mathematical modelling of flow through thin curved pipes...

Linköping Studies in Science and Technology Dissertation No. 1988

Arpan Ghosh M

athematical m

odelling of flow through thin curved pipes w

ith application to hemodynam

ics2019

Mathematical modelling of flow through thin curved pipes with application to hemodynamics

Arpan Ghosh

Linköping Studies in Science and Technology. Dissertations. No. 1988

Mathematical modelling of flow through thincurved pipes with application to

hemodynamics

Arpan Ghosh

Division of Mathematics and Applied MathematicsDepartment of Mathematics

Linköping UniversitySE-581 83 Linköping, Sweden

Linköping 2019

Linköping Studies in Science and Technology. Dissertations. No. 1988

Mathematical modelling of flow through thin curved pipes with application tohemodynamics

Copyright c© Arpan Ghosh, 2019Division of Mathematics and Applied MathematicsDepartment of MathematicsLinköping UniversitySE-581 83, Linköping, [email protected]

https://liu.se/en/organisation/liu/mai/mtm

ISSN 0345-7524ISBN 978-91-7685-073-2Printed by LiU-Tryck, Linköping, Sweden 2019

Abstract

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

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

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

The one-dimensional model described in the second paper is used to derive a simplifiedone-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 bythe surrounding muscle tissue. Numerical simulations are presented which demonstratedifferent characteristics associated with a false aneurysm.

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

Populärvetenskaplig sammanfattning

En av de viktigaste processerna i en mänsklig kropp är blodomloppet. Det är utsatt förmånga risker och därför är det viktigt att kunna diagnotisera fel i blodomloppet i god tid.Att formulera en enkel men realistisk matematisk modell för blodflödet som kan hjälpa tillmed diagnotisering är därmed viktigt. I den här avhandlingen, matematisk modellering avinkompressibelt flöde genom smala rör med tanke p̊a blodkärl har studerats. Avhandlingenär en sammansättning av fyra artiklar.

I den första artikeln presenteras en tv̊adimensionell modell som beskriver det elastiskabeteendet hos väggen av ett krökt flexibelt rör. Väggen antas ha en laminatstrukturbest̊aende av flera lager av varierande tjocklek och elastiska egenskaper. Väggen antas ocks̊avara mycket mindre i tjocklek jämfört med kanalens bredd vilken till̊ats variera längs röret.Den tv̊adimensionella modellen inneh̊aller p̊averkan av omgivande material. Modellen harskapats via en procedur som kallas för dimensionreduktion där krökning och vridning avrörets axel samt laminatstrukturen hos väggen utgör de viktigaste utmaningarna. I artikelnpresenteras ocks̊a n̊agra exempel p̊a modellen för enkelformer av röret.

I den andra artikeln presenteras en endimensionell modell som beskriver blodflödet genomett måttligt krökt och elastiskt blodkärl. Modellen byggs p̊a antagandet att kärlet ärtunt jämfört med dess längd. Kärlväggen modelleras av den tv̊adimensionella modellensom presenteras i den första artikeln. Navier-Stokes ekvationer beskriver flöden av vätskoroch de används här för att modellera flödet genom kanalen. Omgivande muskelvävnaderoch närvaro av yttre krafter tas med i beräkningen. Resultat av numeriska simuleringarpresenteras för att markera p̊averkan av olika faktorer p̊a blodflödet.

Den endimensionella modellen som beskrivs i den andra artikeln används för att härleda enförenklad endimensionell modell av en falsk aneurysm som utgör föremålet för den tredjeartikeln. En falsk aneurysm är en ackumulering av blod utanför ett blodkärl men begränsatav den omgivande muskelvävnaden. Det är ett allvarligt medicinskt tillst̊and som måsteövervakas och under vissa omständigheter behandlas br̊adskande, vilket medför behovetav korrekt diagnos. Numeriska simuleringar presenteras som visar olika egenskaper hos enfalsk aneurysm.

Reynolds ekvation beskriver tryckfördelning i tunna hinnor av vätskor. Stokes ekvationerbeskriver l̊angsamt flytande vätskor. I den sista artikeln presenteras en modifierad Reynold-sekvation tillsammans med dess härledning fr̊an Stokes ekvationer. Ekvationen reglerar detstabila flödet av en vätska med l̊agt Reynolds-tal genom ett krökt, smalt rör. Kanalen ansesha stor krökning och vridning. Approximation av hastighet och tryck hos vätskan inuti

iv 0 Populärvetenskaplig sammanfattning

kanalen har konstruerats. En motivering till approximationerna ges tillsammans med enjämförelse med ett enklare fall.

