Hemodynamical Flows

by Galdi, Giovanni P.; Rannacher, Rolf; Robertson, Anne M.; Turek, Stefan

• ISBN13: 9783764378059
• ISBN10: 3764378050
• eBook ISBN(s): 9783764378066
• Edition: 1st
• Format: Paperback
• Copyright: 2008-05-04
• Publisher: Birkhauser

Summary

This book surveys results on the physical and mathematical modeling as well as the numerical simulation of hemodynamical flows, i.e., of fluid and structural mechanical processes occurring in the human blood circuit. The topics treated are continuum mechanical description, choice of suitable liquid and wall models, mathematical analysis of coupled models, numerical methods for flow simulation, parameter identification and model calibration, fluid-solid interaction, mathematical analysis of piping systems, particle transport in channels and pipes, artificial boundary conditions, and many more. Hemodynamics is an area of active current research, and this book provides an entry into the field for graduate students and researchers. It has grown out of a series of lectures given by the authors at the Oberwolfach Research Institute in November, 2005.

Table of Contents

Preface	p. x
Review of Relevant Continuum Mechanics	p. 1
Kinematics	p. 3
Description of motion of material points in a body	p. 3
Referential and spatial descriptions	p. 4
Deformation gradient and measures of stretch and strain	p. 5
Velocity gradient and the rate of deformation tensor	p. 7
Special motions	p. 8
Governing equations	p. 9
The transport theorem	p. 9
Conservation of mass	p. 10
Balance of linear momentum	p. 11
Balance of angular momentum	p. 13
Mechanical energy equation	p. 14
Balance of energy	p. 14
Restrictions on constitutive equations	p. 15
Nonlinear viscous fluids	p. 18
Restrictions due to invariance requirements	p. 19
Restrictions on the Reiner-Rivlin equation due to behavior of real fluids	p. 19
Restrictions on generalized Newtonian fluids due to thermodynamic considerations	p. 20
Examples of generalized Newtonian fluids	p. 20
Yield stress "fluids"	p. 21
Bingham model	p. 22
Modified Bingham model	p. 24
Herschel-Bulkley model	p. 24
Casson model	p. 25
Viscoelastic fluids	p. 25
Simple fluids	p. 26
Approximations for simple fluids with fading memory	p. 31
Finite viscoelastic models	p. 34
Thixotropic fluids	p. 36
Rheometrical flows	p. 38
Viscometric flows	p. 38
Periodic flows	p. 45
Non-periodic unsteady flows	p. 51
Rheometers	p. 53
Couette rheometer	p. 54
Cone and plate rheometer	p. 54
Capillary rheometer	p. 55
Nonlinear elastic solids	p. 55
Introduction to hyperelastic materials	p. 56
Invariance restrictions	p. 57
Example: incompressible, isotropic hyperelastic materials	p. 58
References	p. 59
Hemorheology	p. 63
Blood components	p. 66
Plasma	p. 67
Red blood cells (Erythrocytes)	p. 67
White blood cells (Leukocytes)	p. 68
Platelets (Thrombocytes)	p. 69
Relevant parameters for flow in the human cardiovascular system	p. 69
Multiphase behavior of blood in shear flows	p. 72
Low shear behavior: aggregation and disaggregation of erythrocytes	p. 73
High shear rate behavior: deformation, tumbling and realignment of erythrocytes	p. 75
Spatial distribution of erythrocytes in shear flows	p. 76
Outcomes of non-homogeneous distribution of erythrocytes	p. 77
Platelet activation and blood coagulation	p. 78
Vasoconstriction	p. 78
Primary hemostasis	p. 78
Secondary hemostasis or clot formation	p. 79
Wall repair and clot dissolution	p. 79
Models of activation and blood coagulation	p. 79
Special considerations in rheometry of blood	p. 80
Inhomogeneous distribution of particles	p. 80
Thixotropy	p. 82
Biochemical effects	p. 82
Other considerations	p. 82
Viscosity of whole blood	p. 83
Nomenclature for blood viscosity	p. 83
Experimental data for whole blood viscosity	p. 84
Significance of shear thinning in the circulatory system	p. 86
Viscosity models for blood	p. 89
Dependence of blood viscosity on factors other than shear rate	p. 90
Yield stress behavior of blood	p. 96
Yield stress data for blood	p. 97
Yield stress constitutive models for blood	p. 99
Viscoelasticity of blood	p. 100
Measurements of blood viscoelasticity	p. 100
Dependence of viscoelasticity on other factors	p. 105
Significance of viscoelasticity in the circulatory system	p. 107
