Начально-краевая задача о течении в кровеносном сосуде

Abstract

The solution of the initial boundary value problem for a system of two quasi-linear hyperbolic equations, which describes a rotationally symmetric vortex-free flow of a viscous incompressible fluid in an infinite cylindrical domain (pipe, blood vessel) is constructed. It is assumed that the side surface of the domain is a free (soft, compliant) boundary on which the kinematic condition is set. The dynamic condition at the boundary is modeled by some constitutive relation – the dependence of the pressure P on the cross-sectional area S of the cylindrical domain. To study the flow an asymptotic model based on the shallow water theory is used. When constructing the solution of the initial-boundary (as well as initial and boundary) problem for a system of two quasilinear partial differential equations of the first order, the hodograph method based on the conservation law is used. A variant of the polynomial constitutive relation is chosen for the study: P~S2β (β>0). In the case of the initial data problem which specified at the initial time the Riemann-Green function, allowing to construct an implicit analytical solution, and an algorithm (numerical) for constructing an explicit solution of the original problem are given. The main attention is focused on the special case P~S2, for which all the formulas necessary to construct a solution are written in explicit form. For P~S2, several variants of conditions (initial and boundary value) are considered for the initial boundary value problem, which allow us to trace the evolution of the solution in detail. Numerical calculations demonstrating the motion of shock waves and wave fronts are given. The results obtained, in practice, can be used to describe flows in blood vessels, as well as reliable tests for testing compu tational algorithms intended to solve such problems, in particular, systems of quasi-linear hyperbolic equations.