Acknowledgements

First and foremost, I would like to thank my supervisor Vladimir Kozlov for his unwaveringsupport and guidance. He has always answered my questions with patience and I amgrateful to him for his invaluable teachings. Next, I would like to thank my co-supervisorsMatts Karlsson and David Rule for their support and encouragement. David has been atrue teacher to me, and I thank him for the numerous discussions about mathematics andbeyond.

I am sincerely grateful to my co-authors Sergei Nazarov and Fredrik Berntsson for theircollaborations. It is truly a pleasure working with them.

The last five years have been a great learning period and I thank all my teachers who havecontributed to my learning experience. I am also grateful to all my colleagues and especiallythe fellow PhD students at the department who have made for a pleasant workplace.

A few of the colleagues became wonderful friends to me over the years and I would like totake this opportunity to express my appreciation for their friendships. I thank Jolanta foralways being a great company whenever we meet. My gratitude towards Samira for thoseweekly bike rides and the competitions. My appreciation towards Sonja for appreciatingmy jokes. Many thanks to Evgeniy for the drunk discussions and a great friendship. Iam eternally grateful to Anna for her kind gestures and for sharing her passion for sciencewith me. Big thanks to kompis Mikael for pushing me towards sporting adventures andfor the laughs about the legends. And finally, a special thanks to Alexandra for being thebest of friends whenever possible.

The final words of appreciation go to my family. I am grateful to Bishnupada Ghosh forguiding and inspiring me since my childhood. Lastly, I am thankful to my parents Lalkamaland Chhaya whose sacrifices and hard work made it possible for me to come this far.

List of Papers

I. A. Ghosh, V. A. Kozlov, S. A. Nazarov, and D. Rule. A two-dimensional model of thethin laminar wall of a curvilinear flexible pipe. The Quarterly Journal of Mechanicsand Applied Mathematics, 71(3):349-367, 2018.Athuor’s contribution: Writing the manuscript except Appendix A, major parts ofthe computations and analysis of the equations.

II. F. Berntsson, A. Ghosh, V.A. Kozlov, and S.A. Nazarov. A one dimensional modelof blood flow through a curvilinear artery. Applied Mathematical Modelling, 63:633-643, 2018.Athuor’s contribution: Writing the manuscript, numerical simulations, computationsand analysis of the equations.

III. F. Berntsson, A. Ghosh, M. Karlsson, V.A. Kozlov, and S.A. Nazarov. A one-dimensional asymptotic model of a false aneurysm. Submitted.Athuor’s contribution: Writing the manuscript, numerical simulations and computa-tions.

IV. A. Ghosh, V. A. Kozlov, and S. A. Nazarov. Modified Reynolds equation for steadyflow through a curved pipe. arXiv e-prints, page arXiv: 1901.01953, January 2019.Athuor’s contribution: Writing the manuscript, major parts of the computations andanalysis of the equations.

Contents

Abstract i

Populärvetenskaplig sammanfattning iii

Acknowledgements v

List of Papers vii

1 Introduction 1

1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Elements of continuum mechanics 3

2.1 Theory of linear elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Fluid mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Curvilinear coordinates 7

3.1 Rotation minimizing frame . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Non-orthogonal frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 Volume and area elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Asymptotic analysis 13

4.1 Asymptotic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2 Method of slow variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

References 15

INCLUDED PAPERS

I. A two-dimensional model of the thin laminar wall of a curvilinear flexible pipe 17

II. A one dimensional model of blood flow through a curvilinear artery . . . . . 39

x CONTENTS

III. A one-dimensional asymptotic model of a false aneurysm . . . . . . . . . . . 53

IV. Modified Reynolds equation for steady flow through a curved pipe . . . . . . 69

1

Introduction

Pipes transporting fluids are ubiquitous in our surroundings and hence a significant amountof research has been devoted to them. Fluid-flow through rigid pipes has been studiedquite extensively, although, much of the literature is about straight pipes. In comparison,research on curved elastic pipes is not as abundant. The study of flow through elasticpipes is quite significant due to the numerous areas of application. Elastic tubes are foundin various artificial mechanisms such as fuel or water transmission, vehicular systems, fireextinguishers etc. With the development of new, strong and light materials, the study offlexible tubes with thin, laminar walls acquire more importance. Moreover, elastic tubesalso occur naturally in biological systems such as the respiratory system, the digestivesystem, the renal system, the cardiovascular system etc, see [5].

