Stokes flow applied to the sedimentation of a red blood cell
Authors:
M. Hadjinicolaou, G. Kamvyssas and E. Protopapas
Journal:
Quart. Appl. Math. 73 (2015), 511-523
MSC (2010):
Primary 35Q35, 76D07, 92C05
DOI:
https://doi.org/10.1090/qam/1390
Published electronically:
June 12, 2015
MathSciNet review:
3400756
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The sedimentation of a red blood cell (RBC) through the blood plasma is described as the translation of a rigid inverted prolate spheroid through a quiescent unbounded viscous fluid. The inverted spheroid is moving with constant velocity along its axis of symmetry. The physical characteristics of the RBC and the blood plasma allow us to consider this problem as a Stokes flow problem. The Kelvin inversion method and the concept of semiseparation of variables for the Stokes operators, which we used for solving the Stokes flow past an RBC, are also employed here. The stream function is then expressed as a series expansion in terms of Gegenbauer functions of even order. Through this, important hydrodynamic quantities such as the drag force, the drag coefficient and the terminal settling velocity of the RBC are calculated. The celebrated Stokes formula for the drag force exerted on a sphere is now expanded in order to account for the shape deformation of an RBC. Sample streamlines are depicted showing the dependence of these quantities on the geometrical characteristics of the RBC and also of any inverted prolate spheroid. The obtained results seem to be directly applicable in medical tests such as the Erythrocyte Sedimentation Rate (ESR).
References
- K. Tsubota, S. Wada, and T. Yamaguchi, Particle Method for Computer Simulation of Red Blood Cell Motion in Blood Flow, Computer Methods and Programs in Biomedicine 83 (2006), 139-146.
- T. Zien, Hydrodynamics of Bolus FlowβAn Analytic Approach to Blood Flow in Capillaries, Bulletin of Mathematical Biophysics 31 (1969), 681-694.
- P. Bagchi, Mesoscale Simulation of Blood Flow in Small Vessels, Biophysical Journal 92 (2007), 1858-1877.
- H. Noguchi and G. Gompper, Shape Transitions of Fluid Vesicles and Red Blood Cells in Capillary Flows, Proc. Natl. Acad. Sci. USA 102 (2005), 14159-14164.
- J. L. McWhirter, H. Noguchi and G. Gompper, Flow-Inducted Clustering and Alignment of Red Blood Cells in Microchannels, Proc. Natl. Acad. Sci. USA 106 (2009), 6039-6043.
- G. Dassios, M. Hadjinicolaou, and E. Protopapas, Blood plasma flow past a red blood cell: mathematical modelling and analytical treatment, Math. Methods Appl. Sci. 35 (2012), no. 13, 1547β1563. MR 2957516, DOI https://doi.org/10.1002/mma.2540
- S. Oka, A physical theory of erythrocyte sedimentation, Biorheology 22 (1985), 315-321.
- A. J. Reuben and A. G. Shannon, Some problems in the mathematical modelling of erythrocyte sedimentation, IMA J. Math. Appl. Med. Biol. 7 (1990), no. 3, 145β156. MR 1085599
- V. I. Yamaikina and V. Ivashkevich, A mathematical model of erythrocyte sedimentation in capillaries, J. Eng. Phys. Thermophys. 72 (1999), 55-61.
- J. Tan, B.-D. Lee, L. Polo-Parada and S. Sengupta, Kinetically limited differential centrifugation as an inexpensive and readily available alternative to centrifugal elutriation, BioTechniques 53 (2002), no. 2, 104-108.
- Y. Sugii, R. Okuda, K. Okamoto, and H. Madarame, Velocity Measurement of Both Red Blood Cells and Plasma of in vitro Blood Flow using High-Speed Micro PIV Technique, Institute of Physics Publishing Meas. Sci. Technol. 16 (2005), 1126-1130.
- J. A. Davis, D. W. Inglis, K. J. Morton, D. A. Lawrence, L. R. Huang, S. Y. Chou, J. C. Sturm, and R. H. Austin, Deterministic Hydrodynamics: Taking Blood Apart, PNAS 103 (2003), no 40, 14779-14784.
