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A one dimensional model of blood flow through a curvilinear artery
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-2681-8965
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
St Petersburg State Univ, Russia; Inst Problems Mech Engn RAS, Russia.
2018 (English)In: Applied Mathematical Modelling, ISSN 0307-904X, E-ISSN 1872-8480, Vol. 63, p. 633-643Article in journal (Refereed) Published
Abstract [en]

We present a one-dimensional model describing the blood flow through a moderately curved and elastic blood vessel. We use an existing two dimensional model of the vessel wall along with Navier-Stokes equations to model the flow through the channel while taking factors, namely, surrounding muscle tissue and presence of external forces other than gravity into account. Our model is obtained via a dimension reduction procedure based on the assumption of thinness of the vessel relative to its length. Results of numerical simulations are presented to highlight the influence of different factors on the blood flow. (C) 2018 Elsevier Inc. All rights reserved.

Place, publisher, year, edition, pages
ELSEVIER SCIENCE INC , 2018. Vol. 63, p. 633-643
Keywords [en]
Blood flow; Curvilinear vessel; Asymptotic analysis; Dimension reduction; Numerical simulation
National Category
Fluid Mechanics and Acoustics
Identifiers
URN: urn:nbn:se:liu:diva-151627DOI: 10.1016/j.apm.2018.07.019ISI: 000444362800034OAI: oai:DiVA.org:liu-151627DiVA, id: diva2:1254398
Note

Funding Agencies|Russian Foundation of Basic Research [18-01-00325]

Available from: 2018-10-09 Created: 2018-10-09 Last updated: 2019-04-16
In thesis
1. Mathematical modelling of flow through thin curved pipes with application to hemodynamics
Open this publication in new window or tab >>Mathematical modelling of flow through thin curved pipes with application to hemodynamics
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The problem of mathematical modelling of incompressible flows with low velocities through narrow curvilinear pipes is addressed in this thesis. The main motivation for this modelling task is to eventually model the human circulatory system in a simple way that can facilitate the medical practitioners to efficiently diagnose any abnormality in the system. The thesis comprises of four articles.

In the first article, a two-dimensional model describing the elastic behaviour of the wall of a thin, curved,  exible pipe is presented. The wall is assumed to have a laminate structure consisting of several anisotropic layers of varying thickness. The width of the channel is allowed to vary along the pipe. The two-dimensional model takes the interactions of the wall with any surrounding material and the  fluid  flow into account and is obtained through a dimension reduction procedure. Examples of canonical shapes of pipes and their walls are provided with explicit systems of differential equations at the end.

In the second article, a one-dimensional model describing the blood flow through a moderately curved, elastic blood vessel is presented. The two-dimensional model presented in the first paper is used to model the vessel wall while linearized Navier-Stokes equations are used to model the  flow through the channel. Surrounding muscle tissues and presence of external forces other than gravity are taken into account. The model is again obtained via a dimension reduction procedure based on the assumption of thinness of the vessel relative to its length. Results of numerical simulations are presented to highlight the influence of different factors on the blood flow.

The one-dimensional model described in the second paper is used to derive a simplified one-dimensional model of a false aneurysm which forms the subject of the third article. A false aneurysm is an accumulation of blood outside a blood vessel but confined by the surrounding muscle tissue. Numerical simulations are presented which demonstrate different characteristics associated with a false aneurysm.

In the final article, a modified Reynolds equation, along with its derivation from Stokes equations through asymptotic methods, is presented. The equation governs the steady flow of a fluid with low Reynolds number through a narrow, curvilinear tube. The channel considered may have large curvature and torsion. Approximations of the velocity and the pressure of the fluid inside the channel are constructed. These approximations satisfy a modified Poiseuille equation. A justification for the approximations is provided along with a comparison with a simpler case.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2019. p. 16
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1988
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-156346 (URN)10.3384/diss.diva-156346 (DOI)9789176850732 (ISBN)
Public defence
2019-05-09, Nobel (BL32), B-huset, Campus Valla, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2019-04-16 Created: 2019-04-16 Last updated: 2019-04-17Bibliographically approved

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The full text will be freely available from 2020-07-17 13:46
Available from 2020-07-17 13:46

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