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Mathematical modeling of blood flow through a deformable thin-walled vesselan analytical and quantitative investigation of the advection and diffusion contribution in theglobal and local equations of motion

Ritzen, R.G.P.

Award date:2012

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Den Dolech 2, 5612 AZ EindhovenP.O. Box 513, 5600 MB EindhovenThe Netherlandswww.tue.nl

AuthorR.G.P. (Rob) Ritzen

DateAugust 16, 2012

Where innovation starts

Mathematical modeling of blood flow through adeformable thin-walled vessel

An analytical and quantitative investigation of the advection and diffusioncontribution in the global and local equations of motion

R.G.P. (Rob) Ritzen

Where innovation starts

Department of Mathematics andComputer Science

Den Dolech 2, 5612 AZ EindhovenP.O. Box 513, 5600 MB EindhovenThe Netherlandswww.win.tue.nl

AuthorR.G.P. (Rob) Ritzen

Graduation committeeProf. dr. ir. F.N. (Frans) van de VosseDr. ir. A.A.F. (Fons) van de VenDr. ir. A.S. (Arris) TijsselingDr. A. (Adrian) Muntean

Under supervision ofDr. ir. A.A.F. (Fons) van de Ven

DateAugust 16, 2012

Mathematical modeling of bloodflow through a deformablethin-walled vessel

An analytical and quantitative investigation of theadvection and diffusion contribution in the globaland local equations of motion

Master’s thesis

Industrial and Applied MathematicsComputational Science and Engineering

Abstract

In hemodynamics, the interaction of the blood flow through an arterial vessel andthe geometrical and mechanical properties of the vessel is studied. Very often,numerical methods are used to determine important flow characteristics, suchas the pressure distribution, in estimating the behavior of blood flow. This is avery useful method, often based on measurements of patient specific parameters,in predicting and preventing complications in cardiovascular surgery or in theapplication of cardiovascular equipment. Mathematical investigation of the orderof magnitude of the flow characteristics can help to reduce the computationalcosts in the numerical analysis of the blood flow.

In this project, the flow of blood through a thin-walled, linearly elastic de-formable vessel is modeled. The assumption of linearly elastic behavior of thevessel wall gives rise to an affine constitutive relationship between the cross-sectional surface of the vessel and the pressure distribution. The blood, as afluid, is regarded as an incompressible Newtonian fluid. We distinguish betweenthe global one-dimensional equations of motion and the local three-dimensionalequations of motion, describing the relation between flow characteristics like thevolumetric flow rate, the pressure distribution and the axial velocity. Of specialinterest is the order of relevance of the contributions of radial diffusion, i.e. theshear stress at the wall of the vessel, and nonlinear advection, which for the globalequations of motion are assumed to be of the forms as proposed by Bessemsin [1]. In contrast with most studies in this area, where periodic solutions arestudied, we focus on the transient solutions by imposing a single-pulse inlet flow.Moreover, the solutions obtained from the global and local equations of motion areas far as possible analytical and they are compared which each other, based onplots of the solutions for a set of representative arteries, both on quantitative andqualitative features.

A parameter of great relevance in this study is the dimensionless compli-ance (in this report denoted by δ), which equals the ratio of the characteristicspeed of the blood flow and the elastic wave speed of the wall (for flexural waves).This parameter is always less than one, and often much smaller than one. Forarteries having a small compliance, i.e. for a compliance much smaller than one,we observe that radial diffusion is of important influence. For arteries with largervalues of the compliance, close to but still less than one, we still observe the effectof the radial diffusion term, however the flow is now dominated by the effectsof the spatial and time derivatives of the volumetric flow rate and the pressuredistribution. We expect that the influence of the nonlinear advection term is almostalways small compared to the effect of all other terms in the equations of motion.However, we can not draw any conclusions about this statement based on theresults in this project, due to the asymptotic assumptions we make in deriving thesolutions for the nonlinear advection contribution, which lead to large variances inthe qualitative and quantitative aspects of the solution. This holds specifically ifthe compliance is not too small (i.e. close to one).

Technische Universiteit Eindhoven University of Technology

2 Mathematical modeling of blood flow through a deformable thin-walled vessel


“Just as the body cannot exist without blood, so the soul needsthe matchless and pure strength of faith”

Mahatma Gandhi, 1869-1948

Mathematical modeling of blood flow through a deformable thin-walled vessel 3


Dankwoord

Dit verslag is het resultaat van mijn afstudeerproject voor de opleiding Industrial and AppliedMathematics met specialisatie Computational Science and Engineering aan de Technische Uni-versiteit Eindhoven. Ik sluit een bijzondere studietijd af en hoop hiermee een goede basis teleggen voor de toekomst. Met veel plezier maak ik gebruik van de gelegenheid een aantalmensen te bedanken die een belangrijke rol hebben gespeeld in de totstandkoming van dit resul-taat.

Allereerst wil ik de afstudeercommissie, bestaande uit Fons van de Ven, Frans van de Vosse,Adrian Muntean (plaatsvervanger van leerstoelhouder Mark Peletier) en Arris Tijsseling, be-danken voor hun eindeloze wijsheid en de bereidwilligheid mijn afstudeerverslag te bestuderenen te beoordelen. Een bijzonder woord van dank gaat hierbij uit naar Fons van de Ven. Fons, ikheb genoten van de wijze waarop en het geduld waarmee jij mij hebt begeleid bij het tot standbrengen van dit eindresultaat. Ik kijk met genoegen terug op de vele gesprekken die we hebbengevoerd over dit afstudeerproject en de adviezen die je mij hierbij hebt gegeven. Natuurlijk denkik hierbij ook terug aan de gesprekken die we hebben gehad aan de bar bij GEWIS, vaak onderhet genot van een biertje, over de wijze waarop wij, ieder op onze eigen manier, genieten van hetleven. Nogmaals heel erg bedankt!

Uiteraard wil ik ook graag mijn studiegenoten bedanken. Een speciaal woord van dank is voorde mensen van dispuut GELIMBO, met wie ik tijdens mijn studie een bijzondere vriendschap hebopgebouwd en met wie ik heel veel bijzondere momenten heb mogen delen. Verder ben ik dankverschuldigd aan mijn vrienden van het “HBO-schakelprogramma”, de collega’s van Muziekge-bouw Frits Philips, mijn musicalvrienden (in het bijzonder de mensen van Moulin Rouge de Mu-sical) en iedereen die afgelopen tijd aan mij heeft gevraagd “Rob, ben je al klaar met afstuderen?”.

Tenslotte wil ik mijn familie bedanken. Zij hebben mij gevormd tot wie ik nu ben en tijdens mijnstudie hebben ze mij op alle fronten gesteund, mijn keuzes gerespecteerd en mij het vertrouwengegeven om het afstuderen tot een goed einde te brengen. In het bijzonder dank ik: Laura, voorhet “groene rakkertje”, die mij heel veel vervelende treinreizen naar Valkenburg heeft bespaard;Pap, voor de geweldige manier waarop je chef was van klusteam Euterpestraat; Mandy, voor alleopmerkingen die je er zonder bij na te denken uitgooit en waar ik altijd zo hard om moet lachen;en tenslotte Mam, vooral voor wie je bent, maar natuurlijk ook “veur ut doon van de wesj”. Julliezijn top!

Rob Ritzen16 augustus 2012



Contents

Nomenclature 9

1 Introduction 13

1.1 Background of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Problem approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Outline of the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.5 Results and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Biological background 17

2.1 The human circulatory system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 The heart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.2 The systemic circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 The vessel wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Blood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Mathematical model 25

3.1 Geometry of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Mechanical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 One-dimensional equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.5 Aims and proposed solution tactics . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 The global equations of motion 33

4.1 Scaling for a thin-walled vessel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 Basic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2.1 Solution for constant wave velocity . . . . . . . . . . . . . . . . . . . . . . . 35

4.2.2 General case for variable wave velocity . . . . . . . . . . . . . . . . . . . . 36



4.2.3 The shell equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 Inclusion of radial diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.4 Inclusion of advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.5 Preliminary conclusions from this chapter . . . . . . . . . . . . . . . . . . . . . . . 47

5 The local equations of motion 49

5.1 Problem set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.2 Relaxed solution for inlet flow in a rigid vessel . . . . . . . . . . . . . . . . . . . . . 53

5.3 Zeroth order solution for the axial velocity . . . . . . . . . . . . . . . . . . . . . . . 56

5.4 An approximative zeroth order solution . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.4.1 Derivation of the approximative flow characteristics . . . . . . . . . . . . . . 62

5.4.2 The accuracy of the approximative solution . . . . . . . . . . . . . . . . . . 64


6 Results and discussion 67

6.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.2 Comments on and discussion of the results . . . . . . . . . . . . . . . . . . . . . . 69


7 Conclusion and recommendation 81

7.1 Recapitulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.2 Main achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

7.3 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

A Linear elasticity theory for a thin-walled vessel 87

A.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

A.2 Derivation of the compliance C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

A.3 The shell equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

B Inverse Laplace transformation 95

B.1 Inverse Laplace transform (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

B.2 Inverse Laplace transform (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

C Fourier-Bessel series 101

C.1 Example (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

C.2 Example (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

D Global equations of motion with radial diffusion 103

List of Figures 109

List of Tables 110



Nomenclature

Greek symbols

α Womersley number.

δ Dimensionless compliance parameter (Mach number).

ε Quotient of Ra and L.

εi j Components of the deformation tensor (Appendix A).

0 Characteristic.

0a Local nonlinear advection contribution.

γa Global nonlinear advection contribution.

λk Positive zeros of J0.

µ Dynamic viscosity.

ν Kinematic viscosity. (Only in Section 4.2.3 and Appendix A ν

represents the Poisson ratio)

ρ Density.

σk Positive zeros of J2.

σi j Components of the stress tensor (Appendix A).

τw Shear force at the wall.

ω Pulsating (radial) blood frequency.

Roman symbols (1)

A Cross-sectional area.

A0 Undeformed cross-sectional area.

C Compliance.

c0 Unperturbed wave speed in thin-walled elastic vessel wall.

c Actual wave speed in thin-walled elastic vessel wall.

D Plate constant.

E Young’s modulus.



Roman symbols (2)

h Thickness of the vessel wall.

L Typical length scale in axial direction.

Lv Length of an arterie.

Mz Bending moment.

Nθ Normal force in axial direction.

Nθ Normal force in azimuthal direction.

n Outward normal vector.

P Characteristic pressure.

p Pressure.

p0 Reference pressure.

Q0 Maximum of the initial volumetric flow rate.

Qz Shear force.

q Volumetric flow rate.

qi Initial volumetric flow rate.

R Internal radius.

Ra Typical length scale in radial direction.

1R Relative change of internal radius.

Re Reynolds number.

r Radial coordinate.

Sr Strouhal number.

T Typical time scale.

t Time variable.

U Radial displacement of the vessel wall.

V Characteristic axial velocity.

v Velocity vector with components (vr , vθ , vz).

v Mean axial velocity.

w Alternative notation for vz .

x Position vector.

z Axial coordinate.



Miscellaneous

Ci Contour element.

D Domain.

H Heaviside function.

In Modified Bessel function of first kind of order n.

ı Imaginary unit.

Jn Bessel function of first kind of order n.

Kn Modified Bessel function of second kind of order n.

L Laplace transform.

L−1 Inverse Laplace transform.

O Asymptotic order symbol (“big-O”)..= Equality up to leading order.



Chapter 1Introduction

This report is entitled “Mathematical modeling of blood flow through a deformable vessel:An analytical and quantitative investigation of the advection and diffusion contribution inthe global and local equations of motion” and it is the result of a final project in orderto obtain a master’s degree in Industrial and Applied Mathematics. In this first chapterwe introduce the problem of the project in full detail and shortly recapitulate the resultsobtained.

1.1 Background of the problem

Fluid mechanics, one of the many specialisms in physics where mathematics plays an importantrole, finds its applications in many practical situations. A prominent example of a problem, comingfrom a subspecialism of this area of research called cardiovascular fluid mechanics, is the model-ing of blood flow through a deformable vessel. The wall of a blood vessel deforms under influenceof the local pressure and the shear stress at the wall. Therefore, this modeling project must becaptured under the name fluid-structure interactions. Since the shape of the wall is essentiallyunknown, this class of problems is more complex then the purely elastic or purely fluid ones. Thestudy of the flow of blood through an arterial vessel, or more specific, of the relationship of flowcharacteristics as pressure and flow wave forms with the geometrical and mechanical propertiesof the vessel, such as its radius, wall thickness, and elastic properties (stiffness/compliance), iscaptured by the name hemodynamics. The blood flow through the human circulatory system,consisting of arteries and veins, is driven by pressure waves generated by the heart. Changes inthe geometry of the arteries due to stresses on the vessel wall result in changes of pressure andflow-rate of the blood flow. Drastic changes in for instance the blood pressure can be indicationsfor diseases like aortic stenoses.

Often studies of the blood flow through the human circulatory system are based on physicalmodels in which more or less severe approximations are made. Such approximations can bebased on a chosen shape of the velocity profile in the vessel; see e.g. Bessems et al. [10], or onan empirical relation between the pressure gradient and the flow rate; see, e.g. Stergiopulos etal. [14]. Often these physical approaches lead to one-dimensional models relating the pressure,flow rate and cross-sectional area to each other; Stergiopulos [14], Bessems [10], [1]. Formally,these 1-D models are obtained by integration of the 3-D hydrodynamical (Navier-Stokes) equa-tions over the cross-section of the vessel, see [10]. Computer simulations are the common toolto obtain explicit numerical results, as done by Stergiopulos [14], Giannopapa [9], Van Geel [13],Olufsen et al. [6] and Bessems [1]. In most of these papers, periodic flow processes are consid-



ered. However, in many of these processes, not only the periodic part of the flow is relevant, butalso the transient one. For instance, in a so-called lock-exchange problem, in which, by suddenlyopening a valve, blood flows initially into an empty vessel, the transient problem is of main inter-est. Therefore, in this project we will focus on the transient flow, rather than the periodic flow.

We are convinced that an analytical mathematical model can be of great use to support the phys-ical modeling mentioned above. Therefore, we will try to solve the 3-D hydrodynamical equationsfor the flow in an elastically deformable vessel by as much as possible analytical means. For this,we will first construct a mathematical model, based on certain, physically motivated, assump-tions, and then try to solve the thus obtained system of partial differential equations and initialand boundary conditions in a consistent and accurate way. We will do this with great accuracy,but, on the other hand, we must realize that the desired accuracy to support the physical models,is not too high. This holds the more as the experimental accuracy in blood-flow measurements isat most of the order of 10 percent.

1.2 Problem description

The basic problem that we consider in this project is the sudden inlet flow of blood into a semi-infinite circular cylindrical deformable vessel. Blood is modeled as an incompressible Newtonianfluid. The vessel is a linearly elastic, thin-walled cylinder. The flow of blood is described bythe classical hydrodynamical equations: equation of continuity (or incompressibility condition),and equations of motion (Navier-Stokes equations). The wall is treated as a thin-walled, linearlyelastic tube under (space and time dependent) internal pressure. The interaction between fluid(blood) and solid (wall) is via the normal stress of the fluid on the wall. The problem is essentially3-D, but we assume it to be rotationally symmetric.

One of the main goals of this project is to estimate the order of magnitude of the diffusion andadvection terms in the equations of motion for the dynamical blood flow through the vessel andto indicate under what conditions these effects can, or can not, be neglected. We distinguishbetween two different types of equations of motion:

1. LOCAL 3-D EQUATIONS OF MOTION. A set of partial differential equations, based on theclassical Navier-Stokes equations, describing the relation between the velocity vector andthe pressure gradient.

2. GLOBAL 1-D EQUATIONS OF MOTION. A set of partial differential equations describingthe relationship between the flow rate, the cross-sectional area and the pressure gradient.These equations are the integrated version of the local equations of motion.

We will solve the global equations of motion for three cases: (i) we neglect both the advectionand viscous diffusion term, (ii) we neglect the advection term, (iii) we neglect the viscous diffusionterm. The local equations of motion will be solved up to leading order of the dimensionlesscompliance parameter, which is equal to the ratio of the characteristic flow velocity of the bloodand the elastic wave speed of the vessel wall. We will try to find as much as possible fullyanalytical solutions for all these cases. This will finally result in a classification of the order ofrelevance of the diffusion and advection for a range of representative human arteries.

1.3 Problem approach

To answer the questions in the previous section, we construct a (simplified) model and we tryto solve this with mathematical tools. We model the flow of blood through an arterie as the flow



through a semi-infinite vessel and we assume the flow to be rotationally symmetric. For thevessel we assume a state of plane strain (no axial elongation). Moreover, we take the vessel"thin-walled", meaning that the thickness of the vessel wall is much smaller than the internal ra-dius of the vessel. We describe the vessel wall as a linearly elastic medium. Thus, we considerthe vessel wall as a linearly elastic thin-walled tube, and this enables us to derive a constitutiveequation for the radial displacement of the wall in terms of the blood pressure. On top of this, weassume here, following Bessems [1], that also the wavelength of the deformations of the vesselis much larger than the internal radius of the vessel.

To describe the flow of a fluid through a vessel we need a balance of mass (equation of continuity)and a balance of momentum (equation of motion). We assume the blood to be an incompress-ible Newtonian fluid. With this assumption, the balance of mass and the balance of momentumturn into the well-known Navier-Stokes equations as the governing equations of motion and, withsome dimensional analysis, we arrive at a dimensionless system of equations describing the re-lation between the pressure gradient, the volumetric flow rate and the velocity vector. We referto this set of equations as the local 3-D equations of motion. The global 1-D equations of motioncan be obtained by integrating the local ones over the cross-section of the vessel.

Next, we try to solve both the local and global equations of motion. We focus on transient solu-tions, which means that the (time-dependent) solution is depending on the initial state of the flow,and thus depends on the initial conditions; this in contrast to periodic solutions. To this end, wewill prescribe a single pulse inlet-flow function as initial boundary condition. Furthermore, we tryto solve the equations analytically as far as possible.

In solving the global equations of motion we distinguish three cases: (i) we neglect both the ad-vection and viscous diffusion term, (ii) we neglect the advection term, (iii) we neglect the viscousdiffusion term. For the first case the system of equations of motion reduces to a wave-equation-like system. For this reason, it seems reasonable to use the method of characteristics to solvethe system. As a basis for investigating the effect in cases (ii) and (iii), we follow Bessems etal. [10] in using an expression for both the advection and the diffusion term in the equations ofmotion. Another way to solve these kind of problems, is to use Laplace transformations. We optfor this method in solving case (ii). It will turn out that the characteristics found in (i) also pop uphere. In case (iii), we switch back to characteristics. However, because of the non-linearity in thepressure distribution, we have to use an asymptotic expression for the pressure function and wesolve the system up to leading order.

To solve the local equations of motion we introduce asymptotic expansions for the pressure-,flow rate- and velocity-function. This enables us to solve the system up to leading order, againcompletely analytically. To this end, we assume a so-called Fourier-Bessel series for the velocity-function and we use the method of Laplace transformations to solve the system. At the end, wewill compare the solutions found here with the ones found for the global equations of motion, andwe hope to find some similarity.

1.4 Outline of the report

As basis structure for this report we follow the classification as described in the problem descrip-tion (Section 1.2). We started with a general introduction, in which we stated the formulation andthe goal of this project and in which we shortly introduced the aspects related to the problem, thatwill be discussed in the core chapters of this report. In the second chapter a short introduction ofthe main biological aspects, related to the problem, is presented. We start with a description ofthe human circulation system, which is the complex network of the heart, arteries and veins andwhich transports blood through the body. This is followed by a section about the structure of a



vessel wall and a physiological description of blood as a fluid.

In the third chapter the equations of motion for the flow through a tube will be formulated andused as a model for blood flow through an artery or a vein. Starting from the full Navier-Stokesequations as a description of the flow of a fluid, we arrive via a dimensional analysis based on aset of data from the real human circulatory system at two nonlinear partial differential equationsdepending on the flow velocity, the blood pressure, the flow rate and some flow characteristicparameters. We will refer to these equations as the local equations of motion. To arrive at theso-called global equations of motion, we integrate the local equations of motions over the cross-section of vessel.

In Chapter 4 we try to find analytical solutions for the global equations of motion, where wedistinguish the three different cases as denoted with (i)-(iii) in Section 1.2. In Chapter 5 we in-troduce asymptotic expansions for the flow quantities and solve the local equations of motion upto leading order of the dimensionless compliance parameter. With the solution obtained here weestimate flow parameters for the advective flow and the diffusive flow and compare them with theparameters used for the global solutions in Chapter 4.

Numerical results from Chapters 4 and 5 will be presented graphically in Chapter 6. To thisend, we have to choose an expression for the inlet flow function. The parameters, based onreal measurements and needed to make the plots, are borrowed from literature and tabled in thebeginning of Chapter 6. Finally, in Chapter 7 we recapitulate all results from the previous threechapters and combine them to come to a conclusion and preferably also to a classification for therelevance of the advection and diffusion contribution in some representative human arteries.

Some notes on the elastic behavior of the vessel wall can be found in Appendix A. The otherappendices mainly contain extensive computations to support the subresults of the core chap-ters.

1.5 Results and conclusions

From the expressions obtained for the volumetric flow rate, the pressure distribution and the axialflow velocity, we made plots for the different cases studied for the global equations of motionand for the zeroth order approximations obtained for the local equations of motion. Here, wechoose a specific expression for the inlet flow function and we opt for three representative humanarteries and corresponding parameters on which the plots are based. Next to this, we listed thepercentage differences between the solutions from the global equations of motion and the localones. We discussed in detail the plots and the table and based on this discussion we drew ourconclusions.

Summarizing, we state that, whenever the dimensionless compliance parameter is “large”, i.e.still less than, but close to one, the effect of the shear stress at the wall is noticeable, specificallyin the local 3-D model, however the flow is mainly dominated by the time and spatial derivativesof the volumetric flow function and the pressure distribution. For small values of the dimension-less compliance parameter, the radial diffusion term is of significant order. Although we stronglybelieve that the influence of the advection term is almost always negligible with respect to theother terms in the equations of motion in all cases, we can not conclude this from the resultsobtained. Here, the neglect of higher order terms of the dimensionless compliance parametercauses large fluctuations in the amplitude and large phase differences for the solution from theglobal equations of motion.



Chapter 2Biological background

As we are going to study the flow of blood through a vessel, it is important to get insightinto the biological structure of the blood flow through the human body. Therefore weprovide some biological background of the problem in this chapter. First the system ofarteries and veins, through which the blood will flow, will be described. Secondly, bloodas a fluid will be described. Finally, since we are dealing with a fluid-solid interactionproblem, we describe the mechanical properties of a vessel wall. The text of this chapteris mainly based on [3] and [12]. The figures are taken from [5].

2.1 The human circulatory system

The human circulatory system enables the blood to flow through the whole body and to transportall kinds of substances over large distances via a complex network of arteries and veins. Thehuman circulation system, as well as the circulation system for other vertebrates, can be splitinto two main separated parts, the pulmonary circulation and the systemic circulation. In thepulmonary circulation deoxygenated blood will be pumped from the right part of the heart via thepulmonary arterie to the lungs. In the lungs the red blood cells will react with the oxygen and allwaste materials will be transported to the air. Via the pulmonary veins the oxygenated blood willgo back to the left part of the heart. In the systemic circulation oxygenated blood is transportedaway from the heart to the rest of the body, where oxygen is taken up by the organs. The deoxy-genated blood is then returned to the right atrium of the heart via the inferior and superior venacava.

For a schematic representation of the human circulatory system see Figure 2.1 and remark thatleft and right are taken with respect to the body itself (so opposite in the figure).

2.1.1 The heart

The heart is situated in the chest of the human body and consists of two parts separated by aninternal wall. Each of these two parts has an atrium and a ventricle , so the heart has four cham-bers in total. Deoxygenated blood from the body enters the right atrium via the inferior vena cavaand the superior vena cava. From the right atrium the blood flows to the right ventricle where it willbe pumped into the pulmonary arterie. Now the pulmonary circulation starts. After this circulationoxygenated blood reenters the heart via the pulmonary veins into the left atrium. From the leftatrium the blood flows via the left ventricle into the aorta en starts the systemic circulation. See



Figure 2.1: Schematic representation of the human circulatory system. Oxygenated and deoxy-genated blood is transported trough the red and blue part of the circulatory system, respectively.



Figure 2.2: Schematic representation of the human heart (unfolded).

Figure 2.2 for a schematic representation of the heart.

Both the right and left atrium and the right and left ventricle are separated by the atrioventricular(tricuspid or mitral) valve. At the beginning of the pulmonary artery as well as at the beginningof the aorta there are seminular valves in order to regulate the inflow of blood into the pulmonaryand systemic circulation.

The heart can be regarded as a hollow muscle. Along the wall of the heart blood is transportedvia the coronary arteries and the coronary veins in order to transport oxygen to the heart muscletissue and expel waste materials from the heart muscle tissue. The coronary veins end directlyinto the right ventricle.

