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THE INFLUENCE OF DRIVING FUNCTION ON FLOW DRIVEN BY PUMPING WITHOUT VALVES

  • Jung, Eun-Ok (DEPARTMENT OF MATHEMATICS, KONKUK UNIVERSITY)
  • Received : 2011.03.21
  • Accepted : 2011.03.28
  • Published : 2011.06.25

Abstract

Fluid dynamics driven by pumping without valves (valveless pumping) shows interesting physics. Especially, the driving function to generate valveless pump mechanism is one of important factors. We consider a closed system of valveless pump which consists of flexible tube part and stiffer part. Fluid and structure (elastic tube) interaction motions are generated by the periodic compress-and-release actions on an asymmetric location of the elastic loop of tubing. In this work, we demonstrate how important the driving forcing function affects a net flow in the valveless circulatory system and investigate which parameter set of the system gives a more efficient net flow around the loop.

Acknowledgement

Supported by : National Research Foundation of Korea

References

  1. K. M. ARTHURS, L. C. MOORE, C. S. PESKIN, AND ET AL., MODELING ARTERIOLAR FLOW AND MASS TRANSPORT USING THE IMMERSED BOUNDARY METHOD, J. Comput. Phys., 147 (1998), pp. 402-440. https://doi.org/10.1006/jcph.1998.6097
  2. C. BEATTIE, A. D. GUERCI, T. HALL, A. M. BORKON, W. BAUMGARTNER, R. S. STUART, J. PETERS, H. HALPERIN, AND J. L. ROBOTHAM, Mechanisms of blood flow during pneumatic vest cardiopulmonary resuscitation, J. Appl. Physiol., 70 (1991), pp. 454-465. https://doi.org/10.1152/jappl.1991.70.1.454
  3. R. P. BEYER, A computational model of the cochlea using the immersed boundary method, J. Comput. Phys., 98 (1992), pp. 145-162. https://doi.org/10.1016/0021-9991(92)90180-7
  4. D. C. BOTTINO, Modeling Viscoelastic Networks and Cell Deformation in the Context of the Immersed Boundary Method, J. Comput. Phys., 147 (1998), pp. 86-113. https://doi.org/10.1006/jcph.1998.6074
  5. R. CORTEZ AND M. MINION, The Blob Projection Method for Immersed Boundary Problems, J. Comput. Phys., 161 (2000), pp. 428-453. https://doi.org/10.1006/jcph.2000.6502
  6. J. M. CRILEY, J. T. NIEMANN, J. P. ROSBOROUGH, S. UNG, AND J. SUZUKI, The heart is a conduit in CPR, Crit. Care Med., 9 (1981), pp. 373. https://doi.org/10.1097/00003246-198105000-00010
  7. R. DILLON, L. J. FAUCI, AND D. GAVER III, A microscale model of bacterial swimming, chemotaxis and substrate transport, Journal of Theoretical Biology, 177 (1995), pp. 325-340. https://doi.org/10.1006/jtbi.1995.0251
  8. R. DILLON, L. J. FAUCI, A. L. FOGELSON, AND D. GAVER III, Modeling biofilm processes using the immersed boundary method, J. Comput. Phys., 129 (1) (1996), pp. 57-73. https://doi.org/10.1006/jcph.1996.0233
  9. L. J. FAUCI AND C. S. PESKIN, A computational model of aquatic animal locomotion, J. Comput. Phys., 77 (1988), pp. 85-108. https://doi.org/10.1016/0021-9991(88)90158-1
  10. L. J. FAUCI, Peristaltic pumping of solid particles, Computers and Fluids, 21 (1992), pp. 583-598. https://doi.org/10.1016/0045-7930(92)90008-J
  11. L J. FAUCI AND A. L. FOGELSON, Truncated Newton methods and the modeling of complex immersed elastic structures, Communications on Pure and Applied Mathematics, 46 (1993), pp. 787-818. https://doi.org/10.1002/cpa.3160460602
  12. L J. FAUCI AND A. MCDONALD, Sperm motility in the presence of boundaries, Bulletin of Mathematical Biology, 57 (5) (1995), pp. 679-699. https://doi.org/10.1007/BF02461846
  13. A. L. FOGELSON, A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting, J. Comput. Phys., 56 (1984), pp. 111-134. https://doi.org/10.1016/0021-9991(84)90086-X
  14. A. L. FOGELSON AND C. S. PESKIN, A fast numerical method for solving the three-dimensional Stoke's equations in the presence of suspended particles, J. Comput. Phys., 79 (1988), pp. 50-69. https://doi.org/10.1016/0021-9991(88)90003-4
  15. J. GUNTZIG, S. NOLTE, P. SCHAD, AND R. PFANKUCHEN, Die Lymphdrainage von Cornea, Limbus und Conjunctiva Klin, MbL Augenheilkunde, 190 (1987), pp. 491-495. https://doi.org/10.1055/s-2008-1050441
  16. H. R. HALPERIN, J. E. TSITLIK, R. BEYAR, N. CHANDRA, AND A. D. GUERCI, Intrathoracic pressure fluctuations move blood during CPR: comparison of hemodynamic data with predictions from a mathematical model, Ann. Biomed. Eng., 15 (1987), pp. 385-403. https://doi.org/10.1007/BF02584292
  17. W. HARVEY, Exercitatio Anatomica de Motu Cordis et Sanguinis in Animalibus, Frankford, (1987), Caput 14.
