Analysis of the 3D non-linear Stokes problem coupled to transport-diffusion for shear-thinning heterogeneous microscale flows, applications to digital rock physics and mucociliary clearance

¹ Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, INSA, GMM, 135 Avenue de Rangueil, 31077 Toulouse, France
² CNRS, University Pau & Pays Adour, E2S UPPA, Laboratoire de Mathématiques et Applications de Pau (LMAP), UMR 5142, IPRA, 64000 Pau, France.
³ Université de Lyon, ENI Saint Etienne, LTDS, UMR CNRS 5513, 58 Rue Jean Parot, 42023 Saint-Etienne Cedex 2, France

^* Corresponding author: philippe.poncet@univ-pau.fr

Received: 19 July 2018
Accepted: 21 February 2019

Abstract

This study provides the analysis of the generalized 3D Stokes problem in a time dependent domain, modeling a solid in motion. The fluid viscosity is a non-linear function of the shear-rate and depends on a transported and diffused quantity. This is a natural model of flow at very low Reynolds numbers, typically at the microscale, involving a miscible, heterogeneous and shear-thinning incompressible fluid filling a complex geometry in motion. This one-way coupling is meaningful when the action produced by a solid in motion has a dominant effect on the fluid. Several mathematical aspects are developed. The penalized version of this problem is introduced, involving the penalization of the solid in a deformable motion but defined in a simple geometry (a periodic domain and/or between planes), which is of crucial interest for many numerical methods. All the equations of this partial differential system are analyzed separately, and then the coupled model is shown to be well-posed and to converge toward the solution of the initial problem. In order to illustrate the pertinence of such models, two meaningful micrometer scale real-life problems are presented: on the one hand, the dynamics of a polymer percolating the pores of a real rock and miscible in water; on the other hand, the dynamics of the strongly heterogeneous mucus bio-film, covering the human lungs surface, propelled by the vibrating ciliated cells. For both these examples the mathematical hypothesis are satisfied.