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Mathematical modelling of flow through thin curved pipes with application to hemodynamics
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
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: urn:nbn:se:liu:diva-156346DOI: 10.3384/diss.diva-156346ISBN: 9789176850732 (print)OAI: oai:DiVA.org:liu-156346DiVA, id: diva2:1305438
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
List of papers
1. A TWO-DIMENSIONAL MODEL OF THE THIN LAMINAR WALL OF A CURVILINEAR FLEXIBLE PIPE
Open this publication in new window or tab >>A TWO-DIMENSIONAL MODEL OF THE THIN LAMINAR WALL OF A CURVILINEAR FLEXIBLE PIPE
2018 (English)In: Quarterly Journal of Mechanics and Applied Mathematics, ISSN 0033-5614, E-ISSN 1464-3855, Vol. 71, no 3, p. 349-367Article in journal (Refereed) Published
Abstract [en]

We present a two-dimensional model describing the elastic behaviour of the wall of a curved flexible pipe. The wall has a laminate structure consisting of several anisotropic layers of varying thickness and is assumed to be much smaller in thickness than the radius of the channel which itself is allowed to vary. Our two-dimensional model takes the interaction of the wall with any surrounding or supporting material and the fluid flow into account and is obtained via a dimension reduction procedure. The curvature and twist of the pipes axis as well as the anisotropy of the laminate wall present the main challenges in applying the dimension reduction procedure so plenty of examples of canonical shapes of pipes and their walls are supplied with explicit systems of differential equations at the end.

Place, publisher, year, edition, pages
OXFORD UNIV PRESS, 2018
National Category
Other Electrical Engineering, Electronic Engineering, Information Engineering
Identifiers
urn:nbn:se:liu:diva-150869 (URN)10.1093/qjmam/hby009 (DOI)000441808700006 ()
Note

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

Available from: 2018-09-06 Created: 2018-09-06 Last updated: 2019-04-16
2. A one dimensional model of blood flow through a curvilinear artery
Open this publication in new window or tab >>A one dimensional model of blood flow through a curvilinear artery
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
Keywords
Blood flow; Curvilinear vessel; Asymptotic analysis; Dimension reduction; Numerical simulation
National Category
Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:liu:diva-151627 (URN)10.1016/j.apm.2018.07.019 (DOI)000444362800034 ()
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

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