The circulatory system is one of the most important systems in the human body. It per-forms a number of essential tasks like supplying nutrition throughout the body, regulationof body temperature and fighting diseases. It is also susceptible to various kinds of dis-eases. Simulating the system with a reasonable model can be of great help in managingproblems arising in the system. A good model can lead to early diagnosis of a problemand thereby enable a better management of the issue.

Substantial efforts have been made to model blood flow through blood vessels, see, forexample, the monograph [3]. The complex arrangement of elastic tissues forming a bloodvessel poses an increased difficulty in accurately modeling the interaction of blood flowwith the vessel wall. The vessel walls have a laminate structure consisting of three layersof tissues (called adventitia, media and intima) having different composition and elasticproperties, see [2]. Various one-dimensional models have been previously presented byemploying simplifying assumptions on the geometry and structure of the blood vessels.For example, the models presented by [4, 8, 11, 14, 15] are based on mass and momentumconservation laws together with a linear stationary tube law, while [12, 17] present modelsbased on existing shell models coupled with Navier-Stokes equations. Navier equationscoupled with Navier-Stokes equations have also been used to model blood flow, see [3, 13,16]. However, majority of these works are based on cylinders having straight referenceaxes.

This thesis is aimed at the mathematical modelling of fluid flow through thin, curvilinearand compliant pipes. To model blood flow through curved vessels is the main motivation

2 1 Introduction

behind this study. This work consists of a collection of four articles, each addressing adifferent aspect of the modelling project.

1.1 Outline

This dissertation is comprised of two parts. The first part consists of four chapters whichprovide the fundamental concepts that have been used in the research that is being pre-sented in this thesis. Chapter 1 introduces the subject matter of the thesis and providesthe underlying motivation for the work presented. The dissertation is about the modellingof interaction between a fluid and a solid in a curved domain. In Chapter 2, some fun-damental concepts from continuum mechanics are presented. The chapter is divided intotwo sections, the first of which is devoted to the theory of linear elasticity while the secondaddresses the theory of mechanics of fluids. Chapter 3 is dedicated to the description ofthe geometrical framework in which the modelling tasks have been performed. Lastly, inChapter 4, a brief overview of the asymptotic method used for the modelling assignmentis presented.

The second part consists of four research articles that have been produced as a result ofthe research. In Paper I, the derivation a two-dimensional model of the laminated, elasticwall of a curvilinear pipe is presented. Some examples of the model in the cases of simplegeometric configurations are also provided in this paper. Paper II provides the derivationof a one-dimensional model of blood flow through a curved artery. A simple numericalscheme to solve the model equations is discussed and the results of some simulations withartificial data are also presented in this paper. Paper III presents an application of theone-dimensional model to the modelling of false aneurysms. In Paper IV, generalizations ofReynolds equation and Poiseuille equation are presented in the context of a thin curvilinearchannel. Justification of the approximations generated by these equations is also providedin this paper.

2

Elements of continuum mechanics

The model equations for the elastic wall of a pipe are derived from the principles of me-chanics of linear elastic solids whereas the principles of fluid mechanics provide the modelequations describing the flow in the channel inside the pipe. In this chapter, a brief overviewof the concepts from continuum mechanics are presented which have been used in the pa-pers included in this thesis. For a detailed introduction to the subject matter, the readeris referred to [7].

2.1 Theory of linear elasticity

Elasticity is the ability of a body to regain its original shape after being subjected to adeformation by some forces. The main quantity of interest for us in the modelling of adeformed elastic body is the displacement vector field, denoted here by u, with respect to itsreference state. The displacement gradient is the second order tensor ∇u. The deformationgradient quantifies the relative change in infinitesimal vectors under a deformation withrespect to the reference state. It is given as

D = I +∇u.

Here I is the identity tensor.

The relative change in infinitesimal lengths in a deformed body with respect to its referenceconfiguration is quantified by the so called the Cauchy-Green deformation tensor and isgiven by

C = DTD = I +∇u +∇uT +∇uT∇u.

The tensor (C − I)/2 is termed as the Lagrange strain tensor. It provides a measure ofdeformation in a body by means of the relative displacements of points in the body withrespect to its reference configuration.

If the deformations in an elastic body are infinitesimal, then the tensor C can be approxi-mated by I + 2ε where

ε =1

2(∇u +∇uT ).

4 2 Elements of continuum mechanics

The tensor ε is called the infinitesimal strain tensor.

Stress is a measure of the forces exerted by particles on their adjacent particles across aninfinitesimal surface in a body. The Cauchy stress vector at a point across a surface passingthrough the point is defined as

t =dF

dS

where dF is the infinitesimal resultant force across the infinitesimal surface dS containingthe point.

