Exact solutions and conservation lawsof a one-dimensional PDE model for a blood vessel

TL;DR

This paper derives explicit traveling wave solutions and new conservation laws for a 1D blood flow model, analyzing shock waves, wave-front pulses, and the effects of viscosity on conservation properties.

Contribution

It provides the first explicit quadrature solutions for the model's traveling waves and introduces three novel conservation laws for inviscid blood flow.

Findings

Explicit traveling wave solutions including shock waves and pulses.

Three new conservation laws for inviscid flows.

Viscous flows replace conservation laws with balance equations including dissipation.

Abstract

Two aspects of a widely used 1D model of blood flow in a single blood vessel are studied by symmetry analysis, where the variables in the model are the blood pressure and the cross-section area of the blood vessel. As one main result, all travelling wave solutions are found by explicit quadrature of the model. The features, behaviour, and boundary conditions for these solutions are discussed. Solutions of interest include shock waves and sharp wave-front pulses for the pressure and the blood flow. Another main result is that three new conservation laws are derived for inviscid flows. Compared to the well-known conservation laws in 1D compressible fluid flow, they describe generalized momentum and generalized axial and volumetric energies. For viscous flows, these conservation laws get replaced by conservation balance equations which contain a dissipative term proportional to the friction coefficient in the model.