Three different main stages can be distinguished in the function of the heart during a cycle.The contraction and the relaxation of the cardiac muscles is called systole and diastole, respec-tively. During the first phase of the cycle, blood is pumped from the atrium into the ventricle underinfluence of the pressure increase in the atrium due to the systole of the atrium. The ventricleis now completely relaxed. Next, a contraction in the ventricle takes place. The pressure in theventricle increases until it exceeds the pressure in the aorta. Bloods now flows directly into thesystemic circulation. Finally the pressure in both the ventricle and the aorta drops down and alsothe ventricle is in diastole. These three stages are represented in Figure 2.3. Clearly, the phasesare not equal in length.

2.1.2 The systemic circulation

We can distinguish three different parts in the systemic circulation, namely the arterial system,the capillary system and the venous system.

The arterial system transports oxygenated blood from the left ventricle into the systemic circula-tion towards tissues. This starts with transport via arteries . One of the most important arteriesin the arterial system is the aorta. This main artery is able to buffer blood by use of internalvalves to regulate the pulsating blood flow. The blood pressure in the arterial system is relativelyhigh. In general, the arteries are named after the tissue they are leading to. The pressure in thearterial system can be kept at high level because at its end the arteries strongly divide into two



Figure 2.3: The heart cycle for a heart having rate frequency of 75 beats per minute.

or more smaller branches and so increase the total resistance to the blood. These branches arecalled small arteries or arterioles. The smooth muscle cells (see Section 2.2) in the wall of thesearterioles are able to change the diameter of the vessel and, while doing so, meeting the needsof the specific part of the body. In the arterial system, viscous forces play an important role in theblood flow since there are significant variations in the characteristic velocities and length scalesof the arteries and arterioles.

From the arterioles blood flows into the capillary system. This is a network of very small vesselssituated around a certain tissue. The diameter of these vessels is even that small, that modelingthe blood as a homogenous fluid is not possible and so differs strongly from modeling the bloodflow in the arterial and the venous system. In the capillary system, oxygen is exchanged from thered blood cells to the tissue with all other kinds of nutrients and waste materials. The blood flowin the capillary system is also called micro circulation.

Finally, in the venous system deoxygenated blood is transported back from the capillary sys-tem to the right atrium. The blood from the capillaries is collected in the small veins, also calledvenules. These venules merge into veins, which are generally called after the tissue they arecoming from. The blood flows via the vena cava back to the heart. The vena vaca is, just likethe aorta, able to store blood and so enables the heart to regulate the flow into the arteries. Thetotal volume of the venous system is much larger than the volume of the arterial system andconsequently the characteristic velocity of the blood is lower than the one in the arterial system.Moreover, the pressure in the venous system is that low that gravitational forces can play a rolein the flow of the blood.

The changes in blood pressure and velocity due to changes in characteristic lengths (diameter)of the different types of vessels are graphically shown in Figure 2.4.

2.2 The vessel wall

The vessel wall plays an important role in the pressure distribution of the blood in the vessel,since the vessel wall deforms under influence of the internal pressure. The deformation of thevessel wall is determined by the mechanical properties of the vessel wall.

We distinguish three different layers in the vessel wall (see Figure 2.5). The first one is the



Figure 2.4: Changes of the blood pressure and velocity due to changes in characteristic lengthsof the vessel.

Figure 2.5: Schematic representation of an artery wall, a vein wall and a capillary wall.



innermost layer, which is called the tunica intimal. The tunica intimal is divided into three differentstructures. The endothelial cells cover the innersurface of the vessel wall and are therefore indirect contact with the blood that flows through the vessel. These endothelial cells play an impor-tant role in the growth and regeneration of the vessel. The internal elastic membrane consistsof collagenous bundles and elastic fibrils. These elastic fibrils are build of elastin, which is abiological material that provides the elastic behavior of the vessel wall. The collagenous bundlesare build of collagen, a protein that gives strength to the vessel wall and which in turn is build ofthree chains of amino-acids. Since this membrane is relatively small, it does not have significantinfluence on the mechanical properties of the vessel wall. The endothelial cells and the internalelastic membrane are separated by a connective tissue.

The second layer of the vessel wall is the middle one, called the tunica media. The tunica mediais the thickest one of the three layers and consists of two typical structures, namely elastic fibrilsand involuntary muscle fibrils, or also called smooth muscle cells. Especially for small arteries,the smooth muscle cells are dominating the mechanical properties of the vessel wall and are ableto regulate the local blood flow. The smooth muscle cells are placed in the large network of elasticfibrils. This structure gives the vessel wall strength and elasticity. In large arteries, such as theaorta, the smooth muscle cells are almost absent and the tunica media mainly consists of elasticfibrils. Therefore, these kind of vessels are often referred to as elastic arteries. The opposite istrue for arteries situated towards the periphery, very often referred to as muscular arteries.

Finally, the last and outermost layer is the tunica adventitia. The thickness of this layer candiffer from one to the other artery. It consist of two different structures, as can be seen in Figure2.5. The external elastic membrane consist just as the internal elastic membrane of collagenousbundles and elastic fibrils. The connective tissue serves to connect the vessel to the surroundingtissue.

As becomes clear from the above classification, the geometry of a vessel wall is very complex.The elasticity of the vessel wall is determined by all constituents of the wall and is representedby the so-called Young’s modulus (see Table 2.1). This parameter clearly varies a lot. In the restof this report we will assume the vessel wall to be thin, which means that the ratio between thediameter of the vessel and the thickness of the vessel wall is small. In Appendix A we will showthat this assumption enables us to regard the vessel wall as a homogeneous medium and to uselinear elasticity theory in modeling the behavior of the vessel wall. In this case we use an overalland commonly accepted Young’s modulus equal to 0.5 · 106Pa.

Constituent Young’s modulus [MPa]

Elastin ±0.5

Collagen ±0.5 · 103 (stretched)

Smooth muscle cells ±0.1 (relaxed)

±2.0 (activated)

Table 2.1: The Young’s modulus for the three main constituents of the vessel wall.

2.3 Blood

Blood takes care of the transport of, amongst others, oxygen en waste products in the humanbody. It consists of a fluid called blood plasma and some solid substances. These substancescan be divided into platelets, red and white blood cells.



Figure 2.6: Schematic representation of the main components of blood.

The blood plasma, which constitutes approximately 55% of the blood, consists of water and dis-solved proteins. The plasma transports many substances such as glucose, vitamins, oxygen,minerals, carbon dioxide and of course platelets and blood cells. Some of these substancescan dissolve easily into the plasma (for example glucose), while others need to be coupled to aplasma-protein to dissolve into the blood plasma. Blood plasma plays an important role in reg-ulating the internal human body climate via a homeostatic regulation system. On average, theplasma has a constant temperature of 38 ◦C and a pH-value of about 7.4.

The platelets and the red and white blood cells arise from a blood stem cell. These stem cellscan be found mainly in the red bone marrow of for example the shoulder bones and ribs.

The red blood cells (erythrocytes) look like small circular plates that are less thick towards thecenter of the plate. These cells only live for 4 months. Consequently, the red bone marrow pro-duces red blood cells continuously (approximately 2 million cells per second). The red blood cellscontain hemoglobin, which is essential in the transport of oxygen and carbon dioxide and givesthe blood its red color. People who suffer from anemia have an insufficient level of hemoglobin,probably because they have a shortage of iron minerals in their nutrition.

The white blood cells (leukocytes) can change their form thus enabling them to diffuse through avessel wall. The red bone marrow produces three types of white blood cells: lymphocytes, gran-ulocytes and monocytes. Lymphocytes can make antibodies against different types of foreignmaterial, granulocytes are able to encapsulate bacteria, while monocytes have the same functionas granulocytes, but these cells can also remove dead cells after an infection. People who sufferfrom leukemia have an overdosis of white blood cells because of a malfunction of the stem cells.



Finally, the platelets (thrombocytes) are parts of fragmented cells. The platelets play an importantrole in blood clotting. For example, they can react on substances coming from an injured vesselwall. This reaction leads to a mesh of fibrin onto which red blood cells collect and clot. Peoplewho suffer from hemophilia have a genetic disorder in the level of platelets. The composition ofblood is summarized in Figure 2.6.

In the main body of this report we will regard the blood as a whole (plasma and cells) and modelit as an incompressible Newtonian fluid. The material parameters of this fluid are given in Table2.2.

Parameter Notation Unit Value

Density ρ kg/m3 1.05 · 103

Kinematic viscosity ν m2/s 3.5 · 10−6

Dynamic viscosity η kg/(m · s) 3.675 · 10−3

Table 2.2: Values for the material parameters of blood, regarded as a Newtonian fluid.



Chapter 3Mathematical model

In this chapter the flow of blood through a deformable (elastic) thin-walled vessel is in-vestigated. As in many (bio)mechanical problems it is necessary to make simplifyingassumptions and to find the equations of motion in order to analyse the different types ofbehavior in a given configuration. In this chapter the equations of motion for the bloodflow and the relevant boundary conditions are formulated. The equations are scaled in or-der to estimate the relative relevance of the different terms in these equations. This leadsto some, generally valid, simplification of the system of equations. Finally, the local 3-Dequations are integrated over the cross-section to obtain the global 1-D equations.

3.1 Geometry of the model

For describing the blood flow through a (deformable or corrugated) vessel a cylindrical coordinatesystem (r, θ, z) is introduced. A schematic representation of the configuration is given in Figure3.1. We assume the flow to be rotationally symmetric, i.e. vθ = 0 and the flow variables areindependent of θ . This leads us to the following set of unknowns:

vr = vr (r, z, t), vz = vz(r, z, t), p = p(r, z, t), (3.1)

for 0 ≤ r ≤ R(z, t), t > 0 and z ∈ [0,∞). Here, vr is the radial velocity, vz the axial velocity, p thepressure and R the current radius of the vessel at time t and axial position z. If Ra is the typicallength scale in radial direction and L the typical length scale in axial direction, then we assumeRa to be much smaller than L. For later purposes we define the parameter ε to be equal to Ra/L.This means that ε � 1.

zr

θ

R

Figure 3.1: Schematic representation of a thin-walled vessel



3.2 Mechanical model

The flow of a fluid is described by the equation of continuity and the momentum equation. Thefluid (i.e. blood) is modeled here as an incompressible Newtonian fluid. Using the correspondingconstitutive equation in the momentum equation, or equation of motion, this equation becomesthe Navier-Stokes equation. These equations reduce for our rotationally symmetric configurationto:

1r∂

∂r(rvr )+ ∂vz

∂z= 0, (3.2)

∂vr

∂t+ vr

∂vr

∂r+ vz

∂vr

∂z= − 1

ρ

∂p∂r+ ν

[1r∂

∂r

(r∂vr

∂r

)+ ∂

2vr

∂z2 −vr

r2

], (3.3)

∂vz

∂t+ vr

∂vz

∂r+ vz

∂vz

∂z= − 1

ρ

∂p∂z+ ν

[1r∂

∂r

(r∂vz

∂r

)+ ∂

2vz

∂z2

]. (3.4)

Equation (3.2) states the conservation of mass (equation of continuity) for an incompressible fluid,and (3.3) and (3.4) represent the r - and z-component of the momentum equation, respectively.

3.2.1 Boundary conditions

Two boundary conditions at the wall r = R(z, t) are given by

(v, t) = 0, (v,n) = ∂R∂t, (3.5)

where t is the axial tangential vector and n the outward normal vector to the wall. If we define γto be the angle between the radial direction and the normal vector to the wall, then n and t canbe expressed as

n = sin(γ )ez + cos(γ )er , t = cos(γ )ez − sin(γ )er . (3.6)

Substitution of (3.6) into (3.5) yields

(v, t) = vz cos(γ )− vr sin(γ ) = 0, (3.7)

(v,n) = vz sin(γ )+ vr cos(γ ) = ∂R∂t. (3.8)

From (3.7) and (3.8) we find

vr = ∂R∂t

cos(γ ), vz = ∂R∂t

sin(γ ). (3.9)

Since

tan(γ ) = ∂R∂z= O

(Ra

L

)= O(ε)� 1, (3.10)

by assumption, we have γ = O(ε) and then, neglecting all O(ε)-effects, we obtain

vr (R, z, t) = ∂R∂t, vz(R, z, t) = 0. (3.11)

These are the boundary conditions we shall use in the sequel.



3.3 Scaling

Let us introduce the following dimensionless variables:

r := rRa, z := z

L, t := t

T, (3.12)

where Ra , the typical length scale in radial direction is the inner radius of the undeformed vessel,L the typical length scale in z-direction and T a typical time scale depending on the frequencyω of the pulsating blood flow through T = ω−1. Also we define the dimensionless velocities andpressure by

vr := vr

Vr, vz := vz

Vz, p := p

Pc, (3.13)

with Vr , Vz and Pc specified further on. Introducing the dimensionless variables into (3.2), weobtain

1r R2

a

∂

∂ r(r Ra Vr vr )+ Vz

L∂vz

∂ z= 0. (3.14)

To make the two terms in (3.14) of the same order of magnitude we take (LVr/Ra Vz) = 1, whichleads to

Vz = V, Vr = Ra

LVz = εV, (3.15)

or

vr = εV vr , vz = V vz, (3.16)

and so (3.2) becomes in dimensionless form

1r∂

∂ r(r vr )+ ∂vz

∂ z= 0. (3.17)

Here, V is the characteristic speed for the axial flow through the vessel (e.g. the averagedaxial velocity). Substituting the dimensionless variables into (3.3) and (3.4) and multiplying by(R2

a/εV ν) and (R2a/V ν), respectively, we arrive at

R2a

T ν∂vr

∂ t+ RaεV

νvr∂vr

∂ r+ Ra V

νvz∂vr

∂ z= − 1

ρ

Ra Pc

ενV∂ p∂ r+ 1

r∂

∂ r

(r∂vr

∂ r

)+ ε2 ∂

2vr

∂ z2 −vr

r2 , (3.18)

R2a

νT∂vz

∂ t+ εV Ra

νvr∂vz

∂ r+ εV Ra

νvz∂vz

∂ z= − 1

ρ

R2a Pc

LV ν∂ p∂ z+ 1

r∂

∂ r

(r∂vz

∂ r

)+ ε2 ∂

2vz

∂ z2 . (3.19)

We introduce the following two dimensionless numbers:

Re = Ra Vν, α = Ra

√1

T ν= Ra

√ω

ν, (3.20)

referred to as the Reynolds number and the Womersley number, respectively.

We now have two possibilities to scale the pressure (gradient):

1. We assume that the axial pressure gradient scales with the axial viscous term, suggestingthat (R2

a Pc/ρνV L) should be chosen equal to 1, yielding

Pc = P(v)v =ρνV L

R2a

⇒ p = ρνV LR2

ap. (3.21)

The justification for this choice is the requirement that the advection terms in (3.19) remainof O(1), or εV Ra/ν = V L/ν = O(1), for ε � 1.



2. We assume that the axial pressure gradient scales with the axial advection term, suggestingthat (εV Ra/ν) = (R2

a Pc/ρνV L) should be chosen, yielding

Pc = P(c)c = ρV 2 ⇒ p = ρV 2 ˆp. (3.22)

The justification for this choice is the requirement that, after dividing (3.19) by εV Ra/ν, theviscous term remains of O(1), or ν/εV Ra = ν/V L = O(1), for ε � 1.

Hence, the choice for the two options 1. and 2. is determined by the order of magnitude of εRe.Here, and in first instance, we opt for the first choice. With the use of p and the two dimensionlessnumbers Re and α, the equations of motion become

1r∂

∂ r(r vr )+ ∂vz

∂ z= 0, (3.23)

ε2α2 ∂vr

∂ t+ ε3Revr

∂vr

∂ r+ ε2Revz

∂vr

∂ z= −∂ p

∂ r+ ε2 1

r∂

∂ r

(r∂vr

∂ r

)+ ε4 ∂

2vr

∂ z2 − ε2 vr

r2 , (3.24)

α2 ∂vz

∂ t+ εRevr

∂vz

∂ r+ εRevz

∂vz

∂ z= −∂ p

∂ z+ 1

r∂

∂ r

(r∂vz

∂ r

)+ ε2 ∂

2vz

∂ z2 . (3.25)

As said before, option 1. is justified if εRe = O(1). However, for larger arteries, e.g. the aorta,εRe� 1, and then we have to use option 2. In that case (3.25) becomes

Sr∂vz

∂ t+ vr

∂vz

∂ r+ vz

∂vz

∂ z= −∂

ˆp∂ z+ 1εRe

[1r∂

∂ r

(r∂vz

∂ r

)+ ε2 ∂

2vz

∂ z2

], (3.26)

where Sr is the Strouhal number defined as

Sr = α2

εRe= ωL

V, (3.27)

which is always O(1), for ε � 1, in all our applications. Considering (3.24), we note that ifεRe = O(1) (option 1) and α = O(1), then ∂ p/∂ r = O(ε); this holds for not too large arteries. Onthe other hand, for larger arteries such as the aorta, εRe becomes O(ε−1), and also α = O(ε−1)

then. In that case we have to apply option 2, and then again we conclude that ∂p/∂r = O(ε) (notefor instance that ε2α2 → ε2Sr = O(ε2)). Hence, for all types of arteries we have

∂ p∂ r= O

(ε2)= 0, (3.28)

for ε→ 0.

Equation (3.23) is not influenced at all by the scaling, so it keeps its original form. Finally (3.25)shows different behavior for either small or larger arteries: for the small ones the advective termscan be neglected (diffusion dominated flow ), whereas for the larger ones the viscous term isnegligible (advection dominated flow ). To keep our analysis general (for large and small arter-ies), we neglect neither the advection nor the viscous term. However, we see that the contributionof the term ∂2vz/∂z2 in the viscous term is always O(ε2), compared to the radial diffusion termand therefore we will consistently neglect this term, unless explicitly stated otherwise.

Summarizing, accounting for the results of our scaling, we arrive at the following reduced, again



dimensional, system of equations, as we will use it in the rest of this report,

1r∂

∂r(rvr )+ ∂vz

∂z= 0, (3.29)

∂p∂r= 0 ⇒ p = p(z, t), (3.30)

∂vz

∂t+ vr

∂vz

∂r+ vz

∂vz

∂z= − 1

ρ

∂p∂z+ ν

r∂

∂r

(r∂vz

∂r

). (3.31)

To conclude this section, we take a short look at the boundary conditions, especially (3.11) for vr ,which with vr = εV vr , R = Ra R and t = T t becomes

vr = Ra

εV T∂ R∂ t= L

V T∂ R∂ t, (3.32)

for r = R. This would suggest a time scale T = L/V , while up to here we have used T =ω−1. However, at this point there seems to be no need to change this time scale, because theirappears to be never a ‘large order of magnitude’ difference between these two time scales, sinceSr = ωL/V is always strictly of O(1), for ε � 1.

3.4 One-dimensional equations of motion

In the previous section the local equations of motion were presented. In this section the globalversion of these equations is derived. To do so, the equations are integrated over the cross-sectional surface, resulting in equations only depending on z and t .

1. For the equation of continuity:

∫ 2π

0

∫ R(z,t)

0

{1r∂

∂r(rvr )+ ∂vz

∂z

}r dr dθ =

2π∫ R(z,t)

0

{1r∂

∂r(rvr )+ ∂vz

∂z

}r dr =

2π[rvr]R(z,t)

0 + 2π∂

∂z

∫ R(z,t)

0rvz dr − 2π

[rvz

∂R∂z

]

R(z,t)=

π∂

∂t(R2(z, t))+ ∂

∂z(A(z, t)vz(z, t)) =

∂

∂t(A(z, t))+ ∂

∂z(A(z, t)vz(z, t)) = 0, (3.33)

where in the third step we have used the boundary conditions (3.11). Here, A = πR2 is thecross-sectional area at (z, t) and vz the mean velocity at (z, t), defined as

vz(z, t) = 2πA(z, t)

∫ R(z,t)

0rvz(r, z, t) dr. (3.34)



2. For the left-hand side of the z-component of the Navier-Stokes equation:

2π∫ R(z,t)

0

{∂vz

∂t+ vr

∂vz

∂r+ vz

∂vz

∂z

}r dr =

2π∫ R(z,t)

0

{r∂vz

∂t+ ∂

∂r(rvrvz)− vz

∂

∂r(rvr )+ rvz

∂vz

∂z

}dr =

2π∫ R(z,t)

0

{r∂vz

∂t+ ∂

∂r(rvrvz)+ 2rvz

∂vz

∂z

}dr =

2π

{∂

∂t

∫ R(z,t)

0rvz dr −

[rvz

∂R∂t

]

R(z,t)

}+ 2π

[rvzvr

]R(z,t)0

+ 2π

{∂

∂z

∫ R(z,t)

0rv2

z dr −[

rv2z∂R∂z

]

R(z,t)

}=

∂

∂t(A(z, t)vz(z, t))+ ∂γa

∂z, (3.35)

where in the second step the equation of continuity (3.2) is used. Here, following [10], wehave introduced γ by

γa = γa(z, t) = 2π∫ R(z,t)

0rv2

z (r, z, t) dr. (3.36)

Hence, γa represents the contribution of the nonlinear advection term.

For the right-hand side of the z-component of the Navier-Stokes equation:

2π∫ R(z,t)

0

{− 1ρ

∂p∂z+ ν 1

r∂

∂r

(r∂vz

∂r

)}r dr =

− πρ

∂p∂z

R2(z, t)+ 2πν[

r∂vz

∂r

]R(z,t)

0=

− 1ρ

∂p∂z

A(z, t)− τw(z, t), (3.37)

where

τw(z, t) = −2πνR(z, t)∂vz

∂r

∣∣∣∣R(z,t)

, (3.38)

the shear force per unit of length on the fluid at the wall of the vessel. Just as the nonlinearadvection term in (3.35), the shear term τw can not directly be expressed in terms of theglobal quantities p(z, t) and A(z, t), because for this we need to know the precise distributionof vz(r, z, t) over the cross-section of the flow. We come back to this subject in Chapter 5 ofthis report.

Before recapitulating the two 1-D, or global, equations of motion, we introduce the volumetric flowrate q(z, t) through the vessel by

q(z, t) = A(z, t)v(z, t) = 2π∫ R(z,t)

0rvz(r, z, t) dr. (3.39)



Then, the equations become

∂A∂t+ ∂q∂z= 0, (3.40)

∂q∂t+ ∂γa

∂z+ Aρ

∂p∂z+ τw = 0. (3.41)

We note that, assuming for a moment γa and τw known, this system constitutes of two equationsfor three unknowns A, p and q. What we still need is a constitutive equation expressing A interms of p and representing the elasticity of the vessel wall. This relation is derived in AppendixA and describes the linear elastic behavior of the thin-walled vessel wall, and reads

A(z, t) = A0 + C(p(z, t)− p0), (3.42)

where A0 = πR2a , the area of the undeformed cross section of the vessel and C the compliance

of the wall. For the deformed radius of the wall we then obtain

R(z, t) =√

1π

A(z, t) =√

R2a +

Cπ(p(z, t)− p0)

.= Ra

(1+ C

2πR2a(p(z, t)− p0)

), (3.43)

where the latter step is in accordance with linear elasticity (here, .= means linearization).

In Chapter 4 we will describe and define (the geometry of) the problem in full detail, such that wecan pose a set of boundary and initial conditions which together with the equations of motion, asgiven in (3.40) and (3.41), form a well-posed mathematical description of our problem.

3.5 Aims and proposed solution tactics

The complexity of both the local 3-D system (3.29)-(3.31) and the global 1-D system (3.40)-(3.41)reduces drastically if both radial diffusion (represented by τw) and advection (represented by γa)are neglected. However, in practical situations this is almost never allowed. On the other hand,in many cases one of the two effects may be discarded. For instance, when the radial diffusionis important (as in smaller arteries) advection is negligible and oppositely (for larger arteries). Itis seldom that both effects are equally relevant. Axial diffusion is almost always negligible. In theremaining of this report we will try to estimate the order of magnitude of the two effects; radialdiffusion and advection, and to indicate under what conditions these effects can, or can not, beneglected.

In the next chapter, we will analyse the global 1-D system. We will solve the three cases: (i)τw = γa = 0, (ii) τw 6= 0, γa = 0 (iii) τw = 0, γa 6= 0. We will try to find as much as possible fullyanalytical solutions for these cases.

In Chapter 5, we will analyse the local 3-D system. Here, we will employ asymptotic expan-sions for the flow characteristics in order to find a leading order solution for the axial flow velocity.Our aim is to find expressions for τw and γa , which we then, at its turn, can use in the analytical1-D results of Chapter 4. All this will finally result in a classification of the order of relevance ofthe diffusion and advection for a range of typical human arteries.



Chapter 4The global equations of motion

In the previous chapter we derived a general mathematical model for blood flow througha deformable vessel and we distinguished two different types of equations of motion,namely the global and the local ones. The geometry of the problem is the same assketched in Section 3.1, in so far that we here consider a semi-infinite vessel (z ≥ 0).At the entrance of the vessel, the inlet flow rate is prescribed, starting at t = 0. We arespecially interested in the transient part of the solution. From now on we focus on thin-walled vessels only, i.e. vessels for which the ratio of wall thickness and radius is small.We investigate separately the effects of the viscous diffusion term and the advection in theglobal equations of motion for a thin-walled vessel. In order to estimate the importance ofboth effects, we start from a simple example where the effects are absent. As far as pos-sible, we try to find analytical solutions of the governing equations of motion. The resultsof this chapter will graphically be presented in Chapter 6 where we impose a certain giveninlet flow.

