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Mathematical Modelling to Simulate Biological Fluid Flow in a Collapsible Tube

Received: 26 December 2017     Accepted: 11 January 2018     Published: 29 January 2018
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Abstract

The purpose of this paper is to advance a mathematical model for reviewing to simulate biological flows such as blood flow in arteries or veins, flow of urine in urethras and air flow in the bronchial airways. They can also be used to study and prediction of many diseases, as the lung disease (asthma and emphysema), or the cardiovascular diseases (heart stroke), Makinde (2005). In this work, laminar flow of an incompressible viscous fluid through a collapsible tube of circular cross section is considered. Collapsible tubes are easily deformed by negative transmural pressure owing to marked reduction of rigidity. Thus, they show a considerable nonlinearity and reveal various complicated phenomena Our objectives are to study the effect of temperature along the tube as the fluid Prandtl number and Reynolds number increases. We launch the mathematical formulation of the problem. The problem is solved by using power series and perturbation techniques with help of boundary conditions and results are displayed graphically for different flow characteristics, velocity profile.

Published in American Journal of Mechanics and Applications (Volume 6, Issue 1)
DOI 10.11648/j.ajma.20180601.11
Page(s) 1-6
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Power Series, Transmural Pressure, Collapsible Tube

References
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[2] C. D. Bertram and T. J. Pedley, (1982), A mathematical model of unsteady collapsible tube behaviour, J. Biomech. 15, 39–50.
[3] C. D. Bertram and C. J. Raymond, (1991), Measurements of wave speed and compliance in a collapsible tube during self-excited oscillations: A test of the choking hypothesis, Med. Biol. Eng. Comput. 29, 493–500.
[4] C. D. Bertram, C. J. Raymond and T. J. Pedley, (1990), Mapping of instabilities for flow through collapsible tubes of deferring length, J. Fluids. Struct. 4, 125–153.
[5] C. D. Bertram, C. J. Raymond and T. J. Pedley, (1991), Application of nonlinear dynamics concepts to the analysis of self-excited oscillations of a collapsible tube conveying a fluid, J. Fluids. Struck. 5, 391–287.
[6] M. Bonis and C. Ribreau, (1987), Etude de quelques propriétés de l’ecoulement dans une conduite collabable, La Houille Blanche 3/4, 165–173.
[7] R. W. Brower and C. Scholten, (1975), Experimental evidence on the mechanism for the instability of flow in collapsible vessels, Med. Biol. Engng 13, 839–845.
[8] W. A. Contrad, (1969), Pressure-flow relationship in collapsible tubes, IEEE Trans. Bio-Med. Engng BME-16, 284–295.
[9] S. J. Cowley, (1982), Elastic jumps in fluid-filled elastic tubes, J. Fluid Mech. 116, 459–473.
[10] S. J. Cowley, (1983), On the wavetrains associated with elastic jumps on fluid-filled elastic tubes, Q. J. Mech. Appl. Math., 36, 289–312.
[11] C. Domb and M. F. Sykes, (1957), On the susceptibility of a ferromagnetic above the Curie point, Proc. R. Soc. London, Ser. A, 240, 214–228.
[12] P. G. Drazin and Y. Tourigny, (1996), Numerical study of bifurcations by analytic continuation of a function defined by a power series, SIAM J. Appl. Math. 56, 1–18.
[13] D. Elad, R. D. Kamm and A. H. Shapiro, (1987), Choking phenomena in a lung-like model, ASME J. Biomech. Engng 109, 1–9.
[14] J. E. Flaherty, J. B. Keller and S. I. Rubinow, (1972), Post buckling behaviour of elastic tubes and rings with opposite sides in contact, SIAM J. Appl. Math., 23, 446–455.
[15] J. B. Grotberg, (1971), Pulmonary flow and transport phenomena, Ann. Rev. Fluid Mech. 26, 529–571.
[16] A. J. Guttamann, (1989), Asymptotic analysis of power –series expansions, Phase Transitions and Critical Phenomena, C. Domb and J. K. Lebowitz, eds. Academic Press, New York, 1–234.
[17] C. Hunter, C. and B. Guerrieri, (1980), Deducing the properties of singularities of functions from their Taylor series coefficients, SIAM J. Appl. Math., 39, 248–263.
[18] D. L. Hunter and G. A. Baker, (1979), Methods of series analysis III: Integral approximant methods, Phys. Rev. B 19, 3808–3821.
[19] M. Heil, (1997), Stokes flow in collapsible tubes-computational and experiment, J. Fluid Mech., 353, 285–312.
[20] O. D. Makinde, (1999), Extending the utility of perturbation series in problems of laminar flow in porous pipe and a diverging channel, Jour. of Austral. Math. Soc. Ser. B 41, 118–128.
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Cite This Article
  • APA Style