The Cauchy stress principle is an axiom in classical continuum mechanics which statesthat the Cauchy stress vector at a point does not depend on the curvature of the surfaceat the concerned point, rather, only on its unit normal n at that point. This results in thestress vector having the expression

t = σn.

The tensor field σ is known as the Cauchy stress tensor.

The field density of the resultant internal forces in a body is given by the quantity ∇ · σand applying Newton’s second law of motion at each point in a body, one gets for timeparameter t,

∇ · σ = ρd2u

dt2.

Here ρ is the mass density of the body. The equation is called Cauchy’s equation of motion.

A body is called linearly elastic or Hookean if it obeys Hooke’s law, that is, the loadingforce at each point in the body is directly proportional to the relative displacement at thepoint with respect to the reference state of the body. This implies that the stress insidethe body is a linear function of the strain at each point, with no stress in the absence ofstrain. Hence the ‘constitutive equation’ for the body is given as

σ = Tε

where the fourth order tensor field T is called the elasticity tensor.

The constitutive equations along with Cauchy’s equation of motion provide a system ofhyperbolic partial differential equations for the relative displacement field u, namely

1

2∇ · (T(∇u +∇uT )) = ρd

2u

dt2.

The preceding equation forms the basis of our model for the elastic part of the vessel.

2.2 Fluid mechanics 5

2.2 Fluid mechanics

While modelling the flow of a fluid through a channel, the main quantities of interest arethe velocity field, say v and the hydrostatic pressure in the fluid body, say p as functions ofposition vector x and time t. The fluid is characterised by its dynamic viscosity µ, assumedto be constant throughout, which quantifies its resistance to relative motion between layers.

The strain rate tensor is defined as the quantity

1

2(∇v +∇vT ).

We make certain simplifying assumptions about the fluid in consideration.

First, we assume that the fluid is incompressible. This means through any given volume,the total mass flowing in is equal to the total mass flowing out at any instance. In otherwords, the density of the fluid, say ρ, is considered a constant. This results in the continuityequation

∇ · v = 0.

Secondly, the fluid is assumed to be Newtonian. This means the viscous stress, that is thepart of the stress accompanying the strain rate, is directly proportional to the strain ratetensor. For an incompressible fluid, this results in the total stress tensor field being

σ = −pI + µ(∇v +∇vT ).

Note that

d

dtv(x, t) =

∂

∂tv(x, t) +

∂x

∂t· ∇v(x, t) = ∂

∂tv(x, t) + v(x, t) · ∇v(x, t)

and∇ · (∇v +∇vT ) = ∆v

due to v being divergence free.

Let g be the acceleration due to gravity. Then, as in the previous section, by Newton’ssecond law of motion,

ρdv

dt= ∇ · σ + ρg

⇔ ∂v∂t

+ v · ∇v = −1ρ∇p+ µ

ρ∆v + g.

The preceding equation is called the Navier-Stokes equation. As the density of the fluid is

taken to be constant, the pressure p can be replaced by the kinematic pressure q =p

ρand

instead of the dynamic viscosity µ, kinematic viscosity ν =µ

ρcan be used.

6 2 Elements of continuum mechanics

The Reynolds number is a dimensionless quantity signifying the ratio of inertial forces toviscous forces associated with a flow. It also signifies the transition of a flow from laminar toturbulent. A low value of the Reynolds number can be achieved by high values of viscosityor low velocities among other factors. In such cases, the viscous forces are dominating andhence the inertial terms in the Navier-Stokes equation can be neglected to obtain

−ν∆v +∇q = g.

This is called the Stokes equation and the flow described by it is called the Stokes flow.

In our work, the modelling of the flow is based either on the Stokes equation or on thelinearized (neglecting only the non-linear term) Navier-Stokes equation.

3

Curvilinear coordinates

In many modelling problems, the choice of a coordinate system proves to be crucial as itgreatly affects the ease with which we can carry out computations. Sometimes, this choicefor the ‘most suitable’ coordinate system is made simpler by the underlying geometry of themodelling assignment. In this chapter, some coordinate systems suitable to the modellingof curvilinear pipes are presented. The accompanying formulas necessary for calculus inthese coordinate systems are also provided. The articles included in the thesis rely heavilyon versions of the computations presented in this chapter.