- R. Lima, T. Ishikawa, T. Imai et al., Blood Flow Behavior in Microchannels: Past, Current and Future Trends, Chem. Biomed. Eng. (2012), 513-547.
- G. G. Stokes, On the Theories of the Internal Friction of Fluids in Motion and the Equilibrium and Motion of Elastic Solids, Trans. Camp. Phil. Soc. 8 (1845), 287-319.
- G. G. Stokes, On the Effect of the Internal Friction of Fluids on the Motion of Pendulums, Trans. Camp. Phil. Soc. 8 (1845), 8-106.
- J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, Kluwer Academic Publishers, 1991.
- G. Dassios, M. Hadjinicolaou, and A. C. Payatakes, Generalized eigenfunctions and complete semiseparable solutions for Stokes flow in spheroidal coordinates, Quart. Appl. Math. 52 (1994), no. 1, 157β191. MR 1262325, DOI https://doi.org/10.1090/qam/1262325
- G. Dassios, M. Hadjinicolaou, F. A. Coutelieris and A. C. Payatakes, Stokes Flow in Spheroidal Particle-in-cell Models with Happel and Kuwabara Boundary Conditions, International Journal of Engineering Science 33 (1995), 1465-1490.
- M. Hadjinicolaou and E. Protopapas, On the $R$-semiseparation of the Stokes bi-stream operator in inverted prolate spheroidal geometry, Math. Methods Appl. Sci. 37 (2014), no. 2, 207β211. MR 3153095, DOI https://doi.org/10.1002/mma.2841
- G. Green, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. Nottingham: Private Publication, 1928. Also in Mathematical Papers of the Late George Green (N. M. Ferrers, ed.), London: Macmillan, 1871.
- George Dassios and R. E. Kleinman, On Kelvin inversion and low-frequency scattering, SIAM Rev. 31 (1989), no. 4, 565β585. MR 1025482, DOI https://doi.org/10.1137/1031126
- George Dassios and Ralph E. Kleinman, On the capacity and Rayleigh scattering for a class of nonconvex bodies, Quart. J. Mech. Appl. Math. 42 (1989), no. 3, 467β475. MR 1018522, DOI https://doi.org/10.1093/qjmam/42.3.467
- G. Baganis and M. Hadjinicolaou, Analytic solution of an exterior Dirichlet problem in a non-convex domain, IMA J. Appl. Math. 74 (2009), no. 5, 668β684. MR 2549954, DOI https://doi.org/10.1093/imamat/hxp023
- G. Baganis and M. Hadjinicolaou, Analytic solution of an exterior Neumann problem in a non-convex domain, Math. Methods Appl. Sci. 33 (2010), no. 17, 2067β2075. MR 2762318, DOI https://doi.org/10.1002/mma.1316
- George Dassios, The Kelvin transformation in potential theory and Stokes flow, IMA J. Appl. Math. 74 (2009), no. 3, 427β438. MR 2507298, DOI https://doi.org/10.1093/imamat/hxn027
- G. Baganis, G. Dassios, M. Hadjinicolaou, and E. Protopapas, The Kelvin transformation as a tool for analyzing problems in medicine and technology, Math. Methods Appl. Sci. 37 (2014), no. 2, 194β199. MR 3153093, DOI https://doi.org/10.1002/mma.2903
- M. R. Ismailov, A. N. Shevchuk, H. Khusanov, Mathematical model describing erythrocyte sedimentation rate. Implications for blood viscocity changes in traumatic shock and crush syndrome, Biomedical Engineering OnLine, 2005, DOI 10.1186/1475-925X-4-24.
- K. Cha, F. E. Brown, and W. D. Wilmore, A new bioelectrical impedance method for measurement of the erythrocyte sedimentation rate, Physiol. Meas. 15 (1994), 499-508.
- N. N. Lebedev, Special functions and their applications, Dover Publications, Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman; Unabridged and corrected republication. MR 0350075
- P. Moon and D. E. Spencer, Field theory handbook, 2nd ed., Springer-Verlag, Berlin, 1988. Including coordinate systems, differential equations and their solutions. MR 947546
- L. E. Payne and W. H. Pell, The Stokes flow problem for a class of axially symmetric bodies, J. Fluid Mech. 7 (1960), 529β549. MR 115471, DOI https://doi.org/10.1017/S002211206000027X
References
- K. Tsubota, S. Wada, and T. Yamaguchi, Particle Method for Computer Simulation of Red Blood Cell Motion in Blood Flow, Computer Methods and Programs in Biomedicine 83 (2006), 139-146.