Viscoelastic constitutive models	p. 107
Disease states and mechanical properties of blood	p. 108
Increased hematocrit	p. 108
Plasma hyperviscosity	p. 109
Hyperaggregation of red blood cells	p. 109
Decreased RBC deformability	p. 109
Gender and the mechanical properties of blood	p. 110
References	p. 112
Mathematical Problems in Classical and Non-Newtonian Fluid Mechanics	p. 121
Problems in the pipe flow of a Navier-Stokes liquid	p. 125
Fully developed flows	p. 125
The entry flow problem	p. 142
Mathematical modeling of a piping system. Unbounded domain approach	p. 148
Mathematical modeling of a piping system. Bounded domain approach with "do-nothing" boundary conditions	p. 171
Problems in non-Newtonian fluid mechanics	p. 193
Why non-Newtonian models?	p. 193
Problems related to generalized Newtonian models	p. 196
Problems related to viscoelastic liquid models	p. 220
Some results in the mathematical theory of second-order fluids 226
Some results in the mathematical theory of Oldroyd-B fluids and related models	p. 232
Problems in liquid-particle interaction	p. 236
Sedimentation of symmetric particles in viscoelastic liquid	p. 238
Shape-tilting phenomenon	p. 245
Motion of a disk in the shear flow of a liquid in a horizontal channel	p. 247
References	p. 266
Methods for Numerical Flow Simulation	p. 275
Finite-element methods for the simulation of viscous flow	p. 276
The Navier-Stokes equations	p. 276
Discretization of space	p. 278
The stationary algebraic problems	p. 284
Discretization of time	p. 284
The quasi-stationary algebraic problems	p. 288
Numerical simulation of pipe flow	p. 289
Variational `open' boundary conditions	p. 289
Problems with the `do-nothing' boundary condition	p. 292
The problem of well-posedness	p. 293
The closure problem	p. 294
Mesh adaptation and model calibration	p. 295
Principles of a-posteriori error estimation	p. 296
The dual weighted residual (DWR) method	p. 298
Model problems and practical aspects	p. 304
Hyperbolic model case: transport problem	p. 310
Application to flow models	p. 315
Application in optimal flow control	p. 320
Application in hydrodynamic stability analysis	p. 321
Calibration of flow models	p. 323
Current work and further development	p. 328
References	p. 328
Numerics of Fluid-Structure Interaction	p. 333
Fluid-`single rigid body' interaction	p. 334
Model setup in the body frame	p. 335
The stationary free-fall problem	p. 337
Numerical approximation	p. 339
The issue of domain truncation	p. 343
Toward economical meshes	p. 344
Hydrodynamic stability	p. 350
Dynamics of non-stationary free fall	p. 350
Open problems and further development	p. 351
Fluid-`many rigid bodies-wall' interaction	p. 352
The stress-DLM method	p. 352
The fractional-step scheme	p. 353
Open problems and further development	p. 356
Fluid-`elastic structure' interaction	p. 357
Solution methods for FSI problems	p. 357
Variational formulation	p. 358
Numerical approximation	p. 363
Mesh adaptation	p. 365
Stationary test case `Elastic Flow Cavity'	p. 369
Non-stationary test case `Vibrating Thin Plate'	p. 371
Open problems and further development	p. 373
References	p. 375
Numerical Techniques for Multiphase Flow with Liquid-Solid Interaction	p. 379
Numerical methods for incompressible flow	p. 380
Introduction	p. 380
Discretization of the Navier-Stokes equations in time	p. 382
Discretization of the Navier-Stokes equations in space	p. 387
Pressure Schur complement solvers	p. 394
Global MPSC approach	p. 397
Local MPSC approach	p. 401
Multilevel solution strategy	p. 403
Coupling with scalar equations	p. 405
Coupling with k - ¿ turbulence model	p. 408
Adaptive time-step control	p. 413
Some numerical examples	p. 415
Application to more complex flow models	p. 418
Conclusions	p. 429
FSI for fluid - elastic solid configurations	p. 431
Overview	p. 431
Continuum description	p. 433
Fluid structure interaction problem formulation	p. 438
Applications	p. 447
FSI benchmark	p. 451
Conclusion	p. 454
Numerical techniques for fluid-rigid solid configurations	p. 457
Introduction	p. 457
Description of the physical models	p. 460
Moving-mesh method	p. 462
Numerical method	p. 464
Numerical results	p. 471
Conclusions	p. 485
References	p. 492
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