Mathematical models for blood flow in elastic vessels: Theory and numerical analysis

Abstract

In this thesis we study model equations that describe the propagation of pulsatile flow in elastic vessels. Since dealing with the Navier-Stokes equations is a very difficult task, we derive new asymptotic weakly non-linear and weakly-dispersive Boussinesq systems. Properties of the these systems, such as the well-posedness, and existence of travelling waves are being explored. Finally, we discretize some of the new model equations using finite difference methods and we demonstrate their applicability to blood flow problems. First we introduce the basic equations that describe f luid flow in elastic vessels and previously derived systems. We also review previously derived model equations for fluid flow in elastic tubes. We start with the description of the equations of motion of elastic vessel. Then wederive asymptotically Boussinesq systems for fluid flow in elastic vessels. Because these systems are weakly non-linear and weakly dispersive we expect then to have solitary waves as special solutions. We explore some possibilities by construction analytical solutions. After that we continue the derivation of the previous chapter. We derive a general system where the horizontal velocity is evaluated at any distance from the center of the tube. Special emphasis is paid on the case of constant radius vessels. We also derive unidirectional models and obtain the dissipative Boussinesq system by taking the viscosity effects into account. There is also an alternative derivation of the general system when considering the equations of potential flow. We show that the two different derivations lead to the same system. The alternative derivation is based on asymptotic series expansions. Then we develop finite difference methods for the numerical solution of the BBM equation and for the classical Boussinesq system studied in the previous chapters. Finally, we demonstrate the application of the new models to blood flow problems. By performing several numerical simulations.