4.1 Scaling for a thin-walled vessel

The specific problem we consider in this chapter is a suddenly started inflow of a given amountof fluid into a thin-walled deformable semi-infinite vessel, which is initially already filled with fluid.The flow into the vessel starts at t = 0. So, for t < 0, the flow rate is zero, the pressure is equal tothe reference pressure p0 and the vessel is still undeformed with area A0. The flow rate throughthe initial cross-section at z = 0, is prescribed and given by qi (t). For finite times, the pressuregradient goes to zero for z going to infinity. We consider here a given inlet flow rate of pulse-type(so not periodic), only unequal to zero during a finite period of time after t = 0. We are especiallyinterested in the transient behavior of the flow. This is in contrast to most of literature in whichperiodic behavior is studied.

We start this chapter with a rigorous scaling analysis of the global equations of motion, as statedin (3.40) and (3.41), adapted for the case of a linearly elastic thin-walled vessel. For these kindof vessels we can assume that the cross-sectional area A of the deformed vessel is an affinefunction of the pressure p, described by the constitutive equation

A(z, t) = A0 + C(p(z, t)− p0), (4.1)

where A0 is the area in the undeformed reference state, when p = p0, and C > 0 is the compli-ance of the wall. This relation, the derivation of which is given in Appendix A, characterizes the



linear elastic behavior of the thin-walled vessel wall. Substituting (4.1) into (3.40) and (3.41), weobtain, for z > 0 and t > 0,

C∂p∂t+ ∂q∂z= 0, (4.2)

∂q∂t+ ∂γa

∂z+ A0

ρ

(1+ C

A0(p − p0)

)∂p∂z+ τw = 0, (4.3)

where γa and τw are given by (3.36) and (3.38), respectively. The boundary and initial conditionsowing to the inlet problem described above are

p(z, 0) = q(z, 0) = 0 for z > 0,

q(0, t) = qi (t) for t > 0,

p(z, t)→ 0 for z→∞, t > 0.

(4.4)

To scale the dimensional equations (4.2) and (4.3), we define dimensionless variables, pressureand flow rate in the following way:

t := T t, z := Lz, p := p0 + P p, q := Q0q, (4.5)

where Q0 is the maximum of qi (t) for t > 0. Substituting these dimensionless quantities into(4.2) and (4.3) leads us to C P L/Q0T = 1, and A0 PT/ρQ0L = 1, or, after elimination of T/L, to(A0/ρC)(C P/Q0)

2 = 1, yielding

P = Q0

C

√ρCA0= Q0

Cc0= ρc0V, (4.6)

where c0 = √A0/ρC , the unperturbed wave velocity for pressure waves in the undeformed thin-walled wall and V = Q0/A0, the characteristic velocity of the flow. We note that for the semi-infinite tube L is undetermined, but from C P L/Q0T = 1 and with T = 1/ω it follows that L = c0/ω

is an appropriate characteristic length for this problem. Finally, we introduce the dimensionlessnonlinear advection and diffusion term in a natural sense as follows:

γa = A0 V 2 γa, τw = Q0 ω τw, (4.7)

Observe that the actual (perturbed) wave velocity of the elastic wall is given by

c = c(z, t) =√

A(z, t)ρC

=√

A0 + C(p(z, t)− p0)

ρC= c0

√1+ δ p(z, t), (4.8)

where δ = Q0/(A0c0) = V/c0. Thus the system (4.2) and (4.3) can be written in dimensionlessformulation as

∂ p∂ t+ ∂q∂ z= 0, (4.9)

∂q∂ t+ δ ∂γa

∂ z+ (1+ δ p

) ∂ p∂ z+ τw = 0. (4.10)

The transformation of the boundary and initial conditions to their dimensionless form is trivial. Inthe sequel we will omit the ‘hats’ for the dimensionless quantities.



4.2 Basic example

In this basic example we neglect both the advection an diffusion term (γa = τw = 0). The globalequations of motion, as stated in (4.9) and (4.10), then simplify to

∂p∂t+ ∂q∂z= 0, (4.11)

∂q∂t+ (1+ δp)

∂p∂z= 0. (4.12)

We assume these equations to hold for z > 0 (semi-infinite vessel) and t > 0, and we impose thefollowing boundary and initial conditions:

p(z, 0) = q(z, 0) = 0 for z > 0,

q(0, t) = qi (t) for t > 0,

p(z, t)→ 0 for z→∞, t > 0

(4.13)

for a given inflow qi (t), with qi (0) = 0. The system (4.11) - (4.12) consists of two 1-D, nonlinear,wave equations for p and q. This system of equations can be solved using the method of char-acteristics. To this end, let us assume that z = z(γ ) and t = t (γ ) and so p(z, t) = P(γ ) andq(z, t) = Q(γ ). The system of equations now transforms into

dPdγ

∂γ

∂t+ dQ

dγ∂γ

∂z= 0, (4.14)

dQdγ

∂γ

∂t+ (1+ δP)

dPdγ

∂γ

∂z= 0. (4.15)

For this system to have a non-trivial solution, the determinant of the coefficient matrix has to bezero, i.e. (

∂γ

∂t

)2

− (1+ δP)(∂γ

∂z

)2

= 0, (4.16)

or

c2 = (1+ δP) = γ 2t

γ 2z

⇒ γt

γz= ±c, (4.17)

where we used the notation γt and γz for the derivative of γ with respect to t and to z, respectively.We can now define two characteristics, say 01 and 02, along which successively γ2 (referring to−c) or γ1 (referring to +c) are constant.

4.2.1 Solution for constant wave velocity

As a first step, we consider a further simplification in which we assume the compliance C verysmall (i.e. the wall is very stiff), implying that then A ≈ A0, and consequently δ = 0. In that casethe system of two wave equations becomes linear,

∂γ1

∂t− ∂γ1

∂z= 0⇒ γ1 = t + z along 01, (4.18)

∂γ2

∂t+ ∂γ2

∂z= 0⇒ γ2 = t − z along 02; (4.19)

see Figure 4.1, (a). This gives rise to the following expressions for p(z, t) and q(z, t):

p(z, t) = P1 (t + z)+ P2 (t − z) , (4.20)

q(z, t) = Q1 (t + z)+ Q2 (t − z) . (4.21)



t0z

t

Ŵ2,0

Ŵ2,1

Ŵ2,0

Ŵ1,1

Ŵ1,2

Ŵ1,3

t0z

t

Ŵ2,0Ŵ2,1

Ŵ2,0

Ŵ1,1

Ŵ1,2

Ŵ1,3

(a) (b)

Figure 4.1: Sketch of the characteristics in the (z, t)-plane (a) for constant c = c0; (b) for variablec = c(z, t).

Substituting these expressions into (4.11), we get

P ′1 (t + z)+ P ′2 (t − z)+ Q′1 (t + z)− Q′2 (t − z) = 0, (4.22)

yielding

Q1 (t + z) = P1 (t + z)+ Q1,0, (4.23)

Q2 (t − z) = P2 (t − z)+ Q2,0. (4.24)

From the initial conditions (4.13), it follows immediately that Q1(t + z) = P1(t + z) = Q1,0 = 0, forall z > 0, t > 0, and that Q2(t − z) = P2(t − z) = Q2,0 = 0, for all z > t . This means that thereis only an ongoing wave (propagating in positive z-direction), because there is no reflection in asemi-infinite vessel, and that this wave is only non-zero in the region 0 < z < t . The condition atinfinite is then already satisfied, while the boundary condition at z = 0 yields P2(t) = qi (t), for allt > 0, implying that P2(t − z) = qi (t − z), for 0 < z < t . Thus, we arrive at the following explicitsolution for this dimensionless linear system:

p(z, t) = q(z, t) = qi (t − z) H (t − z) , (4.25)

for z > 0, t > 0, where H(·) is the Heaviside function.

4.2.2 General case for variable wave velocity

We now return to the general case in which c = c(z, t) = √1+ δp is still unknown. In the sameway as for the case δ = 0, we introduce a variable γ such that p(z, t) = P(γ ) and q(z, t) = Q(γ ).This results in defining the two characteristics:

01 ={γ1, γ2 ∈ R|γ2(z, t) = γ2 is constant

}, (4.26)


}, (4.27)

where γ1 describes waves traveling in negative z-direction and γ2 waves traveling in positive z-direction. See Figure 4.1 (b), for a sketch of the characteristics. Again, γ1 and γ2 satisfy (4.18) and



(4.19). Since for a semi-infinite vessel no backward traveling waves exist, we infer that p = P(γ2)

and q = Q(γ2). This means that along 01, P(γ2) and Q(γ2) are both constant.

From (4.14) and (4.17)2 (−sign) it follows that

dPdγ2

∂γ2

∂t+ dQ

dγ2

∂γ2

∂z=[− dP

dγ2c(γ2)+ dQ

dγ2

]∂γ2

∂z= 0, (4.28)

implying that (when γ2,z 6= 0)dQdγ2= c(γ2)

dPdγ2

. (4.29)

We can integrate (4.29) to obtain

Q(γ2) = Q(0)+∫ γ2

0c(γ )

dPdγ

dγ = Q(0)+∫ P(γ2)

P(0)

√1+ δP dP (4.30)

= Q(0)+ 23δ

{[1+ δP(γ2)

] 32 − [1+ δP(0)]

32

}= 2

3δ

{[1+ δP(γ2)

] 32 − 1

}, (4.31)

because P = Q = 0 for γ2 = 0. Inverting this relation, we obtain

P(γ2) = 1δ

{[1+ 3δ

2Q(γ2)

] 23 − 1

}. (4.32)

Let us take a closer look at the specific characteristic 01 that intersects the t-axis at t = t0, forarbitrary t0 > 0 (see Figure 4.1). This means that γ2 = γ2(0, t0) = γ2(t0). Because P and Q areconstant along 01, we know that also c(γ1) =

√1+ δP(γ2) is constant. This leads to

0 = dγ2 = ∂γ2

∂tdt + ∂γ2

∂zdz = ∂γ2

∂t

[dt − dz

c

], along 01, (4.33)

from which follows that

t − zc= constant = t0, along 01, (4.34)

where c = c(γ2(0, t0)) = c(t0). Finally, we still need to satisfy the initial condition q(0, t) = qi (t).To this end, we consider the specific characteristic 02 that intersects the t-axis in t = t0. Alongeach 02, q(z, t) = Q(γ2), and hence in (z, t) = (0, t0) we get

Q(γ2(t0)) = q(0, t0) = qi (t0), for every t0 > 0, (4.35)

and then from (4.32)

P(γ2(t0)) = p(0, t0) = 1δ

{[1+ 3δ

2qi (t0)

] 23 − 1

}, (4.36)

and, moreover,

c(t0) =√

1+ δp(0, t0) =(

1+ 32δqi (t0)

) 13. (4.37)

Substituting this expression into (4.34), we obtain

t − z(

1+ 32δqi (t0)

) 13= t0, (4.38)



from which t0 can be solved as t0 = T0(z, t). An explicit numerical solution is obtained withMatlab.

Summarizing, we note that the solutions for p and q are

q(z, t) = qi (T0(z, t))H(t − z), (4.39)

p(z, t) = 1δ

{[1+ 3δ

2qi (T0(z, t))

] 23 − 1

}H(t − z). (4.40)

We characterize the deformation of the vessel wall by its relative displacement 1R, defined as

1R(z, t) = R(z, t)− Ra

Ra= R(z, t)− 1. (4.41)

According to (4.1) the deformed area A(z, t) = πR2(z, t) is a local and linear (in accordance withthe linear elasticity theory assumed here) function of the pressure, i.e.

A(z, t) = πR2(z, t) = πR2a + C(p(z, t)− p0) = πR2

a

(1+ C P

πR2a

p(z, t)), (4.42)

or, with C P/πR2a = Q0/c0πR2

a = V/c0 = δ,

R(z, t) =√

1+ δ p(z, t) .= 1+ 12δ p(z, t), (4.43)

where the latter linearization is completely in consistency with the basic assumptions of linearelasticity theory. Hence, assuming for a moment that p is strictly of O(1), we note that a neces-sary condition for linear elasticity is here δ � 1.

Using the linear elasticity form of (4.43), we obtain from (4.41)

1R(z, t) = 12δ p(z, t), (4.44)

with p(z, t) = p(z, t) as given by (4.40).

NOTE: In this section we have derived solutions for q(z, t) and p(z, t) valid for arbi-trary values of δ. However, since we have adopted linear elasticity already before, itis in principle inconsistent to employ these results for arbitrary large δ. Since in linearelasticity all terms of O(δ2) are consistently neglected with respect tot the terms ofO(δ), the same must be done with the results in this section.

Thus, consistently neglecting O(δ2)-terms, leads us to the following (linear in δ) re-sults for q(z, t) and p(z, t) from (4.39) and (4.40):

q(z, t) = qi (T0(z, t))H(t − z), (4.45)

p(z, t) =(

qi (T0(z, t))− 14δq2

i (t − z))

H(t − z), (4.46)

while the relation for T0(z, t), (4.38), reduces to

t −(

1− 12δqi (t0)

)z = t0. (4.47)

This relation can also be found as a special case of the solution obtained in the forth-coming Section 4.4.



4.2.3 The shell equation

The local relation (4.1) relates the deflection of the vessel wall in a point z directly to the pres-sure in the same point z. A direct consequence of this is that when the internal pressure in thevessel makes a sudden jump (e.g. at a valve), the deflection follows this jump and thus becomesdiscontinuous. Mechanically, this means that the vessel wall will rupture, which is not realistic.The reason for this anomaly is that the local formula does not account for the effect (the stiffness)of neighboring parts of the vessel wall near z. A model that does account for this non-local ef-fect, and thus yields a smooth curve for the deflection, is described by the shell equation for athin-walled tube, which is derived in Appendix A, and results in (A.48):

∂4U∂z4 (z, t)+ 4β4U (z, t) = 1

Dp(z, t), (4.48)

where U (z, t) is the radial displacement, p(z, t) ≈ p(z, t) the internal pressure, D = (Eh3)/(12(1−ν2)) the plate constant and β4 = 3/(Rah)2. Note that here ν is Poisson’s ratio, which we givefrom here on the fixed value ν = 0.5, because we assume the vessel wall incompressible, soD = Eh3/9. The complete derivation of the shell equation can be found in Appendix A. Thesolution of this equation can be split into a homogeneous part and a particular part. For thehomogeneous solution, we solve the corresponding characteristic equation to find

Uh(z, t) = c1e(1+i)βz + c2e(1−i)βz + c3e(i−1)βz + c4e−(1+i)βz, (4.49)

where c1, ..., c4 are still functions of t . This homogeneous solution delivers us the fundamentalsystem of eigenfunctions that we can use to obtain a particular solution by the method of variationof constants. We choose the integration bounds in such a way that the particular solution Up(z, t)goes to zero for z → ∞, provided that also p(z, t) goes to zero for z → ∞. We choose the totalsolution U (z, t) equal to this particular solution (for reasons that we will explain soon hereafter)and thus obtain the non-local result

U (z, t) = 18β3 D

[∫ ∞

z(cos(β(ζ − z))+ sin(β(ζ − z)))e−β(ζ−z) p(ζ, t) dζ+

∫ z

0(cos(β(ζ − z))− sin(β(ζ − z)))e−β(z−ζ ) p(ζ, t) dζ

]. (4.50)

We call this a non-local result, because the displacement U in the point z depends here onthe pressure in all points of the wall near z. If p(z, t) has a finite support, i.e. if we can writep(z, t) = p(z, t)H(l − z), for arbitrary l > 0 (possibly even with l = l(t)), with H(·) the Heavisidefunction, then (4.50) can be written as

U (z, t) = 18β3 D

[∫ l

z(cos(β(ζ − z))+ sin(β(ζ − z)))e−β(ζ−z) p(ζ, t) dζ+

∫ z

0(cos(β(ζ − z))− sin(β(ζ − z)))e−β(z−ζ ) p(ζ, t) dζ

], (4.51)

for 0 < z < l, and

U (z, t) = 18β3 D

∫ l

0(cos(β(ζ − z))− sin(β(ζ − z)))e−β(z−ζ ) p(ζ, t) dζ, (4.52)

for z > l.

Finally, we note that for the thin-walled arteries we consider here the parameter β is very large,which, more precisely, means that Lβ � 1 (where L is the characteristic length for the range of



z). For very large β (Lβ → ∞), β exp(−βζ) behaves like the Dirac delta function δ(ζ ), for ζ ≥ 0.This means that then (4.50) reduces to

U (z, t) = 18β4 D

[∫ ∞

z(cos(β(ζ − z))+ sin(β(ζ − z)))δ(ζ − z)p(ζ, t) dζ+

∫ z

0(cos(β(ζ − z))− sin(β(ζ − z)))δ(z − ζ )p(ζ, t) dζ

]

= p(z, t)4β4 D

. (4.53)

Hence, we conclude that for large β the particular solution (4.53) tends to the same expressionas (4.48) in case 1/4β4 → 0 (i.e. after the neglect of (∂4U/∂z4)/4β4), namely

U (z, t) = p(z, t)4β4 D

= 3R2a

4Ehp(z, t), (4.54)

which at its turn completely corresponds to the simplified (affine) relation between A(z, t) andp(z, t) according to (4.1) as we shall show:

A(z, t) = π(Ra +U (z, t))2 = πR2a + 2πRaU (z, t)+ ... = πR2

a + Cp(z, t), (4.55)

or

U (z, t) = C2πRa

p(z, t) = 2R2a

4Ehp(z, t), (4.56)

with C according to (A.19), or in dimensionless notation,

U (z, t) = U (z, t)Ra

= 12δ p(z, t), (4.57)

in accordance with (4.44).

4.3 Inclusion of radial diffusion

In this section we investigate the influence of the radial diffusion term τw as introduced in Section3.4. This term represents the effect of the shear stress at the wall on the flow. To focus onthe effect of radial diffusion purely, we neglect both the advection term (∂γa/∂z = 0) and thecompliance distribution in the coefficient of ∂p/∂z (δ = 0). Moreover, we assume the diffusionterm τw to be a linear combination of the flow rate q and the pressure gradient ∂p/∂z, as derivedby Bessems et al. in [4]. So

τw(z, t) = τ1q(z, t)+ τ2∂p∂z(z, t), (4.58)

with τ1,2 ∈ R. We first step back to the dimensional equations of motion and substitute (4.58)for τw. As posed before, we leave out the advection term and the compliance contribution. Wethus arrive at (note that the first-order effect of the deformation of the wall is incorporated in thecoefficient C being greater than zero)

C∂p∂t+ ∂q∂z= 0, (4.59)

∂q∂t+ A0

ρ

∂p∂z+ τ1 q + τ2

∂p∂z= 0. (4.60)



Introducing the scaled problem parameters as in (4.5), these equations of motion transform into

C L∗ P∗

T Q0

∂p∂t+ ∂q∂z= 0, (4.61)

∂q∂t+ (A0 + τ2ρ)T P∗

Q0 ρ L∗∂p∂z+ τ1 T q = 0. (4.62)

Note that we added a ∗ to P and L to denote that these parameters will (slightly) differ from theexpressions found in Section 4.1 due to the effect of τ2. If we choose

P = Q1

C c∗0, L = c∗0

ω, τ = T τ1, (4.63)

where c∗0 =√(A0 + τ2ρ)/C ρ, the modified wave velocity, we see that the equations of motion

transform into

∂p∂t+ ∂q∂z= 0, (4.64)

∂q∂t+ ∂p∂z+ τq = 0, (4.65)

with boundary and initial conditions

p(z, 0) = q(z, 0) = 0 for z > 0,

q(0, t) = qi (t) for t > 0,

p(z, t)→ 0 for z→∞, t > 0

(4.66)

We try to solve the system (4.64) - (4.65) by using Laplace transformation. Tot this end, weintroduce the Laplace transforms of p(z, t) and q(z, t) as follows:

P (z; s) =∫ ∞

0p(z, t)e−st dt, (4.67)

Q(z; s) =∫ ∞

0q(z, t)e−st dt. (4.68)

If we apply the Laplace transform to the system (4.64) and (4.65) and the boundary conditions(4.66), with the above definitions of the transforms of p(z, t) and q(z, t), we are left with

s P + dQ

dz= 0, (4.69)

dP

dz+ (τ + s)Q = 0, (4.70)

P (z; s)→ 0 (z→∞), (4.71)

Q(0; s) =∫ ∞

0qi (t)e−st dt = Qi (s). (4.72)

The solution of this system of ordinary differential equations can easily be found by eliminatingP (z; s) from the second equation using the first one, which results in a differential equation interms of Q(z; s) only. The solutions are given by

Q(z; s) = Qi (s)e−σ(s)z, σ (s) =√(τ + s)s, Re(σ ) ≥ 0, (4.73)

P (z; s) = σ(s)s

Qi (s)e−σ(s)z . (4.74)



Taking the inverse Laplace transform of the solution obtained above, we find for (0 ≤ z ≤ t) that

q(z, t) = limT→∞

12π ı

∫ c+ıT

c−ıTQ(z; s)ets ds = lim

T→∞1

2π ı

∫ c+ıT

c−ıTQi (s)e−σ(s)zets ds

=∫ t

0qi (t − x) f (z, x)dx, (4.75)

where

f (z, t) = limT→∞

12π ı

∫ c+ıT

c−ıTe−σ(s)zets ds, (4.76)

the inverse Laplace transform of exp(−σ(s)z). We used two properties of the inverse Laplacetransform, stating that if in general F (s)↔ f (t) and G(s)↔ g(t), it holds that

F (s)/s ↔ (H ∗ f )(t), (4.77)

F (s)G(s)↔ ( f ∗ g)(t), (4.78)

where H(·) is the Heaviside function.

It remains to determine f (z, t). This can be done by some simple mathematical manipulationand using a table for inverse Laplace transforms (see [7]). This leads to

f (z, t) = − ∂∂z

{lim

T→∞1

2π ı

∫ c+ıT

c−ıT

e−z√(τ+s)s

√(τ + s)s

ets ds

}

= −e−12 τ t ∂

∂z

{lim

T→∞1

2π ı

∫ c+ıT

c−ıT

e−z√

p2−(τ /2)2√

p2 − (τ /2)2 etpdp

}

= −e−2τ t ∂

∂z

[I0

(2τ√

t2 − z2)

H(t − z)]

= e−2τ t

δ(t − z)+ 2τ z

I1

(2τ√

t2 − z2)

√t2 − z2

H(t − z)

, (4.79)

where we used the substitutions s = p − τ /2 and consequently c = c + τ /2 in the first line andτ = τ /4 in the second line. Combining (4.75) and (4.79), we finally arrive at

q(z, t) =∫ t

0qi (t − x) e−2τ x

δ(x − z)+ 2τ z

I1

(2τ√

x2 − z2)

√x2 − z2

H(x − z)

dx

= qi (t − z)e−2τ z + 2τ z∫ t−z

0

I1

(2τ√(t − s)2 − z2

)

√(t − s)2 − z2

e−2τ (t−s)qi (s) ds, (4.80)

for 0 ≤ z ≤ t .

To solve the equation for p(z, t) we use the resulting solution of q(z, t) in (4.80) and we in in-troduce the auxiliary function B(z, t) as

B(z, t) =∫ t

zI0

(2τ√

x2 − z2)

Qi (t − x)e−2τ x dx H(t − z), (4.81)

where

Qi (t) =∫ t

0qi (ξ) dξ. (4.82)



With this definition of the function B(z, t) we rewrite the expression for q(z, t) in (4.80) to

q(z, t) = − ∂∂z

∫ t

zI0

(2τ√

x2 − z2)

qi (t − x)e−2τ x dx = − ∂2B∂z∂t

(z, t). (4.83)

Using the original equation given in (4.65), we see that

∂p∂z= −

(∂q∂t+ 4τq

)=(∂

∂t+ 4τ

)∂2B∂z∂t

= ∂

∂z

(∂

∂t+ 4τ

)∂B∂t, (4.84)

resulting in

p(z, t) =(∂

∂t+ 4τ

)∂B∂t=(∂

∂t+ 4τ

)∫ t

zI0

(2τ√

x2 − z2)

qi (t − x)e−2τ x dx

=∫ t

zI0

(2τ√

x2 − z2) (

qi (t − x)+ 4τqi (t − x))

e−2τ x dx

=∫ t−z

0I0

(2τ√(t − s)2 − z2

) (qi (s)+ 4τqi (s)

)e−2τ (t−s) ds, (4.85)

for 0 ≤ z ≤ t .

NOTE: ForA(x, z) = I0

(2τ√

x2 − z2)

(4.86)

it holds that∂2A

∂x2 −∂2A

∂z2 = 4τ 2A. (4.87)

With this relation one easily checks that the obtained solutions for p and q also satisfy(4.64).

So, summarizing we see that for 0 ≤ z ≤ t

q(z, t) = qi (t − z)e−2τ z + 2τ z∫ t−z

0

I1

(2τ√(t − s)2 − z2

)

√(t − s)2 − z2

e−2τ (t−s)qi (s) ds, (4.88)

p(z, t) =∫ t−z

0I0

(2τ√(t − s)2 − z2

) (qi (s)+ 4τqi (s)

)e−2τ (t−s) ds. (4.89)

The corresponding asymptotic expansions with respect to τ , for 0 < t � 1, are given by:

q(z, t) =qi (t − z)− 2τ zqi (t − z)+ 2τ 2(

z2qi (t − z)+ z∫ t−z

0qi (s) ds

)+O

(τ 3), (4.90)

p(z, t) =qi (t − z)+ 2τ∫ t−z

0(2qi (s)− (t − s)qi (s)) ds+

τ 2∫ t−z

0

((3(t − s)2 − z2

)qi (s)− 8(t − s)q(s)

)ds +O

(τ 3). (4.91)

An alternative derivation of p(z, t) and q(z, t), using a more elegant fundamental mathematicalapproach, can be found in Appendix B.



