4.7 Article

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

Journal

CHAOS SOLITONS & FRACTALS
Volume 170, Issue -, Pages -

Publisher

PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.chaos.2023.113360

Keywords

Conservation laws; Exact solutions; Blood flow; Boundary conditions

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This study focuses on two aspects of a widely used 1D model of blood flow in a single blood vessel through symmetry analysis. The main results include the discovery of all travelling wave solutions through explicit quadrature and the derivation of three new conservation laws for inviscid flows. These solutions exhibit shock waves and sharp wave-front pulses for the blood pressure and flow. However, for viscous flows, the conservation laws are replaced by conservation balance equations with dissipative terms proportional to the friction coefficient in the model.
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.

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