Flow of Incompressible Viscous Fluid Through a Tube with Non-Linear Elastic Membrane Insertion: Applications

Abstract

The study of viscous flow in tubes with deformable walls is of specific interest to the biomedical technology industry as well as the medicine and biology professions in connection with atherosclerosis, artery replacement by a graft, clotting, etc. In line with this philosophy, this paper describes numerically the behavior of a viscous incompressible fluid through a tube with a non-linear elastic membrane insertion for various geometries of the tube and initial conditions. The emphasis is given to the applications. Six different test cases, with regard to the problem of atherosclerosis, are examined. The membrane insertion in the solid tube is studied for a linear as well as a non-linear material. The linear elastic material follows a linear elastic strain-energy density function. However, the deformation is large. The non-linear elastic material follows the type strain-energy density function of Fung (1993). The fluid is described through a Navier-Stokes code coupled with a system of non-linear equations that describe membrane deformation. The objective of this paper is to outline the similarities and differences between linear and non-linear membrane insertions in influencing fluid-membrane interaction in all six test cases. It is revealed that although the maximum deformed radius of the axisymmetric membrane is almost the same for all cases, the deformed shape of the membrane is quite different resulting in different regions of the circulation regions. In fact, the non-linear membrane exhibits a very restricted region of the circulation of the flow, which is important for understanding the cause of thrombosis.