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Cardiovascular Fluid Mechanics - Lecture Notes 8W090 Cardiovascular Fluid Mechanics - lecture notes 8W090 - F.N. van de Vosse (2013) Eindhoven University of Technology department of Biomedical Engineering Preface i Preface As cardiovascular disease is a major cause of death in the western world, knowledge of cardiovascular pathologies, including heart valve failure and atherosclerosis is of great importance. This knowledge can only be gathered after well understanding the circulation of the blood. Also for the development and usage of diagnostic tech- niques, like ultrasound and magnetic resonance assessment of blood flow and vessel wall displacement, knowledge of the fluid mechanics of the circulatory system is in- dispensable. Moreover, awareness of cardiovascular fluid mechanics is of great help in endovascular treatment of diseased arteries, the design of vascular prostheses that can replace these arteries when treatment is not successful, and in the development of prosthetic heart valves. Finally, development and innovation of extra-corporal systems strongly relies on insight into cardiovascular fluid mechanics. The lecture notes focus on fluid mechanical phenomena that occur in the human cardiovascular system and aim to contribute to better understanding of the circulatory system. ii Cardiovascular Fluid Mechanics - lecture notes In the introductory part of these notes a short overview of the circulatory system with respect to blood flow and pressure will be given. In chapter 1 a simple model of the vascular system will be presented despite the fact that the fluid mechanics of the cardiovascular system is complex due to the non-linear and non-homogeneous rheological properties of blood and arterial wall, the complex geometry and the pulsatile flow properties. An important part, chapter 2, is dedicated to the description of Newtonian flow in straight, curved and bifurcating, rigid tubes. With the aid of characteristic di- mensionless parameters the flow phenomena will be classified and related to specific physiological phenomena in the cardiovascular system. In this way difference be- tween flow in the large arteries and flow in the micro-circulation and veins and the difference between flow in straight and curved arteries will be elucidated. It will be shown that the flow in branched tubes shows a strong resemblance to the flow in curved tubes. Although flow patterns as derived from rigid tube models do give a good approx- imation of those that can be found in the vascular system, they will not provide information on pressure pulses and wall motion. In order to obtain this informa- tion a short introduction to vessel wall mechanics will be given and models for wall motion of distensible tubes as a function of a time dependent pressure load will be derived in chapter 3. The flow in distensible tubes is determined by wave propagation of the pressure pulse. The main characteristics of the wave propagation including attenuation and reflection of waves at geometrical transitions are treated in chapter 4, using a one- dimensional wave propagation model. As blood is a fluid consisting of blood cells suspended in plasma its rheological properties differ from that of a Newtonian fluid. In chapter 5 constitutive equations for Newtonian flow, generalized Newtonian flow, viscoelastic flow and the flow of suspensions will be dealt with. It will be shown that the viscosity of blood is shear and history dependent as a result of the presence of deformation and aggregation of the red blood cells that are suspended in plasma. The importance of non-Newtonian properties of blood for the flow in large and medium sized arteries will be discussed. Finally in chapter 6 the importance of the rheological (non-Newtonian) properties of blood, and especially its particulate character, for the flow in the micro-circulation will be elucidated. Velocity profiles as a function of the ratio between the vessel diameter and the diameter of red blood cells will be derived. In the appendix A, a short review of the equations governing fluid mechanics is given. This includes the main concepts determining the constitutive equations for both fluids and solids. Using limiting values of the non-dimensional parameters, simplifications of these equations will be derived in subsequent chapters. In order to obtain a better understanding of the physical meaning, many of the mathematical models that are treated are implemented in MATLAB. Descriptions of these implementations are available in a separate manuscript: ’Cardiovascular Fluid Mechanics - computational models’. Contents 1 General introduction 1 1.1 Introduction................................ 1 1.2 Thecardiovascularsystem. 2 1.2.1 Theheart............................. 3 1.2.2 Thesystemiccirculation. 7 1.3 Pressure and flow in the cardiovascular system . ... 