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A Lattice Boltzmann and Immersed Boundary Scheme for Model Blood Flow in Constricted Pipes: Part 1 - Steady Flow

Published online by Cambridge University Press:  03 June 2015

S. C. Fu ,
W. W. F. Leung  and
R. M. C. So
S. C. Fu*
Affiliation:
Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
W. W. F. Leung*
Affiliation:
Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
R. M. C. So*
Affiliation:
Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
*
Corresponding author.Email:mescfu@ust.hk
Email:Wallace.Leung@inet.polyu.edu.hk
Email:mmmcso@polyu.edu.hk
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Abstract

Hemodynamics is a complex problem with several distinct characteristics; fluid is non-Newtonian, flow is pulsatile in nature, flow is three-dimensional due to cholesterol/plague built up, and blood vessel wall is elastic. In order to simulate this type of flows accurately, any proposed numerical scheme has to be able to replicate these characteristics correctly, efficiently, as well as individually and collectively. Since the equations of the finite difference lattice Boltzmann method (FDLBM) are hyperbolic, and can be solved using Cartesian grids locally, explicitly and efficiently on parallel computers, a program of study to develop a viable FDLBM numerical scheme that can mimic these characteristics individually in any model blood flow problem was initiated. The present objective is to first develop a steady FDLBM with an immersed boundary (IB) method to model blood flow in stenoic artery over a range of Reynolds numbers. The resulting equations in the FDLBM/IB numerical scheme can still be solved using Cartesian grids; thus, changing complex artery geometry can be treated without resorting to grid generation. The FDLBM/IB numerical scheme is validated against known data and is then used to study Newtonian and non-Newtonian fluid flow through constricted tubes. The investigation aims to gain insight into the constricted flow behavior and the non-Newtonian fluid effect on this behavior.

