Homogenization of Biomechanical Models for Plant Tissues

Abstract

Abstract. In this paper homogenization of a mathematical model for plant tissue biomechanics is presented. The microscopic model constitutes a strongly coupled system of reaction-diffusionconvection equations for chemical processes in plant cells, the equations of poroelasticity for elastic deformations of plant cell walls and middle lamella, and Stokes equations for fluid flow inside the cells. The chemical process in cells and the elastic properties of cell walls and middle lamella are coupled because elastic moduli depend on densities involved in chemical reactions, whereas chemical reactions depend on mechanical stresses. Using homogenization techniques, we derive rigorously a macroscopic model for plant biomechanics. To pass to the limit in the nonlinear reaction terms, which depend on elastic strain, we prove the strong two-scale convergence of the displacement gradient and velocity field.

Advection-dominated transport past isolated disordered sinks: stepping beyond homogenization

“…We can now revisit the expansion (9) and use evidence that terms Da C 1 or Da 2 C 2 become comparable in magnitude to C H as evidence of the breakdown of the approximation. In 1D, based on the estimates in (18), the restriction Pe max(1, √ Da) must be extended to Pe max(1, √ Da, ρ 1/2 Da), which holds along the distinguished limit Pe ∼ Da for arbitrarily large Pe. The parameter Da 2 ρ/ Pe, measuring the magnitude (relative to C H ) of the fluctuation variance and the correction to the mean, takes the value 0.05 in Figure 1 (with Pe → ∞, but with Da ρ 1/2 / Pe = S 1 /ρ 1/2 ; see Appendix A) and Figures 4(a) and 5(a).…”

“…This allowed accurate predictions of concentration means (Figure 5a) in 1D and 2D, and of variance in 1D (Figure 4); the over-prediction of solute variance in 2D would likely be corrected by use of the regularised sink distribution, albeit using more expensive quadrature. Cruder analytical estimates were achieved by neglecting any upstream influence of one sink on another (18,19,20).…”

“…This allows an asymptotic two-scale expansion to be developed, with a unit-cell problem (with periodic boundary conditions) being solved in order to provide a description of slowly varying (homogenized) variables at the macroscale. While this approach has been extended to accommodate slow spatial variation of the microscale field [4,9] and developed for a variety of applications [5,11,19,10,15,18], it is less adaptable to characterising situations where the microscale exhibits appreciable spatial disorder.…”

Smoothed particle hydrodynamics for root growth mechanics

“…A way to bring together these two aspects is to formulate a multi-scale approach, combining several levels of description and allowing them to interact. Several propositions exist and among them, gene-regulated network combined to growth [4,31,58,71,27], averaging approaches through analytical homogenisation [1,35,65,79,82], and the incorporation of a representation of individual cells in a continuous formulation of tissue deformation [5,13,42,48,53,99].…”

Stochastic Unfolding and Homogenization of Spring Network Models

“…In some cases, this method facilitates a more straightforward, and operator theory flavored, analysis of periodic homogenization problems. In recent years periodic unfolding has been applied to a large variety of multiscale problems; e.g., see [11,17,37,38,34,43,10,29,42,21]. For a systematic investigation of two-scale calculus associated with the use of the periodic unfolding method we refer to [14,44,45,37,12].…”