The aim is to choose a suitable coordinate system that simplifies the computations evenin the case of the most general geometry of the tube. As is the case with the subsequentarticles, it is assumed that a canonical Cartesian coordinate system describes the spacecontaining the vessel and a ‘centre’ curve of the vessel is known and given by an arc-length perameterised curve c ∈ C2([0, L],R3) for some positive real L that represents thetotal length of the considered tube. Without loss of generality, we may assume the initialconditions

c(0) = (0, 0, 0)T and c′(0) = (0, 0, 1)T . (3.1)

Here and henceforth, if a function, say f , depends only on one variable, we denote itsderivative by f ′. Let the arc-length parameter be denoted by s. We use the centre curveto develop the required coordinate frames.

3.1 Rotation minimizing frame

At first, we need to build a right handed coordinate frame at each point c(s). The coor-dinate system should be the cylindrical coordinate system in the special case of a straightcylinder, that is, when c′′(s) ≡ 0. It is natural to take one of the coordinate directions tobe c′(s) and the other two to be perpendicular to it. A natural choice is to use the Frenet-Serret frame. This has the obvious problem that it is not defined when the curvaturevanishes.

Let e1 be one such unit vector perpendicular to c′ at each s. We could use the Frenet-Serret

frame to define e1(θ, s) = cos θN(s) − sin θB(s) where N and B are the unit normal andthe unit binormal of the curve c, if we assume that the curve has non-vanishing curvature.

8 3 Curvilinear coordinates

Consider the surface S(θ, s) = c(s) + rδe1(θ, s) for some rδ > 0. The surface S is a pipearound the centre curve so that every point on it has a constant distance rδ from the centrecurve. Then,

∂S(θ, s)

∂θ= − sin θN(s)− cos θB(s),

∂S(θ, s)

∂s= cos θ(−κ(s)c′(s) + τ(s)B(s)) + τ(s)N(s),

where κ and τ are the curvature and torsion respectively of the curve c. Hence

∂S(θ, s)

∂θ· ∂S(θ, s)

∂s= −τ(s).

As the partial derivatives of S with respect to θ and s give us the coordinate directionsfor the respective parameters, we conclude that the corresponding coordinate lines do notintersect at right angles when c has nonzero torsion. For this reason, we reject this choiceof coordinate frame.

A better choice proves to be one that is based on the requirement that the change ine1 as we travel along the central curve, should be coplanar with e1 and c

′(s), that is,∂se1 · (e1 × c′) = 0. This prevents the coordinate lines corresponding to s from ‘wrappingaround’ the tubular surface due to torsion of the centre curve. Since e1 is a unit vector,we have

∂se1 · e1 =1

2∂s‖e1(s)‖2 = 0.

Also, as c′ is perpendicular to e1 at each s, it follows that

∂s(e1 · c′) = 0⇔ ∂se1 · c′ = −c′′ · e1.

Therefore, the coplanarity condition on the vector ∂se1 with the orthonormal vectors{c′, e1} is equivalent to

∂se1 = (∂se1 · c′)c′ + (∂se1 · e1)e1 = −(c′′ · e1)c′.

We choose the initial value of e1 at s = 0 to be (cos θ, sin θ, 0)T for some θ ∈ [0, 2π]. Then

we obtain the following initial value problem that defines e1

∂se1(θ, s) = −(c′′(s) · e1(θ, s))c′(s) and e1(θ, 0) = (cos θ, sin θ, 0)T . (3.2)

Defining e2(θ, s) = c′(s) × e1(θ, s), the triple {e1(θ, s), e2(θ, s), c′} forms an orthonormal

frame at each point c(s) for a given angle θ. As a result, we have

∂se2(θ, s) = c′′(s)× e1(θ, s) + c′(s)× ∂se1(θ, s) = c′′(s)× (e2(θ, s)× c′(s))

= c′′(s) · c′(s)e2(θ, s)− c′′(s) · e2(θ, s)c′(s) = −c′′(s) · e2(θ, s)c′(s).

3.1 Rotation minimizing frame 9

The initial condition for e2 reads

e2(θ, 0) = c′(0)× e1(θ, 0) = (− sin θ, cos θ, 0)T .

So we obtain

∂se2(θ, s) = −(c′′(s) · e2(θ, s))c′(s) and e2(θ, 0) = (− sin θ, cos θ, 0)T . (3.3)

The equations (3.2) and (3.3) ensure that the frame {c′, e1, e2} is a so called ‘rotationminimizing frame’, see [1, 6].

One can define a rotation-matrix valued function R so that ei = R(s)ei(θ, 0) for i = 1, 2.Then it is readily obtained that

∂θe1(θ, s) = e2(θ, s) and ∂θe2(θ, s) = −e1(θ, s).

