We study the weak solvability of a macroscopic, quasilinear reaction–diffusion system posed in a $2D$ porous medium which undergoes microstructural problems. The solid matrix of this porous medium is assumed to be made out of circles of not-necessarily uniform radius. The growth or shrinkage of these circles, which are governed by an ODE, has direct feedback to the macroscopic diffusivity via an additional elliptic cell problem.

The reaction–diffusion system describes the macroscopic diffusion, aggregation, and deposition of populations of colloidal particles of various sizes inside a porous media made of prescribed arrangement of balls. The mathematical analysis of this two-scale problem relies on a suitable application of Schauder’s fixed point theorem which also provides a convergent algorithm for an iteration method to compute finite difference approximations of smooth solutions to our multiscale model. Numerical simulations illustrate the behavior of the local concentration of the colloidal populations close to clogging situations.

我们研究了宏观准线性反应扩散系统的弱可解性 $2D$存在微观结构问题的多孔介质。假定这种多孔介质的固体基质由半径不一定均匀的圆组成。这些圆的增长或收缩由 ODE 控制，通过额外的椭圆细胞问题直接反馈到宏观扩散率。

反应扩散系统描述了在由指定排列的球组成的多孔介质中各种尺寸的胶体粒子群的宏观扩散、聚集和沉积。这个两尺度问题的数学分析依赖于 Schauder 不动点定理的适当应用，该定理还为迭代方法提供了收敛算法，以计算我们的多尺度模型的平滑解的有限差分近似。数值模拟说明接近堵塞情况的胶体群的局部浓度的行为。