Keywords

76Z0576M2065N06Finite difference methodlattice Boltzmann methodimmersed boundary methodblood flowconstricted pipe
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 1 , July 2013 , pp. 126 - 152
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]M. J, Davies. and A. C., Thomas, 1985, Plaque Fissuring - The Cause of Acute Myocardial Infarction, Sudden Ischemic Death, and Crescendo Angina, British Heart Journal, 53(4), pp. 363–373.Google Scholar
[2]V., Fuster, B., Stein, J. A., Ambrose, L., Badimon, J. J., Badimon and J. H., Chesebro, 1990, Atherosclerotic Plaque Rupture and Thrombosis, Circulation, Supplement II, 82(3), pp. II–47-II-59.Google Scholar
[3]A. P., Burke, A., Farb, G. T., Malcom, Y. H., Liang, J. E., Smialek and R., Virmani, 1999, Plaque Rupture and Sudden Death Related to Exertion in Men with Coronary Artery Disease, Journal of the American Medical Association, 281(10), pp. 921–926.Google Scholar
[4]S., Chenand G. D., Doolen, 1998, Lattice Boltzmann Method for Fluid Flows, Annual Review of Fluid Mechanics, 30, pp. 329–364.Google Scholar
[5]D. A., Wolf-Gladrow, 2000, Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction, Chapter 5, Springer-Verlag.Google Scholar
[6]J. D., Sterling and S., Chen, 1996, Stability Analysis of Lattice Boltzmann Methods, Journal of Computational Physics, 123(1), pp. 196–206.Google Scholar
[7]G., Wellein, P., Lammers, G., Hager, S., Donath and T., Zeiser, 2005, Towards Optimal Performance for Lattice Boltzmann Applications on Terascale Computers, in Parallel Computational Fluid Dynamics: Theory and Applications (Editors: Deane, A.et al.), Proceedings of the 2005 International Conference on Parallel Computational Fluid Dynamics, May 24–27, College Park, MD, USA, pp. 31–40.Google Scholar
[8]G., Wellein, T., Zeiser, G., Hager and S, Donath, 2006, On the Single Processor Performance for Simple Lattice Boltzmann Kernels, Computers and Fluids, 35, pp. 910–919.Google Scholar
[9]M., Krafczyk, M., Cerrolzaz, M., Schulz and E., Rank, 1998, Analysis of 3D Transient Blood Flow Passing Through an Artificial Aortic Valve by Lattice-Boltzmann Methods, Journal of Biomechanics, 31(5), pp. 453–462.Google Scholar
[10]R., Ouared, B., Chopard, B., Stahl, D. A., R?fenacht, H., Yilmaz, and G., Courbebaisse, 2008, Thrombosis Modelling in Intracranial Aneurysms: A Lattice Boltzmann Numerical Algorithm, Computer Physics Communications, 179(1-3), pp. 128–131.Google Scholar
[11]J., Bernsdorf, S. E., Harrison, S. M., Smith, P. V., Lawford, and D. R., Hose, 2008, Applying the Lattice Boltzmann Technique to Biofluids: A Novel Approach to Simulate Blood Coagulation, Computers and Mathematics with Applications, 55(7), pp. 1408–1414.Google Scholar
[12]D., Wang, and J., Bernsdorf, 2009, Lattice Boltzmann Simulation of Steady Non-Newtonian Blood Flow in a 3D Generic Stenosis Case, Computer and Mathematics with Applications, 58(5), pp. 1030–1034.Google Scholar
[13]C., Chen, H, Chen, D., Freed, R., Shock, I., Staroselsky, R., Zhang, A., Co?kun ?., P. H., Stone, and C. L., Feldman, 2006, Simulation of Blood Flow Using Extended Boltzmann Kinetic Approach, Physica A, 362(1), pp. 174–181.Google Scholar
[14]J., Boyd, J. M., Buick, J. A., Cosgrove, and P., Stansell, 2004, Application of the Lattice Boltz-mann Method to Arterial Flow Simulation: Investigation of Boundary Conditions for Complex Arterial Geometries, Australasian Physical & Engineering Sciences in Medicine, 27(4), pp. 207–212.Google Scholar
[15]C. S., Peskin, 1977, Numerical Analysisof Blood Flow in the Heart, Journalof Computational Physics, 25(3), pp. 220–252.Google Scholar
[16]R., Mittal and G., Iaccarino, 2005, Immersed Boundary Methods, Annual Review of Fluid Mechanics, 37, pp. 239–261.Google Scholar
[17]S. C., Fu, W. W. F., Leung and R. M. C., So, 2009, A Lattice Boltzmann Based Numerical Scheme for Microchannel Flows, Journal of Fluids Engineering, 131(August), Paper 081401(11 pages).Google Scholar
[18]S. C., Fu and R. M. C., So, 2009, Modeled Boltzmann Equation and the Constant Density Assumption, AIAA Journal, 47(12), pp. 3038–3042.Google Scholar
[19]X., He and L. S., Luo, 1997, Lattice Boltzmann Model for the Incompressible Navier-Stokes Equation, Journal of Statistical Physics, 88(3/4), pp. 927–944.Google Scholar
[20]S. C., Fu, R. M. C., So and W. W. F., Leung, 2011, A Discrete Flux Scheme for Aerodynamic and Hydrodynamic Flows, Communications in Computational Physics 9(5), 1257–1283.Google Scholar
[21]J. M., Siegel, C. P., Markou, D. N., Ku and S. R., Hanson, 1994, A Scaling Law for Wall Shear Rate through anArterial Stenosis, Journal of Biomechanical Engineering, 116(4), pp. 446–451.Google Scholar
