# Mathematical and numerical modeling of flow and transport in fluid–poroelastic structure interaction

Wang, Xing (2022) Mathematical and numerical modeling of flow and transport in fluid–poroelastic structure interaction. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

 Preview
PDF

## Abstract

The thesis focuses on the analysis and simulation of fluid–poroelastic structure interaction (FPSI), which describes the interaction between free fluid and a neighboring poroelastic medium through an interface. The Stokes or time-dependent Navier-Stokes equations govern the flow in the fluid region. The Biot system describes the flow within the poroelastic media and the structure deformation. The two regions are coupled via mass conservation, balance of normal stress, balance of momentum, and the Beaver-Joseph-Saffman slip with friction condition. A mixed Darcy formulation is employed, and a pressure Lagrange multiplier on interface is introduced to impose weakly continuity of flux. The Stokes–Biot model is further coupled with an advection-diffusion transport equation for the solute concentration.

First, we discuss a Lagrange multiplier method for the fully dynamic Navier-Stokes–Biot model. The existence, uniqueness, and stability of the solution are obtained. We further study the well-posedness of a fully discrete scheme approximating the model based on suitable finite element spaces and derive error estimates. Numerical simulations illustrate the order of convergence and the feasibility of the numerical method to model a benchmark problem involving Newtonian blood flow.

We extend the Newtonian model to a non-Newtonian fluid via the Carreau-Yasuda model for shear-thinning rheology. We study the effect of poroelasticity of blood vessels and non- Newtonian blood rheology on important clinical markers such as wall shear stress and relative residence time. We further conduct numerical experiments illustrating the blood flow in stenotic vessels.

Finally, we discuss a two-way coupled Stokes–Biot–transport model. The convective term in the transport equation, which depends on the fluid velocity and the concentration-
dependent viscosity, makes the coupled system non-linear. We show well-posedness for a linearized formulation. Next, the convergence of a fixed-point iterative algorithm is analyzed to obtain a solution of the original non-linear model. We study the well-posedness of the non-linear formulation and the finite element approximation. Error estimates are derived for the semi-discrete problem. A series of computational experiments are conducted to illustrate the theoretical convergence rates and to explore the method’s applicability to model fluid flow and solute transport within networks of channels.

## Share

Citation/Export: Select format... Citation - Text Citation - HTML Endnote BibTex Dublin Core OpenURL MARC (ISO 2709) METS MODS EP3 XML Reference Manager Refer

## Details

Item Type: University of Pittsburgh ETD
Status: Unpublished
Creators/Authors:
Wang, Xingxiw117@pitt.eduxiw117
ETD Committee:
Committee ChairYotov, Ivanyotov@math.pitt.edu
Committee MemberLayton, Williamwjl@pitt.edu
Committee MemberTrenchea, Catalintrenchea@pitt.edu
Committee MemberCaucao, Sergioscaucao@ucsc.cl
Date: 12 October 2022
Date Type: Publication
Defense Date: 25 July 2022
Approval Date: 12 October 2022
Submission Date: 30 July 2022
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 212
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: FPSI, Navier-Stokes–Biot model, coupled flow and transport, non-linear convective transport, Lagrange multiplier, mixed Darcy formulation, convergence analysis.
Date Deposited: 12 Oct 2022 20:27