A two-scale Stefan problem arising in a model for tree sap exudation

Abstract

The study of tree sap exudation, in which a (leafless) tree generates elevated stem pressure in response to repeated daily freeze-thaw cycles, gives rise to an interesting multiscale problem involving heat and multiphase liquid/gas transport. The pressure generation mechanism is a cellular-level process that is governed by differential equations for sap transport through porous cell membranes, phase change, heat transport, and generation of osmotic pressure. By assuming a periodic cellular structure based on an appropriate reference cell, we derive an homogenized heat equation governing the global temperature on the scale of the tree stem, with all the remaining physics relegated to equations defined on the reference cell. We derive a corresponding strong formulation of the limit problem and use it to design an efficient numerical solution algorithm. Numerical simulations are then performed to validate the results and draw conclusions regarding the phenomenon of sap exudation, which is of great importance in trees such as sugar maple and a few other related species. The particular form of our homogenized temperature equation is obtained using periodic homogenization techniques with two-scale convergence, which we investigate theoretically in the context of a simpler two-phase Stefan-type problem corresponding to a periodic array of melting cylindrical ice bars with a constant thermal diffusion coefficient. For this reduced model, we prove results on existence, uniqueness and convergence of the two-scale limit solution in the weak form, clearly identifying the missing pieces required to extend the proofs to the fully nonlinear sap exudation model. Numerical simulations of the reduced equations are then compared with results from the complete sap exudation model.

Experimental and computational comparison of freeze-thaw induced pressure generation in red and sugar maple

“…We now summarize the essential aspects of the periodic homogenization procedure, for which complete mathematical details are provided in Konrad et al (2017). For technical reasons, the reference cell Y is separated into two sub-regions (Y 2 and Y 1 ) pictured in Figure 4a, where Y 2 refers to the fiber plus the inner portion of the vessel where heat diffuses slowly, whereas Y 1 is the remaining outer portion of the vessel where diffusion is fast compared to Y 2 .…”

“…which sets the outer stem surface temperature equal to a given ambient air temperature and is what ultimately drives the freeze-thaw process. Complete details of the homogenization procedure can be found in Konrad et al (2017), and we also refer the interested reader to the work of Chavarría-Krauser and Ptashnyk (2013) who applied periodic homogenization to a related problem involving osmotic transport in non-woody plants.…”

“…The parameter values appearing in these equations are taken mostly from previous work (Graf et al, 2015;Konrad et al, 2017) that focused on comparisons to experimental data from black walnut. These parameters are summarized in Table 1, with a few small adjustments for red/sugar maple as indicated.…”