We conclude this section by deriving an expression for our dimensionless coefficient τ . To thisend, we follow Bessems [1], who defines the dimensional radial diffusion term as

τw(z, t) = 2πRa

ρ

(2η

(1− ζc)Ra A0q(z, t)+ Ra (ζc − 1)

4∂p∂z(z, t)

), (4.92)

where ζc is the square of the dimensionless core-diameter and defined as

ζc = max

0,

(1−√

2α

)2 , (4.93)

with α = ra√ω/ν, the Womersley number according to (3.20). All other quantities involved are as

introduced before. So we see that

τ1 = 4νR2

a (1− ζc), τ2 = A0 (ζc − 1)

2ρ. (4.94)

If we define ϑ = √(1+ ζc) /2, we derive with the definition of τ2 that c∗0 = c0 ϑ and so P∗ = P/ϑand L∗ = L ϑ . Here, c0, P and L are the originally defined parameters from Section 4.1. Finally,we arrive for τ at

τ = τ1

4ω= ν

ωR2a(1− ζc)

= 1α2 (1− ζc)

. (4.95)

We conclude by noting that the formula for ζc, (4.93), according to Bessems et al, [1], only holdsfor relatively large α, implying that ζc is close to one. Hence, also the parameter ϑ ≈ 1 (in factϑ = 1+O(α−1)).

4.4 Inclusion of advection

In this section we will investigate the contribution of the advection term to the flow. We start fromthe dimensionless equation of motion given in (4.9) and (4.10). To specifically investigate theeffect of the advection term in (4.10), we neglect the radial diffusion term (τw = 0). Moreover,following Bessems et al. [3], we assume the nonlinear advection to be of the form

γa(z, t) = γa1 q2(z, t)+ γa2 q(z, t)∂p∂z(z, t)+ γa3

(∂p∂z(z, t)

)2

, (4.96)

where γa1,2,3 are real positive constants. Once more, we follow the choice of Bessems et al. byputting γa2 = γa3 ≈ 0 yielding γa(z, t) = γa1 q2(z, t). As we will see, this choice for γa(z, t) allowsus to also incorporate the compliance contribution in the equations of motion.

We thus obtain

∂p∂t+ ∂q∂z= 0, (4.97)

∂q∂t+ 2δγa1q

∂q∂z+ (1+ δp)

∂p∂z= 0, (4.98)


p(z, 0) = q(z, 0) = 0 for z > 0,

q(0, t) = qi (t) for t > 0,

p(z, t)→ 0 for z→∞, t > 0

(4.99)



We can solve this system of equations in a completely analogous way as done for the system inSection 4.2.2, but we assume δ2 → 0 and solve the system up to the first order in δ. So, againwe introduce a variable γ in such a way that p(z, t) = P(γ ) and q(z, t) = Q(γ ) and we define thetwo characteristics:


}, (4.100)


}, (4.101)

where γ1 describes waves traveling in negative z-direction and γ2 waves traveling in positive z-direction. Substitution of p(z, t) = P(γ ) and q(z, t) = Q(γ ) in (4.97) and (4.98) gives (in matrixnotation)

∂γ

∂t∂γ

∂z

(1+ δP)∂γ

∂z∂γ

∂t+ 2δγa1 Q

∂γ

∂z

∂P∂γ∂Q∂γ

= 0, (4.102)

leading to the solvability condition

γ 2t + 2δγa1 Qγtγz − (1+ δP) γ 2

z = 0. (4.103)

As solution for this condition we find

γt

γz= ±

√δ2γ 2

a1Q2 + 1+ δP − δγa1 Q, (4.104)

where the +sign and the −sign refer to a solution along 01 and 02, respectively. Along 02 (so γ1is constant) we see that (4.102)1 after substitution of (4.104) leads to

dPdγ2

∂γ2

∂t+ dQ

dγ2

∂γ2

∂z=[− dP

dγ2

(√δ2γ 2

a1Q2 + 1+ δP + δγa1 Q

)+ dQ

dγ2

]∂γ2

∂z= 0, (4.105)

implying that (when γ2,z 6= 0, with O(δ2)→ 0)

dQdγ2=(

1+ 12δP + δγa1 Q

)dPdγ2

. (4.106)

Considering Q(γ2) = Q(P), we obtain an ordinary differential equation for (4.106)

dQdP− δγa1 Q = 1+ 1

2δP (4.107)

Q(0) = 0, (4.108)

having solution (again up to O(δ))

Q(P) = 1+ 2γa1

2γ 2a1δ

(eδγa1 P − 1

)− 1

2γa1

P

= 1+ 2γa1

2γ 2a1δ

(1+ δγa1 P + 1

2δ2γ 2

a1P2 − 1

)− 1

2γa1

P

= P + 14

(1+ 2γa1

)δP, (4.109)

or equivalently as a function of γ2

P(γ2)+ 14

(1+ 2γa1

)δP(γ2)− Q(γ2) = 0. (4.110)



Since we want to solve the system (4.102) up to order O(δ), we assume P(γ2) = P0(γ2)+ δP1(γ2)

and together with (4.110) we find that

P0(γ2) = Q(γ2), (4.111)

P1(γ2) = −14(1+ 2γa1)Q

2(γ2), (4.112)

and so we obtain an inverted relation of (4.109)

P(γ2) = Q(γ2)− 14δ(1+ 2γa1)Q

2(γ2). (4.113)

Along 01 we know that P and Q are both constant, which means that (4.104) along 01 (+sign) isconstant. We take a specific characteristic on 01 that intersects the t-axis at t = t0, for arbitraryt0 > 0. This means that γ2 = γ2(0, t0) = γ2(t0) is constant. This leads to

0 = dγ2 = ∂γ2

∂tdt + ∂γ2

∂zdz = ∂γ2

∂t

dt − dz√

δ2γ 2a1

Q2 + 1+ δP + δγa1 Q

, (4.114)

from which follows that

t − z√δ2γ 2

a1Q2 + 1+ δP + δγa1 Q

= t0, (4.115)

along 01. Finally, we consider the specific characteristic on 02 that intersects the t-axis in t = t0.In the point (z, t) = (0, t0) we have γ2 = γ2(t0) and so we get

Q(γ2(t0)) = q(0, t0) = qi (t0), , (4.116)

and consequentially

P(γ2(t0)) = qi (t0)− 14δ(1+ 2γa1)q

2i (t0), (4.117)

for every t0 > 0. Moreover, we see that

t0 = T0(z, t) = t − z√δ2γ 2

a1Q2(γ2(t0))+ 1+ δP(γ2(t0))+ δγa1 Q(γ2(t0))

= t − z√δ2γ 2

a1q2

1 (t0)+ 1+ δ(

qi (t0)− 14δ(1+ 2γa1)q

2i (t0)

)+ δγa1qi (t0)

. (4.118)

Summarizing, we note that the solutions for p and q are

q(z, t) = qi (T0(z, t))H(t − z), (4.119)

p(z, t) =(

qi (T0(z, t))− 14δ(1+ 2γa1)q

2i (T0(z, t))

)H(t − z), (4.120)

with T0(z, t) the solution of (4.118); this solution is obtained numerically with for example Matlab.

Note: If we put γa1 equal to zero in the obtained solutions (4.119) and (4.120), we seethat the result is in accordance with the linearized solutions of Section 4.2.2 (equations(4.45) and (4.46)).



To conclude this section, we take a closer look at the constant γa1 . Bessems et al. ([3]) introducesγa(z, t) (dimensional) as (with γa2 = γa3 = 0)

γa(z, t) = δ1

A0q2(z, t), (4.121)

whereδ1(ζc) = 2− 2ζc (1− ln (ζc))

(1− ζc)2 , (4.122)

and were ζc as defined in the previous section. From this expression and with use of (4.121) wecan easily determine our dimensionless coefficient γa1

γa = A0V 2γa = A0V 2γa1 q2 = δ1

A0q2 = δ1 Q2

0A0

q2, (4.123)

leading to

γa1 =δ1 Q2

0

A20V 2= δ1. (4.124)

4.5 Preliminary conclusions from this chapter

In this chapter we have found closed-form solutions for the global, one dimensional, inlet-flowproblem for a semi-infinite deformable vessel. In order to specifically investigate the effects ofadvection and diffusion, we have solved the following three partial problems

1. No advection, no diffusion: investigates especially the effect of the deformation of the vesselwall.

2. Diffusion, no advection: investigates the effect of the shear wall stress on the flow.

3. Advection, no diffusion: investigates the effect of the nonlinear advection terms.

To enable the calculations, we borrowed explicit expressions for the terms mentioned above fromBessems et al. [3]. In the three-dimensional analysis, to be presented in the next chapter, wehope to find a justification for these choices or alternative expressions that can also be used ina more or less analogous way to obtain closed-form solutions. In Chapter 6, we give explicitnumerical presentations of the results of this, and next, chapter, which can serve to quantify thedifferent effects considered here. We hope that this will lead to definitive conclusions regardingthe classification of the orders of magnitudes of the effects.



Chapter 5The local equations of motion

In this chapter, we consider the same inlet-flow as in the preceding chapter, but now in alocal, i.e. three-dimensional, formulation. We solve the equations of motions up to leadingorder by imposing asymptotic expansions for the axial velocity, the flow-rate and the pres-sure function in terms of the small parameter δ, the dimensionless compliance of the wall.This leads to a zeroth order and a first-order set of equations of motions and enables usto derive zeroth-order expressions for the flow characteristics; especially the shear forceand the nonlinear advection contribution. We start from a simplified situation in which thevessel is undeformable and that serves as a preparation step in solving the equations ofmotion for a deformable vessel. Again, we are looking for as much as possible analyticalsolutions. To this end, we impose a certain approximation to the axial velocity. This ap-proximation also affects all other quantities and therefore we need to discuss the accuracyof the approximation we make. As for the results of Chapter 4, the results of this chapterwill graphically be presented in Chapter 6.

5.1 Problem set-up

In this chapter, we investigate the inlet flow in a three-dimensional semi-infinite thin-walled ves-sel of initial (undeformed) radius Ra . As in the previous chapter we use a cylindrical coordi-nate system and assume the flow to be rotationally symmetric. This means that the velocityv = v(r, z, t) = vr (r, z, t)er + vz(r, z, t)ez and the pressure p = p(r, z, t) for 0 < r < Ra and0 < z <∞.

The general equations describing the inlet flow problem are (see (3.29)-(3.31))

1r∂

∂r(rvr )+ ∂vz

∂z= 0, (5.1)

∂p∂r= 0 ⇒ p = p(z, t), (5.2)

∂vz

∂t+ vr

∂vz

∂r+ vz

∂vz

∂z= − 1

ρ

∂p∂z+ ν

r∂

∂r

(r∂vz

∂r

). (5.3)

Moreover, for the problem we consider here, the initial and boundary conditions are

v(r, z, 0) = p(z, 0) = 0, (5.4)



and

vr (R, z, t) = ∂R∂t(z, t), vz(R, z, t) = 0, (5.5)

at r = R = R(z, t), the deformed inner radius of the vessel, respectively. At the inlet z = 0, weonly describe the total influx q(0, t) = qi (t), but we choose the distribution in accordance with theobtained solution form; we refer to this as the relaxed problem. Finally, we require the solution(vr , vz, p − p0) to go to zero for z→∞.

This system will be supplemented by the equations of the global formulation, which will furtheron be used to obtain the local solution (we note here that C > 0 for a deformable wall, but C = 0for a rigid wall). For these equations we refer to (4.2) and (4.3). These equations hold for z > 0,t > 0, and read

C∂p∂t+ ∂q∂z= 0, (5.6)

∂q∂t+ ∂γa

∂z+ A0

ρ

(1+ C

A0p)∂p∂z+ τw = 0, (5.7)

with

q(z, t) = 2π∫ R(z,t)

0rvz(r, z, t) dr,

γa = γa(z, t) = 2π∫ R(z,t)

0rv2

z (r, z, t) dr,

τw(z, t) = −2πνR(z, t)∂vz

∂r

∣∣∣∣R(z,t)

,

R(z, t) = Ra

(1+ C

2πR2a

p(z, t)),

(5.8)

andp(z, 0) = q(z, 0) = 0 for z > 0,

R(z, 0) = Ra for z > 0,

q(0, t) = qi (t) for t > 0,prescribed

p(z, t)→ 0 for z→∞, t > 0.

(5.9)

First, we make these equations dimensionless, using the same dimensionless quantities as in-troduced in Chapters 3 and 3. To keep up the readability, we repeat these definitions here:

t := ω−1 t, z := Lz = εRa z, r := Ra r , R := Ra R,

vr := εV vr , vz := V vz, p := P p, q := Q0q,(5.10)

where ω = 2π and

L := c0

ω, ε := Ra

L� 1, V := Q0

πR2a, Q0 := max

t>0qi (t), P := ρc0V, (5.11)

and, moreover, we introduce the (small) parameter δ by (δ must be small, as it represents thedimensionless compliance, and because we applied linear elasticity theory to calculate the defor-mations of the wall; in linear elasticity theory, O(δ2)-terms are consistently neglected):

δ = Vc0� 1. (5.12)



Secondly, we eliminate R through

R(z, t) =√

1+ δp(z, t) = 1+ 12δp(z, t)+O

(δ2), (5.13)

and vr by means of

vr (r, z, t) = −1r

∫ r

0x∂vz

∂z(x, z, t) dx, (5.14)

which follows by integrating (5.1), using the regularity of vr in r = 0. With this relation, theboundary condition (5.9), is trivially satisfied, as we can show as follows:

vr (R, z, t) = − 1R(z, t)

∫ R(z,t)

0x∂vz

∂z(x, z, t) dx

= − 1R(z, t)

{∂

∂z

∫ R(z,t)

0xvz(x, z, t) dx − R(z, t)vz (R(z, t), z, t)

∂R∂z

}

= − 12R(z, t)

∂q∂z(z, t) = 1

2R(z, t)∂p∂t(z, t) = 1

δ

∂R∂t(z, t), (5.15)

according to (5.13)1.

Finally, we introduce the advection function 0a as

0a(r, z, t) = vr∂vz

∂r+ vz

∂vz

∂z

={−1

r

∫ r

0x∂vz

∂z(x, z, t) dx

}∂vz

∂r+ vz

∂vz

∂z

= −1r∂

∂r

{vz

∫ r

0x∂vz

∂z(x, z, t) dx

}+ 1

rvzr

∂vz

∂z+ vz

∂vz

∂z

= −1r∂

∂r

{vz

∫ r

0x∂vz

∂z(x, z, t) dx

}+ ∂v

2z

∂z. (5.16)

With all this we arrive at the following system of equations for the unknowns vz(r, z, t), p(z, t) andq(z, t), all in dimensionless form

∂p∂t+ ∂q∂z= 0, (5.17)

∂q∂t+ δ ∂γa

∂z+ (1+ δp)

∂p∂z+ τw = 0, (5.18)

∂vz

∂t+ δ0a − νr

∂

∂r

(r∂vz

∂r

)= −∂p

∂z, (5.19)

with

q(z, t) = 2∫ R(z,t)

0rvz(r, z, t) dr,

γa = γa(z, t) = 2∫ R(z,t)

0rv2

z dr,

τw(z, t) = −2νR(z, t)∂vz

∂r

∣∣∣∣R(z,t)

,

R(z, t) = 1+ 12δp(z, t),

(5.20)



where ν = ν/(ωR2a) and

p(z, 0) = q(z, 0) = vz(r, z, 0) = 0 for z > 0,

vz(R, z, t) = 0 for z > 0, t > 0

q(0, t) = qi (t) for t > 0, : prescribed

p(z, t)→ 0 for z→∞, t > 0.

(5.21)

Since the constitutive equation for R(z, t) is only accurate up to O(δ) (i.e. O(δ2) → 0), it seemsconsistent to neglect all O(δ2)-terms in the above system. To this end, we expand all variablesasymptotically as follows:

vz(r, z, t; δ) = vz,0(r, z, t)+ δvz,1(r, z, t)+O(δ2), (5.22)

p(z, t; δ) = p0(z, t)+ δp1(z, t)+O(δ2), (5.23)

q(z, t; δ) = q0(z, t)+ δq1(z, t)+O(δ2). (5.24)

We substitute these relations into (5.17) - (5.21), develop them in powers of δ, split up the systemin one of O

(δ0) and one of O

(δ1) and neglect all terms of O(δ2). This finally amounts to the

following systems of equations.

1. Zeroth-order system:

∂p0

∂t+ ∂q0

∂z= 0, (5.25)

∂q0

∂t+ ∂p0

∂z+ τw,0 = 0, (5.26)

∂vz,0

∂t− ν

r∂

∂r

(r∂vz,0

∂r

)= −∂p0

∂z, (5.27)

with

q0(z, t) = 2∫ 1

0rvz,0(r, z, t) dr,

τw,0(z, t) = −2ν∂vz,0

∂r

∣∣∣∣r=1,

(5.28)

andp0(z, 0) = q0(z, 0) = vz,0(r, z, 0) = 0 for z > 0,

vz,0(1, z, t) = 0 for z > 0, t > 0

q0(0, t) = qi (t) for t > 0, : prescribed

p0(z, t)→ 0 for z→∞, t > 0.

(5.29)

2. First-order system:

∂p1

∂t+ ∂q1

∂z= 0, (5.30)

∂q1

∂t+ ∂γa,0

∂z+ ∂p1

∂z+ p0

∂p0

∂z+ τw,1 = 0, (5.31)

∂vz,1

∂t+ 0a,0 − νr

∂

∂r

(r∂vz,1

∂r

)= −∂p1

∂z, (5.32)



with

q1(z, t) = 2∫ 1

0rvz,1(r, z, t) dr,

γa,0 = γa,0(z, t) = 2∫ 1

0rv2

z,0 dr,

τw,1(z, t) = −2ν{∂vz,1

∂r+ 1

2p0(z, t)

(∂vz,0

∂r+ ∂

2vz,0

∂r2

)} ∣∣∣∣r=1,

0a,0 = ∂2vz,0

∂z2 − vz,0∂vz,0

∂z,

(5.33)

andp1(z, 0) = q1(z, 0) = vz,1(r, z, 0) = 0 for z > 0,

vz,1(1, z, t) = 14ν

p0(z, t)τw,0(z, t) for z > 0, t > 0,

q1(0, t) = 0 for t > 0,

p1(z, t)→ 0 for z→∞, t > 0.

(5.34)

5.2 Relaxed solution for inlet flow in a rigid vessel

In this section, we consider the problem of a sudden (from t = 0 on) inlet flow in a semi-infinite(z > 0) rigid vessel, of radius R (≡ Ra). The fluid is initially (for t < 0) in rest. We prescribethe total inlet flow qi (t) at z = 0, but not its exact distribution. Instead, we consider the relaxedsolution, which means that we assume that the local (a function of r ) inflow boundary conditionis such that the resulting flow in the tube is independent of z. We consider this, very simplified,problem here to get already some grip on the more complicated problem to be solved in the nextsection and in the hope that some aspects of the solution derived here can also be used in thenext section.

For a rigid vessel, the compliance C = 0, and then (5.6) yields

∂q∂z= 0 ⇒ q = q(t) = qi (t), (5.35)

in accordance with (5.9)3. As a consequence, also the axial velocity vz is independent of z (bythe way, the radial velocity vr = 0, here). Finally, the pressure can be written as

p = p(z, t) = pi (t)−5(t)z, (5.36)

where the inlet pressure pi is further irrelevant. With all this, and with vz = vz(r, t) := w(r, t) andδ = 0, (5.18) and (5.19) yield

dqdt+ τw = −∂p

∂z= 5(t), (5.37)

and

∂w

∂t− ν

r∂

∂r

(r∂w

∂r

)= dq

dt+ τw, (5.38)

where

τw = −2ν∂w

∂r(1, t). (5.39)



Equation (5.38) can be solved by Laplace transforms with respect to the time t , defined by:

W(r; s) = L {w(r, t), t, s} =∫ ∞

0w(r, t)e−st dt, (5.40)

Q(s) = L {q(t), t, s} =∫ ∞

0q(t)e−st dt, (5.41)

T (s) = L {τw(t), t, s} =∫ ∞

0τw(t)e−st dt. (5.42)

By taking the Laplace transform of (5.38), we find an ordinary differential equation in terms of theradius r , for 0 ≤ r < 1,

νd2W

dr2 +ν

rdW

dr− sW = −T − sQ. (5.43)

The solution of this differential equation can be split into a homogenous part and a particular part.By putting the right-hand side of the differential equation equal to zero, we see that the solutionof the homogenous part reads

Whom(r; s) = c1 I0

(r√

sν

)+ c2 K0

(r√

sν

). (5.44)

Here, I0(·) is the modified Bessel-function of the first kind of order 0, K0(·) the modified Bessel-function of the second kind of order 0, and c1, c2 constants in R. By observing that both Q and Tare independent of r , the particular solution of the differential equation is simply given by

Wpar(r; s) = Q(s)+ T (s)s. (5.45)

The condition that the solution should be bounded for r = 0, leads to c2 = 0. From the boundarycondition w(1, t) = 0, and so W(1; s) = 0, we see that

c1 = − 1

I0

(√sν

)(

Q(s)+ T (s)s

). (5.46)

As the total solution for the Laplace transform of the radial velocity w(r, t) we thus find

W(r; s) =(

Q(s)+ T (s)s

)1−

I0

(r√

sν

)

I0

(√sν

)

. (5.47)

The definition of τw gives

T (s) = −2νdW

dr(1, s) = 2ν

√sν

I1

(√sν

)

I0

(√sν

)(

Q(s)+ T (s)s

), (5.48)

and so (with I0(z)− (2/z)I1(z) = I2(z))

T (s) = 2ν√

sν

I1

(√sν

)

I2

(√sν

)Q(s). (5.49)

Finally, the Laplace transform W can be written as

W(r; s) = 1s

I0

(√sν

)− I0

(r√

sν

)

I2

(√sν

) sQ(s) = sQ(s)3(r; s), (5.50)



with

3(r; s) = 1s

I0

(√sν

)− I0

(r√

sν

)

I2

(√sν

) . (5.51)

Taking the inverse Laplace transform with respect to s, and using the convolution theorem forLaplace transforms, we find the solution w(r, t) of (5.38) as

w(r, t) = L−1 {sQ(s)3(r; s), s, t} = (λ ∗ qi )(r, t)

=∫ t

0λ(r, t − τ)qi (τ ) dτ, (5.52)

where we have used that q(t) = qi (t), and where the function λ(r, t) is the inverse Laplacetransform of 3(r; s), i.e.

λ(r, t) = L−1 {3(r, s), s, t} = 12π ı

limz→∞

∫ c+ı z

c−ı z3(r, s)est ds. (5.53)

The above integral can be computed by determining the residues of the function 3(r, s)est and byapplying the l’Hôpital rule. This is done in the following way:

1. The function 3(r; s) has a singularity for s = 0. For s → 0 a Laurent series for 3 can becomputed. This leads to

3(r; s) = 2(1− r2)

s+O(1). (5.54)

The residue of 3(r; s) in s = 0 is given by the coefficient of s−1, i.e.

Ress=03(r; s) = 2(

1− r2). (5.55)

2. Other singularities of 3(r; s) can be found by computing the zeros of the function

N (s) = I2

(√sν

). (5.56)

This function does not have any singularities in the positive real plane, except for s = 0.To investigate the singularities in the negative real plane, denoted with sk (k = 1, 2, ...), weintroduce the variable σ , according to

√s/ν = ıσ . This results in

N (s) = I2 (ıσ) = −J2 (σ ) , (5.57)

and so3(r; s) = 1

νσ 2J0 (σ )− J0 (rσ)

J2 (σ )= 3(r; σ). (5.58)

The zeros σk of N (σ ), corresponding to sk , are taken on the positive real axis and can becomputed by numerical methods. Using that

N ′(s) = 12ν

√ν

s

(I1

(√sν

)− 2

√ν

sI2

(√sν

))

= 12νσ

(J1(σ )− 2

σJ2(σ )

), (5.59)

or, with J2(σk) = 0,

N ′(sk) = 12ν

J1 (σk)

σk= 1

4νJ0 (σk) , (5.60)



where we used the recurrence relation (2α/x)Jα(x) = Jα−1(x) + Jα+1(x) in the final step.We find with use of the l’Hôpital rule that

Ress=sk3(r; s) = −4σ 2

k

J0 (σk)− J0 (rσk)

J0 (σk), (5.61)

NOTE: There are no complex valued zeros of N (s).

