Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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MULTIDIMENSIONAL OPEN SYSTEM FOR VALVELESS PUMPING

  • JUNG, EUNOK (Department of Mathematics Konkuk University) ;
  • KIM, DO WAN (Department of Mathematics Inha University) ;
  • LEE, JONGGUL (Department of Mathematics Konkuk University) ;
  • LEE, WANHO (National Institute for Mathematical Sciences)
  • Received : 2014.09.29
  • Published : 2015.11.30
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Abstract

In this study, we present a multidimensional open system for valveless pumping (VP). This system consists of an elastic tube connected to two open tanks filled with a fluid under gravity. The two-dimensional elastic tube model is constructed based on the immersed boundary method, and the tank model is governed by a system of ordinary differential equations based on the work-energy principle. The flows into and out of the elastic tube are modeled in terms of the source/sink patches inside the tube. The fluid dynamics of this system is generated by the periodic compress-and-release action applied to an asymmetric region of the elastic tube. We have developed an algorithm to couple these partial differential equations and ordinary differential equations using the pressure-flow relationship and the linearity of the discretized Navier-Stokes equations. We have observed the most important feature of VP, namely, the existence of a unidirectional net flow in the system. Our computations are focused on the factors that strongly influence the occurrence of unidirectional flows, for example, the frequency, compression duration, and location of pumping. Based on these investigations, some case studies are performed to observe the details of the ow features.

