Homogenization of biomechanical models of plant tissues with randomly distributed cells

Abstract

In this paper homogenization of a mathematical model for biomechanics of a plant tissue with randomly distributed cells is considered. Mechanical properties of a plant tissue are modelled by a strongly coupled system of reaction-diffusion-convection equations for chemical processes in plant cells and cell walls, the equations of poroelasticity for elastic deformations of plant cell walls and middle lamella, and the Stokes equations for fluid flow inside the cells. The nonlinear coupling between the mechanics and chemistry is given by the dependence of elastic properties of plant tissue on densities of chemical substances as well as by the dependence of chemical reactions on mechanical stresses present in a tissue. Using techniques of stochastic homogenization we derive rigorously macroscopic model for plant tissue biomechanics with random distribution of cells. Strong stochastic two-scale convergence is shown to pass to the limit in the non-linear reaction terms. Appropriate meaning of the boundary terms is introduced to define the macroscopic equations with flux boundary conditions and transmission conditions on the microscopic scale.

Stochastic homogenization on perforated domains I -- Extension operators

“…with C independent from Q. Similarly, one obtains for the trace T that [24] T u L p (Q∩∂P) ≤ C u L p (Q∩P) + ∇u L p (Q∩P) .…”

“…In contrast, the homogenization on randomly perforated domains is still open to a large extend. Recent results focus on minimally smooth domains [9,24] or on decreasing size of the perforations when the smallness parameter tends to zero [8] (and references therein). The main issue in homogenization on perforated domains compared to classical homogenization problems is compactness.…”

Upscaling of a reaction-diffusion-convection problem with exploding non-linear drift

“…To obtain the value of B * and some information on the oscillating structure of u 1 , we introduce a cell problem related to ( 26)- (28). The structure of problem ( 26)-( 28) and jointly with equation ( 29) allow us to look for u 1 in the form…”

“…From a totally different perspective, it would be interesting to study how this type of twoscale asymptotics with drift can cope with an eventual stochasticity either in the geometry of the material (e.g. in the distribution and/or choice of shapes of the obstacles) or in the dynamics of the problem; see [28] and [22] for remotely related works.…”

Stochastic Homogenization on Irregularly Perforated Domains

“…Given ε > 0, we consider for a bounded domain Q ⊂ R d perforated by a random set G ε and write Q ε := Q\G ε . Typically, G ε ≈ εG where G is a stationary random set and G ε is additionally regularized close to ∂Q [GK15, FHL20,PP20]. We then study the following PDE on Q ε for the time interval I = [0, T ]:…”