Multi-Scale Modeling and Simulation of Transport Processes in an Elastically Deformable Perforated Medium

TL;DR

This paper develops a multi-scale model for transport in elastic perforated media, incorporating cyclic deformations like lung tissue movement, and validates it through numerical simulations to understand deformation effects on transport.

Contribution

It introduces a novel upscaled model for transport in deformable perforated media using two-scale asymptotic expansion, coupling deformation with diffusion.

Findings

Effective model captures deformation effects on transport.

Numerical simulations validate the model.

Deformation significantly influences transport processes.

Abstract

In this paper, we derive an effective model for transport processes in periodically perforated elastic media, taking into account, e.g., cyclic elastic deformations as they occur in lung tissue due to respiratory movement. The underlying microscopic problem couples the deformation of the domain with a diffusion process within a mixed \textit{Lagrangian}/\textit{Eulerian} formulation. After a transformation of the diffusion problem onto the fixed domain, we use the formal method of two-scale asymptotic expansion to derive the upscaled model, which is nonlinearly coupled through effective coefficients. The effective model is implemented and validated using an application-inspired model problem. Numerical solutions for both, cell problems and macroscopic equations, are investigated and interpreted. We use simulations to qualitatively determine the effect of the deformation on the transport process.