Altogether this gives for λ(r, t)

λ(r, t) =∞∑

k=0

Ressk

(3(r, s)est)

= 2(

1− r2)−∞∑

k=1

4σ 2

k


J0 (σk)e−νσ

2k t . (5.62)

To conclude we find for the axial velocity w(r, t) the expression

w(r, t) =∫ t

0λ(r, t − τ)qi (τ ) dτ

=2(

1− r2) ∫ t

0qi (τ ) dτ −

∫ t

0

{ ∞∑

k=1

4σ 2

k


J0 (σk)e−νσ

2k (t−τ)

}qi (τ ) dτ

=2(

1− r2)

qi (t)−∞∑

k=1

4σ 2

k


J0 (σk)

{∫ t

0e−νσ

2k (t−τ)qi (τ ) dτ

}. (5.63)

For later purposes and to keep up the readability of this result, we here introduce the followingshort-hand notations (for k = 1, 2, ...):

Jk(r) = 4σ 2

k


J0 (σk), (5.64)

8k(t) =∫ t

0e−νσ

2k τ qi (t − τ) dτ. (5.65)

Note that we changed the integration variable in the definition of 8k(t) with respect to (5.63) inorder to make the time derivative of the sum of 8k(t) converge. Moreover, note here that for agiven, and not too complicated, expression for qi (t), these integrals can be calculated analytically.And so we finally arrive at

w(r, t) = 2(

1− r2)

qi (t)−∞∑

k=1

Jk(r)8k(t). (5.66)

5.3 Zeroth order solution for the axial velocity

Our next step will be to find a zeroth order solution for a sudden inlet flow in a semi-infinite, flexible(so R = R(z, t) = 1 + δ/2p(z, t)) vessel, as described by the system under item 1. on page 52.One should realize that the flexibility of the vessel wall is already incorporated by the inclusion ofthe first term in (5.6) or (5.68). For the sake of clarity, we will omit the 0 index in the sequel. The



problem is described by the system

∂w

∂t− ν

r∂

∂r

(r∂w

∂r

)= 5(z, t) := −∂p

∂z, (5.67)

∂p∂t+ ∂q∂z= 0, (5.68)

∂q∂t+ ∂p∂z+ τw = 0, (5.69)


p(z, 0) = q(z, 0) = vz(r, z, 0) = 0 for z > 0,

vz(1, z, t) = 0 for z > 0, t > 0,

q(0, t) = qi (t) for t > 0, qi (t) is prescribed,

p(z, t)→ 0 for z→∞, t > 0.

(5.70)

Note that5(z, t) = −∂p

∂z= ∂q∂t+ τw. (5.71)

We suggest three possible methods to solve the above system of equations by expressing thesolution in terms of 5(z, t) or q(z, t):

1. The first option is to use the technique of separation of variables by introducing a Fourier-Bessel series (in analogy with the standard Fourier serie) for the axial velocity. This leadsto the general form

w(r, z, t) = 2∞∑

k=1

J0 (λkr)λk J1 (λk)

�k (z, t) , (5.72)

where

�k (z, t) =∫ t

z5(z, τ ) e−νλ

2k (t−τ) dτ H(t − z), (5.73)

and where λk(k = 1, 2, ...) are the consecutive zeros of the zeroth-order Bessel function ofthe first kind J0(·).

2. The second option is to follow the method used in the previous section, but with the pre-scribed qi (t) now replaced by the unknown q(z, t). The result is (for 0 < z < t)

w(r, z, t) = 2(

1− r2)

q(z, t)−∞∑

k=1

Jk(r){∫ t

0e−νσ

2k (t−τ)q(z, τ ) dτ

}. (5.74)

3. The third option is again to use separation of variables, but this time by expressing the axialvelocity as a finite sum of Zernike polynomials. We find (again for 0 < z < t)

w(r, z, t) = (1− Z2(K+1) (r))

q(z, t)+K∑

k=1

wk (z, t)(Z2k (r)− Z2(K+1)(r)

), (5.75)

with q(z, t) and wk(z, t) still unknown.

Remark that in all three options there stil is a lot of work to be done, since we need to expressthe solution in terms of the initial inlet flow function qi (t) instead of q(z, t). In this section we optfor the first solution method to work out in all detail.



We start by introducing a Fourier-Bessel series for w(r, z, t), i.e.

w(r, z, t) =∞∑

k=1

wk(z, t)J0 (λkr) , (5.76)

where wk(z, t) are Fourier-Bessel coefficients and λk the consecutive roots of J0 (·). If we substi-tute (5.76) into (5.67) and rewrite the right-hand side of that same equation by means of

1 =∞∑

k=1

2J0 (λkr)λk J1 (λk)

, (5.77)

we arrive at∞∑

k=1

{∂wk

∂tJ0 (λkr)+ νλ2

k J0 (λkr) wk (z, t)}=∞∑

k=1

5(z, t)2J0 (λkr)λk J1 (λk)

, (5.78)

where we used that1r

ddr

(r

dJ0 (λkr)dr

)= −λ2

k J0 (λkr) . (5.79)

Since (5.78) should hold for every k, we see that, after dividing by J0 (λkr), we are left with

∂wk

∂t+ νλ2

kwk = 25λk J1 (λk)

, for all k = 1, 2, ... (5.80)

This equation has as general solution

wk (z, t) = e−νλ2k t[

ck(z)+ 2λk J1 (λk)

∫ t

z5(z, τ ) eνλ

2kτ dτ

]. (5.81)

Since w(z, r, t) = 0 for t = 0 (see (5.70)1), we conclude that ck = 0 and thus

w(r, z, t) = 2∞∑

k=1


�k (z, t) , (5.82)

with �k(z, t) as defined in (5.73). This solution is indeed the same as the one suggested in (5.82).

Our next goal is to determine5(z, t), and consequently �k (z, t), in terms of the inlet flow functionqi (t). First of all we derive with (5.82) and the definitions for q(z, t) and τw(z, t) in Section 5.1,

q(z, t) = 2∫ 1

0rw(r, z, t) dr = 4

∞∑

k=1

1λ2

k�k (z, t) , (5.83)

τw(z, t) = −2ν∂w

∂r(1, z, t) = 4ν

∞∑

k=1

�k (z, t) . (5.84)

With these expressions and (5.68), we find a relation for 5(z, t):

∂5

∂t= ∂2q∂z2 = 4

∞∑

k=1

1λ2

k

∂2�k

∂z2 . (5.85)

NOTE: The equation (5.69) is automatically full-filled since∞∑

k=1

1λ2

k= 1

4. (5.86)

A derivation of the latter equality, as well as a derivation of (5.77), can be found inAppendix C.



We will solve (5.85) by means of Laplace transforms. For this, we change the coordinate systemfrom (z, t) to (η, ξ), with η = z and ξ = t − z, such that

5(z, t) = 5(z, t − z) = 5(η, ξ). (5.87)

This gives

�k (z, t) =∫ t

z5(z, τ ) e−νλ

2k (t−τ) dτ

=∫ ξ+η

η

5(η, τ) e−νλ2k (ξ+η−τ) dτ

=∫ ξ

05(η, y + η) e−νλ

2k (ξ−y) dy

=∫ ξ

05 (η, y) e−νλ

2k (ξ−y) dy = �k(η, ξ). (5.88)

Rewriting (5.85) in terms op 5 and �k , we obtain

∂5

∂ξ= 4

∞∑

k=1

1λ2

k

(∂2

∂η2 − 2∂2

∂η∂ξ+ ∂2

∂ξ2

)�k . (5.89)

We will solve (5.89) for 5 and at the end we go back to 5. We define the Laplace transform of5(η, ξ) by

P (η; s) = L(5 (η, ξ) ; ξ, s

)=∫ ∞

05 (η, ξ) e−sξ dξ. (5.90)

Applying the Laplace transform (with respect to ξ ) to both sides of (5.89) and using the abovedefinition, we arrive at

sP (η; s)− 5(η, 0) = 4∞∑

k=1

1λ2

k

(∂2

∂η2 Ok(η; s)− 2∂

∂η

(sOk(η; s)− �k(η, 0)

)+

s2Ok(η; s)− s�k(η, 0)− ∂�k

∂ξ(η, 0)

), (5.91)

whereOk(η; s) = L

(�k(η, ξ), ξ, s

)= P (η; s)

s + νλ2k. (5.92)

From (5.88) we immediately see that �k(η, 0) = 0. From this same identity we can derive that∂ξ �k(η, 0) = 5(η, 0), which means that these two terms cancel out. This reduces (5.91) to

∂2

∂η2 P (η; s)− 2s∂

∂ηP (η; s)+

(s2 − ρ2(s)

)P (η; s) = 0, (5.93)

with

ρ(s) =√

s∑∞

k=14

λ2k(s+νλ2

k)=

√√√√√√s2I0

(√sν

)

I2

(√sν

) ; (5.94)

see Appendix C for the derivation of the last step in (5.94). The general solution of this equation,meeting the condition P → 0 for η→∞, is

P (η; s) = P0(s)e−(ρ(s)−s)η, (5.95)



where the function P0(s) is still to be determined. To this end, we propose a boundary conditionto 5(η; s). We know that 50(ξ) := 5(0, ξ) := 5(0, t) = 50(t). With (5.73) and (5.83) we find

qi (t) = limz↓0

q(z, t) =∞∑

k=1

4λ2

k

∫ t

050(τ )e−νλ

2k (t−τ) dτ. (5.96)

The Laplace transform of (6.1) yields

Qi (s) = P0(s)∞∑

k=1

4λ2

k(s + νλ2

k) = s

ρ2(s)P0(s), (5.97)

with Qi (s) the Laplace transform of qi (t). So, we conclude that

P (η; s) = ρ2(s)s

Qi (s)esηe−ρ(s)η. (5.98)

Moreover, we find that

W(r, η; s) = L (w (r, η, ξ) ; ξ, s) = 2∞∑

k=1


L(�k (η, ξ) ; ξ, s

)

= P (η, s)∞∑

k=1

2J0 (λkr)λk(s + νλ2

k)

J1 (λk)

= ρ2(s)s

I0

(√sν

)− I0

(r√

sν

)

s I0

(√sν

) Qi (s)esηe−ρ(s)η; (5.99)

the derivation of the last step of (5.99) can be found in Appendix C. With (5.94) we derive

ρ2(s)s

I0

(√sν

)− I0

(r√

sν

)

s I0

(√sν

) =I0

(√sν

)− I0

(r√

sν

)

I2

(√sν

) = s3(r; s) , (5.100)

with 3(·, ·) as defined in (5.51). Substitution of (5.100) in (5.99) yields

W(r, η; s) = 3(r; s) sQi (s)esηe−ρ(s)η (5.101)

Inverse transformation of (5.101) results in (analogous to the procedure in (5.51) - (5.66) in thepreceding section)

w(r, η, ξ) = L−1{3(r; s) sQi (s)esηe−ρ(s)η

}

=∫ ξ

0L−1 {3(r; s) sQi (s)} (x)L−1

{esηe−ρ(s)η

}(ξ − x) dx

=∫ ξ

0g (r, x) f (η, ξ + η − x) dx =

∫ t−z

0g (r, x) f (z, t − x) dx (5.102)

=∫ t

zg (r, t − x) f (z, x) dx = w(r, z, t) (5.103)

where

g(r, t) = L−1 {3(r; s) sQi (s)} (x) =∫ t

0λ (r, t − τ) qi (τ ) dτ

= 2(

1− r2)

qi (t)−∞∑

k=1

Jk(r)8k(t), (5.104)



which is identical to (5.66), where λ(·, ·) the inverse transformation of 3(·, ·) (as in (5.62)), and

f (z, t) = L−1{

e−ρ(s)z; s, t}. (5.105)

The evaluation of the latter inverse Laplace transform can be found in Appendix B.

5.4 An approximative zeroth order solution

In the previous section we derived an exact solution for the zeroth order approximation of theaxial velocity. This solution is given by

w(r, z, t) =∫ t

zg (r, t − x) f (z, x) dx, (5.106)

where

g(r, t) = 2(

1− r2)

qi (t)−∞∑

k=1

Jk(r)8k(t), (5.107)

andf (z, t) = L−1

{e−ρ(s)z; s, t

}. (5.108)

It turns out that the obtained solution for the inverse Laplace transform f (z, t), however exact, israther complex, and in the form of an infinite series containing integrals that can only be calcu-lated numerically. Of course, we can substitute this series-representation for f (z, t), truncatedafter a finite number of terms, into (5.103) and compute w(r, z, t) for any given qi (t).

However, for the sake of simplicity, we opt here for an alternative approach in which we usean approximation for f (z, t), which seems us useful for practical purposes and certainly for smallvalues of ν, when the boundary layer at the wall is most steep. For larger values of ν the flowprofile starts to resemble more and more the classical, quadratic, Poiseuille profile. An extra ad-vantage inherent to the approximation for f (z, t) that we shall derive below is that it enables us,as we will show further on, to compare this solution with the one of Bessems er al., [10], that wasused in the preceding chapter. We proceed with the derivation of this approximation.

Observe that, if F (z, s) = e−ρ(s)z , F satisfies

d2F

dz2 − ρ2(s)F = 0, F (0, s) = 1, (5.109)

whereρ2(s) = s

4∑∞

k=11

λ2k(νλ2

k+s). (5.110)

Substituting (5.110) into (5.109) and using (5.86), leads us to

∞∑

k=1

1λ2

k

[1

νλ2k + s

d2F

dz2 − sF

]= 0. (5.111)

For the first approximation, we only take into account the contribution for k = 1. So F ≈ F (1),where F (1) is the solution of

d2F (1)

dz2 −(νλ2

1 + s)

sF (1) = 0, F (1)(0, s) = 1. (5.112)



So

F (1)(z; s) = e−√

s(s+νλ2

1)

z, (5.113)

and, analogous to (4.79), but with τ → νλ21/4,

f (1)(z, t) = L−1{F (1)(z; s); s, t

}

= e−12 νλ

21t

δ(t − z)+ 1

2νλ2

1zI1

(12 νλ

21

√t2 − z2

)

√t2 − z2

H(t − z)

. (5.114)

Notice that for ν ↓ 0 the exact solution for f (z, t) and the approximated one f (1)(z, t) coincide,and thus, it seems reasonable to assume that f (1)(z, t) is a good approximation for f (z, t) forsmaller values of ν (say for νλ2

1 � 1).

5.4.1 Derivation of the approximative flow characteristics

We now substitute the approximation f (1)(z, t), (5.114), for f (z, t) into (5.103), but we keep the fullsolution for g(r, t). Again, to keep the final result readable, we introduce the number α1 = νλ2

1/2and following short-hand notation:

A(z, t) =I1

(α1√

t2 − z2)

√t2 − z2

. (5.115)

With these abbreviations, (5.103) results in (with w(r, z, t) ≈ w(1)(r, z, t)), and for 0 < z < t

w(1)(r, z, t) =∫ t

zg(r, t − x) f (1)(z, x) dx

= e−α1z

{2(

1− r2)

qi (t − z)−∞∑

k=1

Jk(r)8k(t − z)

}

+ 2α1z(

1− r2) ∫ t

zA(z, x)qi (t − x)e−α1x dx

− α1z∞∑

k=1

Jk(r){∫ t

zA(z, x)8k(t − x)e−α1x dx

}(5.116)

If we define the functions H(z, t) and Hk(z, t) (for k = 1, 2, ...) as

H(z, t) =∫ t

zA(z, x)qi (t − x)e−α1x dx, (5.117)

Hk(z, t) =∫ t

zA(z, x)8k(t − x)e−α1x dx, (5.118)

then the final result for w(1)(r, z, t) reads

w(1)(r, z, t) = e−α1z

{2(

1− r2)

qi (t − z)−∞∑

k=1

Jk(r)8k(t − z)

}+

α1z

{2(

1− r2)

H(z, t)−∞∑

k=1

Jk(r)Hk(z, t)

}. (5.119)



The goal is to compare the solutions for the flow characteristics in the global equations of motionwith the ones in the local case. With the approximative expression w(1)(r, z, t) for the zeroth orderaxial flow velocity, we are able to also derive zeroth order expressions for q(z, t) and p(z, t), butespecially expressions for τw(z, t) and γa(z, t). Using the definitions given in (5.28) leads us, for0 < z < t , to

q(z, t) = e−α1zqi (t − z)+ α1zH(z, t), (5.120)

τw(z, t) = 8ν(e−α1zqi (t − z)+ α1zH(z, t)

)+ 4ν∞∑

k=1

(e−α1z8k(t − z)+ α1zHk(z, t)

)(5.121)

Remark the resemblance between the solution for q(z, t) with the one we found in (4.88) in Sec-tion 4.3. Since we make an approximation for f (z, t), we need to choose p(z, t) to satisfy (5.68) or(5.69). From the observation that q(z, t) resembles the solution from Chapter 4, it seems reason-able to opt for the first possibility and so we choose to take also for p(z, t) the same comparableform as the solution for p(z, t) in Section 4.3. So we find

p(z, t) =∫ t

zI0

(α1

√x2 − z2

)(qi (t − x)+ 2α1qi (t − x)) e−α1x dx . (5.122)

Finally, we can use definition (5.33)2 to derive a zeroth order expression for γa(z, t). Note that thisdefinition is depending on the zeroth order expression for the axial velocity but is part of the firstorder system (due to the δ coefficient of ∂γa/∂z in 5.18). Since

(w(1)(r, z, t)

)2 =(

2(

1− r2) (

e−α1zqi (t − z)+ α1zH(z, t))

−∞∑

k=1

Jk(r)(e−α1z8k(t − z)+ α1zHk(z, t)

))2

= 4(

1− r2)2

q2(z, t)+( ∞∑

k=1

Jk(r)(e−α1z8k(t − z)+ α1zHk(z, t)

))2

−∞∑

k=1

4(

1− r2)

Jk(r)q(z, t)(e−α1z8k(t − z)+ α1zHk(z, t)

), (5.123)

we find that

γa(z, t) = 43

q2(z, t)+∞∑

k=1

4σ 2

k


)2

− q(z, t)∞∑

k=1

8σ 2

k


), (5.124)

where we used the orthogonality property for Bessel functions for the quadratic term of Jk(r),such that

∫ 1

0rJk(r)Jl(r) dr =

0 for k 6= l,4σ 2

kfor k = l.

(5.125)



1

πt

hh(1)

Figure 5.1: Plots of h(z, t) (numerical solution) and h(1)(z, t) (analytical approximation) for z = 1,0 < t < 2π and ν = 0.03.

5.4.2 The accuracy of the approximative solution

By introducing the approximation f (1)(z, t) for f (z, t) we agreed on also introducing an errorwith respect to the exact solution. In this subsection we try to estimate the difference. One ofour primary goals was to solve all systems analytically as far as possible, which supports thechoice for f (1)(z, t). However, there are numerical methods available for calculating an numer-ical approximation of the exact inverse Laplace transforms. For a first impression, we compareour approximation f (1)(z, t) with a numerical approximation. We opt to use the built-in functioninvlap.m in Matlab, which is based on [2].

Since f (1)(z, t) contains a delta function, we want to avoid the generation of a delta functionin calculating the numerical solution. To this end, we do not compare f (1)(z, t) with f (z, t), buth(1)(z, t) = −∂z f (1)(z, t) with h(z, t) = −∂z f (z, t). The resulting plots, for 0 < t < 2π , z = 1and a typical value for ν(= 0.03), are shown in Figure 5.1. We observe that around t = z thereis a difference between the behavior of both functions, but in general we may conclude that thedifference of both solutions is small compared to the O(1) behavior of the flow characteristics.

An other way to investigate the accuracy of the approximation, is to see in how far the approxima-tive flow characteristics are in agreement with the equations of motion (5.67)-(5.69). In the sequelof this section we omit (1), since it is clear that we are dealing with the approximative solutions.From (5.116) we deduce that

∂w

∂t= e−α1z

{2(

1− r2)

qi (t − z)−∞∑

k=1

Jk(r)8k(t − z)

}+

α1z

{2(

1− r2) ∂H∂t−∞∑

k=1

Jk(r)∂Hk

∂t

}

= ∂q∂t+ 4ν

∞∑

k=1

J0 (σk)− J0 (σkr)J0 (σk)

(e−α1z8k(t − z)+ α1z Ik(z, t)

), (5.126)

where in the last step we used that, according to (5.65),

8k(t) = qi (t)− νσ 2k8k(t), (5.127)



and that for 0 ≤ r < 1∞∑

k=1

Jk(r) = 1− 2r2. (5.128)

Moreover

− νr∂

∂r

(r∂w

∂r

)= 8ν

(e−α1zqi (t − z)+ α1zH(z, t)

)+

ν

r

∞∑

k=1

4J1 (σkr)σk J0 (σk)


)+

ν

∞∑

k=1

2 (J0 (σkr)− J2 (σkr))σk J0 (σk)


)

= 8νq(z, t)+ 4ν∞∑

k=1

J0 (σkr)J0 (σk)


), (5.129)

which, taken together, yields

∂w

∂t− ν

r∂

∂r

(r∂w

∂r

)= ∂q∂t+ 8νq + 4ν

∞∑

k=1


). (5.130)

We immediately conclude that indeed

∂w

∂t− ν

r∂

∂r

(r∂w

∂r

)= ∂q∂t+ τw. (5.131)

From Chapter 4 we know that for the approximative q(z, t) from (5.120) and p(z, t) from (5.122),(5.68) is already satisfied. So, as a criterium for the correctness of our approximation, we will use(5.71). From expression (5.122) for p(z, t) we see that

5(z, t) = −∂p∂z= (qi (t − z)+ 2α1qi (t − z)) e−α1z+

α1z∫ t

zA(z, x) (qi (t − x)+ 2α1qi (t − x)) e−α1x dx

= ∂q∂t(z, t)+ 2α1q(z, t). (5.132)

So, we might conclude that the approximation is permitted whenever the difference of 5(z, t) and∂q/∂t + τw is small, i.e.

(8ν − 2α1

)q(z, t)+ 4ν

∞∑

k=1


) ≈ 0, (5.133)

which means

maxz,t

∣∣∣∣∣(8ν − 2α1

)q(z, t)+ 4ν

∞∑

k=1


)∣∣∣∣∣� max

z,t|5(z, t)| . (5.134)

We will discuss this result in the next chapter, when we choose an explicit function for qi (t).




In this chapter we found an analytical zeroth order (in δ) solution for the axial velocity as part ofthe local 3-D equations of motions describing the flow through a semi-infinite, deformable ves-sel. As part of the solution tactic, we first solved the same problem for a rigid (undeformable)vessel with constant internal radius. The solution for the axial velocity depends on a rather com-plicated inverse Laplace transform, which for reasons of simplification, has been approximatedanalytically. This approximated solution for the axial velocity enabled us to also find zeroth orderapproximated solutions for the pressure distribution, the volumetric flow rate, the radial diffusionterm and the advection term.

In the next chapter we hope to compare the resulting expressions for the approximated flowquantities from this chapter, based on the local 3-D axial velocity, with the solutions from theglobal 1-D equations of motion in the previous chapter. Of special interest will be comparisonof the radial diffusion term and the advection term, which in the global case both are based onBessems et al. [10]. We hope to establish a classification, based on these comparisons, for theflow characteristics for a set of representative human arteries.



Chapter 6Results and discussion

In Chapter 4 we derived solutions for the flow rate and pressure distribution from theglobal equations of motion. The expressions for the radial diffusion and the advection con-tribution are based on Bessems [10]. In Chapter 5 we derived a zeroth order approximationfor the axial velocity from the local equations of motion, which gave rise to expressions forthe flow rate and pressure distribution according to this zeroth-order approximation. Moreimportantly, we were also able to derive the radial diffusion and the advection contributiondue to this expression for the axial velocity. In this chapter we will plot the results from theprevious two chapters for a given expression for the inlet-flow function. We compare theresults from the plots and discuss the differences between the solutions from the globalequations of motion and the solutions from the local ones.

6.1 Results

An overview of all the expressions derived in the previous two chapters are recapitulated in Table6.3. We added the subscripts {c,v ,d ,a } to distinguish between solutions for constant wave veloc-ity, variable wave velocity, radial diffusion and advection, respectively, for the global equation ofmotion and the subscript l the solutions from the local equations of motion.

All solutions derived are expressed in terms of the inlet-flow function q(0, t) = qi (t). In orderto be able to plot the solutions and compare them we need an expression for the scaled inlet-flowfunction qi (t). We choose for

qi (t) = 2t (π − t)(1+ t2

) (√1+ π2 − 1

) H(t) H(π − t), (6.1)

which is zero for t < 0 and t > π and has a maximum value of one. A plot of (6.1) can be found inFigure 6.1. Besides the inlet-flow function, we also introduced a lot of geometrical and physicalparameters, inclusive the dimensionless numbers, for describing the flow through the vessel. Allthese quantities are calculated for a set of seven typical representative human arteries, denotedwith vessel (A)-(H) (leaving out vessel (G), since it is very much the same as vessel (F)), basedon data from [10] and [1]. See Table 6.2, page 73, for all details. The choice of arteries followsthe choice of Bessems [1] and the position of the arteries is schematically presented in Figure6.2. In the analysis in Chapter 4 and 5, the dimensionless compliance δ plays an important role.Therefore, we choose to base our results on three vessels, which correspond to three typicalvalues of this number δ; namely vessel A (δ = 0.12), vessel B (δ = 0.24) and vessel D (δ = 0.04).



1

πt

qi(t)

1

πt

qi(t)

(a) (b)Figure 6.1: Plot of the scaled inlet-flow function as defined in equation 6.1.

