American Journal of Mechanics and Applications
Volume 6, Issue 1, March 2018, Pages: 1-6
Received: Dec. 26, 2017;
Accepted: Jan. 11, 2018;
Published: Jan. 29, 2018
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Muhammad Zeeshan Ashraf, Department of Basic Sciences & Humanities, University of Engineering and Technology, Lahore, Pakistan
Muhamad Riaz Khan, Department of Mathematics, Lahore Garrison University, Lahore, Pakistan
Shahzad Waheed, Department of Mathematics, University Of Lahore, Lahore, Pakistan
Muhammad Ahsan, Department of Mathematics, Hajvery University, Lahore, Pakistan
Saira Hussnain, Department of Mathematics, Hajvery University, Lahore, Pakistan
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.
Muhammad Zeeshan Ashraf,
Muhamad Riaz Khan,
Mathematical Modelling to Simulate Biological Fluid Flow in a Collapsible Tube, American Journal of Mechanics and Applications.
Vol. 6, No. 1,
2018, pp. 1-6.
C. D. Bertram, (1986), Unstable equilibrium behaviour in collapsible tubes, J. Biomech. 19, 61–69.
C. D. Bertram and T. J. Pedley, (1982), A mathematical model of unsteady collapsible tube behaviour, J. Biomech. 15, 39–50.
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.
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.
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.
M. Bonis and C. Ribreau, (1987), Etude de quelques propriétés de l’ecoulement dans une conduite collabable, La Houille Blanche 3/4, 165–173.
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.
W. A. Contrad, (1969), Pressure-flow relationship in collapsible tubes, IEEE Trans. Bio-Med. Engng BME-16, 284–295.
S. J. Cowley, (1982), Elastic jumps in fluid-filled elastic tubes, J. Fluid Mech. 116, 459–473.
S. J. Cowley, (1983), On the wavetrains associated with elastic jumps on fluid-filled elastic tubes, Q. J. Mech. Appl. Math., 36, 289–312.
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.
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.
D. Elad, R. D. Kamm and A. H. Shapiro, (1987), Choking phenomena in a lung-like model, ASME J. Biomech. Engng 109, 1–9.
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.
J. B. Grotberg, (1971), Pulmonary flow and transport phenomena, Ann. Rev. Fluid Mech. 26, 529–571.
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.
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.
D. L. Hunter and G. A. Baker, (1979), Methods of series analysis III: Integral approximant methods, Phys. Rev. B 19, 3808–3821.
M. Heil, (1997), Stokes flow in collapsible tubes-computational and experiment, J. Fluid Mech., 353, 285–312.
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.
O. D. Makinde, (2001), Heat and mass transfer in a pipe with moving surface: Effects of viscosity variation and energy dissipation, Quaestiones Mathematicae, Vol. 24, 97–108.
R. E. Shafer, (1974), On quadratic approximation, SIAM J. Numer. Anal., 11, 447–460.