HEAD-ON COLLISION OF THE SOLITARY WAVES IN FLUID-FILLED ELASTIC TUBES

Year 2018, Volume: 8 Issue: 2, 1 - 13, 01.12.2018

A. E. Özden H. Demiray

Abstract

In the present work, by employing the eld equations given in [15] and the extended PLK method derived in [9], we have studied the head-on collision of solitary waves in arteries. Introducing a set of stretched coordinates which include some unknown functions characterizing the higher order dispersive eects and the trajectory functions to be determined from the removal of possible secularities that might occur in the solu- tion. Expanding these unknown functions and the eld variables into power series of the smallness parameter  and introducing the resulting expansions into the eld equations we obtained the sets of partial dierential equations. By solving these dierential equa- tions and imposing the requirements for the removal of possible secularities we obtained the speed correction terms and the trajectory functions. The results of our calculation show that both the evolution equations and the phase shifts resulting from the head-on collision of solitary waves are quite dierent from those of Xue [15], who employed the incorrect formulation of Su and Mirie [4]. As opposed to the result of previous works on the same subject, in the present work the phase shifts depend on the amplitudes of both colliding waves.

Keywords

Elastic tubes, Solitary waves, Head-on collision, Extended PLK Method.

References

• Korteweg D. J. and de Vries G., (1895), On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves, Phil. Mag., 39, pp.422-443.
• Gardner C. S., Greene J. M., Kruskal M. D. and Miura R. M., (1967), Method for solving KdV equation, Phys. Rev. Lett., 19, pp.1095-1097.
• Zakharov V. E., Manakov S. V., Novikov S. P., and Pitaievski L. P., (1980), Theory of Soliton: The Inverse Problem Method, Nauka, Moscow [English translation(Plenium, New York, 1984)].
• Su C. H. and Mirie R. M., (1980), On head-on collisions between two solitary waves, J. Fluid Mech., , 3, pp.509-525.
• Huang G. and Velarde M.G., (1996), Head-on collision of two concentric cylindrical ion-acoustic solitary waves, Physical Review E, 53, pp.2988-2991.
• Narahara K., Characterization of collision-induced generation of pulses in coupled electrical nonlinear transmission lines, (2014), Japanese Journal of Applied Physics, 53, 067301.
• Demiray H., (2012), Contribution of higher order terms to the nonlinear shallow water waves, TWMS
• J. Appl .and Engr. Math., 2, pp.210-218. El-Tantawy S. A., Moslem W. M.,Sabry R., El-Labany S. K., El-Metwally M. and Schlickeiser R., (2013), Nonplanar solitons collision in ultracold plasmas, Physics of Plasmas, 20, 092126.
• Ozden A.E. and Demiray H., (2015), Re-visiting the head-on collision problem between two solitary waves in shallow water, Int. J. Nonlinear Mechanics, 69, pp.66-70.
• McDonald D. A., (1974), Blood Flow in Arteries, Second Edition, Edward Arnold, London.
• Paquerot J. F. and Lambrakos S. G., (1994), Monovariable representation of blood flow in a large elastic artery, Phys. Rev. E, 49, pp.3432-3439.
• Sakanishi A., Hasegawa M. and Ushiyama Y., (1996), Pressure pulse wave for blood flow in the aorta from the viewpoint of Toda lattice, Phys. Lett. A, 2219, pp.395-399.
• Antar N. and Demiray H., (1999), Weakly nonlinear waves in a prestressed thin elastic tube containing a viscous fluid, Int. J. Engr. Sci., 37, pp.1859-1876.
• Malfliet W. and Ndayirinde I., (1998), Dressed solitary waves in an elastic tube, Physica D, 123, pp.92-98.
• Xue J. K., Head-on collision of blood solitary waves, (2004), Phys. Lett. A, 331, pp.409-413.
• Demiray H., (2005), Head-on collision of solitary waves in fluid-filled elastic tubes, Appl. Math. Letters, , pp.941-950.
• Demiray H., (2009), Head-on collision of nonlinear waves in a fluid of variable viscosity contained in an elastic tube, Chaos, Solitons and Fractals, 41, pp.1578-1586.
• Duan W. S., Wang B. R. and Wein R. J., (1997), Reflection and transmission of nonlinear blood waves due to arterial branching, Phys. Rev. E, 55, pp.1773-1778.
• Noubissie S. and Woafo P., (2003), Dynamics of solitary blood waves in arteries with prosthesis, Phys. Rev. E, 67, 041911.