Homogenization For CVI Process And Ion Transport In Multicellular Tissues

In this thesis,we consider two problems.The first one is to analyze the macroscopic modeling of Chemical Vapor Infiltration(CVI)process.We establish a nonlinear discrete node-bond network mathematical model for CVI process,a discrete reactiondiffusion equation is coupled with a discrete porosity evolution equation in this model.The coefficients in this model are oscillatory.We establish the homogenization theory of this discrete system and derive error estimation between the solutions of the homogenization equations and the original discrete equations.This work established the mathematical foundation for the multi-scale simulation of CVI process.In the second part,we study ion transport in biological tissues and the corresponding homogenization theory.Biological tissue us ually consists of large amount of cells,and it’s a bi-domain comprised of intracellular region and extracellular region which are separated by cell membrane.Ion concentrations and electric potentials are discontinuous on the two sides of cell membrane.We propose a bi-domain electro-neutral ion transport model.In this model,the dynamics of ion concentrations and electric potential are coupled,and the cell membrane is seen as a capacitor.Then we use two-scale convergence method and unfolding operator to derive the homogenized system of the microscopic model in the case when the diffusion coefficients of different ions are the same.Because of the discontinuity on the membrane,two related macroscopic systems are derived in intracellular and extracellular region under certain circumstances.The capacitor effect of the membrane can also be seen in the macroscopic system.For the case when the diffusion coefficients are different,we use asymptotic expansion method to derive the corresponding homogenized equations.Then we use the macroscopic homogenized system to study the ion transport in smooth muscle tissue,especially three aspects that can affect the system:production rate of agonist IP3,extracellular potassium chloride(KCl)concentration and intracellular calcium source.We build a 3-D cell model when estimating the effective diffusion coefficients in the intracellular region.The results of the numerical simulations show that when these three variables are elevated to some threshold,intracellular calcium concentrations and some other variables will begin to oscillate.And when extracellular KCl concentration is too high,we can observe depolarization phenomenon.However,extracellular variables stay at relatively stable states.Most models studying intracellular calcium oscillations only consider intracellular variables,so lastly we consider a single domain macroscopic model which only consists of intracellular equations.The results of this model and the bi-domain model are quite similar,which in some extent proved that those models that only consider intracellular variables are reasonable.