δ qc qv qd qa pc pv pd pa τw γa

(A) 0.12 +3% + 3% -11% +1% +2% +1% -4% -9% +10% +30%

(B) 0.24 +10% + 10% -17% +9% +7% +1% -4% -19% +5% +18%

(D) 0.04 +43% +43% -13% +43% +26% +24% +5% +21% +6% +93%

Table 6.1: Average percentages for the differences of the solutions for qc,v,d,a(z, t) with respect toql(z, t), pc,v,d,a(z, t) with respect to pl(z, t) and for the global solutions for τw and γa with respectto local ones.

NOTE: In Table 6.2, Lv is the actual length of the vessel (for this we took the sum ofthe values from [10] per artery), and L = c0/ω is the characteristic length for the wavepropagation in a semi-infinite vessel, as it is used in our model. The parameter ε =Ra/L is defined on L, rather than on Lv, and is for all vessels considered less than 5%,which justifies the neglect of O

(ε2)-terms in (3.24) and (3.25), Chapter 3. However,

we note that although Lv < L always, even if we would define ε on Lv, then stillε ≤ 5%, except for case A, for which ε ≈ 36%. In the latter case the aforementionedapproximation becomes critical (albeit not in our semi-infinite model).

In Figure 6.3, on page 75, we plotted the axial velocity as a function of r for certain time t and afew values of z. In this same Figure, but in the other column, we plotted 5(z, t) = −∂p/∂z andthe error, as derived in (5.134). In Figure (6.4)-(6.7), on pages 76-79, the solutions for the flowrate q(z, t), the pressure distribution p(z, t), the shear force τw(z, t) at the wall and the advectiondistribution γa(z, t) are plotted; in column (a) as a function of t for two values of z, and in column(b) as a function of z for two values of t . Except for the plots of the error in Figure 6.3, all plottedquantities are dimensional, whereas the variables r , z and t are dimensionless.

In order to get some inside into the difference between the global solutions and the local so-lutions, we calculated the ratio of the maximum of a global solution and the maximum of thecorresponding local solution and took the average of the results; in all cases there are four re-sults: two from column (a) and two from column (b) in each of the plots. The results from thesecalculations can be found in Table 6.1.



6.2 Comments on and discussion of the results

In this section we comment on the results, as presented in the previous section and we drawconclusions based on the observations we make. Here, we follow the order of the figures onpages 76-79.

1. The axial velocity w(r, z, t) is plotted in Figure 6.3, column (a), as a function of r for t = πand z = {0, 0.1, 0.25, 0.5, 0.75}π . For small values of z (with respect to time t), we see thatthe axial velocity is negative for r → 1, which means that in a small boundary layer near thevessel wall, fluid is flowing backward for small values of the axial coordinate. This effect ismathematically caused by the contribution of the zH term in the expression of w (see Table6.3). For small values of ν, i.e. vessel (A) and (B), the axial velocity profile near by the flowfront, i.e. for z such that t − z � t , resembles that of a plug flow. This is caused by theeffect of the combination of the exp(−α1z)qi and zH term. For vessel (D), the larger valueof ν, a more quadratic Poiseuille-like profile establishes for z near to t . Finally, we noticethat the larger δ gets, the larger the value of the axial velocity at the center of the vessel,i.e. in r = 0, becomes. This is caused by the elasticity of the vessel wall, which is higher forlarger values of δ and leads to a larger cross-sectional area and therefore to a higher flowrate and a higher velocity.

2. In Section 5.4.2 we discussed the accuracy of our solution due to the approximation wemade for the inverse Laplace transform f (z, t). In Figure 6.3, column (b), we plotted 5(z, t)as a function of t , where we choose 5 = −∂p/∂z, and the error defined by (5.134) whichis the difference of 5 = −∂p/∂z and 5 = ∂q/∂t + τw. In all three different cases for δwe conclude that the error is much smaller (less than 15%) than 5(z, t), which means thatthe approximation we made for f (z, t) is suitable. We leave a detailed discussion about thebehavior of5(z, t) out of consideration, since we do not need it to draw the final conclusions.

3. The solutions for the volumetric flow rate q(z, t) are shown graphically in Figure 6.4, wherein column (a) we plotted q(z, t) as a function of t for two values of z, whereas in column(b) q(z, t) is plotted as a function of z for two values of t . The same has been done for thepressure distribution p(z, t) in Figure 6.5. Here, we combine the discussion of the plots forq and p, since much of the observations hold for both cases.

First of all we note, as a trivial observation, that for z = 0 (see column (a)), all solutionsfor q(z, t) are exactly the same, which is in agreement with the solutions we obtain if wesubstitute z = 0 in the solutions for q(z, t) in Table 6.3.

The solutions with constant wave velocity, i.e. qc and pc (red), perform like stationary waves,which means that the wave profiles (which are equal to qi (t)) do not change in time andplace and travel with constant wave speed c0. For both vessel (A) and (B) we concludefrom the plots and from Table 6.1 that the solutions show great resemblance (in both theprofile and the amplitude) with the solutions from the local equations of motion (green). Forvessel (D) (i.e. for small δ), we see that the amplitudes differ a lot and so for this case thesolution with constant wave velocity does not satisfy and we definitely need to include thecontribution of the radial diffusion; notice that for this case ν is relatively large, and hence,induces a large viscous resistance.

NOTE: For solutions with constant wave velocity, we have δ = 0 in (4.11). How-ever, we should realize that this does not mean that the vessel is rigid. Thedeformation of the vessel is still incorporated in C∂p/∂t .

For the case in which the wave velocity is variable, so for the solutions qv and pv (orange),the waves travel with wave speed c, the wave phase increases with increasing δ and theamplitudes of the waves change (attenuation). So, here no qualitative differences but only



quantitative ones are observed. An exception has to be made for the wave front at z = t .Here we see that the derivatives of q and p for the larger values of t , z and δ at the wavefront z = t are infinite. This singular behavior can be explained from the fact that the charac-teristics are not parallel to each other as for δ = 0, but intersect with the frontcharacteristicz = t (the characteristic of t = 0), for δ > 0, while q and p are still nonzero. The result isa shockwave. The irregularity at the wave front is caused by the fact that we neglected allO(ε2)-terms in the derivation of the global equations of motion. If we did incorporate these

terms, then the axiale diffusion, i.e. ∂2vz/∂z2, would still have been present. This term issmall in general, however at the wave front z = t it is significant and gives rise to a boundarylayer in which the discontinuity we observe now, would have been spread out to a smoothprogression in both q and p. Again, in de same way as for the solutions for constant wavevelocity, we state here that the solutions are of the same order, however different in shape,of the solution obtained in Chapter 5.

The solutions in which the radial diffusion term is included, i.e. qd and pd (blue), perform likea traveling wave which is almost completely in phase with the solution from the local equa-tions of motion and has the same profile. This can be expected since the expressions forboth qc and pc are the same as the ones found in Chapter 5, apart from a small differencein the characteristic parameters. In all plots we observe that the amplitude is smaller thanthe amplitude for local solutions, but the difference is, also regarding the results in Table6.1, within an acceptable range for qd (approximately 15 %) and very good for pd (rangingfrom -4% to + 5%). Finally, we should notice that the solutions for pd are still larger thanzero for π < t < 2π in column (a) and for 0 < z < π in column (b). This effect is due to thecontribution of qi in the solution pd (and also pl ) at t = π or z = π , respectively, where thevalue of qi is positive, not zero.

For qa and pa (cyan), i.e. the solutions where we incorporated the advection term, weobserve that the results show a triangular-like profile with fast decreasing amplitude for thelarger values of δ. Just as for the solutions with the variable wave velocity, remarkablediscontinuities at z = t are present. Here, the problem lies in the fact that we neglectedhigher order δ-terms, i.e. terms of O

(δ2), in (4.106), (4.110), but also in the definition of

t0 in (4.118). Due to this assumption, the solution performs a phase difference which getshigher whenever δ is larger. We may conclude, also regarding the results in Table 6.1, thatthe solutions obtained for the inclusion of the advection term, even for the small value of δ,are highly unreliable. Note that in the other cases we were able to find an exact solution, buthere we had to neglect higher order terms of δ to find a solution. We would have found anexact solution when we would have neglected the compliance contribution in (4.98). How-ever, we can not do this since we are then leaving out a term which is of the same order asthe effect we are investigating, namely ∂γa/∂z.

4. In Figure 6.6, the resulting plots for τw according to the solutions in Chapter 4 (blue) and 5(red) can be found as a function of time t for two values of z in column (a) and as a functionof z for two values of t in column (b). In column (a) we see, that for both solutions, the valueof τw is positive in the beginning but gets negative after some time, whereas in column (b)the same effects is visible but in opposite order. Furthermore, we observe that the solutionfrom the global equations of motion shows a discontinuity at z = t , which is due to the factthat ∂p/∂z is discontinuous at z = t . The solution from the local equations of motion doesnot perform this discontinuity.

However, here we observe another important effect. In column (a) for example, we seethat the solution at z = 0 is not equal to zero for t > π , but gradually tends to zero fort > π . This effect is caused again, just as in the discussion for pd , by qi and finds its originin the

∑exp(α1z)8k(t − z) term. Physically this effects seems plausible, when we imagine

that the shear force at the wall fades away gradually after the passing of a single pulse.



Furthermore, we notice from the plots and from Table 6.1 that, maybe except for a smallenvironment near z = t , both the solutions are in agreement with each other in a qualitativebut also quantitative way, i.e. in both the amplitude and the shape of the solution. Theinfluence of δ, or in this context more suitable, the influence of ν only effects the magnitudein the way that the smaller ν gets, the bigger the amplitude of τw (initially) becomes.

5. The solutions for γa are plotted in Figure 6.7, again as a function of t for two values of zin column (a) and as a function of z for two values of t in column (b). The solution for γafrom the global equations of motion (blue) shows the same effects as we described for thesolution qa and pa , i.e. the discontinuity at z = t due to the neglect of the O

(δ2)-terms.

We observe an almost triangular-like behavior and a larger fluctuation in the amplitude forall three values of δ. This in contrast with the red curves, the solutions for γa from thelocal equations of motion, which show a nice smooth traveling wave. We might concludethat, also from Table 6.1, the global solution has not the desired effects and therefore it isunusable to compare and to draw conclusions from it. One final remark has to be made onthe solution of γa from the local equations of motion if we compare it with the solution ofτw from the local equations of motion. We see for vessel (A) and (B) that both quantitiesare almost of the same order, whereas for vessel (D), the small value of δ, the value of γais a factor 10 smaller than the value of τw which implies that for the smaller vessels theradial diffusion is much more important than advection, which is very logical and as can beexpected.

6. As already mentioned before, most of the work done is based on Bessems [1]. So finally, itseems reasonable to compare the results obtained in Figures (6.4)-(6.7) with the results in[1], especially Figures 6.4-6.6, where plots of q, p and τw can be found for the same vessels(A), (B) and (D) as discussed in this report. Unfortunately, no plots of γa are available in [1].However, since we already concluded that our results for γa do no meet our expectations,we don’t feel the urge to to compare them with other investigations, such as the ones inBessems [1].

In we compare the plots for q, we see, although we only regarded a single pulse whereasBessems in [1] regards a periodic pulse, that both figures show great resemblance. Ofcourse, this is expected, since we based our values for Q0 on these pictures. The differ-ence in shape and in time scale is due to the choice we made for the inlet flow function qi ,which is a very simplify representation of the real inlet flow. For the graphs of the pressuredistribution p(z, t) we see that the shape of the graphs show the same behavior. However,the maximum values of p(z, t) in Figure 6.5 and Figure 6.4 of [1] differ about a factor five.One of the reasons could be that we regard a semi-infinite vessel. This means that inour case no reflection of waves will occur. If we would have taken a finite vessel in whichreflection occurs, then demping and strengthening will take place which will increase themaximum value for p(z, t).

If we want to compare the results of Bessems in [1] for τw with with our results, we needto realize that there is a small difference in the definition of τw. Except that there is a dif-ference in sign between both definitions, we also see that our definition of τw compareswith (2πRa/ρ)τw,B , where τw,B is the definition Bessems uses in [1]. Since (2πRa/ρ) =O(10−5), at least for the three vessel (A), (B) and (D) we discuss here, we need to incorpo-

rate a factor 10−5 for the plots of τw in [1]. If we flip the results in Figure 6.6 of [1] aroundthe x-axis, which is due to the difference in sign of both definitions of τw, we see that thesolutions behave alike, even change of sign is present in both result after some time t . Wealso observe the same peak for τw around z = t . Even the amplitudes, when taking intoaccount 2π10−5, are of the same order of magnitude for vessel (A) and (B). The differencefor vessel (D) is approximately a factor 2. All together we may conclude that the results forτw are very accurate.



90 Chapter 6

5

6

12 16

18

25

22

24

27

31 32

38

39

45

43

50

51 44

48 41

4249

46

2335

26

1921

142

13 17

15

20

7

9

10

8

11

34 33

3028

3637

29

134

4047

A

B

C

D

E

FG

H

Figure 6.2: Representation of the 51 main arteries in the human arterial system.The numbers correspond to the artery numbers listed in table 6.1 and the charactersindicate the monitoring sites for visualising the computational results.

Figure 6.2: Representation of the human arterial system. Figure taken from [1], page 90



TYPE (A)A

orta

asce

nden

s

(B)A

orta

thor

acal

is

(C)A

orta

abdo

min

alis

(D)A

.sub

clav

ia

(E)A

.ilia

caex

tern

a

(F)-

(G)A

.fem

oral

is

(H)A

.tib

ialis

ante

rior

Lv [m] 0.04 0.16 0.16 0.10 0.08 0.32 0.34

Ra [10−2m] 1.46 0.77 0.58 0.41 0.29 0.25 0.13

h [10−3m] 1.63 0.99 0.81 0.67 0.55 0.51 0.39

E [106Pa] 0.4 0.4 0.4 0.4 0.4 0.4 1.6

Q0 [10−6m3/s] 420 260 70 14 8 7 2.5

V = Q0/(πR2a) [m/s] 0.63 1.40 0.66 0.27 0.30 0.36 0.47

C = (3πR3a)/(2Eh) [10−9m3sec2/kg] 22.49 5.43 2.84 1.21 0.52 0.36 0.02

c0 =√(πR2

a)/(ρC) [m/s] 5.32 5.71 5.96 6.44 6.94 7.20 17.45

L = c0/ω [m] 0.85 0.91 0.95 1.03 1.10 1.15 2.78

ε = Ra/L [10−2] 1.72 0.85 0.61 0.40 0.26 0.22 0.05

δ = V/c0 [−] 0.12 0.24 0.11 0.04 0.04 0.05 0.03

α = Ra√ω/ν [−] 19.56 10.31 7.77 5.49 3.89 3.35 1.74

Re = (Ra V )/ν [−] 2616 3070 1097 311 251 255 175

Sr = α2/(εRe) [−] 8.49 4.09 8.99 24.30 22.92 20.10 37.07

P = (Q0T )/(LC) [Pa] 3507 8375 4142 1793 2207 2694 8631

D = (Eh3)/9 [10−5Nm] 19.25 4.31 2.36 1.34 0.74 0.59 1.05

β = 4√

3/(Rah)2 [m−1] 270 477 607 794 1042 1166 1848

ζc = (1−√

2/α)2 [−] 0.86 0.74 0.67 0.55 0.40 0.33 0.04

τ = 1/(α2(1− ζc)) [−] 0.02 0.04 0.05 0.07 0.11 0.13 0.34

δ1 (see (4.122)) [−] 1.05 1.10 1.13 1.20 1.29 1.35 1.82

ν = ν/(ωR2a) [10−1] 0.02 0.09 0.17 0.33 0.66 0.89 3.30

ϑ =√

12 (1+ ζc) [−] 0.96 0.93 0.91 0.88 0.84 0.82 0.72

Table 6.2: Table of constants for a set of different arteries. The labels (A)-(H) correspond to thelabels in Figure 6.2. The values for Lv, Ra , h and E are taken from [4] and the values for Q0are estimated from [1], page 95, Figure 6.5. For the values of Lv, the length of the vessel, wetook the sum of the values in [4] per artery and for the values of Ra and h we took the arithmeticmean of the values in [4] per artery. The values for the material parameters of the fluid areρ = 1.05 · 103kg/m3 and ν = 3.5 · 10−6m2/s, while the radial frequency ω = 2π . Finally, we notethat we have used the value 0.5 for the Poisson ratio of the wall, assuming that the material ofthe wall is incompressible.



OVERVIEW SOLUTIONS

GLO

BA

LE

QU

ATIO

NS

OF

MO

TIO

N Constant wave velocity

(4.25): qc(z, t) = qi (t − z) H (t − z)

(4.25): pc(z, t) = qi (t − z) H (t − z)

Variable wave velocity

(4.39): qv(z, t) = qi (T0(z, t))H(t − z)

(4.40): pv(z, t) = 1δ

{[1+ 3δ

2 qi (T0(z, t))] 2

3 − 1}

H(t − z)

with t0 solution of t −(

z/(

1+ 32δqi (t0)

) 13)= t0, (4.38)

Radial diffusion

(4.88): qd(z, t) =[

e−2τ zqi (t − z)+ 2τ z∫ t−z

0

I1

(2τ√(t−s)2−z2)

)

√(t−s)2−z2

e−2τ (t−s)qi (s) ds

]H(t − z)

(4.89): pd(z, t) = ∫ t−z0 I0

(2τ√(t − s)2 − z2)

) (qs + 4τqi (s)

)e−2τ (t−s) ds H(t − z)

where τw = τqd(z, t)+ τ2∂pd/∂z, (4.58)

Advection

(4.119): qa(z, t) = qi (T0(z, t))H(t − z)

(4.120): pa(z, t) =(

qi (T0(z, t))− 14δ(1+ 2γa1

)q2

i (T0(z, t)))

H(t − z)

where γa(z, t) = γa1q2a (z, t), (4.96), t0 solution of t − z + 1

2 z(1+ 2γa1

)qi (t0) = t0, (4.118)

LOC

AL

EQ

UAT

ION

SO

FM

OTI

ON

(5.119): w(r, z, t) = 2(1− r2) {e−α1zqi (t − z)+ α1zH(z, t)

}−∑∞

k=1 Jk(r){e−α1z8k(t − z)+ α1zHk(z, t)

}

(5.120): q(z, t) = (e−α1zqi (t − z)+ α1z I (z, t)

H(t − z))

(5.122): p(z, t) = ∫ tz I0

(α1√

x2 − z2)(qi (t − x)+ 2α1qi (t − x)) e−α1x dx H(t − z)

(5.121): τw(z, t) = 8ν(e−α1zqi (t − z)+ α1zH(z, t)

)+4ν∑∞

k=1(e−α1z8k(t − z)+ α1zHk(z, t)

)

(5.124): γa(z, t) = 43 q2(z, t)+∑∞k=1

4σ 2

k


)2−q(z, t)

∑∞k=1

8σ 2

k


)

where H(z, t), (5.117), Hk(z, t), (5.118), 8k(t) (5.65), and Jk(r), (5.64), as defined inChapter 5.

Table 6.3: An overview of all solutions obtained in Chapter 4 and Chapter 5. Notice the subscriptswe added for the solutions of the global equations of motion in order to mark the differences inthe plots of this chapter.



Vessel A (δ = 0.12, ν = 0.002)

0.25

0.50

0.5 1.0r

w

1.5

3.0

πt

5

0 0.5t

Vessel B (δ = 0.24, ν = 0.009)

0.75

1.50

0.5 1.0r

w

1.5

3.0

πt

5

0 0.5t

Vessel D (δ = 0.04, ν = 0.033)

0.15

0.30

0.5 1.0r

w

1.5

3.0

πt

5

0 0.5t

(a) (b)

Figure 6.3: Column (a) Plots of axial velocity w(r , z, t) for vessels A, B and D as a function of rfor t = π and z = {0, 0.1, 0.25, 0.5, 0.75}π . Column (b) Plots of 5(z, t) (= −∂p/∂z) and the error,defined in (5.134), for vessels A, B and D as a function of t for z = 0 (solid line) and z = π (dashedline). All quantities and variables are dimensionless (in contrast with all other plots)!



Vessel A (δ = 0.12)

200 · 10−6

400 · 10−6

πt

q

0 0.5t

200 · 10−6

400 · 10−6

πz

q

0 1

zLv

Vessel B (δ = 0.24)

130 · 10−6

260 · 10−6

πt

q

0 0.5t

130 · 10−6

260 · 10−6

πz

q

0 1

zLv

Vessel D (δ = 0.04)

6 · 10−6

12 · 10−6

πt

q

0 0.5t

6 · 10−6

12 · 10−6

πz

q

0 1

zLv

(a) (b)

Figure 6.4: Plots of qc(z, t), qv(z, t), qd(z, t), qa(z, t) and ql(z, t) for vessels A, B and D andfor column (a) as a function of t for a set z = 0 (solid line) and z = π (dashed line); and forcolumn (b) as a function of z for a set t = π (solid line) and t = 2π (dashed line). The plottedquantities are dimensional, based on the quantities of Table 6.2, whereas the variables t and zare dimensionless.




1.5 · 103

3.0 · 103

πt

p

0 0.5t

1.5 · 103

3.0 · 103

πz

p

0 1

zLv


4 · 103

8 · 103

πt

p

0 0.5t

4 · 103

8 · 103

πz

p

0 1

zLv


1 · 103

2 · 103

πt

p

0 0.5t

1 · 103

2 · 103

πz

p

0 1

zLv

(a) (b)

Figure 6.5: Plots of pc(z, t), pv(z, t), pd(z, t), pa(z, t) and pl(z, t) for vessels A, B and D andfor column (a) as a function of t for a set z = 0 (solid line) and z = π (dashed line); and forcolumn (b) as a function of z for a set t = π (solid line) and t = 2π (dashed line). The plottedquantities are dimensional, based on the quantities of Table 6.2, whereas the variables t and zare dimensionless.




2 · 10−4

4 · 10−4

πt

τw

0 0.5t

2 · 10−4

4 · 10−4

πz

τw

0 1

zLv


3 · 10−4

6 · 10−4

πt

τw

0 0.5t

3 · 10−4

6 · 10−4

πz

τw

0 1

zLv


3 · 10−5

6 · 10−5

πt

τw

0 0.5t

3 · 10−5

6 · 10−5

πz

τw

0 1

zLv

(a) (b)

Figure 6.6: Plots of τw(z, t) in the global case and the local case for vessels A, B and D and forcolumn (a) as a function of t for a set z = 0 (solid line) and z = π (dashed line); and for column(b) as a function of z for a set t = π (solid line) and t = 2π (dashed line). The plotted quantitiesare dimensional, whereas the variables t and z are dimensionless.




1 · 10−4

2 · 10−4

πt

γa

0 0.5t

1 · 10−4

2 · 10−4

πz

γa

0 1

zLv


2 · 10−4

πt

γa

0 0.5t

2 · 10−4

πz

γa

0 1

zLv


2 · 10−6

4 · 10−6

πt

γa

0 0.5t

2 · 10−6

4 · 10−6

πz

γa

0 1

zLv

(a) (b)

Figure 6.7: Plots of γa(z, t) in the global case and the local case for vessels A, B and D and forcolumn (a) as a function of t for a set z = 0 (solid line) and z = π (dashed line); and for column(b) as a function of z for a set t = π (solid line) and t = 2π (dashed line). The plotted quantitiesare dimensional, whereas the variables t and z are dimensionless.




In this chapter we presented the results from the solutions from Chapter 4 and Chapter 5, i.e.a set of plots of the flow rate q, the pressure distribution p, the shear force at the wall τw andthe advection distribution γa . We distinguished between the results from the global equations ofmotion and the local equations of motion. Next to these plots, we generated a table in which theaverage percentages of the differences from the solutions of the global equations of motion withrespect to the zeroth order approximative solutions from the local equations of motion are listed.Also, a list of parameters for a representative set of arteries and based on real measurements isincluded and is used to generate the results.

We performed a detailed analysis and discussion of the results we gathered. For each of theflow characteristics described above, we performed a qualitative and quantitative investigation inorder to compare the solutions from the global equations of motion with the local ones. At the end,as an important subject of discussion, we also compared our results with the results obtained byBessems in [1], since we based most of this research on this work. As a special remark, wenote here that all our scaled variables, and also their derivatives as far as we could check this,remained of O(1); this confirms the correctness of the scaling we introduced in Chapter 3.

In the next, and final chapter we come to the conclusions of this report. We shortly summarizethe steps taken to come to the results we obtained in this chapter and try to collect all importantissues from the discussion in this chapter, to come to the final conclusion and give answers tothe questions posed in the introduction in Chapter 1. Finally, we will do some recommendationsfor further investigation.



Chapter 7Conclusion and recommendation

In Chapter 6 we presented numerical results from the expressions derived in the foregoingtwo chapters, we also discussed these results in full detail and we drew some temporaryconclusions. In this final chapter we combine the results from Chapter 4 and 5 with theresults and conclusions from the previous chapter and we finally conclude whether we canor can not meat the goals and questions we posed in the problem description in Section1.2. To this end, we start with summarizing the intermediate steps we followed in thisreport, followed by an enumeration of the most important results and conclusions. Andthe end of this chapter we finish with a few suggestions for further investigation of theproblem.

7.1 Recapitulation

In this project we modeled the flow of blood through a semi-infinite deformable vessel, with specialinterest for the order of magnitude of the radial diffusion term and the nonlinear advection term.We assumed the blood to be an incompressible Newtonian fluid and we modeled the vessel as alinearly elastic, thin-walled vessel. From the continuity equation and the Navier-Stokes equations,we derived, based on a dimensional analysis, a set of three-dimensional time-dependent localequations of motion. The global, one-dimensional, equations of motion are obtained by integrat-ing the local ones over the cross-sectional area.

