The Physics of Pulsatile Flow, 2000 Biological and Medical Physics, Biomedical Engineering Series

Classic texts in the field of analysis of flow in blood vessels have been written over the years and what these say is still valid today. However, our knowledge of pathophysiological mechanisms has changed with increasing rapidity over the past 20 years, as has our ability to visualize the three­ dimensional geometry of blood flow and blood flow velocity distribution within the in vivo blood vessels. Consequently, with the increased need to fully exploit the new imaging capabilities and our additional biological knowledge, this book is a welcome addition to our armamentarium used to achieve those new goals. the past pulsatile flow (and consequent wave reflections) was Whereas in often seen as "frosting on the cake" of analysis of blood flow problems or perhaps as an issue that should be understood only in a general sense, our new capabilities and understanding require more accurate analyses of spe­ cific systems, not just of constructs based on statistical data describing a vascular tree. Examples of this new need include the situation where the detailed branching geometry of an arterial tree is known from imaging and it is desired to see to what extent local fluid dynamic characteristics can explain the specific localization of disease such as atherosclerosis, or of the extent to which the heterogeneity of perfusion throughout an organ can be attributed to the vascular tree branching geometry or to the mechani­ cal properties of the vascular walls.

1 Preliminary Concepts.- 1.1 Flow in a Tube.- 1.2 What Is a Fluid?.- 1.3 Microscopic and Macroscopic Scales.- 1.4 What Is Flow?.- 1.5 Eulerian and Lagrangian Velocities.- 1.6 Acceleration in a Flow Field.- 1.7 Is Blood a Newtonian Fluid?.- 1.8 No-Slip Boundary Condition.- 1.9 Laminar and Turbulent Flow.- 1.10 Problems.- 1.11 References and Further Reading.- 2 Equations of Fluid Flow.- 2.1 Introduction.- 2.2 Equations at a Point.- 2.3 Equations and Unknowns.- 2.4 Conservation of Mass: Equation of Continuity.- 2.5 Momentum Equations.- 2.6 Forces on a Fluid Element.- 2.7 Constitutive Equations.- 2.8 Navier-Stokes Equations.- 2.9 Problems.- 2.10 References and Further Reading.- 3 Steady Flow in Tubes.- 3.1 Introduction.- 3.2 Simplified Equations.- 3.3 Steady-State Solution: Poiseuille Flow.- 3.4 Properties of Poiseuille Flow.- 3.5 Balance of Energy Expenditure.- 3.6 Cube Law.- 3.7 Arterial Bifurcation.- 3.8 Arterial Tree.- 3.9 Entry Length.- 3.10 Noncircular Cross Section.- 3.11 Problems.-3.12 References and Further Reading.- 4 Pulsatile Flow in a Rigid Tube.- 4.1 Introduction.- 4.2 Oscillatory Flow Equations.- 4.3 Fourier Analysis.- 4.4 Bessel Equation.- 4.5 Solution of Bessel Equation.- 4.6 Oscillatory Velocity Profiles.- 4.7 Oscillatory Flow Rate.- 4.8 Oscillatory Shear Stress.- 4.9 Pumping Power.- 4.10 Oscillatory Flow at Low Frequency.- 4.11 Oscillatory Flow at High Frequency.- 4.12 Tubes of Elliptic Cross Sections.- 4.13 Problems.- 4.14 References and Further Reading.- 5 Pulsatile Flow in an Elastic Tube.- 5.1 Introduction.- 5.2 Bessel Equations and Solutions.- 5.3 Balance of Forces.- 5.4 Equations of Wall Motion.- 5.5 Coupling with Fluid Motion.- 5.6 Matching at the Tube Wall.- 5.7 Wave Speed.- 5.8 Arbitrary Constants.- 5.9 Flow Properties.- 5.10 Problems.- 5.11 References and Further Reading.- 6 Wave Reflections.- 6.1 Introduction.- 6.2 One-Dimensional Wave Equations.- 6.3 Basic Solution of Wave Equation.- 6.4 Primary Wave Reflections in a Tube.- 6.5 SecondaryWave Reflections in a Tube.- 6.6 Pressure-Flow Relations.- 6.7 Effective Admittance.- 6.8 Vascular Tree Structure.- 6.9 Problems.- 6.10 References and Further Reading.- Appendix B. Solutions to Problems.