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Mat. Sb., 2017, Volume 208, Number 8, Pages 56–105 (Mi msb8748)  

This article is cited in 3 scientific papers (total in 4 papers)

A one-dimensional model of flow in a junction of thin channels, including arterial trees

V. A. Kozlova, S. A. Nazarovbcd

a Department of Mathematics, Linköpings Universitet, Sweden
b St. Petersburg State University, Department of Mathematics and Mechanics
c Peter the Great St. Petersburg Polytechnic University
d Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg

Abstract: We study a Stokes flow in a junction of thin channels (of diameter $O(h)$) for fixed flows of the fluid at the inlet cross-sections and fixed peripheral pressure at the outlet cross-sections. On the basis of the idea of the pressure drop matrix, apart from Neumann conditions (fixed flow) and Dirichlet conditions (fixed pressure) at the outer vertices, the ordinary one-dimensional Reynolds equations on the edges of the graph are equipped with transmission conditions containing a small parameter $h$ at the inner vertices, which are transformed into the classical Kirchhoff conditions as $h\to+0$. We establish that the pre-limit transmission conditions ensure an exponentially small error $O(e^{-\rho/h})$, $\rho>0$, in the calculation of the three-dimensional solution, but the Kirchhoff conditions only give polynomially small error. For the arterial tree, under the assumption that the walls of the blood vessels are rigid, for every bifurcation node a ($2\times2$)-pressure drop matrix appears, and its influence on the transmission conditions is taken into account by means of small variations of the lengths of the graph and by introducing effective lengths of the one-dimensional description of blood vessels whilst keeping the Kirchhoff conditions and exponentially small approximation errors. We discuss concrete forms of arterial bifurcation and available generalizations of the results, in particular, the Navier-Stokes system of equations.
Bibliography: 59 titles.

Keywords: junction of thin channels, bifurcation of a blood vessel, Reynolds equation, modified Kirchhoff conditions, pressure drop matrix, effective length of a one-dimensional image of a blood vessel.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-01-02175-а
Linköping University
S. A. Nazarov's research was supported by the Russian Foundation for Basic Research (grant no. 15-01-02175-a) and by Linköpings Universitet (Sweden).

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English version:
Sbornik: Mathematics, 2017, 208:8, 1138–1186

Bibliographic databases:

UDC: 517.958+539.3(5)+531.3-324
MSC: Primary 76D07; Secondary 76D05, 76Z05, 92C35
Received: 30.05.2016 and 30.11.2016

Citation: V. A. Kozlov, S. A. Nazarov, “A one-dimensional model of flow in a junction of thin channels, including arterial trees”, Mat. Sb., 208:8 (2017), 56–105; Sb. Math., 208:8 (2017), 1138–1186

Citation in format AMSBIB
\by V.~A.~Kozlov, S.~A.~Nazarov
\paper A~one-dimensional model of flow in a~junction of thin channels, including arterial trees
\jour Mat. Sb.
\yr 2017
\vol 208
\issue 8
\pages 56--105
\jour Sb. Math.
\yr 2017
\vol 208
\issue 8
\pages 1138--1186

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    This publication is cited in the following articles:
    1. V. A. Kozlov, S. A. Nazarov, “Model of saccular aneurysm of the bifurcation node of the artery”, J. Math. Sci. (N. Y.), 238:5 (2019), 676–688  mathnet  crossref
    2. V. A. Kozlov, S. A. Nazarov, “Letter to the editors”, Sb. Math., 209:6 (2018), 919–919  mathnet  crossref  crossref  adsnasa  isi  elib
    3. German L. Zavorokhin, “A mathematical model of an arterial bifurcation”, Ural Math. J., 5:1 (2019), 109–126  mathnet  crossref  mathscinet
    4. V. A. Kozlov, S. A. Nazarov, G. L. Zavorokhin, “Pressure drop matrix for a bifurcation with defects”, Eurasian J. Math. Comput. Appl., 7:3 (2019), 33–55  crossref  isi
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