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References

  1. D. Auerbach, W. Moehring, and M. Moser, An analytic approach to the liebau problem of valveless pumping, Cardiovascular Engineering: An International Journal 4 (2004), no. 2, 201-207. https://doi.org/10.1023/B:CARE.0000031549.13354.5e
  2. R. P. Beyer, A computational model of the cochlea using the immersed boundary method, J. Comput. Phys. 98 (1992), no. 1, 145-162. https://doi.org/10.1016/0021-9991(92)90180-7
  3. A. Borzi and G. Propst, Numerical investigation of the liebau phenomenon, Z. Angew. Math. Phys. 54 (2003), no. 6, 1050-1072. https://doi.org/10.1007/s00033-003-1108-x
  4. T. T. Bringley, S. Childress, N. Vandenberghe, and J. Zhang, An experimental investigation and a simple model of a valveless pump, Phys. Fluids 20 (2008), no. 3, 033602; http://dx.doi.org/10.1063/1.2890790
  5. L. J. Fauci, Peristaltic pumping of solid particles, Comput. Fluids 21 (1992), no. 4, 583-598. https://doi.org/10.1016/0045-7930(92)90008-J
  6. L. J. Fauci and C. S. Peskin, A computational model of aquatic animal locomotion, J. Comput. Phys. 77 (1988), no. 1, 85-108. https://doi.org/10.1016/0021-9991(88)90158-1
  7. A. L. Fogelson and C. S. Peskin, A fast numerical method for solving the threedimensional stokes' equations in the presence of suspended particles, J. Comput. Phys. 79 (1988), no. 1, 50-69. https://doi.org/10.1016/0021-9991(88)90003-4
  8. S. Greenberg, D. M. McQueen, and C. S. Peskin, Three-dimensional fluid dynamics in a two-dimensional amount of central memory, Wave motion: theory, modelling, and computation (Berkeley, Calif., 1986), 85-146, Math. Sci. Res. Inst. Publ., 7, Springer, New York, 1987.
  9. B. E. Grith and C. S. Peskin, On the order of accuracy of the immersed boundary method: higher order convergence rates for sucffiently smooth problems, J. Comput. Phys. 208 (2005), no. 1, 75-105. https://doi.org/10.1016/j.jcp.2005.02.011
  10. A. I. Hickerson and M. Gharib, On the resonance of a pliant tube as a mechanism for valveless pumping, J. Fluid Mech. 555 (2006), 141-148. https://doi.org/10.1017/S0022112006009220
  11. E. Jung, 2-D simulations of valveless pumping using the immersed boundary method, PhD thesis, New York University, Graduate School of Arts and Science, 1999.
  12. E. Jung, A mathematical model of valveless pumping: A lumped model with timedependent compliance, resistance, and inertia, Bull. Math. Biol. 69 (2007), no. 7, 2181-2198. https://doi.org/10.1007/s11538-007-9208-y
  13. E. Jung and W. Lee, Lumped parameter models of cardiovascular circulation in normal and arrhythmia cases, J. Korean Math. Soc 43 (2006), no. 4, 885-897. https://doi.org/10.4134/JKMS.2006.43.4.885
  14. E. Jung, S. Lim, W. Lee, and S. Lee, Computational models of valveless pumping using the immersed boundary method, Comput. Methods Appl. Mech. Engrg. 197 (2008), no. 25-28, 2329-2339. https://doi.org/10.1016/j.cma.2008.01.024
  15. E. Jung and C. S. Peskin, Two-dimensional simulations of valveless pumping using the immersed boundary method, SIAM J. Sci. Comput. 23 (2001), no. 1, 19-45. https://doi.org/10.1137/S1064827500366094
  16. Y. Kim, W. Lee, and E. Jung, An immersed boundary heart model coupled with a multicompartment lumped model of the circulatory system, SIAM J. Sci. Comput. 32 (2010), no. 4, 1809-1831. https://doi.org/10.1137/090761963
  17. W. Lee, E. Jung, and S. Lee, Simulations of valveless pumping in an open elastic tube, SIAM J. Sci. Comput. 31 (2009), no. 3, 1901-1925. https://doi.org/10.1137/08071613X
  18. W. Lee, S. Lim, and E. Jung, Dynamical motion driven by periodic forcing on an open elastic tube in fluid, Commun. Comput. Phys. 12 (2012), no. 2, 494-514. https://doi.org/10.4208/cicp.240111.060811s
  19. G. Liebau,  Uber ein ventilloses pumpprinzip, Naturwissenschaften 41 (1954), 327-327.
  20. G. Liebau, Die stromungsprinzipien des herzens, Z. Kreislau orsch 44 (1955), 677-684.
  21. G. Liebau, Die bedeutung der tragheitskrafte fur die dynamik des blutkreislaufs, Z. Kreislaufforsch 46 (1957), 428-438.
  22. S. Lim and E. Jung, Three-dimensional simulations of a closed valveless pump system immersed in a viscous fluid, SIAM J. Appl. Math. 70 (2010), no. 6, 1999-2022. https://doi.org/10.1137/08073620X
  23. S. Lim and C. S. Peskin, Simulations of the whirling instability by the immersed boundary method, SIAM J. Sci. Comput. 25 (2004), no. 6, 2066-2083. https://doi.org/10.1137/S1064827502417477
  24. C. G. Manopoulos, D. S. Mathioulakis, and S. G. Tsangaris, One-dimensional model of valveless pumping in a closed loop and a numerical solution, Phys. Fluids 18 (2006), 017106. https://doi.org/10.1063/1.2165780
  25. D. M. McQueen, C. S. Peskin, and E. L. Yellin, Fluid dynamics of the mitral valve: physiological aspects of a mathematical model, Am. J. Physiol. Heart Circ. Physiol. 242 (1982), H1095-H1110. https://doi.org/10.1152/ajpheart.1982.242.6.H1095
  26. J. T. Ottesen, Valveless pumping in a fluid- filled closed elastic tube-system: onedimensional theory with experimental validation, J. Math. Biol. 46 (2003), no. 4, 309- 332. https://doi.org/10.1007/s00285-002-0179-1
  27. C. S. Peskin, Flow patterns around heart valves: a digital computer method for solving the equations of motion, PhD thesis, Sue Golding Graduate Division of Medical Sciences, Albert Einstein College of Medicine, Yeshiva University, 1972.
  28. C. S. Peskin, Numerical analysis of blood ow in the heart, J. Comput. Phys. 25 (1977), no. 3, 220-252. https://doi.org/10.1016/0021-9991(77)90100-0
  29. C. S. Peskin and D. M. McQueen, A three-dimensional computational method for blood flow in the heart I. Immersed elastic bers in a viscous incompressible fluid, J. Comput. Phys. 81 (1989), no. 2, 372-405. https://doi.org/10.1016/0021-9991(89)90213-1
  30. C. S. Peskin and D. M. McQueen, A general method for the computer simulation of biological systems interacting with fluids, Sympos. Soc. Exp. Biol. 49 (1995), 265-276.
  31. C. S. Peskin and B. F. Printz, Improved volume conservation in the computation of flows with immersed elastic boundaries, J. Comput. Phys. 105 (1993), no. 1, 33-46. https://doi.org/10.1006/jcph.1993.1051
  32. G. Propst, Pumping effects in models of periodically forced flow configurations, Phys. D 217 (2006), no. 2, 193-201. https://doi.org/10.1016/j.physd.2006.04.007
  33. K. A. Rejniak, A single-cell approach in modeling the dynamics of tumor microregions, Math. Biosci. Eng. 2 (2005), no. 3, 643-655. https://doi.org/10.3934/mbe.2005.2.643
  34. D. Rinderknecht, A. I. Hickerson, and M. Gharib, A valveless micro impedance pump driven by electromagnetic actuation, J. Micromech. Microeng. 15 (2005), 861-866. https://doi.org/10.1088/0960-1317/15/4/026
  35. S. J. Shin and H. J. Sung, Three-dimensional simulation of a valveless pump, Int. J. Heat Fluid Flow 31 (2010), no. 5, 942-951. https://doi.org/10.1016/j.ijheatfluidflow.2010.05.001
  36. H. Thomann, A simple pumping mechanism in a valveless tube, Z. Angew. Math. Phys. 29 (1978), no. 2, 169-177. https://doi.org/10.1007/BF01601511
  37. S. Timmermann and J. T. Ottesen, Novel characteristics of valveless pumping, Phys. Fluids 21 (2009), 053601. https://doi.org/10.1063/1.3114603