- T. Zien, Hydrodynamics of Bolus FlowβAn Analytic Approach to Blood Flow in Capillaries, Bulletin of Mathematical Biophysics 31 (1969), 681-694.
- P. Bagchi, Mesoscale Simulation of Blood Flow in Small Vessels, Biophysical Journal 92 (2007), 1858-1877.
- H. Noguchi and G. Gompper, Shape Transitions of Fluid Vesicles and Red Blood Cells in Capillary Flows, Proc. Natl. Acad. Sci. USA 102 (2005), 14159-14164.
- J. L. McWhirter, H. Noguchi and G. Gompper, Flow-Inducted Clustering and Alignment of Red Blood Cells in Microchannels, Proc. Natl. Acad. Sci. USA 106 (2009), 6039-6043.
- G. Dassios, M. Hadjinicolaou, and E. Protopapas, Blood plasma flow past a red blood cell: mathematical modelling and analytical treatment, Math. Methods Appl. Sci. 35 (2012), no. 13, 1547β1563. MR 2957516, DOI https://doi.org/10.1002/mma.2540
- S. Oka, A physical theory of erythrocyte sedimentation, Biorheology 22 (1985), 315-321.
- A. J. Reuben and A. G. Shannon, Some problems in the mathematical modelling of erythrocyte sedimentation, IMA J. Math. Appl. Med. Biol. 7 (1990), no. 3, 145β156. MR 1085599 (92e:92011)
- V. I. Yamaikina and V. Ivashkevich, A mathematical model of erythrocyte sedimentation in capillaries, J. Eng. Phys. Thermophys. 72 (1999), 55-61.
- J. Tan, B.-D. Lee, L. Polo-Parada and S. Sengupta, Kinetically limited differential centrifugation as an inexpensive and readily available alternative to centrifugal elutriation, BioTechniques 53 (2002), no. 2, 104-108.
- Y. Sugii, R. Okuda, K. Okamoto, and H. Madarame, Velocity Measurement of Both Red Blood Cells and Plasma of in vitro Blood Flow using High-Speed Micro PIV Technique, Institute of Physics Publishing Meas. Sci. Technol. 16 (2005), 1126-1130.
- J. A. Davis, D. W. Inglis, K. J. Morton, D. A. Lawrence, L. R. Huang, S. Y. Chou, J. C. Sturm, and R. H. Austin, Deterministic Hydrodynamics: Taking Blood Apart, PNAS 103 (2003), no 40, 14779-14784.
- R. Lima, T. Ishikawa, T. Imai et al., Blood Flow Behavior in Microchannels: Past, Current and Future Trends, Chem. Biomed. Eng. (2012), 513-547.
- G. G. Stokes, On the Theories of the Internal Friction of Fluids in Motion and the Equilibrium and Motion of Elastic Solids, Trans. Camp. Phil. Soc. 8 (1845), 287-319.
- G. G. Stokes, On the Effect of the Internal Friction of Fluids on the Motion of Pendulums, Trans. Camp. Phil. Soc. 8 (1845), 8-106.
- J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, Kluwer Academic Publishers, 1991.
- G. Dassios, M. Hadjinicolaou, and A. C. Payatakes, Generalized eigenfunctions and complete semiseparable solutions for Stokes flow in spheroidal coordinates, Quart. Appl. Math. 52 (1994), no. 1, 157β191. MR 1262325 (95d:76034)
- G. Dassios, M. Hadjinicolaou, F. A. Coutelieris and A. C. Payatakes, Stokes Flow in Spheroidal Particle-in-cell Models with Happel and Kuwabara Boundary Conditions, International Journal of Engineering Science 33 (1995), 1465-1490.