The relations (3.2) and (3.3) can be represented in an analogous way to the Frenet-Serretframe as

∂s

e1e2c

=

0 0 −c′′ · e10 0 −c′′ · e2

c′′ · e1 c′′ · e2 0

e1e2c

. (3.4)

The parameter θ corresponds to the orientation of the vectors e1(θ, s) and e2(θ, s) for givens, with respect to some reference pair of orthogonal vectors in the same disc perpendicularto the corresponding tangent vector c′(s) of the central curve. We assume it to be zerowhen r = 0. Note that in the curvature free case, the orthonormal frame is the same asthe cylindrical coordinate frame.

The relationship between the Cartesian and the new curvilinear coordinate systems areexpressed as

x(r, θ, s) = c(s) + re1(θ, s).

The scale factors associated with the curvilinear coordinates {r, θ, s} are defined as Hi :=‖∂ix‖ for the i-th coordinate. As a result,

Hr ≡ 1, Hθ = r, and Hs = 1− rc′′ · e1. (3.5)

The non-zero components of the metric tensor are

g11 = H2r = 1, g22 = H

2θ = r

2, g33 = H2s = (1− rc′′ · e1)2.

Hence,√

det(g) = r(1− rc′′ · e1).The nabla operator ∇ in the new coordinates is given as

∇ = 1Hr

e1∂r +1

Hθe2∂θ +

1

Hsc′∂s = e1∂1 +

1

re2∂θ +

1

1− rc′′ · e1c′∂s.


Keeping in mind the derivatives of the coordinate unit vectors with respect to the newcoordinate variables, the Laplacian ∆ = ∇ · ∇ is given as

∆ = ∂21 +1

r2∂2θ +

1

r∂1 −

1

1− rc′′ · e1c′′ · (e1∂1 +

1

re2∂θ)

+1

(1− rc′′ · e1)2∂2s +

rc′′ · e1(1− rc′′ · e1)3

∂s.

Consider a vector field v = v1e1 + v2e2 + v3c′. The divergence of v becomes

∇ · v = ∂rv1 +1

r∂θv2 +

1

1− rc′′ · e1∂sv3 +

v1r− c

′′ · v1− rc′′ · e1

.

3.2 Non-orthogonal frame

The reference frame {e1, e2, c′} is sufficient to perform vector calculus with ease whenmodelling the flow through the interior of the pipes. However, it loses its efficiency whenmodelling the laminate wall of the pipe. Due to the customary ways of formulation of thephysical laws, it is desirable to have simple expressions for the normal vectors to each layer.In general, the vector field e1 is not aligned with the normal vectors of the layers due tothe variable width of the channel or that of the wall. In such cases, the computations aresimplified by using a non-orthogonal frame to describe the vector fields.

A detailed presentation of tensor algebra in curvilinear coordinates for application to con-tinuum mechanics can be found in Appendix D of [9].

Let there be new coordinates {q1, q2, q3} having the desired properties. Let us define∂i = ∂/∂q

i for i ∈ {1, 2, 3}. Also, we adopt Einstein’s summation convention, that is,repeated indices, when appearing concurrently at both the top and bottom positions in aterm, are assumed to be summed over the index set, which is {1, 2, 3} in our case.

We now define a set of contravariant basis vectors for tangent vectors. Let xi = ∂ixfor i = 1, 2, 3 and position vector x(q1, q2, q3). We can define (the metric tensor havingcoefficients) gij = xi · xj for i, j = 1, 2, 3. Let g denote the matrix [gij].

We may also define a set of covariant or reciprocal basis vectors for the same space (cf.Appendix D of [9]) which are given as xi = gijxj where g

ij is such that gijgjk = δik with δ

ik

being the Kronecker delta. Note that the contravariant basis vectors have bottom indiceswhile the reciprocal basis vectors have top indices. By definition, xi is perpendicular toboth xj and xk for each permutation (i, j, k) of (1, 2, 3). This means for suitable a choiceof coordinates {q1, q2, q3}, x1 is automatically normal to any surface defined by a fixed q1.This makes it easier for us to formulate the physical laws.

3.3 Volume and area elements 11

In order to express derivatives in a curvilinear system, we need the Christoffel symbolscorresponding to the curvilinear system which are defined as Γijk = x

i · ∂jxk for i, j, k =1, 2, 3. They quantify the change in the coordinate directions with derivation and aresymmetric in the lower indices, i.e., Γijk = Γ

ikj.