In solving the global equations of motion we distinguished three different cases. First we ne-glected both the diffusion and advection term in the set of equations. We solved the resultingsystem using the method of characteristics, where we first considered the case of a constantwave velocity with corresponding parallel characteristics and secondly the case of a variablewave velocity with intersecting characteristics. For the second case, we incorporated the radialdiffusion term but neglected the advection term in the equations of motion. This system is solvedusing Laplace transforms with respect to the time variable t . Finally, we solved the system inwhich we neglected the radial diffusion term but now incorporated the advection term. As forthe first case, we here also used the method of characteristics. However, in this case we had toneglect higher order terms of the dimensionless compliance parameter, i.e. terms of O

(δ2), in

order to find an analytical solution up to leading order.

The local equations are solved up to the zeroth order of δ, by assuming asymptotic expansionsfor the axial velocity, the pressure distribution and the flow rate in positive powers of δ. First wesolved, as an intermediate step, the zeroth order system for the axial velocity in case of a rigid



vessel, i.e. a vessel with constant radius. Secondly, we solved the zeroth order system for thegeneral case, using the solution for the rigid vessel. In both cases we used Laplace transformsas an important part of the solution tactic. The obtained solution for the axial velocity in the gen-eral case contained an inverse Laplace transform, which was difficult to determine analytically.This is why we introduced an approximation for the inverse Laplace transform, resulting in anapproximative zeroth order solution for the axial velocity. From this solution and the definitions forthe flow rate and the shear stress at the wall, we were able to find a zeroth order approximationfor the flow rate and the radial diffusion term, and, by comparison with the results for the globalequations of motion, an expression for the pressure distribution. Even for the advection term wesucceeded to find an approximative solution, however not as a part of the zeroth order system,but as a part of the first order system.

Finally, we chose an expression for the initial flow rate. From this choice, we plotted numer-ical results obtained for the flow characteristics for both the global and the local equations ofmotion and compared the solutions. We added a table in which we listed the average percentagemaximal difference between the solutions from the global equations of motion and zeroth orderapproximative ones from the local equations of motion. We did this in order to get inside into therelevances of the effects described. Based on the plots and the table, we performed an extensiveanalysis and discussion on the effects observed.

7.2 Main achievements

In this project we wanted to investigate the order of relevance of the radial diffusion and the ad-vection term in the equations of motion. Based on the results of Chapter 6 and the discussionperformed in this same chapter, we come to the final conclusions in this section by summarizingthe main achievements obtained.

We saw that the approximation for the inverse Laplace transform, as introduced in Chapter 5,is justified since the resulting error caused by the approximation is of negligible order. Based onthis approximation, solutions for the flow characteristics were derived from the local equationsof motion and compared with the solution for the flow characteristics from global ones. Sincethe dimensionless quantities and their derivatives are of O(1), as we observed from the plots inChapter 6, we conclude that the scaling parameters, as introduced in Section 4.1, are correct.The solutions for the flow rate q and the pressure distribution p without radial diffusion and advec-tion do not differ significantly from the approximative zeroth order solution for the larger values ofthe dimensionless compliance parameter δ. For the small values of δ, the inclusion of the radialdiffusion term becomes more important due to a growing contribution of the viscous resistance.The neglect of the higher order terms of δ in the case where radial diffusion is excluded andnonlinear advection is included, showed large variations in the solutions, mainly differences inphase and amplitude, due to the neglect of higher order terms of δ. Therefore, we conclude thatour solutions for q and p in this specific case are unreliable. In general, only quantitative differ-ences are observed and no qualitative ones; in all cases we observe a traveling wave solutionwith decreasing amplitude. One other important observation was the presence of a irregular dis-continuous behavior at the wave front z = t , caused by the absence of the axial diffusion term asa result of the dimensional analysis. Finally, we noticed a difference of a factor ten with respectto reference [10], Bessems et al., regarding the maximal value of the pressure distribution. Thislast difference needs further investigation.

The plots for the shear stress τw at the wall for both the global an local equations of motionshowed great resemblance in both shape and amplitude for all three representative values ofδ. In all cases we observed two remarkable effects. First of all we saw a discontinuity at thewave front for the global solution, due to the discontinuity of the spatial derivative of the pressure



distribution at the wave front. Secondly, we observed a change of sign for τw after some time tin both the global and local solutions. For the two largest values of δ, our results coincide withthe results found in literature. For the smallest value of δ, there is a difference of a factor twowhich, according to our perspectives, is not too disastrous. As for the solutions for the flow rateand the pressure distribution in the case of inclusion of nonlinear advection, we also observe thedescribed unreliability in the plots for γa in the global solutions. Therefore, it is hard to comparethe solutions for the global equations of motion with the local ones. However, we were able tocompare the local solution for γa with the local solution for τw. For the two larger values of δ bothquantities are of the same order, whereas for the small value of δ, γa is a factor ten smaller thenτw.

Summarizing, we conclude that the solutions obtained for the flow rate q, the pressure distri-bution p and the wall shear stress τw are accurate within an acceptable range. Especially forsmall values of the dimensionless compliance parameter δ, i.e. δ < 0.1, the wall shear stress, asbased on the definition of Bessems in [1], plays an important role. For the nonlinear advectionterm, although we strongly expect that the influence is minimal in all cases for δ < 0.1, it is notpossible to definitely draw this conclusion from the results obtained in this project.

7.3 Recommendations

We may state that the physical behavior of blood flow trough a vessel has been widely studied inall kinds of vessels by means of numerical solution methods. However, we choose to approachthe problem in an, a far as possible, analytical way. In this context there are still some aspectswhich need some more detailed investigation.

As already discussed intensively in Chapter 6, the results obtained for the inclusion of the ad-vection distribution are far from accurate, and thus incomplete. Here, it might for example beuseful to incorporate higher order terms in δ (at least all terms up to O

(δ2)). Another option

would be, although perhaps not completely physically relevant, to investigate what the effect willbe of leaving out the compliance contribution, i.e. the δp term, in the global equations of motion.Here, still some more work has to be done in order to compare it with the results of Chapter 5.

Secondly, we observed that there is a difference of a factor ten in the results for the pressuredistribution as derived by Bessems in [1] and ours. We suggest that it is possibly worthwhile todrop het assumption of a semi-infinite vessel and replace it by a finite vessel. The effect will be areflection at the end of the vessel, which will increase the maximum pressure.

Except from the lack of results for the advection contribution and the suggestion to investigatete effect of a finite vessel instead of a semi-infinite vessel, there are some more interesting waysto extend the model and investigate, again as far as possible in an analytical sense, the effect ofthe extension on the flow characteristics. The possibilities we could think of are for example: (i) amore detailed investigation of the effects of the boundary layer by incorporating the axial diffusionterm ∂2 p/∂z2; (ii) add extra source terms to the global equations of motion which describe forexample phenomena like leakage, aneurysm or branching; (iii) use non-linear elasticity theoryto describe the elastic behavior of the vessel wall, the latter is specifically relevant for δ > 0.1.Remark that this would change the constitutive equation for describing the relation between thecross-sectional area and the (derivatives of the) pressure distribution.



Bibliography

[1] D. Bessems, On the propagation of pressure and flow waves through the patient-specificarterial system, Ph.D. thesis, Eindhoven University of Technology, 2007.

[2] J. Valsa L. Brancik, Approximate formulae for numerical inversion of Laplace transforms,Int. Journal of Numerical Modelling: Electronic Networks, Devices and Fields 11 (1998),153–166.

[3] Landelijke Pedagogische Centra, Biologie voor jou, vol. 5V, Malmberg Uitgeverij, Den Bosch,1990.

[4] N. Westerhof F. Bosman C.J. de Vries and A. Noordergraaf, Analog studies of the humansystematic arterial tree, J. of Biomechanics 2 (1969), 121–143.

[5] dr. G. Verkerk et. al., Binas, informatieboek voor natuurwetenschappen en wiskunde, vijfdedruk ed., Noordhoff Uitgevers, Groningen, 2008.

[6] M. S. Olufsen C.S. Peskin W.Y. Kim E.M. Pedersen A. Nadim and J. Larsen, Numericalsimulation and experimental validation of blood flow in arteries with structured-tree outflowconditions, Annals of Biomedical Engineering 28 (2000), 1281U1299.

[7] Andrei D. Polyanin, Inverse Laplace transforms: Expressions with exponential functions,http://eqworld.ipmnet.ru/, 2005.

[8] I. Rubinstein L. Rubinstein, Partial differential equations in classical mathematical physics,Cambridge University Press, 1993.

[9] C.G. Giannopapa J.M.B. Kroot A.S. Tijsseling M.C.M. Rutten and F.N. van de Vosse, Wavepropagation in thin-walled aortic analogues, Journal of Fluids Engineering : Transactions ofthe ASME 132 (2010).

[10] D. Bessems M. Rutten and F.N. van de Vosse, A wave propagation model of blood flowin large vessels using an approximate velocity profile function, J. of Fluid Mechanics 580(2007), 145–168.

[11] C.J. Tranter, Bessel functions with some physical applications, English Universities Press,1968.

[12] F.N. van de Vosse, Cardiovascular fluid mechanics - lecture notes 8w090, Eindhoven Uni-versity of Technology, Department of Biomedical Engineering, 2009.

[13] M.H.A. van Geel, Computational modeling of blood flow and the role of hypergravity, Master’sthesis, Eindhoven University of Technology (2010).


http://eqworld.ipmnet.ru/


[14] N. Stergiopulos D.F. Young and T.R. Rogge., Computer simulation of arterial flow with appli-cations to arterial aortic stenosos, J. of Biomechanics 25 (1992), 1477–1488.



Appendix ALinear elasticity theory for athin-walled vessel

Consider a linearly elastic thin-walled vessel under internal pressure p = p(z, t). Due tochanges in this pressure the cross-sectional area of the vessel changes from A0, at p = p0to A = A(z, t). The area A is assumed to be an affine function of the the pressure changep − p0, i.e.

A(z, t) = A0 + C(p(z, t)− p0). (A.1)

This appendix provides a derivation of this formula. Also we come up with an expressionfor the compliance C in terms of the physical parameters of the vessel wall. Moreover, wewill derive a non-local equation for the radial displacement of the vessel wall, commonlyrefered to as the shell equation.

A.1 Basic equations

The deformations εi j and the stresses σi j of the vessel wall are related according to Hooke’s law(and its inverse)

σi j = E1+ ν

(εi j + ν

1− 2νδi jεkk

), (A.2)

εi j = 1+ νE

(σi j − ν

1+ ν δi jσkk

), (A.3)

for given Young’s modulus E and Poisson ratio ν. Let Ra be the undeformed mean radius ofthe vessel and R(z, t) = Ra + ur the deformed one, where ur = u(r, z, t) is the displacement ofthe (thin) wall. The axial displacement is uz = w(r, z, t) and the tangential displacement uθ = 0(rotational symmetry). The deformations are given by

εrr = ∂u∂r, εθθ = u

r,

εzz = ∂w

∂z, εr z = 1

2

(∂u∂z+ ∂w∂r

). (A.4)



Hooke’s law, written out in components, reads

σrr = E(1− 2ν)(1+ ν)

((1− ν)εrr + ν(εθθ + εzz)

), (A.5)

σθθ = E(1− 2ν)(1+ ν)

((1− ν)εθθ + ν(εrr + εzz)

), (A.6)

σzz = E(1− 2ν)(1+ ν)

((1− ν)εzz + ν(εrr + εθθ )

), (A.7)

σr z = E(1+ ν)εr z . (A.8)

The equations of equilibrium are

1r∂

∂r(rσrr )+ ∂σr z

∂z− 1

rσθθ = 0, (A.9)

1r∂

∂r(rσr z)+ ∂σzz

∂z= 0. (A.10)

The boundary conditions at the inner side: r = R− = Ra − h/2 and the outer side: r = R+ =Ra + h/2 of the vessel wall of thickness h, with h/Ra � 1, are given by

r = R− : σrr = −p(z, t), σr z = 0, (A.11)

r = R+ : σrr = −p0, σr z = 0. (A.12)

The vessel can be either in a state of plane strain (εzz = 0) or in a state of plane stress (σzz = 0).Since arteries are commonly prestretched with a fixed axial stretch, we opt here for a state ofplane strain.

A.2 Derivation of the compliance C

In relation (A.1) the change of the area A in the point z is directly related to the change in thepressure in the point same z. This implies that the reactions of the neighbouring points arenot accounted for; this is why we call this a local relation. In other words, the vessel is eitherconsidered as a free ring or a tube under uniform pressure. The latter approximation is allowedif the changes in p in axial direction have a typical length scale much larger than the thicknessh (or better, as we will see further on, than

√Rah). This is, in general, true for our problems.

Under these conditions, Laplace’s formula holds, expressing the radial equilibrium of a circularsegement of the wall (see Figure A.1), stating that

pRadθ − 2σθθ sin(

dθ2

)h = 0, (A.13)

or, for dθ → 0,

σθθ = Ra

h(p − p0). (A.14)

Since |σrr | ∼ (p − p0) and Ra/h � 1, we see that |σrr | � |σθθ |, and therefore we neglect σrr inthe sequel. With the deformed radius written as R = Ra + u, and with the use of u/Ra � 1, weobtain for the deformed area A

A = πR2 = π(Ra + u)2 = πR2a + 2πRau + πu2 ≈ A0 + 2πRau. (A.15)



Ra

dθ

h

p0

p

σθθ

σθθ

Figure A.1: Circular segment of a tube under pressure.

From the inverse Hooke’s law (A.3), with σrr ≈ 0, we obtain for the state of plane strain,

εzz = 1E(σzz − νσθθ ) = 0, (A.16)

that σzz = νσθθ . Furthermore, we know that u = Raεθθ . These two results are substituted intoequation (A.15). This leads to

A ≈ A0 + 2πRau = A0 + 2πR2aεθθ = A0 + 2πR2

a1E(σθθ − νσzz)

= A0 + 2πR2a

1E(σθθ − ν2σθθ ) = A0 + 2πR2

a(1− ν2)

Eσθθ

= A0 + 2π(1− ν2)R3a

Eh(p − p0), (A.17)

with use of Laplace’s formula (A.14). This yields for the compliance C in (A.1)

C = 2π(1− ν2)R3a

Eh. (A.18)

Assuming, as is common practice, that the elastic material of the artery wall is incompressible,we take ν = 0.5, to obtain

C = 3πR3a

2Eh. (A.19)

A.3 The shell equation

In this section we will derive a non-local equation which relates the radial displacement u in thepoint z to the pressure p in all neighboring points of z, thus accounting for the stiffness of thenear part of the wall. In contract to the local relation (A.1), this results in a continuous radialdisplacement u even if the pressure is discontinuous. To this end we start with calculating the



global forces and moments. These global forces and moments can be obtained by integratingthe local equations of equilibrium over the wall of the vessel. The result will be a set of equationsthat only depends on the spatial coordinate z and the time t .

1. Integration of (A.9) yields∫ R+

R−

{1r∂

∂r(rσrr )+ ∂σr z

∂z− 1

rσθθ

}r dr =

rσrr

∣∣∣∣R+

R−+ ∂

∂z

∫ R+

R−rσr z dr −

∫ R+

R−σθθ dr =

− R+ p0 + R− p(z, t)+ Ra∂Qz

∂z− Nθ = 0,

or∂Qz

∂z− 1

RaNθ = − p(z, t), (A.20)

where

p(z, t) = R−

Rap(z, t)− R+

Rap0, (A.21)

Qz(z, t) = 1Ra

∫ R+

R−rσr z(r, z, t) dr, (A.22)

Nθ (z, t) =∫ R+

R−σθθ (r, z, t) dr. (A.23)

Here, Qz is the shear force and Nθ the normal force in azimuthal direction per unit of length(“hoop force”) (N/m).

2. Integration of (A.10) gives∫ R+

R−

{1r∂


∂z

}r dr = rσr z

∣∣∣∣R+

R−+ ∂

∂z

∫ R+

R−rσzz dr = Ra

∂Nz

∂z= 0,

or∂Nz

∂z= 0, (A.24)

where

Nz(z, t) = 1Ra

∫ R+

R−rσzz(r, z, t) dr. (A.25)

Here, Nz is the normal force in axial direction per unit of length (N/m).

3. Integration of (A.10) multiplied by (r − Ra) yields∫ R+

R−(r − Ra)

{1r∂


∂z

}r dr

∫ R+

R−

{∂

∂r((r − Ra)rσr z)− rσr z + r(r − Ra)

∂σzz

∂z

}dr =

(r − Ra)rσr z

∣∣∣∣R+

R−+ ∂

∂z

∫ R+

R−r(r − Ra)σzz dr −

∫ R+

R−rσr z dr =

Ra∂Mz

∂z− Ra Qz = 0,



or∂Mz

∂z− Qz = 0, (A.26)

where

Mz(z, t) = 1Ra

∫ R+

R−r(r − Ra)σzz(r, z, t) dr. (A.27)

Here, Mz is the bending moment per unit of length (N/m).

Note that these moments and forces are exact integrals of the local equations of equilibrium. Noapproximation is made with respect to h/Ra . So, these global equations of equilibrium also holdfor thick-walled vessels.

Eliminating Qz from (A.26) in favor of Nθ by means of (A.20), we arrive at

∂2 Mz

∂z2 −1

RaNθ = − p(z, t) (A.28)

In order to come to the shell equation, we assume asymptotic expansions for the displacementsu(r, z, t) and w(r, z, t) in terms of positive powers of the small parameter h/Ra . Since we knowthat (r − Ra)/Ra = O(h/Ra), we arrive at

u(r, z, t) = U0(z, t)+ r − Ra

RaU1(z, t)+ ..., (A.29)

w(r, z, t) = W0(z, t)+ r − Ra

RaW1(z, t)+ ..., (A.30)

where the ... stand for terms which are small in second order of h/Ra with respect to the lead-ing order terms. As said before, in section A.1, we have opted here for a state of plane strain.However, since the bending of the wall under a z-dependent pressure p(z, t) is a problem thatessentially depends on z, it is in principle not right to call this a plane strain problem. What wemean here is that due to the prescribed and fixed axial pre-stretch, there is no extra axial stretchin the mean over the wall. Hence, although there can, and will, be an axial displacement of thewall (due to the bending of the wall), but averaged over the thickness of the wall the resultingmean displacement will be zero. The direct consequence of this assumption is that in (A.30) theleading term W0(z, t) must be taken zero.

We can now express the deformations in (A.4) in terms of Ui and Wi (i = 0, 1) by substitution ofthe asymptotic expansions. This leads to (neglecting the O

((h/Ra)

2) terms)

εθθ = ur= u

Ra

11+ (r − Ra)/Ra

= uRa

(1− r − Ra

Ra

)

= 1Ra

(U0 + r − Ra

RaU1

)(1− r − Ra

Ra

)= U0

Ra+ 1

Ra

r − Ra

Ra(U1 −U0), (A.31)

εzz = ∂w

∂z= r − Ra

Ra

∂W1

∂z, (A.32)

εr z = 12

(∂u∂z+ ∂w∂r

)= 1

2

(∂U0

∂z+ W1

Ra

)+ 1

2r − Ra

Ra

(∂U1

∂z+ 2W2

Ra

). (A.33)

In accordance with the Kirchhoff hypothesis, which states that normal cross-sections remainnormal after bending, there is no shear in the r z-plane. This means that εr z = 0 and so

2εr z = ∂U0

∂z+ W1

Ra+O(d) = 0, (A.34)



which leads to

W1(z, t) = −Ra∂U0

∂z(z, t). (A.35)

As in the previous section (where it was shown that |σrr | = O(h/2Ra)|σθθ |,on basis of Laplace’sformula) we choose σrr = 0, resulting in

σrr = (1− ν)εrr + ν(εθθ + εzz) = (1− ν)∂u∂r+ ν u

r+ ν ∂w

∂z

= (1− ν)U1

Ra+ νU0

Ra= 0, (A.36)

and so

U1(z, t) = − ν

1− νU0(z, t). (A.37)

Moreover, since σrr = 0, we also find by substitution that (A.6) and (A.7) simplify to

σθθ = E1− ν2 (εθθ + νεzz), (A.38)

σzz = E1− ν2 (νεθθ + εzz). (A.39)

Using the above precalculations we are now able to express σθθ and σzz in terms of the leadin-gorder displacement U0(z, t) and so we arrive at

σθθ = E1− ν2 (εθθ + νεzz)

= E1− ν2

(U0

Ra+ 1

Ra

r − Ra

Ra(U1 −U0)+ ν r − Ra

Ra

∂W1

∂z

)

= E1− ν2

(U0

Ra+ r − Ra

Ra

(1

Ra(U1 −U0)+ ν ∂W1

∂z

))

= E1− ν2

(U0

Ra− r − Ra

Ra

{1

1− νU0

Ra+ νRa

∂2U0

∂z2

}), (A.40)

σzz = E1− ν2 (νεθθ + εzz)

= E1− ν2

(ν

(U0

Ra+ 1

Ra

r − Ra

Ra(U1 −U0)

)+ r − Ra

Ra

∂W1

∂z

)

= E1− ν2

(ν

U0

Ra+ r − Ra

Ra

(ν

Ra(U1 −U0)+ ∂W1

∂z

))

= E1− ν2

(ν

U0

Ra− r − Ra

Ra

{ν

1− νU0

Ra+ Ra

∂2U0

∂z2

}). (A.41)

Note that the expressions for σθθ and σzz are only correct up to an order of (h/2Ra)2. With (A.40)-

(A.41), we calculate the global forces and moments given at the beginning of this section by



simple integration. This leads to

Nθ (z, t) =∫ R+

R−σθθ dr

= E1− ν2

(U0

Ra

∫ R+

R−dr −

[1

1− νU0

Ra+ νRa

∂2U0

∂z2

] ∫ R+

R−

r − Ra

Ra

)dr

= Eh1− ν2

U0(z, t)Ra

, (A.42)

Mz(z, t) = 1Ra

∫ R+

R−r(r − Ra)σzz(r, z, t) dr

= ERa(1− ν2)

(ν

U0

Ra

∫ R+

R−r(r − Ra) dr −

[ν

1− νU0

Ra+ Ra

∂2U0

∂z2

] ∫ R+

R−

r(r − Ra)2

Radr

)

=− Eh3

12(1− ν2)

(ν2

1− νU0(z, t)

R2a+ ∂

2U0

∂z2 (z, t)). (A.43)

We note here that the term U0/R2a is small (i.e. O(h/Ra)) compared to ∂2U0/∂z2. To prove this,

we need a result that will be derived a few lines further on, namely that the characteristic axiallength for the radial displacements due to bending is

√h Ra . This implies that

∂2U0

∂z2 ∼U0

h Ra= Ra

hU0

R2a� U0

R2a, (A.44)

and thus we may approximate (A.43) as

Mz(z, t) = −D∂2U0

∂z2 (z, t), (A.45)

with

D = Eh3

12(1− ν2), (A.46)

the plate constant. Substituting (A.45) and (A.42) into (A.28), we arrive at

− D∂4U0

∂z4 (z, t)− Eh1− ν2

U0(z, t)R2

a= − p(z, t) (A.47)

Introducing the parameter β4 = 3/(Rah)2 we obtain the final version of the shell equation as

∂4U0

∂z4 + 4β4U0 = 1D

p(z, t). (A.48)

This shows that the characteristic length in z-direction is directly related to 1/β ∼ √Rah.



Appendix BInverse Laplace transformation

In Chapter 4, as well as in Chapter 5, we used the concept of Laplace transformation tofind a solution of the global and local equations of motions, respectively. For a functionf (t), t > 0, the Laplace transform F(s), s ∈ C, and its inverse are defined by

F(s) = L { f (t); t, s} :=∫ ∞

0f (t)e−st dt, (B.1)

f (t) = L−1 {F(s); s, t} := limT→∞

12π ı

∫ c+ıT

c−ıTF(s)est ds. (B.2)

In this appendix we derive expressions for the inverse Laplace transform in two cases fromthe above mentioned chapters, where we apply contour-integration to come to a result.

B.1 Inverse Laplace transform (1)

In this section we calculate by hand the inverse Laplace transform, used in equation 4.79 insection 4.3, defined by

G(z, t) = limT→∞

12π ı

∫ c+ıT

c−ıT

e−σ(s)z

σ(s)ets ds, (B.3)

for 0 < z < t with σ(s) = √(τ + s)s. We define a contour C as

C =9⋃

i=0

Ci , (B.4)

where Ci (i = 0, 1, ..., 9) as defined in Figure B.1. For this contour holds∫

C

e−σ(s)z

σ(s)ets ds = 0, (B.5)

since there are no poles within the contour C. This means that

G(z, t) = limT→∞

12π ı

∫

C0

e−σ(s)z

σ(s)ets ds = − lim

T→∞1

2π ı

9∑

i=1

∫

Ci

e−σ(s)z

σ(s)ets ds (B.6)

We will know calculate all integrals on the righthand-side of (B.6) to determine the inverse Laplacetransform. As we will see, only the integrals over C4 and C6 will contribute.



b

b

b b b bbbbb

s-plane

C0

C1

C2

C3C4 C5

C8C7

C6

C9

−τ

c

c + ıT

c − ıT

Figure B.1: Definition sketch of the contour for calculating the inverse Laplace transform (B.3).