9 1.3.1 Pressure and flow waves in arteries . 9 1.3.2 Pressure and flow in the micro-circulation . 13 1.3.3 Pressureandflowinthevenoussystem . 13 1.4 Simplemodelofthevascularsystem . 13 1.4.1 Periodic deformation and flow . 13 1.4.2 Thewindkesselmodel . 14 1.4.3 Vascularimpedance . 15 1.5 Summary ................................. 16 2 Newtonian flow in blood vessels 17 2.1 Introduction................................ 17 2.2 Steady and pulsatile Newtonian flow in straight tubes . ..... 18 2.2.1 Fully developed flow . 18 2.2.2 Entranceflow........................... 25 2.3 Steady and pulsating flow in curved and branched tubes . 27 2.3.1 Steadyflowinacurvedtube . 27 2.3.2 Unsteady fully developed flow in a curved tube . 33 2.3.3 Flowinbranchedtubes . 35 2.4 Summary ................................. 35 3 Mechanics of the vessel wall 37 3.1 Introduction................................ 37 3.2 Morphology................................ 38 3.3 Mechanicalproperties . .. .. .. .. .. .. .. 39 3.4 Incompressible elastic deformation . 42 3.4.1 Deformation of incompressible linear elastic solids ...... 42 3.4.2 Approximation for small strains . 43 3.5 Wallmotion................................ 45 3.6 Summary ................................. 47 iii iv Cardiovascular Fluid Mechanics - lecture notes 4 Wave phenomena in blood vessels 49 4.1 Introduction................................ 49 4.2 Pressureandflow............................. 50 4.3 Fluidflow................................. 51 4.4 Wavepropagation ............................ 53 4.4.1 Derivation of a quasi one-dimensional model . 53 4.4.2 Wave speed and attenuation constant . 56 4.5 Wavereflection .............................. 61 4.5.1 Wave reflection at discrete transitions . 61 4.5.2 Multiple wave reflection: effective admittance . 64 4.5.3 Vascular impedance and cardiac work . 67 4.6 Summary ................................. 68 5 Non-Newtonian flow in blood vessels 69 5.1 Introduction................................ 69 5.2 Mechanicalpropertiesofblood . 70 5.2.1 Morphology............................ 70 5.2.2 Rheological properties of blood . 72 5.3 Newtonianmodels ............................ 75 5.3.1 Constitutive equations . 75 5.3.2 Viscometricresults. 76 5.4 Generalized Newtonian models . 76 5.4.1 Constitutive equations . 76 5.4.2 Viscometricresults. 77 5.5 Viscoelasticmodels............................ 80 5.5.1 Constitutive equations . 80 5.5.2 Viscometricresults. 83 5.6 Rheologyofsuspensions . .. .. .. .. .. .. .. 86 5.6.1 Constitutive equations . 86 5.6.2 Viscometricresults. 88 5.7 Rheologyofwholeblood. .. .. .. .. .. .. .. 88 5.7.1 Experimental observations . 88 5.7.2 Constitutive equations . 90 5.8 Summary ................................. 91 6 Flow patterns in the micro-circulation 93 6.1 Introduction................................ 93 6.2 Flow in small arteries and small veins: Dv > 2Dc ........... 94 6.3 Velocityprofiles.............................. 94 6.3.1 Flow................................ 96 6.3.2 Effective viscosity . 96 6.3.3 Concentration........................... 97 6.3.4 Cellvelocity............................ 98 6.4 Flow in arterioles and venules : Dc < Dv < 2Dc ............ 98 6.4.1 Velocityprofiles. .. .. .. .. .. .. .. 98 6.4.2 Flow................................ 98 Contents v 6.4.3 Effective viscosity . 99 6.4.4 Concentration........................... 99 6.4.5 Cellvelocity............................ 100 6.5 Flow in capillaries: Dv < Dc ....................... 100 6.6 Summary ................................. 102 A Basic equations 103 A.1 Introduction................................ 103 A.2 Thestateofstressanddeformation. 104 A.2.1 Stress ............................... 104 A.2.2 Displacement and deformation . 104 A.2.3 Velocity and rate of deformation . 107 A.2.4 Constitutive equations . 108 A.3 Equationsofmotion . .. .. .. .. .. .. .. 109 A.3.1 Reynolds’transporttheorem . 109 A.3.2 Continuity equation . 110 A.3.3 The momentum equation . 111 A.3.4 Initial and boundary conditions . 112 A.4 Incompressibleviscousflow . 112 A.4.1 Newtonianflow.......................... 112 A.5 Incompressiblein-viscidflow. 113 A.5.1 Irotationalflow . 114 A.5.2 Boundaryconditions . 115 A.6 Incompressibleboundarylayerflow . 115 A.6.1 Newtonian boundary layer flow . 115 A.6.2 Initial and boundary conditions . 116 vi Cardiovascular Fluid Mechanics - lecture notes Chapter 1 General introduction 1.1 Introduction The study of cardiovascular fluid mechanics is only possible with some knowledge of cardiovascular physiology. In this chapter a brief introduction to cardiovascular physiology will be given. Some general aspects of the fluid mechanics of the heart, the arterial system, the micro-circulation and the venous system as well as the most important properties of the vascular tree that determine the pressure and
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