  18. E. JUNG, 2-D simulations of valvelelss pumping using the Immersed Boundary Method, Ph.D. Thesis, Courant Institute of Mathematical Sciences in New York University, 1999.
  19. E. JUNG AND C. S. PESKIN, Two-Dimensional Simulations of Valvelelss Pumping Using the Immersed Boundary Method, SIAM J. Sci. Comput., 23, 1 (2001), pp. 19-45. https://doi.org/10.1137/S1064827500366094
  20. P. J. KILNER, Formed flow, fluid oscillation and the heart as a morphodynamic pump (abstract), European surgical research, 19, suppl 1 (1987), pp. 89-90.
  21. M. C. LAI AND C. S. PESKIN, An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, 160 (2) (2000), pp. 705-719. https://doi.org/10.1006/jcph.2000.6483
  22. R. J. LEVEQUE AND Z. LI, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31 (1994), pp.1019-1044. https://doi.org/10.1137/0731054
  23. G. LIEBAU, Die Bedeutung der Tragheitskrafte fur die Dynamik des Blutkreislaufs, Zs Kreislaufforschung, 46 (1957), pp. 428-438.
  24. G. LIEBAU, Aus welchem bleibt die Blutforderung durch das Herz bei valvularem Versagen erhalten?, Z. f. Kreislaufforschg., 45 (1956), pp. 481-488.
  25. G. LIEBAU, Die Stromungsprinzipien des Herzens, Zs Kreislaufforschung, 44 (1955), pp. 677-684.
  26. G. LIEBAU, Uber ein Ventilloses Pumpprinzip, Naturwissenschsften, 41 (1954), pp. 327-328.
  27. D. M. MCQUEEN, C. S. PESKIN, AND E. L. YELLIN, Fluid dynamics of the mitral valve: physiological aspects of a mathematical model, Am. J. of physiol., 242 (1982), pp. H1095-H1110.
  28. D. M. MCQUEEN AND C. S. PESKIN, Shared-memory parallel vector implementation of the immersed boundary method for the computation of blood flow in the beating mammalian heart, Journal of Supercomputing, 11 (3) (1997), pp. 213-236. https://doi.org/10.1023/A:1007951707260
  29. S. MIYAZAKI, T. KAWAI, AND M. ARARAGI, A piezo-electric pump driven by a flexural progressive wave, IEEE Transactions, pp. 283-288, 1991.
  30. K. L. MOORE, Embryologie, Schattauer Stuttgart, 2nd ed (1985), pp. 340-358.
  31. M. MOSER, J. W.HUANG, G. S. SCHWARZ, T. KENNER, AND A. NOORDERGRAAF, Impedance defined flow, generalisation of William Harvey's concept of the circulation - 370 years later, International Journal of Cardiovascular Medicine and Science, Vol 1, Nos 3/4 (1998), pp. 205-211.
  32. C. S. PESKIN, Flow patterns around heart valves: A digital computer method for solving the equations of motion, Ph.D. Thesis, Albert Einstein College of Medicine, 1972.
  33. C. S. PESKIN, Numerical analysis of blood flow in the heart, J. Comput. Phys., 25 (1977), pp. 220-252. https://doi.org/10.1016/0021-9991(77)90100-0
  34. C. S. PESKIN ET AL., Three dimensional fluid dynamics in a two-dimensional amount of central memory. Wave Motion: Theory, Modeling and Computation, 1 (1987), pp. 85-146.
  35. C. S. PESKIN AND D. M. MCQUEEN, A three-dimensional computational method for blood flow in the heart: Immersed elastic fibers in a viscous incompressible fluid, J. Comput. Phys., 81 (1989), pp. 372-405. https://doi.org/10.1016/0021-9991(89)90213-1
  36. C. S. PESKIN AND D. M. MCQUEEN, A general method for the computer simulation of biological systems interacting with fluids, Symposia of the Society for Experimental Biology, 49 (1995), pp. 265-276.
  37. C. S. PESKIN AND D. M. MCQUEEN, Fluid dynamics of the heart and its valves, Case studies in mathematical modeling - Ecology, Physiology, and Cell Biology, pp. 309-337, 1996.
  38. C. S. PESKIN AND B. F. PRINTZ, Improved volume conservation in the computation of flows with immersed elastic boundaries, J. Comput. Phys., 105 (1993), pp. 33-46. https://doi.org/10.1006/jcph.1993.1051
  39. A. M. ROMA, C. S. PESKIN, AND M. J. BERGER, An adaptive version of the Immersed Boundary Method, J. Comput. Phys., 153 (1999), pp. 509-534. https://doi.org/10.1006/jcph.1999.6293
  40. M. E. ROSAR, A three dimensional model for fluid flow through a collapsible tube, PhD thesis, New York University, 1994.
  41. D. J. RANDALL AND P. S. DAVIE, The Hearts and Heart-like Organs, Academic Press, London, 1 (1980), pp. 51-53.
  42. H. THOMANN, A Simple Pumping Mechanism in a Valveless Tube, Journal of Applied Math. and Phys., 29 (1978), pp. 169-177. https://doi.org/10.1007/BF01601511
  43. J. A. WERNER, M.D., H. L. GREENE, M.D., C. L. JANKO, AND L. A. COBB, M.D., Visualization of cardiac valve motion in man during external chest compression using two-dimensional echocardiography.: implications regarding the mechanism of blood flow, Circulation, 63 (1981), pp. 1417-1421. https://doi.org/10.1161/01.CIR.63.6.1417