[22]H., Huang, V. J., Modi and B. R., Seymour, 1995, Fluid Mechanics of Stenosed Arteries, International Journal of Engineering Science, 33(6), pp. 815–828.Google Scholar
[23]S. C., Fu, 2011, Numerical Simulation of Blood Flow in Stenotic Arteries, PhD thesis, Mechanical Engineering Department, The Hong Kong Polytechnic University, Hung Hom, Hong Kong.Google Scholar
[24]R. P., Beyer and R. J., Leveque, 1992, Analysis of a One-Dimensional Model for the Immersed Boundary Method, SIAM Journal on Numerical Analysis, 29(2), pp. 332–364.Google Scholar
[25]M. C., Lai and C. S., Peskin, 2000, An Immersed Boundary Method with Formal Second-Order Accuracy and Reduced Numerical Viscosity, Journal of Computational Physics, 160(2), pp. 705–719.Google Scholar
[26]Z. G., Feng and E. E., Michaelides, 2004, The Immersed Boundary-lattice Boltzmann Method for Solving Fluid-particles Interaction Problems, Journal of Computational Physics, 195, pp. 602–628.Google Scholar
[27]Z. G., Feng and E. E., Michaelides, 2005, Proteus: a Direct Forcing Method in the Simulations of Particulate Flows, Journal of Computational Physics, 202, pp. 20–51.Google Scholar
[28]X. D., Niu, C., Shu, Y. T., Chew and Y., Peng, 2006, A Momentum Exchange-based Immersed Boundary-lattice Boltzmann Method for Simulating Incompressible Viscous Flows, Physics Letters A, 354, pp. 173–182.Google Scholar
[29]O. E., Strack and B. K., Cook, 2007, Three-Dimensional Immersed Boundary Conditions for Moving Solids in the Lattice-Boltzmann Method, International Journal for Numerical Methods in Fluids, 55, pp. 103–125.Google Scholar
[30]C., Shu, N., Liu and Y. T., Chew, 2007, A Novel Immersed Boundary Velocity Correction-Lattice Boltzmann Method and Its application to Simulate Flow Past a Circular Cylinder, Journal of Computational Physics, 226, pp. 1607–1622.Google Scholar
[31]A. J., Chorin, 1967, A Numerical Method for Solving Incompressible Viscous Flow Problems, Journal of Computational Physics, 2, pp. 12–26.Google Scholar
[32]J. C., Tannehill, D. A., Anderson and R. H., Pletcher, 1997, Computational Fluid Mechanics and Heat Transfer, 2nd ed., Taylor & Francis, Washington, D.C., Chap. 9, pp. 649–776.Google Scholar
[33]E. W. S., Kam, R. M. C., So, and R. C. K., Leung, 2007, Lattice Boltzamann Method Simulation of Aeroacoustics and Nonreflecting Boundary Conditions, AIAA Journal, 45(7), pp. 1703–1712.Google Scholar
[34]M. D., Deshpande, D. P., Giddens and R. F., Mabon, 1976, Steady Laminar Flow Through Modeled Vascular Stenoses, Journal of Biomechanics, 9(4), pp. 165–174.Google Scholar
[35]S. S, Shibeshi and W. E., Collins, 2005, The Rheology of Blood Flow in a Branched Arterial System, Applied Rheology, 15(6), pp. 398–405.Google Scholar
[36]Y. C., Fung, 1993, Biomechanics: Mechanical Properties of Living Tissue, Springer-Verlag, Chap. 3.Google Scholar
[37]F. J. H., Gijsen, Vosse, F. N., van de and J. D., Janssen, 1999, The Influence of the Non-Newtonian Properties of Blood on the Flow in Large Arteries: Steady Flow in a Carotid Bifurcation Model, Journal of Biomechanics, 32, pp. 601–608.Google Scholar
[38]R. B., Bird, R. C., Armstrong and O., Hassager, 1987, Dynamics of Polymeric Liquids, Vol. 1, 2nd ed., Wiley, New York, p. 171.Google Scholar
[39]G.N., Waite 2009, Blood Components, in Medical Physiology, 3rd Edition, Rhoades, R. and Bell, D. (Eds). Lippincott Williams & Wilkings, U.S.A., Chap. 9, p. 171.Google Scholar
[40]R. M., Nerem, 1993, Haemodynamics and the Vascular Endothelium, ASME Journal of Biomechanical Engineering, 115(4B), pp. 510–514.Google Scholar
[41]D. N., Ku, 1997, Blood Flow in Arteries, Annual Review of Fluid Mechanics, 29, pp. 399–434.Google Scholar
[42]D. M., Wootton and D. N., Ku, 1999, Fluid Mechanics of Vascular Systems, Diseases, and Thrombosis, Annual Review of Biomedical Engineering, 1, pp. 299–329.Google Scholar
[43]D. P., Giddens, C. K., Zarins, and S., Glagov, 1993, The Role of Fluid Mechanics in the Localization and Detection of Atherosclerosis, Journal of Biomechanical Engineering, 115(4B), pp. 588–594.Google Scholar
[44]D., Young and F., Tsai, 1973, Flow Characteristics in Models of Arterial Stenoses - I. Steady flow, Journal of Biomechanics, 6(4), pp. 395–402.Google Scholar
[45]J. D., Anderson, 1995, Computational Fluid Dynamics: The Basics with Applications, McGraw-Hill, New York, Ch. 10, pp. 458–459.Google Scholar
[46]S. C., Fu, R. M. C., So and W. W. F., Leung, 2013, A Lattice Boltzmann and Immersed Boundary Scheme for Model Blood Flow in Constricted Pipes: Part 2 - Pulsatile Flow, Communications in Computational Physics, 14(1), pp. 153–173.Google Scholar