion tissue, the method of [Segel, J. theor. Biol. (1970) 29, 233-250] gives approximate solutions by regular perturbation theory. For parameters corresponding to the sieve elements in the phloem, the solutions have boundary layers, which can be determined by singular perturbation theory.