With the help of these relations, we can define the gradient operator as ∇ = xi∂i, seeAppendix E in [9]. We are now in a position to express quantities like gradient anddivergence of tensors in our curvilinear coordinates. For any vector v = vjx

j = vkxk, itsgradient is given as

∇v =(∂ivj − Γkijvk

)xixj =

(∂iv

k + Γkijvj)

xixk.

The divergence of the vector v is given as

∇ · v = gij(∂ivj − Γkijvk

)= ∂iv

i + Γiijvj

=(√

det(g))−1

∂i

(√det(g)vi

)

where det(·) denotes the determinant.The Laplacian is then defined as

∆ = ∇ · ∇ = gij(∂i∂j − Γkij∂k

).

Consequently, the Laplacian of the vector v can be given as

∆v = gij[∂i∂jvk − Γlij∂lvk − Γlik∂jvl − Γlkj∂ivl−(∂iΓ

lkj − ΓlkmΓmij − ΓlmjΓmki

)vl]x

k

= gij[∂i∂jvk − Γlij∂lvk + Γkil∂jvl + Γklj∂ivl

+(∂iΓ

klj − ΓklmΓmij + ΓkmiΓmlj

)vl]xk.

Also, for any rank 2 tensor σ = σijxixj = σijxix

j = σ ji xixj = σijx

ixj, its divergence isexpressed as

∇ · σ = (∂iσik + Γiijσjk + Γkijσij)xk= (∂iσ

ik + Γ

iijσ

jk − Γjikσij)xk

= (∂iσkj − Γlijσ kl + Γkilσ lj )gijxk

= (∂iσjk − Γlijσlk − Γlikσjl)gijxk.

3.3 Volume and area elements

The derivation of the final model equations involve integration of certain scalar or vectorquantities over the entire three-dimensional channel or the two-dimensional surface bound-ing the channel or across a cross-section of the pipe. To enable such operations, the volumeand area elements are required to be specified in the new coordinates.


The volume of a parallelepiped formed by the basis vectors {x1,x2,x3} as sides is given by

det([x1,x2,x3]) =√

det([x1,x2,x3]T [x1,x2,x3]) =√

det(g).

Hence, the infinitesimal volume element with respect to the new variables is

dV =√

det(g)dq1dq2dq3, (3.6)

where the superscript [·]T denotes the transpose.On the other hand, the area of the parallelogram formed by the basis vectors {x2,x3} assides is given by

|x2 × x3| =√|x2|2|x3|2 − (x2 · x3)2 =

√g22g33 − (g23)2 =

√g11 det(g).

Hence, the infinitesimal surface element on a surface with fixed q1 is

dS1 =√g11 det(g)dq2dq3. (3.7)

Similarly for dS2 and dS3.

4

Asymptotic analysis

The central theme in our modelling of thin pipes is the use of asymptotic methods. Suchmethods use a small parameter that occurs in a given problem, to solve the problem. Inthis chapter, some basic concepts of the asymptotic method employed in the included worksare presented. An overview on perturbation methods can be found in [10].

4.1 Asymptotic approximation

Here, the order of a function is defined and subsequently the phrase ‘asymptotic expansionof a function’ is formalized.

Definition 4.1.1. A function f(ε) is said to be of the order of g(ε) and written as f(ε) =O(g(ε)) as ε→ 0 iff

limε→0

∣∣∣∣f(ε)

g(ε)

∣∣∣∣

14 4 Asymptotic analysis

4.2 Method of slow variation

In the case of thin pipes, the natural small parameter (say ε) is its aspect ratio, that is,the ratio of its width to its length. The physical quantities vary rapidly in the transversedirection as compared to the longitudinal direction. Hence a stretching of variables isintroduced for the transverse direction through a scaling by ε−1. As a result, the differentialoperators are also modified.

As an example, let L be a differential operator and v(x, y) be a function satisfying Lv = 0with appropriate boundary conditions. Letting x to be the transversal variable, introducethe stretched variable x′ = ε−1x. Then, in terms of the new variable, we have for someN ∈ Z, a decomposition

L = εN(L0 + εL1 + ε2L2 + . . .).

Meanwhile, a formal asymptotic expansion of v is assumed in the form of

v(x, y) = εM(v0(x′, y) + εv1(x

′, y) + ε2v2(x′, y) + . . .)

for some M ∈ (Z).The last step is to match the coefficients of different orders of ε in the equation obtainedby replacing the expansions in the original equation. In particular, a series of equationsare obtained which have the form

L0v0(x′, y) = 0

L0v1(x′, y) = −L1v0(x′, y)

L0v2(x′, y) = −L2v0(x′, y)− L1v1(x′, y)

...