    Muhammad Zeeshan Ashraf, Muhamad Riaz Khan, Shahzad Waheed, Muhammad Ahsan, Saira Hussnain. (2018). Mathematical Modelling to Simulate Biological Fluid Flow in a Collapsible Tube. American Journal of Mechanics and Applications, 6(1), 1-6. https://doi.org/10.11648/j.ajma.20180601.11

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    ACS Style

    Muhammad Zeeshan Ashraf; Muhamad Riaz Khan; Shahzad Waheed; Muhammad Ahsan; Saira Hussnain. Mathematical Modelling to Simulate Biological Fluid Flow in a Collapsible Tube. Am. J. Mech. Appl. 2018, 6(1), 1-6. doi: 10.11648/j.ajma.20180601.11

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    AMA Style

    Muhammad Zeeshan Ashraf, Muhamad Riaz Khan, Shahzad Waheed, Muhammad Ahsan, Saira Hussnain. Mathematical Modelling to Simulate Biological Fluid Flow in a Collapsible Tube. Am J Mech Appl. 2018;6(1):1-6. doi: 10.11648/j.ajma.20180601.11

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  • @article{10.11648/j.ajma.20180601.11,
      author = {Muhammad Zeeshan Ashraf and Muhamad Riaz Khan and Shahzad Waheed and Muhammad Ahsan and Saira Hussnain},
      title = {Mathematical Modelling to Simulate Biological Fluid Flow in a Collapsible Tube},
      journal = {American Journal of Mechanics and Applications},
      volume = {6},
      number = {1},
      pages = {1-6},
      doi = {10.11648/j.ajma.20180601.11},
      url = {https://doi.org/10.11648/j.ajma.20180601.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajma.20180601.11},
      abstract = {The purpose of this paper is to advance a mathematical model for reviewing to simulate biological flows such as blood flow in arteries or veins, flow of urine in urethras and air flow in the bronchial airways. They can also be used to study and prediction of many diseases, as the lung disease (asthma and emphysema), or the cardiovascular diseases (heart stroke), Makinde (2005). In this work, laminar flow of an incompressible viscous fluid through a collapsible tube of circular cross section is considered. Collapsible tubes are easily deformed by negative transmural pressure owing to marked reduction of rigidity. Thus, they show a considerable nonlinearity and reveal various complicated phenomena Our objectives are to study the effect of temperature along the tube as the fluid Prandtl number and Reynolds number increases. We launch the mathematical formulation of the problem. The problem is solved by using power series and perturbation techniques with help of boundary conditions and results are displayed graphically for different flow characteristics, velocity profile.},
     year = {2018}
    }
    

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    T1  - Mathematical Modelling to Simulate Biological Fluid Flow in a Collapsible Tube
    AU  - Muhammad Zeeshan Ashraf
    AU  - Muhamad Riaz Khan
    AU  - Shahzad Waheed
    AU  - Muhammad Ahsan
    AU  - Saira Hussnain
    Y1  - 2018/01/29
    PY  - 2018
    N1  - https://doi.org/10.11648/j.ajma.20180601.11
    DO  - 10.11648/j.ajma.20180601.11
    T2  - American Journal of Mechanics and Applications
    JF  - American Journal of Mechanics and Applications
    JO  - American Journal of Mechanics and Applications
    SP  - 1
    EP  - 6
    PB  - Science Publishing Group
    SN  - 2376-6131
    UR  - https://doi.org/10.11648/j.ajma.20180601.11
    AB  - The purpose of this paper is to advance a mathematical model for reviewing to simulate biological flows such as blood flow in arteries or veins, flow of urine in urethras and air flow in the bronchial airways. They can also be used to study and prediction of many diseases, as the lung disease (asthma and emphysema), or the cardiovascular diseases (heart stroke), Makinde (2005). In this work, laminar flow of an incompressible viscous fluid through a collapsible tube of circular cross section is considered. Collapsible tubes are easily deformed by negative transmural pressure owing to marked reduction of rigidity. Thus, they show a considerable nonlinearity and reveal various complicated phenomena Our objectives are to study the effect of temperature along the tube as the fluid Prandtl number and Reynolds number increases. We launch the mathematical formulation of the problem. The problem is solved by using power series and perturbation techniques with help of boundary conditions and results are displayed graphically for different flow characteristics, velocity profile.
    VL  - 6
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Author Information
  • Department of Basic Sciences & Humanities, University of Engineering and Technology, Lahore, Pakistan

  • Department of Mathematics, Lahore Garrison University, Lahore, Pakistan

  • Department of Mathematics, University of Lahore, Lahore, Pakistan

  • Department of Mathematics, Hajvery University, Lahore, Pakistan

  • Department of Mathematics, Hajvery University, Lahore, Pakistan

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