Homogenization limit of a model system for interaction of flow, chemical reactions, and mechanics in cell tissues - Jäger, Willi; Mikelić, Andro; Neuss-Radu, Maria (SIAM journal on mathematical analysis, 2011)
In this article we obtain rigorously the homogenization limit for a fluid-structure-reactive flow system. It consists of cell tissue and intercellular liquid, transporting solutes. The cell tissue is assumed to be linearly elastic and deforming with a viscous nonstationary flow. The elastic moduli of the tissue change with cumulative concentration value. In the limit, when the scale parameter goes to zero, we obtain the quasi-static Biot system, coupled with the upscaled reactive flow. Effective Biot's coefficients depend on the reactant concentration. In addition to the weak two-scale convergence results, we prove convergence of the elastic and viscous energies. This then implies a strong two-scale convergence result for the fluid-structure variables. Next we establish the regularity of the solutions for the upscaled equations. To the best of our knowledge, ours is the only known study of the regularity of solutions to the quasi-static Biot system. The regularity is used to prove the uniqueness for the upscaled model.