These equations are successively solved with the corresponding boundary conditions to getthe desired order of approximation of the solution to the original equation.

REFERENCES 15

References

[1] R. L. Bishop. There is more than one way to frame a curve. The American Mathe-matical Monthly, 82:246–251, 1975.

[2] P. Fratzl, editor. Collagen: Structure and Mechanics. Springer US, 2008.

[3] Y. C. Fung. Biomechanics: Circulation. Springer-Verlag New York, 1997.

[4] L. Grinberg, E. Cheever, T. Anor, J. R. Madsen, and G. E. Karniadakis. Modelingblood flow circulation in intracranial arterial networks: A comparative 3d/1d simula-tion study. Annals of Biomedical Engineering, 39(1):297–309, Jan 2011.

[5] J. B. Grotberg and O. E. Jensen. Biofluid mechanics in flexible tubes. In Annualreview of fluid mechanics. Vol. 36, volume 36 of Annu. Rev. Fluid Mech., pages 121–147. Annual Reviews, Palo Alto, CA, 2004.

[6] F. Klok. Two moving coordinate framesfor sweeping along a 3d trajectory. ComputerAided Geometric Design, 3:217–229, 1986.

[7] W. M. Lai, D. Rubin, and E. Krempl. Introduction to Continuum Mechanics.Butterworth-Heinemann, Boston, fourth edition, 2010.

[8] J. Lighthill. Mathematical biofluiddynamics. Society for Industrial and Applied Math-ematics, Philadelphia, Pa., 1975. Based on the lecture course delivered to the Math-ematical Biofluiddynamics Research Conference of the National Science Foundationheld from July 16–20, 1973, at Rensselaer Polytechnic Institute, Troy, New York,Regional Conference Series in Applied Mathematics, No. 17.

[9] A. I. Lurie. Theory of Elasticity. Springer, 2005.

[10] R. M. M. Mattheij, S. W. Rienstra, and J. H. M. ten Thije Boonkkamp. Partialdifferential equations. SIAM Monographs on Mathematical Modeling and Computa-tion. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005.Modeling, analysis, computation.

[11] T. J. Pedley. Mathematical modelling of arterial fluid dynamics. J. Engrg. Math.,47(3-4):419–444, 2003. Mathematical modelling of the cardiovascular system.

[12] A. Quarteroni and L. Formaggia. Mathematical modelling and numerical simulationof the cardiovascular system. In Handbook of numerical analysis. Vol. XII, Handb.Numer. Anal., XII, pages 3–127. North-Holland, Amsterdam, 2004.

[13] A. Quarteroni, M. Tuveri, and A. Veneziani. Computational vascular fluid dynamics:Problems, models and methods. Comput. Vis. Sci., 2(4):163–197, mar 2000.

16 REFERENCES

[14] P. Reymond, Y. Bohraus, F. Perren, F. Lazeyras, and N. Stergiopulos. Validationof a patient-specific one-dimensional model of the systemic arterial tree. AmericanJournal of Physiology - Heart and Circulatory Physiology, 301(3):H1173–H1182, 2011.

[15] C. A. Taylor and M. T. Draney. Experimental and computational methods in cardio-vascular fluid mechanics. In Annual review of fluid mechanics. Vol. 36, volume 36 ofAnnu. Rev. Fluid Mech., pages 197–231. Annual Reviews, Palo Alto, CA, 2004.

[16] S. Čanić and A. Mikelić. Effective equations modeling the flow of a viscous incom-pressible fluid through a long elastic tube arising in the study of blood flow throughsmall arteries. SIAM J. Appl. Dyn. Syst., 2(3):431–463, 2003.

[17] S. Čanić, J. Tambača, G. Guidoboni, A. Mikelić, C. J. Hartley, and D. Rosenstrauch.Modeling viscoelastic behavior of arterial walls and their interaction with pulsatileblood flow. SIAM J. Appl. Math., 67(1):164–193, 2006.

Papers

The papers associated with this thesis have been removed for

copyright reasons. For more details about these see:

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-156346

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-156346

FACULTY OF SCIENCE AND ENGINEERING

Linköping Studies in Science and Technology, Dissertation No. 1988, 2019 Department of Mathematics

Linköping University SE-581 83 Linköping, Sweden

www.liu.se

Arpan Ghosh M

athematical m

odelling of flow through thin curved pipes w

ith application to hemodynam

ics2019

AbstractPopulŁarvetenskaplig sammanfattningAcknowledgementsList of PapersContentsIntroductionElements of continuum mechanicsCurvilinear coordinatesAsymptotic analysisReferencesPapers

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