- M. Hadjinicolaou and E. Protopapas, On the $R$-semiseparation of the Stokes bi-stream operator in inverted prolate spheroidal geometry, Math. Methods Appl. Sci. 37 (2014), no. 2, 207β211. MR 3153095, DOI https://doi.org/10.1002/mma.2841
- G. Green, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. Nottingham: Private Publication, 1928. Also in Mathematical Papers of the Late George Green (N. M. Ferrers, ed.), London: Macmillan, 1871.
- George Dassios and R. E. Kleinman, On Kelvin inversion and low-frequency scattering, SIAM Rev. 31 (1989), no. 4, 565β585. MR 1025482 (90i:35073), DOI https://doi.org/10.1137/1031126
- George Dassios and Ralph E. Kleinman, On the capacity and Rayleigh scattering for a class of nonconvex bodies, Quart. J. Mech. Appl. Math. 42 (1989), no. 3, 467β475. MR 1018522 (90h:35061), DOI https://doi.org/10.1093/qjmam/42.3.467
- G. Baganis and M. Hadjinicolaou, Analytic solution of an exterior Dirichlet problem in a non-convex domain, IMA J. Appl. Math. 74 (2009), no. 5, 668β684. MR 2549954 (2011a:35124), DOI https://doi.org/10.1093/imamat/hxp023
- G. Baganis and M. Hadjinicolaou, Analytic solution of an exterior Neumann problem in a non-convex domain, Math. Methods Appl. Sci. 33 (2010), no. 17, 2067β2075. MR 2762318 (2011m:35053), DOI https://doi.org/10.1002/mma.1316
- George Dassios, The Kelvin transformation in potential theory and Stokes flow, IMA J. Appl. Math. 74 (2009), no. 3, 427β438. MR 2507298 (2010h:76052), DOI https://doi.org/10.1093/imamat/hxn027
- G. Baganis, G. Dassios, M. Hadjinicolaou, and E. Protopapas, The Kelvin transformation as a tool for analyzing problems in medicine and technology, Math. Methods Appl. Sci. 37 (2014), no. 2, 194β199. MR 3153093, DOI https://doi.org/10.1002/mma.2903
- M. R. Ismailov, A. N. Shevchuk, H. Khusanov, Mathematical model describing erythrocyte sedimentation rate. Implications for blood viscocity changes in traumatic shock and crush syndrome, Biomedical Engineering OnLine, 2005, DOI 10.1186/1475-925X-4-24.
- K. Cha, F. E. Brown, and W. D. Wilmore, A new bioelectrical impedance method for measurement of the erythrocyte sedimentation rate, Physiol. Meas. 15 (1994), 499-508.
- N. N. Lebedev, Special functions and their applications, Dover Publications Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman; unabridged and corrected republication. MR 0350075 (50 \#2568)
- P. Moon and D. E. Spencer, Field theory handbook, Including coordinate systems, differential equations and their solutions, 2nd ed., Springer-Verlag, Berlin, 1988. MR 947546 (89i:00026)
- L. E. Payne and E. H. Pell, The Stokes flow problem for a class of axially symmetric bodies, Fluid Mech. 7 (1960), 529. MR 0115471 (22 \#6272)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2010):
35Q35,
76D07,
92C05
Retrieve articles in all journals
with MSC (2010):
35Q35,
76D07,
92C05
Additional Information
M. Hadjinicolaou
Affiliation:
School of Science and Technology, Hellenic Open University, 11 Sahtouri str., GR-26 222 Patras, Greece
Email:
hadjinicolaou@eap.gr
G. Kamvyssas
Affiliation:
Department of Mechanical Engineering, TEI of Western Greece, 1, M. Alexandrou str., Koukouli, GR-26 334 Patras, Greece
Email:
greg@teiwest.gr
E. Protopapas
Affiliation:
School of Science and Technology, Hellenic Open University, 11 Sahtouri str., GR-26 222 Patras, Greece
Email:
lprotopapas@eap.gr
Keywords:
Stokes flow,
red blood cell (RBC),
blood plasma flow,
mathematical model,
Kelvin inversion,
stream function,
drag force,
drag coefficient,
terminal settling velocity,
sedimentation
Received by editor(s):
October 15, 2013
Received by editor(s) in revised form:
November 30, 2013
Published electronically:
June 12, 2015
Article copyright:
© Copyright 2015
Brown University