1. Along C1 we know that s = T eıϕ + c, where T � 1 and ϕ ∈ [π2 , π]. This gives

σ(s) = T eıϕ√

1+ (τ + 2c)e−ıϕT−1 + c(τ + c)e−2ıϕT−2, (B.7)

such that

G1(z, t) = limT→∞

etc

2π

∫ π

π2

e−T eıϕ√

1+(τ+2c)e−ıϕT−1+c(τ+c)e−2ıϕT−2z√

1+ (τ + 2c)e−ıϕT−1 + c(τ + c)e−2ıϕT−2etT eıϕ

dϕ

= limT→∞

etc

2π

∫ π

π2

e(t−z)T cos(ϕ) eı(t−z)T sin(ϕ) dϕ = 0, (B.8)

since (t − z)T > 0 and cos(ϕ) < 0 for π2 ≤ ϕ ≤ π . With the same reasoning (now with

ϕ ∈ [−π,−π2 ]) we see that G9(z, t) = 0.

2. Along C2 we see that s ∈ [c − T,−τ ], which means that σ(s) = √(τ + s)s ∈ R+. The sameholds for C8, but we remark that the integration is in opposite direction. So we find that

G2(z, t)+ G8(z, t) = limT→∞

12π ı

−τ∫

c−T

e−σ(s)z

σ(s)ets ds + lim

T→∞1

2π ı

c−T∫

−τ

e−σ(s)z

σ(s)ets ds = 0. (B.9)

3. Along C3 we know that s = reıϕ − τ , where 0 < r � 1 and ϕ ∈ [0, π]. This gives

σ(s) = e12 ıϕ√r

√reıϕ − τ , (B.10)



such that

G3(z, t) = limr→0

e−τ t

2π ı

∫ π

0

e−e12 ıϕ√r

√reıϕ−τ z

e12 ıϕ√r

√reıϕ − τ

ereıϕ t ıreıϕ dϕ

= limr→0

e−τ t

2π

∫ π

0

e−e12 ıϕ√r

√reıϕ−τ z

√reıϕ − τ ereıϕ t e

12 ıϕ√r dϕ = 0. (B.11)

Consequently also G5(z, t) and G7(z, t) are both equal to zero.

4. Along C4 we have that s ∈ [−τ , 0], such that σ(s) = √(τ + s)s = ı

√−(τ + s)s ∈ C. Thisgives


12π ı

∫ 0

−τe−ı√−(τ+s)sz

ı√−(τ + s)s

ets ds

= − 12π

∫ 0

−τe−ı√−(τ+s)sz

√−(τ + s)sets ds (B.12)

5. Finally, along C6 we have that s ∈ [−τ , 0], such that σ(s) = √(τ + s)s = −ı√−(τ + s)s ∈ C.

This gives


12π ı

∫ −τ

0

eı√−(τ+s)sz

−ı√−(τ + s)s

ets ds

= 12π

∫ −τ

0

eı√−(τ+s)sz

√−(τ + s)sets ds (B.13)

These integrals still seem difficult to evaluate. However, if we change the coordinate system,things get a little easier. To this end, put t = Z cos(X) and z = ı Z sin(X) (such that t2 − z2 = Z2)and change the integration variable s = 1

2 τ (sin(ϕ)− 1). This gives

G4(z, t) = − 12π

∫ π/2

−π/2eτ2 cos(ϕ)Z sin(X) e

τ2 Z cos(X)(sin(ϕ)−1) dϕ

= − 12π

e−τ2 Z cos(X)

∫ π/2

−π/2eτ2 Z sin(ϕ+X) dϕ

= −12

e−τ2 Z cos(X) I0

(τ

2Z). (B.14)

This last result can for example be obtained by Mathematica, or using the general integral repre-sentation for Bessel functions;

In(x) = 1ınπ

∫ π/2

−π/2e−ı(nϕ−ı x sin(ϕ)) dϕ, (B.15)

which for example can be found in [11]. In the same way we find that

G6(z, t) = − 12π

e−τ2 Z cos(X)

∫ π/2

−π/2eτ2 Z sin(ϕ−X) dϕ

= −12

e−τ2 Z cos(X) I0

(τ

2Z)= G4(z, t). (B.16)



NOTE: Since we have taken the contour left from c, the above derivation holds for0 < z < t . For z > t , the contour has to be taken right from c and since there are nopoles or square root singularities within this right halve-side of the s-plane, the resultwould be trivially G ≡ 0.

So, we conclude that

G(z, t) = −G4(z, t)− G6(z, t) = e−τ2 Z cos(X) I0

(τ

2Z)

= e−τ2 t I0

(τ

2

√t2 − z2

)H(t − z), (B.17)

which is equal to the result in Section 4.3 with τ = 4τ .

B.2 Inverse Laplace transform (2)

In this section, we derive an expression for f (z, t), defined in (5.105) as

f (z, t) = L−1{

e−ρ(s)z}= − ∂

∂zL−1

{e−ρ(s)z

ρ(s)

}=: − ∂

∂zh(z, t), (B.18)

where

ρ(s) =

√√√√√√s2I0

(√sν

)

I2

(√sν

) . (B.19)

We proceed with the evaluation of the inverse transform h(z, t); notice the analogy with the previ-ous section. The inverse Laplace transform g(z, t) is defined by

h(z, t) = limT→∞

12π ı

∫ c+ıT

c−ıT

e−ρ(s)z

ρ(s)est ds. (B.20)

Remark the resemblance with the inverse Laplace transform calculated in the previous section.We will evaluate the inverse transformation (B.20) by contour integration; see Figure B.2 for thesketch of the contour. The function ρ(s) has roots for s = −νλ2

k and vertical asymptotes fors = −νσ 2

k , where λk are the positive roots of J0 and σk are the positive roots of J2 (k = 1, 2, ...) Asin the previous section, the part of the contour at infinity, i.e. CI and CI I I , and the circles aroundthe square root singularities s = −νλ2

k , do not contribute to g(z, t).

The only contribution comes from the subsets S±k of C±k (k = 0, 1, 2, ...), where

S±k ={

s∣∣− νλ2

k+1 < s < −νσ 2k

}, (B.21)

with σ0 = 0 and where the imaginary parts of ρ(s) have different sign just above or below thenegative real axis, say

ρ(s) = ∓ı ρ(s)+O(ε), (B.22)

for s ∈ S±k ± εı , 0 < ε � 1, where ρ(s) is a positive real function of s for s ∈ S±k given by

ρ(s) =

√√√√√√−s2I0

(√sν

)

I2

(√sν

) . (B.23)



b

b

b b b b b b b bbbbbbbbb

s-plane

CI I

CI I I

C+1

C+λ1C+0 C0

C−1C−λ1

C−0

CI

C+λk

C−λk

C+λk+1

C−λk+1

C+k

C−k

−νλ2k−νλ2

k+1

−νσ 2k −νλ2

1

c

c + ıT

c − ıT

Figure B.2: Definition sketch of the contour for calculating the inverse Laplace transform (B.20).



The contribution of all these parts S±k to g(z, t) is

limT→∞

12π ı

∞∑

n=0

{∫ −νσ 2n

−νλ2n+1

eı ρ(s)z

−ı ρ(s)est ds +

∫ −νλ2n+1

−νσ 2n

e−ı ρ(s)z

ı ρ(s)est ds

}=

12π

∞∑

n=0

∫ νσ 2n

−νλ2n+1

{eı ρ(s)z

ρ(s)+ e−ı ρ(s)z

ρ(s)

}est ds =

1π

∞∑

n=0

∫ −νσ 2n

−νλ2n+1

cos(ρ(s)z

)

ρ(s)est ds, (B.24)

and so we obtain

h(z, t) = − 1π

∞∑

n=0

∫ −νσ 2n

−νλ2n+1

cos(ρ(s)z

)

ρ(s)est ds H(t − z) (B.25)

Again, as in the calculation of (B.3), because of the absence of poles and/or square root singu-larities in the right halve of the s-plane, the solution for z > t is equal to zero. Therefore, since thesolution is valid for 0 < z < t , we added H(t − z) in (B.25).

Finally we see that

f (z, t) = − ∂∂z

h(z, t) = δ (t − z)− 1π

∞∑

n=0

∫ −νσ 2n

−νλ2n+1

sin(ρ(s)z

)est ds H(t − z). (B.26)



Appendix CFourier-Bessel series

In Section 5.3 we derived an expression for the zeroth order axial flow velocity startingfrom the assumption that the axial flow velocity can be written as a Fourier-Bessel serie.In this appendix apply the concept of Fourier-Bessel series, which is strongly related toFourier series for periodic functions, in two examples that are mentioned in the derivationof the axial flow velocity. The Fourier-Bessel series of f (r), which is defined on the interval(0, a), is given by

f (r) =∞∑

k=1

ck Jn (λkr) , 0 < r < a, (C.1)

with Fourier-Bessel coefficients

ck = 2a2 J 2

n+1 (λka)

∫ a

0r f (r) Jn (λkr) dr, (C.2)

where n > −1 real and λk (k = 1, 2, 3, ..) the positive zeros of Jn (λ) = 0. See [11] for alldetails about Fourier-Bessel series and the derivation of the above definition.

C.1 Example (1)

In this first example we choose f (r) = 1, n = 0 and a = 1. Then we find for corresponding theFourier-Bessel coefficients

ck = 2J 2

1 (λk)

∫ 1

0r J0 (λkr) dr = 2

λ2k J 2

1 (λk)

∫ λk

0z J0 (z) dz

= 2λ2

k J 21 (λk)

[z J1 (z)]λk0 =

2λk J1 (λk)

, (C.3)

and so we see that for 0 < r < 1 we find

1 =∞∑

k=1

2J0 (λkr)λk J1 (λk)

. (C.4)



If we multiply both sides of (C.4) by r and consecutively integrate the result with respect to r , weobtain

∫ 1

0

∞∑

k=1

2r J0 (λkr)λk J1 (λk)

dr =∞∑

k=1

2λk J1 (λk)

∫ 1

0r J0 (λkr) dr =

∞∑

k=1

2λ2

k J1 (λk)[J1 (λkr)]1

0

=∞∑

k=1

2λ2

k=∫ 1

0r dr = 1

2, (C.5)

and so we find that ∞∑

k=1

1λ2

k= 1

4. (C.6)

C.2 Example (2)

In this second example we choose n = 0 and a = 1, just as in the previous example. Let f (r; s)be given by

f (r; s) =I0

(√sν

)− I0

(r√

sν

)

I0

(√sν

) . (C.7)

The Fourier-Bessel coefficients for f (r; s) are given by

ck = 2J 2

1 (λk)

∫ 1

0r

I0

(√sν

)− I0

(r√

sν

)

I0

(√sν

)

J0 (λkr) dr = 2

J 21 (λk)

s J1 (λk)

λk(s + νλ2

k) , (C.8)

and so we find for 0 < r < 1 that

I0

(√sν

)− I0

(r√

sν

)

I0

(√sν

) =∞∑

k=1

2s

λk(s + νλ2

k)

J1 (λk)J0 (λkr) (C.9)

Again, by multiplication of both sides of (C.9) by r and integration of the result with respect to r ,we find

∫ 1

0r

J0

(ı√

sν

)− J0

(ır√

sν

)

J0

(ı√

sν

) dr =I2

(√sν

)

2I0

(√sν

) =

∞∑

k=1

2s

λk(s + νλ2

k)

J1 (λk)

∫ 1

0r J0 (λkr) dr =

∞∑

k=1

2s

λ2k(s + νλ2

k) . (C.10)

And so we find that∞∑

k=1

4λ2

k(s + νλ2

k) =

I2

(√sν

)

s I0

(√sν

) . (C.11)



Appendix DGlobal equations of motion with radialdiffusion

In Section 4.3 we derived a solution for the volumetric flow rate q(z, t) from the global equa-tions of motion, in which we incorporated the radial diffusion term τw and neglected theadvection term ∂zγa . This solution is obtained by introducing Laplace transforms for theflow rate and the pressure distribution. In this appendix we opt for an alternative methodto derive the expression for the flow rate. From a change of the coordinate system, wearrive at a linear case of Cauchy’s problem. In solving the resulting problem we follow themethod executed in [8], which uses contour integration and Green’s divergence theory.

Once more, we recall that the global equations of motion with inclusion of the radial diffusionterm are given by

∂p∂t+ ∂q∂z= 0, (D.1)

∂q∂t+ ∂p∂z+ τq = 0, (D.2)

In Section 4.2 we saw that for the case δ = 0, the characteristic parameters are given by η = t+ zand ξ = t − z. In this perspective, we assume q(z, t) = Q(η, ξ) and p(z, t) = P(η, ξ). Substitutionin (4.64) and (4.65) gives

∂P∂η+ ∂P∂ξ+ ∂Q∂η− ∂Q∂ξ= 0, (D.3)

∂Q∂η+ ∂Q∂ξ+ ∂P∂η− ∂P∂ξ+ τQ = 0. (D.4)

Successively adding and subtracting (D.3) and (D.4), we obtain

2∂P∂η= −2

∂Q∂η− τQ, (D.5)

2∂P∂ξ= 2

∂Q∂ξ+ τQ. (D.6)

Elimination of P by cross-differentiation with respect to ξ and η and subtraction of the resultingequations finally leads to (with τ = τ /4)

∂2 Q∂η∂ξ

+ τ ∂Q∂η+ τ ∂Q

∂ξ= 0, (D.7)



b b

bb

η

ξ

C1

C2

C3

C4D

(0, 0) = O A = (η0, 0)

B = (η0, ξ0)(ξ0, ξ0) = Cξ0

η = ξ

Figure D.1: Sketch of the domain D and the contour C in the (η, ξ)-plane.

Finding the solution Q(η, ξ) of (D.7) is a special case of the so-called Cauchy problem. In orderto find this solution, we define the differential operator and its adjoint corresponding to (D.7) by

L[Q] = Qηξ + τQη + τQξ , (D.8)

M[8] = 8ηξ − τ8η − τ8ξ , (D.9)

where 8 = 8(η, ξ) is a new unknown function. Simple substitution shows that

8L[Q] − QM[8] = (τ 8 Q −8ξ Q)η+ (τ 8 Q +8 Qη

)ξ

=(τ 8 Q + 1

28 Qξ − 1

28ξ Q

)

η

+(τ 8 Q + 1

28 Qη − 1

28η Q

)

ξ

. (D.10)

By using Green’s divergence theorem , stating that

∫∫

D

(Ux + Vy

)dxdy =

∫

C

(U dy − V dx) , (D.11)

where D is the region of integration and C the corresponding closed contour, as defined in FigureD.1, we see that

∫∫

D

(8L[Q] − QM[8]) dηdξ =∫

C

[(τ 8 Q + 1

28 Qξ − 1

28ξ Q

)dξ (D.12)

−(τ 8 Q + 1

28 Qη − 1

28η Q

)dη]. (D.13)

We want to eliminate the derivatives of Q from the contour integral. This can be done by integra-tion by parts.



1. Along C1 we see that

−∫ A

O

(τ 8 Q + 1

28 Qη − 1

28η Q

)dη =

−∫ A

O

(τ 8 Q − 1

28η Q

)dη − 1

2

∫ A

O8 Qη dη =

−∫ A

O

(τ 8 Q − 1

28η Q

)dη − 1

2

(8(A)Q(A)−8(O)Q(O)−

∫ A

O8η Q dη

)=

−∫ A

O

(τ 8−8η

)Q dη − 1

28(A)Q(A)+ 1

28(O)Q(O). (D.14)


∫ B

A

(τ 8 Q + 1

28 Qξ − 1

28ξ Q

)dξ =

∫ B

A

(τ 8 Q − 1

28ξ Q

)dξ + 1

2

∫ B

A8 Qξ dξ =

∫ B

A

(τ 8 Q − 1

28ξ Q

)dξ + 1

2

(8(B)Q(B)−8(A)Q(A)−

∫ B

A8ξ Q dξ

)=

∫ B

A

(τ 8−8ξ

)Q dξ + 1

28(B)Q(B)− 1

28(A)Q(A). (D.15)


−∫ C

B

(τ 8 Q + 1

28 Qη − 1

28η Q

)dη =

−∫ C

B

(τ 8 Q − 1

28η Q

)dη − 1

2

∫ C

B8 Qη dη =

−∫ C

B

(τ 8 Q − 1

28η Q

)dη − 1

2

(8(C)Q(C)−8(B)Q(B)−

∫ C

B8η Q dη

)=

−∫ C

B

(τ 8−8η

)Q dη − 1

28(C)Q(C)+ 1

28(B)Q(B). (D.16)


∫ O

C

(τ 8 Q + 1

28 Qξ − 1

28ξ Q

) ∣∣∣∣η=ξ

dξ −∫ O

C

(τ 8 Q + 1

28 Qη − 1

28η Q

) ∣∣∣∣ξ=η

dη =

∫ O

C

(12∂

∂ξ(8 Q)−8ξ Q

) ∣∣∣∣η=ξ

dξ −∫ O

C

(12∂

∂η(8 Q)−8η Q

) ∣∣∣∣ξ=η

dη =

∫ O

C8η Q

∣∣∣∣ξ=η

dη −∫ O

C8ξ Q

∣∣∣∣η=ξ

dξ + 128(O)Q(O)− 1

28(C)Q(C). (D.17)



So finally we are left with∫∫

D

(8L[Q] − QM[8]) dηdξ =

8(O)Q(O)−8(C)Q(C)+8(B)Q(B)−8(A)Q(A)

−∫ A

O

(τ 8−8η

)Q dη +

∫ B

A

(τ 8−8ξ

)Q dξ

−∫ C

B

(τ 8−8η

)Q dη +

∫ O

C8η Q

∣∣∣∣ξ=η

dη −∫ O

C8ξ Q

∣∣∣∣η=ξ

dξ (D.18)

Along C1 it holds that η = t − z = 0, which means that t = z. From the conditions proposed onq(z, t) we see that Q(η, 0) = 0. So, the integral over the interval (O, A) and the terms 1

28(O)Q(O)and 8(A)Q(A) disappear from (D.18). Moreover, along C4 we have that η = ξ , which means thatt + z = t − z or equivalently z = 0. Since q(0, t) = qi (t), we can substitute Q(·, ·) = qi (·) for theintegral over the interval (C, O) and Q(C) by qi (ξ0). We conclude that

∫∫

D

(8L[Q] − QM[8]) dηdξ =

8(B)Q(B)−8(C)qi (ξ0)+∫ B

A

(τ 8−8ξ

)Q dξ

−∫ C

B

(τ 8−8η

)Q dη +

∫ O

C8η

∣∣∣∣ξ=η

qi (η) dη −∫ O

C8ξ

∣∣∣∣η=ξ

qi (ξ) dξ (D.19)

The next goal is to determine the function 8(η, ξ). The first observation is that M[8] should beequal to zero in order to eliminate Q form the left-hand side of (D.19). Since we want to expressthe solution Q(η, ξ) in the point B in terms of the initial data proposed on Q(η, ξ) on C1 and C4, wewant the integrals over C2 and C3 to disappear from (D.19). This leads to two ordinary differentialequations for two characteristic initial conditions on 8(η, ξ). If we on top of this demand that8(B) = 1, this leads to

τ 8−8ξ = 0, along C2, (D.20)

τ 8−8η = 0, along C3, (D.21)

implying that

8(η0, ξ) = eτ (ξ−ξ0), for ξ ≤ ξ0, (D.22)

8(η, ξ0) = eτ (η−η0), for η ≤ η0. (D.23)

These conditions are satisfied by assuming that

8(η, ξ) = A(η, ξ ; η0, ξ0)eτ (η−η0)eτ (ξ−ξ0). (D.24)

Substitution of (D.24) in the adjoint equation M[8] = 0 leads to the telegraph equation

∂2A

∂η∂ξ− τ 2A = 0, (D.25)

A(η0, ξ ; η0, ξ0) = 1, for ξ ≤ ξ0, (D.26)

A(η, ξ0; η0, ξ0) = 1, for η ≤ η0, (D.27)



where the boundary conditions are such that (D.24) satisfies the characteristic initial conditions(D.22) and (D.23). The solution of (D.25) is the so-called Riemann funtion , given by

A(η, ξ ; η0, ξ0) = I0

(2τ√(η0 − η)(ξ0 − ξ)

), (D.28)

where I0(·) represents the modified Bessel function of order zero.

Since we know A, we are able to express Q in the arbitrary point (η0, ξ0) in terms of the ini-tial data proposed on q(z, t) by substitution of 8 in (D.19). This gives, for 0 ≤ ξ0 ≤ η0,

Q(η0, ξ0) =eτ (ξ0−η0)qi (ξ0)+

τ (η0 − ξ0)

∫ ξ0

0

I1(2τ√(η0 − s)(ξ0 − s)

)√(η0 − s)(ξ0 − s)

e−τ (η0+ξ0−2s)qi (s) ds. (D.29)

From the (η, ξ)-domain we can easily go back to the (z, t)-domain. This gives for q(z, t):

q(z, t) =[e−2τ zqi (t − z)+

2τ z∫ t−z

0

I1

(2τ√(t − s)2 − z2)

)

√(t − s)2 − z2

e−2τ (t−s)qi (s) ds

H(t − z). (D.30)

In a completely analogous way, we can derive for p(z, t).



tekst



List of Figures

2.1 Schematic representation of the human circulatory system. . . . . . . . . . . . . . 182.2 Schematic representation of the human heart. . . . . . . . . . . . . . . . . . . . . 192.3 The heart cycle for a heart having rate frequency of 75 beats per minute. . . . . . 202.4 Changes of the blood pressure and velocity. . . . . . . . . . . . . . . . . . . . . . . 212.5 Schematic representation of an artery wall, a vein wall and a capillary wall. . . . . 212.6 Schematic representation of the main components of blood. . . . . . . . . . . . . . 23

3.1 Schematic representation of a thin-walled vessel . . . . . . . . . . . . . . . . . . . 25

4.1 Sketch of the characteristics in the (z, t)-plane. . . . . . . . . . . . . . . . . . . . . 36

5.1 Plot of h(z, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.1 Plot of the scaled inlet-flow function. . . . . . . . . . . . . . . . . . . . . . . . . . . 686.2 Representation of the human arterial system. . . . . . . . . . . . . . . . . . . . . . 726.3 Plots of the axial velocity and 5(z, t) . . . . . . . . . . . . . . . . . . . . . . . . . . 756.4 Plots of q(z, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.5 Plots of p(z, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.6 Plots of τw(z, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.7 Plots of γa(z, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

A.1 Circular segment of a tube under pressure. . . . . . . . . . . . . . . . . . . . . . . 89

B.1 Definition sketch of the contour (1). . . . . . . . . . . . . . . . . . . . . . . . . . . . 96B.2 Definition sketch of the contour (2). . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

D.1 Sketch of the domain D and the contour C. . . . . . . . . . . . . . . . . . . . . . . 104



List of Tables

2.1 The Young’s modulus for the three main constituents of the vessel wall. . . . . . . 222.2 Values for the material parameters of blood. . . . . . . . . . . . . . . . . . . . . . . 24

6.1 Average percentages differences for q, p, τw and γa . . . . . . . . . . . . . . . . . . 686.2 Table of constants for a set of different arteries. . . . . . . . . . . . . . . . . . . . . 736.3 Solution overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74



Index

accuracy, 69adjoint, 104advection, 44, 71advection dominated flow, 28approximative solution, 62artery, 19atrium, 17attenuation, 69axial diffusion, 70axial velocity, 69

bending moment, 91boundary conditions, 26boundary layer, 69, 70

Cauchy problem, 103, 104compliance, 33, 89

dimensionless, 50constitutive equation, 33

deformation, 87diffusion dominated flow, 28

equations of equilibrium, 88equations of motion

global, 14, 31, 34local, 14, 29, 51

error, 64, 69

first-order system, 52fluid-structure interaction, 13Fourier-Bessel series, 57, 101

coefficient, 101

Green’s divergence theorem, 104

hemodynamics, 13Hooke’s law, 87human circulation system, 17

pulmonary circulation, 17systemic circulation, 17, 19

arterial system, 19capillary system, 19

venous system, 19

inlet-flow function, 67

Kirchhoff hypothesis, 91

Laplace transform, 41, 95inverse, 95

Laplace’s formula, 88lock-exchange problem, 14

method of characteristics, 35micro circulation, 20

Navier-Stokes equation, 26Newtonian fluid, 26nonlinear advection, 30normal force, 90

plate constant, 93Poiseuille flow, 69Poisson’s ratio, 39pressure distribution, 69

radial diffusion, 40relaxed problem, 50, 53residue, 55Reynolds number, 27Riemann function, 107

shear force, 30, 90shear stress, 70shell equation, 39, 93shockwave, 70smooth muscle cells, 22state of plane strain, 88state of plane stress, 88stationary waves, 69stress, 87Strouhal number, 28

Telegraph equation, 106transient, 15



vein, 20ventricle, 17venule, 20volumetric flow rate, 30, 69

wave velocity, 34modified, 41

Womersley number, 27

Young’s modulus, 22, 87

Zernike polynomial, 57zeroth-order system, 52


Eindhoven University of Technology MASTER Mathematical ...het genot van een biertje, over de wijze...

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