A new two-dimensional blood flow model with arbitrary cross sections

Institute of Mathematics, National University of Mexico, Blvd. Juriquilla, 3001 Queretaro, Mexico

^* Corresponding author: cesar@im.unam.mx

Received: 17 October 2022
Accepted: 28 March 2023

Abstract

A new two-dimensional model for blood flows in arteries with arbitrary cross sections is derived. The domain consists of a narrow, large vessel that extends along an axial direction, with cross sections described by radial and angular coordinates. The model consists of a system of balance laws for conservation of mass and balance of momentum in the axial and angular directions. The equations are derived by applying asymptotic analysis to the incompressible Navier-Stokes equations in a moving domain with an elastic membrane, and integrating in the radial direction in each cross section. The resulting model is a system of hyperbolic balance laws with source terms. The main properties of the system are discussed and a positivity-preserving well-balanced central-upwind scheme is presented. The merits of the scheme will be tested in a variety of scenarios. In particular, simulations using an idealized aorta model are shown. We analyze the time evolution of the blood flow under different initial conditions such as perturbations to steady states, which parametrizes a bulging in a vessel’s wall. We consider different situations given by distinct variations in the vessel’s elasticity.